Chapter 4 Image Enhancement in the Frequency Domain

Transcription

1 Chapter 4 Image Enhancement in the Frequency Domain Yinghua He School of Computer Science and Technology Tianjin University

2 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

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4 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

5 The One-Dimensional Fourier Transform and its Inverse The Two-Dimensional Fourier Transform and its Inverse Filtering in the Frequency Domain Correspondence between Filtering in the Spatial and Frequency Domains

6 The Fourier transform Fu of a single variable continuous function f is defined by the equation Where j F u 1 j πu f e 2 d Conversely given Fu we can obtain f by means of the inverse Fourier transform f j πu F u e 2 du These two equations comprise the Fourier transform pair.

7 These equations are easily etended to two variables u and v: F u v j 2π u+ vy f y e ddy And similarly for the inverse transform f y j 2π u+ vy F u v e dudv

8 The Fourier transform of a discrete function of one variable f 012 M-1 is given by the equation M 1 F u 1 f e j2πu / M for M M Similarly given Fu we can obtain the original function back using the inverse DFT: f M 1 u 0 F u e j2πu / M u for M 1

9 The concept of the frequency domain follows directly from Euler s formula: θ e j cosθ + j sinθ Substituting this epression into Eq and using the fact that cos θ cosθ gives us F u M 1 M 1 0 f [ cos 2πu / M j sin 2πu / M ] for u M 1

10 In the analysis of comple numbers we find it convenient sometimes to epress Fu in polar coordinates: Where F u F u e [ ] 2 2 R u I 1/ 2 F u + u jφ u Is called the magnitude or spectrum of the Fourier transform and

11 Is called the phase angle or phase spectrum of the transform. Another quantity that is used in this chapter is the power spectrum defined as the square of the Fourier spectrum: tan 1 u R u I u φ u I u R u F u P +

12 The first value of the sampled function is then f 0. The kth sample gives us f 0 + kδ. When we write fk it is understood that we are utilizing shorthand notation that really means f 0 + kδ.

13 f is then understood to mean Fu is then understood to mean and are inversely related by the epression 0 f f Δ + Δ u u F u F Δ Δ Δ u Δ M u Δ Δ 1

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15 The One-Dimensional Fourier Transform and its Inverse The Two-Dimensional Fourier Transform and its Inverse Filtering in the Frequency Domain Correspondence between Filtering in the Spatial and Frequency Domains

16 The discrete Fourier transform of a function image fy of size M*N is given by the equation Given Fuv we obtain fy via the inverse Fourier transform given by the epression for 012 M-1 and y012 N / / 2 1 M N y N vy M u j e y f MN v u F π / / 2 M u N v N vy M u j e v u F y f π

17 The Fourier spectrum phase angle and power spectrum: Where Ruv and Iuv are the real and imaginary parts of Fuv respectively. [ ] 2 1/ 2 2 v u I v u R v u F + tan 1 v u R v u I v u φ v u I v u R v u F v u P +

18 It is common practice to multiply the input + y image function by 1 prior to computing the Fourier transform. Due to the properties of eponentials it is not difficult to show that I [ ] + y f y 1 F u M / 2 v N / 2 Where I [] denotes the Fourier transform of the argument.

19 The value of the transform at uv00 is If fy is real its Fourier transform is conjugate symmetric; that is From this it follows that M N y y f MN F * v u F v u F v u F v u F

20 The following relationships between samples in the spatial and frequency domains: and Δu 1 MΔ Δv 1 NΔy

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22 The One-Dimensional Fourier Transform and its Inverse The Two-Dimensional Fourier Transform and its Inverse Filtering in the Frequency Domain Correspondence between Filtering in the Spatial and Frequency Domains

23 It consists of the following steps: 1 Multiply the input image by -1 +y to center the transform. 2 Compute Fuv the DFT of the image from 1. 3 Multiply Fuv by a filter function Huv. 4 Compute the inverse DFT of the result in 3. 5 Obtain the real part of the result in 4. 6 Multiply the result in 5 by -1 +y.

