Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

where a1 and a2 are scalars. Above ðx; yÞ is replaced by ½m; n& in the case of a linear discrete-space system.

Suppose that the input at ðx1; y1Þ is the delta function d(x1, y1) (see Appendix A for a discussion of the delta function). The output at location ðx; yÞ is defined as

hðx2; y2; x1; y1Þ is called the impulse response (point-spread function) of the system. The sifting property of the delta function allows an arbitrary input uðx; yÞ to be

expressed as

Now the output can be written as

gðx; yÞ ¼ O½uðx; yÞ&

ð1 ð

¼uðx1; y1ÞO½dðx x1; y y1Þ&dx1dy1

1 ð2:2-6Þ

1ð

¼uðx1; y1Þhðx; y; x1; y1Þdx1dy1

1

This result is known as the superposition integral. Physically, the delta function corresponds to a point source. The superposition integral implies that all we need to know is the response of the system to point sources throughout the field of interest in order to characterize the system.

A linear imaging system is called space invariant or shift invariant if a translation of the input causes the same translation of the output. For a point source at the origin, the output of a shift-invariant system can be written as

If the input is shifted as dð x1; y1Þ, the output of the shift-invariant system must be hðx x1; y y1; 0; 0Þ. This is usually written simply as

By a change of variables, this can also be written as

This is the same as the 2-D convolution of hðx; yÞ with uðx; yÞ to yield gðx; yÞ. It is often written symbolically as

The significance of this result is that a linear shift-invariant (LSI) system is governed by convolution. Hence, the convolution theorem can be used to express the input–output relationship as

where Gð fx; fyÞ, Hð fx; fyÞ, and Uð fx; fyÞ are the Fourier transforms of gðx; yÞ; hðx; yÞ and uðx; yÞ, respectively. The Fourier transform is discussed in the next section. Hð fx; fyÞ given by

is called the transfer function of the system.

In the case of a discrete-space system, the superposition integral becomes the superposition sum, given by

m1¼ 1 n1¼ 1

In the case of a discrete-space LSI system, the convolution integral becomes the convolution sum, given by

m1¼ 1 n1¼ 1

which can also be written as

m1¼ 1 m2¼ 1

The transfer function of a discrete-space LSI system is the discrete-space Fourier transform of the impulse response, given by

m1¼ 1 n1¼ 1

The convolution theorem is stated by Eq. (2.2-12) in this case as well.

2.3CONTINUOUS-SPACE FOURIER TRANSFORM

The property of linearity allows the decomposition of a complex signal into elementary signals often called basis signals. In Fourier analysis, basis signals or functions are sinusoids.

The 1-D Fourier transform of a signal uðtÞ, 1 t 1 is defined as

The inverse Fourier transform is the representation of uðtÞ in terms of the basis functions e j2pft and is given by

Equation (2.3-1)is also referred to as the analysis equation. Equation (2.3-2) is the corresponding synthesis equation.

The multidimensional (MD) Fourier transform belongs to the set of separable unitary transforms as the transformation kernel is separable along each direction. For example, the 2-D transform kernel bðx; y; fx; fyÞ can be written as

bið ; Þ for i equal to 1 or 2 is the 1-D transform kernel.

The 2-D Fourier transform of a signal uðx; yÞ, 1 < x; y < 1 is defined as

where fx and fy are the spatial frequencies corresponding to the x- and y-directions, respectively.

Now, we can write

Uðfx; fyÞ ¼ e pð fx2þfy2Þ=N2

Denoting ðfx; fyÞ as the Fourier transform of dðx; yÞ in the limit as N ! 1, we find

ðfx; fyÞ ¼ lim Uð fx; fyÞ ¼ 1

N!1

2.5PROPERTIES OF THE FOURIER TRANSFORM

The properties of the 2-D Fourier transform are generalizations of the 1-D Fourier transform. We will need the definitions of even and odd signals in 2-D. A signal uðx; yÞ is even (symmetric) if

uðx; yÞ is odd (antisymmetric) if

These definitions really indicate two-fold symmetry. It is possible to extend them to four-fold symmetry in 2-D.

Below we list the properties of the Fourier transform.

Property 1: Linearity

Property 3: Correlation is similar to convolution, and the correlation between u1ðx; yÞ and u2ðx; yÞ is given by