1 Chapter 5 Image Transforms

2 Image Processing for Pattern Recognition Feature Extraction Acquisition Preprocessing Classification Post Processing Scaling Centering Enhancement Filtering (Transform) Binarization (Thresholding) Edge detection Thinning Pixel Feature (Histogram) Boundary Projection Moments Transformation Matching Tree Classification Neural Network

3 Why need transformation? featuresBy image transformation with different basis functions (kernels), image f(x,y) is decomposed into a series expansion of basis functions, which are used as the features for further recognition.

4 Image Transforms Fourier transform Discrete Fourier transform Discrete Cosine transform Hough transform Wavelet transform

Transform t f(t)  F(  ) Transform Input function Basis function Basis function g(t) Operation: Inner Product

Wave transforms Wave transforms use the waves as their basis functions Fourier transform uses sinusoidal waves as its orthogonal basis functions

Transform t f(t) 0 t 0 t 0 t  F(  ) Fourier Transform

f0(x) = 1; f1(x) = sin(x); f2(x) = cos(2x); f3(x) = cos(3x); f4(x) = sin(18x); f(x) = f0(x) + f1(x) + 2  f2(x) - 4  f3(x) + f4(x)

f1 2f2 - 4f3 f4 f0 f0(x) = 1; f1(x) = sin(x); f2(x) = cos(2x); f3(x) = cos(3x); f4(x) = sin(18x); f0(x) = 1; f1(x) = sin(x); f2(x) = cos(2x); f3(x) = cos(3x); f4(x) = sin(18x);

f = f0 + f1 +2f2 - 4f3 + f4

11 Fourier Transforms Fourier integral transform Discrete Fourier transform (DFT) Fast Fourier Transform (FFT)

12 Let f (x) be a continuous function of a real variable x. The Fourier transform of f (x) is Input signal Basic function F(u) is complex: Real component Imaginary component Fourier spectrum: Phase angle:

13 Example:

14 The 2-D Fourier transform of f (x,y) is Fourier spectrum: Phase angle:

15 Example: Input function

16 Input function Spectrum displayed as an intensity function Fourier spectrum

17 Discrete Fourier Transform 1D: 2D: (N=M)

18 Discrete Fourier Transform (cont’) The Fourier spectrum, phase, and energy spectrum of 1D and 2D discrete functions are the same as the continuous case. But unlike the continuous case, both F(u) and F(u,v) always exist in the discrete case.

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22 Some Properties of the 2D Fourier Transform Separability: –The principle advantage of the separability property is that F(u, v) or f(x, y) can be obtained in two steps by successive applications of the 1D FT or its inverse.

23 Some Properties of the 2D Fourier Transform (cont’)

24 Some Properties of the 2D Fourier Transform (cont’) Periodicity and Conjugate Symmetry: –If f(x, y) is real, the FT also exhibits conjugate symmetry:

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26 Some Properties of the 2D Fourier Transform (cont’) Translation: where the double arrow is used to indicate the correspondence between a function and its FT (and vice versa).

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28 Some Properties of the 2D Fourier Transform (cont’) Scaling and Distributivity –FT and its inverse are distributive over addition, but not over multiplication.

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30 Some Properties of the 2D Fourier Transform (cont’) Average Value: Substituting u=v=0 into F(u, v) yields Giving

31 Fast Fourier Transform (FFT) The number of complex multiplications and additions required to implement a 1D discrete Fourier Transform is proportional to N 2. The FFT computation of this is Nlog 2 N. In the 2D case, the number of direct operations is N 4 and the FFT operation is 2N 2 log 2 N. FFT offers considerable computation advantage over direct implementation when N is relatively large (>256).

32 Fast Fourier Transform (cont’)

33 Fourier Transform (FFT) and Fourier Inverse Transform (FFT)

34 Fourier High Pass Filtering

35 Fourier Low Pass Filtering

36 Discrete Cosine Transform The 1-D DCT of a function f (x) is C(u), u = 0, 1, 2, …, N-1 By the DCT, a function f(x) is decomposed into a series expansion of basis functions, which are used as the features

37 The 2-D DCT of an image f (x,y) is C(u,v), u,v = 0, 1, 2, …, N-1 By the DCT, image f(x,y) is decomposed into a series expansion of basis functions, which are used as the features

38 Hough Transform Consider a point (x i, y i ) and the general equation of a straight line in slope- intercept form, y i =ax i +b. There is an infinite number of lines that pass through (x i, y i ), but they all satisfy the above equation for varying values of a and b.

39 Hough Transform (cont’) Consider b=-x i a+y i, and the ab plane (parameter space), then we have the equation of a single line for a fixed pair (x i, y i ).

40 Find the locations of strong peaks in the Hough transform matrix. The locations of these peaks correspond to the location of straight lines in the original image.

41 In this example, the strongest peak in R corresponds to and,. The line perpendicular to that angle and located at x’ is shown below, superimposed in red on the original image. The Radon transform geometry is shown in black.

42 Waves, Wavelets, and Transforms Waves & Wavelets Book and booklet A new word in English - Wavelets

Waves & Wavelets Waves Waves are non-compact support functions Non-compact support function The functions extend to infinity in both directions They are non-zero over their entire domain f(x), x = - , …, 0, …,  f(-  )  0, f(  )  0

Wavelets Wavelets are compact support functions Compact support functions: The functions are in a limited duration f(x)  0, for x = (a, b)

These basis functions vary in position as well as frequency Waves Wavelets Low-frequencyHigh-frequency Position

a scale parameter a translation parameter a is a scale parameter, b is a translation parameter. Wavelet Transform For any f(t)  L 2 (R), the wavelet transform is A function  (t)  R is called a wavelet, if it satisfies where Wave Transform

An example of Wavelet Transform Haar function (mother) Haar baby wavelets  (t) t 1 t t

Transform t f(t) 0 t 0 t 0 t

t Signal Wavelet transform Inverse wavelet transform Time Frequency Musical notation Wavelet components

t f(t) Signal Fourier transform Inverse Fourier transform Time Frequency Musical notation Fourier components

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