Object Recognizing
We will discuss:
• Features
• Classifiers
• Example ‘winning’ system
Object Classes
Class Non-class
Class Non-class
Features and Classifiers
Same features with different classifiersSame classifier with different features
Generic Features
Simple (wavelets) Complex (Geons)
Class-specific Features: Common Building Blocks
Optimal Class Components?
• Large features are too rare
• Small features are found
everywhere
Find features that carry the highest amount of information
Entropy
Entropy:
x = 0 1 H
p = 0.5 0.5 ? 0.1 0.9 0.47 0.01 0.99 0.08
)p(x log )p(x- H i2i
Mutual Information I(C,F)
Class:11010100
Feature:10011100
I(F,C) = H(C) – H(C|F)
Optimal classification features
• Theoretically: maximizing delivered information minimizes classification error
• In practice: informative object components can be identified in training images
Mutual Info vs. Threshold
0.00 20.00 40.00
Detection threshold
Mu
tu
al
Info
forehead
hairline
mouth
eye
nose
nosebridge
long_hairline
chin
twoeyes
Selecting Fragments
Adding a New Fragment(max-min selection)
?
MIΔ
MI = MI [Δ ;class] - MI [ ;class ]Select: Maxi Mink ΔMI (Fi, Fk)
)Min. over existing fragments, Max. over the entire pool(
);(),;(min);(),;( jjiij
i FCMIFFCMIFCMIFFCMI
Highly Informative Face Fragments
Horse-class features
Car-class features
Pictorial features Learned from examples
Fragments with positions
On all detected fragments within their regions
Star model
Detected fragments ‘vote’ for the center location
Find location with maximal vote
Bag of words
ObjectObject Bag of ‘words’Bag of ‘words’
Bag of visual words A large collection of image patches
–
1.Feature detection 1.Feature detection and representationand representation
•Regular grid– & VogelSchiele ,2003
–Fei- ,Fei & Perona2005
Each class has its words historgram
–
–
–
SVM – linear separation in feature space
Optimal Separation
SVMPerceptron
Find a separating plane such that the closest points are as far as possible
Separating line: w ∙ x + b = 0 Far line: w ∙ x + b = +1Their distance: w ∙ ∆x = +1 Separation: |∆x| = 1/|w|Margin: 2/|w|
0+1
-1 The Margin
Max Margin Classification
)Equivalently, usually used
How to solve such constraint optimization ?
The examples are vectors xi
The labels yi are +1 for class, -1 for non-class
Using Lagrange multipliers: Minimize LP =
With αi > 0 the Lagrange multipliers
Minimize Lp :
Set all derivatives to 0:
Also for the αi
Dual formulation: Maximize the Lagrangian w.r.t. the αi and the above conditions. Put into Lp
Dual formulationMathematically equivalent formulation: Can maximize the Lagrangian with respect to the αi
After manipulations – nice concise optimization :
SVM: in simple matrix form
We first find the α. From this we can find: w, b, and the support vectors.
The matrix H is a simple ‘data matrix’: Hij = yiyj <xi∙xj>
Final classification: w∙x + b ∑αi yi <xi x> + b
Because w = ∑αi yi xi Only <xi x> with support vectors are used
Full story – separable case
Or use ∑αi yi <xi x> + b
Quadratic Programming QP
Minimize (with respect to x)
Subject to one or more constraints of the form:
Ax < b (inequality constraints)Ex = d (equality constraints)
The non-separable case
It turns out that we can get a very similar formulation of the problem and solution, if we penalize the incorrect classification in a certain way. The penalty is Cξi where ξi ≥ 0is the distance of the miss-classified point from the respective plane. We now minimize a penalty with the miss-classifications:
Kernel Classification
Using kernels
A kernel K(x,x’) is also associated with a mapping x → φ(x) We can use φ(x) and perform a linear classification in the target
space .
