eece253_08_frequencyfiltering
Post on 27-Oct-2014
10 Views
Preview:
TRANSCRIPT
EECE\CS 253 Image Processing
Richard Alan Peters II
Department of Electrical Engineering and Computer Science
Fall Semester 2007
Lecture NotesLecture Notes: Frequency Filtering
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 send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.
April 7, 2023 2April 7, 2023 2 1999-2007 by Richard Alan Peters II
Convolution Property of the Fourier Transform
.}{
Moreover,
.}{
Then,
).,( and ),( TransformsFourier
have ),( and ),( functionsLet
GFgf
GFgf
vuGvuF
crgcrf
F
F 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
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
* = convolution · = multiplication
April 7, 2023 3April 7, 2023 3 1999-2007 by Richard Alan Peters II
Convolution via Fourier Transform
Image & Mask Transforms
Pixel-wise Product
Inverse Transform
April 7, 2023 4April 7, 2023 4 1999-2007 by Richard Alan Peters II
1. Read the image from a file into a variable, say I.
2. Read in or create the convolution mask, h.
3. Compute the sum of the mask: s = sum(sum(h));
4. If s == 0, set s = 1;
5. Create: H = zeros(size(I));
6. Copy h into the middle of H.
7. Shift H into position: H = ifftshift(H);
8. Take the 2D FT of I and H: FI=fft2(I); FH=fft2(H);
9. Pointwise multiply the FTs: FJ=FI.*FH;
10. Compute the inverse FT: J = real(ifft2(FJ));
11. Normalize the result: J = J/s;
How to Convolve via FT in Matlab
For color images you may need to do each step for each band separately.
For color images you may need to do each step for each band separately.
The mask is usually 1-band
The mask is usually 1-band
April 7, 2023 5April 7, 2023 5 1999-2007 by Richard Alan Peters II
Coordinate Origin of the FFT
Center =(floor(R/2)+1, floor(C/2)+1)
Center =(floor(R/2)+1, floor(C/2)+1)
Even EvenOdd Odd
Image Origin Weight Matrix OriginImage Origin Weight Matrix Origin
After FFT shift After IFFT shiftAfter FFT shift After IFFT shift
April 7, 2023 6April 7, 2023 6 1999-2007 by Richard Alan Peters II
5 6 4
8 9 7
2 3 1
1 2 3
4 5 6
7 8 9
Matlab’s fftshift and ifftshift
J = fftshift(I):
I (1,1) J ( R/2 +1, C/2 +1)
I = ifftshift(J):
J ( R/2 +1, C/2 +1) I (1,1)
where x = floor(x) = the largest integer smaller than x.
1 2 3
4 5 6
7 8 9
5 6 4
8 9 7
2 3 1
April 7, 2023 7April 7, 2023 7 1999-2007 by Richard Alan Peters II
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.The values of the output image are all non-negative.
April 7, 2023 8April 7, 2023 8 1999-2007 by Richard Alan Peters II
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.The values of the output image positive & negative.
April 7, 2023 9April 7, 2023 9 1999-2007 by Richard Alan Peters II
Convolution Property of the Fourier Transform
.
by compute can weThus
.}{
Moreover,
.}{
Then,
).,( and ),( TransformsFourier
have ),( and ),( functionsLet
1 GFgf
gf
GFgf
GFgf
vuGvuF
crgcrf
-
F
F
FThe 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
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
* = convolution · = multiplication
Recall:Recall:
April 7, 2023 10April 7, 2023 10 1999-2007 by Richard Alan Peters II
Ideal Lowpass Filter
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Image size: 512x512FD filter radius: 16
Image size: 512x512FD filter radius: 16
Multiply by this, or …
… convolve by this
April 7, 2023 11April 7, 2023 11 1999-2007 by Richard Alan Peters II
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Ideal Lowpass Filter Image size: 512x512FD filter radius: 8
Image size: 512x512FD filter radius: 8
Multiply by this, or …
… convolve by this
April 7, 2023 12April 7, 2023 12 1999-2007 by Richard Alan Peters II
Power Spectrum and Phase of an Image
Consider the image below:
Consider the image below:
Original ImageOriginal Image Power SpectrumPower Spectrum PhasePhase
April 7, 2023 13April 7, 2023 13 1999-2007 by Richard Alan Peters II
Ideal LPF in FDIdeal LPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum
Ideal Lowpass Filter Image size: 512x512FD filter radius: 16
Image size: 512x512FD filter radius: 16
April 7, 2023 14April 7, 2023 14 1999-2007 by Richard Alan Peters II
Original ImageOriginal ImageFiltered ImageFiltered Image Filtered Power SpectrumFiltered Power Spectrum
Ideal Lowpass Filter Image size: 512x512FD filter radius: 16
Image size: 512x512FD filter radius: 16
April 7, 2023 15April 7, 2023 15 1999-2007 by Richard Alan Peters II
Ideal Highpass Filter
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Image size: 512x512FD notch radius: 16
Image size: 512x512FD notch radius: 16
Multiply by this, or …
Multiply by this, or …
… convolve by this
… convolve by this
April 7, 2023 16April 7, 2023 16 1999-2007 by Richard Alan Peters II
Ideal HPF in FDIdeal HPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum
Ideal Highpass Filter Image size: 512x512FD notch radius: 16
Image size: 512x512FD notch radius: 16
April 7, 2023 17April 7, 2023 17 1999-2007 by Richard Alan Peters II
Original ImageOriginal ImageFiltered Image*Filtered Image*Filtered Power SpectrumFiltered Power Spectrum
Ideal Highpass Filter Image size: 512x512FD notch radius: 16
Image size: 512x512FD notch radius: 16
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 18April 7, 2023 18 1999-2007 by Richard Alan Peters II
Filtered Image*Filtered Image*Positive PixelsPositive Pixels Negative PixelsNegative Pixels
Ideal Highpass Filter Image size: 512x512FD notch radius: 16
Image size: 512x512FD notch radius: 16
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 19April 7, 2023 19 1999-2007 by Richard Alan Peters II
The Uncertainty Relation
FTFT
space frequency
FTFT
space frequency
A small object in space has a large frequency extent and vice-versa.
