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Alec Jacobson 6b245d62ea floor/ceil, slice and sort exampels

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title: libigl Tutorial author: Alec Jacobson, Daniele Pannozo and others date: 20 June 2014 css: style.css html header: <script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>

Introduction

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 live coding and interactive examples, we demonstrate how to accomplish various common geometry processing tasks such as computation of differential quantities and operators, real-time deformation, global parametrization, numerical optimization and mesh repair. Each section of these lecture notes links to a cross-platform example application.

Basic Usage
- 100_FileIO: Example of reading/writing mesh files
- 101_Serialization: Example of using the XML serialization framework
- 102_DrawMesh: Example of plotting a mesh
Chapter 2: Discrete Geometric Quantities and Operators
Chapter 3: Matrices and Linear Algebra
- 301 Slice
- 301 Sort
  - Other Matlab-style functions

Compilation Instructions

All examples depends on glfw, glew and anttweakbar. A copy of the sourcecode of each library is provided together with libigl and they can be precompiled using:

Alec: Is this just compiling the dependencies? Then perhaps rename compile_dependencies_*

sh compile_macosx.sh (MACOSX)
sh compile_linux.sh (LINUX)
compile_windows.bat (Visual Studio 2012)

Every example can be compiled by using the cmake file provided in its folder. On Linux and MacOSX, you can use the provided bash script:

sh ../compile_example.sh

(Optional: compilation with libigl as static library)

By default, libigl is a headers only library, thus it does not require compilation. However, one can precompile libigl as a statically linked library. See ../README.md in the main directory for compilations instructions to produce libigl.a and other libraries. Once compiled, these examples can be compiled using the CMAKE flag -DLIBIGL_USE_STATIC_LIBRARY=ON:

../compile_example.sh -DLIBIGL_USE_STATIC_LIBRARY=ON

Chapter 2: Discrete Geometric Quantities and Operators

This chapter illustrates a few discrete quantities that libigl can compute on a mesh. This also provides an introduction to basic drawing and coloring routines in our example viewer. Finally, we construct popular discrete differential geometry operators.

Normals

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.

Per-face

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.

Per-vertex

Storing normals at vertices, Phong or Gouraud shading will interpolate shading inside mesh triangles to produce smooth(er) renderings. Most techniques for computing per-vertex normals take an average of incident face normals. The techniques vary with respect to their different weighting schemes. Uniform weighting is heavily biased by the discretization choice, where as area-based or angle-based weighting is more forgiving.

The typical half-edge style computation of area-based weights might look something like this:

N.setZero(V.rows(),3);
for(int i : vertices)
{
  for(face : incident_faces(i))
  {
    N.row(i) += face.area * face.normal;
  }
}
N.rowwise().normalize();

Without a half-edge data-structure it may seem at first glance that looping over incident faces---and thus constructing the per-vertex normals---would be inefficient. 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();

Per-corner

Storing normals per-corner is an efficient an 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 a specified dihedral angle (e.g. 20°).

Gaussian Curvature

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 [][#meyer_2003].

Just like the continuous analog, our discrete Gaussian curvature reveals elliptic, hyperbolic and parabolic vertices on the domain.

Curvature Directions

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 call principal directions and are always orthogonal.

Mean curvature is defined simply 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 [][#meyer_2003].

#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_VORONOI,M);
igl::invert_diag(M,Minv);
HN = -Minv*(L*V);
H = (HN.rowwise().squaredNorm()).array().sqrt();

Combined with the angle defect definition of discrete Gaussian curvature, one can define principal curvatures and use least squares fitting to find directions [][#meyer_2003].

Alternatively, a robust method for determining principal curvatures is via quadric fitting [][#pannozo_2010]. In the neighborhood around every vertex, a best-fit quadric is found and principal curvature values and directions are sampled from this quadric. With these in tow, one can compute mean curvature and Gaussian curvature as sums and products respectively.

This is an example of syntax highlighted code:

#include <foo.html>
int main(int argc, char * argv[])
{
  return 0;
}

Gradient

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=0}^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.

$Hat function \phi_i is one at vertex i, zero at all other vertices, and linear on incident triangles.$

Thus gradients of such piecewise linear functions are simply sums of gradients of the hat functions:

$\nabla f(\mathbf{x}) \approx \nabla \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x}), f_i = \sum\limits_{i=0}^n \nabla \phi_i(\mathbf{x}), f_i.$

This reveals that the gradient is a linear function of the vector of f_i values. Because \phi_i are linear in each triangle their gradient 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][#jacobson_thesis_2013]. Libigl's gradMatAlec: check name function computes \mathbf{G} for triangle and tetrahedral meshes:

Laplacian

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 edge {ij}. This oft produced 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));
    }
  }
}

Without a half-edge data-structure it may seem at first glance that looping over one-rings, and thus constructing the Laplacian would be inefficient. However, 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. Alec: cite Ariel reconstruction paper.

The operator applied to mesh vertex positions amounts to smoothing by flowing the surface along the mean curvature normal direction. This is equivalent to minimizing surface area.

Mass matrix

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 [#meyer_2003][]. The inverse of this matrix is also very useful as it transforms integrated quantities into point-wise quantities, e.g.:

\nabla 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).

Alternative construction of Laplacian

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 \nabla f 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 gradient. Naturally, \mathbf{G}^T is n \times md sparse matrix which takes vector values stored at triangle faces to scalar divergence values at vertices.

Chapter 3: Matrices and Linear Algebra

Libigl relies heavily on the Eigen library for dense and sparse linear algebra routines. Besides geometry processing routines, libigl has a few linear algebra routines which bootstrap Eigen and make Eigen feel even more like a high-level algebra library like Matlab.

Slice

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:

VectorXi R,C;
MatrixXd A,B;
...
igl::slice(A,R,C,B);

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);

Sort

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.

Other Matlab-style functions

Libigl implements a variety of other routines with the same api and functionality as common matlab functions.

igl::any_of 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::cross Cross product per-row
igl::dot dot product per-row
igl::find Find subscripts of non-zero entries
igl::floot 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::jet Quantized colors along the rainbow.
igl::kronecker_product Compare to Matlab's kronprod
igl::median Compute the median per column
igl::mode Compute the mode per column
igl::orth Orthogonalization of a basis
igl::speye Identity as sparse matrix

[#meyer_2003]: Mark Meyer and Mathieu Desbrun and Peter Schröder and Alan H. Barr, "Discrete Differential-Geometry Operators for Triangulated 2-Manifolds," 2003. [#pannozo_2010]: Daniele Pannozo, Enrico Puppo, Luigi Rocca, "Efficient Multi-scale Curvature and Crease Estimation," 2010. [#jacobson_thesis_2013]: Alec Jacobson, Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes, 2013. [#kazhdan_2012]: Michael Kazhdan, Jake Solomon, Mirela Ben-Chen, "Can Mean-Curvature Flow Be Made Non-Singular," 2012.

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