Project_01
Image Deblurring
I Can See Clearly Now

Reference:

1. James Nagy, Dianne O'Leary,
   Image Deblurring: I Can See Clearly Now,
   Computing in Science and Engineering,
   Volume 5, Number 3, May/June 2003, pages 82-85.
2. Dianne O'Leary,
   Scientific Computing with Case Studies,
   SIAM, 2008,
   ISBN13: 978-0-898716-66-5,
   LC: QA401.O44.

In our discussions about the SVD, we looked at what happens when a 2x2 matrix A multiplies vectors on the unit circle.

CIRCLE_FORWARD examines what happens when a 2x2 matrix A multiplies vectors on the unit circle.

• circle_forward.m a program which plots the image of the unit circle under the mapping y = A*x.
• circle_forward.png an image of the unit circle under the mapping y = A*x, for the 2x2 matrix A = [ 1, 2; 3, 4].

COMPUTE_SPLUS takes the "S" factor of an SVD decomposition and computes the SPLUS factor used for the pseudoinverse.

• compute_splus.m computes the SPLUS factor from the S factor.

PSEUDOINVERSE computes the pseudoinverse of a matrix. It needs the COMPUTE_SPLUS function.

• pseudoinverse.m computes the pseudoinverse of a matrix.

It is possible to use a Kronecker product without actually forming it; this can be crucial when the Kronecker product would require a huge amount of storage.

The following MATLAB file demonstrates the implicit use of a Kronecker product:

• proj1demo.m, the Kronecker product demo.
• proj1demo_output.txt, the output from a run of the program.

The data for the project includes matrices A and B, and a blurred image G. Recall, from the article, that the matrices A and B are used to represent the big matrix K. K is the Kronecker product of A and B. A and B are of size 256x256, but K is much too big to work with. K will be of order (256*256)x(256*256). We want the SVD of K. Luckily, since K is a Kronecker product, its SVD can be computed by working with the smaller factors A and B, and determining factors UA and UB, SA and SB, VA and VB, such that K's SVD is the Kronecker product of these pairs. Without actually forming K or its SVD, we can carry out the work we need to do.

One way to get the data is as a MATLAB "mat" file:

• proj1data.mat, A, B and G saved in a MATLAB "mat" file.

To get the data, start up MATLAB, and issue the command

        load proj1data.mat
      

        whos
      

        imagesc ( G ), colormap ( gray )
      

The data is also available as an ASCII file. In that case, you must write your own function to transfer the data from the file into your program.

• proj1data.txt, A, B and G saved in an ASCII file.

In creating the ASCII file, the matrix A was written first, then B, then G. Each array is of dimension 256x256. The entries are written by rows. There are 4 numeric entries per line of the file, using the "E" or "exponential" format.

• proj1solution.m, a sample solution.
• origimage.jpg, the original image in JPEG format.
• origimage.mat, the original image in MATLAB "MAT" format.

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