Geodesic Bending Invariants for Surfaces¶

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This tour explores the computation of bending invariants of surfaces.

Bending Invariants¶

Bending invariants replace the position of the vertices in a shape $\Ss$ (2-D or 3-D) by new positions that are insensitive to isometric deformation of the shape. This defines a bending invariant signature that can be used for surface matching.

Bending invariant were introduced in EladKim03. A related method was developped for brain flattening in SchwShWolf89. This method is related to the Isomap algorithm for manifold learning TenSolvLang03.

We assume that $Ss$ has some Riemannian metric, for instance coming from the embedding of a surface in 3-D Euclidian space, or by restriction of the Euclian 2-D space to a 2-D sub-domain (planar shape). One thus can compute the geodesic distance $d(x,x')$ between points $x,x' \in \Ss$.

The bending invariant $\tilde \Ss$ of $\Ss$ is defined as the set of points $Y = (y_i)_j \subset \RR^d$ that are optimized so that the Euclidean distance between points in $Y$ matches as closely the geodesic distance between points in $X$, i.e. $$ \forall i, j, \quad \norm{y_i-y_j} \approx d(x_i,x_j) $$

Multi-dimensional scaling (MDS) is a class of method that aims at computing such a set of points $Y \in \RR^{d \times N}$ in $\RR^d$ such that $$ \forall i, j, \quad \norm{y_i-y_j} \approx \de_{i,j} $$ where $\de \in \RR^{N \times N}$ is a input data matrix. For a detailed overview of MDS, we refer to the book BorgGroe97

In this tour, we apply two specific MDS algorithms (strain and stress minimization) with input $\de_{i,j} = d(x_i,x_j)$.

3-D Surfaces and Geodesic Distances¶

We consider here a syrface $\Ss \subset \RR^3$.

Load a mesh of $N$ vertices that discretizes this surfaces.

Display it.

The geodesic distance map $U(x) = d(x,x_i)$ to a starting point $x_i$ can be computed in $O(N \log(N))$ operations on a mesh of $N$ vertices using the Fast Marching algorithm.

Extract a bunch of geodesic shortest paths from $x_i$ to randomly selected vertices $ (x_j)_{j \in J} $.

Display the distance $U$ on the 3-D mesh together with the geodesic paths.

The geodesic distance matrix $\de \in \RR^{N \times N}$ is defined as $$ \forall i,j=1,\ldots,N, \quad \de_{i,j} = d(x_i,x_j). $$

Exercise 1

Compute the geodesic distance matrix $\de$. It is going to take some of time. progressbar(i,N);

Bending Invariant with Strain Minimization¶

The goal is to compute a set of points $Y = (y_i)_{i=1}^N$ in $\RR^d$, (here we use $d=2$) stored in a matrix $Y \in \RR^{d \times N}$ such that $$ \forall i, j, \quad D^2(Y)_{i,j} \approx \de_{i,j}^2 \qwhereq D^2(Y)_{i,j} = \norm{y_i-y_j}^2. $$

Target dimensionality $d$.

This can be achieved by minimzing a $L^2$ loss $$ \umin{Y} \norm{ D^2(Y)-\de^2 }^2 = \sum_{i<j} \abs{ \norm{y_i-y_j}^2 - \de_{i,j}^2 }^2. $$

Strain minimization consider instead the following weighted $L^2$ loss (so-called strain) $$ \umin{Y \in \RR^{d \times N} } \text{Strain}(Y) = \norm{ J ( D^2(Y)-\de^2 ) J }^2 $$ where $J$ is the so-called centering matrix $$ J_{i,j} = \choice{ 1-1/N \qifq i=j, \\ -1/N \qifq i \neq j. }$$

Using the properties of squared-distance matrices $D^2(Y)$, one can show that $$ \norm{ J ( D^2(Y)-\de^2 ) J }^2 = \norm{ Y Y^* - K }^2 \qwhereq K = - \frac{1}{2} J \de^2 J. $$

The solution to this (non-convex) optimization problem can be computed exactly as the rows of $Y$ being the two leading eigenvectors of $K$ propery rescaled.

Display the bending invariant surface.

Bending Invariant with Stress Minimization¶

The stress functional does not have geometrical meaning. An alternative MDS method directly minimizes a geometric loss, the so-called Stress $$ \umin{Y \in \RR^{d \times N} } \text{Stress}(Y) = \norm{ D(Y)-\de }^2 = \sum_{i<j} \abs{ \norm{y_i-y_j} - \de_{i,j} }^2. $$ It is possible to find a local minimizer of this energy by various descent algorithms, as initially proposed by Kruskal64

Operator to compute the distance matrix $D(Y)$.

The SMACOF (Scaling by majorizing a convex function) algorithm solves at each iterations a quadratic energy, that is guaranteed to diminish the value of the Strain. It was introduced by Leeuw77

It computes iterates $X^{(\ell)}$ as $$ X^{(\ell+1)} = \frac{1}{N} X^{(\ell)} B(D(X^{(\ell)}))^*, $$ where $$ B(D) = \choice{ -\frac{\de_{i,j}}{D_{i,j}} \qifq i \neq j, \\ -\sum_{k} B(D)_{i,k} \qifq i = j. } $$

Initialize the positions for the algorithm.

Operator $B$.

Update the positions.

Exercise 2

Perform the SMACOF iterative algorithm. Save in a variable |s(l)| the values of Stress$( X^{(\ell)} )$.