Image Processing Frequency Filtering Instructor: Juyong Zhang juyong@ustc.edu.cn juyong.

transcript

Image Processing

Frequency Filtering

Instructor: Juyong Zhangjuyong@ustc.edu.cn

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.

The mask is usually 1-band

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Coordinate Origin of the FFT

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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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.

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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

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:

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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:

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

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

space frequency

A small object in space has a large frequency extent and vice-versa.

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space frequency

The Uncertainty Relation

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.

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spacespacefrequency

small extent

large extent

IFTIFT

spacespacefrequency

small extent

large extent

IFTIFT

Ideal Filters Do Not Produce Ideal Results

A sharp cutoff in the frequency 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.

IFTIFT

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One-Dimensional Gaussian

22 2)(

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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.

cR = 512, C = 512

= 257, = 64

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Gaussian LPFGaussian LPF

With a gaussian lowpass filter, the image above …

… is blurred without ringing or ghosting.

Optimal Filter: The Gaussian

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Compare with an “Ideal” LPF

Ideal LPFIdeal LPF

Blurring the image above with an ideal lowpass filter…

…distorts the results with ringing or ghosting.

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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

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 juyong@ustc.edu.cn juyong.

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