MMAT5390 Mathematical Image Processing
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MMAT5390 Mathematical Image Processing
Midterm Examination
You have to answer all five questions. Please show your steps unless otherwise stated.
1. This question is about Haar transformation.
(a) Compute the Haar transform f˜ of the following 4 × 4 image
╱r 0 0 0、
f = 
．0 0 s 0．
where r and s are two non-zero real numbers.
(b) Following (a), show that f can be written as a linear combination of exactly four elementary images under the Haar transformation if and only if r = s or r = −s. Please explain your answer in details.
(c) Following (a) and (b), if r = −s, write f as a linear combination of four elementary images under the Haar transformation.
2. This question is about image decomposition using singular value decomposition
|
(SVD). Consider: ╱0(0) 0(a) A = ．(．) 0 0 ．0 0 |
0 b 0 0 |
0(0)、．
|
where a, b and c are positive real numbers and a ≥ b ≥ c.
(a) Compute the SVD of A. Please show all your steps. (Hint: Compute A本 A and find the eigenvalues of A本 A.)
(b) Write A as a linear combination of eigen-images.
(c) Suppose A is degraded during the transmission process to get the corrupted image 
. Suppose 
is the same as A except for the 4-th row 1-st column entry. In particular, 
(4, 1) = c. Using (a) and (b), find the SVD of 
. Please explain your answer with details.
╱ r 2r
3. Let H = ．(．) 3(3r) 6(r) ．9 3
spread function h
、． be the transformation matrix corresponding to a point
3s s ．
= h(x, α, y, β), where r, s, u, v are all non-zero real numbers.
Prove that h is both separable and shift-invariant if and only if r = s and u = v (here, we do not assume h to be periodic in each variables). Please explain your answer with details.
4. This question studies how a blurry image is degraded from its original image. Let I = (I(m, n))0<h←n<疝 -1 be a N × N image, which is periodically extended. Suppose I˜ be a blurry image given by:
﹔
I˜(x, y) = I(x, y) + r à I(x − k, y + k) for 0 ≤ x, y ≤ N − 1,
à=1
where 0 < r < 1. Denote the discrete Fourier transforms of I and I˜by DFT (I) and DFT (I˜) respectively. Prove that DFT (I˜)(u, v) = H(u, v)DFT (I)(u, v) for some H, where 0 ≤ u, v ≤ N − 1. What is H in terms of N , u, v , r and L? Please derive your answer from the definiton of DFT and show all your steps clearly (including how the changes of variables are applied, indices are shifted and so on). Missing details will lead to mark deduction.
5. Let g = (g(m, n))0<h<L -1 ←0<n<疝 -1 ∈ MLx疝(R) be a M × N image. Assume g is periodically extended. Suppose 
is obtained by translating and rotating g. More specifically,
(m, n) = g(3 − m, 5 − n), where 4 ≤ m ≤ M + 3 and − 4 ≤ n ≤ N − 5.
Express 
in terms of gˆ, where 
is the DFT of 
and gˆ is the DFT of g. Please derive your answer from the definiton of DFT and show all your steps clearly (including how the changes of variables are applied, indices are shifted and so on). Missing details will lead to mark deduction.
2022-04-12