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25 Huv is called a filter is because it suppresses certain frequencies in the transform while leaving others unchanged. The Fourier transform of the output image is given by G u v H u v F u v The filtered image is obtained simply by taking the inverse Fourier transform of Guv: 1 Filtered Image I [ G u v ]

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27 Assuming that the transform has been centered we can do this operation by multiplying all values of Fuv by the filter function: H u v 0 if u v 1 otherwise M / 2 N / 2

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31 The One-Dimensional Fourier Transform and its Inverse The Two-Dimensional Fourier Transform and its Inverse Filtering in the Frequency Domain Correspondence between Filtering in the Spatial and Frequency Domains

32 The discrete convolution of two functions fy and hy of size M*N is denoted by fy*hy and is defined by the epression * M m N n n y m h n m f MN y h y f

33 Letting Fuv and Huv denote the Fourier transforms of fy and hy the following result holds: * v u H v u F y h y f * v u H v u F y h y f

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35 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

36 Basic model for filter in the frequency domain is given by the following equation G u v H u v F u v Where Fuv is the Fourier transform of the image to be smoothed. The objective is to select a filter transfer function Huv that yields Guv by attenuating the high-frequency components of Fuv.

37 Ideal Lowpass Filter Butterworth Lowpass Filters Gaussian Lowpass Filter Additional Eamples of Lowpass Filtering

38 The transfer function of a two-dimensional2- D ideal lowpass filterilpf is: 1 H u v 0 if D u v D D u v > D The distance from any point uv to the center of the Fourier transform is given by Duv [u - M/2 2 + v - N/2 2 ] 1/2. if 0 0

39 Ideal Lowpass Filters

40 Ideal Lowpass Filters Image power Image power P T is obtained by summing the component of the power spectrum at each point uv for u 012 M-1 and v 012 N-1 M 1 N 1 u 0 v 0 P u v A circle of radius r with origin at the center of the frequency rectangle encloses α 100 [ Puv/P T ]

41 Ideal Lowpass Filters

42 Ideal Lowpass Filters

43 Ideal Lowpass Filters As the filter radius increases less and less power is removed resulting in less severe blurring. Fig 4.12c through e are characterized by ringing which becomes finer in teture as the amount of high frequency content removed decreases.

44 Ideal Lowpass Filters Ringing effect occurs along the edges of the filtered spatial domain image illustrated in a Figure. Net slide figure shows the shape of the one-dimensional filter in both the frequency and spatial domains for two different values of D 0. We obtain the shape of the twodimensional filter by rotating these functions about the y-ais.

45 Ideal Lowpass Filters Multiplication in the Fourier domain corresponds to a convolution in the spatial domain. Due to the multiple peaks of the ideal filter in the spatial domain the filtered image produces ringing along intensity edges in

46 Ideal Lowpass Filter Butterworth Lowpass Filters Gaussian Lowpass Filter Additional Eamples of Lowpass Filtering

47 Butterworth Lowpass Filters Ideal filtering simply cuts off the Fourier transform. It is easy to implement however it has the disadvantage of introducing unwanted artifacts ringing into the result. One way of avoiding these 1 artifacts is to use a filter H umatri v a Dcircle u v with a cutoff 2n that is less sharp. 1+ [ ] D 0

48 Butterworth Lowpass Filters

49 Lowpass Filters

50 Butterworth Lowpass Filters

51 Ideal Lowpass Filter Butterworth Lowpass Filters Gaussian Lowpass Filter Additional Eamples of Lowpass Filtering

52 Gaussian Lowpass Filters The form of these filters in two dimensions is given by2 D u v H u v ep 2 2σ H u v ep 2 D u v 2 2D 0 Duv is the distance from the origin of the Fourier transform which we assume has been shifted to the center of the frequency rectangle

53 Gaussian Lowpass Filters

54 Lowpass Filters

55 p Lowpass Filtering Fig shows a sample of tet of poor resolution that may be occurred from fa transmission duplicated material and historical records. Fig 4.19a shows characters in a document have distorted shapes. Many characters are broken. Fig 4.19b shows repaired characters by this simple process using a Gaussian lowpass filter with D 0 80

56 Gaussian Lowpass Filters

57 Ideal Lowpass Filter Butterworth Lowpass Filters Gaussian Lowpass Filter Additional Eamples of Lowpass Filtering

58 Additional Eamples of Lowpass Filtering Fig shows an application of lowpass filtering to produce a smoother softer-looking result from a sharp original. The smoothed images look quite soft and pleasing Fig shows images with prominent scan line. Lowpass filtering a crude but simple way to reduce the effect of these lines. Fig. 4.21b shows an image with D 0 30 Fig. 4.21c shows an image with D 0 10

59 Gaussian Lowpass Filters

60 Gaussian Lowpass Filters

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63 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

64 Sharpening Frequency Domain Filters Image sharpening can be achieved by a highpass filtering process which attenuates the low-frequency components without disturbing high-frequency information. Zero-phase-shift filters: radially symmetric and completely specified by a cross section. H hp u v 1 H u v lp

