03 - Image Transforms

8/8/2019 03 - Image Transforms

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Image Transforms

copyright 2002H. R. Myler

Fourier Transform Pair:

2 ux v

+

( )2( , ) ( , )j ux vy

f x y F u v e dudv

+

=

, ,u v x y e x y

=


Same properties as in 1D apply


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Fourier Transform Pair:

We can show that for a box image of finite size X by Yand of a constant value A:

( ) ( ) ( )2 2

2

2 2

sin sin( , )

X Y

j ux vy

X Y

uX vY F u v Ae dxdy AXY

uX vY

+

= =

The magnitude (spectrum) is:


( ) ( )sin sin( , )

uX vY F u v AXY

uX vY

=

maxima of:cos 2 (ux+vy)



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http://sepwww.stanford.edu/oldsep/hale/FftLab.html


Fast Fourier TransformTo approximate a function by samples, and toapproximate the Fourier integral by the discrete Fouriertransform, requires applying a matrix whose order is the

.

Since multiplying an n x n matrix by a vector costs on theorder of n2 arithmetic operations, the problem gets quicklyworse as the number of sample points increases.

However, if the samples are uniformly spaced, then the


sparse matrices, and the resulting factors can be appliedto a vector in a total of order n log narithmetic operations.

This is the so-called fast Fourier transform or FFT.


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A separable transform can be written as:

1 1

1 2

0 0

( , ) ( , ) ( , ) ( , )M N

x

T u v f x y g x u g y v

= ==

The kernel is symmetric if:

1 1

1 2

0 0

( , ) ( , ) ( , ) ( , )M N

u v

f x y T u v h x u h y v

= =

=


g1(x,u)= g2(y,v)

A separable transform can be computed intwo steps:

1N

1

1

0

( , ) ( , ) ( , )M

x

T u v T x v g x u

=

=

2

0

, , ,y=

=


This means that the transform can becomputed on the rows of the image, andthen on the columns.


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This means that the transform can becomputed on the rows of the image, and

then on the columns.f(x,y) processrows

T(x,v)

T(u,v) processcolumns


2D DFT example

Bright spots

in the corners

Centered


spectrum


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2D DFT examples with enhanced details

Details were enhancedby the logtransformation.

Both spectrum of thetranslated rectangle isidentical to the spectrum

of the initial rectangle(previous slide).

The spectrum of therotated ima e also


rotates by the sameangle

Walsh Transform:

( ) 11

( ) ( )1( , ) 1 i n i

nb x b u

g x u

=

b k z is the kth bit from the binaryrepresentation of z.

e.g., let z=13, then z=1101 binary.=

0i=


3 z =

b 2 z =1b 1 z =0


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TheWalshtransformusessquarewavesas

it's

basis

functions,

and

these

vary

from

1

to+1.

whendiscussingtheWalsh.TheadvantageoftheWalshtransformis

thatitdoesnotrequirefloatingpointmathortranscendentalfunctions.

TheinverseWalshkernelisidenticaltothe


forwardkernel.

Hadamard Transform:

TheHadamardtransformissimilartotheWalshinthatitusessquarewavesof1to+1in

.Thetransformiseasilyderivedfrom

multiplicationoftheimagewiththeHadamardmatrix:

F=


Thelowest

order

Hadamard

matrices

are

given

by:

0 1 2

1 1 1 1

1 1 1 1 1 11 11; ;

1 1 1 1 1 122

1 1 1 1

H H H

= = =


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Anyorderof2NHadamardmatrixcannowbederivedfrom:

22

N N

N

N N

HH H

=


,omitted.

Discrete Cosine Transform(DCT):

g(x,u) = 2N

cos2x+1u

2NKernel:

ThistransformissimplerthantheFourierasitdoesnotrequirecomplexnumbers.TheDCThastheaddedadvantageofmirrorsymmetry.

1 1

0 0

1 1( , ) ( , )cos cos

2 2

M N

x y

F u v f x y x u y vM N

= =

= + +

2D:


IftheDCTisusedforblockencoding,wheretheimageispartitionedintosubpicturespriortotransforming,theDCThaslessedgedegredationbecauseofit'ssymmetryproperties.

Theinversekernelisidenticaltotheforwardform.

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03 - Image Transforms