Transcript EECE\CS 253 Image Processing Lecture Notes: Frequency Lecture Notes Filtering Richard Alan Peters II Department of Electrical Engineering and Computer Science Fall Semester 2011 This work is licensed under.

EECE\CS 253 Image Processing
Lecture Notes:
Frequency
Lecture
Notes Filtering
Richard Alan Peters II
Department of Electrical Engineering and
Computer Science
Fall Semester 2011
This work is licensed under the Creative Commons Attribution-Noncommercial 2.5 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/2.5/ or
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Convolution Property of the Fourier Transform
Let functions f (r , c) and g(r , c) have
Fourier Transforms F(u , v) and G(u , v).
Then,
F {f * g } = F ×G.
Moreover,
F {f ×g } = F * G.
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* = convolution
· = multiplication
The Fourier Transform of a
convolution equals the
product of the Fourier
Transforms. Similarly, the
Fourier Transform of a
convolution is the product of
the Fourier Transforms
1999-2011 by Richard Alan Peters II
2
Convolution via
Fourier Transform
Image & Mask
Transforms
Pixel-wise
Product
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1999-2011 by Richard Alan Peters II
Inverse
Transform
3
How to Convolve via FT in Matlab
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Read the image from a file into a variable, say I.
Read in or create the convolution mask, h. The mask is usually 1-band
Compute the sum of the mask: s = sum(h(:));
If s == 0, set s = 1;
For color images you may
Replace h with h = h/s;
need to do each step for
Create: H = zeros(size(I));
each band separately.
Copy h into the middle of H.
Shift H into position: H = ifftshift(H);
Take the 2D FT of I and H: FI=fft2(I); FH=fft2(H);
Pointwise multiply the FTs: FJ=FI.*FH;
Compute the inverse FT: J = real(ifft2(FJ));
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4
Coordinate Origin of the FFT
Center =
(floor(R/2)+1, floor(C/2)+1)
Even
Odd
Even
Odd
Image Origin
Image Origin
Weight Matrix Origin
Weight Matrix Origin
After FFT shift
After FFT shift
After IFFT shift
After IFFT shift
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Matlab’s fftshift and ifftshift
J = fftshift(I):
5
6
4
8
9
7
I (1,1)  J ( R/2 +1, C/2 +1)
2
I = ifftshift(J):
J ( R/2 +1, C/2 +1)  I (1,1)
3
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
5
6
4
8
9
7
2
3
1
where x = floor(x) = the largest integer smaller than x.
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6
Blurring: Averaging / Lowpass Filtering
Blurring results from:

Pixel averaging in the spatial domain:
–
–

Each pixel in the output is a weighted average of its neighbors.
Is a convolution whose weight matrix sums to 1.
Lowpass filtering in the frequency domain:
–
–
High frequencies are diminished or eliminated
Individual frequency components are multiplied by a nonincreasing
function of  such as 1/ = 1/(u2+v2).
The values of the output image are all non-negative.
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Sharpening: Differencing / Highpass Filtering
Sharpening results from adding to the image, a copy of
itself that has been:

