9

2

Image Formation

2.1 Introduction

An image is a projection of a three-dimensional scene in the object space to a

two-dimensional plane in the image space. An ideal imaging system should

map every point in the object space to a dened point in the image plane,

keeping the relative distances between the points in the image plane the

same as those in the object space. An extended object can be regarded as an

array of point sources. The image so formed should be a faithful reproduc-

tion of the features (size, location, orientation, etc.) of the targets in the object

space to the image space, except for a reduction in the size; that is, the image

should have geometric delity. The imaging optics does this transformation

from object space to image space.

The three basic conditions that an imaging system should satisfy to have a

geometrically perfect image are (Wetherell 1980) as follows:

1. All rays from an object point (x, y) that traverse through the imaging

system should pass through the image point (x’, y’). That is, all rays

from an object point converge precisely to a point in the image plane.

The imaging is then said to be stigmatic.

2. Every element in the object space that lies on a plane normal to the

optical axis must be imaged as an element on a plane normal to the

optical axis in the image space. This implies that an object that lies in

a plane normal to the optical axis will be imaged on a plane normal

to the optical axis in the image space.

3. The image height h must be a constant multiple of the object height,

no matter where the object (x, y) is located in the object plane.

The violation of the rst condition causes image degradations, which

are termed aberrations. The violation of the second condition produces eld

curvature, and the violation of the third condition introduces distortions.

Consequences of these deviations will be explained in Section 2.4. In addi-

tion, the image should faithfully reproduce the relative radiance distribution

of the object space, that is, radiometric delity.

10 Building Earth Observation Cameras

2.2 Electromagnetic Radiation

As electromagnetic (EM) energy forms the basic source for Earth observation

cameras, we shall describe here some of the basic properties of EM radiation,

which help us to understand image formation. Various properties of EM radia-

tion can be deduced mathematically using four differential equations, gener-

ally referred to as Maxwellʼs equations. Readers interested in the mathematical

formulation of EM theory may refer to the work by Born and Wolf (1964). Using

Maxwellʼs equations, it is possible to arrive at a relationship between the velocity

of an EM wave and the properties of the medium. The velocity c

m

in a medium

with an electric permittivity ε and magnetic permeability µ is given as follows:

c

=

ε

1

m

(2.1)

In vacuum,

ε=ε×

=π×

0

−

−

8.85 10 farad/m

410henry/m

12

0

7

Thus, the velocity of the EM radiation in vacuum, c, is given by

c

=

ε

×

−

1

~3 10 ms

00

81

This value is familiar to the readers as the velocity of light. The wave-

lengthλ, frequency ν, and the velocity of the EM wave c are related such that

c =νλ

(2.2)

Other quantities generally associated with wave motion are the period

T(1/ν), wave number k (2π/λ), and angular frequency ω(2πν).

The terms ε and µ in a medium can be written as

ε=εε

r 0

and =

r0

,

where

ε

r

is the relative permittivity (called the dielectric constant) and µ

r

is

the relative permeability. Therefore the velocity of EM radiation in a medium

c

m

can be expressed as,

111

m

r0 r0 rr 00

c =

εε

=

εε

m

rr

c

cc

n

=

ε

=

(2.3)

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