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**Multiclass SVM and Applications in Object Classification**

Yuval Kaminka, Einat Granot Advanced Topics in Computer Vision Seminar Faculty of Mathematics and Computer Science Weizmann Institute May 2007

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**Object Classification**

?

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**Motivation – Human Visual System**

Large Number of Categories (~30,000) Discriminative Process Small Set of Examples Invariance to transformation Similarity to Prototype instead of Features

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**Similarity to Prototypes Vs Features**

No need for Feature Space Easy to enlarge number of categories Includes spatial relation between features No need for feature definition, for example in the tangent distance

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**D( ) , Distance Function Similarity is defined by Distance Function**

Easy to adjust to different types (Shape, Texture) Can include invariance to intra-class transformations

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**Distance Function – simple example**

) = ) = || 2.1, 27, 31, 15, 8 . - || 13, 45, 22.5, 78, 91 ? , , 2.1 27 31 .

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**A Classic Classification Problem**

Training Set S: (X1..Xn), with class label (Y1.. Yn) Given a query image q, determine its label X2 X3 X1 X5 q X4 X6 X7

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Nearest Neighbor (NN) ?

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**K-Nearest Neighbor (KNN)**

? K = 3

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**K-NN Pros Simple, yet outperforms other methods Low Complexity: O(Dּn)**

D - the cost per one distance function calculation No need for Feature Space definition No computational cost for adding new categories n ∞ ==> Error Rate Bayes optimal Bayes Optimal – A classifiers that always classify the classification that will get maximum probability, going over all possible hypothesis

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**K-NN Cons Complete Set Missing Set NN SVM**

P. Vincent et al., K-local hyperplane and convex distance nearest neighbor algorithms, NIPS 2001

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**SVM Two class classification algorithm**

Hyperplane – תת-קבוצה של וקטורים במימד n-1 שמגדיר הפרדה במימד ה-n. Linear Hyperplane – Hyperplane שעובר דרך הראשית Class 1 We’re looking for a hyperplane that best separates the classes Some of the slides on SVM are adapted with permission from Martin Law’s presentation on SVM

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**As far away as possible from the data of both classes**

SVM - Motivation Class 2 Class 2 Class 1 Class 1 As far away as possible from the data of both classes

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**SVM – A learning algorithm**

KNN – simple classification, no training Class 1 Class 2 SVM – a learning algorithm Training – find the hyperplane Classification – label a new query Two Phases:

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**SVM – Training Phase We’re looking for (w,b) that will:**

Class 2 ~b wTx+b=0 Class 1 We’re looking for (w,b) that will: Classify correctly the classes Give maximum margins

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**1. Correct classification**

{x1, ..., xn} our training set wTx+b=0 Class 1 Correct classification: wTxi+b>0 for green, and wTxi+b<0 for red Assume the labels {y1.. yn} are from the set {-1,1}:

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2. Margin maximization Class 2 m Class 1 m = ?

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**2. Margin maximization m We can scale (w,b) (w,b), >0**

|wTz+b| ||w|| Class 2 z m Class 1 We can scale (w,b) (w,b), >0 Won’t change classification: wTx+b>0 wTx+b>0 Get a desired distance: |wTz+b|=a =1/a, |wTz+b|=1

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**SVM as an Optimization Problem**

Maximize margins Correct Classification Solve optimization problem with constraints We can find a1.. an, such that: Langrangian multipliers C.J.C. Burges. A Tutorial on Support Vector Machines for Pattern Recognition, 1998.

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**SVM as an Optimization Problem**

Maximize margins Correct Classification Classic optimization problem with constraints לשנות x ל-w ולתקן למטה ל-xi s.t.

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**SVM as an Optimization Problem**

s.t. There must exist positive a1.. an such that: And in our case: There must exist positive a1.. an such that: gi(x) f(x)

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**Support Vectors xi with ai>0 are called support vectors (SV)**

Class 2 a=0 a>0 a=0 a=0 a>0 a=0 a>0 a=0 a=0 Class 1 xi with ai>0 are called support vectors (SV) w is determined only by the SV

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**Allowing errors We would now like to minimize wTx+b=1 wTx+b=0 wTx+b=-1**

Class 2 wTx+b=1 Class 1 wTx+b=0 wTx+b=-1 We would now like to minimize

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Allowing errors As before we get: Class 2 Class 1

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**SVM – Classification phase**

q Class 1 Compute wTq+b Classify as class 1 if positive, and class 2 otherwise

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**Upgrade SVM We only need to calculate inner products**

In order to find a1.. an we need to calculate xiTxj i,j In order to classify a query q we need to calculate:

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**Feature Expansion f(.) Extended space Input space f(.)**

( 1 , x , y , xy , x2 , y2 ) (x , y) Problem: too expensive!

