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** = **A ^{T}p**. 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).

CS/ECE 545 Staff

Contents ©1997 - 2011 Norman Wittels and Michael A. Gennert