7th October 1999

Maximum Marks 30 ...................... 2 hours

* Solve the following problems:

1. The illumination-reflectance model is described by
   

   where, g(x,y) is the observed image, f(x,y) is the original illumination, and r(x,y) is the reflectance component.

   a) Is H[.] linear? Justify.

   b) Is H[.] homogeneous? Justify.

   c) Is H[.] position-invariant? Justify.

   (3)

2. Image blurring caused by long-term exposure to atmospheric turbulence can be modeled by the transfer function
   

   a) Calculate H(u,v).

   b) What is the inverse filter that will deblur the image?

   (4)

3. A certain x-ray imaging geometry produces a blurring degradation that can be modeled as the convolution of the sensed image with the spatial, circularly symmetric function
   

   where, . Obtain the transfer function of a constrained least squares filter you could use to deblur the images produced by this x-ray system. Assume that the images are square.

   (5)

4. Assume image blur due to zooming motion of a camera. The zooming is occurring around the center of the image. The camera exposure time is 0.1 sec. The radial zoom factor is 1pixel/sec. What is the transfer function of the deblurring filter?

   (3)

5. Appendix B of the course text lists a number of 64 x 64 32-level images. If each of these is run-length coded, which image do you expect will be compressed the most and which the least. Explain your reasoning. Now threshold these images in your mind to convert them to binary level images with two allowed levels, 0 and 1. The threshold should be chosen such that the resulting two-level image best represents the original image. Which image do you expect to be compressed the most with run-length coding? Why? Let this be image A. Assume that the largest run-length allowed is 16. Design the 5 bit run-length code {4 bit run-length, 1 bit for representing gray level of the run}. Make an estimate of the achievable compression for image A using this code for run-length coding. For full credit state your method fully and explicitly.

   (3)

6. Consider a source alphabet A consisting of seven letters. The source probabilities are z={0.49, 0.26, 0.12, 0.04, 0.04, 0.03, 0.02}. What is the entropy of this source. Find the binary Huffman code for A. Find the expected codelength for this encoding.

   (3)

7. Which of these cannot be Huffman codes for any probability assignment? (a) {0,10,11} (b) {00,01,10,110} (c) {01,10}. Give reasons.

   (3)

8. Derive the Lloyd-Max decision and reconstruction levels for L=4 and the uniform probability density function

   p(s)= 1/2A for -A <= s <= A

   and p(s)= 0 otherwise

   (3)

9. For a lossy predictive coder consider the following situation: A modified lossy predictive coder is constructed that has the error input to the predictor changed only at the encoding end. The predictor at the encoding end gets as input e_n instead of e_n (actual error instead of quantized error), all other things remaining the same. The encoder remains unchanged. Make a comparison of the modified lossy predictor with the original lossy predictive coder. Explain your results. For full credit you will need to be thorough in your analysis.

   (3)