## Description

2. We have studied two greedy algorithms for compressive recovery in class – MP and OMP. Your task is to do a google search and find out a research paper that proposes a greedy algorithm for CS recovery that is different from OMP and MP. Write down the algorithm in your report in the form of a simple pseudo-code. State the key theorem from the paper which presents performance bounds for the algorithm, and explain the meaning of the terms involved. If there are multiple theorems, pick the one that states the strongest result.

[20 points]

3. Consider that you learned a dictionary D to sparsely represent a certain class S of images – say handwritten alphabet or digit images. How will you convert D to another dictionary which will sparsely represent the following classes of images? Note that you are not allowed to learn the dictionary all over again, as it is time-consuming.

(a) Class S1 which consists of images obtained by applying a known derivative filter to the images in S.

(b) Class S2 which consists of images obtained by rotating a subset of the images in class S by a known fixed angle α, and the other subset by another known fixed angle β.

(c) Class S3 which consists of images obtained by applying an intensity transformation Inewi (x,y) = to the images in S, where α,β,γ are known.

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(d) Class S4 which consists of images obtained by applying a known blur kernel to the images in S.

(e) Class S5 which consists of images obtained by applying a blur kernel which is known to be a linear combination of blur kernels belonging to a known set B, to the images in S. [4+4+4+4+4=20 points]

4. How will you solve for the minimum of the following objective functions: (1) J(Ar) = kA−Ark2F, where

A is a known m × n matrix of rank greater than r, and Ar is a rank-r matrix, where r < m,r < n.

(2) J(R) = kA−RBk2F, where A and R is constrained to be orthonormal. Note that A and B are both known.

In both cases, explain briefly any one situation in image processing where the solution to such an optimization problem is required. [5+5+5+5=20 points]

5. Read the paper ‘A Signal Processing Perspective on Hyperspectral Unmixing’, a copy of which is placed in the homework folder. Answer the following questions:

(b) In equation 40 of the paper, explain how non-negative matrix factorization is used for hyperspectral unmixing.

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