Fourier Transformation. Sampling. Some slides taken from “Digital Image
Processing: An Algorithmic Introduction using Java”,. Wilhelm Burger and Mark
James ...
3. Lecture Fourier Transformation Sampling Some slides taken from “Digital Image Processing: An Algorithmic Introduction using Java”, Wilhelm Burger and Mark James Burge
Separability ² The 2D DFT can be separated in two 1D DFT's: ¡1 NX ¡1 1 MX F (u; v) = f (x; y)e¡j2¼(ux=M +vy=N ) MN x=0 y=0 2 3 M ¡1 N ¡1 1 X 4 ¡j2¼ux=M 1 X = e f (x; y)e¡j2¼vy=N 5 M x=0 N y=0
² The DFT can be obtained in two successive applications of 1D transforms
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Separability
² First compute the 1D DFT's for all the rows ² Second compute the 1D DFT's for all the columns ² And each of the 1D DFT's can be carried out as FFT of course 3
Separability ² Also the inverse transform can be separated: f (x; y) = =
M ¡1 NX ¡1 X
F (u; v)ej2¼(ux=M +vy=N )
u=0 v=0 2 3 M ¡1 N ¡1 X X 4ej2¼ux=M F (u; v)ej2¼vy=N 5 u=0
v=0
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Example: Separability ² Matlab example using the 2D FFT function: fft2(magic(3)) ans = 45.0000 0 + 0.0000i 0 - 0.0000i
0 13.5000 + 7.7942i 0.0000 + 5.1962i
0 0.0000 - 5.1962i 13.5000 - 7.7942i
² Matlab example using two 1D FFT function calls: fft(fft(magic(3)).').' ans = 45.0000 0 + 0.0000i 0 - 0.0000i
0 13.5000 + 7.7942i 0.0000 + 5.1962i
0 0.0000 - 5.1962i 13.5000 - 7.7942i
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Symmetry and periodicity ² If f (x; y) is real (e.g. an image), its Fourier transform is conjugate symmetric ² Additionally the DFT is in¯nitely periodic
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Symmetry and periodicity ² The same property holds for the 2D case
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Translation in the Fourier domain ² A translation (u0; v0) in the Fourier domain result in F (u ¡ u0 ; v ¡ v0) () f (x; y)ej2¼(u0x+v0 y)=N ² The origin of the Fourier domain is shifted to the point (u0; v0 ) ² A special case is u0 = v0 = N=2, here the exponential term ej¼(x+y) ¡ 1x+y f (x; y)(¡1)x+y () F (u ¡ N=2; v ¡ N=2) ² The origin will be moved from (0; 0) to the centre of the image
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Example: Translation
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Rotation ² A rotation in the spatial domain rotates the Fourier domain by the same angle and vice versa.
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Examples: DFT image scaling. The rectangular pulse in the image function (a–c) creates a strongly oscillating power spectrum (d–f), as in the onedimensional case. Stretching the image causes the spectrum to contract and vice versa.
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Examples: DFT—oriented, repetitive patterns. The image 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.
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Examples: 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).
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Examples: DFT—superposition of image patterns. Strong, oriented subpatterns (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).
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Examples: DFT—natural image patterns. Examples of repetitive structures in natural images (a–c) that are also visible in the corresponding spectrum (d–f).
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Examples: DFT—natural image patterns with no dominant orientation. The repetitive 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).
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Examples: 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.
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Correcting the geometry 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.
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Windowing Effects of periodicity in the 2D spectrum. The discrete Fourier transform is computed under the implicit assumption that the image signal is periodic along both dimensions (top). Large differences in intensity at opposite image borders—here 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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Windowing
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Fourier basis functions ² Fourier transform represents signal in terms of sine and cosine (basis functions) ² Magnitude gives frequency, direction gives orientation
² Fourier basis element e¡j2¼(ux+vy) = cos(2¼(ux + vy)) ¡ j sin(2¼(ux + vy))
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Fourier basis functions ² Here u and v are larger than in the previous slide
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Fourier basis functions ² And larger still
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Fourier basis functions ² Fourier basis elements e¡j2¼(ux+vy) = cos(2¼(ux + vy)) ¡ j sin(2¼(ux + vy))
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Convolution and Correlation ² Convolution: Operator on two sequences that represents a ¯ltering operation ² Correlation: Correlation is a measure of similarity between two signals
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Discrete Convolution ²
1 f (x) ¤ g(x) = M
M ¡1 X m=0
f (m)g(x ¡ m)
² M ¸ A + B ¡ 1 where A is the length of f and B is the length of g ²
1 f (x; y)¤g(x; y) = MN
M ¡1 NX ¡1 X
f (m; n)g(x¡m; y¡n)
m=0 n=0
² A £ B and C £ D are the arrays for f (x; y) and g(x; y) M ¸A+C ¡1 N ¸B+D¡1
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Example: Convolution
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Example: 2D convolution
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Discrete Correlation ² The correlation of two functions f (x) and g(x) is: 1 1D : f (x) ± g(x) = M
M ¡1 X m=0
1 2D : f (x; y)±g(x; y) = MN ²
¤
f ¤ (m)g(x + m)
M ¡1 NX ¡1 X
f ¤(m; n)g(x+m; y+n)
m=0 n=0
is the complex conjugate
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Correlation theorem A correlation in the spatial domain is a multiplication with the complex conjugate in the Fourier domain and vice versa. f (x) ± g(x) , F ¤(u)G(u) and
f (x; y) ± g(x; y) , F ¤ (u; v)G(u; v) f ¤(x)g(x) , F (u) ± G(u)
f ¤(x; y)g(x; y) , F (u; v) ± G(u; v)
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Example: Correlation
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Example: 2D correlation
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Sampling ² Sampling can be described as the multiplication of a signal with a sequence of impulse functions
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Sampling with the impulse function
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The comb function
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The comb function
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Sampling in Fourier domain ² Multiplication of signal with comb function in time domain, is convolution of them in Fourier domain g(x)III(x) , G(u) ¤ III(u)
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Sampling: Fourier domain
A
f(x) 0 X
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x
Reconstruction filters 50
Square pixels
Gaussian reconstruction filter 100
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Image reconstruction: pixelization
square pixels
Harmon & Julesz 1973
Gaussian reconstruction 40
Sampling: Aliasing effect
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Nyquist theorem ² Nyquist theorem: The sampling frequency must be at least twice the highest frequency !s ¸ 2! ² If this is not the case the signal needs to be bandlimited before sampling, e.g. with a lowpass ¯lter
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Aliasing effect ² Sampling without smoothing: Top row shows the images, sampled at every second pixel to get the next; bottom row shows the magnitude spectrum of these images. ² Nyquist criterium not ful¯lled (aliasing artifacts)
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Aliasing effect ² Sampling with smoothing: We get the next image by smoothing the image with a Gaussian with sigma 1 pixel, then sampling at every second pixel.
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