a) Consider a 3 × 3 mask that averages the 4 non-diagonal neighbors of a pixel [m,n], but
excludes the pixel itself. Find the equivalent filter H(u,v) in frequency domain. Is this a low-pass or highpass
filter?
b) For an M ×N image, you are asked to repeatedly apply a Gaussian filter, with H(u,v) = e−(u2+v2)/2D2 for
some D. If you apply the filter repeatedly for K times for a sufficiently large K, what happens to the
resulting image? If the lowest positive real number representable by the computer is , what is the
minimum K to guarantee this behavior (here assume/note that H(u,v) will also be stored digitally in the
computer)?
2) [5 pts] Prove the following matrix relation:
(A + UBV)−1 = A−1 − A−1U B−1 + VA VA−1.
3) [5 pts] MATLAB Exercise: We want to familiarize ourselves with function handles. Write function
handles to calculate the following: a) The image derivative in x direction (the built-in function diff may
be useful), b) the image derivative in y direction, c) the magnitude of the gradient vector in (x,y)
directions, d) discrete cosine transform (you may use the dct2 function for this) of each 8 × 8 distinct
block in the image.
For a, b, c make the derivative operator circulant. Thus, the output should be the same size as the image.
Verify all 4 function handles on the cameraman image(attached).