Pixel-differenced in the spatial domain:
–
–

Each pixel in the output is a difference between itself and a weighted
average of its neighbors.
Is a convolution whose weight matrix sums to 0.
Highpass filtered in the frequency domain:
–
–
High frequencies are enhanced or amplified.
Individual frequency components are multiplied by an increasing
function of  such as  = (u2+v2), where  is a constant.
The values of the output image positive & negative.
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8
Recall:
Convolution Property of the Fourier Transform
Let functions f (r , c) and g( r , c) have
Fourier Transforms F (u , v) and G (u, v).
* = convolution
· = multiplication
Then,
F {f *g } = F×G.
Moreover,
F {f×g } = F * G.
Thus we can compute f *g by
f *g =
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F
-1
{F×G}.
The Fourier Transform of a
convolution equals the
product of the Fourier
Transforms. Similarly, the
Fourier Transform of a
convolution is the product of
the Fourier Transforms
1999-2011 by Richard Alan Peters II
9
Ideal Lowpass Filter
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Ideal Lowpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
FD filter radius: 16
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
11
Ideal Lowpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
FD filter radius: 8
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
12
Consider the
image below:
Power Spectrum and Phase of an Image
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Phase
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Ideal Lowpass Filter
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD filter radius: 16
Ideal LPF in FD
14
Ideal Lowpass Filter
Filtered Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD filter radius: 16
Original Image
15
Ideal Lowpass Filter
Original Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD filter radius: 16
Filtered Image
16
Ideal Highpass Filter
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1999-2011 by Richard Alan Peters II
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Ideal Highpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
FD notch radius: 16
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
18
Ideal Highpass Filter
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch radius: 16
Ideal HPF in FD
19
*signed image:
0 mapped to 128
Ideal Highpass Filter
Filtered Image*
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch radius: 16
Original Image
20
*signed image:
0 mapped to 128
Ideal Highpass Filter
Original Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch radius: 16
Filtered Image*
21
*signed image:
0 mapped to 128
Ideal Highpass Filter
Positive Pixels
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Filtered Image*
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch radius: 16
Negative Pixels
22
Ideal Bandpass Filter
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Ideal Bandpass Filter
A bandpass filter is created by
(1) subtracting a FD radius 2 lowpass filtered image from a FD radius 1 lowpass
filtered image, where 2 < 1, or
(2) convolving the image with a mask that is the difference of the two lowpass masks.
FD LP mask with radius 1
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=
FD LP mask with radius 2
1999-2011 by Richard Alan Peters II
FD BP mask 1 - 2
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*signed image:
0 mapped to 128
Ideal Bandpass Filter
image LPF radius 1
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image LPF radius 2
1999-2011 by Richard Alan Peters II
image BPF radii 1, 2*
25
*signed image:
0 mapped to 128
Ideal Bandpass Filter
filtered image*
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filter power spectrum
1999-2011 by Richard Alan Peters II
original image
26
*signed image: 0
mapped to 128
Ideal Bandpass Filter
original image*
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filter power spectrum
1999-2011 by Richard Alan Peters II
filtered image
27
*signed image:
0 mapped to 128
A Different Ideal Bandpass Filter
original image
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filter power spectrum
1999-2011 by Richard Alan Peters II
filtered image*
28
The Optimal Filter
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The Uncertainty Relation
space
frequency
FT
space
frequency
FT
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If  x  y is the extent of
the object in spaceand
if  u  v is its extent in
frequencythen,
1
 x  y  u  v 
16 2
A small object in space
has a large frequency
extent and vice-versa.
1999-2011 by Richard Alan Peters II
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The Uncertainty Relation
 small extent 
A symmetric pair of
lines in the frequency
domain becomes a
sinusoidal line in the
spatial domain.
 large extent 
frequency
 large extent 
 small extent 
 small extent 
space
IFT
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Recall: a symmetric
pair of impulses in the
frequency domain
becomes a sinusoid in
the spatial domain.
space
IFT
 small extent 
 large extent 
frequency
 large extent 
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Ideal Filters Do Not Produce Ideal Results
IFT
A sharp cutoff in the
frequency domain…
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…causes ringing in the
spatial domain.
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Ideal Filters Do Not Produce Ideal Results
Ideal LPF
Blurring the image above
w/ an ideal lowpass filter…
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…distorts the results with
ringing or ghosting.
1999-2011 by Richard Alan Peters II
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Optimal Filter: The Gaussian
IFT
The Gaussian filter optimizes the uncertainty relation.
It provides the sharpest cutoff with the least ringing.
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One-Dimensional Gaussian
g ( x) 
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1
 2
e
 ( x   ) 2 2 2
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Two-Dimensional Gaussian
R = 512, C = 512
g ( r, c)  g ( r ) g (c)
c


1
 r  c 2
1
 r  c 2
e
e


If  and  are
different for r & c…
( r   r )2 ( y   c )

2 r2
2 c2
2
 c2 ( x   r ) 2   r2 ( y   c ) 2
2 r2 c2
…or if  and  are
the same for r & c.
r
 = 257,  = 64
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g ( r, c)   212 e

