Mesh Segmentationvia

Spectral Embeddingand Contour Analysis

Speaker: Min Meng2007.11.22

Background knowledge

Spectrum of matrix

• Given an nxn matrix M• Eigenvalues

• Eigenvectors

• By definition

• The spectrum of matrix M

1 2( ) { , , , }nM

1 2 n

1 2, , , nv v v

for 1, ,i i iM i n v v

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

1

nT T

i i ii

S V V

v v

1 2[ ]nV v v v

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

(1 ) (1 )T

k kX V V X

Use of spectral method

• Use of eigenprojections

• Project the signal into a different domain

• Mesh compression• Remove high-frequency

• Spectral watermark• Remove low-frequency

TX V X

Mesh laplacians

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

• Mesh laplacians

1

( )

( ) ( )i i ij i jj N i

Lf b w f f

1L B S

Mesh laplacians

• Combinatorial mesh laplacians• Defined by the graph associated with mesh• Adjacency matrix W

• Graph :• Normalized graph:

• Geometric mesh laplacians

1/ 2 1/ 2Q D KD

K D W

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

L D W

1 0

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

0ijW

M D W

Geometric laplacian M

• If an edge e=(i, j)• • Takes a large weight

• Mesh vertices from concave region• Pulled close• Geometric segmentability

0 or 0i jk k <

Contour analysis

• Segmentability analysis

• Sampling points (faces)

Contour extract

Contour Convexity

• Area-based

Struggle with boundary defects

• perimeter-based

• Sensitive to noise

• Combinational measure

(0,1]

1

C

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

,x y O

( , , )d x y O

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

*( )LS

( ) 1 ( )S C

( )LS *L L

( )MS *M M

*( )MS

0.1

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

( , )l mDist f f

Spectral 1D embedding -faces

• Nystrom approximation• Let

• If

• Approximate eigenvector of A

TX U U

Spectral 1D embedding -faces

• Given sample faces ,s tf f

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,

0.1, 0.3, 0.6

0.1, 0.6, 0.3

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

0.03 Manuall

y searche

d

0.1 automatic

Thanks!

Q&A