Problem 1 (10 Points)

Transcription

1 Problem 1 (10 Points) Please explain the following phenomena. The three images shown are the blurred versions of image (a) (Note: vertical bars have width of 5 and height of 100, they are spaced 20 pixels apart.) using square averaging masks of sizes n=23, 25 and 35, respectively. The vertical bars on the left lower part of (b) and (d) are blurred, but a clear separation exists between them. However, the bars have merged in image (c), in spite of the fact that the mask that produced the image is significantly smaller than the mask that produced image (d). Why is this happening? Solution: IN the figure the vertical bars are 5 pixels wide and 20 pixels apart. Figure below shows one scan line in the image subject of this question. The distance in pixels between the onset of one bar and the onset of the next one is 25 pixels. Consider the figure of one scan line shown below, which also shows the cross section of a 25x25 pixel mask. The response of the mask is the average of the pixels that it encompasses. Note that when the mask moves one pixel to the right, it loses one value of the vertical bar on the left but it picks up an identical one on the right, so the response doesn t change. In fact, the number of pixels belonging to the vertical bars and contained within the mask does not change, regardless of where the mask is located (as long as it is contained within the bars, and not near the edges of the set of bars). The fact that the number of bar pixels under the mask does not change is due to the peculiar separation between bars and the width of the lines in relation to the 25-pixel width of the mask. This constant response is the reason no white gaps is seen in the image shown in the problem statement. Note that this constant response does not happen with the 23x23 or 35x35 masks because they are not synchronized with the width of the bars and their separations.

2 Problem 2 (30 Points) We have a 2D sequence f(m,n)=n(m*m+2), where m and n are all integers. The task now is to reconstruct f(x,y) for x and y, which are real numbers in interval [0,1] when x=y, i.e. points on the straight line connecting point A with coordinates (0,0) and point B with coordinates (1,1)), as shown in the figure. Please provide the function f(x,y) when using the following two reconstruction methods: 2.1 (10 points) Using closest point value reconstruction 2.2 (20 points) Using bilinear reconstruction Solution: 2.1 f(x) = 0, when 0<= x <= 1/2 f(x) = 3 when 1/2 < x <= 1 Proof:

3 Note that x = y for any point (x,y) on the line from (0,0) to (1,1). When x<=1/2, (0,0) is the nearest point to (x,x). According to separate rectangle (i.e., nearest neighbor) criteria, f(x,x) should be the same as the value of its nearest point, i.e., f(0,0). Hence, f(x,x) = f(0,0) = 0; Similarly,, when x>=1/2, (1,1) is the nearest point to (x,x), hence f(x,x) = f(1,1) = f(x) = x 2 +2x, 0<=x<=1 Proof: According to the separate triangle (i.e., bilinear) criteria, for any point (x,y), 0<=x,y<=1, f(x,y) = f(0,0) * (1-x) *(1-y) + f(0,1) * (1-x) * y + f(1,0) * x * (1-y) + f(1,1) * x * y, Note that x= y, hence, f(x) = f(0,0) * (1-x) *(1-x) + f(0,1) * (1-x) * x + f(1,0) * x * (1-x) + f(1,1) * x * x, Note that f(m,n) = n(m*m + 2), we know that: f(0,0) = 0, f(0,1) = 2, f(1,0) = 0, f(1,1) = 3, hence, f(x) = 0 * (1-x) *(1-x) + 2 * (1-x) * x + 0 * x * (1-x) + 3 * x * x, = x 2 +2x Problem 3 (30 points) Given the following NxN gray-scale image, where the gray-level value of the k-th vertical bar (from left) is as follows: ( k 1) G( k) = 256, k = 1,...,8 8 N / (10 points.) Sketch the histogram of this image. Use M=16 bins for the histogram uniformly spaced in the range [0, 256). 3.2 (20 points.) Apply the point operator given below to the image. Sketch the histogram of the resulting image. Again use M=16 bins for the histogram.

4 Solution: Problem 4 (30 points) Mix-and-match. Shown in (a) and (b) are an image of the famous swirl skylight in

5 Guggenheim museum and the Fourier transform amplitude of it s luminance, rendered as center-shifted log-amplitude with contrast stretching. Now please match the log-amplitude graph in (c)(d)(e) with their original image among (f)(g)(h). v FFT2 u (a) (b) center-shifted log-amplitude after contrast stretching (c) (d) (e) (f) (g) (h) Solution: (f)-(d), (g)-(e), (h)-( c ) Problem 5 (50 points optional bonus programming problem) Consider a filter defined by a 3x3 convolution array mask: d c e b a b e c d

6 5.1 (20 points) Choose appropriate values for a, b, c, d, and e such that the result of applying this filter to the image baboon blurs the edges. 5.2 (30 points) Now, choose the parameters to enhance the edges in the image baboon. Report your choice of parameters and the resulting images. Baboon:

7 Solution:

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