CS/ECE 545 Homework 7

Complete by Class 9

Projections
Write a program to generate projections along the x- and y-directions of an image. Output a vector of M numbers (projecting along the y-direction for θ = 0) and a vector of N numbers (projecting along the x-direction for θ = π/2). Be sure to use long integers or floating point numbers to avoid numeric overflow.

Run your program on at least 3 images. You may use the Sample Images or other image set. Graph the projections. You can do this in MS Word or Excel or other program. Do not turn in a list of numbers!

Back-Projections
The back-projection b(x,y) of a set of projections p(s,θ) is the image obtained by adding up, for each x and y, those values of p(s,θ) for which f(x,y) projected into (s,θ). Essentially, we put (that is, project!) the values of p(s,θ) back into the (x,y) they came from.

(If you think of projection as a matrix operation, p = Af, then back-projection is just the matrix transpose, b = ATp. This is not the inverse. Usually.)

In the case of projections along the x- and y- axes, the back projection is particularly simple. Because f(x,y) projects along y into p(x,0) and along x into p(y,π/2), we have b(x,y) = p(x,0) + p(y,π/2).

Implement this to back-project the projections from Part 1 above and show the results on your test images. You may have to divide by 2 to prevent numeric overflow. The back-projected images should look very blurry.

Impulse Response
The combination of projection / back-projection is a linear and shift-invariant operation, ignoring edge effects. Therefore we can, and do, ask: What is the impulse response of this particular projector / back-projector combination that uses only 2 projections, that is, if f(x,y) = δ(x,y), what is b(x,y)? You should be able to write this as a fairly simple expression. Hint: Consider creating an image containing only a single point and then see what your projector / back-projector pair does to it.

In general, if there are K projections, describe the impulse response of the projector / back-projector pair. This does not have a simple expression.

Project Team
List your teammates and which project you will work on.

What to hand in
Hand in to class

• Your projection program, including enough comments so we can understand what you have done, with projection
• Your back-projection program, showing the back-projected images.
• Answer the impulse response questions.
• Teammates and project (1 point).

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