# CS6643 Homework3- Compute the Euler number of the foreground Solved

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1.  Compute the Euler number of the foreground below under: (a) 4connectedness definition of connected components. (b) 8-connectedness definition of connected components.

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1. Use the formula 𝐶𝐶 = 𝑃𝑃𝐴𝐴2 to compute the compactness of the foreground object in the binary image below.  Define perimeter to be the sum of crack edges of the foreground object.

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1.  Apply the Boundary Following Algorithm below to the image in the figure below.
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1.  (a) Apply the iterative distance transform algorithm below to the input image

𝑓𝑓(𝑟𝑟, 𝑐𝑐) in the figure below, where the 1’s represent foreground pixels and background pixels are unlabeled (assumed to have a value of 0.)

f 0[r,c] = f [r,c]

f m[r,c] = f 0[r,c]+ min( f m1[u,v])

Stop when 𝑓𝑓𝑚𝑚[𝑟𝑟, 𝑐𝑐] = 𝑓𝑓𝑚𝑚−1[𝑟𝑟, 𝑐𝑐]

Show the image result after each iteration. You must execute step 3 in the algorithm above to know when to stop. (b) From the distance transform, circle those pixels that belong to the medial axis. The image uses the (𝑟𝑟, 𝑐𝑐) coordinate system with the upper-left corner pixel having coordinates (0,0).

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Input image f(r,c)

We would like to use the Hough Transform to detect straight lines in an image by using the line equation 𝑝𝑝 = 𝑥𝑥 ∗ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝑦𝑦 ∗ 𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐. Using subpixel edge localization, two edge points at locations (x,y) = (19.5, 0.0) and (10.0,10.0) have been detected in the image. An accumulator array consisting of 8 × 5 cells has been formed where the horizontal axis is for 𝑐𝑐 and the vertical axis is for A step size of 5 degrees is used for 𝑐𝑐 and a step size of 1 pixel is used for p. The cell at the lower-left corner has range [25,30) for 𝑐𝑐 and [11,12) for 𝑝𝑝, and so on for other cells. For each edge point, compute the p values for 𝑐𝑐 = 27.5, 32.5, 37.5, 42.5 and 47.5 degrees and increment the appropriate cell locations in the array. (a) Show the final content of the accumulator array. (b) What is the equation of the line (with parameters (𝑝𝑝, 𝑐𝑐)) as detected by the Hough Transform method?

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