The Frequency Domain
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Date: 07/21/04
References: [Efford 2000] [Petrou & Bosdogianni,1999] [Gonzalez & Woods 1993]
The frequency domain
o the goal is to represent an image as a weighted sum of sinusoidal functions
o a periodic function - a sinusoid - Eric Weisstein's World of Mathematics - periodic function
(http://mathworld.wolfram.com/PeriodicFunction.html)
 period - length of the cycle, L
 frequency - 1/L
 if L is a distance, 1/L is the spatial frequency
 amplitude - height of the peak, A
 phase - position of the start of a cycle, phi
o images of sinusoidal functions - equation 8.1
 u is the spatial frequency - number of cycles that fit into the width of the image
 u/N is cycles per pixel
o Fourier theory
 any periodic function can be represented as a sum of simple sinusoids
 basis functions - sine and cosine functions of a given frequency
 Fourier series - weighted sum of basis functions - equation 8.2
 n is the frequency of a basis function - number of cycles that fit into one period L
 Fourier series representation of f(x) can be given by two 1D arrays (of infinite size)
of coefficients
 Fourier coefficients - weighting factors for each sine and cosine function
 example - summation of sinusoids can generate discontinuities and homogeneous regions
o 2D Fourier series
 use 2D sine and cosine functions - equation 8.3
 Fourier series representation of f(x,y) can be given by two 2D arrays (of infinite size) of
coefficients
 basis images - figure 8.7
 coefficients determine contributions of each basis image to the representation
 Fourier series (http://mathworld.wolfram.com/FourierSeries.html)
o Fourier transform
 determine the contribution made by any basis function (image) to the representation of a
function (image)
 Fourier transform pair (http://mathworld.wolfram.com/FourierTransform.html)
 exists if f(x) continuous and integrable, F(u) integrable
 easily extended to 2D
o discrete Fourier transform - with images we are operating in the discrete domain
 discrete Fourier transform
(http://mathworld.wolfram.com/DiscreteFourierTransform.html)
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equation 8.4
equation 8.5 - forward transform
 u and v specify spatial frequencies
 for a given u and v, F(u,v) represents the contribution of the corresponding basis
images
 F(u,v) is a complex number
 Fourier coefficients - the set of F(u,v) for all u,v - also called the Fourier transform
 Fourier transform - the process of computing all F(u,v)
 equation 8.6 - inverse transform - errata
 reconstructs the image from the Fourier coefficients
we can convert between the spatial and frequency domains without loss of information
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when we manipulate f(x,y) we are processing the image in the spatial domain
when we manipulate F(x,y) we are processing the image in the frequency domain
spectra of an image
 Fourier spectrum
 compute the magnitude of F(u,v) - equation 8.8
 amplitude of the basis function
 power spectrum - square of the Fourier spectrum
 phase - equation 8.9
we typically manipulate just the Fourier spectrum and leave the phase alone
Fast Fourier Transform
o computation of a 1D discrete Fourier transform (DFT) of N samples requires
 N complex multiplications (http://mathworld.wolfram.com/ComplexMultiplication.html)
 N-1 complex additions (http://mathworld.wolfram.com/ComplexMultiplication.html)
 for each of the N values of u
o a 1D DFT is O(N2)
o a 2D DFT is of an N x N image is O(N4)
o assuming 1 microsecond for a complex multiplication:
 2D DFT with N = 256 takes 71 minutes
 2D DFT with N = 1024 takes 12 days!
o there are two ways to reduce the complexity - we'll use them both
 use the separability of the Fourier transform to implement a 2D DFT as a two-pass algorithm
 reduces complexity of the 2D DFT to O(N3)
 decompose the 1D DFT in such a way to make it O(N log2 N)
 this procedure is known as the fast Fourier transform algorithm (FFT)
 requires N to be a power of 2 - must either crop or pad the data
 an N-point transform can be decomposed into the sum of two N/2-point transforms
 we do this recursively until we are computing 2-point transforms
 we do this O(log2 N) process for each of the N values of u giving complexity O(N
log2 N)
 requires reordering of the input using the bit-reversal rule
 reordering the input and then computing the FFT produces output in the correct order
o Numerical Recipes online - http://lib-www.lanl.gov/numerical/
o Efford’s book page 198
o ImageFFT.java - CD
Properties of the Fourier transform
o periodicity - F(u,v) repeats itself endlessly in both directions
o complex conjugate symmetry - for the amplitude spectrum, negative frequencies are mirror images
of positive frequencies
http://mathworld.wolfram.com/ComplexConjugate.html
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spectrum is more easily interpreted (visually) if we shift the results
Windowing
o there is an implied periodicity of the image
o discontinuities (in gray level) between the right and left (top and bottom) edges affect the spectrum
o a windowing function is applied to cause the pixel values to taper off to zero
Using the ImageFFT class
SpectrumViewer application
SpectralProbe application
Images of spectra
o scaling of spectrum for display purposes
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spectra of periodic patterns
spectra of edges
spectra of simple shapes
Filtering in the frequency domain
o point-wise multiplication of the spectrum by a filter transfer function
o convolution theorem
 convolution in the spatial domain is equivalent to multiplication in the frequency domain
 to perform filtering in the frequency domain given a convolution kernel
 take Fourier transform of the image and the kernel
 multiply (point-wise) the two transforms together
 take inverse Fourier transform of the product
o low pass filtering
 ideal low pass filter
 ringing
 Butterworth low pass filter
o high pass filtering
o band pass and band stop filtering
o removing periodic noise
Filter support in ImageFFT.java
Frequency domain filtering implementation
o ImageFFT.java
o ButterworthLowPassOp.java
Deconvolution
o the degradation of an image due to blurring and noise can be modeled
o point spread function (PSF) - convolution kernel which models the blurring
o our degraded image is produced through a convolution of the correct image and the PSF
o deconvolution of the degraded image would give us the correct image
o thanks to the convolution theorem, we can do deconvolution in the frequency domain
 take the Fourier transform of the degraded image and the kernel
 modulation transfer function (MTF) - the Fourier transform of the PSF
 divide the Fourier transform of the degraded image by the MTF (point-wise)
 take the inverse transform of the resulting deconvolved spectrum to get the deconvolved
image
o division by zero (in the MTF) is a problem
o must also compensate for noise
 threshold on the MTF
 restoration cutoff frequency
References :
1. 2D Frequency Domain Filtering and the 2D DFT
http://www.owlnet.rice.edu/~elec301/Projects01/image_filt/transform.html
2. Image Filtering in the Frequency Domain
http://astronomy.swin.edu.au/~pbourke/analysis/imagefilter/
3. Frequency Filter
http://www.dai.ed.ac.uk/HIPR2/freqfilt.htm
4. Filtering in the Frequency Domain
http://www.netnam.vn/unescocourse/computervision/94.htm
5. The Wiener filter
http://osl.iu.edu/~tveldhui/papers/MAScThesis/node15.html
More Links:
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 Frequency Filters, Deconvolution, Wiener Filtering:
http://www.dai.ed.ac.uk/HIPR2/freqfilt.htm#7
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 Image Restoration, Point Spread Function (PSF):
http://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/VELDHUIZEN/node8.html
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 Wiener Filter:
http://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/VELDHUIZEN/node15.html
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 Wiener Filter:
http://www.khoral.com/contrib/contrib/dip2001/html-dip/c7/s3/front-page.html
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 Research paper; 3D restoration of tomosynthetic images, 3D Regularized Wiener Filtering:
ftp://ftp.mthcsc.wfu.edu/pub/plemmons/tim_icis.ps.gz