It turns out that this can be done directly using kernels and without the mapping, the results are equivalent. The optimal separation in the target space is the same as what we will get using the procedure below. It is similar to the linear case, with
the kernel replacing the dot-product .
Use K(xi, xj)
Use ∑αi yi K<xi x> + b
Summary points
• Linear separation with the largest margin, f(x) = w∙x + b
• Dual formulation, f(x) = ∑αi yi (xi ∙ x) + b
• Natural extension to non-separable classes
• Extension through kernels, f(x) = ∑αi yi K(xi x) + b
Felzenszwalb et al .
• Felzenszwalb, McAllester, Ramanan CVPR 2008. A Discriminatively Trained, Multiscale, Deformable Part Model
Object model using HoG
A bicycle and its ‘root filter ’The root filter is a patch of HoG descriptor Image is partitioned into 8x8 pixel cells In each block we compute a histogram of gradient orientations
Using patches with HoG descriptors and classification by SVM
The filter is searched on a pyramid of HoG descriptors, to deal with unknown scale
Dealing with scale: multi-scale analysis
A part Pi = (Fi, vi, si, ai, bi) .
Fi is filter for the i-th part, vi is the center for a box of possible positions for part i relative to the root position, si the size of this box
ai and bi are two-dimensional vectors specifying coefficients of a quadratic function measuring a score for each possible placement of the i-th part. That is, ai and bi are two numbers each, and the penalty for deviation ∆x, ∆y from the expected location is a1 ∆ x + a2 ∆y + b1 ∆x2 + b2 ∆y2
Adding Parts
Bicycle model: root, parts, spatial map
Person model
The full score of a potential match is: ∑ Fi ∙ Hi + ∑ ai1 xi + ai2 y + bi1x2 + bi2y2
Fi ∙ Hi is the appearance part
xi, yi, is the deviation of part pi from its expected location in the model. This is the spatial part.
Match Score
The score of a match can be expressed as the dot-product of a vector β of coefficients, with the image:
Score = β∙ψ
Using the vectors ψ to train an SVM classifier :β∙ψ > 1 for class examples
β∙ψ < 1 for class examples
Using SVM:
β∙ψ > 1 for class examples β∙ψ < 1 for class examples
However, ψ depends on the placement z, that is, the values of ∆xi, ∆yi
We need to take the best ψ over all placements. In their notation :Classification then uses β∙f > 1
We need to take the best ψ over all placements. In their notation :
Classification then uses β∙f > 1
In analogy to classical SVMs we would like to train from labeled examples D = (<x1, y1i> . . . , <xn, yn>) By optimizing the following objective function,
Finding β, SVM training:
search with gradient descent over the placement. This includes also the levels in the hierarchy. Start with the root filter, find places of high score for it. For these high-scoring locations, each for the optimal placement of the parts at a level with twice the resolution as the root-filter, using GD.
With the optimal placement, use
β∙ψ > 1 for class examples β∙ψ < 1 for class examples
Recognition
• Training -- positive examples with bounding boxes around the objects, and negative examples.
• Learn root filter using SVM
• Define fixed number of parts, at locations of high energy in the root filter HoG
• Use these to start the iterative learning
Hard Negatives
The set M of hard-negatives for a known β and data set DThese are support vector (y ∙ f =1) or misses (y ∙ f < 1)
Optimal SVM training does not need all the examples, hard examples are sufficient. For a given β, use the positive examples + C hard examples Use this data to compute β by standard SVM Iterate (with a new set of C hard examples)
All images contain at least 1 bike
Correct person detections
Difficult images, medium results. About 0.5 precision at 0.5 recall
All images contain at least 1 bird
Average precision :Roughly, AP of 0.3 – in a test with 1000 class images, out of the top 1000 detection, 300 will be true class examples (recall =
precision = 0.3) .
Future Directions
• Dealing with very large number of classes – Imagenet, 15,000 categories, 12 million images
• To consider: human-level performance for at least one class