OPTI 512 Homework 7 Due: November 6, 2019

1. In the spirit of Halloween, we are going to find a ghost using a matched filter. You will

need Matlab or something equivalent to do this problem. The images below are called

Skeletons.jpg and Ghost.jpg, and can be downloaded from the course website.

Do the following to find the ghost in a sea of skeletons:

a) Import the file Skeletons.jpg and convert it to gray scale. This should be a 1200 x

1200 pixel array. What is the mean value of the array? Subtract this mean value

from the array. We will call this zero-mean skeleton array 𝑠𝑠(𝑥𝑥,𝑦𝑦).

I just used the green channel from the image for the gray scale conversion. If the

conversion was done differently, the means values might be slightly different. The

mean value for the Skeletons gray scale image is 205.684.

b) Import the file Ghost.jpg and convert it to gray scale. This should be a 76 x 58 pixel

array. What is the mean value of this array? Subtract this mean value from the array.

We will call this zero-mean ghost array 𝑡𝑡(𝑥𝑥,𝑦𝑦) and use it as the template for the

object we are trying to find in the skeleton array.

I again just used the green channel from the image for the gray scale conversion. If

the conversion was done differently, the means values might be slightly different. The

mean value for the Ghost gray scale image is 212.062.

c) Create an empty 1200 x 1200 pixel array and place the template 𝑡𝑡(𝑥𝑥, 𝑦𝑦) in the middle

of the array We will call this array the ghost array 𝑔𝑔(𝑥𝑥,𝑦𝑦). Display this array in your

code to verify that you did it correctly. Save you ink. No need to put it in the

assignment.

d) We now want to find the cross correlation 𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥, 𝑦𝑦) of the two arrays. This is more

easily done in the Fourier domain. We know that ℱ2𝐷𝐷�𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥, 𝑦𝑦)� = 𝑆𝑆(𝜉𝜉, 𝜂𝜂)𝐺𝐺∗(𝜉𝜉, 𝜂𝜂),

where 𝑆𝑆(𝜉𝜉, 𝜂𝜂) = ℱ2𝐷𝐷{𝑠𝑠(𝑥𝑥, 𝑦𝑦)} and 𝐺𝐺(𝜉𝜉, 𝜂𝜂) = ℱ2𝐷𝐷{𝑔𝑔(𝑥𝑥,𝑦𝑦)}. Knowing this, calculate

the cross correlation as 𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥,𝑦𝑦) = ℱ2𝐷𝐷−1�𝑆𝑆(𝜉𝜉, 𝜂𝜂)𝐺𝐺∗(𝜉𝜉, 𝜂𝜂)�. Plot �𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥,𝑦𝑦)�.

e) What are the x, y coordinates of the maximum value of �𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥,𝑦𝑦)�? Verify that the

ghost is indeed located at this location in the original Sketetons.jpg image.

The maximum value of �𝛾𝛾𝑠𝑠𝑠𝑠(𝑥𝑥,𝑦𝑦)� occurs at the point (1117, 357). Zooming into this

position on the original image shows our ghost.

2. State whether the following functions and band-limited or not. If they are band-limited,

determine the Nyquist frequency.

a) 𝑟𝑟𝑟𝑟𝑟𝑟𝑡𝑡 �𝑥𝑥3�

The Fourier transform is 3𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟(3𝜉𝜉) and the sinc() function has non-zero values out

to infinity, so the function is not band-limited.

b) 𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟 �𝑥𝑥3�

The Fourier transform is 3𝑟𝑟𝑟𝑟𝑟𝑟𝑡𝑡(3𝜉𝜉) which is zero for |𝜉𝜉| > 16 , so the function is

band-limited and the Nyquist frequency 𝑁𝑁𝜉𝜉 = 16.

c) 𝐺𝐺𝐺𝐺𝐺𝐺𝑠𝑠(4𝑥𝑥)

The Fourier transform is 14𝐺𝐺𝐺𝐺𝐺𝐺𝑠𝑠 �𝜉𝜉

4� which has non-zero values out to infinity, so the

function is not band-limited.

d) 𝑡𝑡𝑟𝑟𝑠𝑠 �𝑥𝑥2�

The Fourier transform is 2𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟2(2𝜉𝜉) which has non-zero values out to infinity, so the

function is not band-limited.

