Fundamentals of Digital Image Processing

shadowgram 10.1 shows a typical method of obtaining projections. Each horizontal line shown in

obtained by illuminating an object by penetrating radiation. Figure

this figure is a one-dimensional projection of a horizontal slice of the object. Each pixel on the projected image represents the total absorption of the X-ray along its

path from the source to the detector. By rotating the source-detector assembly around the object, projection views for several different angles can be obtained.

from these projections. Imaging systems that generate such slice views are called CT image reconstruction

The goal of is to obtain an image of a cross section of the object

(computerized tomography) lose resolution along the path of the X-rays. CT restores this resolution by using information from multiple projections. Therefore, image reconstruction from pro­

scanners. Note that in obtaining the projections, we

jections can also be viewed as a special case of image restoration. Transmission Tomography For X-ray CT scanners, a simple model of the detected image is obtained as follows.

at some fixed value of f (x, y) (Fig. 10. 1). Assuming the illumination to consist of an (x, y) infinitely thin parallel beam of X-rays, the intensity of the detected beam is given by z

Let denote the absorption coefficient of the object at a point in a slice

[ {t(x, y) du ]

I= lo exp -

X-rays

Typical projection

Display Source

Detectors

Computer

slice views

Reconstructed cross-section

Figure

10.1 An X-ray CT scanning system.

where /0 is the intensity of the incident beam, L is the path of the ray, and u is the distance along L (Fig. 10.2). Defining the observed signal as

g = In (�)

(10.2) we obtain the linear transformation

g � g(s, 0) = f(x, y) du,

-oo < s < oo, O :s 0 < 1T (10.3)

where (s, 0) represent the coordinates of the X-ray relative to the object. The image reconstruction problem is to determine f(x, y) from g(s, 0). In practice we can only estimate f(x, y) because only a finite number of views of g(s, 0) are available. The

preceding imaging technique is called transmission tomography because the trans­ mission characteristics of the object are being imaged. Figure

10.1 also shows an X-ray CT scan of a dog's thorax, that is, a cross-section slice, reconstructed from 120 such projections. X-ray CT scanners are used in medical imaging and non­ destructive testing of mechanical objects.

Reflection Tomography There are other situations where the detected image is related to the object by a

transformation equivalent to (10.3). For example, in radar imaging we often obtain 432

Image Reconstruction from Projections Chap. 1 0 Image Reconstruction from Projections Chap. 1 0

Figure 10.2 Projection imaging geometry in CT scanning.

a projection of the reflectivity of the object. This is called reflection tomography. For instance, in the FLR imaging geometry of Figure 8.7a, suppose the radar pulse width is infinitesimal (ideally) and the radar altitude (h) is large compared to the minor axis of the antenna half-power ellipse. Then the radar return at ground range r and scan angle <!> can be approximated by (10.3), where f(x, y) represents the

ground reflectivity and L is the straight line parallel to the minor axis of the ellipse and passing through the center point of the shaded area. Other examples are found in spot mode synthetic aperture and CHIRP-doppler radar imaging [10, 36).

Emission Tomography Another form of imaging based on the use of projections is emission tomography,

for example, positron emission tomography (PET), where the emissive properties of isotopes planted within an object are imaged. Medical emission tomography ex­

ploits the fact that certain chemical compounds containing radioactive nuclei have a tendency to affix themselves to specific areas of the body, such as bone, blood,

tumors, and the like. The gamma rays emitted by the decay of the isotopes are detected, from which the location of the chemical and the associated tissue within the body can be determined. In PET, the radioactive nuclei used are such that positrons (positive electrons) are emitted during decay. Near the source of emis­ sion, the positrons combine with an electron to emit two gamma rays in nearly opposite directions. Upon detection of these two rays, a measurement representing

the line integral of the absorption distribution along each path is obtained.

Sec. 1 0. 1 Introduction 433

Magnetic Resonance Imaging Another important situation where the image reconstruction problem arises is in

magnetic resonance imaging (MRI). t Being noninvasive, it is becoming increasingly attractive in medical imaging for measuring (most commonly) the density of protons (that is, hydrogen nuclei) in tissue. This imaging technique is based on the funda­ mental property that protons (and all other nuclei that have an odd number of protons or neutrons) possess a magnetic moment and spin. When placed in a mag­ netic field, the proton precesses about the magnetic field in a manner analogous to a top spinning about the earth's gravitational field. Initially the protons are aligned either parallel or antiparallel to the magnetic field. When an RF signal having an appropriate strength and frequency is applied to the object, the protons absorb

energy, and more of them switch to the antiparallel state. When the applied RF signal is removed, the absorbed energy is reemitted and is detected by an RF receiver. The proton density and environment can be determined from the charac­ teristics of this detected signal. By controlling the applied RF signal and the sur­ rounding magnetic field, these events can be made to occur along only one line within the object. The detected signal is then a function of the line integral of the MRI signal in the object. In fact, it can be shown that the detected signal is the Fourier transform of the projection at a given angle [8, 9].

