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

Скачать файл
Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 915

Скачиваний: 0

ВНИМАНИЕ! Если данный файл нарушает Ваши авторские права, то обязательно сообщите нам.

328 COMPUTERIZED IMAGING II: IMAGE RECONSTRUCTION FROM PROJECTIONS

pðu; yÞ is also called the ray-sum or the ray integral of gðx; yÞ at the angle y. A set of ray integrals forms a projection. Below y will be implicitly treated by writing pðu; yÞ as pðuÞ.

A projection taken along a set of parallel rays as shown in Figure 18.2 is called a parallel projection. If the set of rays emanates from a point source, the resulting projection is called a fan-beam projection. Only the case of parallel projections is considered in this chapter.

The Radon transform can also be expressed in terms of the Dirac-delta function. u can be written as atr where a ¼ ½cos y sin y&t, r ¼ ½x y&t. Then, Eq. (18.2-1 ) is the

same as

ð

pðu; yÞ ¼ gðrÞdðu atrÞdr

ðð

ð

 

Þ

1

 

18:2-3

 

¼gðx; yÞdðu cos y x sin y yÞdxdy

1

The Radon transform has many applications in areas such as computerized tomography, geophysical and multidimensional signal processing. The most common problem is to reconstruct gðx; yÞ from its projections at a finite number of the values of y. The reconstruction is exact if projections for all y are known. This is due to the theorem given in the next section.

18.3THE PROJECTION SLICE THEOREM

The 1-D FT of pðuÞ denoted by Pðf Þ is equal to the central slice at angle y of the 2-D FT of gðx; yÞ denoted by Gðfx; fyÞ:

Pð f Þ ¼ Gð f cos y; f sin yÞ

ð18:3-1Þ

The slice of frequency components in the 2-D frequency plane is visualized in Figure 18.3.

Figure 18.3. The slice of frequencies used in the projection slice theorem.

THE PROJECTION SLICE THEOREM

329

Proof:

1ð

Pðf Þ ¼ pðuÞe j2pftudu

1

ðð1

¼gðu cos y v sin y; u sin y þ v cos yÞe j2pfudvdu

1

In the unrotated coordinate system, this is the same as

ðð1

Pð f Þ ¼

gðx; yÞe j2pð fx cos yþfy sin yÞdxdy

1

¼ Gðf cos y; f sin yÞ

EXAMPLE 18.1 Find the Radon transform of gðx; yÞ ¼ e x2 y2 :

Solution:

ðð1

pðu; yÞ ¼ e x2 y2 dðu x cos y y sin yÞdxdy

1

Since the transformation ðx; yÞ ! ðu; vÞ as given by Eq. (18.2-1) is orthonormal, x2 þ y2 ¼ u2 þ v2. Hence,

1ð

p

u;

yÞ ¼

e u2 e v2 dv

ð

 

 

1

¼ ppe u2

EXAMPLE 18.2 Find the Radon transform of gðx; yÞ if

 

 

g

x; y

ð

1

 

x2

y2Þl 1

x2 þ y2 < 1

 

 

ð

Þ ¼

 

 

 

0

 

 

 

otherwise

 

 

Solution: gðx; dyÞ is nonzero inside the unit circle. Since x2

þ y2

¼ u2 þ v2, the

 

 

 

 

p

 

 

2

 

 

 

 

 

limits of integration with respect to u are

 

p1 u2. Thus,

 

 

 

p u; y

 

 

 

 

1

 

u2

v2 l 1dv

 

 

 

 

 

 

 

1 u

 

 

 

 

 

 

 

 

ð

Þ ¼

 

ð

 

½

 

 

&

 

 

p

1 u2

330 COMPUTERIZED IMAGING II: IMAGE RECONSTRUCTION FROM PROJECTIONS

The following definite integral can be found in [Erdelyi]:

 

ð ða2 t2

Þl 1dt ¼

 

 

1ð Þ

 

 

a

 

