1. Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain

2. 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

4.  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

5.  Applications Heat diffusion Fast Fourier transform (FFT) developed in the late 1950s

6. Introduction to the Fourier Transform and the Frequency Domain  The one-dimensional Fourier transform and its inverse Fourier transform Inverse Fourier transform

7. Two variables Fourier transform Inverse Fourier transform

8.  Discrete Fourier transform (DFT) Original variable Transformed variable

10.  DFT The discrete Fourier transform and its inverse always exist f(x) is finite in the book

11.  Sines and cosines

12.  Time domain  Time components  Frequency domain  Frequency components

13.  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

14.  Polar coordinates Real part Imaginary part

15. Magnitude or spectrum Phase angle or phase spectrum Power spectrum or spectral density

17.  Samples

18.  Some references http://local.wasp.uwa.edu.au/~pbourk e/other/dft/ http://local.wasp.uwa.edu.au/~pbourk e/other/dft/ http://homepages.inf.ed.ac.uk/rbf/HIP R2/fourier.htm http://homepages.inf.ed.ac.uk/rbf/HIP R2/fourier.htm

19.  Examples  test_fft.c test_fft.c  fft.h fft.h  fft.c fft.c  Fig4.03(a).bmp Fig4.03(a).bmp  test_fig2.bmp test_fig2.bmp

20.  The two-dimensional DFT and its inverse

21.  Spatial, or image variables: x, y  Transform, or frequency variables: u, v

22. Magnitude or spectrum Phase angle or phase spectrum Power spectrum or spectral density

23.  Centering  Average gray level F(0,0) is called the dc component of the spectrum

24.  Conjugate symmetric If f(x,y) is real  Relationships between samples in the spatial and frequency domains

25. The separation of spectrum zeros in the u-direction is exactly twice the separation of zeros in the v direction

26.  Filtering in the frequency domain

27. 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

28.  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

29.  Fourier transform of the output image  zero-phase-shift filter Real H(u,v)

30.  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

32.  Set F(0,0) to be zero, a notch filter

34.  Lowpass filter Pass low frequencies, attenuate high frequencies Blurring  Highpass filter Pass high frequencies, attenuate low frequencies Edges, noise

36.  Convolution theorem

37.  Impulse function of strength A

41.  Gaussian filter

43.  Highpass filter

44. Smoothing Frequency-Domain Filterers  Ideal lowpass filters

46. Cutoff frequency Total image power Portion of the total power

49.  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

51.  The filter A dominant component at the origin Concentric, circular components about the center component --- ringing The radius of the center component and the number of circles per unit distance from the origin are inversely proportional to the value of the cutoff frequency of the ideal filter.

52.  Butterworth lowpass filters when

55.  Butterworth lowpass filters Order 1: No ringing Order 2: Imperceptible ringing Higher order: Ringing becomes a significant factor

57.  Gaussain lowpass filters When No ringing

60.  Additional examples of lowpass filters Machine perception Printing and publishing Satellite and aerial images

64. Sharpening Frequency Domain Filters  Highpass filter Spatial filter:  was multiplied by  Then the inverse DFT  The real part of the inverse DFT was multiplied by

67.  Ideal highpass filters

69.  Butterworth highpass filters

71.  Gaussian highpass filters

73.  The Laplacian in the frequency domain

74. After centering

75. Inverse Fourier transform Fourier-transform pair

77. Subtracting the Laplacian from the original image

79.  Unsharp masking, high-boost filtering, and high-frequency emphasis filtering Highpass filtering High-boost filtering

80. Frequency domain

82. High-frequency emphasis  where and

84. Homomorphic Filtering  Illumination and reflectance components  Derivations

85. Or

86.  Frequency domain  Spatial domain

88. Decrease the contribution made by the low frequencies (illumination) Amplify the contribution made by high frequencies (reflectance) Simultaneous dynamic range compression and contrast enhancement

91. Implementation  Translation When and

93.  Distributivity and scaling

94.  Rotation Polar coordinates Rotating by an angle rotates by the same angle

95.  Periodicity and conjugate symmetry Periodicity property

96. Conjugate symmetry Symmetry of the spectrum

98.  Separability

99. where We can compute the 2-D transform by first computing a 1-D transform along each row of the input image, and then computing a 1-D transform along each column of this intermediate result

101.  Computing the inverse Fourier transform using a forward transform algorithm

102. Calculate

103. Inputting into an algorithm designed to compute the forward transform gives the quantity

104. 2-D

105.  More on periodicity: the need for padding Convolution: Flip one of the functions and slide it pass the other