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Image Processing Frequency Filtering Instructor: Juyong Zhang juyong@ustc.edu.cn http://staff.ustc.edu.cn/~juyong

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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 5/5/20152

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Convolution via Fourier Transform Image & Mask Transforms Pixel-wise Product Inverse Transform 5/5/20153

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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 5/5/20154

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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 Odd Image OriginWeight Matrix OriginImage OriginWeight Matrix Origin After FFT shiftAfter IFFT shiftAfter FFT shiftAfter IFFT shift 5/5/20155

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564 897 231 123 456 789 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. 123 456 789 564 897 231 5/5/20156

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Blurring: Averaging / Lowpass Filtering Blurring results from: l 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. l 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/ (u 2 +v 2 ). The values of the output image are all non-negative. 5/5/20157

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Sharpening: Differencing / Highpass Filtering Sharpening results from adding to the image, a copy of itself that has been: l 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. l Highpass filtered in the frequency domain: – High frequencies are enhanced or amplified. – Individual frequency components are multiplied by an increasing function of such as = (u 2 +v 2 ), where is a constant. The values of the output image positive & negative. 5/5/20158

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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: 5/5/20159

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10 Ideal Lowpass Filter

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Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 FD filter radius: 16 Image size: 512x512 FD filter radius: 16 Multiply by this, or … … convolve by this 5/5/201511

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Spatial Representation Central Profile Ideal Lowpass Filter Image size: 512x512 FD filter radius: 8 Image size: 512x512 FD filter radius: 8 Multiply by this, or … … convolve by this 5/5/201512 Fourier Domain Rep.

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Power Spectrum and Phase of an Image Consider the image below: Original Image Power Spectrum Phase 5/5/201513

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Ideal LPF in FD Original Image Power Spectrum Ideal Lowpass Filter Image size: 512x512 FD filter radius: 16 Image size: 512x512 FD filter radius: 16 5/5/201514

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Filtered Power Spectrum Ideal Lowpass Filter Image size: 512x512 FD filter radius: 16 Image size: 512x512 FD filter radius: 16 5/5/201515 Filtered Image Original Image

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5/5/201516 Ideal Highpass Filter

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Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 FD notch radius: 16 Image size: 512x512 FD notch radius: 16 Multiply by this, or … … convolve by this 5/5/201517

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Ideal HPF in FD Original Image Power Spectrum Ideal Highpass Filter Image size: 512x512 FD notch radius: 16 Image size: 512x512 FD notch radius: 16 5/5/201518

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Original Image Filtered Image* Filtered Power Spectrum Ideal Highpass Filter Image size: 512x512 FD notch radius: 16 Image size: 512x512 FD notch radius: 16 5/5/201519 *signed image: 0 mapped to 128

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Filtered Image* Positive Pixels Negative Pixels Ideal Highpass Filter Image size: 512x512 FD notch radius: 16 Image size: 512x512 FD notch radius: 16 *signed image: 0 mapped to 128 5/5/201520

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5/5/201521 Ideal Bandpass Filter

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5/5/201522 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 FD LP mask with radius 2 FD BP mask 1 - 2 - =

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Ideal Bandpass Filter 5/5/201523 *signed image: 0 mapped to 128 image LPF radius 1 image LPF radius 2 image BPF radii 1, 2 *

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Ideal Bandpass Filter original image* filter power spectrum filtered image 5/5/201524 *signed image: 0 mapped to 128

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A Different Ideal Bandpass Filter original image filter power spectrum filtered image* 5/5/201525 *signed image: 0 mapped to 128

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5/5/201526 The Optimal Filter

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The Uncertainty Relation FT spacefrequency A small object in space has a large frequency extent and vice-versa. 5/5/201527 spacefrequency FT

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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. 5/5/201528 space frequency small extent large extent IFT space frequency large extent small extent large extent IFT

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Ideal Filters Do Not Produce Ideal Results A sharp cutoff in the frequency domain… …causes ringing in the spatial domain. IFT 5/5/201529

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Ideal Filters Do Not Produce Ideal Results Ideal LPF Blurring the image above with an ideal lowpass filter… …distorts the results with ringing or ghosting. 5/5/201530

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Optimal Filter: The Gaussian The Gaussian filter optimizes the uncertainty relation. It provides the sharpest cutoff with the least ringing. IFT 5/5/201531

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One-Dimensional Gaussian 5/5/201532

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Two-Dimensional Gaussian If and are different for r & c … …or if and are the same for r & c. r c R = 512, C = 512 = 257, = 64 5/5/201533

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Gaussian LPF With a gaussian lowpass filter, the image above … … is blurred without ringing or ghosting. Optimal Filter: The Gaussian 5/5/201534

