Discriminative and generative methods for bags of features
Download
1 / 40

- PowerPoint PPT Presentation


  • 128 Views
  • Uploaded on

Discriminative and generative methods for bags of features. Zebra. Non-zebra. Many slides adapted from Fei-Fei Li, Rob Fergus, and Antonio Torralba. Image classification.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about '' - calix


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Discriminative and generative methods for bags of features l.jpg
Discriminative and generative methods for bags of features

Zebra

Non-zebra

Many slides adapted from Fei-Fei Li, Rob Fergus, and Antonio Torralba


Image classification l.jpg
Image classification

  • Given the bag-of-features representations of images from different classes, how do we learn a model for distinguishing them?


Discriminative methods l.jpg
Discriminative methods

  • Learn a decision rule (classifier) assigning bag-of-features representations of images to different classes

Decisionboundary

Zebra

Non-zebra


Generative methods l.jpg
Generative methods

  • Model the probability of a bag of features given a class


Classification l.jpg
Classification

  • Assign input vector to one of two or more classes

  • Any decision rule divides input space into decision regions separated by decision boundaries


Nearest neighbor classifier l.jpg
Nearest Neighbor Classifier

  • Assign label of nearest training data point to each test data point

from Duda et al.

Voronoi partitioning of feature space

for 2-category 2-D and 3-D data

Source: D. Lowe


Slide7 l.jpg

K-Nearest Neighbors

  • For a new point, find the k closest points from training data

  • Labels of the k points “vote” to classify

  • Works well provided there is lots of data and the distance function is good

k = 5

Source: D. Lowe


Functions for comparing histograms l.jpg
Functions for comparing histograms

  • L1 distance

  • χ2 distance

  • Quadratic distance (cross-bin)

Jan Puzicha, Yossi Rubner, Carlo Tomasi, Joachim M. Buhmann: Empirical Evaluation of Dissimilarity Measures for Color and Texture. ICCV 1999


Earth mover s distance l.jpg
Earth Mover’s Distance

  • Each image is represented by a signatureS consisting of a set of centers {mi } and weights {wi }

  • Centers can be codewords from universal vocabulary, clusters of features in the image, or individual features (in which case quantization is not required)

  • Earth Mover’s Distance has the formwhere the flowsfij are given by the solution of a transportation problem

Y. Rubner, C. Tomasi, and L. Guibas: A Metric for Distributions with Applications to Image Databases. ICCV 1998


Linear classifiers l.jpg
Linear classifiers

  • Find linear function (hyperplane) to separate positive and negative examples

Which hyperplaneis best?


Support vector machines l.jpg
Support vector machines

  • Find hyperplane that maximizes the margin between the positive and negative examples

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Support vector machines12 l.jpg
Support vector machines

  • Find hyperplane that maximizes the margin between the positive and negative examples

For support, vectors,

Distance between point and hyperplane:

Therefore, the margin is 2 / ||w||

Support vectors

Margin

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Finding the maximum margin hyperplane l.jpg
Finding the maximum margin hyperplane

  • Maximize margin 2/||w||

  • Correctly classify all training data:

  • Quadratic optimization problem:

  • Minimize Subject to yi(w·xi+b) ≥ 1

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Finding the maximum margin hyperplane14 l.jpg
Finding the maximum margin hyperplane

  • Solution:

learnedweight

Support vector

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Finding the maximum margin hyperplane15 l.jpg
Finding the maximum margin hyperplane

  • Solution:b = yi – w·xi for any support vector

  • Classification function (decision boundary):

  • Notice that it relies on an inner product between the testpoint xand the support vectors xi

  • Solving the optimization problem also involvescomputing the inner products xi· xjbetween all pairs oftraining points

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Nonlinear svms l.jpg

x

0

x

0

x2

Nonlinear SVMs

  • Datasets that are linearly separable work out great:

  • But what if the dataset is just too hard?

  • We can map it to a higher-dimensional space:

0

x

Slide credit: Andrew Moore


Nonlinear svms17 l.jpg
Nonlinear SVMs

  • General idea: the original input space can always be mapped to some higher-dimensional feature space where the training set is separable:

Φ: x→φ(x)

Slide credit: Andrew Moore


Nonlinear svms18 l.jpg
Nonlinear SVMs

  • The kernel trick: instead of explicitly computing the lifting transformation φ(x), define a kernel function K such thatK(xi,xjj) = φ(xi )· φ(xj)

  • (to be valid, the kernel function must satisfy Mercer’s condition)

  • This gives a nonlinear decision boundary in the original feature space:

C. Burges, A Tutorial on Support Vector Machines for Pattern Recognition, Data Mining and Knowledge Discovery, 1998


Kernels for bags of features l.jpg
Kernels for bags of features

  • Histogram intersection kernel:

  • Generalized Gaussian kernel:

  • D can be Euclidean distance, χ2distance,Earth Mover’s Distance, etc.

J. Zhang, M. Marszalek, S. Lazebnik, and C. Schmid, Local Features and Kernels for Classifcation of Texture and Object Categories: A Comprehensive Study, IJCV 2007


Pyramid match kernel l.jpg
Pyramid match kernel

  • Fast approximation of Earth Mover’s Distance

  • Weighted sum of histogram intersections at mutliple resolutions (linear in the number of features instead of cubic)

optimal partial matching between sets of features

K. Grauman and T. Darrell.  The Pyramid Match Kernel: Discriminative Classification with Sets of Image Features, ICCV 2005.


