Libigl is an open source C++ library for geometry processing research and development. Dropping the heavy data structures of tradition geometry libraries, libigl is a simple header-only library of encapsulated functions. This combines the rapid prototyping familiar to Matlab or Python programmers with the performance and versatility of C++. The tutorial is a self-contained, hands-on introduction to libigl. Via interactive, step-by-step examples, we demonstrate how to accomplish common geometry processing tasks such as computation of differential quantities and operators, real-time deformation, parametrization, numerical optimization and remeshing. Each section of the lecture notes links to a cross-platform example application.
We introduce libigl with a series of self-contained examples. The purpose of each example is to showcase a feature of libigl while applying to a practical problem in geometry processing. In this chapter, we will present the basic concepts of libigl and introduce a simple mesh viewer that allows to visualize a surface mesh and its attributes. All the tutorial examples are cross-platform and can be compiled on MacOSX, Linux and Windows.
Before getting into the examples, we summarize the main design principles in libigl:
No complex data types. We mostly use matrices and vectors. This greatly favors code reusability and forces the function authors to expose all the parameters used by the algorithm.
Minimal dependencies. We use external libraries only when necessary and we wrap them in a small set of functions.
Header-only. It is straight forward to use our library since it is only one additional include directory in your project. (if you are worried about compilation speed, it is also possible to build the library as a static library)
Function encapsulation. Every function (including its full implementation) is contained in a pair of .h/.cpp files with the same name of the function.
libigl can be downloaded from our github repository or cloned with git:
git clone --recursive https://github.com/libigl/libigl.git
The core libigl functionality only depends on the C++ Standard Library and Eigen.
To build all the examples in the tutorial, you can use the CMakeLists.txt in the tutorial folder:
cd tutorial
mkdir build
cd build
cmake -DCMAKE_BUILD_TYPE=Release ../
make
The examples can also be built independently using the CMakeLists.txt inside each example folder.
Note for linux users: Many linux distributions do not include gcc and the basic development tools in their default installation. On Ubuntu, you need to install the following packages:
sudo apt-get install git
sudo apt-get install build-essential
sudo apt-get install cmake
sudo apt-get install libx11-dev
sudo apt-get install mesa-common-dev libgl1-mesa-dev libglu1-mesa-dev
sudo apt-get install libxrandr-dev
sudo apt-get install libxi-dev
sudo apt-get install libxmu-dev
sudo apt-get install libblas-dev
sudo apt-get install libxinerama-dev
sudo apt-get install libxcursor-dev
Note for windows users: libigl only supports the Microsoft Visual Studio 2015 compiler in 64bit mode. It will not work with a 32bit build and it will not work with older versions of visual studio.
A few examples in Chapter 5 requires the CoMiSo solver. We provide a mirror of CoMISo that works out of the box with libigl. To install it:
cd libigl/external
git clone --recursive https://github.com/libigl/CoMISo.git
You can then build the tutorials again and it libigl will automatically find and compile CoMISo.
Note 1: CoMISo is distributed under the GPL3 license, it does impose restrictions on commercial usage.
Note 2: CoMISo requires a blas implementation. We use the built-in blas in macosx and linux, and we bundle a precompiled binary for VS2015 64 bit. Do NOT compile the tutorials in 32 bit on windows.
We provide a blank project example showing how to use libigl and cmake. Feel free and encouraged to copy or fork this project as a way of starting a new personal project using libigl.
libigl uses the Eigen library to encode vector and matrices. We suggest that you keep the dense and sparse quick reference guides at hand while you read the examples in this tutorial.
A triangular mesh is encoded as a pair of matrices:
Eigen::MatrixXd V;
Eigen::MatrixXi F;
V
is a #N by 3 matrix which stores the coordinates of the vertices. Each
row stores the coordinate of a vertex, with its x,y and z coordinates in the first,
second and third column, respectively. The matrix F
stores the triangle
connectivity: each line of F
denotes a triangle whose 3 vertices are
represented as indices pointing to rows of V
.
Note that the order of the vertex indices in F
determines the orientation of
the triangles and it should thus be consistent for the entire surface.
This simple representation has many advantages:
libigl provides input [output] functions to read [write] many common mesh formats. The IO functions are contained in the files read*.h and write*.h. As a general rule each libigl function is contained in a pair of .h/.cpp files with the same name. By default, the .h files include the corresponding cpp files, making the library header-only.
Reading a mesh from a file requires a single libigl function call:
igl::readOFF(TUTORIAL_SHARED_PATH "/cube.off", V, F);
The function reads the mesh cube.off and it fills the provided V
and F
matrices.
Similarly, a mesh can be written in an OBJ file using:
igl::writeOBJ("cube.obj",V,F);
Example 101 contains a simple mesh converter from OFF to OBJ format.
Libigl provides an glfw-based OpenGL 3.2 viewer to visualize surfaces, their properties and additional debugging information.
The following code (Example 102) is a basic skeleton for all the examples that will be used in the tutorial. It is a standalone application that loads a mesh and uses the viewer to render it.
#include <igl/readOFF.h>
#include <igl/opengl/glfw/Viewer.h>
Eigen::MatrixXd V;
Eigen::MatrixXi F;
int main(int argc, char *argv[])
{
// Load a mesh in OFF format
igl::readOFF(TUTORIAL_SHARED_PATH "/bunny.off", V, F);
// Plot the mesh
igl::opengl::glfw::Viewer viewer;
viewer.data().set_mesh(V, F);
viewer.launch();
}
The function set_mesh
copies the mesh into the viewer.
Viewer.launch()
creates a window, an OpenGL context and it starts the draw loop.
Additional properties can be plotted on the mesh (as we will see later),
and it is possible to extend the viewer with standard OpenGL code.
Please see the documentation in
Viewer.h for more details.
Keyboard and mouse events triggers callbacks that can be registered in the viewer. The viewer supports the following callbacks:
bool (*callback_pre_draw)(Viewer& viewer);
bool (*callback_post_draw)(Viewer& viewer);
bool (*callback_mouse_down)(Viewer& viewer, int button, int modifier);
bool (*callback_mouse_up)(Viewer& viewer, int button, int modifier);
bool (*callback_mouse_move)(Viewer& viewer, int mouse_x, int mouse_y);
bool (*callback_mouse_scroll)(Viewer& viewer, float delta_y);
bool (*callback_key_down)(Viewer& viewer, unsigned char key, int modifiers);
bool (*callback_key_up)(Viewer& viewer, unsigned char key, int modifiers);
A keyboard callback can be used to visualize multiple meshes or different stages of an algorithm, as demonstrated in Example 103, where the keyboard callback changes the visualized mesh depending on the key pressed:
bool key_down(igl::opengl::glfw::Viewer& viewer, unsigned char key, int modifier)
{
if (key == '1')
{
viewer.data().clear();
viewer.data().set_mesh(V1, F1);
viewer.core.align_camera_center(V1,F1);
}
else if (key == '2')
{
viewer.data().clear();
viewer.data().set_mesh(V2, F2);
viewer.core.align_camera_center(V2,F2);
}
return false;
}
The callback is registered in the viewer as follows:
viewer.callback_key_down = &key_down;
Note that the mesh is cleared before using set_mesh. This has to be called every time the number of vertices or faces of the plotted mesh changes. Every callback returns a boolean value that tells the viewer if the event has been handled by the plugin, or if the viewer should process it normally. This is useful, for example, to disable the default mouse event handling if you want to control the camera directly in your code.
The viewer can be extended using plugins, which are classes that implements all the viewer’s callbacks. See the Viewer_plugin for more details.
Colors and normals can be associated to faces or vertices using the set_colors function:
viewer.data().set_colors(C);
C
is a #C by 3 matrix with one RGB color per row. C
must have as many
rows as the number of faces or the number of vertices of the mesh.
Depending on the size of C
, the viewer applies the color to the faces or to
the vertices.
Colors can be used to visualize a scalar function defined on a surface. The
scalar function is converted to colors using a color transfer function, which
maps a scalar value between 0 and 1 to a color. A simple example of a scalar
field defined on a surface is the z coordinate of each point, which can be
extract from our mesh representation by taking the last column of V
(Example 104). The function igl::jet
can be used to
convert it to colors:
Eigen::VectorXd Z = V.col(2);
igl::jet(Z,true,C);
The first row extracts the third column from V
(the z coordinate of each
vertex) and the second calls a libigl functions that converts a scalar field to colors. The second parameter of jet normalizes the scalar field to lie between 0 and 1 before applying the transfer function.
igl::jet
is an example of a standard function in libigl: it takes simple
types and can be easily reused for many different tasks. Not committing to
heavy data structures types favors simplicity, ease of use and reusability.
In addition to plotting the surface, the viewer supports the visualization of points, lines and text labels: these overlays can be very helpful while developing geometric processing algorithms to plot debug information.
viewer.data().add_points(P,Eigen::RowVector3d(r,g,b));
Draws a point of color r,g,b for each row of P. The point is placed at the coordinates specified in each row of P, which is a #P by 3 matrix.
viewer.data().add_edges(P1,P2,Eigen::RowVector3d(r,g,b);
Draws a line of color r,g,b for each row of P1 and P2, which connects the 3D point in to the point in P2. Both P1 and P2 are of size #P by 3.
viewer.data().add_label(p,str);
Draws a label containing the string str at the position p, which is a vector of length 3.
These functions are demonstrate in Example 105 where
the bounding box of a mesh is plotted using lines and points.
Using matrices to encode the mesh and its attributes allows to write short and
efficient code for many operations, avoiding to write for loops. For example,
the bounding box of a mesh can be found by taking the colwise maximum and minimum of V
:
Eigen::Vector3d m = V.colwise().minCoeff();
Eigen::Vector3d M = V.colwise().maxCoeff();
As of version 1.2 the viewer uses a new menu and completely replaces AntTweakBar. It is based on the open-source projects nanovg and nanogui. To extend the default menu of the viewer and to expose more user defined variables you have to define a callback function:
igl::opengl::glfw::Viewer viewer;
bool boolVariable = true;
float floatVariable = 0.1f;
enum Orientation { Up=0,Down,Left,Right } dir = Up;
// Extend viewer menu
viewer.callback_init = [&](igl::opengl::glfw::Viewer& viewer)
{
// Add new group
viewer.ngui->addGroup("New Group");
// Expose a variable directly ...
viewer.ngui->addVariable("float",floatVariable);
// Expose an enumaration type
viewer.ngui->addVariable<Orientation>("Direction",dir)->setItems({"Up","Down","Left","Right"});
// Add a button
viewer.ngui->addButton("Print Hello",[](){ std::cout << "Hello\n"; });
// call to generate menu
viewer.screen->performLayout();
return false;
};
// start viewer
viewer.launch();
If you need a separate new menu window use:
viewer.ngui->addWindow(Eigen::Vector2i(220,10),"New Window");
If you do not want to expose variables directly but rather use the get/set functionality:
// ... or using a custom callback
viewer.ngui->addVariable<bool>("bool",[&](bool val) {
boolVariable = val; // setter
},[&]() {
return boolVariable; // getter
});
Libigl’s igl::opengl::glfw::Viewer
provides basic support for rendering
multiple meshes.
Which mesh is selected is controlled via the viewer.selected_data_index
field. By default it his is set to 0
, so in the typical case of a single mesh
viewer.data()
returns the igl::ViewerData
corresponding to the one
and only mesh.
This chapter illustrates a few discrete quantities that libigl can compute on a mesh and the libigl functions that construct popular discrete differential geometry operators. It also provides an introduction to basic drawing and coloring routines of our viewer.
Surface normals are a basic quantity necessary for rendering a surface. There are a variety of ways to compute and store normals on a triangle mesh. Example 201 demonstrates how to compute and visualize normals with libigl.
Normals are well defined on each triangle of a mesh as the vector orthogonal to triangle’s plane. These piecewise-constant normals produce piecewise-flat renderings: the surface appears non-smooth and reveals its underlying discretization.
Normals can be computed and stored on vertices, and interpolated in the interior of the triangles to produce smooth renderings (Phong shading). Most techniques for computing per-vertex normals take an average of incident face normals. The main difference between these techniques is their weighting scheme: Uniform weighting is heavily biased by the discretization choice, whereas area-based or angle-based weighting is more forgiving.
The typical half-edge style computation of area-based weights has this structure:
N.setZero(V.rows(),3);
for(int i : vertices)
{
for(face : incident_faces(i))
{
N.row(i) += face.area * face.normal;
}
}
N.rowwise().normalize();
At first glance, it might seem inefficient to loop over incident faces—and thus constructing the per-vertex normals— without using an half-edge data structure. However, per-vertex normals may be throwing each face normal to running sums on its corner vertices:
N.setZero(V.rows(),3);
for(int f = 0; f < F.rows();f++)
{
for(int c = 0; c < 3;c++)
{
N.row(F(f,c)) += area(f) * face_normal.row(f);
}
}
N.rowwise().normalize();
Storing normals per-corner is an efficient and convenient way of supporting both smooth and sharp (e.g. creases and corners) rendering. This format is common to OpenGL and the .obj mesh file format. Often such normals are tuned by the mesh designer, but creases and corners can also be computed automatically. Libigl implements a simple scheme which computes corner normals as averages of normals of faces incident on the corresponding vertex which do not deviate by more than a specified dihedral angle (e.g. 20°).
Gaussian curvature on a continuous surface is defined as the product of the principal curvatures:
\(k_G = k_1 k_2.\)
As an intrinsic measure, it depends on the metric and not the surface’s embedding.
Intuitively, Gaussian curvature tells how locally spherical or elliptic the surface is ( \(k_G>0\) ), how locally saddle-shaped or hyperbolic the surface is ( \(k_G<0\) ), or how locally cylindrical or parabolic ( \(k_G=0\) ) the surface is.
