Outline

Outline

• Statistical Modeling and Conceptualization of Visual Patterns

– S. C. Zhu, “Statistical modeling and conceptualization of visual patterns,”

IEEE

Transactions on Pattern Analysis and Machine

Intelligence , vol. 25, no. 6, 1-22, 2003

A Common Framework of Visual Knowledge Representation

• Visual patterns in natural images

– Natural images consist of an overwhelming number of visual patterns

• Generated by very diverse stochastic processes

• Comments

– Any single image normally consists of a few recognizable/segmentable visual patterns

– Scientifically, given that visual patterns are generated by stochastic processes, shall we model the underlying stochastic processes or model visual patterns presented in the observations from the stochastic processes?

April 11, 2020 Computer Vision 2

A Common Framework of Visual Knowledge Representation – cont.



• The image analysis as an image parsing problem

– Parse generic images into their constituent patterns

(according to the underlying stochastic processes)

• Perceptual grouping when applied to points, lines, and curves processes

• Image segmentation when applied to region processes

• Object recognition when applied to high level objects





• Required components for parsing

– Mathematical definitions and models of various visual patterns

• Definitions and models are intrinsically recursive

– Grammars (or called rules)

• Which specifies the relationships among various patterns

• Grammars should be stochastic in nature

– A parsing algorithm


Syntactical Pattern Recognition



• Conceptualization of visual patterns

– The concept of a pattern is an abstraction of some properties decided by certain “visual purposes”

• They are feature statistics computed from

– Raw signals

– Some hidden descriptions inferred from raw signals

– Mathematically, each pattern is equivalent to a set of observable signals governed by a statistical model



• Statistical modeling of visual patterns

– Statistical models are intrinsic representations of visual knowledge and image regularities

• Due to noise and distortion in imaging process?

• Due to noise and distortion in the underlying generative process?

• Due to transformations in the underlying stochastic process?

– Pattern theory



• Statistical modeling of visual patterns

- continued

– Mathematical space for patterns and spaces

• Depends on the forms

– Parametric

– Non-parametric

– Attributed graphs

– Different models

• Descriptive models

– Bottom-up, feature-based models

• Generative models

– Hidden variables for generating images in a top-down manner



• Learning a visual vocabulary

– Hierarchy of visual descriptions for general visual patterns

– Vocabulary of visual description

• Learning from an ensemble of natural images

• Vocabulary is far from enough

– Rich structures in physics

– Large vocabulary in speech and language



• Computational tractability

– Computational heuristics for effective inference of visual patterns

• Discriminative models

– A framework

• Discriminative probabilities are used as proposal probabilities that drive the Markov chain search for fast convergence and mixing

• Generative models are top-down probabilities and the hidden variables to be inferred from posterior probabilities



• Discussion

– Images are generated by rendering 3D objects under some external conditions

• All the images from one object form a low dimensional manifold in a high dimensional image space

• Rendering can be modeled fairly accurately

• Describing a 3D object requires a huge amount of data

– Under this setting

• A visual pattern simply corresponds to the manifold

• Descriptive model attempts to characterize the manifold

• Generative model attempts to learn the 3D objects and the rendering


3D Model-Based Recognition


Literature Survey

• To develop a generic vision system, regularities in images must be modeled

• The study of natural image statistics

– Ecologic influence on visual perception

– Natural images have high-order (i.e., non-Gaussian) structures

• The histograms of Gabor-type filter responses on natural images have high kurtosis

– Histograms of gradient filters are consistent over a range of scales


Natural Image Statistics Example


Analytical Probability Models for Spectral Representation

• Transported generator model

(Grenander and Srivastava, 2000) where

– g i

’s are selected randomly from some generator space G

– the weigths a i

’s are i.i.d. standard normal

– the scales r i

’s are i.i.d. uniform on the interval [0,L]

– the locations z i

’s as samples from a 2D homogenous Poisson process, with a uniform intensity l

, and

– the parameters are assumed to be independent of each other


Analytical Probability Models

- continued

• Define

• Model u by a scaled 

-density



- continued



- continued



- continued


Analysis of Natural Image Components

• Harmonic analysis

– Decomposing various classes of functions by different bases

– Including Fourier transform, wavelet transforms, edgelets, curvelets, and so on

22 April 11, 2020 Computer Vision

Sparse Coding

From S. C. Zhu


Grouping of Natural Image Elements

• Gestalt laws

– Gestalt grouping laws

– Should be interpreted as heuristics rather than deterministic laws

– Nonaccidental property


Illusion


Illusion – cont.


