lecture chapter 4 frequency domain processing

The values of fx are multiplied by sines and cosines of various frequencies © The domain values of u of Fu is appropriately called frequency domain because u determines the frequency of

Chapter4 Frequency Domain

Processing

Image Enhancement in the

Frequency Domain

Introduction to the Fourier Transform

« Any function that periodically repeats itself can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient (Fourier series)

+» Even functions that are not periodic (but whose area under the curve is finite) can be expressed as the integral of sines and/or cosines multiplied by a weighting function (Fourier transform)

» The advent of digital computation and the “discovery” of fast

revolutionized the field of signal processing, and allowed for the first time practical processing and meaningful interpretation of

importance.

Frequency Domain Processing

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1D Fourier transform and Its inverse

2D Fourier transform and Its inverse

Discrete Fourier Transform

(Discrete Fourier transform, DFT)

© The multiplier 1/M in front of the Fourier transform

sometimes is placed in front of the Inverse instead

© Or, both equations are multiplied by 1/VM

Discrete Fourier Transform

© Each term of the Fourier transform is composed of the sum of all values

fo the function f(x) The values of f(x) are multiplied by sines and cosines

of various frequencies

© The domain (values of u) of F(u) is appropriately called frequency

domain because u determines the frequency of the components of the transform

Discrete Fourier Transform

The Fourier transform may be viewed as a *‘mathematical prism” that separates

a function into various components based on frequency content

In the analysis of complex numbers, it is convinient to express F(u) in polar

2D DFT and Inverse DFT (IDFT)

x, y: image pixel position

u, VU: spatial frequency

Filtering in frequency domain

+ Low-pass filters: preserve low frequencies, useful for noise suppression

useful for edge detection

Filtering in frequency domain

Computing and visualizing DFT

original Image Fourier spectrum

Computing and visualizing DFT

Computing and visualizing DFT

original Image Fourier tranform fft2

ifft2

Computing and visualizing DFT

subplot(2, 2, 1), mmshow(]); title(original Image’, 'FontSize',14)

subplot(2, 2, 2);imshow(F);title(‘Fourier tranform fft2",

'FontSize',14)

subplot(2, 2, [3 4]); imshow(F0);title( Centered spectrum:

ifft2', ‘'FontSize',14)

Filtering in the frequency domain

ƒ(x,y)*ñh(x, y) © Alu,v)F (u,v)

The convolution theorem:

+» Linear spatial convolution is by convolving f(x,y) and

h(x,y) The same result is obtained in the frequency domain

by multiplying F(u,v) and H(u,yv)

« The basic idea in frequency domain is to select a filter transfer function that modifies F(u,v) in a specified manner + A filter is called low pass filter if it attenuates the high

frequency components to F(u,v) while leaving the low frequencies relatively unchanged.

Filtering in the frequency domain

-0.0000 0.0000 -0.0000 -0.0000

0.0000 0.0000

1.0000 2.0000 3.0000 0.0000

6.0000 -0.0000 9.0000 0

-0.0000 0.0000 0.0000 0.0000

-0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000

+ It is more efficient to use the Fourier transform than

a spatial filter for a large filter

* The Fourier Transform also allows us to isolate and process particular image frequencies and so perform low-pass and high-pass filtering with a great degree

of precision.

Paddedsize function

function PQ = paddedsize(AB, CD, PARAM)

% PQ(1) = PQ(2) = 24nextpow2(2*m), where m is MAX({[AB CD])

if nargin ==

PQ = 2*AB; elseif nargin == 2&~ischar(CD)

PQ = AB + CD - 1; PQ =2 * ceil(PQ / 2);

elseif nargin == 2

m = max(AB); % Maximum dimension

% Find power-of-2 at least twice m

Paddedsize function

function H = Ipfilter(tyoe, M, N, DO, n)

function [U, V] = dftuv(M, N)

function [U, V] = dftuv(M, N)

%DETUV Computes meshgrid frequency matrices

% [U, V] = DF TUV(M, N) computes mesherid frequency matrices U and

% V.U and V are useful for computing frequency-domain filter

% functions that can be used with DF TFILT U and V are both M-by-N

% Set up range of variables

% Compute the meshørid arrays

[Y, U] = meshgrid(v, u);

Application of function g = aftfilt(f, H)

Obtain Frequency Domain Filters from Spatial Filters

Consider the image, f, of size 600 X 600 pixels shown in

following figure In what follows, we generate the frequency

domain filter, H, corresponding to the Sobel spatial filter that

enhance vertical edges We then compare the result of

filtering f in the spatial domain with the Sobel mask process

in the frequency domain.

