For that we will use a method called probabilistic latent semantic analysis. pLSA can be thought of as a matrix decomposition. Here is our term-document matrix (documents, words) and we want to find topic vectors common to all documents and mixture coefficients P(z|d) specific to each document. Note that these are all probabilites which sum to one. So a column here is here expressed as a convex combination of the topic vectors. What we would like to see is that the topics correspond to objects. So an image will be expressed as a mixture of different objects and backgrounds.
An images is a mixture of learned topic vectors and mixing weights. Fit the model and use the learned mixing coeficents to classify an image. In particular assign to the highest P(z|d)
We use maximum likelihood approach to fit parameters of the model. We write down the likelihood of the whole collection and maximize it using EM. M,N goes over all visual words and all documents. P(w|d) factorizes as on previous slide the entries in those matrices are our parameters. n(w,d) are the observed counts in our data
Cvpr2007 tutorial bag_of_words
Part 1: Bag-of-words modelsby Li Fei-Fei (Princeton)
Analogy to documents China is forecasting a trade surplus of $90bnOf all the sensory impressions proceeding to (£51bn) to $100bn this year, a threefoldthe brain, the visual experiences are the increase on 2004s $32bn. The Commercedominant ones. Our perception of the world Ministry said the surplus would be created byaround us is based essentially on the a predicted 30% jump in exports to $750bn,messages that reach the brain from our eyes. compared with a 18% rise in imports toFor a long time it was thought that the retinal $660bn. The figures are likely to furtherimage was transmitted pointbrain, to visual sensory, by point China, trade, annoy the US, which has long argued thatcenters in the brain; the cerebral cortex was visual, perception,a movie screen, so to speak, upon which the surplus, commerce, Chinas exports are unfairly helped by a retinal, cerebral cortex, deliberately undervalued yuan. Beijing exports, imports, US,image in the eye was projected. Through the agrees the surplus is too high, but says thediscoveries of Hubelcell, optical eye, and Wiesel we now yuan, bank, domestic, yuan is only one factor. Bank of Chinaknow that behind the origin of the visual nerve, imageperception in the brain there is a considerably foreign, increase, governor Zhou Xiaochuan said the country also needed to do more to boost domesticmore complicated course Wiesel By Hubel, of events. trade, value demand so more goods stayed within thefollowing the visual impulses along their path country. China increased the value of theto the various cell layers of the optical cortex, yuan against the dollar by 2.1% in July andHubel and Wiesel have been able to permitted it to trade within a narrow band, butdemonstrate that the message about the the US wants the yuan to be allowed to tradeimage falling on the retina undergoes a step- freely. However, Beijing has made it clearwise analysis in a system of nerve cells that it will take its time and tread carefullystored in columns. In this system each cell before allowing the yuan to rise further inhas its specific function and is responsible for value.a specific detail in the pattern of the retinalimage.
A clarification: definition of “BoW”• Looser definition – Independent features
A clarification: definition of “BoW”• Looser definition – Independent features• Stricter definition – Independent features – histogram representation
2 generative models1. Naïve Bayes classifier – Csurka Bray, Dance & Fan, 20042. Hierarchical Bayesian text models (pLSA and LDA) – Background: Hoffman 2001, Blei, Ng & Jordan, 2004 – Object categorization: Sivic et al. 2005, Sudderth et al. 2005 – Natural scene categorization: Fei-Fei et al. 2005
First, some notations• wn: each patch in an image – wn = [0,0,…1,…,0,0]T• w: a collection of all N patches in an image – w = [w1,w2,…,wN]• dj: the jth image in an image collection• c: category of the image• z: theme or topic of the patch
Case #1: the Naïve Bayes model c w N N c ∗ = arg max p (c | w) ∝ p (c) p ( w | c) = p(c)∏ p( wn | c) c n =1Object class Prior prob. of Image likelihood decision the object classes given the class Csurka et al. 2004
d z w Case #2: the pLSA model ND K p ( wi | d j ) = ∑ p ( wi | z k ) p ( z k | d j ) k =1Observed codeword Codeword distributions Theme distributions distributions per theme (topic) per image Slide credit: Josef Sivic
Case #2: Recognition using pLSA ∗ z = arg max p ( z | d ) z Slide credit: Josef Sivic
Case #2: Learning the pLSA parameters Observed counts of word i in document j Maximize likelihood of data using EM M … number of codewords N … number of images Slide credit: Josef Sivic
