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EE
SPRING 2000
COMPUTER PROJECT 2
PART A: ORTHOGONALIZATION
We consider two different stochastic aspects of orthogonalization. Prewhitening is used to
convert colored noise into white noise thereby simplifying discrimination architectures such as
the maximum likelihood ratio. Edge enhancement is often thought of as orthogonalizing
deterministic images. In fact, the SOBEL edge enhancement is a correlation technique that is
optimum for detecting lines corrupted by additive white Gaussian noise.
1. Prewhitening Colored Noise.
Generate three 1024x1 Gaussian random vectors. Their elements are iid, zero mean with
variance
2
=1. Form three new vectors from these such that:
t 1
=3g 1
+2g 2
+g 3
t 2
=g 1
+3g 2
+2g 3
t 3
=2g 1
+g 2
+3g 3
a. Analytically determine C tt
= E{T
T
T} in terms of C =
3 1 2
2 3 1
1 2 3
where T=[t 1
, t 2
, t 3
].
b. Using the concept of eigenvectors and eigenvalues, determine a weighting matrix W such that
Z = T W and C ZZ
= E{Z
T
Z} = I where W is 3x3.
c. Use matlab to solve for W and empirically estimate C ZZ
d. Is W C
e. C is a circulent matrix. One solution for its Nx1 eigenvectors are
T
N
N k
j
N
k
j
N
k
j
k
e e e e
2 2 2 2 ( 1 )
0
where k = 0,1...(N-1). The eigenvector matrix is then
0 1 1
N
T
Verify that will diagonalize C.
2. Edge Enhancement
Edge Enhancement can be used to decorrelate two different images. This is true for images
where the edges carry the discriminating information. One of the most common edge
enhancement techniques is known as SOBEL enhancement. Interestingly, very few researchers
realize that this technique is optimum in output Signal-to-Noise Ratio (SNR) for edges corrupted
by additive white Gaussian noise (AWGN). The technique is correlation based and the bank of
correlation filters are combined in the “largest of” architecture. The filters are in the form of
3 3 kernels which are individually convolved with the input. The output convolutions are
elementwise combined by taking the “largest of” or maximum value to be the final element
output. The kernels are:
S
1
1 2 1
0 0 0
1 2 1
S
2
2 1 0
1 0 1
0 1 2
S
3
1 0 1
2 0 2
1 0 1
S
4
0 1 2
1 0 1
2 1 0
S
5
1 2 1
0 0 0
1 2 1
S
6
2 1 0
1 0 1
0 1 2
S
7
1 0 1
2 0 2
1 0 1
S
8
0 1 2
1 0 1
2 1 0
where half the kernels are negatives of the other half such that
S S
5 1
S S
6 2
S S
7 3
S S
8 4
Each kernel is convolved with the input image s.t.
Y a b S a b X a b
n n
, , ,
,
The output result is
Z a , b max Y a , b , Y a , b , Y a , b
1 2 8
,
Because half of the responses are negatives of the other half, the output result can be simplified
to be the maximum of the absolute value of four of the kernels. Using fft2 and ifft2, show the
matlab code for convolving each of the kernels (don't forget to zero pad, and don't forget to
convert Y n
to real values after the fft2 based convolution). Also show how you would get the
result from the first 4 kernels.
PART B: BINARY DISCRIMINATION
1. Frequency Shift Keying.
the symbols information across a bandwidth of frequencies making the demodulation less
sensitive to a single frequency jamming signal then the FSK technique.
Repeat B.1, but replace
s t
1
and
s t
2
with two iid. pseudo-random sequences. Each
sequence should have a different seed and their distribution should be bipolar
s 1 , 1
, with
P ( 1 ) P ( 1 ) 0. 5
and therefore 0 mean.
PART C: 2-D DETECTION WITH NOISE
1. Test Image Selection.
Choose two 64 64 sections of the Mandelbrot set. One section will represent the target image
that you want to detect and the other will be a clutter image that you want to suppress. Give the
instructor, by email, the coordinates of these two images.
The auto-regressive formula, for each pixel position is
If magnitude of Z >2 then encode the x , y pixel shade as n.
Let
real C x
imag C y
and give the coordinates for
x y
ll ll
, as lower left corner and
x y
ur ur
, as the upper right corner of the image window.
2. Training Set Generation.
Rotate the original target and clutter images by 45 degrees to obtain training sets of 8 images
each. The trick to rotation is to select an output matrix element and then rotate it backwards to
the closest input matrix element. This method eliminates “pin holes” which are pixels with no
assigned value. Pin holes will occur if the input matrix elements are mapped to the closest
output matrix. The rotation transformation from input to output locations by angle
is given
as:
x x
y y
x x
y y
out out center
out out center
in in center
in in center
,
,
,
,
cos sin
sin cos
By multiplying the above equation by the inverse transformation matrix we obtain the
transformation from output back to input as
x x
y y
x x
y y
in in center
in in center
out out center
out out center
,
,
,
,
cos sin
sin cos
The pseudo code for rotating an
M N
image
A
by and storing the result in
B
is given as:
x x
N
in , center out , center
1
2
y y
M
in , center out , center
1
2
for m=1 to M
Z Z C
n n
2
1
for n=1 to N
x
out
n
y
out
m
x x x y y x
in out out center out out center in center
, , ,
cos sin
y x x y y y
in out out center out out center in center
, , ,
sin cos
Truncate
x y
in in
, to be integers
if
1 x N
in
and
1 y M
in
then
B m n A y x
in in
, ,
else
B m , n
background shade
Once the images are rotated, then the training images should be SOBEL edge enhanced.
Typically, the edge enhancement should follow the rotation to more accurately model an actual
system where the target object will be of arbitrary rotation when imaged and then edge enhanced
by the computer. Let the edge enhanced image of A be X.
3. Test Set Generation.
The test image is formed by first augmenting all the target and clutter images together in a
checker board pattern to form a 256 256 image. A noise image is added to this signal image.
The noise image is white gaussian noise attenuated from left to right from 0 noise level to a
noise level equivalent to a Noise-to-Signal Ratio (NSR) of 2. The NSR is defined as
NSR n n X a b
a b
2
2
1
64
1
64
where noise element variance is
2
1
2
1
n n
N
.
The value of N is 4=(256/64) so n corresponds to each of the four 256 64 partitions. Each
partition has its own NSR(n) where n=1,2,3,.
4. Filter Correlation Test.
You will need to zero pad the 64 64 filter impulse responses to be 256 256 before
correlating them with the test image. Be sure to use the 2-D FFT to perform the correlation
process.
a. Form an Linear Phase Coefficient Composite Filter (LPCCF) set from the target training set
and show a 3-D mesh representation of the correlation intensity response with the test image
with noise, for filter orders
k 0 , 1 , 2 and 3
.
b. Form a Minimum Average Correlation Energy (MACE) filter and repeat 4a.
REFERENCES
