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10-信息光学2.pdf
#2 Review of Linear Systems
and Fourier Transforms
1
Systems
Imaging
p1(x1, y1) System S{p1(x1, y1)}
= p2 (x2 , y2 )
S{ }
A system accepts an input signal and produces an output signal.
Mathematically, a system can be described using an operator S{ } that
maps a set of input functions onto a set of output functions.
For imaging systems, the inputs and outputs are generally two
dimensional complex-valued functions.
2
Examples of linear and nonlinear systems
Linear System
Multiply by 5
S{p(x1, y1) + q(x1, y1)} = 5p(x1, y1) + 5q(x1, y1)
Linear since the
input signals interact
Nonlinear independently
System
Square
2 2
S{p(x1, y1) + q(x1, y1)} = p (x1, y1) +q (x1, y1) + 2 p(x1, y1)q(x1, y1)
Not linear since the input signals
interact with one another in this
term. 3
Linear systems satisfy superposition
and scaling properties
Suppose we have a signal that can be composed of a sum of
“elementary” functions.
Response to an individual
elementary function:
Response to an input signal
composed of these scaled
elementary functions
(inputted at the same time
into the system):
S{ap(x1, y1) + bq(x1, y1)} = aS{p(x1, y1)}+ bS{q(x1, y1)}
where a, b are constants (can be complex-valued)
4
Properties of Linear Systems
The system treats each of the elementary
functions p(x1,y1) and q(x1,y1)
independently.
S{ap(x1, y1) +
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