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Mesh Segmentationvia
Spectral Embeddingand Contour Analysis
Speaker: Min Meng2007.11.22
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Background knowledge
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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
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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
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Spectral method
• Solve the problem by manipulating
• Eigenvalues• Eigenvectors• Eigenspace projections• Combination of these quantities
• Which derived from an appropriately defined linear operator
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Use of spectral method
• Use of eigenvalues
• Global shape descriptors
• Graph and shape matching
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Use of spectral method
• Use of eigenvectors
• Spectral embedding
• K-D embedding
(1 ) (1 )T
k kX V V X
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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
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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
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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
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Overview
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Outline
• 2D Spectral embedding - vertices
• 2D Contour analysis
• 1D Spectral embedding - faces line search with salience
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2D Spectral projections-point
• Graph laplacian L• Structural segmentability
• Geometric laplacian M• Geometrical segmentability
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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
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Graph laplacian L
• Spectral projection• Branch is retained• Capture structural segmentability
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Geometric laplacian M
• Geometric matrix W• For edge e=(i, j)
• Others
• Geometric laplacian M
0ijW
M D W
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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 <
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Contour analysis
• Segmentability analysis
• Sampling points (faces)
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Contour extract
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Contour Convexity
• Area-based
Struggle with boundary defects
• perimeter-based
• Sensitive to noise
• Combinational measure
(0,1]
1
C
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Contour Convexity
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Convexity and Segmentability
• Not exactly the same concept
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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
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Multidimensional scaling (MDS)
• Provide a visual representation of the pattern of proximities
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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
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Iterations of spectral cut
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Sampling points (faces)
• Integrated bending score (IBS)
• I is inner distance• E is Euclidean distance
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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
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Salience-guided spectral cut
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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
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Spectral 1D embedding -faces
• Nystrom approximation• Let
• If
• Approximate eigenvector of A
TX U U
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Spectral 1D embedding -faces
• Given sample faces ,s tf f
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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
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salient cut: line search
• Part salience
• When L used,• When M used,
0.1, 0.3, 0.6
0.1, 0.6, 0.3
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Experimental results
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L-embedding
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Pro.
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Segmentability analysis :automatic• Graph laplacian - L• Geometric laplacian - M• MDS based on inner distance
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Robustness of sampling
• Two samples reside on different parts
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Cor.
• Segmentation measure• Salience measure
0.03 Manuall
y searche
d
0.1 automatic
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Thanks!
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Q&A