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## Digital Image Processing

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**Digital Image Processing**Chapter 4: Image Enhancement in the Frequency Domain**Background**• The French mathematiian Jean Baptiste Joseph Fourier • Born in 1768 • Published Fourier series in 1822 • Fourier’s ideas were met with skepticism • Fourier series • Any periodical function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient**Fourier transform**• Functions can be expressed as the integral of sines and/or cosines multiplied by a weighting function • Functions expressed in either a Fourier series or transform can be reconstructed completely via an inverse process with no loss of information**Applications**• Heat diffusion • Fast Fourier transform (FFT) developed in the late 1950s**Introduction to the Fourier Transform and the Frequency**Domain • The one-dimensional Fourier transform and its inverse • Fourier transform • Inverse Fourier transform**Two variables**• Fourier transform • Inverse Fourier transform**Discrete Fourier transform (DFT)**• Original variable • Transformed variable**DFT**• The discrete Fourier transform and its inverse always exist • f(x) is finite in the book**Time domain**• Time components • Frequency domain • Frequency components**Fourier transform and a glass prism**• Prism • Separates light into various color components, each depending on its wavelength (or frequency) content • Fourier transform • Separates a function into various components, also based on frequency content • Mathematical prism**Polar coordinates**• Real part • Imaginary part**Magnitude or spectrum**• Phase angle or phase spectrum • Power spectrum or spectral density**Some references**• http://local.wasp.uwa.edu.au/~pbourke/other/dft/ • http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm**Examples**• test_fft.c • fft.h • fft.c • Fig4.03(a).bmp • test_fig2.bmp**Spatial, or image variables: x, y**• Transform, or frequency variables: u, v**Magnitude or spectrum**• Phase angle or phase spectrum • Power spectrum or spectral density**Centering**• Average gray level • F(0,0) is called the dc component of the spectrum**Conjugate symmetric**• If f(x,y) is real • Relationships between samples in the spatial and frequency domains**The separation of spectrum zeros in the u-direction is**exactly twice the separation of zeros in the v direction**Strong edges that run approximately at +45 degree, and -45**degree • The inclination off horizontal of the long white element is related to a vertical component that is off-axis slightly to the left • The zeros in the vertical frequency component correspond to the narrow vertical span of the oxide protrusions**Basics of filtering in the frequency domain**• 1. Multiply the input image by to center the transform • 2. Compute F(u,v) • 3. Multiply F(u,v) by a filter function H(u,v) • 4. Compute the inverse DFT • 5. Obtain the real part • 6. Multiply the result by**Fourier transform of the output image**• zero-phase-shift filter • Real H(u,v)**Inverse Fourier transform of G(u,v)**• The imaginary components of the inverse transform should all be zero • When the input image and the filter function are real**Lowpass filter**• Pass low frequencies, attenuate high frequencies • Blurring • Highpass filter • Pass high frequencies, attenuate low frequencies • Edges, noise**Smoothing Frequency-Domain Filterers**• Ideal lowpass filters**Cutoff frequency**• Total image power • Portion of the total power**Blurring and ringing properties**• Filter • Convolution • : Spatial filter • was multiplied by • Then the inverse DFT • The real part of the inverse DFT was multiplied by