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- ECSE 4540: Introduction to Image Processing Homework #2

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1. (15 points.) Sketch 4 images that all have the histogram given by:

D p(I(x, y) = D)

0 0.25

128 0.25

255 0.5

Now, suppose each of these images was blurred using an averaging filter such as ones(9)/81. Will the

images have the same histogram after this operation? Why?

2. (15 points.) Suppose an image’s histogram has two peaks at intensity values 90 and 180. Determine

(i.e., give an explicit form for) a linear stretch function f (D) so that the first peak is mapped to the new

intensity value of 30, the second peak is mapped to the new intensity value of 210, and all the output intensity values are clipped to the range [0, 255]. Sketch the function f (D). What is the largest intensity that

gets mapped to 0, and the smallest intensity that gets mapped to 255?

3. (15 points.) Consider an image I with intensities between 0 and 255 such that the number of pixels having

intensity n is equal to 2n +1. After applying histogram equalization to the image, what new intensity value

are pixels with intensity 100 mapped to? For this problem, it’s OK to think of the intensity distribution as a

continuous function and solve the problem with calculus. Sketch the function that maps input intensities

to output intensities. Hint: you should first figure out how many total pixels there must be in the image.

You may need to remember/derive the formula for the finite sum PN−1

n=0

n.

4. (15 points.) Consider the “mystery image” at http://www.ecse.rpi.edu/~rjradke/4540/mystery2.

png. It appears to be a bunch of solid rectangles on a grey background. Apply histogram equalization

and describe what you see. Then, apply local histogram equalization with an 80 × 80 neighborhood. This

can be accomplished with the blockproc command; see help blockproc for how to set up and use this

function. How do the global and local histogram equalization results differ, and why?

5. (10 points.) Determine the single 2×2 similarity transformation that sequentially (a) rotates the image

clockwise by 30 degrees and (b) flips it horizontally (i.e., across the x axis). Here, we assume the transformation is applied in Cartesian coordinates. Will the answer be the same if (a) and (b) are switched?

6. (15 points.) It was determined that a given affine transformation x

0 = Ax +b maps (1, 1) to (0, 1), (-1, 1) to

(-1, 0), and (1, -1) to (2, 0). Compute A and b; show your work! You may need Matlab.

7. (15 points.) Suppose we transform the image described by I(x, y) =

j

255 exp−

³

(x−250)2+(y−250)2

102

´k using a

given transformation T , so that T

−1

(128, 64) = (251.7, 252.2). If we create the new image I

0 using bilinear

interpolation, what is its intensity at (128, 64)? (You’ll need a calculator or Matlab for this problem. Note

that b·c means to round down to the nearest integer. Assume we’re using image coordinates, not Cartesian

coordinates.)

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