65 Sharpening Frequency Domain Filters Fig shows typical 3-D plots image representations and cross sections for these filters IHPF BHPF GHPF. Fig illustrates what these filters look like in the spatial domain. A spatial representation of a frequency domain filter is obtained by 1multiplying Huv by -1 u+v for centering 2computing the inverse DFT 3 multiplying the real part of the inverse DFT by -1 +y

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68 Sharpening Frequency Domain Filters Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters The Laplacian in the Frequency Domain Unsharp Masking High-Boost Filtering and High-Frequency Emphasis Filtering

69 Ideal Highpass Filters A 2-D ideal highpass filter IHPF is defined as 0 H u v 1 D u v D u v > D 0 is the cutoff distance measured. This filter is the opposite of the ideal lowpass filter. if if D D 0 0

70 Ideal Highpass Filters Fig. 4.24a is so severe that it produced distorted thickened object boundaries. Edges on the top three circles do not show well. The result for D 0 80 is more of what a high pass-filtered image should look like. The edges are much cleaner and less distorted and the smaller objects have been filtered properly.

71 Ideal Highpass Filters

72 Sharpening Frequency Domain Filters Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters The Laplacian in the Frequency Domain Unsharp Masking High-Boost Filtering and High-Frequency Emphasis Filtering

73 Butterworth Highpass Filters The transfer function of the Butterworth highpass filter BHPF of order n and will cutoff frequency locus at distance D 0 from the origin is given by H u v 1 D 1+ [ D u v 0 n ] 2 High-frequency emphasis: Adding a constant to a highpass filter to preserve the low-frequency components.

74 Butterworth Highpass Filters Fig. 4.25: The boundary is much less distorted than in Fig even for the smallest value of cut off frequency. Since the center spot sizes of the IHPF and the BHPF are similar the performance of the two filters I terms of filtering the smaller objects is comparable. The transition into higher values of cutoff frequencies is much smoother with the BHPF

75 Butterworth Highpass Filters

76 Sharpening Frequency Domain Filters Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters The Laplacian in the Frequency Domain Unsharp Masking High-Boost Filtering and High-Frequency Emphasis Filtering

77 Gaussian Highpass Filters The transfer function of the Gaussian Highpass Filters GHPF with cutoff frequency locus at distance D 0 from the origin is given by 2 D u v H u v 1 ep 2 2D 0

78 Gaussian Highpass Filters Fig. 4.26: As epected the results obtained are smoother than with the previous two filters. Even the filtering of the smaller objects and thin bars cleaner with the Gaussian filter.

79 Gaussian Highpass Filters

80 Sharpening Frequency Domain Filters Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters The Laplacian in the Frequency Domain Unsharp Masking High-Boost Filtering and High-Frequency Emphasis Filtering

81 Laplacian in the Frequency Domain It can be shown that: I [ 2 f y ] u 2 + v 2 Fuv The Laplacian can be implemented in the frequency domain by using the filter Shift to center H u v u 2 [ u + v M 2 / v N / 2 2 ].

82 Laplacian in the Frequency Domain The laplacian-filtered image in the spatial domain is obtain by computing the inverse Fourier Transform of HuvFuv 2 f y I 1 { [ u M / v N / 2 2 ] F u v}.

83 Laplacian in the Frequency Domain Fig. 4.27a is a 3-D perspective plot of H u v u 2 + v 2 [ u M / The function is center at M/2N/2 and its value at the top of the dome is zero. All other values are negative. Fig. 4.27b shows Huv as an image also centered. Fig. 4.27c is the Laplacian in the spatial domain v N / 2 2 ].

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85 Laplacian in the Frequency Domain Fig. 4.28a is the same image in as Fig. 3.40a. Fig. 4.28b shows the result of filtering this image in the frequency domain using f y I { [ u M / 2 + v N / 2 ] F u v}. Fig. 4.28c show the scaled image for display only Fig. 4.28d should be compared with Fig which shows eactly the same sequence of steps but computed using only spatial domain techniques. The results are identical for all practical purposes.

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87 Sharpening Frequency Domain Filters Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters The Laplacian in the Frequency Domain Unsharp Masking High-Boost Filtering and High-Frequency Emphasis Filtering

88 Unsharp Masking High-Boost Filtering Unsharp masking: f hp y fy - f lp y High boost filtering: f hb y Afy - f lp y f hb y A-1fy + f hp y H hb uv A-1 + H hp uv

89 Unsharp Masking High-Boost Filtering Fig b is a highpass filterd iamge. The image in Fig. 4.29c was obtained using f hb y A-1fy + f hp y with A 2. This image is sharper but still too dark. Fig. 4.29d was obtained with A 2.7 which in effect means that the input image was multiplied by 1.7 before the Laplacian was subtracted from it.