 

 

 

 

a2l 1pp l

 

a

 

 

 

 

 

 

l þ

 

 

 

 

 

 

 

 

 

 

2

 

Utilizing this result above yields

 

 

 

 

 

 

 

p u; y

8

ð

1Þ

ð1 u2Þl 1=2

1 < u < 1

 

>

pp

l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð Þ ¼

<

 

 

 

 

 

 

 

 

 

 

>

l þ

2

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

:

 

 

 

 

 

 

 

 

 

 

0otherwise

18.4THE INVERSE RADON TRANSFORM

The 2-D FT representation of gðx; yÞ is given by

 

gðx; yÞ ¼

ðð Gðfx; fyÞej2pðfxxþfyyÞdfxdfy

ð18:4-1Þ

 

 

 

 

1

 

 

 

 

 

1

 

Converting ð fx; fyÞ to polar coordinates ð f ; yÞ gives

 

p

1

 

 

 

 

gðx; yÞ ¼ ð

ð

Gðf cos y; f sin yÞej2pf ðx cos yþy sin yÞjf jdf dy

ð18:4-2Þ

0

1

 

 

 

 

Since Pðf Þ ¼ Gðf cos y;

f sin yÞ, this is the same as

 

 

 

p

1

fPðf Þsgnðf Þej2pf ðx cos yþy sin yÞdf dy

 

gðx; yÞ ¼ ð

ð

ð18:4-3Þ

 

 

0

1

 

 

By convolution theorem, with u ¼ x cos y þ y sin y, the first integral is given by

1

 

 

 

 

 

 

e j2pfudf

 

1

 

@

 

 

 

 

 

1

 

 

fP

 

f

 

sgn

f

 

 

 

 

p u

 

 

 

 

ð

Þ

Þ

¼

2jp @u

Þ

jpu

ð

 

 

ð

 

ð

1

 

 

 

 

 

 

 

 

 

 

1

 

 

 

ð18:4-4Þ

 

 

 

 

 

 

 

 

¼ 2p2

ð

 

 

@@ðttÞ u t dt

 

 

 

 

 

 

 

 

 

1

 

 

 

 

p

1

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

PROPERTIES OF THE RADON TRANSFORM

331

The last result is the Hilbert transform of 21p @@u pðuÞ. Let

 

 

1

@ @ðttÞ u t dt

ð18:4-5Þ

^pðuÞ ¼ 2p2 ð

 

 

1

 

p

1

 

 

 

 

1

 

 

 

 

^pðuÞ is called the filtered projection. The inverse Radon transform becomes

 

p

 

 

 

 

 

 

gðx; yÞ ¼ ð0

^pðuÞdy

 

 

ð18:4-6Þ

 

p

 

 

 

 

 

¼ ð0

^pðx cos y þ y sin yÞdy

 

This is equivalent to the back-projection of ^pð This means the projections for all y are needed for not band-limited.

uÞ along the angular direction y. perfect reconstruction if gðx; yÞ is

18.5PROPERTIES OF THE RADON TRANSFORM

The Radon transform has a number of properties that are very useful in applications. The most important ones are summarized below. When needed, pðuÞ will be written as pgðuÞ to show that the corresponding signal is gðx; yÞ.

Property 1: Linearity

If hðx; yÞ ¼ g1ðx; yÞ þ g2ðx; yÞ;

 

 

phðuÞ ¼ pg1 ðuÞ þ pg2 ðuÞ

ð18:5-1Þ

Property 2: Periodicity

 

 

pðu; yÞ ¼ pðu; y þ 2pkÞ; k an integer

ð18:5-2Þ

Property 3: Mass conservation

ð pðuÞdu

 

ðð gðx; yÞdxdy ¼

ð18:5-3Þ

1

1

 

1

1

 

Property 4: Symmetry

 

 

pðu; yÞ ¼ pð u; y pÞ

ð18:5-4Þ

Смотрите также файлы