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

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 MatchingGeneral approach:

Define a function that takes in two models and returns a measure of their proximity.

D , D ,M1 M1 M3M2

M1 is closer to M2 than it is to M3

Shape DescriptorsShape Descriptor:

A structured abstraction of a 3D model that is well suited to the challenges of shape matching

Descriptors

3D Models

Matching with DescriptorsPreprocessingCompute database descriptors

Run-Time

3D Query ShapeDescriptor

3D Database

BestMatches

Run-TimeCompute query descriptor

3D Database

BestMatches

Run-TimeCompute query descriptorCompare query descriptor to database descriptors

3D Database

BestMatches

Run-TimeCompute query descriptorCompare query descriptor to database descriptorsReturn best Match(es)

3D Database

BestMatches

Shape Matching ChallengeNeed shape descriptor that is:Concise to store

– Quick to compute

– Efficient to match

– Discriminating

3D Database

BestMatches

Shape Matching ChallengeNeed shape descriptor that is:

– Concise to storeQuick to compute

– Discriminating

3D Database

BestMatches

– Concise to store

– Quick to computeEfficient to match

– Discriminating

3D Database

BestMatches

– Efficient to matchDiscriminating

3D Database

BestMatches

– Discriminating Invariant to transformations

– Invariant to deformations

– Insensitive to noise

– Insensitive to topology

– Robust to degeneracies

Different Transformations(translation, scale, rotation, mirror)

– Discriminating

– Invariant to transformations Invariant to deformations

– Robust to degeneraciesDifferent Articulated Poses

– Discriminating

– Invariant to transformations

– Invariant to deformations Insensitive to noise

– Robust to degeneracies

Scanned Surface

Image courtesy ofRamamoorthi et al.

– Discriminating

– Insensitive to noise Insensitive to topology

– Robust to degeneraciesDifferent Tessellations

Different Genus

Images courtesy of Viewpoint & Stanford

– Discriminating

– Insensitive to topologyRobust to degeneracies

No Bottom!

&*Q?@#A%!

Images courtesy of Utah & De Espona

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.

Properties:– Structured representation– Easy to compare– Generalizes the

matching problem

Models represented as points in a fixed dimensional vector space

General Approaches:– Retrieval– Clustering– Compression– Hierarchical

representation

Models represented as points in a fixed dimensional vector space

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 Models Sorted Models

D(Q,Mi

MQDMQD ji

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 WV, the projection onto W, written W, is the map that sends vV to the nearest vector in W.

If {w1,…,wm} is an orthonormal basis for W, then:

iiW wwvv

Tensor Algebra

Definition: The inner product of two n-dimensional vectors v={v1,…,vn} and w={w1,…,wn}, written v,w, is the scalar value defined by:

vvwvwv t

Tensor Algebra

Definition: The outer product of two n-dimensional vectors v={v1,…,vn} and w={w1,…,wn}, written vw, is the matrix defined by:

Tensor Algebra

Definition: The transpose of an mxn matrix M, written Mt, is the nxm matrix with:

Property: For any two vectors v and w, the transpose has the property:

ijjit MM ,,

wvMMwv t ,,

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={p1,…,pk} is the initial n-dimensional point set, and {w1,…,wm} is an orthonormal basis for the m-dimensional subspace, we will compress the point set by sending:

mwpwpp ,,,, 1

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={p1,…,pk}, in an n-dimensional vector space, determine how the vectors are distributedacross different directions. pi

Define the VarP as the function:

giving the variance of thepoint set P in direction v(assume |v|=1).

iiP pvpvVar

More generally, for a subspace WV, define the variance of P in the subspace W as:

If {w1,…,wm} is an orthonormal basis for W, then:

iiWP pWVar

jjiP wVarwpWVar

2)(,)(

Example:

The variance in the direction v1

is large, while the variance inthe direction v2 is small.

If we want to compressdown to one dimension, weshould project the points onto v1

Covariance Matrix

Definition: The covariance matrix MP, of a point set P={p1,…,pk} is the symmetric matrix which is the sum of the outer products of the pi:

tiiP ppM

Covariance Matrix

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

vMvvpvVar pt

Covariance Matrix

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

Proof:

vMvvpvVar pt

vVarpv

vppvvppv

vppvvMv

Singular Value Decomposition

Theorem: Every symmetric matrix M can be written out as the product:

where O is a rotation/reflection matrix (OOt=Id) and D is a diagonal matrix with the property:

Implication:Given a point set P, we can compute the covariance matrix of the point set, MP, and express the matrix in terms of its SVD factorization:

where {v1,…,vn} is an orthonormal basis and i is the variance of the point set in direction vi.

1 with

Compression:

The vector subspace spanned by {v1,…,vm} 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.

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.

Hierarchical Matching:

SVD expresses the initial vectors in terms of the eigenbasis:

Because there is more variance in v1 than in v2, more variance in v2 than in v3, etc. this gives a hierarchical representation of the data so that coarse comparisons can be performed by comparing only the first m coefficients.

nn vvpvvpp ,, 11

Efficient to match?

Preprocessing:• Compute SVD factorization• Transform database descriptors

Run-Time:

3. Transform Query

Efficient to match?

4. Low resolution sort

0.040 0.052 0.103 0.6610.430

Distance to Query

Query Database

Efficient to match?

5. Update closest matches

6. ResortQuery

0.229 0.052 0.103 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.2290.141 0.103 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.2290.1410.189 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.2290.230 0.189 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.229 0.2300.200 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.229 0.230=0.289 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.334 0.230 =0.289 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.334=0.301 =0.289 0.6610.430

Database

Distance to Query

Efficient to match?

6. ResortQuery

0.334=0.301=0.289 0.6610.430

Database

Distance to Query

Theorem: Every symmetric matrix M can be written out as the product:

where O is a rotation/reflection matrix (OOt=Id) and D is a diagonal matrix with the property:

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

wvSince 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.

Proof (Step 1):

Let F(v) be the function on the unit sphere (||v||=1) defined by:

FF((vv))

MvvvF t)(

Proof (Step 1):

Then F must have a maximumat some point v0. vv00

FF((vv00))MvvvF t)(

Proof (Step 1):

Then F must have a maximumat some point v0.

Then F(v0)=v0.

MvvvF t)(

FF((vv00))

If F has a maximum at some point v0 then F(v0)=v0.If w0 is on the sphere, next to v0, then w0-v0 is nearly perpendicular to v0. And for any small vector w1 perpendicular to v0, v0+ w1 is nearly on the sphere

vv00ww

00 -v-v00

If F has a maximum at some point v0 then F(v0)=v0.For small values of w0 close to v0, we have:

For v0 to be a maximum, we must have:

for all w0 near v0.

Thus, F(v0) must be perpendicular to all vectors that are perpendicular to v0, and hence must itself be a multiple of v0.

00000 ),()()( vwvFvFwF

0),( 000 vwvF

Proof (Step 1):

Then F(v0)=v0.

MvvvF t)(

FF((vv00))

Proof (Step 1):

Then F(v0)=v0.

But F(v)=2Mv

MvvvF t)(

FF((vv00))

v0 is an eigenvector of M.

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.

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.

0,,, wvwMvMwv

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

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