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Compare with an “Ideal” LPF Ideal LPF Blurring the image above with an ideal lowpass filter… …distorts the results with ringing or ghosting. 5/5/201535

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5/5/201536 Gaussian Lowpass Filter

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Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 SD filter sigma = 8 Image size: 512x512 SD filter sigma = 8 Gaussian Lowpass Filter Multiply by this, or … … convolve by this 5/5/201537

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Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 SD filter sigma = 2 Image size: 512x512 SD filter sigma = 2 Multiply by this, or … … convolve by this 5/5/201538 Gaussian Lowpass Filter

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Gaussian LPF in FD Original Image Power Spectrum Image size: 512x512 SD filter sigma = 8 Image size: 512x512 SD filter sigma = 8 5/5/201539 Gaussian Lowpass Filter

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Filtered Image Original Image Filtered Power Spectrum Image size: 512x512 SD filter sigma = 8 Image size: 512x512 SD filter sigma = 8 5/5/201540 Gaussian Lowpass Filter

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5/5/201541 Gaussian Highpass Filter

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Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 FD notch sigma = 8 Image size: 512x512 FD notch sigma = 8 Multiply by this, or … … convolve by this 5/5/201542

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Gaussian HPF in FD Original Image Power Spectrum Gaussian Highpass Filter Image size: 512x512 FD notch sigma = 8 Image size: 512x512 FD notch sigma = 8 5/5/201543

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Filtered Image* Filtered Power Spectrum Gaussian Highpass Filter Image size: 512x512 FD notch sigma = 8 Image size: 512x512 FD notch sigma = 8 *signed image: 0 mapped to 128 Original Image 5/5/201544

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Negative Pixels Positive Pixels Gaussian Highpass Filter Image size: 512x512 FD notch sigma = 8 Image size: 512x512 FD notch sigma = 8 Filtered Image* 5/5/201545 *signed image: 0 mapped to 128

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Another Gaussian Highpass Filter original image filter power spectrum filtered image * 5/5/201546 *signed image: 0 mapped to 128

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5/5/201547 Gaussian Bandpass Filter

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5/5/201548 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 FD LP mask with radius 2 FD BP mask 1 - 2 - =

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Ideal Bandpass Filter original image filter power spectrum filtered image* 5/5/201549 *signed image: 0 mapped to 128

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Gaussian Bandpass Filter 5/5/201550 *signed image: 0 mapped to 128 image LPF radius 1 image LPF radius 2 image BPF radii 1, 2 *

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Gaussian Bandpass Filter Fourier Domain Rep. Spatial Representation Central Profile Image size: 512x512 sigma = 2 - sigma = 8 Image size: 512x512 sigma = 2 - sigma = 8 5/5/201551

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Gaussian BPF in FD Original Image Power Spectrum Gaussian Bandpass Filter Image size: 512x512 sigma = 2 - sigma = 8 Image size: 512x512 sigma = 2 - sigma = 8 5/5/201552

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Original Image Filtered Image * Filtered Power Spectrum Gaussian Bandpass Filter Image size: 512x512 sigma = 2 - sigma = 8 Image size: 512x512 sigma = 2 - sigma = 8 5/5/201553 *signed image: 0 mapped to 128

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Filtered Image* Negative Pixels Filtered Image Positive Pixels Gaussian Bandpass Filter Image size: 512x512 sigma = 2 - sigma = 8 Image size: 512x512 sigma = 2 - sigma = 8 5/5/201554 *signed image: 0 mapped to 128

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5/5/201555 Ideal vs. Gaussian Filters

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Original Image Ideal HPF * Ideal LPF Ideal Lowpass and Highpass Filters 5/5/201556 *signed image: 0 mapped to 128

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Original Image Gaussian HPF* Gaussian LPF 5/5/201557 *signed image: 0 mapped to 128 Gaussian Lowpass and Highpass Filters

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Original Image Gaussian BPF * Ideal BPF * 5/5/201558 *signed image: 0 mapped to 128 Ideal and Gaussian Bandpass Filters

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Original Image Ideal BPF* Gaussian BPF* 5/5/201559 *signed image: 0 mapped to 128 Gaussian and Ideal Bandpass Filters

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5/5/201560 Effects on Power Spectrum

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Power Spectrum and Phase of an Image original image power spectrum phase 5/5/201561

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Power Spectrum and Phase of a Blurred Image 5/5/201562 blurred image power spectrum phase

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Power Spectrum and Phase of an Image original image power spectrum phase 5/5/201563

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Power Spectrum and Phase of a Sharpened Image 5/5/201564 power spectrum phase sharpened image

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5 May 201565 Thanks!

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