e) 𝑠𝑠𝑠𝑠𝑠𝑠𝑟𝑟2 �𝑥𝑥2�

The Fourier transform is 2𝑡𝑡𝑟𝑟𝑠𝑠(2𝜉𝜉) which is zero for |𝜉𝜉| > 12 , so the function is band-

limited and the Nyquist frequency 𝑁𝑁𝜉𝜉 = 12.

f) 𝛿𝛿(𝑥𝑥)

The Fourier transform is 1 which is non-zero values out to infinity, so the function is

not band-limited.

g) 𝑟𝑟𝑐𝑐𝑠𝑠(5𝜋𝜋𝑥𝑥)

The Fourier transform is

12�𝛿𝛿 �𝜉𝜉 −

52� + 𝛿𝛿 �𝜉𝜉 +

52��

which is zero for |𝜉𝜉| > 52 , so the function is band-limited and the Nyquist frequency

𝑁𝑁𝜉𝜉 = 52.

3. Download the file phase.dat from the course website. You will need Matlab or something

equivalent to do this problem. This is a binary file containing 1024 x 1024 consecutive

double values corresponding to a 1024 x 1024 array of phase values Φ(𝑥𝑥, 𝑦𝑦). For this

problem, phase contrast techniques will be illustrated. Do the following:

a) Use the following Matlab code snippet below to aid in importing the data. You will

need to edit the file path accordingly.

fid = fopen('C:\Users\jschw\Dropbox\Class\OPTI 512 Linear Systems, Fourier

Transforms\Homeworks\phase.dat','rb');

phase = fread(fid, [1024 1024], 'double');

imshow(phase,[])

Plot the phase pattern Φ(𝑥𝑥, 𝑦𝑦).

b) Create the complex array 𝑓𝑓(𝑥𝑥,𝑦𝑦) = 𝑟𝑟𝑥𝑥𝑒𝑒[−𝑠𝑠Φ(𝑥𝑥,𝑦𝑦)]. This will be the input to our

phase contrast system. Sensors and our eyes can only record the squared magnitude of

complex signals or |𝑓𝑓(𝑥𝑥,𝑦𝑦)|2. What is |𝑓𝑓(𝑥𝑥, 𝑦𝑦)|2 for this input?

|𝑓𝑓(𝑥𝑥,𝑦𝑦)|2 = 𝑟𝑟𝑥𝑥𝑒𝑒[−𝑠𝑠Φ(𝑥𝑥,𝑦𝑦)]𝑟𝑟𝑥𝑥𝑒𝑒[𝑠𝑠Φ(𝑥𝑥,𝑦𝑦)] = 1 everywhere, which means the object

is transparent.

c) The transfer function for this system is

𝐻𝐻(𝜉𝜉, 𝜂𝜂) = �𝑟𝑟𝑥𝑥𝑒𝑒 �𝑠𝑠𝜋𝜋2� 𝑓𝑓𝑐𝑐𝑟𝑟 𝜉𝜉 = 𝜂𝜂 = 0

1 𝑐𝑐𝑡𝑡ℎ𝑟𝑟𝑟𝑟𝑒𝑒𝑠𝑠𝑠𝑠𝑟𝑟,

so calculate 𝐹𝐹(𝜉𝜉, 𝜂𝜂) using the FFT routines in Matlab and then multiply 𝐹𝐹(0,0) by

𝑟𝑟𝑥𝑥𝑒𝑒 �𝑠𝑠 𝜋𝜋2� to create the output 𝐺𝐺(𝜉𝜉, 𝜂𝜂). Be careful with the shifting that occurs with the

FFT and also note that Matlab starts its arrays with a value of 1 instead of 0. Finally,

calculate output of the phase contrast system with the inverse FFT routines to get

𝑔𝑔(𝑥𝑥,𝑦𝑦). Plot |𝑔𝑔(𝑥𝑥,𝑦𝑦)|2.

The point here is that by simply multiplying the point 𝐹𝐹(0,0) by 𝑟𝑟𝑥𝑥𝑒𝑒 �𝑠𝑠 𝜋𝜋2�, we were

able to convert a pure phase input that is invisible to us since |𝑓𝑓(𝑥𝑥,𝑦𝑦)|2 is constant

into an output |𝑔𝑔(𝑥𝑥,𝑦𝑦)|2 that depicts the phase pattern.