image enhancement frequency domain methods

 We can therefore directly design a transfer function Hu,v and implement the enhancement in the frequency domain as follows: Gu,v = Hu,v*Fu,v 3 Enhanced Image Transfer Function Give

  Prof Duong Anh Duc

 The concept of filtering is easier to visualize in the frequency domain Therefore, enhancement of image f(m,n) can be done in the frequency domain, based on its DFT F(u,v)

 This is particularly useful, if the spatial extent of the point-spread

sequence h(m,n) is large In this case, the convolution

g(m,n) = h(m,n)*f(m,n)

may be computationally unattractive

2

Enhanced Image

PSS

Given Image

 We can therefore directly design a transfer function H(u,v) and

implement the enhancement in the frequency domain as follows:

G(u,v) = H(u,v)*F(u,v)

3

Enhanced Image

Transfer Function

Given Image

 Given a 1-d sequence s[k], k = {…,-1,0,1,2,…,}

 Fourier transform

 Fourier transform is periodic with 2

 Inverse Fourier transform

4

 How is the Fourier transform of a sequence s[k] related to the Fourier transform of the continuous signal

 Continuous-time Fourier transform

5

 Given a 2-d matrix of image samples

s[m,n], m,n  Z2

 Fourier transform

 Fourier transform is 2-periodic both in  x and  y

 Inverse Fourier transform

6

 How is the Fourier transform of a sequence s[m,n] related to the

Fourier transform of the continuous signal

 Continuous-space 2D Fourier transform

7

|F(u,v)| displayed as image

f(x,y)

9

|F(u,v)| displayed in 3-D

10 Image Magnitude Spectrum

11 Image Magnitude Spectrum

12 Image Magnitude Spectrum

 As the size of the box increases in spatial domain, the corresponding

“size” in the frequency domain decreases

13

|F(u,v)|

f(x,y)

15

16

|G(u,v)|

g(x,y)

17

 Image formed from magnitude

spectrum of Rice and phase

spectrum of Camera man

18

 Image formed from magnitude

spectrum of Camera man and

phase spectrum of Rice

19

 For discrete images of finite extent, the analogous Fourier transform is the DFT

 We will first study this for the 1-D case, which is easier to visualize

 Suppose { f(0), f(1), …, f(N – 1)} is a sequence/ vector/1-D image

of length N Its N-point DFT is defined as

 Inverse DFT (note the normalization):

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 Example: Let f(n) = {1, -1 ,2,3 } (Note that N=4)

21

 F(u) is complex even though f(n) is real This is typical

 Implementing the DFT directly requires O(N2) computations, where N

is the length of the sequence

 There is a much more efficient implementation of the DFT using the

Fast Fourier Transform (FFT) algorithm This is not a new transform (as the name suggests) but just an efficient algorithm to compute the DFT

22

 The FFT works best when N = 2m (or is the power of some integer

base/radix) The radix-2 algorithm is most commonly used

 The computational complexity of the radix-2 FFT algorithm is Nlog(N)

adds and ½Nlog(N) multiplies So it is an Nlog(N) algorithm

 In MATLAB, the command fft implements this algorithm (for 1-D

case)

23

 The Fourier transform is suitable for continuous-domain images, which maybe of infinite extent

 For discrete images of finite extent, the analogous Fourier transform is the 2-D DFT

24

 Suppose f(m,n), m = 0,1,2,…M – 1, n = 0,1,2,…N – 1, is a discrete

N  M image Its 2-D DFT F(u,v) is defined as:

 Inverse DFT is defined as:

25

 For discrete images of finite extent, the analogous Fourier transform is the 2-D DFT

 Note about normalization: The normalization by MN is different than that in text We will use the one above since it is more widely used The Matlab function fft2 implements the DFT as defined above

26

 Most often we have M=N (square image) and in that case, we define a unitary DFT as follows:

 We will refer to the above as just DFT (drop unitary) for simplicity

27

In matlab, if f and h are matrices representing two images,

conv2(f, h) gives the 2D-convolution of images f and h

 Linearity (Distributivity and Scaling): This holds inboth discrete and

continuous-domains

o DFT of the sum of two images is the sum of their individual DFTs

o DFT of a scaled image is the DFT of the original image scaled by the same factor

30

 Spatial scaling (only for continuous-domain):

o If a, b > 1, image “shrinks” and the spectrum “expands.”