In the discrete setting, one definition for a “discrete Gaussian curvature” on a triangle mesh is via a vertex’s angular deficit:
\(k_G(v_i) = 2π - \sum\limits_{j\in N(i)}θ_{ij},\)
where \(N(i)\) are the triangles incident on vertex \(i\) and \(θ_{ij}\) is the angle at vertex \(i\) in triangle \(j\) (1).
Just like the continuous analog, our discrete Gaussian curvature reveals elliptic, hyperbolic and parabolic vertices on the domain, as demonstrated in Example 202.
The two principal curvatures \((k_1,k_2)\) at a point on a surface measure how much the surface bends in different directions. The directions of maximum and minimum (signed) bending are called principal directions and are always orthogonal.
Mean curvature is defined as the average of principal curvatures:
\(H = \frac{1}{2}(k_1 + k_2).\)
One way to extract mean curvature is by examining the Laplace-Beltrami operator applied to the surface positions. The result is a so-called mean-curvature normal:
\(-\Delta \mathbf{x} = H \mathbf{n}.\)
It is easy to compute this on a discrete triangle mesh in libigl using the cotangent Laplace-Beltrami operator (1).
#include <igl/cotmatrix.h>
#include <igl/massmatrix.h>
#include <igl/invert_diag.h>
...
MatrixXd HN;
SparseMatrix<double> L,M,Minv;
igl::cotmatrix(V,F,L);
igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_VORONOI,M);
igl::invert_diag(M,Minv);
HN = -Minv*(L*V);
H = HN.rowwise().norm(); //up to sign
Combined with the angle defect definition of discrete Gaussian curvature, one can define principal curvatures and use least squares fitting to find directions (1).
Alternatively, a robust method for determining principal curvatures is via quadric fitting (2). In the neighborhood around every vertex, a best-fit quadric is found and principal curvature values and directions are analytically computed on this quadric (Example 203).
Scalar functions on a surface can be discretized as a piecewise linear function with values defined at each mesh vertex:
\(f(\mathbf{x}) \approx \sum\limits_{i=1}^n \phi_i(\mathbf{x})\, f_i,\)
where \(\phi_i\) is a piecewise linear hat function defined by the mesh so that for each triangle \(\phi_i\) is the linear function which is one only at vertex \(i\) and zero at the other corners.
Thus gradients of such piecewise linear functions are simply sums of gradients of the hat functions:
\(\nabla f(\mathbf{x}) \approx \nabla \sum\limits_{i=1}^n \phi_i(\mathbf{x})\, f_i = \sum\limits_{i=1}^n \nabla \phi_i(\mathbf{x})\, f_i.\)
This reveals that the gradient is a linear function of the vector of \(f_i\) values. Because the \(\phi_i\) are linear in each triangle, their gradients are constant in each triangle. Thus our discrete gradient operator can be written as a matrix multiplication taking vertex values to triangle values:
\(\nabla f \approx \mathbf{G}\,\mathbf{f},\)
where \(\mathbf{f}\) is \(n\times 1\) and \(\mathbf{G}\) is an \(md\times n\) sparse
matrix. This matrix \(\mathbf{G}\) can be derived geometrically, e.g.
(ch. 2, 3).
Libigl’s grad
function computes \(\mathbf{G}\) for
triangle and tetrahedral meshes (Example 204):
The discrete Laplacian is an essential geometry processing tool. Many interpretations and flavors of the Laplace and Laplace-Beltrami operator exist.
In open Euclidean space, the Laplace operator is the usual divergence of gradient (or equivalently the Laplacian of a function is the trace of its Hessian):
\(\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.\)
The Laplace-Beltrami operator generalizes this to surfaces.
When considering piecewise-linear functions on a triangle mesh, a discrete Laplacian may be derived in a variety of ways. The most popular in geometry processing is the so-called “cotangent Laplacian” \(\mathbf{L}\), arising simultaneously from FEM, DEC and applying divergence theorem to vertex one-rings. As a linear operator taking vertex values to vertex values, the Laplacian \(\mathbf{L}\) is a \(n\times n\) matrix with elements:
\(L_{ij} = \begin{cases}j \in N(i) &\cot \alpha_{ij} + \cot \beta_{ij},\\ j \notin N(i) & 0,\\ i = j & -\sum\limits_{k\neq i} L_{ik}, \end{cases}\)
where \(N(i)\) are the vertices adjacent to (neighboring) vertex \(i\), and \(\alpha_{ij},\beta_{ij}\) are the angles opposite to edge \({ij}\). This formula leads to a typical half-edge style implementation for constructing \(\mathbf{L}\):
for(int i : vertices)
{
for(int j : one_ring(i))
{
for(int k : triangle_on_edge(i,j))
{
L(i,j) = cot(angle(i,j,k));
L(i,i) -= cot(angle(i,j,k));
}
}
}
Similarly as before, it may seem to loop over one-rings without having an half-edge data structure. However, this is not the case, since the Laplacian may be built by summing together contributions for each triangle, much in spirit with its FEM discretization of the Dirichlet energy (sum of squared gradients):
for(triangle t : triangles)
{
for(edge i,j : t)
{
L(i,j) += cot(angle(i,j,k));
L(j,i) += cot(angle(i,j,k));
L(i,i) -= cot(angle(i,j,k));
L(j,j) -= cot(angle(i,j,k));
}
}
Libigl implements discrete “cotangent” Laplacians for triangles meshes and tetrahedral meshes, building both with fast geometric rules rather than “by the book” FEM construction which involves many (small) matrix inversions, cf. (4)[].
The operator applied to mesh vertex positions amounts to smoothing by flowing the surface along the mean curvature normal direction (Example 205). Note that this is equivalent to minimizing surface area.
The mass matrix \(\mathbf{M}\) is another \(n \times n\) matrix which takes vertex values to vertex values. From an FEM point of view, it is a discretization of the inner-product: it accounts for the area around each vertex. Consequently, \(\mathbf{M}\) is often a diagonal matrix, such that \(M_{ii}\) is the barycentric or voronoi area around vertex \(i\) in the mesh (1)[]. The inverse of this matrix is also very useful as it transforms integrated quantities into point-wise quantities, e.g.:
\(\Delta f \approx \mathbf{M}^{-1} \mathbf{L} \mathbf{f}.\)
In general, when encountering squared quantities integrated over the surface, the mass matrix will be used as the discretization of the inner product when sampling function values at vertices:
\(\int_S x\, y\ dA \approx \mathbf{x}^T\mathbf{M}\,\mathbf{y}.\)
An alternative mass matrix \(\mathbf{T}\) is a \(md \times md\) matrix which takes triangle vector values to triangle vector values. This matrix represents an inner-product accounting for the area associated with each triangle (i.e. the triangles true area).
An alternative construction of the discrete cotangent Laplacian is by “squaring” the discrete gradient operator. This may be derived by applying Green’s identity (ignoring boundary conditions for the moment):
\(\int_S \|\nabla f\|^2 dA = \int_S f \Delta f dA\)
Or in matrix form which is immediately translatable to code:
\(\mathbf{f}^T \mathbf{G}^T \mathbf{T} \mathbf{G} \mathbf{f} = \mathbf{f}^T \mathbf{M} \mathbf{M}^{-1} \mathbf{L} \mathbf{f} = \mathbf{f}^T \mathbf{L} \mathbf{f}.\)
So we have that \(\mathbf{L} = \mathbf{G}^T \mathbf{T} \mathbf{G}\). This also hints that we may consider \(\mathbf{G}^T\) as a discrete divergence operator, since the Laplacian is the divergence of the gradient. Naturally, \(\mathbf{G}^T\) is a \(n \times md\) sparse matrix which takes vector values stored at triangle faces to scalar divergence values at vertices.
The discrete geodesic distance between two points is the length of the shortest path between then restricted to the surface. For triangle meshes, such a path is made of a set of segments which can be either edges of the mesh or crossing a triangle.
Libigl includes a wrapper for the exact geodesic algorithm (6) developed by Danil Kirsanov (https://code.google.com/archive/p/geodesic/), exposing it through an Eigen-based API. The function
igl::exact_geodesic(V,F,VS,FS,VT,FT,d);
computes the closest geodesic distances of each vertex in VT or face in FT, from the source vertices VS or faces FS of the input mesh V,F. The output is writted in the vector d, which lists first the distances for the vertices in VT, and then for the faces in FT. For example, if you want to compute the distance from the vertex with id vid
, to all vertices of F you can use:
Eigen::VectorXi VS,FS,VT,FT;
// The selected vertex is the source
VS.resize(1);
VS << vid;
// All vertices are the targets
VT.setLinSpaced(V.rows(),0,V.rows()-1);
Eigen::VectorXd d;
igl::exact_geodesic(V,F,VS,FS,VT,FT,d);
Libigl relies heavily on the Eigen library for dense and sparse linear algebra routines. Besides geometry processing routines, libigl has linear algebra routines which bootstrap Eigen and make it feel even more similar to a high-level algebra library such as Matlab.
A very familiar and powerful routine in Matlab is array slicing. This allows reading from or writing to a possibly non-contiguous sub-matrix. Let’s consider the Matlab code:
B = A(R,C);
If A
is a \(m \times n\) matrix and R
is a \(j\)-long list of row-indices
(between 1 and \(m\)) and C
is a \(k\)-long list of column-indices, then as a
result B
will be a \(j \times k\) matrix drawing elements from A
according to
R
and C
. In libigl, the same functionality is provided by the slice
function (Example 301):
VectorXi R,C;
MatrixXd A,B;
...
igl::slice(A,R,C,B);
Note that A
and B
could also be sparse matrices.
Similarly, consider the Matlab code:
A(R,C) = B;
Now, the selection is on the left-hand side so the \(j \times k\) matrix B
is
being written into the submatrix of A
determined by R
and C
. This
functionality is provided in libigl using slice_into
:
igl::slice_into(B,R,C,A);
Matlab and other higher-level languages make it very easy to extract indices of sorting and comparison routines. For example in Matlab, one can write:
[Y,I] = sort(X,1,'ascend');
so if X
is a \(m \times n\) matrix then Y
will also be an \(m \times n\) matrix
with entries sorted along dimension 1
in 'ascend'
ing order. The second
output I
is a \(m \times n\) matrix of indices such that Y(i,j) =
X(I(i,j),j);
. That is, I
reveals how X
is sorted into Y
.
This same functionality is supported in libigl:
igl::sort(X,1,true,Y,I);
Similarly, sorting entire rows can be accomplished in Matlab using:
[Y,I] = sortrows(X,'ascend');
where now I
is a \(m\) vector of indices such that Y = X(I,:)
.
In libigl, this is supported with
igl::sortrows(X,true,Y,I);
where again I
reveals the index of sort so that it can be reproduced with
igl::slice(X,I,1,Y)
.
Analogous functions are available in libigl for: max
, min
, and unique
.
Libigl implements a variety of other routines with the same api and functionality as common Matlab functions.
Name | Description |
---|---|
igl::all |
Whether all elements are non-zero (true) |
igl::any |
Whether any elements are non-zero (true) |
igl::cat |
Concatenate two matrices (especially useful for dealing with Eigen sparse matrices) |
igl::ceil |
Round entries up to nearest integer |
igl::cumsum |
Cumulative sum of matrix elements |
igl::colon |
Act like Matlab’s : , similar to Eigen’s LinSpaced |
igl::components |
Connected components of graph (cf. Matlab’s graphconncomp ) |
igl::count |
Count non-zeros in rows or columns |
igl::cross |
Cross product per-row |
igl::cumsum |
Cumulative summation |
igl::dot |
dot product per-row |
igl::eigs |
Solve sparse eigen value problem |
igl::find |
Find subscripts of non-zero entries |
igl::floor |
Round entries down to nearest integer |
igl::histc |
Counting occurrences for building a histogram |
igl::hsv_to_rgb |
Convert HSV colors to RGB (cf. Matlab’s hsv2rgb ) |
igl::intersect |
Set intersection of matrix elements. |
igl::isdiag |
Determine whether matrix is diagonal |
igl::ismember |
Determine whether elements in A occur in B |
igl::jet |
Quantized colors along the rainbow. |
igl::max |
Compute maximum entry per row or column |
igl::median |
Compute the median per column |
igl::min |
Compute minimum entry per row or column |
igl::mod |
Compute per element modulo |
igl::mode |
Compute the mode per column |
igl::null |
Compute the null space basis of a matrix |
igl::nchoosek |
Compute all k-size combinations of n-long vector |
igl::orth |
Orthogonalization of a basis |
igl::parula |
Generate a quantized colormap from blue to yellow |
igl::pinv |
Compute Moore-Penrose pseudoinverse |
igl::randperm |
Generate a random permutation of [0,…,n-1] |
igl::rgb_to_hsv |
Convert RGB colors to HSV (cf. Matlab’s rgb2hsv ) |
igl::repmat |
Repeat a matrix along columns and rows |
igl::round |
Per-element round to whole number |
igl::setdiff |
Set difference of matrix elements |
igl::setunion |
Set union of matrix elements |
igl::setxor |
Set exclusive “or” of matrix elements |
igl::slice |
Slice parts of matrix using index lists: (cf. Matlab’s B = A(I,J) ) |
igl::slice_mask |
Slice parts of matrix using boolean masks: (cf. Matlab’s B = A(M,N) ) |
igl::slice_into |
Slice left-hand side of matrix assignment using index lists (cf. Matlab’s B(I,J) = A ) |
igl::sort |
Sort elements or rows of matrix |
igl::speye |
Identity as sparse matrix |
igl::sum |
Sum along columns or rows (of sparse matrix) |
igl::unique |
Extract unique elements or rows of matrix |
A common linear system in geometry processing is the Laplace equation:
\(∆z = 0\)
subject to some boundary conditions, for example Dirichlet boundary conditions (fixed value):
\(\left.z\right|_{\partial{S}} = z_{bc}\)
In the discrete setting, the linear system can be written as:
\(\mathbf{L} \mathbf{z} = \mathbf{0}\)
where \(\mathbf{L}\) is the \(n \times n\) discrete Laplacian and \(\mathbf{z}\) is a vector of per-vertex values. Most of \(\mathbf{z}\) correspond to interior vertices and are unknown, but some of \(\mathbf{z}\) represent values at boundary vertices. Their values are known so we may move their corresponding terms to the right-hand side.
