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 dened 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×
×
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 λ
(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
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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