90 Unsharp Masking High-Boost Filtering Fig. 4.29d is not as sharp as Fig. 3.43d. The reason for this that a frequency domain representation of the Laplacian is closer to the mask that ecludes the diagonal neighbors [Fig. 4.27f]. It is known that a mask that includes the diagonal neighbors produces slightly sharper results. They do become evident for images with larger features.

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92 High-Frequency Emphasis Filtering Sometimes it is advantageous to accentuate the contribution to enhancement made by the high-frequency component of an image. We multiply a high pass filter function by a constant and add an offset so that the zero frequency term is not eliminated by the filter. H hfe uv a+bh hp uv where a>0 and b>a. [ Typical a [ ] and b [1.52.0] ]

93 High-Frequency Emphasis Filtering Fig. 4.30a shows a chest X-ray with a narrow range of gray levels. Our objective is to sharpen the image. Fig 4.30c shows image using HFE with a 0.5 and b 2.0. Although the image is still dark the gray level tonality due to the low frequency components was not lost. Fig. 4.30d shows image that is been performing histogram equalization.

94 High-Frequency Emphasis Filtering

95 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

96 Homomorphic Filtering We can view an image fy as a product of two components: f 0 0 y i y r y < < i r y y < < iy: illumination. It is determined by the illumination source. ry: reflectance or transmissivity. It is determined by the characteristics of imaged objects. 1

97 Homomorphic Filtering In some images the quality of the image has reduced because of non-uniform illumination. Homomorphic filtering can be used to perform illumination correction. y i y r y f The above equation cannot be used directly in order to operate separately on the frequency components of illumination and reflectance.

98 v u F v u F v u Z r i + y r y i y f y z ln ln ln + ep 0 0 ' ' y r y i y s y g y r y i y s + Homomorphic Filtering v u Z v u H v u S ln : DFT : Huv : DFT -1 : ep :

99 Homomorphic Filtering By separating the illumination and reflectance components homomorphic filter can then operate on them separately. Illumination component of an image generally has slow variations while the reflectance component vary abruptly. By removing the low frequencies highpass filtering the effects of illumination can be removed.

100 Homomorphic Filtering A good idea of control can be gained over the illumination and reflectance components with a homomorphic filter. This control requires specification of a filter function Huv that affects the low and high frequency components of the Fourier transform in different ways. Fig shows a cross section of such a filter. If the parameters γ L and γ H are chosen so that γ L <1and γ H >1. The curve in Fig can be approimated using modified Gaussian highpass filter: H u v 2 γ γ [1 ep c D u v / D ] + γ H L 2 0 L

101 Homomorphic Filtering Fig is typical of the results that can be obtained with the homomorphic filtering function in Fig Fig. 4.33b shows the result of processing this image by homomorphic filtering with γ L 0.5 γ H 2.0 in the filter function of Fig A reduction of dynamic range in the brightness together with an increase in contrast brought out the details of objects inside the shelter and balanced the gray levels of the outside wall. The enhanced image also is sharper.

102 Homomorphic Filtering

103 Homomorphic Filtering

104 Background Introduction to the Fourier Transform and the Frequency Domain Smoothing Frequency-Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering Implementation

105 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

106 The Fourier transform pair has the following translation properties: and / / v v u u F e y f N y v M u j + π / / N vy M u j e v u F y y f + π

107 When u 0 M / 2 and v 0 N / 2 it follows that e j 2 0 π u0 / M + v y / N jπ + y e 1 + y In this case Eq becomes f + y y 1 F u M / 2 v N / 2 and similarly f M / 2 y N / 2 F u v 1 u+ v

108 Distributivity snd scaling From the definition of the Fourier transform it follows that And in general that The Fourier transfoem is distributive over addition but nor over multiplication.