Conceptually, this is very easy if we have sorted \(\mathbf{z}\) so that interior vertices come first and then boundary vertices:
\[\left(\begin{array}{cc} \mathbf{L}_{in,in} & \mathbf{L}_{in,b}\\ \mathbf{L}_{b,in} & \mathbf{L}_{b,b}\end{array}\right) \left(\begin{array}{c} \mathbf{z}_{in}\\ \mathbf{z}_{b}\end{array}\right) = \left(\begin{array}{c} \mathbf{0}_{in}\\ \mathbf{z}_{bc}\end{array}\right)\]
The bottom block of equations is no longer meaningful so we’ll only consider the top block:
\[\left(\begin{array}{cc} \mathbf{L}_{in,in} & \mathbf{L}_{in,b}\end{array}\right) \left(\begin{array}{c} \mathbf{z}_{in}\\ \mathbf{z}_{b}\end{array}\right) = \mathbf{0}_{in}\]
We can move the known values to the right-hand side:
\[\mathbf{L}_{in,in} \mathbf{z}_{in} = - \mathbf{L}_{in,b} \mathbf{z}_{b}\]
Finally we can solve this equation for the unknown values at interior vertices \(\mathbf{z}_{in}\).
However, our vertices will often not be sorted in this way. One option would be to sort V
,
then proceed as above and then unsort the solution Z
to match V
. However,
this solution is not very general.
With array slicing no explicit sort is needed. Instead we can slice-out
submatrix blocks (\(\mathbf{L}_{in,in}\), \(\mathbf{L}_{in,b}\), etc.) and follow
the linear algebra above directly. Then we can slice the solution into the
rows of Z
corresponding to the interior vertices (Example 303).
The same Laplace equation may be equivalently derived by minimizing Dirichlet energy subject to the same boundary conditions:
\(\mathop{\text{minimize }}_z \frac{1}{2}\int\limits_S \|\nabla z\|^2 dA\)
On our discrete mesh, recall that this becomes
\(\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{G}^T \mathbf{D} \mathbf{G} \mathbf{z} \rightarrow \mathop{\text{minimize }}_\mathbf{z} \mathbf{z}^T \mathbf{L} \mathbf{z}\)
The general problem of minimizing some energy over a mesh subject to fixed value boundary conditions is so wide spread that libigl has a dedicated api for solving such systems.
Let us consider a general quadratic minimization problem subject to different common constraints:
\[\mathop{\text{minimize }}_\mathbf{z} \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} + \mathbf{z}^T \mathbf{B} + \text{constant},\]
subject to
\[\mathbf{z}_b = \mathbf{z}_{bc} \text{ and } \mathbf{A}_{eq} \mathbf{z} = \mathbf{B}_{eq},\]
where
This specification is overly general as we could write \(\mathbf{z}_b = \mathbf{z}_{bc}\) as rows of \(\mathbf{A}_{eq} \mathbf{z} = \mathbf{B}_{eq}\), but these fixed value constraints appear so often that they merit a dedicated place in the API.
In libigl, solving such quadratic optimization problems is split into two routines: precomputation and solve. Precomputation only depends on the quadratic coefficients, known value indices and linear constraint coefficients:
igl::min_quad_with_fixed_data mqwf;
igl::min_quad_with_fixed_precompute(Q,b,Aeq,true,mqwf);
The output is a struct mqwf
which contains the system matrix factorization
and is used during solving with arbitrary linear terms, known values, and
constraint in the right-hand sides:
igl::min_quad_with_fixed_solve(mqwf,B,bc,Beq,Z);
The output Z
is a \(n \times 1\) vector of solutions with fixed values
correctly placed to match the mesh vertices V
.
We saw above that min_quad_with_fixed_*
in libigl provides a compact way to
solve general quadratic programs. Let’s consider another example, this time
with active linear equality constraints. Specifically let’s solve the
bi-Laplace equation
or equivalently minimize the Laplace energy:
\[\Delta^2 z = 0 \leftrightarrow \mathop{\text{minimize }}\limits_z \frac{1}{2} \int\limits_S (\Delta z)^2 dA\]
subject to fixed value constraints and a linear equality constraint:
\(z_{a} = 1, z_{b} = -1\) and \(z_{c} = z_{d}\).
Notice that we can rewrite the last constraint in the familiar form from above:
\(z_{c} - z_{d} = 0.\)
Now we can assembly Aeq
as a \(1 \times n\) sparse matrix with a coefficient
\(1\) in the column corresponding to vertex \(c\) and a \(-1\) at \(d\). The right-hand
side Beq
is simply zero.
Internally, min_quad_with_fixed_*
solves using the Lagrange Multiplier
method. This method adds additional variables for each linear constraint (in
general a \(m \times 1\) vector of variables \(\lambda\)) and then solves the
saddle problem:
\[\mathop{\text{find saddle }}_{\mathbf{z},\lambda}\, \frac{1}{2}\mathbf{z}^T \mathbf{Q} \mathbf{z} + \mathbf{z}^T \mathbf{B} + \text{constant} + \lambda^T\left(\mathbf{A}_{eq} \mathbf{z} - \mathbf{B}_{eq}\right)\]
This can be rewritten in a more familiar form by stacking \(\mathbf{z}\) and \(\lambda\) into one \((m+n) \times 1\) vector of unknowns:
$$\mathop{\text{find saddle }}_{\mathbf{z},\lambda}, \frac{1}{2} \left( \mathbf{z}^{T} \lambda^{T} \right) \left( \begin{array}{cc} \mathbf{Q} & \mathbf{A}{eq}^{T}\ \mathbf{A}{eq} & 0 \end{array} \right) \left( \begin{array}{c} \mathbf{z}\ \lambda \end{array} \right) + \left( \mathbf{z}^{T} \lambda^{T} \right) \left( \begin{array}{c} \mathbf{B}\ -\mathbf{B}_{eq} \end{array} \right)
Differentiating with respect to \(\left( \mathbf{z}^T \lambda^T \right)\) reveals
a linear system and we can solve for \(\mathbf{z}\) and \(\lambda\). The only
difference from the straight quadratic minimization system, is that this
saddle problem system will not be positive definite. Thus, we must use a
different factorization technique (LDLT rather than LLT): libigl’s
min_quad_with_fixed_precompute
automatically chooses the correct solver in
the presence of linear equality constraints (Example 304).
We can generalize the quadratic optimization in the previous section even more by allowing inequality constraints. Specifically box constraints (lower and upper bounds):
\(\mathbf{l} \le \mathbf{z} \le \mathbf{u},\)
where \(\mathbf{l},\mathbf{u}\) are \(n \times 1\) vectors of lower and upper bounds and general linear inequality constraints:
\(\mathbf{A}_{ieq} \mathbf{z} \le \mathbf{B}_{ieq},\)
where \(\mathbf{A}_{ieq}\) is a \(k \times n\) matrix of linear coefficients and \(\mathbf{B}_{ieq}\) is a \(k \times 1\) matrix of constraint right-hand sides.
Again, we are overly general as the box constraints could be written as rows of the linear inequality constraints, but bounds appear frequently enough to merit a dedicated api.
Libigl implements its own active set routine for solving quadratric programs (QPs). This algorithm works by iteratively “activating” violated inequality constraints by enforcing them as equalities and “deactivating” constraints which are no longer needed.
After deciding which constraints are active at each iteration, the problem reduces to a quadratic minimization subject to linear equality constraints, and the method from the previous section is invoked. This is repeated until convergence.
Currently the implementation is efficient for box constraints and sparse non-overlapping linear inequality constraints.
Unlike alternative interior-point methods, the active set method benefits from a warm-start (initial guess for the solution vector \(\mathbf{z}\)).
igl::active_set_params as;
// Z is optional initial guess and output
igl::active_set(Q,B,b,bc,Aeq,Beq,Aieq,Bieq,lx,ux,as,Z);
Libigl has rudimentary support for extracting eigen pairs of a generalized eigen value problem:
\(Ax = \lambda B x\)
where \(A\) is a sparse symmetric matrix and \(B\) is a sparse positive definite matrix. Most commonly in geometry processing, we let \(A=L\) the cotangent Laplacian and \(B=M\) the per-vertex mass matrix (e.g. (8)[]). Typically applications will make use of the low frequency eigen modes. Analogous to the Fourier decomposition, a function \(f\) on a surface can be represented via its spectral decomposition of the eigen modes of the Laplace-Beltrami:
\(f = \sum\limits_{i=1}^\infty a_i \phi_i\)
where each \(\phi_i\) is an eigen function satisfying: \(\Delta \phi_i = \lambda_i \phi_i\) and \(a_i\) are scalar coefficients. For a discrete triangle mesh, a completely analogous decomposition exists, albeit with finite sum:
\(\mathbf{f} = \sum\limits_{i=1}^n a_i \phi_i\)
where now a column vector of values at vertices \(\mathbf{f} \in \mathcal{R}^n\) specifies a piecewise linear function and \(\phi_i \in \mathcal{R}^n\) is an eigen vector satisfying:
\(\mathbf{L} \phi_i = \lambda_i \mathbf{M} \phi_i\).
Note that Vallet & Levy (8)[] propose solving a symmetrized standard eigen problem \(\mathbf{M}^{-1/2}\mathbf{L}\mathbf{M}^{-1/2} \phi_i = \lambda_i \phi_i\). Libigl implements a generalized eigen problem solver so this unnecessary symmetrization can be avoided.
Often the sum above is truncated to the first \(k\) eigen vectors. If the low frequency modes are chosen, i.e. those corresponding to small \(\lambda_i\) values, then this truncation effectively regularizes \(\mathbf{f}\) to smooth, slowly changing functions over the mesh (e.g. (9)[]). Modal analysis and model subspaces have been used frequently in real-time deformation (e.g. (10)[]).
In Example 306), the first 5 eigen vectors
of the discrete Laplace-Beltrami operator are computed and displayed in
pseudo-color atop the beetle. Eigen vectors are computed using igl::eigs
(mirroring MATLAB’s eigs
). The 5 eigen vectors are placed into the columns
of U
and the eigen values are placed into the entries of S
:
SparseMatrix<double> L,M;
igl::cotmatrix(V,F,L);
igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_DEFAULT,M);
Eigen::MatrixXd U;
Eigen::VectorXd S;
igl::eigs(L,M,5,igl::EIGS_TYPE_SM,U,S);
Modern mesh-based shape deformation methods satisfy user deformation constraints at handles (selected vertices or regions on the mesh) and propagate these handle deformations to the rest of shape smoothly and without removing or distorting details. Libigl provides implementations of a variety of state-of-the-art deformation techniques, ranging from quadratic mesh-based energy minimizers, to skinning methods, to non-linear elasticity-inspired techniques.
The period of research between 2000 and 2010 produced a collection of techniques that cast the problem of handle-based shape deformation as a quadratic energy minimization problem or equivalently the solution to a linear partial differential equation.
There are many flavors of these techniques, but a prototypical subset are those that consider solutions to the bi-Laplace equation, that is a biharmonic function (11)[]. This fourth-order PDE provides sufficient flexibility in boundary conditions to ensure \(C^1\) continuity at handle constraints (in the limit under refinement) (12)[].
Let us first begin our discussion of biharmonic deformation, by considering biharmonic surfaces. We will casually define biharmonic surfaces as surface whose position functions are biharmonic with respect to some initial parameterization:
\(\Delta^2 \mathbf{x}' = 0\)
and subject to some handle constraints, conceptualized as “boundary conditions”:
\(\mathbf{x}'_{b} = \mathbf{x}_{bc}.\)
where \(\mathbf{x}'\) is the unknown 3D position of a point on the surface. So we are asking that the bi-Laplacian of each of spatial coordinate function to be zero.
In libigl, one can solve a biharmonic problem with igl::harmonic
and setting \(k=2\) (bi-harmonic):
// U_bc contains deformation of boundary vertices b
igl::harmonic(V,F,b,U_bc,2,U);
This produces a smooth surface that interpolates the handle constraints, but all original details on the surface will be smoothed away. Most obviously, if the original surface is not already biharmonic, then giving all handles the identity deformation (keeping them at their rest positions) will not reproduce the original surface. Rather, the result will be the biharmonic surface that does interpolate those handle positions.
Thus, we may conclude that this is not an intuitive technique for shape deformation.
Now we know that one useful property for a deformation technique is “rest pose reproduction”: applying no deformation to the handles should apply no deformation to the shape.
To guarantee this by construction we can work with deformation fields (ie. displacements) \(\mathbf{d}\) rather than directly with positions \(\mathbf{x}\). Then the deformed positions can be recovered as
\(\mathbf{x}' = \mathbf{x}+\mathbf{d}.\)
A smooth deformation field \(\mathbf{d}\) which interpolates the deformation fields of the handle constraints will impose a smooth deformed shape \(\mathbf{x}'\). Naturally, we consider biharmonic deformation fields:
\(\Delta^2 \mathbf{d} = 0\)
subject to the same handle constraints, but rewritten in terms of their implied deformation field at the boundary (handles):
\(\mathbf{d}_b = \mathbf{x}_{bc} - \mathbf{x}_b.\)
Again we can use igl::harmonic
with \(k=2\), but this time solve for the
deformation field and then recover the deformed positions:
// U_bc contains deformation of boundary vertices b
D_bc = U_bc - igl::slice(V,b,1);
igl::harmonic(V,F,b,D_bc,2,D);
U = V+D;
Biharmonic functions (whether positions or displacements) are solutions to the bi-Laplace equation, but also minimizers of the “Laplacian energy”. For example, for displacements \(\mathbf{d}\), the energy reads
\(\int\limits_S \|\Delta \mathbf{d}\|^2 dA,\)
where we define \(\Delta \mathbf{d}\) to simply apply the Laplacian coordinate-wise.