A small object in space has a large frequency extent and vice-versa.
April 7, 2023 20April 7, 2023 20 1999-2007 by Richard Alan Peters II
The Uncertainty Relation
Recall: a symmetric pair of impulses in the frequency domain becomes a sinusoid in the spatial domain.
Recall: a symmetric pair of impulses in the frequency domain becomes a sinusoid in the spatial domain.
A symmetric pair of lines in the frequency domain becomes a sinusoidal line in the spatial domain.
A symmetric pair of lines in the frequency domain becomes a sinusoidal line in the spatial domain.
IFTIFT
spacefrequency
la
rge
ext
en
t
small extent
sm
all e
xten
t
large extent
IFTIFT
spacefrequency
small extent
s
ma
ll e
xte
nt
la
rge
exte
nt
large extent
April 7, 2023 21April 7, 2023 21 1999-2007 by Richard Alan Peters II
Ideal Filters Do Not Produce Ideal Results
A sharp cutoff in the frequency domain…
A sharp cutoff in the frequency domain…
…causes ringing in the spatial domain.
…causes ringing in the spatial domain.
IFTIFT
April 7, 2023 22April 7, 2023 22 1999-2007 by Richard Alan Peters II
Ideal Filters Do Not Produce Ideal Results
Ideal LPFIdeal LPF
Blurring the image above w/ an ideal lowpass filter…
Blurring the image above w/ an ideal lowpass filter…
…distorts the results with ringing or ghosting.
…distorts the results with ringing or ghosting.
April 7, 2023 23April 7, 2023 23 1999-2007 by Richard Alan Peters II
Optimal Filter: The Gaussian
The Gaussian filter optimizes the uncertainty relation. It provides the sharpest cutoff with the least ringing.
The Gaussian filter optimizes the uncertainty relation. It provides the sharpest cutoff with the least ringing.
IFTIFT
April 7, 2023 24April 7, 2023 24 1999-2007 by Richard Alan Peters II
One-Dimensional Gaussian
22 2)(
21)(
xexg
April 7, 2023 25April 7, 2023 25 1999-2007 by Richard Alan Peters II
Two-Dimensional GaussianIf and are different for r & c…
If and are different for r & c…
…or if and are the same for r & c.
…or if and are the same for r & c.
r
cR = 512, C = 512
= 257, = 64
April 7, 2023 26April 7, 2023 26 1999-2007 by Richard Alan Peters II
Gaussian LPFGaussian LPF
With a gaussian lowpass filter, the image above …
With a gaussian lowpass filter, the image above …
… is blurred without ringing or ghosting.
… is blurred without ringing or ghosting.