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**Solution: The Kernel Trick**

We only need to calculate inner products f( ) f(.) Find a kernel function K such that:

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**The Kernel Trick We only need to calculate inner products**

In order to find a1.. an we need to calculate xiTxj i,j Build a kernel matrix MnXn: M[i,j]= (xi)T(xj)=K(xi,xj) In order to classify a query q we need to calculate wTq+b:

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**Inner product Distance Function**

We only need to calculate inner products In our case: convert to distance function Parallelogram law: ||u+v||^2+||u-v||^2=2||u||^2+2||v||^2 From “origin” Pairwise distance

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**Inner product Distance Function**

Use the fact that we only need to calculate inner products In order to find a1.. an we need to calculate xiTxj i,j Build a distance matrix DnXn: D[i,j] = xiTxj = 1/2ּ[d(xi,0)+d(xj,0)-d(xi,xj)] In order to classify a query q we need to calculate wTq+b:

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**SVM Pros and Cons Pros: Easy to integrate different distance functions**

Fast classification of new objects (depends on SV) Good performance even with small set of examples Cons: Slow training ( O(n2), n=# of vectors in training set ) Separates only 2 classes להזכיר שהחיסרון הראשון "נעלם" כאשר מדובר על סט קטן של דוגמאות

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**Multiclass SVM Extend SVM for multi-classes separation**

Nc = number of classes Class 2 Class 1 Class 5 Class 4 Class 3

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**Two approaches Class 1 Class 2 Class 3 Class 4**

1-vs-rest 1-vs-1 DAGSVM Combine multi-binary-classifiers Generate one function based on single optimization problem

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1-vs-rest Class 2 Class 1 Class 4 Class 3

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1-vs-rest w2 w1 Class 2 Class 1 w3 w4 Nc classifiers Class 3 Class 4

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**1-vs-rest Class 2 Class 1 Class 3 Class 4 w2 w1 w3 w4**

~ Similarity(q,SV3) q ~ Similarity(q,SV2) w1Tq+b1 ~ Similarity(q,SV1) ~ Similarity(q,SV4) Class 3 Class 4

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**argmax1≤i ≤Nc{Sim(q,SVi)}**

1-vs-rest w2 w1 Class 2 Class 1 w3 w4 q Label(q)= argmax1≤i ≤Nc{Sim(q,SVi)} Class 3 Class 4

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**1-vs-rest After training we’ll have Nc decision functions:**

fi(x)=wiTx+bi Class of query object q is determined by: argmax1≤i ≤Nc{ wiTx+bi } Pros: Only Nc classifiers to be trained and tested Cons: Every classifier use all vectors for training No bound on generalization error

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**1-vs-rest Complexity For training:**

Nc classifiers, each using n vectors for finding hyperplane For classifying new objects: Nc classifiers, each is tested once, M=max number of SV

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1-vs-1 Class 2 Class 1 Class 4 Class 3

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**1-vs-1 Nc(Nc-1)/2 classifiers Class 2 Class 1 Class 4 Class 3 W1,2**

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**1-vs-1 with Max Wins ☺ ☺ ☺ ☺ ☺ ☺ Class 2 Class 1 Class 4 Class 3 W1,2**

q W2,3 ~ 2 or 4 ? Sign(w1,2Tq+b1,2) ~ 1 or 2 ? W1,3 ~ 1 or 3 ? W2,4 ~ 1 or 4 ? ~ 3 or 4 ? W3,4 ~ 2 or 3 ? Class 4 Class 3 ☺ ☺

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**1-vs-1 with Max Wins ☺ ☺ ☺ ☺ ☺ ☺ Class 2 Class 1 Class 4 Class 3 W1,2**

q W2,3 W1,3 W2,4 W3,4 Class 4 Class 3 ☺ ☺

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1-vs-1 with Max Wins After training we’ll have Nc(Nc-1)/2 decision functions: fij(x)=sign(wijTx+bij) Class of query object x is determined by max-votes Pros: Every classifier use a small set of vectors for training Cons: Nc(Nc-1)/2 classifiers to be trained and tested No bound on generalization error