By linearity of the Laplace(-Beltrami) operator we can reexpress this energy in terms of the original positions \(\mathbf{x}\) and the unknown positions \(\mathbf{x}' = \mathbf{x} - \mathbf{d}\):
\(\int\limits_S \|\Delta (\mathbf{x}' - \mathbf{x})\|^2 dA = \int\limits_S \|\Delta \mathbf{x}' - \Delta \mathbf{x})\|^2 dA.\)
In the early work of Sorkine et al., the quantities \(\Delta \mathbf{x}'\) and \(\Delta \mathbf{x}\) were dubbed “differential coordinates” (13)[]. Their deformations (without linearized rotations) is thus equivalent to biharmonic deformation fields.
We can generalize biharmonic deformation by considering different powers of the Laplacian, resulting in a series of PDEs of the form:
\(\Delta^k \mathbf{d} = 0.\)
with \(k\in{1,2,3,\dots}\). The choice of \(k\) determines the level of continuity at the handles. In particular, \(k=1\) implies \(C^0\) at the boundary, \(k=2\) implies \(C^1\), \(k=3\) implies \(C^2\) and in general \(k\) implies \(C^{k-1}\).
int k = 2;// or 1,3,4,...
igl::harmonic(V,F,b,bc,k,Z);
In computer animation, shape deformation is often referred to as “skinning”. Constraints are posed as relative rotations of internal rigid “bones” inside a character. The deformation method, or skinning method, determines how the surface of the character (i.e. its skin) should move as a function of the bone rotations.
The most popular technique is linear blend skinning. Each point on the shape computes its new location as a linear combination of bone transformations:
\(\mathbf{x}' = \sum\limits_{i = 1}^m w_i(\mathbf{x}) \mathbf{T}_i \left(\begin{array}{c}\mathbf{x}_i\\1\end{array}\right),\)
where \(w_i(\mathbf{x})\) is the scalar weight function of the ith bone evaluated at \(\mathbf{x}\) and \(\mathbf{T}_i\) is the bone transformation as a \(4 \times 3\) matrix.
This formula is embarassingly parallel (computation at one point does not depend on shared data need by computation at another point). It is often implemented as a vertex shader. The weights and rest positions for each vertex are sent as vertex shader attributes and bone transformations are sent as uniforms. Then vertices are transformed within the vertex shader, just in time for rendering.
As the skinning formula is linear (hence its name), we can write it as matrix multiplication:
\(\mathbf{X}' = \mathbf{M} \mathbf{T},\)
where \(\mathbf{X}'\) is \(n \times 3\) stack of deformed positions as row vectors, \(\mathbf{M}\) is a \(n \times m\cdot dim\) matrix containing weights and rest positions and \(\mathbf{T}\) is a \(m\cdot (dim+1) \times dim\) stack of transposed bone transformations.
Traditionally, the weight functions \(w_j\) are painted manually by skilled rigging professionals. Modern techniques now exist to compute weight functions automatically given the shape and a description of the skeleton (or in general any handle structure such as a cage, collection of points, selected regions, etc.).
Bounded biharmonic weights are one such technique that casts weight computation as a constrained optimization problem (14)[]. The weights enforce smoothness by minimizing the familiar Laplacian energy:
\(\sum\limits_{i = 1}^m \int_S (\Delta w_i)^2 dA\)
subject to constraints which enforce interpolation of handle constraints:
\(w_i(\mathbf{x}) = \begin{cases} 1 & \text{ if } \mathbf{x} \in H_i\\ 0 & \text{ otherwise } \end{cases},\)
where \(H_i\) is the ith handle, and constraints which enforce non-negativity, parition of unity and encourage sparsity:
\(0\le w_i \le 1\) and \(\sum\limits_{i=1}^m w_i = 1.\)
This is a quadratic programming problem and libigl solves it using its active set solver or by calling out to Mosek.
Even with high quality weights, linear blend skinning is limited. In particular, it suffers from known artifacts stemming from blending rotations as as matrices: a weight combination of rotation matrices is not necessarily a rotation. Consider an equal blend between rotating by \(-\pi/2\) and by \(\pi/2\) about the \(z\)-axis. Intuitively one might expect to get the identity matrix, but instead the blend is a degenerate matrix scaling the \(x\) and \(y\) coordinates by zero:
\(0.5\left(\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&1\end{array}\right)+ 0.5\left(\begin{array}{ccc}0&1&0\\-1&0&0\\0&0&1\end{array}\right)= \left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&1\end{array}\right)\)
In practice, this means the shape shrinks and collapses in regions where bone weights overlap: near joints.
Dual quaternion skinning presents a solution (15). This method represents rigid transformations as a pair of unit quaternions, \(\hat{\mathbf{q}}\). The linear blend skinning formula is replaced with a linear blend of dual quaternions:
\(\mathbf{x}' = \cfrac{\sum\limits_{i=1}^m w_i(\mathbf{x})\hat{\mathbf{q}_i}} {\left\|\sum\limits_{i=1}^m w_i(\mathbf{x})\hat{\mathbf{q}_i}\right\|} \mathbf{x},\)
where \(\hat{\mathbf{q}_i}\) is the dual quaternion representation of the rigid transformation of bone \(i\). The normalization forces the result of the linear blending to again be a unit dual quaternion and thus also a rigid transformation.
Like linear blend skinning, dual quaternion skinning is best performed in the
vertex shader. The only difference being that bone transformations are sent as
dual quaternions rather than affine transformation matrices. Libigl supports
CPU-side dual quaternion skinning with the igl::dqs
function, which takes a
more traditional representation of rigid transformations as input and
internally converts to the dual quaternion representation before blending:
// vQ is a list of rotations as quaternions
// vT is a list of translations
igl::dqs(V,W,vQ,vT,U);
Skinning and other linear methods for deformation are inherently limited. Difficult arises especially when large rotations are imposed by the handle constraints.
In the context of energy-minimization approaches, the problem stems from comparing positions (our displacements) in the coordinate frame of the undeformed shape. These quadratic energies are at best invariant to global rotations of the entire shape, but not smoothly varying local rotations. Thus linear techniques will not produce non-trivial bending and twisting.
Furthermore, when considering solid shapes (e.g. discretized with tetrahedral meshes) linear methods struggle to maintain local volume, and they often suffer from shrinking and bulging artifacts.
Non-linear deformation techniques present a solution to these problems. They work by comparing the deformation of a mesh vertex to its rest position rotated to a new coordinate frame which best matches the deformation. The non-linearity stems from the mutual dependence of the deformation and the best-fit rotation. These techniques are often labeled “as-rigid-as-possible” as they penalize the sum of all local deformations' deviations from rotations.
To arrive at such an energy, let’s consider a simple per-triangle energy:
\(E_\text{linear}(\mathbf{X}') = \sum\limits_{t \in T} a_t \sum\limits_{\{i,j\} \in t} w_{ij} \left\| \left(\mathbf{x}'_i - \mathbf{x}'_j\right) - \left(\mathbf{x}_i - \mathbf{x}_j\right)\right\|^2\)
where \(\mathbf{X}'\) are the mesh’s unknown deformed vertex positions, \(t\) is a triangle in a list of triangles \(T\), \(a_t\) is the area of triangle \(t\) and \(\{i,j\}\) is an edge in triangle \(t\). Thus, this energy measures the norm of change between an edge vector in the original mesh \(\left(\mathbf{x}_i - \mathbf{x}_j\right)\) and the unknown mesh \(\left(\mathbf{x}'_i - \mathbf{x}'_j\right)\).
This energy is not rotation invariant. If we rotate the mesh by 90 degrees the change in edge vectors not aligned with the axis of rotation will be large, despite the overall deformation being perfectly rigid.
So, the “as-rigid-as-possible” solution is to append auxiliary variables \(\mathbf{R}_t\) for each triangle \(t\) which are constrained to be rotations. Then the energy is rewritten, this time comparing deformed edge vectors to their rotated rest counterparts:
$E_\text{arap}(\mathbf{X}',{\mathbf{R}1,\dots,\mathbf{R}{|T|}}) = \sum\limits_{t \in T} a_t \sum\limits_{{i,j}
\in t} w_{ij} \left| \left(\mathbf{x}‘_i - \mathbf{x}’_j\right)- \mathbf{R}_t\left(\mathbf{x}_i - \mathbf{x}_j\right)\right|^{2}.$
The separation into the primary vertex position variables \(\mathbf{X}'\) and the rotations \(\{\mathbf{R}_1,\dots,\mathbf{R}_{|T|}\}\) lead to strategy for optimization, too. If the rotations \(\{\mathbf{R}_1,\dots,\mathbf{R}_{|T|}\}\) are held fixed then the energy is quadratic in the remaining variables \(\mathbf{X}'\) and can be optimized by solving a (sparse) global linear system. Alternatively, if \(\mathbf{X}'\) are held fixed then each rotation is the solution to a localized Procrustes problem (found via \(3 \times 3\) SVD or polar decompostion). These two steps—local and global—each weakly decrease the energy, thus we may safely iterate them until convergence.
The different flavors of “as-rigid-as-possible” depend on the dimension and codimension of the domain and the edge-sets \(T\). The proposed surface manipulation technique by Sorkine and Alexa (16)[], considers \(T\) to be the set of sets of edges emanating from each vertex (spokes). Later, Chao et al. derived the relationship between “as-rigid-as-possible” mesh energies and co-rotational elasticity considering 0-codimension elements as edge-sets: triangles in 2D and tetrahedra in 3D (17)[]. They also showed how Sorkine and Alexa’s edge-sets are not a discretization of a continuous energy, proposing instead edge-sets for surfaces containing all edges of elements incident on a vertex (spokes and rims). They show that this amounts to measuring bending, albeit in a discretization-dependent way.
Libigl, supports these common flavors. Selecting one is a matter of setting the energy type before the precompuation phase:
igl::ARAPData arap_data;
arap_data.energy = igl::ARAP_ENERGY_TYPE_SPOKES;
//arap_data.energy = igl::ARAP_ENERGY_TYPE_SPOKES_AND_RIMS;
//arap_data.energy = igl::ARAP_ENERGY_TYPE_ELEMENTS; //triangles or tets
igl::arap_precomputation(V,F,dim,b,arap_data);
Just like igl::min_quad_with_fixed_*
, this precomputation phase only depends
on the mesh, fixed vertex indices b
and the energy parameters. To solve with
certain constraints on the positions of vertices in b
, we may call:
igl::arap_solve(bc,arap_data,U);
which uses U
as an initial guess and then computes the solution into it.
Libigl’s implementation of as-rigid-as-possible deformation takes advantage of the highly optimized singular value decomposition code from McAdams et al. (18)[] which leverages SSE intrinsics.
The concept of local rigidity will be revisited shortly in the context of surface parameterization.
Non-linear optimization is, unsurprisingly, slower than its linear cousins. In the case of the as-rigid-as-possible optimization, the bottleneck is typically the large number of polar decompositions necessary to recover best fit rotations for each edge-set (i.e. for each triangle, tetrahedron, or vertex cell). Even if this code is optimized, the number of primary degrees of freedom is tied to the discretization level, despite the deformations' low frequency behavior.
This invites two routes toward fast non-linear optimization. First, is it necessary (or even advantageous) to find so many best-fit rotations? Second, can we reduce the degrees of freedom to better reflect the frequency of the desired deformations.
Taken in turn, these optimizations culminate in a method which optimizes over the space of linear blend skinning deformations spanned by high-quality weights (i.e. manually painted ones or bounded biharmonic weights). This space is a low-dimensional subspace of all possible mesh deformations, captured by writing linear blend skinning in matrix form:
\(\mathbf{X}' = \mathbf{M}\mathbf{T}\)
where the mesh vertex positions in the \(n \times 3\) matrix \(\mathbf{X}'\) are replaced by a linear combination of a small number of degrees of freedom in the \((3+1)m \times 3\) stack of transposed “handle” transformations. Swapping in \(\mathbf{M}\mathbf{T}\) for \(\mathbf{X}'\) in the ARAP energies above immediately sees performance gains during the global solve step as \(m << n\).
The complexity of the local step—fitting rotations—is still bound to the original mesh discretization. However, if the skinning is well behaved, we can make the assumption that places on the shape with similar skinning weights will deform similarly and thus imply similar best-fit rotations. Therefore, we cluster edge-sets according to their representation in weight-space: where a vertex \(\mathbf{x}\) takes the coordinates \([w_1(\mathbf{x}),w_2(\mathbf{x}),\dots,w_m(\mathbf{x})]\). The number of clustered edge-sets show diminishing returns on the deformation quality so we may choose a small number of clusters, proportional to the number of skinning weight functions (rather than the number of discrete mesh vertices).
This proposed deformation model (19)[], can simultaneously be seen as a fast, subspace optimization for ARAP and as an automatic method for finding the best skinning transformation degrees of freedom.
A variety of user interfaces are supported via linear equality constraints on the skinning transformations associated with handles. To fix a transformation entirely we simply add the constraint:
\(\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right) \mathbf{T}_i^T = \hat{\mathbf{T}}_i^T,\)
where \(\hat{\mathbf{T}}_i^T\) is the \((3+1) \times 3\) transposed fixed transformation for handle \(i\).
To fix only the origin of a handle, we add a constraint requiring the transformation to interpolate a point in space (typically the centroid of all points with \(w_i = 1\):
\(\mathbf{c}'^T\mathbf{T}_i^T = \mathbf{c}^T,\)
where \(\mathbf{c}^T\) is the \(1 \times (3+1)\) position of the point at rest in transposed homogeneous coordinates, and \(\mathbf{c}'^T\) the point given by the user.
We can similarly fix just the linear part of the transformation at a handle, freeing the translation component (producing a “chickenhead” effect):
\(\left(\begin{array}{cccc} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{array}\right) \mathbf{T}_i^T = \hat{\mathbf{L}}_i^T,\)
where \(\hat{\mathbf{L}}_i^T\) is the fixed \(3 \times 3\) linear part of the transformation at handle \(i\).