Projection-based Image Processing In the foregoing CT problems, the projection-space coordinates (s, 0) arise nat­

urally because of the data gathering mechanics. This coordinate system plays an important role in many other image processing applications unrelated to CT. For example, the Hough transform, useful for detection of straight-line segments of polygonal shapes (see Section 9.5), is a representation of a straight line in the projection space. Also, two-dimensional linear shift invariant filters can be realized by a set of decoupled one-dimensional filters by working in the projection space. Other applications where projections are useful are in image segmentation (see Example 9.8), geometrical analysis of objects [11] and in image processing applica­ tions requiring transformations between polar and rectangular coordinates.

We are now ready to discuss the Radon transform, which provides the mathe­ matical framework necessary for going back and forth between the spatial coor­ dinates (x, y) and the projection-space coordinates ( s, 0).

1 0.2 THE RADON TRANSFORM [12, 13] Definition The Radon transform of a function f(x, y ), denoted as g (s, 0), is defined as its line

integral along a line inclined at an angle 0 from the y-axis and at a distance s from

t Also called nuclear magnetic resonance (NMR) imaging. To emphasize its noninvasive features, the word nuclear is being dropped by manufacturers of such imaging systems to avoid confusion with nuclear reactions associated with nuclear energy and radioactivity.

434 Image Reconstruction from Projections Chap. 1 0 434 Image Reconstruction from Projections Chap. 1 0

g(s, e) � rAt = Jr t(x, y)8(x cos e + y sin e - s) dx dy,

(10.4) -oo < s < oo, 0 $ 0 < 1T

The symbol rA , denoting the Radon transform operator, is also called the projection operator. The function g(s, 0), the Radon transform of f(x, y), is the one-dimen­

sional projection of f(x, y) at an angle 0. In the rotated coordinate system (s, u), where

s = x cos 0 + y sin 0 x = s cos 0 - u sin 0

(10.5) u = -x sin 0 + y cos 0

or

y = s sin 0 + u cos 0 (10.4) can be expressed as

g(s, 0) = r _., f(s cos 0 - u sin 0, s sin e + u cos 0) du,

< s < oo, 0 $ 0 < 1T The quantity g (s, 0) is also called a ray-sum, since it represents the summation of

f(x, y) along a ray at a distance s and at an angle 0. The Radon transform maps the spatial domain (x, y) to the domain (s, 0). Each point in the (s, 0) space corresponds to a line in the spatial domain (x, y). Note that (s, 0) are not the polar coordinates of (x, y ) . In fact, if ( r, <I>) are the polar coordinates of (x, y), that is,

(10.7) then from Fig. 10.3a

x = r cos<!>, y = r sin <I>

(10.8) For a fixed point (r, <!>), this equation gives the locus of all the points in ( s, 0), which

s = r cos(0 - <!>)

is a sinusoid as shown in Fig. 10.3b. Recall from section 9.5 that the coordinate pair (s, 0) is also the Hough transform of the straight line in Fig. 10.3a.

Example 10.1 Consider a plane wave, f(x, y) = exp[j27T(4x + 3y)]. Then its projection function is

g(s, 0) = f., exp[j87T(s cos 0 - u sin 0)] exp[j67T(s sin 0 + u cos 0)] du

exp[j27Ts(4 cos 0 + 3 sin 0)] f., exp[-j27Tu(4 sin 0 - 3 cos 0)] du

= exp[j27TS (4 COS 0 + 3 sin 0))8( 4 sin 0 - 3 COS 0) = a)ejlO-rrs 8(0 - <j>) where <!> = tan-1a). Here we have used the identity

8[f(0)J ""

- 0k)

wheref'(0) � df(0)/d0 and 0h k = 1, 2, . . . , are the roots of/(0).

Sec. 1 0.2 The Radon Transform 435 Sec. 1 0.2 The Radon Transform 435

-r

(a) Spatial domain (x, y)

(b) The point P maps into a sinusoid in the (s, 8) plane

(c) An image and its Radon transform

Figure 10.3 Spatial and Radon transform domains.

