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Tomasz Malisiewicz tomasz@cmu Advanced Machine Perception February 2006

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A Bayesian Hierarchical Model for Learning Natural Scene Categories L. Fei-Fei and P. Perona. CVPR 2005 Discovering objects and their location in images J. Sivic, B. Russell, A. Efros, A. Zisserman and B. Freeman. ICCV 2005

Tomasz Malisiewicz

tomasz@cmu.edu

Advanced Machine Perception

February 2006

Graphical Models: Recent Trend in Machine Learning

Describing Visual Scenes using

Transformed Dirichlet Processes.

E. Sudderth, A. Torralba, W. Freeman,

and A. Willsky. NIPS, Dec. 2005.

Outline

- Goals of both vision papers
- Techniques from statistical text modeling

- pLSA vs LDA

- Scene Classification via LDA
- Object Discovery via pLSA

Goal: Learn and Recognize Natural Scene Categories

Classify a scene without first extracting

objects

Other techniques we know of:

-Global frequency (Oliva and Torralba)

-Texton Histogram (Renninger, Malik et al)

Goal: Discover Object Categories

- Discover what objects are present in a collection of images in an unsupervised way
- Find those same objects in novel images
- Determine what local image features correspond to what objects; segmenting the image

Enter the world of Statistical Text Modeling

- D. Blei, A. Ng, and M. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, January 2003.
- Bag-of-words approaches: the order of words in a document can be neglected
- Graphical Model Fun

Bag-of-words

- A document is a collection of M words
- A corpus (collection of documents) is summarized in a term-document matrix

Bag of ‘words’

1990: Latent Semantic Analysis (LSA)

- Goal: map high-dimensional count vectors to a lower dimensional representation to reveal semantic relations between words
- The lower dimensional space is called the latent semantic space
- Dim( latent space ) = K

topics

topics

words

NxM

NxK

KxK

KxM

topics

topics

=

x

x

documents

documents

1990: Latent Semantic Analysis (LSA)- D = {d1,…,dN} N documents
- W = {w1,…,wM} M words
- Nij = #(di,wj) NxM co-occurrence term-document matrix

topics

topics

words

NxM

NxK

KxK

KxM

topics

topics

=

x

x

documents

documents

What did we just do?Singular Value Decomposition

LSA summary

- SVD on term-document matrix
- Approximate N by thresholding all but the largest K singular values in W to zero
- Produces rank-K optimal approximation to N in the L2-matrix or Frobenius norm sense

According to this superposition

principle, LSA is unable to capture

multiple senses of a word

LSA and Polysemy- Polysemy: the ambiguity of an individual word or phrase that can be used (in different contexts) to express two or more different meanings
- Under the LSA model, the coordinates of a word in latent space can be written as a linear superposition of the coordinates of the documents that contain the word

Problems with LSA

- LSA does not define a properly normalized probability distribution
- No obvious interpretation of the directions in the latent space
- From statistics, the utilization of L2 norm in LSA corresponds to a Gaussian Error assumption which is hard to justify in the context of count variables
- Polysemy problem

pLSA to the rescue

- Probabilistic Latent Semantic Analysis
- pLSA relies on the likelihood function of multinomial sampling and aims at an explicit maximization of the predictive power of the model

Decomposition into Probabilities!

Observed word

distributions

Topic distributions

per document

word distributions

per topic

pLSA to the rescue

Slide credit: Josef Sivic

Observed counts of word i in document j

Unlike LSA, pLSA does not minimize any type of ‘squared deviation.’

The parameters are estimated in a probabilistically sound way.

Maximize likelihood of data using EM.

Minimize KL divergence between empirical

distribution and model

Slide credit: Josef Sivic

EM for pLSA (training on a corpus)

- E-step: compute posterior probabilities for the latent variables
- M-step: maximize the expected complete data log-likelihood

d

w

Graphical View of pLSA- pLSA is a generative model
- Select a document di with prob P(di)
- Pick latent class zk with prob P(zk|di)
- Generate word wj with prob P(wj|zk)

Observed variables

Latent variables

Plates

How does pLSA deal with previously unseen documents?

- “Folding-in” Heuristic
- First train on Corpus to obtain
- Now re-run same training EM algorithm, but don’t re-estimate and let D={dunseen}

Problems with pLSA

- Not a well-defined generative model of documents; d is a dummy index into the list of documents in the training set (as many values as documents)
- No natural way to assign probability to a previously unseen document
- Number of parameters to be estimated grows with size of training set

pLSA

LDA to the rescue- Latent Dirichlet Allocation treats the topic mixture weights as a k-parameter hidden random variable and places a Dirichlet prior on the multinomial mixing weights
- Dirichlet distribution is conjugate to the multinomial distribution (most natural prior to choose: the posterior distribution is also a Dirichlet!)

