**************Use Matlab************

PROBLEM 4 The goal of this problem is to study some of the properties of the discrete cosine transform. We want to compare between the upper left corner and the bottom right corner in terms of the information contained in the DCT, and the possibility to compress the image using the DCT and its properties. When you display images, be sure that they the same physical size on paper (not pixel size). This will allow you to compare in terms of the quality.

1) Apply the DCT to an image of your choice. To display the DCT, you can use instruction: imshow(log(abs(ImageDCT)),[]) colormap(jet), colorbar (2) where ImageDCT is the DCT of your image. We assume that the size of the original image is (n1, n2).

2) Now we want to reduce the size of ImageDCT by keeping only the top left corner as shown in figure 2. We do this twice: ImageDCTToPLeft1: We keep (n1/2, n2/2) elements from the top left of ImageDCT. 2 ImageDCTToPLeft2: We keep (n1/4, n2/4) elements from the top left of ImageDCT. Write code to do this.

3) Perform the inverse DCT on these matrices (ImageDCTToPLeft1, ImageDCTToPLeft2), show and discuss your results. Hint: If you are using Matlab, you can use command imshow(K,[ ]) to display the images where K is the inverse DCT.

4) Now we keep only the bottom right corner of the matrix representing ImageDCT as shown in figure 2. Again, we do this twice: ImageDCTBottomRight1: We keep (n1/2, n2/2) elements from the bottom right of ImageDCT. ImageDCTBottomRight2: We keep (n1/4, n2/4) elements from the bottom right of ImageDCT. Write code to do this.

5) Perform the inverse DCT on these matrices (ImageDCTBottomRight1, ImageDCTBottomRight2), show, and discuss your results. Hint: If you are using Matlab, you can use the command imshow(K,[ ]) to display the images where K is the inverse DCT. 6) Discuss and compare the results of questions 3 and 5.