EE 4780
Bilateral Filter
Bilateral Filter
K is the normalization constant
K
x N

e  I ( y )  I ( x ) 
2
/ 2 r2  y  x / 2 d2
2
e
y x N
Intensity (range)
proximity
ˆI ( x)  1
K
x N

Spatial (domain)
proximity
 I ( y )  I ( x )  / 2 r2  y  x / 2 d2
2
2
e
e
I ( y)
y x N
Bahadir K. Gunturk
N is a fixed value used to define the spatial
neighborhood of the filter
2
Bilateral Filter
– Matlab implementation
n=1:1:500; % Generate a vector from 1 to 500; the increment is 1.
I0=zeros(size(n)); % Generate a vector of zeros; the size of the vector is equal to the size of n.
I0(1:250)=15; I0(251:end)=10; % Set the first 250 values to 15, and the rest to 10.
I = I0 + 0.5*randn(size(I0)); % 0.5 is the standard deviation of the noise
figure;
subplot(2,1,1); plot(n,I0); axis ([190 310 6 18]); title('Original signal');
subplot(2,1,2); plot(n,I); axis ([190 310 6 18]); title('Noisy signal');
Bahadir K. Gunturk
3
Bilateral Filter
– Matlab implementation
sigma_d=10;
N=round(4*sigma_d); % N determines the spatial neighborhood
sigma_r=1.3;
d = -N:1:N;
weights_d = exp(-d.*d/(2*sigma_d*sigma_d));
ˆI ( x)  1
K
x N

 I ( y )  I ( x )  / 2 r2  y  x / 2 d2
2
2
e
e
y x N
The weights depend on the spatial
distance (to the center pixel x) only;
therefore, it is calculated once and
saved.
Bahadir K. Gunturk
4
I ( y)
Bilateral Filter
– Matlab implementation
sigma_d=10;
N=round(4*sigma_d); % N determines the spatial neighborhood
sigma_r=1.3;
d = -N:1:N;
weights_d = exp(-d.*d/(2*sigma_d*sigma_d));
ˆI ( x)  1
K
x N

 I ( y )  I ( x )  / 2 r2  y  x / 2 d2
2
2
e
e
I ( y)
y x N
x=260; % An example
pixels = I(x-N:x+N); % Put the pixels within the neighborhood of the center pixel into a vector.
weights = weights_d .* exp(-(pixels-I(x)).*(pixels-I(x))/(2*sigma_r*sigma_r)) + 0.0001;
Add a small number in
case weights=0;
weights = weights./sum(weights);
Bahadir K. Gunturk
5
Bilateral Filter
– Matlab implementation
sigma_d=10;
N=round(4*sigma_d); % N determines the spatial neighborhood
sigma_r=1.3;
d = -N:1:N;
weights_d = exp(-d.*d/(2*sigma_d*sigma_d));
ˆI ( x)  1
K
x N

 I ( y )  I ( x )  / 2 r2  y  x / 2 d2
2
2
e
e
y x N
x=260;
pixels = I(x-N:x+N); % Put the pixels within the neighborhood of the center pixel into a vector.
weights = weights_d .* exp(-(pixels-I(x)).*(pixels-I(x))/(2*sigma_r*sigma_r)) + 0.0001;
weights = weights./sum(weights); % Normalize the weights so that its sum is equal to 1.
I_output(x) = sum(weights.*pixels);
Bahadir K. Gunturk
6
I ( y)
Bilateral Filter
figure; plot([x-N:x+N],weights)
Bahadir K. Gunturk
7
Bilateral Filter
– Matlab implementation
d = -N:1:N;
weights_d = exp(-d.*d/(2*sigma_d*sigma_d));
% Repeat for all pixels
I_output = I;
for i=1+N:length(I)-N, % Be careful with the borders; do not exceed the dimensions.
pixels = I(i-N:i+N);
weights = weights_d .* exp(-(pixels-I(i)).*(pixels-I(i))/(2*sigma_r*sigma_r)) + 0.0001;
weights = weights./sum(weights);
I_output(i) = sum(weights.*pixels);
end
figure; plot(n,I_output);
Bahadir K. Gunturk
8
Bilateral Filter
Input
Gaussian
 d  10
Bilateral
 d  10
 r  1.3
Bahadir K. Gunturk
9
Bilateral Filter vs. Gaussian LPF
d  2
 r  10
Gaussian
MSE=49.8
MSE=100.0
 n  10
 r  30
MSE=30.3
sigma_d=10
MSE=99.57
 r  50
Bahadir K. Gunturk
MSE=42.5
10
Wiener Filter
Y  X W
Noisy
image
Original
image
Wiener Filter
Noise
2

x
Xˆ  2
Y
2
 x w
Signal variance
Noise variance
When sigma_x << sigma_w, (noise is very large), X goes to 0.
When sigma_x >> sigma_w, (noise is very small), X goes to Y.
Bahadir K. Gunturk
11
Wiener Filter
 x2
is estimated by
ˆ x2  y 2   w2
Estimate manually by
looking at the variance in a
smooth region.
Since variance is nonnegative, it is modified as
ˆ x2  max[ 0, y 2   w2 ]
Estimate signal variance locally:
ˆ x2  max[0,
Bahadir K. Gunturk
1
N2
2
2
y


 i w]
i
N
N
12
Wiener Filter
Noisy, =10
Denoised (3x3neighborhood)
Mean Squared Error is 56
wiener2 in Matlab
Bahadir K. Gunturk
13
Image Enhancement
 1 1 1
 1 8 1


 1 1 1
This is an high-pass filter.
It removes low-frequency components.
Bahadir K. Gunturk
14
Image Enhancement

High-boost or high-frequency-emphasis filter

Sharpens the image but does not remove the low-frequency
components unlike high-pass filtering
Bahadir K. Gunturk
15
Image Enhancement

High-boost or high-frequency-emphasis filter


High pass = Original – low pass
High boost = A*(Original) + High pass
 Part of the low-frequency components are added back to
the high frequency components
Bahadir K. Gunturk
16
Image Enhancement
 1 1 1
 1 8 1


 1 1 1
 1 1 1
 1 9 1


 1 1 1
A high-pass filter
A high-boost filter
Bahadir K. Gunturk
17
Image Enhancement

High-boost or high-frequency-emphasis filter
Bahadir K. Gunturk
18
Spatial Filtering
Bahadir K. Gunturk
19