Sparse and Redundant Modeling of Image Content Using an Image-Signature-Dictionary

Authors: Michal Aharon and Michael Elad
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Introduction of Sparse Representation

Background

• In sparse and redundant modeling, a signal $y \in R^n$ is represented as a linear combination of some atoms taken from a dictionary $D \in R^{n*m}$ which contains $m$ atoms $d_i \in R^n$. A representation of the signal $y$ is then any vector $x \in R^n$ satisfying $y = Dx$

• $m > n$: the representation is redundant.

• The solutions to $y = Dx$ are infinite and we prefer the sparest one, i.e, the one with the smallest ${||x||}_0$

• The task of computing a representation of a signal can be formly described by

$$min_x{\|x\|}_0 \qquad s.t. \qquad y=Dx$$

• However, the constrait above is often relaxed and replaced by ${||y-Dx||}_2 \le \epsilon$. This allows for additive noise and model deviations

Problem

• Solving this problem was proved to be an NP-hard problem

• Two approximation techniques :

  • Matching-pursuit (MP) methods that find the solution one entry at a time in a greedy way

  • Basis-pursuit (BP) algorithm that replaces the $L_0$ norm by the $L_1$ norm

• Two types of dictionary

  • Prespecified dictionaries

  • Dictionaries obtained by learning procedure

This paper

• The image-signature-dictionary (ISD), is an 2D image in which each patch can serve as a representing atom

• A near shift-invariant property is obtained, due to the overlap between atoms extracted from the ISD in nearby locations

• By taking patches of varying sizes, near scale-invariance is potentially obtained and exploited

The Structure of ISD

$$y = \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}C_{[k,l]}d_s = Dx$$

• $D_s \in R^{\sqrt{m}*\sqrt{m}} :$ signature dictionary

• $y’ \in R^{\sqrt{n}*\sqrt{n}} :$ imasge patch and its single column form $y \in R^n$

• $d_s \in R^m :$ a vector obtained by a column lexicographic ordering of the ISD

• $C_{[k,l]} :$ the linear operator that extract a patch of size $\sqrt{n}*\sqrt{n}$ from the $D_s$ in location $[k,l]$

• $C_{[k,l]}d_s :$ the extracted patch as a column vector

• $x$ is the concatenation of the $m$ coefficients in the array $x_{[k,l]}$

ISD Training

Energy Function

We assume that each patch is represented by a fixed number of atoms, L. The energy function that the ISD is expected to minimize :

$$\hat{d_s} = Arg min_{d_s} \sum_{i=1}^N \epsilon_i^2(d_s)$$

s.t.

$$\epsilon_i^2(d_s) = min_x {\|y_i-\sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}C_{[k,l]}d_s\|}_2^2$$

s.t.

$${\|x\|}_0 \le L,\qquad 1 \le i \le N$$

Two stages in training

• The sparse coding stage: find $\hat{x_i}$

  Assuming that $d_s$ is known and fixed, we can solve the inner optimization problem and find the sparsest representation $\hat{x_i}$ for each example $y_i$

  $$\hat{x_i} = Argmin_x {\|y_i-\sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}C_{[k,l]}d_s\|}_2^2 \qquad s.t. \qquad \|x_o\| \le L$$

  This problem can be solved by any pursuit algorithm, such as a greedy method——the orthogonal matching pursuit (OMP)

  The OMP selects at each stage an atom from the dictionary that best resembles the residual. After each such selection, the signal is back-projected onto the set of chosen atoms, and the new residual signal is calculated (Don’t understand)

• Dictionary update stage: $\hat{d_s}$

  Assuming that the sparse representation vectors $\hat{x_i}$ have been computed, we seek the best ISD $d_s$ to minimize:

  $$E(d_s) = \sum_{i=1}^N {\|y_i-\sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]}d_s\|}_2^2$$

  The gradient of the error expression:

  $$\frac{\partial E(d_s)}{\partial d_s} = Rd_s - p$$

  where

  $$R = \sum_{i=1}^N {\lgroup \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]} \rgroup}^T * {\lgroup \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]} \rgroup} \in R^{m*m}$$ $$p = \sum_{i=1}^N {\lgroup \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]} \rgroup}^T*y_i \in R^m$$

  Thus, the optimal ISD is obtained simply by:

  $$\hat{d_s} = R^{-1}p$$

• Stochastic gradient appoach (SG)

  • The algorithm above updates the $x_i$ for all examples {y_i}_{i=1}^N and then updates the ISD

  • Another way is to update the ISD after the computation of each $x_i$ (SG):

  $$\hat{d_s^{new}} = \hat{d_s^{lod}} - \mu \frac{\partial E(d_s^{lod})}{\partial d_s^{lod}} = \hat{d_s^{lod}} - \mu {\lgroup \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]} \rgroup}^T {\lgroup \sum_{k=1}^{\sqrt{m}}\sum_{l=1}^{\sqrt{m}}x_{[k,l]}^iC_{[k,l]}\hat{d_s^{old}}-y_i \rgroup}$$

  The step-size parameter $\mu$ is typically set as $\mu = \frac{\mu_0}{Iteration}$

Treatment of the mean

• Remove the mean in training and testing data

• Redefine the $C_{[k,l]}$ to also remove the mean of the extrated patch

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