Digital Image Processing_ ch3 enhancement freq-domain

Chapter 3

Frequency Transforms &

Image processing in the Frequency Domain

Introduction

The spatial domain refers to the representation of an image as the array of gray-level intensity.

The electromagnetic spectrum consist of sinusoidel waves of different wavelengths (frequencies).

The frequency content of an image refers to the rate at which the gray levels change in the image

Rapidly changing brightness values correspond to high frequency terms, slowly changing brightness values correspond to low frequency terms

The Fourier transform is a mathematical tool that analyses a signal (e.g. images) into its spectral components depending on its wavelength (i.e. frequency) content.

Fourier Transforms In 1822, Jean B. Fourier has shown that any function f(x) that

have bounded area with the x-axis can be expressed as a linear combination of sine and/or cosine waves of different frequencies.

This is also applicable functions of 2 variables, e.g. images.

+

+

+

=

Q. Can we recover the different frequencies of this signal?

Fourier Transforms Fourier transform converts a signal from time (or space) domain

to frequency domain

Illustration of Fourier Theory for images

Every row is Sine wave of frequncy 1

Sine wave with frequncy 3Sine wave with frequncy 2

Mixed waves with frequncy 5, 2 &1Combined waves frequncy 1+2+3

MATLAB generated images MATLAB can be used to generate images with patterns of and

desired rate of change of brightness. For this we need to use trigonometric functions of 2 variables as

indicated by the following example code:clear all;A=zeros(256,256);B=A;for i=1:1:256 for j=1:1:256 A(i,j)=a1*sin(pi*(a2*i+a3*j)/m);

B(i,j)=b1cos(pi*(b2*i+b3*j)/n); //m and n are to be powers of 2.

endendC=A+B;imshow(C);imwrite(C, 'SinoPattern2.bmp')

Images generated from Sinoside function

Fourier Transform - Definition

The Discrete Fourier Transform (DFT) of f(x) is defined as:

1

0

1

0

) 2

sin()() 2

cos()(M

u

M

u M

xuuI

M

xuuRf(x)

The u values u = 0, 1, ..., M-1) is the frequency domain of f(x).

One can use Excel to implement Fourier tranforms.

1

0

1

0

) 2

sin()(1

) 2

cos()(1

M

x

M

x M

xuxf

MM

xuxf

MF(u)

And the inverse DFT (IDFT) is defined as:

Fourier Transform - continued

.)()(

,)/2sin()(1

,)/2cos()(1

1

0

1

0

uIuR F(u)

MuxxfM

I(u)

MuxxfM

R(u)

M

x

M

x

i.e.

part.imaginary the called

and part, real the called

Unlike f(x), F(u) is a complex valued function, i.e. is a pair of functions involving trigonometric functions:

F is represented by its MAGNITUDE and PHASE rather that its REAL and IMAGINARY parts,

where: MAGNITUDE(u) = SQRT( R(u)^2+IMAGINARY(u)^2 )

The phase angle of the transform is:

).)(

)((tan)( 1

uR

uIu

PHASE(u) = ATAN( IMAGINARY(u)/REAL(u) )

The 2-dimensional DFT The DFT of a digitised function f(x,y) (i.e. an image) is defined as:

Note that, F(0,0) = the average value of f(x,y) and is refered to as the DC component of the spectrum.

It is a common practice to multiply the image f(x,y) by (-1)x+y. In this case, the DFT of (f(x,y)(-1)x+y) has its origin located at the centre of the image, i.e. at (u,v)=(M/2,N/2).

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

)/ / ( 2sin(),(

)/ / ( 2cos(),(

.before asmanner similar ain defined is DFT inverse theand

)/ / ( 2sin(),(1

)/ / ( 2cos(),(1

,

),(N

u

M

v

N

u

M

v

M

x

N

y

M

x

N

y

NyvMxuvuI

NyvMxuvuR

NyvMxuyxfMN

NyvMxuyxfMN

v)F(u

yxf

The Fourier spectrum – in 2D

The Fourier spectrum – in 2D

• The original image contains two principal features: edges run approximately at ±45ο .

• The Fourier spectrum shows important components in the same directions.

Fourier Spectrum

Log enhanced version of Fourier Spectrum

Original imageInverse Fourier

The FTs also tend to have bright lines that are perpendicular to lines in the original letter. If the letter has circular segments, then so does the FT.

Filtering in the Frequency Domain – Scheme

The Notch filter

Original image

Note that the edges stand out more than before filtering. When the average value is 0, some values of the filtered image are negative, but for display purposes pixel values are shifted.

Image after Notch filter application

A simple filter that forces the average image value to become 0.

The average value of an image f(x,y) is the DC component of the

DFT spectrum i.e. F(0,0). The Notch filter is defined as follows:

otherwise.