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**1-vs-1 Complexity For training:**

Assume that every class contains ~ n/Nc instances Nc(Nc-1)/2 classifiers, each using ~2n/Nc vectors: For classifying new objects: Nc(Nc-1)/2 classifiers, each is tested once, M as before

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**What did we have so far? 1-vs-1 1-vs-rest Nc(Nc-1)/2 Nc**

Class 1 Class 2 Class 3 Class 4 Class 1 Class 2 Class 3 Class 4 1-vs-1 1-vs-rest Nc(Nc-1)/2 Nc # of classifiers (each need to be trained and tested) ~2n/Nc n (all vectors) # of vectors for training (per classifier) No bound on generalization error להזכיר שכשהאימון נעשה על מס' דוגמאות קטן זה אמנם יתרון מבחינת סיבוכיות, אך יכול להיות חסרון מבחינת ביצועים

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**DAGSVM 1-vs-1 Decision DAG (DDAG) 4 1 2 3**

3 4 2 3 4 1 2 1 2 3 2 3 not 1 not 2 not 3 not 4 4 1 2 3 Class 1 Class 2 Class 3 Class 4 W1,2 W1,3 W1,4 W2,3 W3,4 W2,4 J. C. Platt et al., Large margin DAGs for multiclass classification. NIPS, 1999.

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**Binary decision function Nc(Nc-1)/2 internal nodes**

DDAG on Nc Classes Single root node 1 vs 4 3 vs 4 2 vs 4 1 vs 3 2 vs 3 1 vs 2 3 4 2 3 4 1 2 1 2 3 2 3 not 1 not 2 not 3 not 4 4 1 2 3 In every node: Binary decision function Nc(Nc-1)/2 internal nodes DAG Nc leaves, one per class

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**Building the DDAG 1 2 3 4 change list order no affect on results 4 3 2**

1 vs 4 change list order no affect on results not 1 not 4 2 3 4 2 vs 4 1 2 3 1 vs 3 not 2 not 4 not 1 not 3 2 3 3 vs 4 2 vs 3 1 vs 2 3 4 1 2 4 3 2 1

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**Classification using DDAG**

1 vs 4 W1,2 ~ 1 or 2 ? q Class 2 Class 1 ~ 1 or 4 ? not 1 not 4 W1,4 ~ 1 or 3 ? 2 3 4 2 vs 4 1 2 3 1 vs 3 W1,3 W2,3 W2,4 W3,4 not 2 not 4 not 1 not 3 בהנחה שה-classes ניתנים להפרדה והשוליים שמתקבלים אכן גדולים אזי הגיוני "להיפטר" מה-class שלא בחרנו לסווג אליה בכל פעם. 3 4 2 3 3 vs 4 2 vs 3 1 vs 2 1 2 Class 4 Class 3 4 3 2 1

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**DAGSVM Pros: Only Nc-1 classifiers to be tested**

Every classifier uses a small set of vectors for training Bound on generalization error (~margins size) Cons: Less vectors for training worse classifier? Nc(Nc-1)/2 classifiers to be trained

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**DAGSVM Complexity For training:**

Assume that every class contains ~n/Nc instances Nc(Nc-1)/2 classifiers, each using ~2n/Nc vectors: For classifying new objects: Nc-1 classifiers, each is tested once M = max number of SV

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**Classification complexity**

Multiclass SVM DAGSVM 1-vs-1 1-vs-rest Nc # of classifiers O(Dּn2) O(DּNcn2) Training complexity O(M2ּNc) O(M2ּNc2) O(M1ּNc) Classification complexity

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**Multiclass SVM comparison**

Classification Training

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**Multiclass SVM - Summary**

Training: Classification: Error rates: Bound of generalization error - only on DAGSVM In practice – 1-vs-1 and DAGSVM The “one big optimization” methods Similar error rates Very slow training – limited to small data sets 1-vs-rest DAGSVM / 1-vs-1 O(DּNcּn2) O(Dּn2) 1-vs-1 DAGSVM / 1-vs-rest O(DּMּNc2) O(DּMּNc)

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**So what do we have? Nearest Neighbor (KNN) SVM Fast**

