SD475 Digital Image Processing – Midterm Examination ...dclausi/SD575/Exams/SD475 Midterm...SD475 Digital Image Processing – Midterm Examination October 25, 2012 9:45am – 11:15am
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SD475 Digital Image Processing – Midterm Examination October 25, 2012 9:45am – 11:15am Prof. D. A. Clausi Aids: calculator, crib sheet (both sides of standard paper, no solutions to problems). Submit crib sheet with solution booklet. Total Marks: 45 1. [Total: 15 marks] SMOOTHING. A Gaussian low pass filter (LPF) h 1 (x) can be used to smooth high frequency noise in an image. Consider its 1-d implementation: h 1 (x) = 2 5 . 0 1 1 2 1 − σ π σ x e (a) [1 mark] What is the primary drawback to using any LPF for smoothing high frequency noise? (b) [2 marks] Would an ideal “box” spatial filter be more appropriate for smoothing an image? Why or why not? (c) [2 marks] Sketch a continuous input signal f(x) = 1 + 5u(x) plus additive Gaussian noise with σ n = 0.2, where u(x) is the unit step. (d) [4 marks] Sketch the outputs when f(x) is filtered using h 1 (x) for (i) σ 1 = 0.1 and (ii) σ 1 = 3. (e) [2 marks] Does h 1 (x) retain the DC gain in the image? Why or why not? (f) [4 marks] Is h 1 (x) (i) a linear filter? (ii) a causal filter? For each case, provide an explanation. 2. [10 marks] POINT OPERATIONS. Assume an image has a distribution p f (r) = 1 - cos(2πr), 0<=r<=1. (a) [2 marks] Sketch p f (r). Labels axes properly. (b) [4 marks] Derive the transformation function that would be used to flatten this image's histogram. (c) [2 marks] Sketch the transformation function. (d) [2 marks] Based on visual inspection of the two sketches in (a) and (c), explain why your transformation derived in (b) would flatten p f (r). 3. [20 marks] LAPLACIAN. In class, we used the first difference operators to generate the Laplacian. However, any edge detector can be used for this purpose. (a) [5 marks] What is the Laplacian operator if you use the Sobel edge detector to obtain the horizontal and vertical first differences? (b) [2 marks] Using the Laplacian operator from (a), determine the 1-Laplacian operator. (c) [2 marks] What type of a filter is a Laplacian in image processing? Does your derivation in (a) satisfy the characteristics for such a filter? Explain. (d) [2 marks] What type of a filter is a 1-Laplacian in image processing? Does your derivation in (b) satisfy the conditions for such a filter? Explain. (e) [3 marks] We could take the 2D DFT of the 1-Laplacian, but this is a bit too much work. To simplify things, just derive the discrete-time Fourier transform (DTFT) (i.e., H(e j2πu )) using the middle row of the 2D mask as a 1d filter. (Note that the middle row alone is not a 1-Laplacian!). (f) [4 marks] Sketch H(e j2πu ). What effect would this filter have on an image? (g) [2 marks] Does H(e j2πu ) have a non-zero phase component? Why or why not? Note: if you are unable to do 3(a), then you can request and I will give you the solution for 3(a) so that you can do the rest of question 3 properly. This will forfeit your 5 marks allocated to 3(a). • However, if you are able to derive the solution for 3(a) after I have given you the correct solution (clearly showing all steps, no guessing!), you can retain up to 2 marks for 3(a).
SD475 Digital Image Processing – Midterm Examination October 25,
2012 9:45am – 11:15am
Prof. D. A. Clausi Aids: calculator, crib sheet (both sides of
standard paper, no solutions to problems). Submit crib sheet
with
solution booklet. Total Marks: 45 1. [Total: 15 marks]
SMOOTHING. A Gaussian low pass filter (LPF) h1(x) can be used to
smooth high frequency noise in
an image. Consider its 1-d implementation:
h1(x) =
25.0
1
1
21
−
σ
πσ
x
e
(a) [1 mark] What is the primary drawback to using any LPF for
smoothing high frequency noise? (b) [2 marks] Would an ideal “box”
spatial filter be more appropriate for smoothing an image? Why or
why not? (c) [2 marks] Sketch a continuous input signal f(x) = 1 +
5u(x) plus additive Gaussian noise with σn = 0.2, where u(x)
is the unit step. (d) [4 marks] Sketch the outputs when f(x) is
filtered using h1(x) for (i) σ1 = 0.1 and (ii) σ1 = 3. (e) [2
marks] Does h1(x) retain the DC gain in the image? Why or why not?
(f) [4 marks] Is h1(x) (i) a linear filter? (ii) a causal filter?
For each case, provide an explanation.
2. [10 marks] POINT OPERATIONS. Assume an image has a
distribution pf(r) = 1 - cos(2πr), 0
