Statistics and Shape Statistics and Shape AnalysisAnalysis

By Marc SobelBy Marc Sobel

Shape similarityShape similarity Humans recognize shapes via both local and Humans recognize shapes via both local and

global features. global features. (i) matching local features between (i) matching local features between

shapes like curvature, distance to centroid shapes like curvature, distance to centroid can be statistically modeled via building can be statistically modeled via building statistics and parameters to reflect the statistics and parameters to reflect the matching.matching.

(ii) matching the relationship between (ii) matching the relationship between global features of shapes (are they both global features of shapes (are they both apples or not?) apples or not?)

Incorporating both local and Incorporating both local and global features in shape global features in shape

matchingmatching How can we incorporate both local How can we incorporate both local

and global features in shape and global features in shape matching?matching?

An obvious paradigm is to model An obvious paradigm is to model global features as governed by global features as governed by priors, and local features given global priors, and local features given global features as a likelihood. features as a likelihood.

Definitions and NotationDefinitions and Notation

Let uLet u11,…,u,…,un n be the vertices of one shape and vbe the vertices of one shape and v11,…,v,…,vm m the vertices of another shape. We’d like to biuld the vertices of another shape. We’d like to biuld correspondences between the vertices which properly correspondences between the vertices which properly reflect the relationship between the shapes. We use reflect the relationship between the shapes. We use the notation (uthe notation (uii,v,vjj) for a correspondence of this type. ) for a correspondence of this type. We use the terminology for a We use the terminology for a particle consisting of a set of such correspondences. particle consisting of a set of such correspondences.

Let XLet Xi,l i,l be the l’th local feature measure for vertex i of be the l’th local feature measure for vertex i of the first shape and Ythe first shape and Yj,lj,l the l’th local feature measure the l’th local feature measure for vertex j of the second shape. For now assume for vertex j of the second shape. For now assume these feature measures are observed. these feature measures are observed.

We’d like to biuld a particle which reflects the local We’d like to biuld a particle which reflects the local and global features of interest.and global features of interest.

1 1( , ),..., ( , )

k ki j i ju v u vP

Contiguity: An important Contiguity: An important global feature.global feature.

If shapes result from one another via If shapes result from one another via rotation and scaling then the order of rotation and scaling then the order of shape 1 correspondence points should shape 1 correspondence points should match the order of shape 2 match the order of shape 2 correspondence points: i.e., if (icorrespondence points: i.e., if (i11,j,j11) is ) is one correspondence and (ione correspondence and (i22,j,j22) is another, ) is another, then either ithen either i11<i<i2 2 and jand j11<j<j22 or i or i11>i>i2 2 and and jj11>j>j22. We can incorporate this into a prior. . We can incorporate this into a prior.

Notation:Notation:

We have that:We have that:

1:

,

'set of correspondences up to time t';

'set of variable weights'

t

i j

P

Simple LikelihoodSimple Likelihood

Based on the observed features we form Based on the observed features we form weight statistics:weight statistics:

Let W denote the weight matrix associated Let W denote the weight matrix associated with the features. with the features.

Therefore given that a correspondence ‘C’ Therefore given that a correspondence ‘C’ belongs in the ‘true’ set of belongs in the ‘true’ set of correspondences, we write the simple correspondences, we write the simple likelihood in the form,likelihood in the form,

, ; i jW w

,(( , ) | , ) exp i ji j C w ωL P

, ,,

1

r i l j li j

l

X Yw

Complicated LikelihoodsComplicated Likelihoods

At stage t, putting At stage t, putting ωω as the parameter, as the parameter, we we define the likelihood:define the likelihood:

1:

1: 1: , ,( , ) [ , ]