Suitable for multi-class Easy to integrate different distance functions Problematic with few samples SVM Good performance even with small set of examples No natural extension to multi-class Slow to train Class 1 Class 2

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**SVM KNN - From coarse to fine**

Suggestion Hybrid system KNN SVM Zhang et al, SVM-KNN: Discriminative Nearest Neighbor Classification for Visual Category Recognition, 2006

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**SVM KNN – General Algorithm**

Calculate distance from query to training images Query image Class 1 Class 2 Class 3 Training images and query

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**SVM KNN – General Algorithm**

Calculate distance from query to training images Pick K nearest neighbors Query image Class 1 Class 2 Class 3 Training images and query

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**SVM KNN – General Algorithm**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Query image Class 1 Class 2 Class 3 SVM works well with few samples Training images and query

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**SVM KNN – General Algorithm**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! Query image Class 1 Class 2 Class 3 Query image Class 2 Training images and query

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**Training + Classification**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Classic process: Training Classification SVM-KNN Coarse Classification Final classification

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**Details Details Details**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Calculating distance is a heavy task Compute crude distance – faster Finding Kpotential images Ignore all other images Compute accurate distance Only relative to the Kpotential images L2 Accurate Kpotential

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**Details Details Details**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Complexity: Crude distance Accurate distance L2 Accurate Kpotential

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**Details Details Details**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM If K neighbors are from the same class Done

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**Details Details Details**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Construct pairwise inner product matrix Improvement – cache distance calculation

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**Details Details Details**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Selected SVM: DAGSVM (faster) Complexity: 1 vs 4 3 vs 4 2 vs 4 1 vs 3 2 vs 3 1 vs 2

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**Complexity Total complexity DAGSVM training complexity**

Calculate distance from query to training images Pick K nearest neighbors Run SVM Label ! KNN SVM Total complexity DAGSVM training complexity

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**SVM KNN – continuum Defining an SVM-KNN continuum: NN SVM**

K = n (#images) NN KNN SVM SVM More than MAJ Biological motivation Human visual system

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**SVM KNN Summary Similarity to prototypes**

Combining Advantages from both methods NN – Fast, suitable for multiclass SVM – performs well with few samples and classes Compatible with many types of distance functions Biological motivation: Human visual system Discriminative process

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**Outline Motivation and Introduction Classification Algorithms**

K-Nearest neighbors (KNN) SVM Multiclass SVM DAGSVM SVM-KNN Results - A taste of the distance Shape distance (shape context, tangent) Texture (texton histograms)

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**D( ) = ?? , Distance functions Shape Texture Query image**

Class 1 Class 2 Class 3 Training images and query Shape Texture D( , ) = ??

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**Understanding the need - Shape**

Well, which is it?? Capturing the shape Distance 1: Shape context Distance 2: Tangent distance query

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**Distance 1: Shape context**

Find point correspondences Estimate transformation Distance correspondence quality transformation quality prototype query Belongie et al., Shape matching and object recognition using shape contexts, IEEE Trans. (2002)

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**Find correspondences Detector - Use edge points**

Descriptor - Create “Landscape” Relationship to other edge points Histogram of orientations and distances Count = 5 Count = 6 prototype query

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**Find correspondence Detector - Use edge points**

Descriptor - Create “Landscape” Relationship to other edge points Histogram of orientations and distances Matching compare histograms ( ) prototype query

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**Distance 1: Shape context**

Find point correspondences Estimate transformation Distance correspondence quality transformation (quality, magnitude) prototype query

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**MNIST – Digit DB 70,000 handwritten digits Each image 28x28**

Us postal service

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**MNIST results Human error rate – 0.2% Better methods exist < 1%**

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**Distance 2: Tangent distance**

Distance includes invariance to small changes small rotations translations thickening Prototype query Taking the original image and allowing small rotations Simard et al., Transformation invariance in pattern recognition-tangent distance and tangent propagation. Neural Networks (1998)

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**Space induced by rotation**

Rotation function α=1 α=0 But – this space might be nonlinear therefore we actually look at a linear approximation Dimension = 1 α= -1 α= -2 Pixel space

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**Tangent distance – Visual intuition**

SQ The Tangent SP Prototype Image Desired distance But – calculating distance between non linear curves can be difficult Solution: Use linear approximation The Tangent P Q Query Image Euclidian distance (L2) Pixel space

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**Tangent Distance - General**

For every image, create surface allowing transformations Rotations Translations Thickness, etc. Find a linear approximation - the tangent plane Distance Calculate distance between linear planes Has efficient solutions 7 dimensions

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**USPS – digit DB 9298 handwritten digits taken from mail envelopes**

Each image 16x16 Us postal service

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**USPS results Human error rate – 2.5% For L2 – For tangent not optimal**

Q Human error rate – 2.5% For L2 – not optimal DAGSVM has similar results For tangent NN similar results DAGSVM similar to SVMKNN but SVM KNN is faster According to the paper on tangent distance, it received a 2.5% with NN using tangent distance.

