digital image processing fundaments

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ing

(fax)DR. ROBERT A. SCHOWENGERDT [email protected]

ECE 425

Image Science and Engineer

Spring Semester 2000

Course Notes

Robert A. Schowengerdt

[email protected]

(520) 621-2706 (voice), (520) 621-8076

ECE402

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DR. ROBERT A. SCHOWENGERDT [email protected]

DEFINITIONS

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tectionDR. ROBERT A. SCHOWENGERDT [email protected]

Image science

The theory of optical image formation and de

Includes elements of:

optics

radiometry

linear systems

statistics

vision

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ission, storage DR. ROBERT A. SCHOWENGERDT [email protected]

Image engineering

The technologies of image acquisition, transmand display

Includes elements of:

detectors

signal processing

data compression

image processing

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rn life

ing, digital

oto scanners and

otics)

n monitoring)

l models, DR. ROBERT A. SCHOWENGERDT [email protected]

OVERVIEW

Electronic imaging systems pervade mode

Examples

Document processing (scanning, storage, printlibraries, WWW)

Consumer products (HDTV, digital cameras, phprinters)

Machine vision (quality control inspection, rob

Medical imaging (disease diagnosis, medicatio

Scientific visualization (complex mathematicainteractive graphics)

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monitoring,

g)DR. ROBERT A. SCHOWENGERDT [email protected]

Remote sensing (earth science, environmentalweather)

Military (reconnaisance, surveillance, targetin

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of systems:

e total

e the DR. ROBERT A. SCHOWENGERDT [email protected]

THE SYSTEMS APPROACH

A perceived image is the result of a chain

optics

detector

coding/decoding

display

human vision

Each can be considered a subsystem of thelectronic imaging system

The engineering design goal is to optimiz

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ion to that of

ing system DR. ROBERT A. SCHOWENGERDT [email protected]

performance of each subsystem in relatthe others and the total system

This course covers the tools used for imaganalysis, design and evaluation

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systems

ctronic

humanvision

subsystem

tina*

neuralnetwork

brainDR. ROBERT A. SCHOWENGERDT [email protected]

An imaging system consists of several sub

* points of signal transduction, optical ele

lightsource scene

imageacquisitionsubsystem

transmissionsubsystem

displaysubsystem*

optics detector* electronics

coder decoder

optics

re

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with:

tics)

ystems)DR. ROBERT A. SCHOWENGERDT [email protected]

For optical components, were concerned

size and location of the image (geometrical op

intensity of the image (radiometry)

contrast and sharpness of the image (linear s

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ed with:

ers, A/D DR. ROBERT A. SCHOWENGERDT [email protected]

For electronic components, were concern

image sampling and quantization (analog filtconverters, coding)

image processing (digital signal processing)

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L TOOLS

s

ier TransformDR. ROBERT A. SCHOWENGERDT [email protected]

SECTION I MATHEMATICAMathematics Background

Convolution and Fourier Transforms

Linear Filtering and Sampling

Two-dimensional Functions and Operation

Discrete Fourier Transform and Fast Four

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DDR. ROBERT A. SCHOWENGERDT [email protected]

MATHEMATICS BACKGROUN

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ier analysis and

, joined by a


Complex Notation

Complex arithmetic will be necessary for Fouroptics

Complex numbers consist of two real numbersphasor relationship

where

c is a complex number,

a is the real part of c

b is the imaginary part of c

j is

Phasor relationship

c a jb+=

1

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b

arealpart

imaginarypart

c

A


The amplitude A of c

The phase of c

Can write c as which, by Eulers Theorem,

A a2

b2

+=

b a( )atan=

c Aej

=

c A cos j sin+( )=

AaA--- j

bA---+ =

a jb+=

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ing relations:


Using Eulers Theorem, we can derive the follow

? Using Eulers Theorem, show that

( )sin 12 j----- e

je

j( )=

( )cos 12--- e

je

j+( )=

sin( )2 cos( )2+ 1=

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ite and its

area

is ce in plots.

ea

n

n

x

x x0 b( )]DR. ROBERT A. SCHOWENGERDT [email protected]

Simple Functions

Delta function and its relatives

delta

NOTE: The delta functions amplitude is infin1. The amplitude is shown as 1 for convenien

? Write the equation that defines the arof a delta function as 1.

? Review the definition of delta functioin terms of the limit of conventional functions, such as the rectangle functio

even delta pair

x x0( )

1

x0x

1x0 = 0

x x0

b-------------- b x x0 b+( ) +[=

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x+ b

x x0 b( ) ]

x+ bDR. ROBERT A. SCHOWENGERDT [email protected]

odd delta pair

x

|b|

b- b

|b|

x0 x0 - b

x0 = 0 x0 0

x x0

b-------------- b x x0 b+( )[=

x

|b|

b- b

x0

x0 - b

x0 = 0

-|b|

|b|

-|b|

x0 0

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xx0+2b

. . .DR. ROBERT A. SCHOWENGERDT [email protected]

comb (shah) comb x x0b

-------------- b x x0 nb( )n =

=

x

|b|

b- b

x0 = 0

0 2b- 2b

|b|

x0+bx0-b 0x0-2b x0

. . .. . . . . .x0 0

Even delta pair, odd

delta pair and comb

functions are all scaled by b

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a particular shift

alue of the ble)DR. ROBERT A. SCHOWENGERDT [email protected]

Use of the function

sifting

NOTE: Sifting is a convolution, evaluated for

Finds the value of a function at a specific vindependent variable (similar to a look-up ta

sampling

f ( ) x0( ) d

f x0( ) constant= =

f x( ) x x0( ) f x0( ) x x0( )=

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mined by the value independent

x

)DR. ROBERT A. SCHOWENGERDT [email protected]

NOTE: Sampling is a mulitplication

Output is a delta function, with area deterof the function at the specified value of the variable.

uniform sampling

xx0 x0

x

x =f(x)

f(x0)

x0

1b----- f x( )comb

x x0

b-------------- f x0 nb+( ) x x0 nb(

n =

=

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tain amplitude of

xb x0+2b

. . .

f x x0( )=

xDR. ROBERT A. SCHOWENGERDT [email protected]

NOTE: Must divide comb function by |b| to ref(x).

NOTE: f(x) modulates the comb function.

shifting

replicating

xx0+bx0-b 0 x0+2bx0-2b x0

. . .. . .x0+x0-b 0x0-2b x0

. . .

1/|b|

g x( ) f x( ) x x0( ) f ( ) x x0 ( )d

= =

xx0

x

=f(x)

x0

g(x)1

g x( ) 1b----- f x( ) comb

x x0

b-------------- =

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of f(x)

xx0+bx0

. . .

x b 1 2>x b 1 2=x b 1 2

digital image processing fundaments

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