(( , ) | , ) expt

t t i j u vu v DIAG i j

i j C w w

ωP

L P

Simple and ComplicatedSimple and ComplicatedPriorsPriors

Model a prior for all sets of correspondences Model a prior for all sets of correspondences which are strongly contiguous:which are strongly contiguous:

a) a simple prior (we use a) a simple prior (we use ωω for the weight for the weight variable)variable)

b) I] a prior giving more weight to diagonals than b) I] a prior giving more weight to diagonals than

other correspondences.other correspondences. II] we can define such a prior sequentially II] we can define such a prior sequentially

based on the fact that based on the fact that

, ,( ) (mutually independent)i j i jw

Complicated PriorComplicated Prior

Put Put

With ‘DIAG[i,j]’ referring to the positively With ‘DIAG[i,j]’ referring to the positively oriented diagonal to which (i,j) belong. oriented diagonal to which (i,j) belong.

1:

, ,( , ) [ , ], 1: 1:

,

&if | ,

&if =t

i j u vu v DIAG i ji j t t

i j

w w

w

w PL P

Simulating the Posterior Simulating the Posterior Distribution: Simple PriorDistribution: Simple Prior

We would like to simulate the posterior We would like to simulate the posterior distribution of contiguous distribution of contiguous correspondences. We do this by calculating correspondences. We do this by calculating the weights:the weights:

1 ,

, ,

exp ;

( )

t i j t

i j i j

W w W

w w

Simulating the Posterior Simulating the Posterior Distribution: Complicated PriorDistribution: Complicated Prior Here we simulate:Here we simulate:

1:

1 ,

, , ,( , ) [ , ]

exp ;

( )t

t i j t

i j i j u vu v DIAG i j

W W

w w

P

A Simpler ModelA Simpler Model

Define the posterior probabilities:Define the posterior probabilities:

For parameter For parameter λλ, described below., described below.

1:

,1: 1 1:

,( , )

exp{ }( , ) |

exp{ }t

i jt t

u vu v

wP i j

w

P

P P

Weights for the simpler Weights for the simpler modelmodel

The weights for the simpler model are The weights for the simpler model are particularly easy:particularly easy:

Choosing Choosing λλ tending to infinity properly, we tending to infinity properly, we get convergence to the MAP estimator of get convergence to the MAP estimator of the simple particle filter. the simple particle filter.

1:

,1

,( , )

exp{ }

exp{ }t

i jt t

u vu v

wW W

w

P

Shape Similarity: A More Shape Similarity: A More complicated model employing complicated model employing

curvature and distance curvature and distance parameters parameters We have:We have:

, , ,

, , ,

;i l i l i l

j l j l j l

X

Y e

Simple LikelihoodSimple Likelihood

Based on the observed features we form Based on the observed features we form weight parameters:weight parameters:

Let W denote the weight matrix associated Let W denote the weight matrix associated with the features. with the features.

Therefore given that a correspondence ‘C’ Therefore given that a correspondence ‘C’ belongs in the ‘true’ set of belongs in the ‘true’ set of correspondences, we write the likelihood in correspondences, we write the likelihood in the form,the form,

, ; i jW w

, ,,

1

r i l j li j

l

Particle LikelihoodParticle Likelihood

We write the likelihood in the form:We write the likelihood in the form:

, , , ,

( , ) 1

( | , )

r i l i l i l i l

i j l l l

C

X Y v

X,Y μ, ν

P

L P

Particle PriorParticle Prior

We assume standard priors for the mu’s We assume standard priors for the mu’s and nu’s. We also assume a prior for the and nu’s. We also assume a prior for the set of contiguous correspondences.set of contiguous correspondences.

The particle is updated as follows: define,The particle is updated as follows: define,

1:

, ,1:

( , ) 1t

r j l i lt

i j l l

P

Particle PriorParticle Prior

At stage t we have particles,At stage t we have particles,

Their weights are given by:Their weights are given by:

(1) ( ),..., kt tP P

1 1

( ) ( )1

1

( , | , )

| ( , ),..., ( , )r r r r

t t

i j i jr rtt

t i j i j

X Y vW W

v v