And lastly we can allow the user to entirely free the transformation’s degrees of freedom, delegating the optimization to find the best possible values for all elements. To do this, we simply abstain from adding a corresponding constraint.
Being a subspace method, an immediate disadvantage is the reduced degrees of freedom. This brings performance, but in some situations limits behavior too much. In such cases one can use the skinning subspace to build an effective clustering of rotation edge-sets for a traditional ARAP optimization: forgoing the subspace substitution. This has an two-fold effect. The cost of the rotation fitting, local step drastically reduces, and the deformations are “regularized” according the clusters. From a high level point of view, if the clusters are derived from skinning weights, then they will discourage bending, especially along isolines of the weight functions. If handles are not known in advance, one could also cluster according to a “geodesic embedding” like the biharmonic distance embedding.
In this light, we can think of the “spokes+rims” style surface ARAP as a (slight and redundant) clustering of the per-triangle edge-sets.
Linear blend skinning (as above) deforms a mesh by propagating full affine transformations at handles (bones, points, regions, etc.) to the rest of the shape via weights. Another deformation framework, called “generalized barycentric coordinates”, is a special case of linear blend skinning (20)[]: transformations are restricted to pure translations and weights are required to retain affine precision. This latter requirement means that we can write the rest-position of any vertex in the mesh as the weighted combination of the control handle locations:
\(\mathbf{x} = \sum\limits_{i=1}^m w_i(\mathbf{x}) * \mathbf{c}_i,\)
where \(\mathbf{c}_i\) is the rest position of the $i$th control point. This simplifies the deformation formula at run-time. We can simply take the new position of each point of the shape to be the weighted combination of the translated control point positions:
\(\mathbf{x}' = \sum\limits_{i=1}^m w_i(\mathbf{x}) * \mathbf{c}_i'.\)
There are many different flavors of “generalized barycentric coordinates” (see table in “Automatic Methods” section, (20)[]). The vague goal of “generalized barycentric coordinates” is to capture as many properties of simplicial barycentric coordinates (e.g. for triangles in 2D and tetrahedral in 3D) for larger sets of points or polyhedra. Some generalized barycentric coordinates can be computed in closed form; others require optimization-based precomputation. Nearly all flavors require connectivity information describing how the control points form a external polyhedron around the input shape: a cage. However, a recent techinique does not require a cage (21)[]. This method ensures affine precision during optimization over weights of a smoothness energy with affine functions in its kernel:
\(\mathop{\text{min}}_\mathbf{W}\,\, \text{trace}(\frac{1}{2}\mathbf{W}^T \mathbf{A} \mathbf{W}), \text{subject to: } \mathbf{C} = \mathbf{W}\mathbf{C}\)
subject to interpolation constraints at selected vertices. If \(\mathbf{A}\) has affine functions in its kernel—that is, if \(\mathbf{A}\mathbf{V} = 0\)—then the weights \(\mathbf{W}\) will retain affine precision and we’ll have that:
\(\mathbf{V} = \mathbf{W}\mathbf{C}\)
the matrix form of the equality above. The proposed way to define \(\mathbf{A}\)
is to construct a matrix \(\mathbf{K}\) that measures the Laplacian at all
interior vertices and at all boundary vertices. The usual definition of the
discrete Laplacian (e.g. what libigl returns from igl::cotmatrix
), measures
the Laplacian of a function for interior vertices, but measures the Laplacian
of a function minus the normal derivative of a function for boundary
vertices. Thus, we can let:
\(\mathbf{K} = \mathbf{L} + \mathbf{N}\)
where \(\mathbf{L}\) is the usual Laplacian and \(\mathbf{N}\) is matrix that computes normal derivatives of a piecewise-linear function at boundary vertices of a mesh. Then \(\mathbf{A}\) is taken as quadratic form computing the square of the integral-average of \(\mathbf{K}\) applied to a function and integrated over the mesh:
\(\mathbf{A} = (\mathbf{M}^{-1}\mathbf{K})^2_\mathbf{M} = \mathbf{K}^T \mathbf{M}^{-1} \mathbf{K}.\)
Since the Laplacian \(\mathbf{K}\) is a second-order derivative it measures zero on affine functions, thus \(\mathbf{A}\) has affine functions in its null space. A short derivation proves that this implies \(\mathbf{W}\) will be affine precise (see (21)[]).
Minimizers of this “squared Laplacian” energy are in some sense discrete biharmonic functions. Thus they’re dubbed “biharmonic coordinates” (not the same as bounded biharmonic weights, which are not generalized barycentric coordinates).
In libigl, one can compute biharmonic coordinates given a mesh (V,F)
and a
list S
of selected control points or control regions (which act like skinning
handles):
igl::biharmonic_coordinates(V,F,S,W);
In computer graphics, we denote as surface parametrization a map from the surface to \(\mathbf{R}^2\). It is usually encoded by a new set of 2D coordinates for each vertex of the mesh (and possibly also by a new set of faces in one to one correspondence with the faces of the original surface). Note that this definition is the inverse of the classical differential geometry definition.
A parametrization has many applications, ranging from texture mapping to surface remeshing. Many algorithms have been proposed, and they can be broadly divided in four families:
Single patch, fixed boundary: these algorithm can parametrize a disk-like part of the surface given fixed 2D positions for its boundary. These algorithms are efficient and simple, but they usually produce high-distortion maps due to the fixed boundary.
Single patch, free boundary: these algorithms let the boundary deform freely, greatly reducing the map distortion. Care should be taken to prevent the border to self-intersect.
Global parametrization: these algorithms work on meshes with arbitrary genus. They initially cut the mesh in multiple patches that can be separately parametrized. The generated maps are discontinuous on the cuts (often referred as seams).
Global seamless parametrization: these are global parametrization algorithm that hides the seams, making the parametrization “continuous”, under specific assumptions that we will discuss later.
Harmonic parametrization (22)[] is a single patch, fixed boundary parametrization algorithm that computes the 2D coordinates of the flattened mesh as two harmonic functions.
The algorithm is divided in 3 steps:
The algorithm can be coded using libigl as follows:
Eigen::VectorXi bnd;
igl::boundary_loop(V,F,bnd);
Eigen::MatrixXd bnd_uv;
igl::map_vertices_to_circle(V,bnd,bnd_uv);
igl::harmonic(V,F,bnd,bnd_uv,1,V_uv);
where bnd
contains the indices of the boundary vertices, bnd_uv their position on the UV plane, and “1” denotes that we want to compute an harmonic function (2 will be for biharmonic, 3 for triharmonic, etc.). Note that each of the three
functions is designed to be reusable in other parametrization algorithms.
A UV parametrization can be visualized in the viewer with:
viewer.data().set_uv(V_uv);
The UV coordinates are then used to apply a procedural checkerboard texture to the mesh (Example 501).
Least squares conformal maps parametrization (23)[] minimizes the conformal (angular) distortion of the parametrization. Differently from harmonic parametrization, it does not need to have a fixed boundary.
LSCM minimizes the following energy:
\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \int_X \frac{1}{2}| \nabla \mathbf{u}^{\perp} - \nabla \mathbf{v} |^2 dA \]
which can be rewritten in matrix form as (24)[]:
\[ E_{LSCM}(\mathbf{u},\mathbf{v}) = \frac{1}{2} [\mathbf{u},\mathbf{v}]^t (L_c - 2A) [\mathbf{u},\mathbf{v}] \]
where \(L_c\) is the cotangent Laplacian matrix and \(A\) is a matrix such that \([\mathbf{u},\mathbf{v}]^t A [\mathbf{u},\mathbf{v}]\) is equal to the vector area of the mesh.
Using libigl, this matrix energy can be written in a few lines of code. The
cotangent matrix can be computed using igl::cotmatrix
:
SparseMatrix<double> L;
igl::cotmatrix(V,F,L);
Note that we want to apply the Laplacian matrix to the u and v coordinates at the same time, thus we need to extend it taking the left Kronecker product with a 2x2 identity matrix:
SparseMatrix<double> L_flat;
igl::repdiag(L,2,L_flat);
The area matrix is computed with igl::vector_area_matrix
:
SparseMatrix<double> A;
igl::vector_area_matrix(F,A);
The final energy matrix is \(L_{flat} - 2A\). Note that in this case we do not need to fix the boundary. To remove the null space of the energy and make the minimum unique, it is sufficient to fix two arbitrary vertices to two arbitrary positions. The full source code is provided in Example 502.
As-rigid-as-possible parametrization (25)[] is a powerful single-patch, non-linear algorithm to compute a parametrization that strives to preserve distances (and thus angles). The idea is very similar to ARAP surface deformation: each triangle is mapped to the plane trying to preserve its original shape, up to a rigid rotation.
The algorithm can be implemented reusing the functions discussed in the
deformation chapter: igl::arap_precomputation
and igl::arap_solve
. The only
difference is that the optimization has to be done in 2D instead of 3D and that
we need to compute a starting point. While for 3D deformation the optimization
is bootstrapped with the original mesh, this is not the case for ARAP
parametrization since the starting point must be a 2D mesh. In Example
503, we initialize the optimization with harmonic
parametrization. Similarly to LSCM, the boundary is free to deform to minimize
the distortion.
The design of tangent fields is a basic tool used to design guidance fields for uniform quadrilateral and hexahedral remeshing. Libigl contains an implementation of all the state-of-the-art algorithms to design N-RoSy fields and their generalizations.
In libigl, tangent unit-length vector fields are piece-wise constant on the faces of a triangle mesh, and they are described by one or more vectors per-face. The function
igl::nrosy(V,F,b,bc,b_soft,b_soft_weight,bc_soft,N,0.5,
output_field,output_singularities);
creates a smooth unit-length vector field (N=1) starting from a sparse set of constrained faces, whose indices are listed in b and their constrained value is specified in bc. The functions supports soft_constraints (b_soft, b_soft_weight, bc_soft), and returns the interpolated field for each face of the triangle mesh (output_field), plus the singularities of the field (output_singularities).
The singularities are vertices where the field vanishes (highlighted in red in
the figure above). igl::nrosy
can also generate N-RoSy fields (26)[],
which are a generalization of vector fields where in every face the vector is
defined up to a constant rotation of \(2\pi / N\). As can be observed in
the following figure, the singularities of the fields generated with different
N are of different types and they appear in different positions.
We demonstrate how to call and plot N-RoSy fields in Example
504, where the degree of the field can be change
pressing the number keys. igl::nrosy
implements the algorithm proposed in
(27)[]. N-RoSy fields can also be interpolated with many other algorithms,
see the library libdirectional for
a reference implementation of the most popular ones. For a complete categorization
of fields used in various applications see Vaxman et al. 2016 (28).
The previous parametrization methods were focusing on creating parametrizations of surface patches aimed at texture mapping or baking of other surface properties such as normals and high-frequency details. Global, seamless parametrization aims at parametrizing complex shapes with a parametrization that is aligned with a given set of directions for the purpose of surface remeshing. In libigl, we provide a reference implementation of the pipeline proposed in the mixed integer quadrangulation paper (27)[].
The first step involves the design of a 4-RoSy field (sometimes called cross field) that describes the alignment of the edges of the desired quadrilateral remeshing. The field constraints are usually manually specified or extracted from the principal curvature directions. In [Example 506], we simply fix one face in a random direction.
Given the cross field, we now want to cut the surface so that it becomes homeomorphic to a disk. While this could be done directly on the cross-field, we opt to perform this operation on its bisector field (a copy of the field rotated by 45 degrees) since it is more stable and generic. Working on the bisectors allow us to take as input generalized, non-orthogonal and non-unit length cross fields.
We thus rotate the field,
and we remove the rotation ambiguity by assigning to each face a u and a v direction. The assignment is done with a breadth-first search starting from a random face.
You can imagine this process as combing an hairy surface: you will be able to comb part of it, but at some point you will not be able to consistently comb the entire surface (Hairy ball theorem). The discontinuities in the combing define the cut graph:
Finally, we rotate the combed field by 45 degrees to undo the initial degrees rotation:
The combed cross field can be seen as the ideal Jacobian of the parametrization that will be computed in the next section.
The mesh is cut along the seams and a parametrization is computed trying to find two scalar functions whose gradient matches the combed cross field directions. This is a classical Poisson problem, that is solved minimizing the following quadratic energy:
\[ E(\mathbf{u},\mathbf{v}) = |\nabla \mathbf{u} - X_u|^2 + |\nabla \mathbf{v} - X_v|^2 \]
where \(X_u\) and \(X_u\) denotes the combed cross field. Solving this problem generates a parametrization whose u and v isolines are aligned with the input cross field.
We hide the seams by adding integer constraints to the Poisson problem that align the isolines on both sides of each seam (27).
Note that this parametrization can only be used for remeshing purposes, since it contains many overlaps.
A quad mesh can be extracted from this parametrization using libQEx (not included in libigl). The full pipeline is implemented in Example 505.
Anisotropic and non-uniform quad remeshing is important to concentrate the elements in the regions with more details. It is possible to extend the MIQ quad meshing framework to generate anisotropic quad meshes using a mesh deformation approach (29)[].
The input of the anisotropic remeshing algorithm is a sparse set of constraints
that define the shape and scale of the desired quads. This can be encoded as a
frame field, which is a pair of non-orthogonal and non-unit length vectors. The
frame field can be interpolated by decomposing it in a 4-RoSy field and a
unique affine transformation. The two parts can then be interpolated
separately, using igl::nrosy
for the cross field, and an harmonic interpolant
for the affine part.
After the interpolation, the surface is warped to transform each frame into an orthogonal and unit length cross (i.e. removing the scaling and skewness from the frame). This deformation defines a new embedding (and a new metric) for the surface.
The deformed surface can the be isotropically remeshed using the MIQ algorithm that has been presented in the previous section.
