Kaihua Zhang

Lei Zhang

(PolyU, Hong Kong)

Ming-Hsuan Yang

(UC Merced, California, U.S.A. )

Real-Time Compressive Tracking

European Conference on Computer Vision, 2012 (ECCV 2012)

Outline

• Introduction

•Random projection

•Classifier construction and update

• Experiments

•Conclusion

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Outline

• Introduction

•Random projection

•Classifier construction and update

• Experiments

•Conclusion

3

Introduction

• Propose an effective and efficient tracking algorithm with an appearance model based on features extracted in the compressed domain.

• Our appearance model employs non-adaptive random projections that preserve the structure of the image feature space of objects

• Compress samples of foreground targets and the background using the same sparse measurement matrix

• The tracking task is formulated as a binary classification via a naive Bayes classifier with online update in the compressed domain

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Compressed sensing

• Compressed sensing is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems (Emmanuel Candès, Terence Tao, and David Donoho)

• There are two conditions under which recovery is possible

• The first one is sparsity which requires the signal to be sparse in some domain

• The second one is incoherence which is applied through the restricted isometry property (RIP) which is sufficient for sparse signals.

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Random Measurement Matrix

high-dimensional space

lower-dimensional space

Main components of compressive tracking

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Frame(t) Samples

Classifier

Sparse projection

matrix

Compressed vectors

Image features

Main components of compressive tracking(2)

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Sample with maximal classifier response

Frame(t+1)

Classifier

Frame(t+1)Sparse measurement

matrix

Multiscale Image features

Compressed vectors

Outline

• Introduction

•Random projection

•Classifier construction and update

• Experiments

•Conclusion

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Random projection

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A matrix R whose rows have unit length projects data from the high-dimensional image space x to a lower-dimensional space v

v = Rx where nm

Random projection(2)

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Intuitive interpretation of JL lemma linear map f: ->

𝑹𝒏

Random projection(3)

• We write the Euclidean distance between two data vectors and in the original large-dimensional space as

• After the random projection, approximated by the scaled Euclidean distance of these vectors in the reduced space:

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where d is the original and k the reduced dimensionality of the data set

Random projection(4)

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(Johnson-Lindenstrauss lemma)

Given 0<<1, a set X of d points in and number n > 8ln(m)/,there is a linear map f: -> such that

The Johnson-Lindenstrauss Lemma

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• Selecting matrix A that provide the desired result are:

Random Measurement Matrix

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𝑹𝒎

𝑹𝒏

Construct random matrix R such that JL lemma

We adopt a very sparse random matrix with entries defined as

This type of matrix with s =3 satisfies the JL-lemma by Achlioptas

Random Measurement Matrix(2)

• In this work, we set s = m/4 which makes a very sparse random matrix

• For each row of R, only about c, c ≤ 4, entries need to be computed

• Therefore, the computational complexity is only O(cn) which is very low

• Furthermore, we only need to store the nonzero entries of R which makes the memory requirement also very light.

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Multiscale filter

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Haar-like features

• Haar-like features are digital image features used in object recognition

• A simple rectangular Haar-like feature can be defined as the difference of the sum of pixels of areas inside the rectangle

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Dimensionality Reduction

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Graphical representation of compressing a high-dimensional vector x to a low-dimensional vector v x х = 𝑣 𝑖=∑

𝑗

𝑟 𝑖𝑗 𝑥 𝑗

Negative entry Positive entry

Zero entry

Analysis of Low-Dimensional Compressive Features

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Outline

• Introduction

•Random projection

•Classifier construction and update

• Experiments

•Conclusion

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Classifier Construction

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For each sample z its low-dimensional representaion

Assume all elements in v are independently distributed model them with a naïve Bayes classifier

=

Classifier Update

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Contitional distributions and in the classifier H(v) are assumed to Gaussian distributed with four parameters

) )

The scalar parameters are incrementally updated:

+

where

Algorithm : Compressive Tracking

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Outline

• Introduction

•Random projection

•Classifier construction and update

•Experiments

•Conclusion

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Experiments

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http://www4.comp.polyu.edu.hk/~cslzhang/CT/CT.htm

Experiments(2)

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Experiments(3)

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Outline

• Introduction

•Random projection

•Classifier construction and update

• Experiments

•Conclusion

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Conclusion

• In this paper, we proposed a simple yet robust tracking algorithm with an ap- pearance model based on non-adaptive random projections that preserve the structure of original image space

• Numerous experiments with state-of-the-art algorithms on challenging sequences demonstrated that the proposed algorithm performs well in terms of accuracy, robustness, and speed.

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