Optimal Filter: The Gaussian
April 7, 2023 27April 7, 2023 27 1999-2007 by Richard Alan Peters II
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Image size: 512x512SD filter sigma = 8
Image size: 512x512SD filter sigma = 8Gaussian Lowpass Filter
Multiply by this, or …
Multiply by this, or …
… convolve by this
… convolve by this
April 7, 2023 28April 7, 2023 28 1999-2007 by Richard Alan Peters II
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Gaussian Lowpass Filter Image size: 512x512SD filter sigma = 2
Image size: 512x512SD filter sigma = 2
Multiply by this, or …
Multiply by this, or …
… convolve by this
… convolve by this
April 7, 2023 29April 7, 2023 29 1999-2007 by Richard Alan Peters II
Gaussian LPF in FDGaussian LPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum
Image size: 512x512SD filter sigma = 8
Image size: 512x512SD filter sigma = 8Gaussian Lowpass Filter
April 7, 2023 30April 7, 2023 30 1999-2007 by Richard Alan Peters II
Original ImageOriginal ImageFiltered ImageFiltered Image Filtered Power SpectrumFiltered Power Spectrum
Gaussian Lowpass Filter Image size: 512x512SD filter sigma = 8
Image size: 512x512SD filter sigma = 8
April 7, 2023 31April 7, 2023 31 1999-2007 by Richard Alan Peters II
Gaussian Highpass Filter
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Image size: 512x512FD notch sigma = 8
Image size: 512x512FD notch sigma = 8
Multiply by this, or …
Multiply by this, or …
… convolve by this
… convolve by this
April 7, 2023 32April 7, 2023 32 1999-2007 by Richard Alan Peters II
Gaussian HPF in FDGaussian HPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum
Gaussian Highpass Filter Image size: 512x512FD notch sigma = 8
Image size: 512x512FD notch sigma = 8
April 7, 2023 33April 7, 2023 33 1999-2007 by Richard Alan Peters II
Original ImageOriginal ImageFiltered Power SpectrumFiltered Power Spectrum
Gaussian Highpass Filter Image size: 512x512FD notch sigma = 8
Image size: 512x512FD notch sigma = 8
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
Filtered Image*Filtered Image*
April 7, 2023 34April 7, 2023 34 1999-2007 by Richard Alan Peters II
Negative PixelsNegative PixelsFiltered ImageFiltered ImagePositive PixelsPositive Pixels
Gaussian Highpass Filter Image size: 512x512FD notch sigma = 8
Image size: 512x512FD notch sigma = 8
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
Filtered Image*Filtered Image*
April 7, 2023 35April 7, 2023 35 1999-2007 by Richard Alan Peters II
Original ImageOriginal Image Ideal HPF*Ideal HPF*Ideal LPFIdeal LPF
Comparison of Ideal and Gaussian Filters
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 36April 7, 2023 36 1999-2007 by Richard Alan Peters II
Original ImageOriginal Image Gaussian HPF*Gaussian HPF*Gaussian LPFGaussian LPF
Comparison of Ideal and Gaussian Filters
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 37April 7, 2023 37 1999-2007 by Richard Alan Peters II
Another Highpass Filter
original imageoriginal image filter power spectrumfilter power spectrum filtered image*filtered image*
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 38April 7, 2023 38 1999-2007 by Richard Alan Peters II
Ideal Bandpass Filter
original imageoriginal image filter power spectrumfilter power spectrum filtered image*filtered image*
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 39April 7, 2023 39 1999-2007 by Richard Alan Peters II
Gaussian Bandpass Filter
Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile
Image size: 512x512sigma = 2 - sigma = 8
Image size: 512x512sigma = 2 - sigma = 8
April 7, 2023 40April 7, 2023 40 1999-2007 by Richard Alan Peters II
Gaussian BPF in FDGaussian BPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum
Gaussian Bandpass Filter Image size: 512x512sigma = 2 - sigma = 8
Image size: 512x512sigma = 2 - sigma = 8
April 7, 2023 41April 7, 2023 41 1999-2007 by Richard Alan Peters II
Original ImageOriginal ImageFiltered Image*Filtered Image*Filtered Power SpectrumFiltered Power Spectrum
Gaussian Bandpass Filter Image size: 512x512sigma = 2 - sigma = 8
Image size: 512x512sigma = 2 - sigma = 8
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 42April 7, 2023 42 1999-2007 by Richard Alan Peters II
Filtered Image*Filtered Image*Negative PixelsNegative PixelsFiltered ImageFiltered ImagePositive PixelsPositive Pixels
Gaussian Bandpass Filter Image size: 512x512sigma = 2 - sigma = 8
Image size: 512x512sigma = 2 - sigma = 8
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 43April 7, 2023 43 1999-2007 by Richard Alan Peters II
Original ImageOriginal Image Gaussian BPF*Gaussian BPF*Ideal BPF*Ideal BPF*
Comparison of Ideal and Gaussian Filters
*signed image; 0 mapped to 128
*signed image; 0 mapped to 128
April 7, 2023 44April 7, 2023 44 1999-2007 by Richard Alan Peters II
Power Spectrum and Phase of a Blurred Image
blurred imageblurred image power spectrumpower spectrum phasephase
April 7, 2023 45April 7, 2023 45 1999-2007 by Richard Alan Peters II
Power Spectrum and Phase of an Image
original imageoriginal image power spectrumpower spectrum phasephase
April 7, 2023 46April 7, 2023 46 1999-2007 by Richard Alan Peters II
Power Spectrum and Phase of a Sharpened Image
sharpened imagesharpened image power spectrumpower spectrum phasephase
top related