CS4770 Pattern Recognition
			      Monsoon 2006

			      Assignment 2
		   Karhunen-Loeve Transform using PCA
			 Due date: Sep 19, 2006


1. Read about the Karhunen-Loeve Transform (KLT) which is the most
   optimal compression scheme for different data such as images.
   KLT is a data-dependent compression scheme unlike JPEG and other
   schemes and hence is not very popular though it gives the best
   results.

2. KLT is based on eigendecomposition of the image or data. For this
   assignment, you have to compress images using KLT and study its
   performance.

3. For KLT-based compression, treat the entire image as an N x N array
   of numbers. (Convert all images to gray-level and deal with the
   intensities. Ignore colour!) Treat the array like a matrix and
   compute its eigenvalues and eigenvectors. Arrange the eigenvalues in
   decreasing order.

4. Select top  K  eigenvalues and corresponding eigenvectors to represent
   the image. Represent the image using its projection to the reduced
   eigenmatrix (of size K x N).

5. Recover the full image by undoing the projection using K
   eigenvectors. There will be a loss which may or may not be visible.
   You can compute the mean-squared error by summing the squares of the
   pixel-wise differences and dividing by (N*N). [Those who want to can
   also compute the PSNR or the peak signal to noise ratio.]

6. Repeat 4 and 5 for different values of K. If K is low, high
   compression is obtained at the expense of quality and vice versa.
   Repeat this for the noiseless and noisy versions of the images given
   to you.

7. Compress the given image with JPEG controlling the compression ratio
   using the Quality Factor of JPEG. (You can use the "convert" command
   from the ImageMagick library to do this and control the quality
   factor.)

8. Plot the error vs compression ratio curve for both JPEG and 
   the KLT compressed images and discuss the results.


The TA will have lots of supporting material on the webpage. Don't try
to write the eigenanalysis function yourself. Use Octave or Matlab for
it.  If you need any help with Octave or Matlab in which the project can
be done, contact him.

Each of you have to submit a report (in HTML perferrably) that explains
the KLT=based compression, give results on different images as well as
give the plots etc. It should be like a report that analyzes the issue
critically. More credits to those who do more work!