3. Lecture Fourier Transformation Sampling Some slides taken from - - PowerPoint PPT Presentation
3. Lecture Fourier Transformation Sampling Some slides taken from Digital Image Processing: An Algorithmic Introduction using Java, Wilhelm Burger and Mark James Burge Separability The 2D DFT can be separated in two 1D DFT's: M 1 N
Some slides taken from “Digital Image Processing: An Algorithmic Introduction using Java”, Wilhelm Burger and Mark James Burge
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M¡1 X x=0 N¡1 X y=0
M¡1 X x=0 2 4e¡j2¼ux=M 1
N¡1 X y=0
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M¡1 X u=0 N¡1 X v=0
M¡1 X u=0 2 4ej2¼ux=M N¡1 X v=0
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fft2(magic(3)) ans = 45.0000 0 0 0 + 0.0000i 13.5000 + 7.7942i 0.0000 - 5.1962i 0 - 0.0000i 0.0000 + 5.1962i 13.5000 - 7.7942i fft(fft(magic(3)).').' ans = 45.0000 0 0 0 + 0.0000i 13.5000 + 7.7942i 0.0000 - 5.1962i 0 - 0.0000i 0.0000 + 5.1962i 13.5000 - 7.7942i
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11 DFT image scaling. The rectangular pulse in the image function (a–c) creates a strongly
(d–f), as in the one- dimensional case. Stretching the image causes the spectrum to contract and vice versa.
12 DFT—oriented, repetitive
function (a–c) contains patterns with three dominant orientations, which appear as pairs of corresponding frequency spots in the spectrum (c–f). Enlarging the image causes the spectrum to contract.
13 DFT —image rotation. The original image (a) is rotated by 15deg (b) and 30deg (c). The corresponding (squared) spectrum turns in the same direction and by exactly the same amount (d–f).
14 DFT—superposition of image patterns. Strong,
(a–c) are easy to identify in the corresponding spectrum (d–f). Notice the broadband effects caused by straight structures, such as the dark beam on the wall in (b, e).
15 DFT—natural image
repetitive structures in natural images (a–c) that are also visible in the corresponding spectrum (d–f).
16 DFT—natural image patterns with no dominant
patterns contained in these images (a–c) have no common orientation or sufficiently regular spacing to stand out locally in the orresponding Fourier spectra (d–f).
17 DFT of a print pattern. The regular diagonally oriented raster pattern (a, b) is clearly visible in the corresponding power spectrum (c). It is possible (at least in principle) to remove such patterns by erasing these peaks in the Fourier spectrum and reconstructing the smoothed image from the modified spectrum using the inverse DFT.
18 Correcting the geometry of the 2D spectrum. Original image (a) with dominant oriented patterns that show up as clear peaks in the corresponding spectrum (b). Because the image and the spectrum are not square (M = N), orientations in the image are not the same as in the actual spectrum (b). After the spectrum is scaled to square size (c), we can clearly observe that the cylinders of this (Harley-Davidson V-Rod) engine are really spaced at a 60◦ angle.
19 Effects of periodicity in the 2D
transform is computed under the implicit assumption that the image signal is periodic along both dimensions (top). Large differences in intensity at
most notably in the vertical direction— lead to broad-band signal components that in this case appear as a bright line along the spectrum’s vertical axis (bottom).
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M¡1 X m=0
M¡1 X m=0 N¡1 X n=0
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M¡1 X m=0
M¡1 X m=0 N¡1 X n=0
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A X x f(x)
Square pixels
spatial Fourier spatial Fourier spatial Fourier spatial Fourier
Bilinear interpolation Gaussian reconstruction filter Perfect reconstruction filter
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Harmon & Julesz 1973
square pixels Gaussian reconstruction
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