N/2)(M/2, v)(u, if

1

0),(

vuH

Low-pass and High-pass filtering

Low frequencies in the DFT spectrum correspond to image values over smooth areas, while high frequencies correspond to detailed features such as edges & noise.

A filter that suppresses high frequencies but allows low ones is called Low-pass filter, while a filter that reduces low frequencies and allows high ones is called High-pass filter.

Examples of such filters are obtained from circular Gaussian functions of 2 variables (see next slide)

.)e()v,u(H

,e)v,u(H

/)vu(

/)vu(

filter Highpass- 12

1

filter, Lowpass - 2

1

222

222

22

22

Low-pass & High-pass filtering - Example

Low pass filtering

High pass filtering

Low pass filtering results in blurring effects, while High pass filtering results in sharper edges.

Wavelet Transforms

Wavelet analysis allows the use of long time intervals for more precise low-frequency information, and shorter intervals for high-frequency information.

A wavelet (i.e. small wave) is a mathematical function used to analyse a continuous-time signal into different frequency components at different resolution scale.

A wavelet transform of a function is a representation of f wavelets. The wavelets are scaled and translated copies of a finite-length or fast-decaying oscillating waveform (t), known as the mother wavelet.

There are many wavelet filters to choose from. Here we only discuss the Discrete Wavelet Transform.

Wavelet Transforms -Properties

The Wavelet transform is a short time anlysis tool of finite energy quasi-stationary signals at multi-resolutions.

The Discrete wavelet transform (DWT) provide a compact representation of a signal’s frequency commponents with strong spatial support.

DWT decomposes a signal into frequency subbands at different scales from which it can be perfectly recontructed.

2D-signals such as images can be decomposed in many different ways.

The Haar Wavelet

It can be implemented using a simple filter:

If X={x1,x2,x3,x4 ,x5 ,x6 ,x7 ,x8 } is a time-signal of length 8, then the Haar wavelet decomposes X into an aproximation subband containing the Low frequencies and a detail subband containing the high frequencies:

Low= {x2+x1, x4+x3 , x6+x5 , x8+x7 }/2

High= {x2-x1, x4-x3 , x6-x5 , x8-x7 }/2

The Haar wavelet

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-2 0 2 4 6 8 10

The Haar wavelet is a discontinuous, and resembles a step function.

It is a crude version of the Truncated cosine.

Haar Wavelet – in MATLAB

Wavelet Decomposition of Images

1 stage Transformation

After 2 stages

Original Image

…

A Haar wavelet decompose images first on the rows and then on the columns resulting in 4 subbands, the LL-subband which an approximation of the original image while the other subbands contain the missing details

The LL-subband output from any stage can be decomposed further.

Different Decomposition Schemes.

The previous 2 decomposition scheme is known as the Pyrimad scheme, whereby at successive stages only the LL subband is wavelet transdormed.

Other decomposition schemes include:

The standard scheme – At every stage all the image is wavelet transformd

The wavelet packet – After stage 1, a non-LL subband is transformed only if it satisfied certain condition.

The Quincux – During each stage, the columns decomposition is only applied on the L-subband

LL subband

0

100

200

300

400

500

600

700

1 25 49 73 97 121 145 169 193 217 241

HL subband

0

1000

2000

3000

4000

5000

6000

1 25 49 73 97 121 145 169 193 217 241

LH subband

010002000300040005000600070008000

1 25 49 73 97 121 145 169 193 217 241

HH subband

0

1000

2000

3000

4000

5000

6000

1 25 49 73 97 121 145 169 193 217 241

Statistical Properties of Wavelet subbands

Original pixels distribution of Mandrill

0

500

1000

1500

2000

2500

3000

1 20 39 58 77 96 115

134

153

172

191

210

229

248

coefficient value

frequ

ency

The distribution of the LL-subband approximate that of the original but all non-LL subbands have a Laplacian distribution. This remains valid at all depths.

Applications of Wavelet Transforms

The list of applications is growing fast. These include:

Image and video Compression

Feature detection and recognition

Image denoising

Face Recognition

Signal interpulation

Most applications benefit from the statistical propererty of the non-LL subbands (The laplacian distribution of the wavelet coefficients in these subbands).

Wavelet-based Feature detection Non-LL subbands of a wavelet decomposed image contains high

frequencies (i.e. image features) which are highlighted. These significant coefficients are the furthest away from the mean.

Thresholding reveals the main features.

Horizontal features

Vertical features

28

Extracting significant coefficients

Best value for sigma is related to the STD

0

NSC

SC

- +

HH1 LH1

HL1 HH2

HL2

LH2

Feature enhancement method using wavelet sub-band segmentation

29

End of Chapter 3