Shape Analysis and Retrieval Statistical Shape Descriptors Notes courtesy of Funk et al., SIGGRAPH 2004

Outline: Shape Descriptors Statistical Shape Descriptors Singular Value Decomposition (SVD)

Shape Matching General approach: Define a function that takes in two models and returns a measure of their proximity. D,D, M1M1 M1M1 M3M3 M2M2 M 1 is closer to M 2 than it is to M 3

Shape Descriptors Shape Descriptor: A structured abstraction of a 3D model that is well suited to the challenges of shape matching Descriptors 3D Models D, D,

Matching with Descriptors Preprocessing  Compute database descriptors Run-Time 3D Query Shape Descriptor 3D Database Best Matches

Matching with Descriptors Preprocessing  Compute database descriptors Run-Time  Compute query descriptor 3D Query Shape Descriptor 3D Database Best Matches

Matching with Descriptors Preprocessing  Compute database descriptors Run-Time  Compute query descriptor  Compare query descriptor to database descriptors 3D Query Shape Descriptor 3D Database Best Matches

Matching with Descriptors Preprocessing  Compute database descriptors Run-Time  Compute query descriptor  Compare query descriptor to database descriptors  Return best Match(es) 3D Query Shape Descriptor 3D Database Best Matches

Shape Matching Challenge Need shape descriptor that is:  Concise to store –Quick to compute –Efficient to match –Discriminating 3D Query Shape Descriptor 3D Database Best Matches

Shape Matching Challenge Need shape descriptor that is: –Concise to store  Quick to compute –Efficient to match –Discriminating 3D Query Shape Descriptor 3D Database Best Matches

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute  Efficient to match –Discriminating 3D Query Shape Descriptor 3D Database Best Matches

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match  Discriminating 3D Query Shape Descriptor 3D Database Best Matches

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match –Discriminating  Invariant to transformations –Invariant to deformations –Insensitive to noise –Insensitive to topology –Robust to degeneracies Different Transformations (translation, scale, rotation, mirror)

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match –Discriminating –Invariant to transformations  Invariant to deformations –Insensitive to noise –Insensitive to topology –Robust to degeneracies Different Articulated Poses

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match –Discriminating –Invariant to transformations –Invariant to deformations  Insensitive to noise –Insensitive to topology –Robust to degeneracies Scanned Surface Image courtesy of Ramamoorthi et al.

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match –Discriminating –Invariant to transformations –Invariant to deformations –Insensitive to noise  Insensitive to topology –Robust to degeneracies Different Tessellations Different Genus Images courtesy of Viewpoint & Stanford

Shape Matching Challenge Need shape descriptor that is: –Concise to store –Quick to compute –Efficient to match –Discriminating –Invariant to transformations –Invariant to deformations –Insensitive to noise –Insensitive to topology  Robust to degeneracies No Bottom! Images courtesy of Utah & De Espona

Outline: Shape Descriptors Statistical Shape Descriptors Singular Value Decomposition (SVD)

Statistical Shape Descriptors Challenge: Want a simple shape descriptor that is easy to compare and gives a continuous measure of the similarity between two models. Solution: Represent each model by a vector and define the distance between models as the distance between corresponding vectors.

Statistical Shape Descriptors Properties: –Structured representation –Easy to compare –Generalizes the matching problem Models represented as points in a fixed dimensional vector space

Statistical Shape Descriptors General Approaches: –Retrieval –Clustering –Compression –Hierarchical representation Models represented as points in a fixed dimensional vector space

Outline: Shape Descriptors Statistical Shape Descriptors Singular Value Decomposition (SVD)

Complexity of Retrieval Given a query: Compute the distance to each database model Sort the database models by proximity Return the closest matches ~ Best Match(es) 3D Query Database ModelsSorted Models D(Q,M i ) Q M1M1 M2M2 MkMk M1M1 M2M2 MkMk ~ ~ ~ M1M1 ~ M2M2

Complexity of Retrieval If there are k models in the database and each model is represented by an n-dimensional vector: Computing the distance to each database model: –O(k n) time Sort the database models by proximity: –O(k logk) time If n is large, retrieval will be prohibitively slow.

Algebra Definition: Given a vector space V and a subspace W  V, the projection onto W, written  W, is the map that sends v  V to the nearest vector in W. If {w 1,…,w m } is an orthonormal basis for W, then:

Tensor Algebra Definition: The inner product of two n-dimensional vectors v={v 1,…,v n } and w={w 1,…,w n }, written  v,w , is the scalar value defined by:

Tensor Algebra Definition: The outer product of two n-dimensional vectors v={v 1,…,v n } and w={w 1,…,w n }, written v  w, is the matrix defined by:

Tensor Algebra Definition: The transpose of an mxn matrix M, written M t, is the nxm matrix with: Property: For any two vectors v and w, the transpose has the property:

SVD Compression Key Idea: Given a collection of vectors in n-dimensional space, find a good m-dimensional subspace (m<<n) in which to represent the vectors.

SVD Compression Specifically: If P={p 1,…,p k } is the initial n-dimensional point set, and {w 1,…,w m } is an orthonormal basis for the m-dimensional subspace, we will compress the point set by sending:

SVD Compression Challenge: To find the m-dimensional subspace that will best capture the initial point information.

