261 EE The Fourier Transform and its Applications
Fall 2011
Problem Set Nine
Due Friday, December 9
1. (20 points) 2D Fourier Transforms
Find the 2D Fourier Transforms of:
(a) sin 2ax1 sin 2bx2
(b) e2i(ax+by) cos(2cx)
(c) cos(2 (ax + by )) Hint: Use the addition formula for the cosine.
2. Linear Transformations (20 points)
Consider a 2D rectangular function (x, y ):
This 3D representation is depicted by the following 2D image, where white corresponds to 1,
and black to 0.
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Student Version of MATLAB
This 2D rectangular function is subjected to 3 dierent linear transformations. The following
images (A, B, C) are obtained:
A:
Student Version of MATLAB
B:
Student Version of MATLAB
C:
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Student Version of MATLAB
(a) Each of the gures is a result of a horizontal shear. If |k1 | > |k2 | > |k3 |, match the
following linear transformations with gures A, B and C.
1 k1
01
1 k2
01
Student Version of MATLAB
1 k3
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(b) The 2D Fourier transform of each gure is taken, and the following magnitude plots I, II,
and III, below, are obtained. Match gures A, B, and C with their corresponding Fourier
transforms. Explain your reasoning. (Hint: Look at the orientation of the crests.)
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I:
II:
Student Version of MATLAB
III:
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Student Version of MATLAB
3. (15 points) 2D Discrete Fourier Transform Let f be a M N matrix (you can think of f
as an M N image). The 2D DFT of f is given by the following formula:
M 1 N 1
f [m, n]Nln Mkm
F f [k, l] =
m=0 n=0
where
N = e2i/N ,
M = e2i/M .
Student Version of MATLAB
Independent of the problems to follow, we comment that the 2D DFT is separable in the
following sense. Let fm be the mth row of the matrix f : its a vector of length N. Let F fm
be the 1D DFT of fm then:
N 1
N 1
fm [n]Nln
F fm [l] =
n=0
f [m, n]Nln
=
n=0
Then, from the given formula for the 2D DFT, we can easily see that:
M 1
F fm [l]Mkm
F f [k, l] =
m=0
In other words, this is a 1D DFT of the vector of length M which consists of the lt h entry in
each vector F fm .
We can take advantage of the separability by rst computing the 1D DFT of the rows of f
and then doing another 1D DFT to nd the [k, l] entry in F f .
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(a) Modulation: Let g be an M N image obtained from f in the following way: g [m, n] =
mk nl
f [m, n]M 0 N 0 where k0 and l0 are integers. What is F g in terms of F f ?
(b) 2D convolution Given 2 matrices f and g of size M N , their 2D convolution is given
by:
M 1 N 1
(f g )[m, n] =
f [u, v ]g [m u, n v ]
u=0 v =0
Show that F (f g ) = F f F g , where on the right we mean the M N matrix whose
elements are the products of the corresponding elements of F f and F g .
Hint: Write out the expression for F (f g ). Youll get a nasty looking expression with
four nested sums. But swap the order of summation, change the variable of summation
(analogous to changing the variable of integration) and use the fact that the 1D DFT
can be computed by summing over any N (or M ) consecutive indices.
4. (15 points) Watermarking an image A common application of the 2D Fourier Transform
is the compression of images. The rst JPEG compression standard heavily quantizes the
high frequency components within an image because the human eye is poor at discriminating
between signals with high spatial frequencies.
Another application of this property is hiding one signal inside another without experiencing
a decrease in the overall quality of the initial signal. (For example, the French television
standard SECAM used to encode the sound information in the high frequencies of the image
spectrum to save transmission bandwidth.) This procedure is called digital watermarking,
which is what this problem is about.
The transform commonly used in image processing is a modied version of the DFT, called
the Discrete Cosine Transform ; well use it in this problem. The basic principle of the DCT
is to replicate the initial signal in all dimensions and obtain an even signal that has twice the
size of the original signal in all dimensions and take the DFT of this. The output of the DFT
is then truncated to keep only the same amount of samples as initially present. Given that
we take the DFT on real and even signals, the output is then also real and even. Truncating
the signal does not delete any information, and this technique thus has the advantage of
outputting only real numbers. (You can nd many descriptions of the DCT with a little
Googling, and I hope to include a discussion in future versions of the reader.)