Ambiguous Figure


Statistical Modeling of Natural Image Patterns

• Synthesis-by-analysis


Analog from Speech Recognition


Modeling of Natural Image Patterns

• Shape-from-X problems are fundamentally ill-posed

• Markov random field models

• Deformable templates for objects

– Inhomogeneous MRF models on graphs


Four Categories of Statistical Models

• Descriptive models

– Constructed based on statistical descriptions of the image ensembles

– Homogeneous models

• Statistics are assumed to be the same for all elements in the graph

– Inhomogeneous models

• The elements of the underlying graph are labeled and different features and statistics are used at different sites


Variants of Descriptive Models

• Casual Markov models

– By imposing a partial ordering among the vertices of the graph, the joint probability can be factorized as a product of conditional probabilities

– Belief propagation networks

• Pseudo-descriptive models


Generative Models

• Use of hidden variables that can “explain away” the strong dependency in observed images

– This requires a vocabulary

– Grammars to generate images from hidden variables

• Note that generative models can not be separated from descriptive models

– The description of hidden variables requires descriptive models


Discriminative Models

• Approximation of posterior probabilities of hidden variables based on local features

– Can be seen as importance proposal probabilities


An Example


Problem formation

Input: a set of images

S

 {

I

1

, I

2

, ...

, I n

} ~ f (I)

Output: a probability model p ( I)

 f (I)

Here, f (I) represents the ensemble of images in a given domain, we shall discuss the relationship between ensemble and probability later.

Computer Vision 36 April 11, 2020

Problem formation

The model p approaches the true density p *

 arg min p

 

D ( f || p )

The Kullback-Leibler Divergence

D ( f || p )

  f f ( I)

( I) log p ( I )



E f

[log f ( I)] E f d I

[log p ( I)]


Maximum Likelihood Estimate p *

 argmax p

 

E f

[log p ( I)]

 argmax p

  i n 



1 log p ( I i

)

 


Model Pursuit



1

,



2

, ...

,

 n

  * p f

1. What is

 -the family of models ?

2. How do we augment the space

?


Two Choices of Models

1. The exponential family – descriptive models

--- Characterize images by features and statistics

2. The mixture family -- generative models

--- Characterize images by hidden variables

Features are deterministic mathematical transforms of an image.

Hidden variables are stochastic and are inferred from an image.

F



F(I) vs.

H ~ p (H | I)


I: Descriptive Models

Step 1: extracting image features/statistics as transforms



1

( I),



2

( I), ...

,

 k

(I)

For example: histograms of Gabor filter responses.

Other features/statistics: Gabors, geometry, Gestalt laws, faces.


I.I: Descriptive Models

Step 2: using features/statistics to constrain the model

Two cases:

1. On infinite lattice Z 2 --- an equivalence class.

2. On any finite lattice --- a conditional probability model.

image space on Z 2

April 11, 2020 Computer Vision image space on lattice

L

42

I.I Descriptive Model on Finite Lattice

Modeling by maximum entropy: p

 arg max

 p (I) log p (I) d I

Subject to:

E p

[

 m

( I)]



E f

[

 m

( I)]



1 n

  j m

( I j

)

Remark: p and f have the same projected marginal statistics.


Minimax Entropy Learning p *

 arg min p

 

D ( f || p )

 arg min p

 

E f

[

log p ]

For a Gibbs (max. entropy) model p , this leads to the minimax entropy principle

(Zhu,Wu, Mumford 96,97) p *

 arg min



{ max

β entropy ( p ( I ;

β

))

}


FRAME Model

• FRAME model

– Filtering, random field, and maximum entropy

– A well-defined mathematical model for textures by combining filtering and random field models



The FRAME model

(Zhu, Wu, Mumford, 1996) p ( I ; β) 

1 z

(



) exp

{ k  j



1

β j

 j

( I)

}

This includes all Markov random field models.

