# ECSE4540 Homework#2 solution

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1. (15points.) Sketch4imagesthatallhavethehistogramgivenby: 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 imageshavethesamehistogramafterthisoperation? 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 getsmappedto0,andthesmallestintensitythatgetsmappedto255?
3. (15points.) Consideranimage I withintensitiesbetween0and255suchthatthenumberofpixelshaving intensityn isequalto2n+1. Afterapplyinghistogramequalizationtotheimage,whatnewintensityvalue arepixelswithintensity100mappedto? Forthisproblem,it’sOKtothinkoftheintensitydistributionasa 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. Youmayneedtoremember/derivetheformulafortheﬁnitesumPN−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. Howdotheglobalandlocalhistogramequalizationresultsdiffer,andwhy?
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 transformationisappliedinCartesiancoordinates. Willtheanswerbethesameif(a)and(b)areswitched?
6. (15 points.) It was determined that a given afﬁne transformation x0=Ax+b maps (1, 1) to (0, 1), (-1, 1) to (-1,0),and(1,-1)to(2,0). Compute A andb;showyourwork! YoumayneedMatlab. 7. (15 points.) Suppose we transform the image described by I(x,y)=j255exp−³(x−250)2+(y−250)2 102 ´kusing agiven transformation T, so that T−1(128, 64)=(251.7, 252.2). If we create the new image I0 using bilinear interpolation, what is its intensity at (128, 64)? (You’ll need a calculator or Matlab for this problem. Note thatb·cmeanstorounddowntothenearestinteger. Assumewe’reusingimagecoordinates,notCartesian coordinates.)