Demo: feature detection• Output of crude feature detector – Find edges – Draw points randomly from edge set – Draw from uniform distribution to get scale
Demo: learnt parameters• Learning the model: do_plsa(‘config_file_1’)• Evaluate and visualize the model: do_plsa_evaluation(‘config_file_1’) Codeword distributions Theme distributions per theme (topic) per image p(w | z ) p( z | d )
Discriminative methods based on ‘bag of words’ representation Decision Zebra boundary Non-zebra
Discriminative methods based on ‘bag of words’ representation• Grauman & Darrell, 2005, 2006: – SVM w/ Pyramid Match kernels• Others – Csurka, Bray, Dance & Fan, 2004 – Serre & Poggio, 2005
Summary: Pyramid match kernel optimal partial matching between sets of features Grauman & Darrell, 2005, Slide credit: Kristen Grauman
Pyramid Match (Grauman & Darrell 2005)Histogramintersection Slide credit: Kristen Grauman
Pyramid Match (Grauman & Darrell 2005)Histogramintersection matches at this level matches at previous level Difference in histogram intersections across levels counts number of new pairs matched Slide credit: Kristen Grauman
Pyramid match kernel histogram pyramids number of newly matched pairs at level imeasure of difficulty of a match at level i • Weights inversely proportional to bin size • Normalize kernel values to avoid favoring large sets Slide credit: Kristen Grauman
Example pyramid match Level 0 Slide credit: Kristen Grauman
Example pyramid match Level 1 Slide credit: Kristen Grauman
Example pyramid match Level 2 Slide credit: Kristen Grauman
Example pyramid matchpyramid matchoptimal match Slide credit: Kristen Grauman
Summary: Pyramid match kernel optimal partial matching between sets of featuresdifficulty of a match at level i number of new matches at level i Slide credit: Kristen Grauman
Object recognition results• ETH-80 database 8 object classes (Eichhorn and Chapelle 2004)• Features: – Harris detector – PCA-SIFT descriptor, d=10 Kernel Complexity Recognition rate Match [Wallraven et al.] 84% Bhattacharyya affinity 85% [Kondor & Jebara] Pyramid match 84% Slide credit: Kristen Grauman
Object recognition results• Caltech objects database 101 object classes• Features: – SIFT detector – PCA-SIFT descriptor, d=10• 30 training images / class• 43% recognition rate (1% chance performance)• 0.002 seconds per match Slide credit: Kristen Grauman
Invariance issues• Scale and rotation• Occlusion• Translation• View point (in theory) – Codewords: detector and descriptor – Theme distributions: different view points Fergus, Fei-Fei, Perona & Zisserman, 2005
Model properties Of all the sensory impressions proceeding to the brain, the visual experiences are the dominant ones. Our perception of the world around us is based essentially on the messages that reach the brain from our eyes. For a long time it was thought that the retinal image was transmitted pointbrain, to visual sensory, by point centers in the brain; the cerebral cortex was• Intuitive visual, perception, a movie screen, so to speak, upon which the retinal, cerebral cortex, image in the eye was projected. Through the – Analogy to documents discoveries of Hubelcell, optical eye, and Wiesel we now know that behind the origin of the visual nerve, image perception in the brain there is a considerably more complicated course Wiesel By Hubel, of events. following the visual impulses along their path to the various cell layers of the optical cortex, Hubel and Wiesel have been able to demonstrate that the message about the image falling on the retina undergoes a step- wise analysis in a system of nerve cells stored in columns. In this system each cell has its specific function and is responsible for a specific detail in the pattern of the retinal image.
Model properties Sivic, Russell, Efros, Freeman, Zisserman, 2005• Intuitive• generative models Dataset Incremental learning model – Convenient for weakly- or un-supervised, Classification incremental training – Prior information – Flexibility (e.g. HDP) Li, Wang & Fei-Fei, CVPR 2007
Model properties• Intuitive• generative models• Discriminative method – Computationally efficient Grauman et al. CVPR 2005
Model properties• Intuitive• generative models• Discriminative method• Learning and recognition relatively fast – Compare to other methods
Weakness of the model• No rigorous geometric information of the object components• It’s intuitive to most of us that objects are made of parts – no such information• Not extensively tested yet for – View point invariance – Scale invariance• Segmentation and localization unclear
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