Corpus-Level parameters in LDA

- Alpha and beta are corpus-level documents that are sampled once in the corpus creating generative model (outside of the plates!)
- Alpha and beta must be estimated before we can find the topic mixing proportions belonging to a previously unseen document

LDA

1

2

K

z1

z2

z3

zN

z1

z2

z3

zN

z1

z2

z3

zN

w1

w2

w3

wN

w1

w2

w3

wN

w1

w2

w3

wN

b

Getting rid of platesThanks to Jonathan Huang for the un-plated LDA graphic

Inference in LDA

- Inference = estimation of document-level parameters
- Intractable to compute must employ approximate inference

Approximate Inference in LDA

- Variational Methods: Use Jensen’s inequality to obtain a lower bound on the log likelihood that is indexed by a set of variational parameters
- Optimal Variational Parameters (document-specific) are obtained by minimizing the KL divergence between the variational distribution and the true posterior

Variational Methods are one way of doing this.

Gibbs sampling (MCMC) is another way.

Variational distribution

Look at some P(w|z) produced by LDA

- Show some pLSI and LDA results applied to text
- An LDA project by Tomasz Malisiewicz and Jonathan Huang
- Search for the word ‘drive’

pLSA and LDA applied to Images

- How can one apply these techniques to the images?

text models

z

d

w

N

D

z

c

w

N

D

Probabilistic Latent Semantic Analysis (pLSA)

Hoffman, 2001

Latent Dirichlet Allocation (LDA)

Blei et al., 2001

text models

z

d

w

N

D

“face”

Probabilistic Latent Semantic Analysis (pLSA)

Sivic et al. ICCV 2005

text models

“beach”

z

c

w

N

D

Latent Dirichlet Allocation (LDA)

Fei-Fei et al. ICCV 2005

How to Generate an Image?

Choose a scene (mountain, beach, …)

Given scene generate an intermediate

probability vector over ‘themes’

For each word:

Determine current theme from mixture

of themes

Draw a codeword from that theme

Inference

- How to make decision on a novel image
- Integrate over latent variables to get:
- Approximate Variational Inference (not easy, but Gibbs sampling is supposed to be easier)

Codebook

- 174 Local Image Patches
- Detection:

Evenly Sampled Grid

Random Sampling

Saliency Detector

Lowe’s DoG Detector

- Representation:

Normalized 11x11 gray values

128-dim SIFT

Results: Average performance 64%

- Confusion Matrix

100 training examples and 50 test examples

Rank statistic test:the probability of a test scene correctly

belong to one of the top N most probable categories

Summary of detection and representation choices

- SIFT outperforms pixel gray values
- Sliding grid, which creates the largest number of patches, does best

Visual Words

- Vector Quantized SIFT descriptors computed in regions
- Regions come from elliptical shape adaptation around interest point, and from the maximally stable regions of Matas et al.
- Both are elliptical regions at twice their detected scale

K-means clustering of 300K regions

to get about 1K clusters for each of

Shape Adapted and Maximally Stable

regions

…

Building a VocabularyVector quantization

Slide credit: Josef Sivic

pLSA Training

- Sanity Check: Remember what quantities must be estimated?

Results #1: Topic Discovery

- This is just the training stage
- Obtain P(zk|dj) for each image, then classify image as containing object k according to the max of P(zk|dj) over k

4 object categories

Plus background

Results #2: Classifying New Images

- Object Categories learned on a corpus, then object categories found in new image

Anybody remember how this is done?

Remember the index d in

the graphical model

How does pLSA deal with previously unseen documents?

- “Folding-in” Heuristic
- First train on Corpus to obtain
- Now re-run same training EM algorithm, but don’t re-estimate and let D={dunseen}

Results #2: Classifying New Images

- Train on one set and test on another

Results #3: Segmentation

- Localization and Segmentation of Object
- For a word occurrence in a particular document we can examine the probability of different topics
- Find words with P(zk|dj,wi) > .8

Results #3: Segmentation

Note: words shown are not the most probable words

for a topic, but instead they are words that have a high

probability of occurring in a topic AND high probability of

occurring in the image

Results #3: Segmentation and Doublets

- Two class image dataset consisting of half the faces (218 images) and backgrounds (217 images)
- A 4 topic pLSA model is learned for all training faces and training backgrounds with 3 fixed background topics, i.e. one (face) topic is learned in addition to the three fixed background topics
- A doublet vocabulary is then formed from the top 100 visual words of the face topic. A second 4 topic pLSA model is then learned for the combined vocabulary of singlets and doublets with the background topics fixed.

Conclusions

- Showed how both papers use bag-of-words approaches
- We’re now ready to become experts on generative models like pLSA and LDA
- Graphical Model Fun! (Carlos Guestrin teaches Graphical Models)

Are you really into Graphical Models?

- Describing Visual Scenes using Transformed Dirichlet Processes. E. Sudderth, A. Torralba, W. Freeman, and A. Willsky. NIPS, Dec. 2005.

References

- A Bayesian Hierarchical Model for Learning Natural Scene Categories, Fei Fei Li et al
- Describing Visual Scenes using Transformed Dirichlet Processes, Sudderth et al
- Discovering objects and their location in images, Sivic et al
- Latent Dirichlet Allocation, Blei et al
- Unsupervised Learning by Probabilistic Latent Semantic Analysis, T. Hoffman

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