The UV coordinates of the deformed surface can then be used to transport the parametrization to the original surface, where the isolines will trace a quad mesh whose elements are similar to the shape prescribed in the input frame field.
Our implementation (Example 506) uses MIQ to generate the UV parametrization, but other algorithms could be applied: the only desiderata is that the generated quad mesh should be as isotropic as possible.
A quad mesh can be transformed in a planar quad mesh with Shape-Up (30), a local/global approach that uses the global step to enforce surface continuity and the local step to enforce planarity.
Example 507 planarizes a quad mesh until it satisfies a user-given planarity threshold.
An additional positive side effect of using matrices as basic types is that it is easy to exchange data between libigl and other software and libraries.
Geometry processing applications often require a considerable amount of computational time and/or manual input. Serializing the state of the application is a simple strategy to greatly increase the development efficiency. It allows to quickly start debugging just before the crash happens, avoiding to wait for the precomputation to take place every time and it also makes your experiments reproducible, allowing to quickly test algorithms variants on the same input data.
Serialization is often not considered in geometry processing due to the extreme difficulty in serializing pointer-based data structured, such as an half-edge data structure (OpenMesh, CGAL), or a pointer based indexed structure (VCG).
In libigl, serialization is much simpler, since the majority of the functions use basic types, and pointers are used in very rare cases (usually to interface with external libraries). Libigl bundles a simple and self-contained binary and XML serialization framework, that drastically reduces the overhead required to add serialization to your applications.
To de-/serialize a set of variables use the following method:
#include "igl/serialize.h"
bool b = true;
unsigned int num = 10;
std::vector<float> vec = {0.1,0.002,5.3};
// use overwrite = true for the first serialization to create or overwrite an
// existing file
igl::serialize(b,"B","filename",true);
// append following serialization to existing file
igl::serialize(num,"Number","filename");
igl::serialize(vec,"VectorName","filename");
// deserialize back to variables
igl::deserialize(b,"B","filename");
igl::deserialize(num,"Number","filename");
igl::deserialize(vec,"VectorName","filename");
Currently all fundamental data types (bool, int, float, double, …) are
supported, as well as std::string, basic STL
containers, dense and sparse
Eigen matrices and nestings of those. Some limitations apply to pointers.
Currently, loops or many to one type of link structures are not handled
correctly. Each pointer is assumed to point to a different independent object.
Uninitialized pointers must be set to nullptr
before de-/serialization to
avoid memory leaks. Cross-platform issues like little-, big-endianess is
currently not supported. To make user defined types serializable, just derive
from igl::Serializable
and trivially implementing the InitSerialization
method.
Assume that the state of your application is a mesh and a set of integer ids:
#include "igl/serialize.h"
struct State : public igl::Serializable
{
Eigen::MatrixXd V;
Eigen::MatrixXi F;
std::vector<int> ids;
void InitSerialization()
{
this->Add(V , "V");
this->Add(F , "F");
this->Add(ids, "ids");
}
};
If you need more control over the serialization of your types, you can override
the following functions or directly inherit from the interface
igl::SerializableBase
.
bool Serializable::PreSerialization() const;
void Serializable::PostSerialization() const;
bool Serializable::PreDeserialization();
void Serializable::PostDeserialization();
Alternatively, if you want a non-intrusive way of serializing your state you can overload the following functions:
namespace igl
{
namespace serialization
{
template <> inline void serialize(const State& obj,std::vector<char>& buffer)
{
::igl::serialize(obj.V,std::string("V"),buffer);
::igl::serialize(obj.F,std::string("F"),buffer);
::igl::serialize(obj.ids,std::string("ids"),buffer);
}
template <> inline void deserialize(State& obj,const std::vector<char>& buffer)
{
::igl::deserialize(obj.V,std::string("V"),buffer);
::igl::deserialize(obj.F,std::string("F"),buffer);
::igl::deserialize(obj.ids,std::string("ids"),buffer);
}
}
}
Equivalently, you can use the following macros:
SERIALIZE_TYPE(State,
SERIALIZE_MEMBER(V)
SERIALIZE_MEMBER(F)
SERIALIZE_MEMBER_NAME(ids,"ids")
)
All the former code is for binary serialization which is especially useful if
you have to handle larger data where the loading and saving times become more
important. For cases where you want to read and edit the serialized data by
hand we provide a serialization to XML files which is based on the library
tinyxml2. There you also have the
option to create a partial binary serialization of your data by using the binary
parameter, exposed in the function serialize_xml()
:
#include "igl/xml/serialize_xml.h"
int number;
// binary = false, overwrite = true
igl::serialize_xml(vec,"VectorXML",xmlFile,false,true);
// binary = true, overwrite = true
igl::serialize_xml(vec,"VectorBin",xmlFile,true,true);
igl::deserialize_xml(vec,"VectorXML",xmlFile);
igl::deserialize_xml(vec,"VectorBin",xmlFile);
For user defined types derive from XMLSerializable
.
The code snippets above are extracted from Example 601. We strongly suggest that you make the entire state of your application always serializable since it will save you a lot of troubles when you will be preparing figures for a scientific report. It is very common to have to do small changes to figures, and being able to serialize the entire state just before you take screenshots will save you many painful hours before a submission deadline.
Libigl can be interfaced with Matlab to offload numerically heavy computation to a Matlab script. The major advantage of this approach is that you will be able to develop efficient and complex user-interfaces in C++, while exploring the syntax and fast protototyping features of matlab. In particular, the use of an external Matlab script in a libigl application allows to change the Matlab code while the C++ application is running, greatly increasing coding efficiency.
We demonstrate how to integrate Matlab in a libigl application in Example 602. The example uses Matlab to compute the Eigenfunctions of the discrete Laplacian operator, relying on libigl for mesh IO, visualization and for computing the Laplacian operator.
Libigl can connect to an existing instance of Matlab (or launching a new one on Linux/MacOSX) using:
igl::mlinit(&engine);
The cotangent Laplacian is computed using igl::cotmatrix and uploaded to the Matlab workspace:
igl::cotmatrix(V,F,L);
igl::mlsetmatrix(&engine,"L",L);
It is now possible to use any Matlab function on the data. For example, we can see the sparsity pattern of L using spy:
igl::mleval(&engine,"spy(L)");
The results of Matlab computations can be returned back to the C++ application
igl::mleval(&engine,"[EV,~] = eigs(-L,10,'sm')");
igl::mlgetmatrix(&engine,"EV",EV);
and plotted using the libigl viewer.
To aid debugging, libigl also supplies functions to write Matlab .mat
“Workspaces”. This C++ snippet saves a mesh and it’s sparse Laplacian matrix to
a file:
igl::readOFF(TUTORIAL_SHARED_PATH "/fertility.off", V, F);
igl::cotmatrix(V,F,L);
igl::MatlabWorkspace mw;
mw.save(V,"V");
mw.save_index(F,"F");
mw.save(L,"L");
mw.write("fertility.mat");
Then this workspace can be loaded into a Matlab IDE:
load fertility.mat
The igl::MatlabWorkspace
depends on Matlab libraries to compile and run,
but—in contrast to the engine routines above—will avoid launching a Matlab
instance upon execution.
Eigen supplies a sophisticated API for printing its matrix types to the screen. Libigl has wrapped up a particularly useful formatting which makes it simple to copy standard output from a C++ program into a Matlab IDE. The code:
igl::readOFF(TUTORIAL_SHARED_PATH "/2triangles.off", V, F);
igl::cotmatrix(V,F,L);
std::cout<<igl::matlab_format(V,"V")<<std::endl;
std::cout<<igl::matlab_format((F.array()+1).eval(),"F")<<std::endl;
std::cout<<igl::matlab_format(L,"L")<<std::endl;
produces the output:
V = [
0 0 0
1 0 0
1 1 1
2 1 0
];
F = [
1 2 3
2 4 3
];
LIJV = [
1 1 -0.7071067811865476
2 1 0.7071067811865475
3 1 1.570092458683775e-16
1 2 0.7071067811865475
2 2 -1.638010440969447
3 2 0.6422285251880865
4 2 0.2886751345948129
1 3 1.570092458683775e-16
2 3 0.6422285251880865
3 3 -0.9309036597828995
4 3 0.2886751345948129
2 4 0.2886751345948129
3 4 0.2886751345948129
4 4 -0.5773502691896258
];
L = sparse(LIJV(:,1),LIJV(:,2),LIJV(:,3));
which is easily copied and pasted into Matlab for debugging, etc.
It is also possible to call libigl functions from matlab, compiling them as MEX functions. This can be used to offload to C++ code the computationally intensive parts of a Matlab application.
We provide a wrapper for igl::readOBJ
in Example 603.
We plan to provide wrappers for all our functions in the future, if you are
interested in this feature (or if you want to help implementing it) please let
us know.
The generation of high-quality triangle and tetrahedral meshes is a very common task in geometry processing. We provide wrappers in libigl to triangle and Tetgen.
A triangle mesh with a given boundary can be created with:
igl::triangulate(V,E,H,V2,F2,"a0.005q");
where E
is a set of boundary edges (#E by 2), H
is a set of 2D positions of
points contained in holes of the triangulation (#H by 2) and (V2
,F2
) is the
generated triangulation. Additional parameters can be passed to triangle
, to
control the quality: "a0.005q"
enforces a bound on the maximal area of the
triangles and a minimal angle of 20 degrees. In Example
604, the interior of a square (excluded a smaller square
in its interior) is triangulated.
Similarly, the interior of a closed manifold surface can be tetrahedralized
using the function igl::tetrahedralize
which wraps the Tetgen library (Example
605):
igl::tetrahedralize(V,F,"pq1.414", TV,TT,TF);
Ambient occlusion is a rendering technique used to calculate the exposure of each point in a surface to ambient lighting. It is usually encoded as a scalar (normalized between 0 and 1) associated with the vertice of a mesh.
Formally, ambient occlusion is defined as:
\[ A_p = \frac{1}{\pi} \int_\omega V_{p,\omega}(n \cdot \omega) d\omega \]
where \(V_{p,\omega}\) is the visibility function at p, defined to be zero if p is occluded in the direction \(\omega\) and one otherwise, and \(d\omega\) is the infinitesimal solid angle step of the integration variable \(\omega\).
The integral is usually approximated by casting rays in random directions around each vertex. This approximation can be computed using the function:
igl::ambient_occlusion(V,F,V_samples,N_samples,500,AO);
that given a scene described in V
and F
, computes the ambient occlusion of
the points in V_samples
whose associated normals are N_samples
. The
number of casted rays can be controlled (usually at least 300–500 rays are
required to get a smooth result) and the result is returned in AO
, as a
single scalar for each sample.
Ambient occlusion can be used to darken the surface colors, as shown in Example 606
Libigl supports read and writing to .png files via the stb image code.
With the viewer used in this tutorial, it is possible to render the scene in a
memory buffer using the function, igl::opengl::ViewerCore::draw_buffer
:
// Allocate temporary buffers for 1280x800 image
Eigen::Matrix<unsigned char,Eigen::Dynamic,Eigen::Dynamic> R(1280,800);
Eigen::Matrix<unsigned char,Eigen::Dynamic,Eigen::Dynamic> G(1280,800);
Eigen::Matrix<unsigned char,Eigen::Dynamic,Eigen::Dynamic> B(1280,800);
Eigen::Matrix<unsigned char,Eigen::Dynamic,Eigen::Dynamic> A(1280,800);
// Draw the scene in the buffers
viewer.core.draw_buffer(viewer.data,viewer.opengl,false,R,G,B,A);
// Save it to a PNG
igl::png::writePNG(R,G,B,A,"out.png");
In Example 607 a scene is rendered in a temporary png and used to texture a quadrilateral.
Extreme deformations or parametrizations with high-distortion might flip elements. This is undesirable in many applications, and it is possible to avoid it by introducing a non-linear constraints that guarantees that the area of every element remain positive.
Libigl can be used to compute Locally Injective Maps (31)[] using a variety of deformation energies. A simple deformation of a 2D grid is computed in Example 608.
Constructive solid geometry (CSG) is a technique to define a complex surface as the result of a number of set operations on solid regions of space: union, intersection, set difference, symmetric difference, complement. Typically, CSG libraries represent the inputs and outputs to these operations implicitly: the solid \(A\) is defined as the open set of points \(\mathbf{x}\) for which some function \(a(\mathbf{x})\) “returns true”. The surface of this shape is the closure of all points \(x\) in \(A\).
With this sort of representation, boolean operations are straightforward. For example, the union of solids \(A\) and \(B\) is simply
$A \cup B = {\mathbf{x} \left.\right|
a(\mathbf{x}) \text{ or } b(\mathbf{x})},$
the intersection is
$A \cap B = {\mathbf{x} \left.\right|
a(\mathbf{x}) \text{ and } b(\mathbf{x})},$
the difference \(A\) minus \(B\) is
$A \setminus B = {\mathbf{x} \left.\right|
a(\mathbf{x}) \text{ and not } b(\mathbf{x})},$
and the symmetric difference (XOR) is
$A \triangle B = {\mathbf{x} \left.\right|
\text{either } a(\mathbf{x}) \text{ or } b(\mathbf{x}) \text{ but not both }}.$
Stringing together many of these operations, one can design quite complex shapes. A typical CSG library might only keep explicit base-case representations of canonical shapes: half-spaces, quadrics, etc.
In libigl, we do currently not have an implicit surface representation. Instead we expect our users to be working with explicit triangle mesh boundary representations of solid shapes. CSG operations are much hard to compute robustly with boundary representations, but are nonetheless useful.
To compute a boolean operation on a triangle mesh with vertices VA
and
triangles FA
and another mesh VB
and FB
, libigl first computes a unified
“mesh arrangement” (see (32)[]) with vertices V
and triangles F
where all triangle-triangle
intersections have been “resolved”. That is, edges and vertices are added
exactly at the intersection lines, so the resulting non-manifold mesh (V,F)
has no self-intersections.