Variance of the Point Set Given a collection of points P={p 1,…,p k }, in an n-dimensional vector space, determine how the vectors are distributed across different directions. pipi p1p1 p2p2 pkpk

Variance of the Point Set Define the Var P as the function: giving the variance of the point set P in direction v (assume |v|=1). pipi p1p1 p2p2 pkpk

Variance of the Point Set More generally, for a subspace W  V, define the variance of P in the subspace W as: If {w 1,…,w m } is an orthonormal basis for W, then:

Variance of the Point Set Example: The variance in the direction v 1 is large, while the variance in the direction v 2 is small.  If we want to compress down to one dimension, we should project the points onto v 1 pipi p2p2 p1p1 v1v1v1v1 v2v2v2v2 pkpk

Covariance Matrix Definition: The covariance matrix M P, of a point set P={p 1,…,p k } is the symmetric matrix which is the sum of the outer products of the p i :

Covariance Matrix Theorem: The variance of the point set P in a direction v is equal to:

Covariance Matrix Theorem: The variance of the point set P in a direction v is equal to: Proof:

Singular Value Decomposition Theorem: Every symmetric matrix M can be written out as the product: where O is a rotation/reflection matrix (OO t =Id) and D is a diagonal matrix with the property:

Singular Value Decomposition Implication: Given a point set P, we can compute the covariance matrix of the point set, M P, and express the matrix in terms of its SVD factorization: where {v 1,…,v n } is an orthonormal basis and i is the variance of the point set in direction v i.

Singular Value Decomposition Compression: The vector subspace spanned by {v 1,…,v m } is the vector sub-space that maximizes the variance in the initial point set P. If m is too small, then too much information is discarded and there will be a loss in retrieval accuracy.

Singular Value Decomposition Hierarchical Matching: First coarsely compare the query to database vectors. If {query is coarsely similar to target} –Refine the comparison Else –Do not refine O(k n) matching becomes O(k m) with m<<n and no loss of retrieval accuracy.

Singular Value Decomposition Hierarchical Matching: SVD expresses the initial vectors in terms of the eigenbasis: Because there is more variance in v 1 than in v 2, more variance in v 2 than in v 3, etc. this gives a hierarchical representation of the data so that coarse comparisons can be performed by comparing only the first m coefficients.

Efficient to match? Preprocessing: Compute SVD factorization Transform database descriptors Run-Time: 3.Transform Query SVD Query

Efficient to match? 4.Low resolution sort      Distance to Query Query Database

Efficient to match? 5.Update closest matches 6.Resort Query      Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query      Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query      Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query      Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query      Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query   =0.289   Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query   =0.289   Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query  =0.301 =0.289   Database Distance to Query

Efficient to match? 5.Update closest matches 6.Resort Query  =0.301 =0.289   Database Distance to Query

Singular Value Decomposition Theorem: Every symmetric matrix M can be written out as the product: where O is a rotation/reflection matrix (OO t =Id) and D is a diagonal matrix with the property:

Singular Value Decomposition Proof: 1.Every symmetric matrix has at least one real eigenvector v. 2.If v is an eigenvector and w is perpendicular to v then Mw is also perpendicular to v. v wvwv Since M maps the subspace of vectors perpendicular to v back into itself, we can look at the restriction of M to the subspace and iterate to get the next eigenvector.

Singular Value Decomposition Proof (Step 1): Let F(v) be the function on the unit sphere (||v||=1) defined by: v F(v)F(v)F(v)F(v)

Singular Value Decomposition Proof (Step 1): Let F(v) be the function on the unit sphere (||v||=1) defined by: Then F must have a maximum at some point v 0. v0v0v0v0 F(v0)F(v0)F(v0)F(v0)

Singular Value Decomposition Proof (Step 1): Let F(v) be the function on the unit sphere (||v||=1) defined by: Then F must have a maximum at some point v 0. Then  F(v 0 )= v 0. v0v0v0v0 F(v0)F(v0)F(v0)F(v0)

Singular Value Decomposition If F has a maximum at some point v 0 then  F(v 0 )= v 0. If w 0 is on the sphere, next to v 0, then w 0 -v 0 is nearly perpendicular to v 0. And for any small vector w 1 perpendicular to v 0, v 0 + w 1 is nearly on the sphere v0v0v0v0 w0w0w0w0 v0v0v0v0 w 0 -v 0 w0w0w0w0

Singular Value Decomposition If F has a maximum at some point v 0 then  F(v 0 )= v 0. For small values of w 0 close to v 0, we have: For v 0 to be a maximum, we must have: for all w 0 near v 0. Thus,  F(v 0 ) must be perpendicular to all vectors that are perpendicular to v 0, and hence must itself be a multiple of v 0.

Singular Value Decomposition Proof (Step 1): Let F(v) be the function on the unit sphere (||v||=1) defined by: Then F must have a maximum at some point v 0. Then  F(v 0 )= v 0. v0v0v0v0 F(v0)F(v0)F(v0)F(v0)

Singular Value Decomposition Proof (Step 1): Let F(v) be the function on the unit sphere (||v||=1) defined by: Then F must have a maximum at some point v 0. Then  F(v 0 )= v 0. But  F(v)=2Mv v0v0v0v0 F(v0)F(v0)F(v0)F(v0)  v 0 is an eigenvector of M.

Singular Value Decomposition Proof: 1.Every symmetric matrix has at least one eigenvector v. 2.If v is an eigenvector and w is perpendicular to v then Mw is also perpendicular to v.

Singular Value Decomposition Proof (Step 2): If w is perpendicular to v, then  v,w  =0. Since M is symmetric: so that Mw is also perpendicular to v.