Remark: all known exponential models are from maxent., and maxent was proposed in Physics ( Jaynes, 1957 ). The nice thing is that it provides a parametric model integrating features.



Two learning phases:

1. Choose information bearing features

-- augmenting the probability family.



1

,



2

, ...

,

 n

  *

2. Compute the parameter

L by MLE

-- learning within a family.


Maximum Entropy

• Maximum entropy

– Is an important principle in statistics for constructing a probability distribution on a set of random variables

– Suppose the available information is the expectations of some known functions

 n

(x), that is

E [



( x )]

  

( x ) p ( x ) dx

  for n



1,  , N p n n n

– Let  be the set of all probability distributions p ( x ) which satisfy the constraints

 

{ p ( x ) | E p

[

 n

( x )]

  n

, n



1 ,  , N }


Maximum Entropy – cont.

• Maximum Entropy – continued

– According to the maximum entropy principle, a good choice of the probability distribution is the one that has the maximum entropy p

*

( x subject to

)

 arg max



 p ( x ) log p ( x ) dx



E p

[

 n

( x )]

   n

( x ) p ( x ) dx

  n for n



1,  , N

 p ( x ) dx



1




– By Lagrange multipliers, the solution for p(x) is p ( x ;

L

)



1

Z (

L

) exp  n

N 



1 l n

 n

( x )

– where

L 

( l

1

, l

2

,  , l

N

) and Z(

L

)

  exp n

N 



1 l n

 n

( x )  dx




– L 

( l

1

, l

2

,  , l

N

) are determined by the constraints

– But a closed form solution is not available general

• Numerical solutions d l n dt



E p ( x ;

L

)

[

 n

( x )]

 n n



1 , 2 ,  , N




– The solutions are guaranteed to exist and be unique by the following properties


Minimax Entropy Learning

(cont.)

Intuitive interpretation of minimax entropy.


Learning A High Dimensional Density


Toy Example I


Toy Example II


FRAME Model

• Texture modeling

– The features can be anything you want  n

(x)

– Histograms of filter responses are a good feature for textures


FRAME Model – cont.

• The FRAME algorithm

– Initialization

Input a texture image I obs

Select a group of K filters S

K

={F (1) , F (2) , ...., F (K) }

Compute {H obs( a

) , a

= 1, ....., K}

Initialize l i

( a

) 

0 , i



1,2,  , L a 

1,2,  , K

Initialize I syn as a uniform white noise image



• The FRAME algorithm – continued

– The algorithm

Repeat calculate H syn( a

)

E p ( I ;

L

K

, S

K

)

( H

, a

=1,..., K from I syn and use it as

( a

)

) l i

( a d l i

( a

)

/ dt



E p ( I ;

L

K

, S

K

)

[ H

( a

)

]

-

H obs ( a

)

Apply Gibbs sampler to flip I syn for w sweeps until

1

2 i

L 



1

| H i obs ( a

) -

H i syn ( a

)

|

  for a 

1, 2,  , K



• The Gibbs sampler



• Filter selection

– In practice, we want a small number of “good” filters

– One way to do that is to choose filters that carry the most information

• In other words, minimum entropy



• Filter selection algorithm

– Initialization




Descriptive Models

– cont.


Existing Texture Features


Existing Feature Statistics


Most General Feature Statistics


Julesz Ensemble – cont.

• Definition

– Given a set of normalized statistics on lattice L h



{ h ( a

)

:, a 

1 , 2 ,  K } a Julesz ensemble



( h ) is the limit of the following set as

L 

Z 2 and H



{ h } under some boundary conditions



L

( H )



{ I : h ( I )



H }


Julesz Ensemble – cont.

• Feature selection

– A feature can be selected from a large set of features through information gain, or the decrease in entropy


Example: 2D Flexible Shapes


A Random Field for 2D Shape

The neighborhood

Co-linearity, co-circularity, proximity, parallelism, symmetry, …


A Descriptive Shape Model

Random 2D shapes sampled from a Gibbs model.