109 Distributivity snd scaling For two scalars a and b af y af u v and 1 f a by F / ab u / a v b

110 Rotation If we introduction the polar coordinates r cos θ y r sinθ u ω cosϕ v ω sinϕ Then f y and F u v become f γ θ and F ω ϕ. Direct substitution into definition of the Fourier transform yields f γ θ + θ 0 F ω ϕ + θ 0

111 Periodicity and conjugate symmetry The discrete Fourier transform has the following periodicity properties: The inverse transform also is periodic: N v M u F N v u F v M u F v u F N y M f N y f y M f y f

112 Conjugate symmetry F u v F * u v The spectrum also is symmetric F u v F u v

113 Separability The discrete Fourier transform in Eq can be epressed in the separable form where 1 0 / / / M M u j N y N vy j M M u j e v F M e y f N e M v u F π π π 1 0 / 2 1 N y N vy j e y f N v F π

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116 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

117 The 1-DFourier transforms: and / 2 1 M M u j e f M u F π 1 0 / 2 M M u j e u F f π

118 Taking the comple conjugate of Eq and dividing both sides by M yields A similar analysis for two variables yields: / / 2 * * 1 1 M u N v N vy M u j e v u F MN y f M π 1 0 / 2 * * 1 1 M u M u j e u F M f M π

119 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

120 Figure 4.36 illustrates the significance of periodicity. The left column of this figure shows convolution computed using the 1-D version of Eq : * M m m h m f M h f

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122 This procedure yields etended or padded functions given by and P A A f f e P B B g g e 0 1 0

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124 Suppose that we have two images fy and hy of sizes A*B and C*D respectively. Wraparound error in 2-D convolution is avoided by choosing: and P A + C 1 Q B + D 1

125 The periodic sequences are formed by etending fy and hy as follows: and Q y B or P A B y and A y f y f e Q y D or P C D y and C y h y h e

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128 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

129 The discrete convolution of two functions fy and hy of size M*N is denoted by fy*hy and is defined by the epression: The convolution theorem consists of the following relationships between the two functions and their Fourier transforms: and * M m N n n y m h n m f MN y h y f * v u H v u F y h y f * v u H v u F y h y f

130 The correlation of two function fy and hy is defined as: where f* denotes the comple conjugate of f * 1 M m N n n y m h n m f MN y h y f o

131 There is a correlation theorem analogous to the convolution theorem. Let Fuv and Huv denote the Fourier transforms of fy and hy. An analogous result is that correlation in the frequency domain reduces to multiplication in the spatial domain; that is * v u H v u F y h y f o * v u H v u F y h y f o

132 The autocorrelation theorem: f y o f y F u v 2 Similarly f y 2 F u v o F u v

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134 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

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139 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

140 The FFT algorithm developed in this section is based on the so-called successive doubling method. For notational convenience we epress Eq in the form F u M 1 0 Where And M is assumed to be of the form or n M 2 1 M W M e f j2π / M u W M M 2K

141 Substitution of Eq into Eq yields Using K u K K u K K u K W f K W f K W f K u F u K u K W W K u K u K K u K W W f K W f K u F

142 Defining and F F 1 K 1 u even u f 2 W K K 0 1 K 1 u odd u f W K K u M u Because W M WM and [ ] u F u F u W F u even odd 2K + u+ M W u 2M W 2M 1 2 u [ F u F u W ] F u + K even odd 2K

143 Continuing this argument for any positive integer value of n leads to recursive epressions for the number of multiplications and additions required to implement the FFT: and m n 2m n n 1 n 1 n a n 2a n n 1

144 The computational advantage of the FFT over a direct implementation of the 1-D DFT is defined as 2 M C M M log M 2 M log M 2 n Because it is assumed that M 2 we can epress Eq in terms of n: C n n 2 n

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146 Some Additional Properties of the 2-D Fourier Transform Computing the Inverse Fourier Transform Using a Forword Transform Algorithm More on Periodicity: the Need for Padding The Convolution and Correlation Theorems Summary of Properties of the 2-D Fourier Tandform The Fast Fourier Transform Some Comments on Filter Design

147 All the filters discussed in this chapter are specified in equation form. In order to use the filters we simply sample the equation for the desired values of uv. This process results in the filter function Huv. In all our eamples this function was multiplied by the DFT of the input image and the inverse DFT was computed. All forward and inverse Fourier transforms in this chapter were computed with an FFT algorithm.

148 The approach to filtering discussed in this chapter is focused strictly on fundamentals the focus being specifically to eplain the effects of filtering in the frequency domain as clearly as possible. We know of no better way to do that than to treat filtering the way we did here. One can view this development as the basis for "prototyping" a filter. In other words given a problem for which we want to find a filter the frequency domain approach is an ideal tool for eperimenting quickly and with full control over filter parameters. Once a filter for a specific application has been found it often is of interest to implement the filter directly in the spatial domain using firmware and/or hardware.