frequency domain methods

Image EnhancementFrequency domain methods

Prof. Duong Anh Duc

Image Enhancement: Frequency domain methods

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.

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EnhancedImage

PSS

Given Image

Frequency domain methods

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)

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EnhancedImage

Transfer Function

Given Image

1-d Fourier transform of a sequence

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

Fourier transform is periodic with 2 Inverse Fourier transform

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1-d Fourier transform of a sequence

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

Continuous-time Fourier transform

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2-d Fourier transform of a digital image

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

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2-d Fourier transform of a digital image

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

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Fourier Transform Example

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|F(u,v)| displayed as imagef(x,y)

Fourier Transform Example

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

Fourier Transform Example Image

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Image Magnitude Spectrum


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As the size of the box increases in spatial domain, the corresponding “size” in the frequency domain decreases.

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Fourier Transform of “Rice” Image

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|F(u,v)|f(x,y)

Fourier Transform of “Rice” Image

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F(u,v)

Fourier Transform of “Camera Man” Image

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|G(u,v)|g(x,y)

Fourier Transform of “Camera Man” Image

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G(u,v)

Image formed from magnitude spectrum of Rice and phase spectrum of Camera man

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Importance of Phase Information in Images

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Image formed from magnitude spectrum of Camera man and phase spectrum of Rice

Importance of Phase Information in Images

1-D Discrete Fourier Transform (DFT)

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)

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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.

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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).

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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.

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

NM image. Its 2-D DFT F(u,v) is defined as:

Inverse DFT is defined as:

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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.

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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.

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Convolution Example

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Convolution Example

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

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

Properties of DFT

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.

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Properties of DFT

Spatial scaling (only for continuous-domain):

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

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Properties of DFT

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.

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Properties of DFT

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

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Properties of DFT

o Similarly,

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Properties of DFT

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)So if

g(x,y) = f(x,y)*h(x,y)

is the output of an LTI transformation with PSF h(x,y) to an input image

f(x,y), then

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

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Properties of DFT

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.

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Properties of DFT

Correlation: In continuous-space, correlation between two images

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

Therefore,

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Properties of DFT

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.

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Properties of DFT

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).

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Properties of DFT

Convolution property for discrete images: Suppose

f(m,n), m = 0,1,2,…M–1, n = 0,1,2,…N–1 is an NM image and

h(m,n), m = 0,1,2,…K–1, n = 0,1,2,…L–1 is an NM image.

then

g(m,n) = f(m,n)*h(m,n) is a (M+K–1)(N+L–1) image.

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Properties of DFT

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

g(m, n) has that dimension).

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)

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Properties of DFT

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.

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Properties of DFT

Translation: (discrete and continuous case):

Note that

so f(m, n) and f(m–m0, n–n0) have the same magnitude spectrum

but different phase spectrum. Similarly,

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Properties of DFT

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.

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Properties of DFT

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)

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“Centered” Magnitude Spectrum

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f(m,n)

“Centered” Magnitude Spectrum

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|F(u–N/2,v–N/2)||F(u,v)|

Properties of DFT

Average value: The average pixel value in an image:

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

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Properties of DFT

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.

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Properties of DFT

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Lowpass and Hipass Filters

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Frequency domain Filters vsSpatial Filters

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Lowpass filtering

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 low-frequency content of its Fourier transform.

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!

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Ideal Lowpass filtering

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For simplicity, we will consider only those filters that are real and radially symmetric.

An ideal lowpass filter with cutoff frequency r0:

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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.

56Ideal LPF with r0 = 57

Ideal LPF examples

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Ideal LPF with r0 = 57 Original Image

Ideal LPF examples

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Ideal LPF with r0 = 26 Ideal LPF with r0= 36

Ideal LPF examples

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.

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Choice of cutoff frequency in ideal LPF

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.

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Choice of cutoff frequency in ideal LPF

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) .

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Butterworth lowpass filter

A two-dimensional Butterworth lowpass filter has transfer function:

n: filter order, r0: cutoff frequency

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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.

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Butterworth LPF examples

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LPF with r0= 18 Original Image

Butterworth LPF examples

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LPF with r0= 10 LPF with r0= 13

Butterworth LPF example: False contouring

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Image with false contouring due to insufficient bits used

for quantization

Lowpass filtered version of previous image

Butterworth LPF example: Noise filtering

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Original Image Noisy Image

Butterworth LPF example: Noise filtering

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LPF Image

Gaussian Low pass filters

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.

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22, vuvuD

22 2,, vuDevuH

Gaussian Low pass filters

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Highpass filtering

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 low-frequency content of its Fourier transform.

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!

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Highpass filtering

For simplicity, we will consider only those filters that are real and radially symmetric.

An ideal highpass filter with cutoff frequency r0:

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Highpass filtering

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.

75Ideal HPF with r0= 36

Ideal HPF examples

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Ideal HPF with r0= 18 Original Image

Ideal HPF examples

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Ideal HPF with r0= 26 Ideal HPF with r0= 36

Ideal HPF examples

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.

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Butterworth highpass filter

A two-dimensional Butterworth highpass filter has transfer function:

n: filter order, r0: cutoff frequency

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Butterworth HPF with r0 = 47 and 2

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Butterworth highpass filter

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

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Butterworth HPF examples

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HPF with r0= 47 Original Image

Butterworth HPF examples

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HPF with r0= 81 HPF with r0= 36

Gaussian High pass filters

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.

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22, vuvuD

22 2,1, vuDevuH

Spatial representations

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frequency domain methods

Education

d fourier

discrete n

y inverse fourier

d dft fu

n square image

n z2 fourier

analogous fourier transform

image magnitude spectrum