ain Filters from Spatial Filters

Obtain Frequency Domain Filters from Spatial Filters

Obtain Frequency Domain Filters from Spatial Filters

Obtain Frequency Domain Filters from Spatial Filters

Obtain Frequency Domain Filters from Spatial Filters

Obtain Frequency Domain Filters from Spatial Filters

Smoothing Frequency-Domain Filters

+ The basic model for filtering in the frequency domain

where F(u,v): the Fourier transform of the image to be

« There are several standard forms of lowpass filters (LPF)

— Ideal lowpass filter

— Butterworth lowpass filter

— Gaussian lowpass filter

Ideal Lowpass Filters (ILPFs)

The simplest lowpass filter 1s a filter that “‘cuts off’ all high-

frequency components of the Fourier transform that are at a

distance greater than a specified distance D, from the origin of

Ideal Lowpass Filters (ILPFs)

ILPF is a type of “nonphysical” filters and can't

be realized with electronic components and is not

very practical

ideal Lowpass Filters (ILPFs)

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Butterworth Lowpass Filters (BLPFs) with order n

between order n and

The BLPF may be viewed as a transition between ILPF AND

GLPF, BLPF of order 2 is a good compromise between

effective lowpass _ fitering and acceptable ringing characteristics

Gaussian Lowpass Filters (FLPFs)

Gaussian Lowpass Filters (FLPFs)

Gaussian Lowpass Filters (FLPFs)

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Gaussian Lowpass Filters (FLPFs)

Gaussian Lowpass Filters (FLPFs)

DO=10; H=exp(-(U.42+V.%2)/(2*(D0“%2))); ø=dftfilt Œ, H);

subplot(2, 3, 1), mmshow(ø.| |); tide( D0=10”, 'FontSize’, 16)

DŨ=1S; H=exp(-(U.^2+YV.^2)/(2*(D0^2))); ø1=dftfit (f., H);

subplot(2, 3, 2),mshow(ø1.|]); tie( D0=1S, 'FontSize’, 16)

D0=30; H=exp(-(U.^2+YV.^2)/(2*(D0^2))); ø2=dftfit (f H);

subplot(2, 3, 3),mshow(ø2.[]); tie( D0=30”, 'EontS1ze', 16)

DO0=60; H=exp(-(U.42+V.%2)/(2*(D0%2))); g3=dithilt , H);

subplot(2, 3, 4),mshow(ø3.|]) ; tile( D0=60", 'FontSize', 16)

DO0=80; H=exp(-(U.42+V.%2)/(2*(D0%2))); g4=dithilt i, H);

subplot(2, 3, 5),mshow(ø4.| |); title "D0=80'," 'EontS1ze', 16)

D0=230; H=exp(-(U.^2+VY.^2)/(2*(D0^2))); ø5=dftilt ( H);

subplot(2, 3, 6),mshow(ø5.[]); tile(D0=200", Kon(tS1ze, 16)

Gaussian Lowpass Filters (FLPFs)

D0=30

Wireframe ano) Surface Plotting

+ Basic syntax:

mesh (H)

+ This function draws a wireframe for x=1:M, y=1:N,where

[M,.N]Z=size(H) If M and N are large, plot typically are

unacceptable So usually using:

mesh (H(1:K:end,1:K:end))

+ MATLAB plots mesh fihures in color , by default The

command

colormap({[000]) “%set the wireframe to black

4 Grid and axes can be turned off by

grid off

axis off

Wireframe and surface Plotting

» The view point (location of the observer) is controlled by

Wireframe and surface Plotting

+ To determine the current viewing geometry , we type

>> [az, el] = view;

» To set the viewpoint to the default values, we type

Wireframe and surface Plotting

>>view(-45,60)

Figure (c) shows the result of

>>view(-25,0)

Wireframe and surface Plotting

Sometimes it is desirable to plot a function as a surface instead of as a

wirefram Function surf does this

>> surf (H)

This function produces a plot identical to mesh; To convert the color

to gray, we use the command

Wireframe and surface Plotting

Wireframe and suriace Plotting

When the objective is to plot an analytic function of two variables, we

use meshgrid to generate the coordinates and from these we generate the discrete (sampled) matrix to use in mesh or surf For example, to plot

Basic Highpass Filtering

Given the transfer function H/, (u,v) of a lowpass filter, we obtain the

transfer function of the corresponding highpass filter by using the simple relation

H,,(u,v)=1-H, (u,v)

function H= hpfilter (type,M,N, DO,n)

% HPFILTER Compute frequency domain highpass

High Frequency Domain Filter

highpass filters <=> lowpass filters

H,,, (u,v) =l—H, u,v)

High Frequency Domain Filter

Ideal highpass filter

Butterworth highpass filter =»

Gaussian highpass filter

filter H (u,v) = 1 if Dtu,v)>D, :

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High Frequency Domain Filter

High Frequency Domain Filter