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ing

(fax)DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

ECE 425

Image Science and Engineer

Spring Semester 2000

Course Notes

Robert A. Schowengerdt

schowengerdt@ece.arizona.edu

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

ECE402

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DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

DEFINITIONS

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tectionDR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

Remote sensing (earth science, environmentalweather)

Military (reconnaisance, surveillance, targetin

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of systems:

e total

e the DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

MATHEMATICS BACKGROUN

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ier analysis and

, joined by a

DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

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

DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

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:

DR. ROBERT A. SCHOWENGERDT schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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 schowengerdt@ece.arizona.edu

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