18-793 Image and Video Processing
Submission instructions.
Summer 2020
Submissions are due on Thursday 09/24 at 10.00pm ET
Please upload scans of your solution in GradeScope (via Canvas)
Homework 3
Instructions
Please solve all non-MATLAB problems using only paper and pen, without resorting to
a computer.
Please show all necessary steps to get the final answer. However, there is no need to be overly elaborate. Crisp and complete answers.
For all MATLAB problems, include all code written to generate solutions.
Please post all questions on the discussion board on the Piazza course website.
If you feel some information is missing, you are welcome to make reasonable assumptions and proceed. Sometimes the omissions are intentional. Needless to say, only reasonable assumptions will be accepted.
1. (Some elementary radon transforms and properties) Consider an image f(x,y) and its radon transform r(θ, α).
a) If f (x, y) is rotationally-invariant, show that r(θ, α) = r(α), i.e., the radon transform is not a function of the projection angle θ.
1 −α2 b)Letr(α)=√2πe 2 .Derivef(x,y).
2. (Scaling and Radon Transforms) Let the Radon transform of the image i1(x, y) be r1(α, θ). Let i2(x, y) = i1(ax, ay), be a scaled image.
Find an expression for r2(α,θ), the Radon transform of i2(x,y), in terms of r1.
3. (Translations and Radon Transforms) Let the Radon transform of an image i1(x,y) be
r1(α,θ). Leti2(x,y)=i1(x−a,y−b).
Find an expression for r2(α,θ), the Radon transform of i2(x,y), in terms of r1 and other
quantities.
4. (Implement filtered backprojection) In hw03.mat, you are given radon transform mea- surements in the variable rad, corresponding to values in alpha and theta. Implement filtered backprojection.
Notes:
2 Homework 3
(a) You are restricted to basic commands like fft, fft2, conv, conv2, meshgrid, interp2, and other basic commands. Specifically, you cannot use commands that do radon inversion like iradon.
(b) Follow the steps laid out in the lecture: (1) 1D Fourier transform of line integral, (2) Ramp filtering, (3) Inverse FT, and finally (4) Backprojection.
(c) Attend recitation
Problems below wont be graded. We wont release solutions as well. We are happy to verify yours, if you post on piazza, and discuss them
5. Suppose that image f (x, y) has radon transform rf (α, θ), and the image g(x, y) has radon transform rg (α, θ). Suppose that rg (α, θ) = d rf (α, θ). Derive an expression relating
dθ
g(x,y) to f(x,y).
6. Suppose that image f (x, y) has radon transform rf (α, θ), and the image g(x, y) has radon
transform rg (α, θ). Suppose that g(x, y) = d f (x, y). Derive an expression relating rg in dx
terms of rf .
7. You are given the radon transform of an image. Derive an analytical expression for the image. It is ok if the values in your expression are approximate.
-150
-100
-50
0
50
100
150
1.5
1
0.5
0
0 20 40 60 80 100 120 140 160 theta in [degrees]
alpha