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**Understanding Texture**

Texture samples How to represent Texture??

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**Texture representation**

Represent using responses to a filter bank Texture patch Filter bank – 48 filters Filter responses for pixel P1 Filter responses for pixel 0.1 0.8 . 0.3 P2 0.6 Filter responses for pixel -0.4 -0.7 . 0.17 P3 48 Motivation – V1 -0.2 . …. 0.4

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**Correspond to pixels of one image**

Introducing Textons Filter responses – points in 48 dimensional space A texture patch – spatially repeating Representation is redundant Select representative responses (K-means) Correspond to pixels of one image Texture patch P1 P2 P3 Filter responses in 48-dimensional space Textons ! T. Leung, J. Malik Representing and recognizing the visual appearance of materials using three-dimensional textons (2001)

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**“Building blocks“ for all textures**

Universal textons “Building blocks“ for all textures Prototype textures Filter bank Texton Filter responses in 48-dim space T1 T2 T3 T4

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**Distance 3: of Texton histograms**

For a query texture Create filter responses Build texton histogram (using universal textons) Query texture Filter bank Filter responses in 48-dim space T1 T2 T3 T4 T1 T2 T3 T4 Query Texton histogram

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**Distance 3: of Texton histograms**

For a query texture Create texton histogram Build texton histogram (using universal textons) Distance compare histograms ( ) Prototype textures Query texture Query Texton histogram Prototype Texton histogram T1 T2 T3 T4 T1 T2 T3 T4

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**CUReT – texture DB 61 textures Different view points**

Different illuminations

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CUReT Results T1 T2 T3 T4 (comparing texton histograms)

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**Caltech-101 DB 102 categories Distance function**

variations in color, pose, illumination Distance function combination of texture and shape 2 algorithms Algo. A, Algo. B Samples from the Caltech-101 DB

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**Caltech-101 Results 66% correct Correct rate (%) Algo. B:**

(15 training images) 66% correct Correct rate (%) Algo. B: Using only DAGSVM (no KNN) Still a long way to go…

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**Motivation – Human Visual System**

Large Number of Categories (~30,000) Discriminative Process Small Set of Examples Invariance to transformation Similarity to Prototype instead of Features

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**Summary Popular methods NN SVM DAGSVM - extension to multi-class SVM**

The hybrid method – SVM KNN Motivated by human perception (??) Improved complexity Better methods exist? A taste of the distance Shape, Texture Results classification method distance function Class 1 Class 2 1 vs 4 3 vs 4 2 vs 4 1 vs 3 2 vs 3 1 vs 2 P Q T1 T2 T3 T4

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References H. Zhang, A. C. Berg, M. Maire and J. Malik. SVM-KNN: Discriminative Nearest Neighbor Classification for Visual Category Recognition. IEEE, Vol. 2, pages , 2006. P. Vincent and Y. Bengio. K-local hyperplane and convex distance nearest neighbor algorithms. NIPS, pages , 2001. J. C. Platt, N. Cristianini, and J. Shawe-Taylor. Large margin DAGs for multiclass classification. NIPS, pages , 1999. C. Hsu and C. Lin. A comparison of methods for multiclass support vector machines. IEEE, Vol. 13, pages , 2002. T. Leung and J. Malik. Representing and recognizing the visual appearance of materials using three-dimensional textons. Int. J. Computation Vision, 43(1):29-44, 2001. P. Simard, Y. LeCun, J. S. Denker, and B. Victorri. Transformation invariance in pattern recognition-tangent distance and tangent propagation. Neural Networks: Tricks of the Trade, pages , 1998. S. Belongie, J. Malik, and J. Puzicha. Shape matching and object recognition using shape contexts. IEEE, Vol. 24, pages , 2002.

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