Question

Problem 3. Extensive experimental evidence in the area of image compression has shown that the Discrete Cosine Transform (DCT) of image patches is a very good approximation to their PCA. It is also well known that all but one of the DCT coefficients (features) have zero mean, and only one has non-zero mean. The latter is the so-called DC coefficient because it results from projecting the image patch into the vector 1 = (1, 1, , 1)T and, therefore, is proportional to the average (DC) value of the patch. In this problem, we are going to explore the connection between the DCT and PCA to explain this fact. For this, we are going to assume the following. . An image patch is a collection of randon variables X = {Xi, ,X64} which are identically distributed Py (z) = f(x), Yi E {1, . . . , 64), where f(x) is a common probability density function The pixels in the image patch are correlated, according to the correlation coefficient Elx, XI This obviously implies that we do not have an iid sample.

Answer 1