linear filtering 2

8/13/2019 Linear Filtering 2

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contd.

Linear Filtering and Convolution

Ideal Low-pass Filter


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Butterworth LP Filter


Butterworth LP

Filter, 2nd Order


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Gaussian


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Gaussian

Variable Cutoffs


Ideal highpass filter

Butterworth highpass filter

Gaussian highpass filter

0

0

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Dv)D(u,if0),( vuH

nvuDvuH

2

0 ),(D1

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2

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Ideal


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Butterworth


Gaussian


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Example of series operations


Image sharpening is frequently used to improve

the appearance of images by increasing

delineation between different regions of possibly

limited contrast. The goal is to improve high spatial frequency

definition with minimal disturbance to the contrast

information.


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Deconvolution

Deconvolution is exactly what it sounds like: the

undoing of convolution.

This means that instead of mixing two signals in

convolution, we are isolating them.

This is used for analyzing the characteristics of the

input signal and the impulse response when given

only the output of the system.

Given y(t) = x(t) * h(t), the system should isolatex(t) and h(t) so that we may study each

individually.

Ideal Deconvolution System


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Deconvolution

Instead of producing one system that outputs both

the convolved signals, it will be easier to consider

separate systems that output one signal at a time.

Each system here is annihilating the undesired

signal, and is called a homomorphic filter

Homomorphic Filters Homomorphic filters have been used in signal

processing methods since the 1970s.

A Homomorphic filter is a system which accepts asignal composed of two components and returns thesignal with one of the components removed.

A frequently used method is to convert the convolutionof two signals into a sum, and then implement ahomomorphic filter to remove signal components.


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In frequency space, one can use a filter that

reduces DC by 50%, then increases the filter

function linearly to 4.0 96 pixels from DC and

remains constant thereafter.

In this case, the total power remains unchanged.


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MATLAB Syntax

Convolution is an example of a neighborhood function.

Dilation and median filtering are also examples of

neighborhood functions.

MATLAB provides functions for implementing

neighborhood functions across images using a single

command.

MATLAB Syntax

Convolution kernels may be defined manually, or

derived from MATLAB library.

As an example, one may define a 3x3 smoothing filter:h1=[0 .5 0; .5 1 .5; 0 .5 0];

Another approach is to use the fspecial function:

h1= fspecial(sobel);


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MATLAB Syntax

To perform the convolution with your filter and an

input image (A), use the conv2 function:

B = conv2(A,h1);

Where B is the output image, and h1 is the filter

kernel.

MATLAB Syntax

MATLAB also provides for higher dimension

convolution using the convn function which is

useful, for example, when working with multiple

RGB arrays. For example if you have 5 RGB images, instead of

calling conv2 15 times, a 4D array can be

assembled and convn called once.


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MATLAB Syntax

To determine the rank of a matrix, A:

b = rank(A);

If the rank = 1, use singular value decompositionto find the row and column vectors:

[u,s,v]=svd(A);

kcol = u(:,1) * sqrt(s(1));

krow = conj(v(:,1))' * sqrt(s(1));

If the process was performed correctly, thenkcol krow

should give you back the original kernel.

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