(a) the Download image watermarked from the course website. This image has been watermarked (i.e. another smaller image was hidden in it) using the following scheme:
- The image has been decomposed in 8 8 blocks.
- We computed the DCT on each of these blocks.
- In each block, the bottom-right 2 2 block were erased and replaced by the amplitudes of the corresponding 2 2 block of the hidden image, divided by 10. Thus,
the dimensions of the hidden image are equal to a quarter of those from the original
image.
Write a Matlab program that reads the original image, and reconstructs the hidden
image inside it. Please turn in your code and the discovered image.
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Use the functions dct, idct, imread and imwrite. Go through Matlab help to get
some information about those functions.
If we dene the matrix
D = dct(I8 )
then the 2-D Discrete Cosine Transform of an 8 8 block A is given by the following
equation:
A = DADT .
What/Who/Where is the hidden signal?
(b) What happens if you reiterate the process on the image you just discovered?
5. (20 points) Echo planar MRI
CT and MRI scans are ubiquitous in society today. What does the doctor order when you
have a head injury? CT scan. Knee injury? MRI. Neither of these imaging modalities would
be possible without Fourier transforms. In this problem, we will look at a simple MR image
reconstruction example.
It turns out that the data acquired from an MRI scan are samples of the Fourier transform of
the image. The Fourier transform of the image is referred to as k -space, and dierent readout
techniques sample k -space along dierent trajectories. To recover the image, the trajectory
must cover all of k -space, but one of the beauties of MRI is the exibility in how to traverse
k -space. A simple trajectory is echo planar imaging (EPI), which alternately samples k -space
from left to right and then right to left as it moves down each row.
ky
ky
>
<
<
1>
2 ><
<
>
<
<
>
>
kx
kx
<
<
<
<
(a) Single shot
<
<
(b) Two shot
The EPI trajectory zig-zags from top to bottom, lling up k -space.
(a) Download epiMRI.mat from the course website and load it into Matlab. The single shot
EPI readout data k is a vector of length 256 256 = 65536 samples. Populate a matrix
K of size 256 256 with the readout data so that the matrix K is the Fourier transform
of the image, which is 256 256 pixels. Reconstruct the MR image (can you tell what
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it is a scan of?). Remember to use fftshift so that Matlab takes the inverse 2DFT
(ifft2) properly.
(b) Suppose your trajectory is a two shot interleaved EPI sequence. That is, you rst acquire
trajectory (1), then you acquire trajectory (2). Reconstruct the second image from the
readout data ki, where the rst row is the data for trajectory (1) and the second row is
the data for trajectory (2). What happens if you only use the rst shot (and use zeros
for the second shot)? What about if you only use the second shot (and use zeros for
the rst shot)? Why is the rst shot image nonnegative, whereas the second shot image
is positive and negative? What happens when you add these two images together, and
what basic property of the FT gives us this result?
Hint: To explain the artifact in the image from the rst shot, look back at Hw 7, #4.
You may also nd the command colorbar useful.
Note: There are many other trajectories, including spiral, interleaved spiral, and radial
projections. Because the sampling is not on a Cartesian grid, one way to reconstruct
the image is to map the acquired data to the Cartesian grid through an operation called
gridding. Then the inverse 2DFT can be applied. Projection sampling is very important
for CT imaging, because a CT scan gives you the data to ll in k -space along the radial
spokes. In addition to gridding, there is an alternative reconstruction technique that involves ltering each projection (spoke) and backprojecting (i.e., smearing) each ltered
projection across a blank image. Believe it or not, the sum of these ltered, backprojected images reconstructs the image! If you are interested in medical imaging, I (Adam)
highly recommend the EE369A/B Medical Imaging Systems series, which is oered in
Winter/Spring 2011 (a shameless plug, I know). Guess what the only prerequesite is
EE261!
ky
ky
<
<
<
k
>
x
<
ky
k
k
x
x
>
>
(c) Spiral
(d) Interleaved spiral
(e) Projection
Figure 1: Various other trajectories that cover k -space.
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