Review discriminative methods l.jpg
Review: Discriminative methods

  • Nearest-neighbor and k-nearest-neighbor classifiers

    • L1 distance, χ2 distance, quadratic distance, Earth Mover’s Distance

  • Support vector machines

    • Linear classifiers

    • Margin maximization

    • The kernel trick

    • Kernel functions: histogram intersection, generalized Gaussian, pyramid match

  • Of course, there are many other classifiers out there

    • Neural networks, boosting, decision trees, …


Summary svms for image classification l.jpg
Summary: SVMs for image classification

  • Pick an image representation (in our case, bag of features)

  • Pick a kernel function for that representation

  • Compute the matrix of kernel values between every pair of training examples

  • Feed the kernel matrix into your favorite SVM solver to obtain support vectors and weights

  • At test time: compute kernel values for your test example and each support vector, and combine them with the learned weights to get the value of the decision function


What about multi class svms l.jpg
What about multi-class SVMs?

  • Unfortunately, there is no “definitive” multi-class SVM formulation

  • In practice, we have to obtain a multi-class SVM by combining multiple two-class SVMs

  • One vs. others

    • Traning: learn an SVM for each class vs. the others

    • Testing: apply each SVM to test example and assign to it the class of the SVM that returns the highest decision value

  • One vs. one

    • Training: learn an SVM for each pair of classes

    • Testing: each learned SVM “votes” for a class to assign to the test example


Svms pros and cons l.jpg
SVMs: Pros and cons

  • Pros

    • Many publicly available SVM packages:http://www.kernel-machines.org/software

    • Kernel-based framework is very powerful, flexible

    • SVMs work very well in practice, even with very small training sample sizes

  • Cons

    • No “direct” multi-class SVM, must combine two-class SVMs

    • Computation, memory

      • During training time, must compute matrix of kernel values for every pair of examples

      • Learning can take a very long time for large-scale problems


Generative methods25 l.jpg
Generative methods

  • Model the probability distribution that produced a given bag of features

  • We will cover two models, both inspired by text document analysis:

    • Naïve Bayes

    • Probabilistic Latent Semantic Analysis


Slide26 l.jpg

The Naïve Bayes model

  • Assume that each feature is conditionally independent given the class

Csurka et al. 2004


Slide27 l.jpg

The Naïve Bayes model

  • Assume that each feature is conditionally independent given the class

Likelihood of ith visual word

given the class

MAP

decision

Prior prob. of

the object classes

Estimated by empiricalfrequencies of visual wordsin images from a given class

Csurka et al. 2004


Slide28 l.jpg

The Naïve Bayes model

  • Assume that each feature is conditionally independent given the class

  • “Graphical model”:

c

w

N

Csurka et al. 2004


Slide29 l.jpg

Probabilistic Latent Semantic Analysis

zebra

“visual topics”

grass

tree

T. Hofmann, Probabilistic Latent Semantic Analysis, UAI 1999


Slide30 l.jpg

Probabilistic Latent Semantic Analysis

= α1

+ α2

+ α3

tree

New image

grass

zebra

T. Hofmann, Probabilistic Latent Semantic Analysis, UAI 1999


Slide31 l.jpg

z

d

w

Probabilistic Latent Semantic Analysis

  • Unsupervised technique

  • Two-level generative model: a document is a mixture of topics, and each topic has its own characteristic word distribution

document

topic

word

P(z|d)

P(w|z)

T. Hofmann, Probabilistic Latent Semantic Analysis, UAI 1999


Slide32 l.jpg

z

d

w

Probabilistic Latent Semantic Analysis

  • Unsupervised technique

  • Two-level generative model: a document is a mixture of topics, and each topic has its own characteristic word distribution

T. Hofmann, Probabilistic Latent Semantic Analysis, UAI 1999


Slide33 l.jpg

z

d

w

“face”

pLSA for images

Document = image, topic = class, word = quantized feature

J. Sivic, B. Russell, A. Efros, A. Zisserman, B. Freeman, Discovering Objects and their Location in Images, ICCV 2005


Slide34 l.jpg

The pLSA model

Probability of word i giventopic k (unknown)

Probability of word i in document j(known)

Probability oftopic k givendocument j(unknown)


Slide35 l.jpg

The pLSA model

documents

topics

documents

p(wi|dj)

p(wi|zk)

p(zk|dj)

words

words

topics

=

Class distributions

per image

(K×N)

Observed codeword

distributions

(M×N)

Codeword distributions

per topic (class)

(M×K)


Slide36 l.jpg

Learning pLSA parameters

Maximize likelihood of data using EM:

Observed counts of word i in document j

M … number of codewords

N … number of images

Slide credit: Josef Sivic


Slide37 l.jpg

Recognition

  • Finding the most likely topic (class) for an image:


Slide38 l.jpg

Recognition

  • Finding the most likely topic (class) for an image:

  • Finding the most likely topic (class) for a visual word in a given image:


Topic discovery in images l.jpg
Topic discovery in images

J. Sivic, B. Russell, A. Efros, A. Zisserman, B. Freeman, Discovering Objects and their Location in Images, ICCV 2005


Summary generative models l.jpg
Summary: Generative models

  • Naïve Bayes

    • Unigram models in document analysis

    • Assumes conditional independence of words given class

    • Parameter estimation: frequency counting

  • Probabilistic Latent Semantic Analysis

    • Unsupervised technique

    • Each document is a mixture of topics (image is a mixture of classes)

    • Can be thought of as matrix decomposition

    • Parameter estimation: EM


ad