Then libigl labels each “cell” bounded by surfaces of the arrangement according to its winding number vector: winding number with respect to each input mesh \((w_A,w_B)\). Finally, according to the desired operation (e.g. union, intersection) the boundary of the corresponding cells are extracted.
Calling libigl’s boolean operations is simple. To compute the union of
(VA,FA)
and (VB,FB)
into a new mesh (VC,FC)
, use:
igl::copyleft::cgal::mesh_boolean(VA,FA,VB,FB,MESH_BOOLEAN_TYPE_UNION,VC,FC);
The following figure shows each boolean operation on two meshes.
The union, symmetric difference and “resolve” have the same outward appearance, but differ in their treatment of internal structures. The union has no internal surfaces: the triangles are not included in the output. The symmetric difference is the same set of triangles as the “resolve”, but internal surfaces have been reversed in orientation, indicating that the solid result of the operation. The “resolve” operation is not really a boolean operation, it is simply the result of resolving all intersections and gluing together coincident vertices, maintaining original triangle orientations.
Libigl also provides a wrapper igl::copyleft::cork::mesh_boolean
to the
cork, which is typically faster, but is not
always robust.
The previous section discusses using
igl::copyleft::cgal::mesh_boolean
to compute the result of a single boolean
operation on two input triangle meshes. When employing constructive solid
geometry (CSG) as a modeling paradigm, shapes are represented as the result of
many such binary operations. The sequence is stored in a binary tree.
Libigl uses exact arithmetic internally to construct the intermediary boolean
results robustly. “Rounding” this result to floating point (even double
precision) would cause problems if re-injected into a further boolean
operation. To facilitate CSG tree operations and encourage callers not to
call igl::copyleft::cgal::mesh_boolean
multiple times explicitly, libigl implements
a class igl::copyleft::cgal::CSGTree
. Leaf nodes of this class are simply “solid”
meshes (otherwise good input to igl::copyleft::cgal::mesh_boolean
). Interior nodes
of the tree combine two children with a boolean operation. Using the intializer
list constructor it is easy to hard-code specific tree constructions. Here’s an
example taking the intersection of a cube A and sphere B minus the union
of three cylinders:
// Compute result of (A ∩ B) \ ((C ∪ D) ∪ E)
igl::copyleft::cgal::CSGTree<MatrixXi> CSGTree =
{{{VA,FA},{VB,FB},"i"},{{{VC,FC},{VD,FD},"u"},{VE,FE},"u"},"m"};
Example 610 computes each intermediary CSG result and then the final composite.
Libigl contains a wide variety of geometry processing tools and functions for dealing with meshes and the linear algebra related to them: far too many to discuss in this introductory tutorial. We’ve pulled out a couple of the interesting functions in this chapter to highlight.
Libigl contains various mesh statistics, including face angles, face areas and the detection of singular vertices, which are vertices with more or less than 6 neighbours in triangulations or 4 in quadrangulations.
The example Statistics computes these quantities and does a basic statistic analysis that allows to estimate the isometry and regularity of a mesh:
Irregular vertices:
136/2400 (5.67%)
Areas (Min/Max)/Avg_Area Sigma:
0.01/5.33 (0.87)
Angles in degrees (Min/Max) Sigma:
17.21/171.79 (15.36)
The first row contains the number and percentage of irregular vertices, which is particularly important for quadrilateral meshes when they are used to define subdivision surfaces: every singular point will result in a point of the surface that is only C^{1}.
The second row reports the area of the minimal element, maximal element and the standard deviation. These numbers are normalized by the mean area, so in the example above 5.33 max area means that the biggest face is 5 times larger than the average face. An ideal isotropic mesh would have both min and max area close to 1.
The third row measures the face angles, which should be close to 60 degrees (90 for quads) in a perfectly regular triangulation. For FEM purposes, the closer the angles are to 60 degrees the more stable will the optimization be. In this case, it is clear that the mesh is of bad quality and it will probably result in artifacts if used for solving PDEs.
The problem of tetrahedralizing the interior of closed watertight surface mesh is a difficult, but well-posed problem (see our Tetgen wrappers). But black-box tet-meshers like TetGen will refuse input triangle meshes with self-intersections, open boundaries, non-manifold edges from multiple connected components. The problem is two-fold: self-intersections present contradictory facet constraints and self-intersections/open-boundaries/non-manifold edges make the problem of determining inside from outside ill-posed without further assumptions.
The first problem is easily solved by “resolving” all self-intersections.
That is, meshing intersecting triangles so that intersects occur exactly at
edges and vertices. This is accomplished using igl::selfintersect
.
TetGen can usually tetrahedralize the convex hull of this “resolved” mesh, and then the problem becomes determining which of these tets are inside the input mesh and which are outside. That is, which should be kept and which should be removed.
The “Generalized Winding Number” is a robust method for determined
inside and outside for troublesome meshes (33)[]. The generalized
winding number with respect to (V,F)
at some point \(\mathbf{p} \in
\mathcal{R}^3\) is defined as scalar function:
\[w(\mathbf{p}) = \sum\limits_{f_i\in F} \frac{1}{4\pi}\Omega_{f_i}(\mathbf{p})\]
where \(\Omega_{f_i}\) is the solid angle subtended by \(f_i\) (the ith face in
F
) at the point \(\mathbf{p}\). This solid angle contribution is a simple,
closed-form expression involving atan2
and some dot-products.
If (V,F)
does form a closed watertight surface, then \(w(\mathbf{p})=1\) if
\(\mathbf{p}\) lies inside (V,F)
and \(w(\mathbf{p})=0\) if outside (V,F)
. If
(V,F)
is closed but overlaps itself then \(w(\mathbf{p})\) is an integer value
counting how many (signed) times (V,F)
wraps around \(\mathbf{p}\). Finally,
if (V,F)
is not closed or not even manifold (but at least consistently
oriented), then \(w(\mathbf{p})\) tends smoothly toward 1 as \(\mathbf{p}\) is
more inside (V,F)
, and toward 0 as \(\mathbf{p}\) is more outside.
The study of mesh simplification or decimation is nearly as old as meshes themselves. Given a high resolution mesh with too many triangles, find a “well approximating” low resolution mesh with far fewer triangles. By now there are a variety of different paradigms for solving this problem and state-of-the-art methods are fairly advanced.
One family of mesh decimation methods operates by successively remove elements from the mesh. In particular, Hoppe advocates for successively remove or rather collapsing edges (34)[]. The generic form of this technique is to construct a sequence of n meshes from the initial high-resolution mesh \(M_0\) to the lowest resolution mesh \(M_n\) by collapsing a single edge:
\(M_0 \mathop{\longrightarrow}_\text{edge collapse} M_1 \mathop{\longrightarrow}_\text{edge collapse} \dots \mathop{\longrightarrow}_\text{edge collapse} M_{n-1} \mathop{\longrightarrow}_\text{edge collapse} M_n.\)
Hoppe’s original method and subsequent follow-up works propose various ways to choose the next edge to collapse in this sequence. Using a cost-based paradigm, one can maintain a priority queue of edges based on their “cost” (how much “worse” will my approximation be if I remove this edge?). The cheapest edge is collapsed and costs of neighboring edges are updated.
In order to maintain the topology (e.g. if the mesh is combinatorially as sphere or a torus etc.), one should assign infinite cost to edges whose collapse would alter the mesh topology. Indeed this happens if and only if the number of mutual neighbors of the endpoints of the collapsing edge is not exactly two!
If there exists a third shared vertex, then another face will be removed, but 2 edges will be removed. This can result in unwanted holes or non-manifold “flaps”.
There is also a one-off condition that no edges of a tetrahedron should be collapsed.
Because libigl (purposefully) does not center its implementations around a dynamic mesh data structure (e.g. half-edge datastructure), support for topology changes are limited. Nonetheless, libigl has support for isolated edge collapses, sequences of edge-collapses (each in O(log) time) and priority queue based decimation.
The simplest is igl::decimation
. By calling
igl::decimate(V,F,1000,U,G);
the mesh (V,F)
will be decimated to a new mesh (U,G)
so that G
has at
most 1000
faces. This uses default (naive) criteria for determining the cost
of an edge collapse and the placement of the merged vertex. Shortest edges are
collapsed first, and merged vertices are placed at edge midpoints.
One can also provide function handles (c++
lambda functions are convenient
here) cost_and_placement
and stopping_condition
for determining the
cost/placement of an edge collapse and the stopping condition respectively. For
example, the default version above is implemented as:
igl::decimate(V,F,shortest_edge_and_midpoint,max_m,U,G);
where shortest_edge_and_midpoint
assign the edge’s length as cost and its
midpoint as the merged vertex placement and max_m
counts the current number
of faces (valid collapses decrease count by 2) and returns true
if the count
drops below m=1000
.
One can also scratch deeper inside the decimation loop and call
igl::collapse_edge
directly. In order to operate efficiently, this routine
needs more than the usual (V,F)
mesh representation. We need E
a list of
edge indices, where E.row(i) --> [s,d]
; we need EMAP
which maps the
“half”-edges of each triangle in F
to its corresponding edge in E
so that
E.row(EMAP(f+i*F.rows)) --> [s,d]
if the edge across from the ith corner of the
fth face is [s,d]
(up to orientation); we need EF
and EI
which keep track
of the faces incident on each edge and across from which corner of those faces
the edges appears, so that EF(e,o) = f
and EI(e,o) = i
means that the edge
E.row(e) --> [s,d]
appears in the fth face across from its ith corner (for
o=0
the edge orientations should match, for o=1
the orientations are
opposite).
When a collapse occurs, the sizes of the F
,E
, etc. matrices do not change.
Rather rows corresponding to “removed” faces and edges are set to a special
constant value IGL_COLLAPSE_EDGE_NULL
. Doing this ensures that we’re able to
remove edges in truly constant time O(1).
Conveniently
IGL_COLLAPSE_EDGE_NULL==0
. This means most OPENGL style renderings ofF
will simply draw a bunch of 0-area triangles at the first vertex.
The following will collapse the first edge and place its merged vertex at the origin:
igl::collapse_edge(0,RowVector3d(0,0,0),V,F,E,EMAP,EF,EI);
If valid, then V
,F
,E
,EF
,EI
are adjusted accordingly.
This is powerful, but low level. To build a decimator around this you’d need to keep track which edges are left to collapse and which to collapse next. Fortunately, libigl also exposes a priority queue based edge collapse with function handles to adjust costs and placements.
The priority queue is implemented as a (ordered) set Q
or (cost,edge index)
pairs and a list of iterators Qit
so that Qit[e]
reveals the iterator in
Q
corresponding to the eth edge. Placements are stored in a #E list of
positions C
. When the following is called:
igl::collapse_edge(cost_and_placement,V,F,E,EMAP,EF,EI,Q,Qit,C);
the lowest cost edge collapse according to Q
is attempted. If valid, then
V
,F
,etc. are adjusted accordingly and that edge is “popped” from Q
. Using
Qit
its neighboring edges are also popped from Q
and re-inserted after
updating their costs according to cost_and_placement
, new placements are
remembered in C
. If not valid, then the edge is “popped” from Q
and
reinserted with infinite cost.
The Example 703 demonstrates using this priority queue based approach with the simple shortest-edge-midpoint cost/placement strategy discussed above.
In the Generalized Winding Number section, we
examined a robust method for determining whether points lie inside or outside
of a given triangle soup mesh. Libigl complements this algorithm with
accelerated signed and unsigned distance queries and “in element” queries for
planar triangle meshes and 3D tetrahedral meshes. These routines make use of
libigl’s general purpose axis-aligned bounding box hierarchy (igl/AABB.h
).
This class is lightweight and—by design—does not store a copy of the mesh
(taking it as inputs to its member functions instead).
For tetrahedral meshes, this is useful for “in element” or “point location”
queries: given a point \(\mathbf{q}\in\mathcal{R}^3\) and a tetrahedral mesh
\((V,T)\) determine in which tetrahedron \(\mathbf{q}\) lies. This is accomplished
in libigl for a tet mesh V,T
and a list of query points in the rows of Q
via the igl::in_element()
:
// Initialize AABB tree
igl::AABB<MatrixXd,3> tree;
tree.init(V,T);
VectorXi I;
igl::in_element(V,T,Q,tree,I);
the resulting vector I
is a list of indices into T
revealing the first
tetrahedron found to contain the corresponding point in Q
.
For overlapping meshes, a point \(\mathbf{q}\) may belong to more than one
tetrahedron. In those cases, one can find them all (not just the first) by
using the igl::in_element
overload with a SparseMatrix
as the output:
SparseMatrix<int> I;
igl::in_element(V,T,Q,tree,I);
now each row of I
reveals whether each tet contains the corresponding row in
Q
: I(q,e)!=0
means that point q
is in element e
.
For Triangle meshes, we use the AABB tree to accelerate point-mesh closest
point queries: given a mesh \((V,F)\) and a query point
\(\mathbf{q}\in\mathcal{R}^3\) find the closest point \(\mathbf{c} \in (V,F)\)
(where \(\mathbf{c}\) is not necessarily a vertex of \((V,F)\)). This is
accomplished for a triangle mesh V,F
and a list of points in the rows of P
via igl::point_mesh_squared_distance
:
VectorXd sqrD;
VectorXi I;
MatrixXd C;
igl::point_mesh_squared_distance(P,V,F,sqrD,I,C);
the output sqrD
contains the (unsigned) squared distance from each point in
P
to its closest point given in C
which lies on the element in F
given by
I
(e.g. from which one could recover barycentric coordinates, using
igl::barycentric_coordinates
).
If the mesh V,F
is static, but the point set P
is changing dynamically then
it’s best to reuse the AABB hierarchy that’s being built during
igl::point_mesh_squared_distance
:
igl::AABB tree;
tree.init(V,F);
tree.squared_distance(V,F,P,sqrD,I,C);
... // P changes, but (V,F) does not
tree.squared_distance(V,F,P,sqrD,I,C);
Finally, from the closest point or the winding number it’s possible to sign
this distance. In igl::signed_distance
we provide two methods for signing:
the so-called “pseudo-normal test” (35)[] and the generalized
winding number (33)[].