(Zhu, 1999)


A Descriptive Shape Model

Random 2D shapes sampled from a Gibbs model.


Example: Face Modeling


Generative Models

• Use of hidden variables that can “explain away” the strong dependency in observed images

– This requires a vocabulary

– Grammars to generate images from hidden variables

• Note that generative models can not be separated from descriptive models

– The description of hidden variables requires descriptive models


Generative Models

– cont.


Philosophy of Generative Models

?

World structure H f ( H, I) observer I

1

, I

2

, ...

, I n

~ f ( I)

April 11, 2020

F

Features Hidden variables



F(I) vs.

H ~ p (H | I)

Computer Vision 77

Example of Generative Model: image coding

I

 i k 



1

α i

ψ i

 n

Random variables

Parameters: wavelets

Assumptions:

1. Overcomplete basis

2. High kurtosis for iid a

, e.g.

p (

α

)

 exp

{ λ

|

α

|

}


A Generative Model

(Zhu and Guo, 2000)

( T

1

, Ψ

1

)

( T

2

, Ψ

2

) noise

Image I occlusion occlusion additive


Example: Texton map

One layer of hidden variables: the texton map

T

 { n , (x i

, y i

,

θ i

, s i

,

α i

), i



1,2,..., n

}


Learning with Generative Model p ( I obs

,



)

  p ( I obs

| H ,



) p ( H ;

β )

 

( 

1

,   k

; d H

β

1

,  β k

)

1. A generative model from H to I

I



I (T

1

,

Ψ

1

)

 I (T

2

,

Ψ

2

)

   I (T k

,

Ψ k

)

 n

2. A descriptive model for H.

k p ( H ;

β

)

 p ( T i

;

β i

) i





1


Learning with Generative Model p *

 arg min p

 

D( f || p )

 arg max p

  log p ( I obs

;



)

 

Learning by MLE:

 log p ( I obs

;



)

 

 

(

 log p ( I obs

 

| H ;



)

 k  i



1

 log p ( T i

 β i

;

β i

)

) p ( H | I obs

;



) dH

1. Regression, fitting.

2. Minimax entropy learning

3. Stochastic inference


Stochastic Inference by DDMCMC

Goal: sampling H ~ p (H | I obs

;

)

Method: a symphony algorithm by data driven

Markov chain Monte Carlo .

(Zhu, Zhang and Tu 1999)

1. Posterior probability p (H | I obs ;

)

2. Importance proposal probability density q (H | I) computer vision pattern recognition


Example of A Generative Model

An observed image:


Data Clustering

The saliency maps used as proposal probabilities




A Descriptive Model for Texton Map




Data Clustering


A Descriptive Model on Texton Map






A Descriptive Model for Texton Map


Outline • Statistical Modeling and Conceptualization of Visual Patterns

Syntactical Pattern Recognition

3D Model-Based Recognition

Literature Survey

Natural Image Statistics Example

Sparse Coding

Grouping of Natural Image Elements

Illusion

Illusion – cont.

Ambiguous Figure

Analog from Speech Recognition

Modeling of Natural Image Patterns

Four Categories of Statistical Models

Variants of Descriptive Models

Generative Models

Discriminative Models

An Example

Problem formation

Problem formation

Model Pursuit

FRAME Model

Maximum Entropy

Maximum Entropy – cont.

Maximum Entropy – cont.

Maximum Entropy – cont.

Maximum Entropy – cont.

Learning A High Dimensional Density

Toy Example I

Toy Example II

FRAME Model

FRAME Model – cont.

FRAME Model – cont.

FRAME Model – cont.

FRAME Model – cont.

FRAME Model – cont.

FRAME Model – cont.

Descriptive Models

Existing Feature Statistics

Most General Feature Statistics

Julesz Ensemble – cont.

Julesz Ensemble – cont.

Example: 2D Flexible Shapes

A Random Field for 2D Shape

Example: Face Modeling

Generative Models

Generative Models

Data Clustering

Data Clustering

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