Mesh Segmentation via Spectral Embedding and Contour Analysis Speaker: Min Meng

Background knowledge

Spectrum of matrix Given an nxn matrix M Eigenvalues Eigenvectors By definition The spectrum of matrix M

The Spectral Theorem Let S be a real symmetric matrix of dimension n, the eigendecomposition of S Where are diagonal matrix of eigenvalues are eigenvectors are real, V are orthogonal

Spectral method Solve the problem by manipulating Eigenvalues Eigenvectors Eigenspace projections Combination of these quantities Which derived from an appropriately defined linear operator

Use of spectral method Use of eigenvalues Global shape descriptors Graph and shape matching

Use of spectral method Use of eigenvectors Spectral embedding K-D embedding

Use of spectral method Use of eigenprojections Project the signal into a different domain Mesh compression Remove high-frequency Spectral watermark Remove low-frequency

Mesh laplacians Mesh laplacian operators Linear operators Act on functions defined on a mesh Mesh laplacians

Combinatorial mesh laplacians Defined by the graph associated with mesh Adjacency matrix W Graph : Normalized graph: Geometric mesh laplacians

Overview

Outline 2D Spectral embedding - vertices 2D Contour analysis 1D Spectral embedding - faces line search with salience

2D Spectral projections-point Graph laplacian L Structural segmentability Geometric laplacian M Geometrical segmentability

Graph laplacian L Adjacency matrix W, graph laplacian L L is positive semi-definite and symmetric Its smallest eigenvalue Corresponding eigenvector v is constant vector Choose k=3 to spectral 2D embedding

Graph laplacian L Spectral projection Branch is retained Capture structural segmentability

Geometric laplacian M Geometric matrix W For edge e=(i, j) Others Geometric laplacian M

If an edge e=(i, j) Takes a large weight Mesh vertices from concave region Pulled close Geometric segmentability

Contour analysis Segmentability analysis Sampling points (faces)

Contour extract

Contour Convexity Area-based Struggle with boundary defects perimeter-based Sensitive to noise Combinational measure

Contour Convexity

Convexity and Segmentability Not exactly the same concept

Inner distance Consider two points Inner distance defined as the length of the shortest path connecting them within O Insensitive to shape bending

Multidimensional scaling (MDS) Provide a visual representation of the pattern of proximities

Segmentability analysis Segmentability score Four steps : If return Compute embedding of via MDS if return If return Compute embedding of via MDS if return

Iterations of spectral cut

Sampling points (faces) Integrated bending score (IBS) I is inner distance E is Euclidean distance

Sampling points (faces) Two samples The first sample s1, maximizes IBS The second s2, has largest distance from s1 Sample points reside on different parts

Salience-guided spectral cut

Spectral 1D embedding -faces Compute matrix A Adjacent faces Construct the dual graph of mesh is the shortest path between their dual vertices

Spectral 1D embedding -faces Nystrom approximation Let If Approximate eigenvector of A

Spectral 1D embedding -faces Given sample faces

salient cut: line search Part salience Sub-mesh M, the part Q Vs : part size Vc : cut strength Vp : part protrusiveness Require an appropriate weighting between three factors

salient cut: line search Part salience When L used, When M used,

Experimental results

L-embedding

Pro.

Segmentability analysis : automatic Graph laplacian - L Geometric laplacian - M MDS based on inner distance

Robustness of sampling Two samples reside on different parts

Cor. Segmentation measure Salience measure Manually searched automatic

Thanks!

Q&A