31

 Periodicity (only for discrete case): The DFT and its inverse are

periodic (in both the dimensions), with period N

F(u,v) = F(u+N,v) = F(u,v+N) = F(u+N,v+N)

o Similarly,

is also N-periodic in m and n

32

 Separability (both continuous and discrete): Decomposition of 2D DFT into 1D DFTs

33

o Similarly,

34

 Convolution: In continuous-space, Fourier transform of the convolution

is the product of the Four transforms

F[f(x,y)*h(x,y)] = F(u,v) H(u,v)

o In other words, output spectrum G(u,v) is the product of the input spectrum F(u,v) and the transfer function H(u,v)

o So the FT can be used as a computational tool to simplify the

convolution operation

36

 Correlation: In continuous-space, correlation between two images

f(x,y) and h(x,y) is defined as:

 Therefore,

37

 rff(x,y) is usually called the auto-correlation of image f(x,y) (with

itself) and rff(x,y) is called the crosscorrelation between f(x,y) and

h(x,y)

 Roughly speaking, rfh(x,y) measures the degree of similarity between images f(x,y) and h(x,y) Large values of rfh(x,y) would indicate that the images are very similar

38

 This is usually used in template matching, where h(x,y) is a template shape whose presence we want to detect in the image f(x,y)

 Locations where rfh(x,y) is high (peaks of the crosscorrelation

function) are most likely to be the location of shape h(x,y) in image

f(x,y)

39

 Convolution property for discrete images: Suppose

So if we want a convolution property for discrete images - something like

g(m,n) = f(m,n)*h(m,n)

we need to have G(u, v) to be of size (M+K–1)(N+L–1) (since

Therefore, we should require that F(u, v) and H(u, v) also have the same dimension, i.e (M+K–1)(N+L–1)

41

So we zero-pad the images f(m, n), h(m, n), so that they are of size

(M+K–1 )(N+L–1) Let fe(m,n) and he(m,n) be the zero-padded (or extended images)

Take their 2D-DFTs to obtain F(u, v) and H(u, v), each of size

(M+K–1)(N+L– 1) Then

 Similar comments hold for correlation of discrete images as well

42

 Translation: (discrete and continuous case):

Note that

but different phase spectrum

Similarly,

43

 Conjugate Symmetry: If f(m, n) is real, then F(u, v) is conjugate

symmetric, i.e

 Therefore, we usually display F(u–N/2,v–N/2), instead of F(u, v),

since it is easier to visualize the symmetry of the spectrum in this case

 This is done in Matlab using the fftshift command

44

 Multiplication: (In continuous-domain) This is the dual of the

convolution property Multiplication of two images corresponds to

convolving their spectra

F[f(x,y)h(x,y)] = F(u,v) H(u,v)

45

f(m,n)

47

|F(u–N/2,v–N/2)|

|F(u,v)|

 Average value: The average pixel value in an image:

 Notice that (substitute u = v = 0 in the definition):

48

 Differentiation: (Only in continuous-domain): Derivatives are normally used for detecting edged in an image An edge is the boundary of an object and denotes an abrupt change in grayvalue Hence it is a region with high value of derivative

49

 Edges and sharp transitions in grayvalues in an image contribute

significantly to high-frequency content of its Fourier transform

 Regions of relatively uniform grayvalues in an image contribute to frequency content of its Fourier transform

low- Hence, an image can be smoothed in the Frequency domain by

attenuating the high-frequency content of its Fourier transform

This would be a lowpass filter!

