Image Enhancement in the Frequency Domain Chaiwoot Boonyasiriwat November 9, 2020
Discrete Fourier Transform ▪ DFT and IDFT of a signal x ( n 1 ), n 1 = 1, …, N 1, are ▪ DFT and IDFT of a 2D signal x ( n 1 ,n 2 ) are 2
Discrete Fourier Transform ▪ Complex sinusoidal signals and their power spectra. 3 Russ and Neal (2016, p. 328)
Image Reconstruction (a) Light micrograph of a stained skeletal muscle. (b) Power spectrum of (a). (c) Image reconstructed using the first three harmonics marked by the green dots. 4 Russ and Neal (2016, p. 331)
Image Reconstruction (a) Transmission electron microscopy (TEM) image atomic lattice of silicon. (b) Power spectrum of (a). (c) Dominant peaks in (b). (d) Image reconstructed using the dominant peaks in (c). 5 Russ and Neal (2016, p. 332)
Frequency-Selection Filters Original Low-pass filter High-pass filter 6 Russ and Neal (2016, p. 339)
Tapering To avoid the Gibbs phenomenon, the edge of the window should be tapered linearly or smoothly. 7 Russ and Neal (2016, p. 339)
Periodic Noise Removal “Periodic noise in an image is isolated into a few points in the frequency domain. So, the noise can be removed by removing these points in the frequency domain.” 8 Russ and Neal (2016, p. 345-348)
2D Convolution ▪ Applying a filter to an image in the spatial domain is the 2D convolution of an image with the filter kernel. ▪ A filter can be applied more efficiently in the frequency domain by the multiplication of the Fourier transforms of the image and filter kernel. ▪ The convolution of signals x and y can be performed using DFT and IDFT as follows. 9 Russ and Neal (2016, p. 352)
2D Convolution a*b F[a] F[b] F -1 [F[a]F[b]] 10 Russ and Neal (2016, p. 354)
Wiener Deconvolution ▪ Let X , Y , and H be the DFT of x , y , and h . ▪ Here h is the smoothing filter kernel, x is the original image, and y is the filtered image given by ▪ Suppose we know y and h . We can estimate the orginal image as follows. where H * is the complex conjugate of H , and is a small constant to avoid division by a very small number when H 2 is close to zero. 11 Russ and Neal (2016, p. 352)
Wiener Deconvolution (a) Original blurred image (b) Normal deconvolution with = 0 (c) Wiener deconvolution with a small (d) Wiener deconvolution with large enough to suppress noise 12 Russ and Neal (2016, p. 360)
2D Cross-Correlation ▪ Cross-correlation is frequently used in image processing as a tool for pattern matching, e.g., searching for vehicles from traffic camera videos, or tracking the motion of storms in weather satellite images. ▪ The 2D cross-correlation of two continuous functions g and h is given by ▪ The cross-correlation is more efficiently performed in the frequency domain by where G = F [ g ], H = F [ h ], and H * is complex conjugate of H . 13 Russ and Neal (2016, p. 368-369)
2D Cross-Correlation (a) Original image with bubbles to be located. (b) Cross-correlation results showing spikes at the bubble positions. (c) The result of a top-hat filter superimposed on the cross- correlation image. 14 Russ and Neal (2016, p. 370)
2D Cross-Correlation (a) and (b) Schmidt wide-field camera images taken several days apart. (c) Absolute difference image after automatic alignment by cross-correlation, showing a moving comet. 15 Russ and Neal (2016, p. 371)
2D Cross-Correlation (a) and (b) original light microscope images at different focal depths (top and bottom focus) (c) Superposition of the two images (edge enhanced) in red and green channels to show lateral image shift (d) Extended focus result after alignment of all images in the set. 16 Russ and Neal (2016, p. 371)
2D Autocorrelation ▪ Autocorrelation is a cross-correlation of an image with itself. It is typically used to find repetitive structures. (a) Image of latex spheres self-organized in an approximately hexagonal array. (b) Autocorrelation image showing the pattern of neighbor distance and direction. (c) Measurement of the circularly averaged radial profile. 17
References ▪ J. C. Russ and F. B. Neal, 2016, The Image Processing Handbook, 7 th edition, CRC Press.
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