( r   )2  ( c   )2
2 2
1999-2011 by Richard Alan Peters II
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Optimal Filter: The Gaussian
Gaussian LPF
With a gaussian lowpass
filter, the image above …
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… is blurred without ringing
or ghosting.
1999-2011 by Richard Alan Peters II
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Compare with an “Ideal” LPF
Ideal LPF
Blurring the image above w/
an ideal lowpass filter…
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…distorts the results with
ringing or ghosting.
1999-2011 by Richard Alan Peters II
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Gaussian Lowpass Filter
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Gaussian Lowpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
SD filter sigma = 8
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
40
Gaussian Lowpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
SD filter sigma = 2
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
41
Gaussian Lowpass Filter
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
SD filter sigma = 8
Gaussian LPF in FD
42
Gaussian Lowpass Filter
Filtered Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
SD filter sigma = 8
Original Image
43
Gaussian Lowpass Filter
Filtered Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
SD filter sigma = 8
Original Image
44
Gaussian Highpass Filter
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1999-2011 by Richard Alan Peters II
45
Gaussian Highpass Filter
Multiply by
this, or …
Fourier Domain Rep.
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Image size: 512x512
FD notch sigma = 8
… convolve
by this
Spatial Representation
1999-2011 by Richard Alan Peters II
Central Profile
46
Gaussian Highpass Filter
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch sigma = 8
Gaussian HPF in FD
47
*signed image:
0 mapped to 128
Gaussian Highpass Filter
Filtered Image*
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch sigma = 8
Original Image
48
*signed image:
0 mapped to 128
Gaussian Highpass Filter
Original Image
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch sigma = 8
Filtered Image*
49
*signed image:
0 mapped to 128
Gaussian Highpass Filter
Positive Pixels
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Filtered Image*
1999-2011 by Richard Alan Peters II
Image size: 512x512
FD notch sigma = 8
Negative Pixels
50
*signed image:
0 mapped to 128
Another Gaussian Highpass Filter
original image
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filter power spectrum
1999-2011 by Richard Alan Peters II
filtered image*
51
Gaussian Bandpass Filter
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1999-2011 by Richard Alan Peters II
52
Gaussian Bandpass Filter
A bandpass filter is created by
(1) subtracting a FD radius 2 lowpass filtered image from a FD radius 1 lowpass
filtered image, where 2 < 1, or
(2) convolving the image with a mask that is the difference of the two lowpass masks.
FD LP mask with radius 1
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=
FD LP mask with radius 2
1999-2011 by Richard Alan Peters II
FD BP mask 1 - 2
53
*signed image:
0 mapped to 128
Gaussian Bandpass Filter
image LPF radius 1
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image LPF radius 2
1999-2011 by Richard Alan Peters II
image BPF radii 1, 2*
54
*signed image:
0 mapped to 128
Gaussian Bandpass Filter
filtered image*
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filter power spectrum
1999-2011 by Richard Alan Peters II
original image
55
*signed image:
0 mapped to 128
Ideal Bandpass Filter
original image
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filter power spectrum
1999-2011 by Richard Alan Peters II
filtered image*
56
Gaussian Bandpass Filter
Fourier Domain Rep.
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Spatial Representation
1999-2011 by Richard Alan Peters II
Image size: 512x512
sigma = 2 - sigma = 8
Central Profile
57
Gaussian Bandpass Filter
Original Image
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Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
sigma = 2 - sigma = 8
Gaussian BPF in FD
58
*signed image:
0 mapped to 128
Gaussian Bandpass Filter
Filtered Image*
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Filtered Power Spectrum
1999-2011 by Richard Alan Peters II
Image size: 512x512
sigma = 2 - sigma = 8
Original Image
59
*signed image:
0 mapped to 128
Gaussian Bandpass Filter
Positive Image
Pixels
Filtered
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Filtered Image*
1999-2011 by Richard Alan Peters II
Image size: 512x512
sigma = 2 - sigma = 8
Negative Pixels
60
Ideal vs. Gaussian Filters
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*signed image:
0 mapped to 128
Ideal Lowpass and Highpass Filters
Ideal LPF
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Original Image
1999-2011 by Richard Alan Peters II
Ideal HPF*
62
*signed image:
0 mapped to 128
Gaussian Lowpass and Highpass Filters
Gaussian LPF
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Original Image
1999-2011 by Richard Alan Peters II
Gaussian HPF*
63
*signed image:
0 mapped to 128
Ideal and Gaussian Bandpass Filters
Ideal BPF*
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Original Image
1999-2011 by Richard Alan Peters II
Gaussian BPF*
64
*signed image:
0 mapped to 128
Gaussian and Ideal Bandpass Filters
Gaussian BPF*
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Original Image
1999-2011 by Richard Alan Peters II
Ideal BPF*
65
Effects on Power Spectrum
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Power Spectrum and Phase of an Image
original image
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power spectrum
1999-2011 by Richard Alan Peters II
phase
67
Power Spectrum and Phase of a Blurred Image
blurred image
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power spectrum
1999-2011 by Richard Alan Peters II
phase
68
Power Spectrum and Phase of an Image
original image
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power spectrum
1999-2011 by Richard Alan Peters II
phase
69
Power Spectrum and Phase of a Sharpened Image
sharpened image
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power spectrum
1999-2011 by Richard Alan Peters II
phase
70