53

 For simplicity, we will consider only those filters that are real and

radially symmetric

 An ideal lowpass filter with cutoff frequency r0:

55

 Note that the origin (0, 0) is at the center and not the corner of the

image (recall the “fftshift” operation)

 The abrupt transition from 1 to 0 of the transfer function H(u,v) cannot

be realized in practice, using electronic components However, it can

be simulated on a computer

56 Ideal LPF with r0 = 57

57 Ideal LPF with r0 = 57

Original Image

58 Ideal LPF with r0 = 26

Ideal LPF with r0= 36

 Notice the severe ringing effect in the blurred images, which is a

characteristic of ideal filters It is due to the discontinuity in the filter

transfer function

59

 The cutoff frequency r0 of the ideal LPF determines the amount of

frequency components passed by the filter

 Smaller the value of r0, more the number of image components

eliminated by the filter

 In general, the value of r0 is chosen such that most components of

interest are passed through, while most components not of interest are eliminated

 Usually, this is a set of conflicting requirements We will see some

details of this is image restoration

 A useful way to establish a set of standard cut-off frequencies is to

compute circles which enclose a specified fraction of the total image power

60

 Suppose

where is the total image power

 Consider a circle of radius =r0(a) as a cutoff frequency with respect to

a threshold a such that

 We can then fix a threshold a and obtain an appropriate cutoff

frequency r0(a)

61

 A two-dimensional Butterworth lowpass filter has transfer function:

n: filter order, r0: cutoff frequency

62

 Frequency response does not have a sharp transition as in the ideal LPF

 This is more appropriate for image smoothing than the ideal LPF, since this not introduce ringing

65

LPF with r0= 18 Original Image

67 LPF with r0= 10

LPF with r0= 13

68

Image with false contouring

due to insufficient bits used

for quantization

Lowpass filtered version of

previous image

69 Original Image Noisy Image

70 LPF Image

 The form of a Gaussian lowpass filter in two-dimensions is given by

where

is the distance from the origin in the frequency plane

 The parameter s measures the spread or dispersion of the Gaussian curve Larger the value of s, larger the cutoff frequency and milder the filtering

 When s = D(u, v), the filter is down to 0.607 of its maximum value of

1

71

, v u v u

72

 Edges and sharp transitions in grayvalues in an image contribute

significantly to high-frequency content of its Fourier transform

 Regions of relatively uniform grayvalues in an image contribute to frequency content of its Fourier transform

low- Hence, image sharpening in the Frequency domain can be done by attenuating the low-frequency content of its Fourier transform This

would be a highpass filter!

73

 For simplicity, we will consider only those filters that are real and

radially symmetric

 An ideal highpass filter with cutoff frequency r0:

74

 Note that the origin (0, 0) is at the center and not the corner of the

image (recall the “fftshift” operation)

 The abrupt transition from 1 to 0 of the transfer function H(u,v) cannot

be realized in practice, using electronic components However, it can

be simulated on a computer

75 Ideal HPF with r0= 36

76 Ideal HPF with r0= 18

Original Image

77 Ideal HPF with r0= 26

Ideal HPF with r0= 36

 Notice the severe ringing effect in the output images, which is a

characteristic of ideal filters It is due to the discontinuity in the filter

transfer function

78

 A two-dimensional Butterworth highpass filter has transfer function:

n: filter order, r0: cutoff frequency

79

 Frequency response does not have a sharp transition as in the ideal HPF

 This is more appropriate for image sharpening than the ideal HPF,

since this not introduce ringing

81

HPF with r0= 47 Original Image

83 HPF with r0= 81

HPF with r0= 36

 The form of a Gaussian lowpass filter in two-dimensions is given by

where

is the distance from the origin in the frequency plane

 The parameter s measures the spread or dispersion of the Gaussian curve Larger the value of s, larger the cutoff frequency and more

severe the filtering

84

, v u v u

2 ,

1 , v e D u v u

85