Image Processing Frequency Filtering Instructor: Juyong Zhang [email protected] juyong.

Image Processing

Frequency Filtering

Instructor: Juyong [email protected]

http://staff.ustc.edu.cn/~juyong

mailto:[email protected]

http://staff.ustc.edu.cn/~juyong

Convolution Property of the Fourier Transform

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

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Convolution via Fourier Transform

Image & Mask Transforms

Pixel-wise Product

Inverse Transform

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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(h(:));

4. If s == 0, set s = 1;

5. Replace h with h = h/s;

6. Create: H = zeros(size(I));

7. Copy h into the middle of H.

8. Shift H into position: H = ifftshift(H);

9. Take the 2D FT of I and H: FI=fft2(I); FH=fft2(H);

10. Pointwise multiply the FTs: FJ=FI.*FH;

11. Compute the inverse FT: J = real(ifft2(FJ));

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

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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

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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

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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.

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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.The values of the output image positive & negative.

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Convolution Property of the Fourier Transform



* = convolution · = multiplication

Recall:Recall:

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Ideal Lowpass Filter

Ideal Lowpass Filter

Fourier Domain Rep.Fourier Domain Rep. Spatial RepresentationSpatial Representation Central ProfileCentral Profile

Image size: 512x512FD filter radius: 16


Multiply by this, or …

… convolve by this

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Spatial RepresentationSpatial Representation Central ProfileCentral Profile

Ideal Lowpass Filter Image size: 512x512FD filter radius: 8




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Fourier Domain Rep.Fourier Domain Rep.

Power Spectrum and Phase of an Image

Consider the image below:

Consider the image below:

Original ImageOriginal Image Power SpectrumPower Spectrum PhasePhase

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Ideal LPF in FDIdeal LPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum



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Filtered Power SpectrumFiltered Power Spectrum



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Filtered ImageFiltered ImageOriginal ImageOriginal Image

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Ideal Highpass Filter

Ideal Highpass Filter


Image size: 512x512FD notch radius: 16






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Ideal HPF in FDIdeal HPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum

Ideal Highpass Filter Image size: 512x512FD notch radius: 16


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Original ImageOriginal Image Filtered Image*Filtered Image*Filtered Power SpectrumFiltered Power Spectrum



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*signed image: 0 mapped to 128


Filtered Image*Filtered Image*Positive PixelsPositive Pixels Negative PixelsNegative Pixels





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Ideal Bandpass Filter


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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 1FD LP mask with radius 1 FD LP mask with radius 2

FD LP mask with radius 2 FD BP mask 1 - 2FD BP mask 1 - 2

- =


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image LPF radius 1image LPF radius 1 image LPF radius 2

image LPF radius 2 image BPF radii 1, 2*image BPF radii 1, 2*


original image*original image* filter power spectrumfilter power spectrum filtered imagefiltered image

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A Different Ideal Bandpass Filter

original imageoriginal image filter power spectrumfilter power spectrum filtered image*filtered image*

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The Optimal Filter

The Uncertainty Relation

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.

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space frequency

FTFT

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.

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spacespacefrequency

small extent

s

ma

ll e

xte

nt

la

rge

exte

nt

large extent

IFTIFT

spacespacefrequency

la

rge

ext

en

t

small extent

sm

all e

xten

t

large extent

IFTIFT

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

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Ideal Filters Do Not Produce Ideal Results

Ideal LPFIdeal LPF

Blurring the image above with an ideal lowpass filter…


…distorts the results with ringing or ghosting.


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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

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One-Dimensional Gaussian

22 2)(

21)(

xexg

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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

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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

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Compare with an “Ideal” LPF

Ideal LPFIdeal LPF





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Gaussian Lowpass Filter


Image size: 512x512SD filter sigma = 8

Image size: 512x512SD filter sigma = 8Gaussian Lowpass Filter





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Gaussian LPF in FDGaussian LPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum



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Filtered ImageFiltered ImageOriginal ImageOriginal Image Filtered Power SpectrumFiltered Power Spectrum



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Gaussian Highpass Filter

Gaussian Highpass Filter


Image size: 512x512FD notch sigma = 8






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Gaussian HPF in FDGaussian HPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum

Gaussian Highpass Filter Image size: 512x512FD notch sigma = 8


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Filtered Image*Filtered Image*Filtered Power SpectrumFiltered Power Spectrum





Original ImageOriginal Image

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Negative PixelsNegative PixelsPositive PixelsPositive Pixels



Filtered Image*Filtered Image*

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Another Gaussian Highpass Filter


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Gaussian Bandpass Filter


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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 1FD LP mask with radius 1 FD LP mask with radius 2

FD LP mask with radius 2 FD BP mask 1 - 2FD BP mask 1 - 2

- =



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image LPF radius 1image LPF radius 1 image LPF radius 2

image LPF radius 2 image BPF radii 1, 2*image BPF radii 1, 2*



Image size: 512x512sigma = 2 - sigma = 8


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Gaussian BPF in FDGaussian BPF in FDOriginal ImageOriginal Image Power SpectrumPower Spectrum

Gaussian Bandpass Filter Image size: 512x512sigma = 2 - sigma = 8


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Original ImageOriginal ImageFiltered Image*Filtered Image*Filtered Power SpectrumFiltered Power Spectrum



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Filtered Image*Filtered Image* Negative PixelsNegative PixelsFiltered ImageFiltered ImagePositive PixelsPositive Pixels



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Ideal vs. Gaussian Filters

Original ImageOriginal Image Ideal HPF*Ideal HPF*Ideal LPFIdeal LPF

Ideal Lowpass and Highpass Filters

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Original ImageOriginal Image Gaussian HPF*Gaussian HPF*Gaussian LPFGaussian LPF

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Gaussian Lowpass and Highpass Filters

Original ImageOriginal Image Gaussian BPF*Gaussian BPF*Ideal BPF*Ideal BPF*

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Ideal and Gaussian Bandpass Filters

Original ImageOriginal Image Ideal BPF*Ideal BPF*Gaussian BPF*Gaussian BPF*

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Gaussian and Ideal Bandpass Filters

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Effects on Power Spectrum


original imageoriginal image power spectrumpower spectrum phasephase

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Power Spectrum and Phase of a Blurred Image

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blurred imageblurred image power spectrumpower spectrum phasephase


original imageoriginal image power spectrumpower spectrum phasephase

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Power Spectrum and Phase of a Sharpened Image

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power spectrumpower spectrum phasephasesharpened imagesharpened image

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Thanks!

Image Processing Frequency Filtering Instructor: Juyong Zhang [email protected] juyong.

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