The pseudo-normal test (see also igl::pseudonormal_test
) assumes the input
mesh is a watertight (closed, non-self-intersecting, manifold) mesh. Then given
a query point \(\mathbf{q}\) and its closest point \(\mathbf{c} \in (V,F)\), it
carefully chooses an outward normal \(\mathbf{n}\) at \(\mathbf{c}\) so that
\(\text{sign}(\mathbf{q}-\mathbf{c})\cdot \mathbf{n}\) reveals whether
\(\mathbf{q}\) is inside \((V,F)\): -1, or outside: +1. This is a fast \(O(1)\) test
once \(\mathbf{c}\) is located, but may fail if V,F
is not watertight.
An alternative is to use the generalized winding
number to determine the sign. This is very robust to
unclean meshes V,F
but slower: something like \(O(\sqrt{n})\) once \(\mathbf{c}\)
is located.
In either case, the interface via igl::signed_distance
is:
// Choose type of signing to use
igl::SignedDistanceType type = SIGNED_DISTANCE_TYPE_PSEUDONORMAL;
igl::signed_distance(P,V,F,sign_type,S,I,C,N);
the outputs are as above for igl::point_mesh_squared_distance
but now S
contains signed (unsquared) distances and the extra output N
(only set when
type == SIGNED_DISTANCE_TYPE_PSEUDON
) contains the normals used for signing
with the pseudo-normal test.
Often 3D data is captured as scalar field defined over space \(f(\mathbf{x}) : \mathcal{R}^3 \rightarrow \mathcal{R}\). Lurking within this field, iso-surfaces of the scalar field are often salient geometric objects. The iso-surface at value \(v\) is composed of all points \(\mathbf{x}\) in \(\mathcal{R}^3\) such that \(f(\mathbf{x}) = v\). A core problem in geometry processing is to extract an iso-surface as a triangle mesh for further mesh-based processing or visualization. This is referred to as iso-contouring.
“Marching Cubes” (36) is a famous method for iso-contouring tri-linear functions \(f\) on a regular lattice (aka grid). The core idea of this method is to contour the iso-surface passing through each cell (if it does at all) with a predefined topology (aka connectivity) chosen from a look up table depending on the function values at each vertex of the cell. The method iterates (“marches”) over all cells (“cubes”) in the grid and stitches together the final, watertight mesh.
In libigl, igl::marching_cubes
constructs a triangle mesh (V,F)
from an
input scalar field S
sampled at vertex locations GV
of a nx
by ny
by
nz
regular grid:
igl::marching_cubes(S,GV,nx,ny,nz,V,F);
Models from the web occasionally arrive unorientated in the sense that the orderings of each triangles vertices do not consistently agree. Determining a consistent facet orientation for a mesh is essential for two-sided lighting (e.g., a cloth with red velvet on one side and gold silk on the other side) and for inside-outside determination(e.g., using generalized winding numbers).
For (open) surfaces representing two-sided sheets, libigl provides a routine to
force consistent orientations within each orientable patch
(igl::orientable_patches
) of a mesh:
igl::bfs_orient(F,FF,C);
This simple routine will use breadth-first search on each patch of the mesh to
enforce a consistent facet orientation in the output faces FF
.
For (closed or nearly closed) surfaces representing the boundary of a solid object, libigl provides a routine to reorient faces so that the vertex ordering corresponds to a counter-clockwise ordering of the vertices with a right-hand-rule normal pointing outward. This method (37)[] assumes that most of the universe is empty. That is, most points in space are outside of the solid object than inside. Points are sampled over surface patches. For each sample point, rays are shot into both hemispheres to compute average of the (distance weighted) ambient occlusion on each side. A patch is oriented so that the outward side is less occluded (lighter, i.e., facing more void space).
igl::embree::reorient_facets_raycast(V,F,FF,I);
The boolean vector I
reveals which rows of F
have been flipped in FF
.
The swept volume \(S\) of a moving solid object \(A\) can be defined as any point in space such that at one moment in time the point lies inside the solid. In other words, it is the union of the solid object transformed by the rigid motion \(f(t)\) over time:
\(S = \bigcup \limits_{t\in [0,1]} f(t) A.\)
The surface of the swept volume of a solid bounded by a triangle mesh undergoing a rigid motion with non-trivial rotation is not a surface exactly representably by triangle mesh: it will be a piecewise-ruled surface.
To see this, consider the surface swept by a single edge’s line segment as it performs a screw motion.
This means that if we’d like to the surface of the swept volume of a triangle mesh undergoing a rigid motion and we’d like the output to be another triangle mesh, then we’re going to have to be happy with some amount of approximation error.
With this in mind, the simplest method for computing an approximate swept volume is by exploiting an alternative definition of the swept volume based on signed distances:
$S = \left{ \mathbf{p} \middle| d(\mathbf{p},\partial S) < 0 \right} = \left{ \mathbf{p}
\middle|
\min\limits_{t \in [0,1]} d(\mathbf{p},f(t) \partial A) < 0 \right}$
If \(\partial A\) is a triangle mesh, then we can approximate this by 1) discretizing time at a finite step of steps \([0,\Delta t,2\Delta t, \dots, 1]\) and by 2) discretizing space with a regular grid and representing the distance field using trilinear interpolation of grid values. Finally the output mesh, \(\partial S\) is approximated by contouring using Marching Cubes (36).
This method is similar to one described by Schroeder et al. in 1994 (38), and the one used in conjunction with boolean operations by Garg et al. 2016 (39).
In libigl, if your input solid’s surface is represented by (V,F)
then the
output surface mesh will be (SV,SF)
after calling:
igl::copyleft::swept_volume(V,F,num_time_steps,grid_size,isolevel,SV,SF);
The isolevel
parameter can be set to zero to approximate the exact swept
volume, greater than zero to approximate a positive offset of the swept volume
or less than zero to approximate a negative offset.
Picking vertices and faces using the mouse is very common in geometry processing applications. While this might seem a simple operation, its implementation is not straightforward. Libigl contains a function that solves this problem using the Embree raycaster. Its usage is demonstrated in Example 708:
bool hit = igl::unproject_onto_mesh(
Vector2f(x,y),
F,
viewer.core.view * viewer.core.model,
viewer.core.proj,
viewer.core.viewport,
*ei,
fid,
vid);
This function casts a ray from the view plane in the view direction. Variables
x
and y
are
the mouse screen coordinates; view
, model
, proj
are the view, model and
projection matrix respectively; viewport
is the viewport in OpenGL format;
ei
contains a Bounding Volume
Hierarchy constructed
by Embree, and fid
and vid
are the picked face and vertex, respectively.
Vector fields on surfaces are commonly visualized by tracing [streamlines] (https://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines). Libigl
supports the seeding and tracing of streamlines, for both simple vector fields
and for N-rosy fields. The seeds for the streamlines are initialized using streamlines_init
,
and the lines are traced using streamlines_next
. Each call to streamlines_next
extends
each line by one triangle, allowing interactive rendering of the traced lines, as demonstrated
in Example 709.
The Scalable Locally Injective Maps (40) algorithm allows to compute locally injective maps on massive datasets. The algorithm shares many similarities with ARAP, but uses a reweighting scheme to minimize arbitrary distortion energies, including those that prevent the introduction of flips.
Example 710 contains three demos: (1) an example of large scale 2D parametrization, (2) an example of 2D deformation with soft constraints, and (3) an example of 3D deformation with soft constraints. The implementation in libigl is self-contained and relies on Eigen for the solution of the linear system used in the global step. An optimized version that relies on Pardiso is available here.
Given a coarse mesh (aka cage) with vertices V
and faces F
, one can createa
higher-resolution mesh with more vertices and faces by subdividing every
face. That is, each coarse triangle in the input is replaced by many smaller
triangles. Libigl has three different methods for subdividing a triangle mesh.
An “in plane” subdivision method will not change the point set or carrier surface of the mesh. New vertices are added on the planes of existing triangles and vertices surviving from the original mesh are not moved.
By adding new faces, a subdivision algorithm changes the combinatorics of the mesh. The change in combinatorics and the formula for positioning the high-resolution vertices is called the “subdivision rule”.
For example, in the in plane subdivision method of igl::upsample
, vertices
are added at the midpoint of every edge: \(v_{ab} = \frac{1}{2}(v_a + v_b)\) and
each triangle \((i_a,i_b,i_c)\) is replaced with four triangles:
\((i_a,i_{ab},i_{ca})\), \((i_b,i_{bc},i_{ab})\), \((i_{ab},i_{bc},i_{ca})\), and
\((i_{bc},i_{c},i_{ca})\). This process may be applied recursively, resulting in
a finer and finer mesh.
The subdivision method of igl::loop
is not in plane. The vertices of the
refined mesh are moved to weight combinations of their neighbors: the mesh is
smoothed as it is refined (41). This and other smooth subdivision
methods can be understood as generalizations of spline curves to surfaces. In
particular the Loop subdivision method will converge to a \(C^1\) surface as we
consider the limit of recursive applications of subdivision. Away from
“irregular” or “extraordinary” vertices (vertices of the original cage with
valence not equal to 6), the surface is \(C^2\). The combinatorics (connectivity
and number of faces) of igl::loop
and igl::upsample
are identical: the only
difference is that the vertices have been smoothed in igl::loop
.
Finally, libigl also implements a form of in plane “false barycentric
subdivision” in igl::false_barycentric_subdivision
. This method simply adds
the barycenter of every triangle as a new vertex \(v_{abc}\) and replaces each
triangle with three triangles \((i_a,i_b,i_{abc})\), \((i_b,i_c,i_{abc})\), and
\((i_c,i_a,i_{abc})\). In contrast to igl::upsample
, this method will create
triangles with smaller and smaller internal angles and new vertices will sample
the carrier surfaces with extreme bias.
A noisy function \(f\) defined on a surface \(\Omega\) can be smoothed using an energy minimization that balances a smoothing term \(E_S\) with a quadratic fitting term:
\(u = \operatorname{argmin}_u \alpha E_S(u) + (1-\alpha)\int_\Omega ||u-f||^2 dx\)
The parameter \(\alpha\) determines how aggressively the function is smoothed.
A classical choice for the smoothness energy is the Laplacian energy of the
function with zero Neumann boundary conditions, which is a form of the
biharmonic energy. It is constructed using the cotangent Laplacian L
and
the mass matrix M
: QL = L'*(M\L)
. Because of the implicit zero Neumann
boundary conditions however, the function behavior is significantly warped at
the boundary if \(f\) does not have zero normal gradient at the boundary.
In #[stein_2017] it is suggested to use the Biharmonic energy with natural
Hessian boundary conditions instead, which corresponds to the hessian energy
with the matrix QH = H'*(M2\H)
, where H
is a finite element Hessian and
M2
is a stacked mass matrix. The matrices H
and QH
are implemented in
libigl as igl::hessian
and igl::hessian_energy
respectively. An example
of how to use the function is given in Example 712.
In the following image the differences between the Laplacian energy with zero Neumann boundary conditions and the Hessian energy can be clearly seen: whereas the zero Neumann boundary condition in the third image bias the isolines of the function to be perpendicular to the boundary, the Hessian energy gives an unbiased result.
Our input is a set of points \(P_0\) (not necessarily part of any mesh), and a set of constraints \(S=\left\{S_1,S_2,...S_m\right\}\), where each constraint is defined on a different, and sparse, subset of \(P_0\). We wish to create a new set of points \(P\) that are close to the original set \(P_0\) (each point with corresponding indices), while adhering to the constraints. Other objectives, such as smoothness, can be employed. The constraints can be nonlinear, which makes the problem nonconvex, difficult, and without a guaranteed global optimum. A very popular lightweight approach to such problems is a local-global iterative algorithm, comprising these two steps:
For iteration \(k\):
The version we implement in libigl is the general version described by (30), and is in two files: <igl/shapeup.h>
and <igl/shapeup_local_projections.h>
. A demo implementing regularity constraints (creating a mesh in which each face is as regular as possible) is in Example 713.
The local step is instantiated by a function of type igl::shapeup_projection_function
. The global step is done by two functions: igl::shapeup_precomputation()
, which precomputes the matrices required for the global step, and igl::shapeup_solve()
, which solves the problem, according to the initial solution \(P_0\) and the input local projection function. The data struct igl::ShapeUpData
contains the information necessary to run the algorithm, and can be configured; for instance, the self-explanatory variable Maxiterations
.
The global step minimizes the following energy:
\[E_{total}=\lambda_{shape}E_{shape}+\lambda_{close}E_{close}+\lambda_{smooth}E_{smooth},\]
where the \(\lambda\) coefficients are encoded in igl::ShapeUpData
, and can be updated prior to calling igl::shapeup_precomputation()
. The \(E_{shape}\) component is the integration energy (fitting \(P_k\) to the local projections). The \(E_{close}\) component is adherence to positional constraints, given by b
and bc
parameters. The \(E_{smooth}\) component is an optional objective function, to minimize differences (in the Dirichlet sense) between points, encodes by “edges” in parameter E
. Both \(E_{close}\) and \(E_{shape}\) are also weighted by wClose
and wShape
function parameters, respectively.
Libigl is in active development, and we plan to focus on the following features in the next months:
A better and more consistent documentation, plus extending this tutorial to cover more libigl features.
Implement a mixed-integer solver which only uses Eigen to remove the dependency on CoMiSo.
Improve the robustness and performance of the active set QP solver. In particular, handle linearly dependent constraints.
Implement more mesh analysis functions, including structural analysis for masonry and 3D-printability analysis.
Increase support for point clouds and general polygonal meshes.
What would you like to see in libigl? Contact us! or post a feature request.
We encourage you to contribute to the library and to report problems and bugs. The best way to contribute new feature or bug fixes is to fork the libigl repository and to open a pull request on our github repository.
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