Notation In order to avoid confusion between functions defined in different coordinates, we

adopt the following notation. Let II be the space of functions defined on JR.2, where JR denotes the real line. The two-dimensional Fourier transform pair for a function

f(x, y) E '// is denoted by the relation .7z

(10. 10) In polar coordinates we write

f(x, y) �-� F(b , b)

436 Image Reconstruction from Projections Chap. 1 0

(10.11) The inner product in 'lt is defined as

Pp (E, 0) = F(E cos

sin 0, E 0)

(10.12) Let Vbe the space of functions defined on Rx [O, ir]. The one-dimensional Fourier

11/11: � (f,f )a

transform of a function g (s, 0) E V is defined with respect to the variable s and is indicated as

(10.13) The inner product in V is defined as

g(s, .71 0)

G (E, 0)

(10.14) For simplicity we will generally consider 'it and V to be spaces of real functions.

ligl!: � (g, g)�

The notation

(10.15) will be used to denote the Radon transform of f(x , y), where it will be understood

g = 9lf

that / E 'lt,g E V. Properties of the Radon Transform

The Radon transform is linear and has several useful properties (Table 10.1), which can be summarized as follows. The projections g(s, 0) are space-limited in s if the

object f ( x, y) is space-limited in (x, y ), and are periodic in 0 with period 2ir. A

translation of f(x, y) results in the shift of g(s, 0) by a distance equal to the pro- TABLE 10.1 Properties of the Radon Transform

Function Radon Transform

( f(x, y) = fp r, <!>) g(s, 0)

a1f1 (x, y) azf2 (x, y) a2g2 (s, 0)

a1 g1 (s, 0)

1 Linearity:

2 Space limitedness:

f(x,y) >2, IYI >2 g(s, 0) = Isl> Dv'2 2

= 0, lxl

f(x, y) g(s, 0)=g(-s, 0±TI)

3 Symmetry:

f(x, y) g(s, 0) k = integer g(s, 0

4 Periodicity: = + 2kTI),

6 f (x Rotation by y ( r, <I> + 00) g(s + 9o)

5 Shift: - Xo , - Yo) - Xo cos 9 - Yo sin

00: fP

g(s, 0

f(ax, ay)

Scaling: i= 0

Jlllg(as, 0), a

8 Mass conservation:

Jf t(x,y)dxdy = f �g(s, 0)ds,

Ve

Sec. 1 0.2 The Radon Transform 437

- 1 .0 -0.8 -0.6

Figure 10.4 (a) Head phantom model; (b ) constant-density ellipise,f(x, y) = fo for (x2/a2) + (y2!b2) :s l.

438 Image Reconstruction from Projections Chap. 1 0

TABLE 10.2 Head Phantom Components; (x, y) are the coordinates of the center

of the ellipse. The densities indicated are relative to the density of water [ 1 8] .

Inclination Density Ellipse

Major

Minor

(degrees) f; ( x, y ) a 0.0000

semiaxis

semi axis

0.00 -0.9800 c 0.2200

- 18.00 -0.0200 d -0.2200

18.00 -0.0200 e 0.0000

h -0.0800

jection of the translation vector on the line s =x cos 0+y sin 0. A rotation of the object by an angle 00 causes a translation of its Radon transform in the variable 0. A scaling of the (x, y) coordinates of f(x, y) results in scaling of the s coordinate together with an amplitude scaling of g(s, 0). Finally, the total mass of a distribution

f(x, y) is preserved by g(s, 0) for all 0.

Example 10.2 Computer generation of projections of a phantom In the development and evaluation of reconstruction algorithms, it is useful to simulate

projection data corresponding to an idealized object. Figure 10.4a shows an object composed of ellipses, which is intended to model the human head [

18, 21 ] . Table 10.2 gives the parameters of the component ellipses. For the ellipse shown in Fig. 10.4b, the

projection at an angle e is given by

lsl > sm where s;,, = a2 cos2 0 + b2 sin2 0. Using the superposition, translation, and rotation

properties of the Radon transform, the projection function for the object of Fig. 10.4a can be calculated (see Fig.

10. 13a ) .

10.3 THE BACK-PROJECTION OPERATOR Definition

Associated with the Radon transform is the back-projection operator QJ , which is defined as

b(x, y) � QJg = rg(x cos 0 + y sin 6, 0) d0

Sec. 1 0.3 The Back-Projection Operator

The quantity b (x, y) is called the back projection of g (s, 0). In polar coordinates it can be written as

(10.17) Back projection represents the accumulation of the ray-sums of all of the rays that

b (x, y) = bp (r, <!>) = r 0 g(r cos(0 - <J>), 0) d0

pass through the point (x, y) or (r, <!>). For example, if

g(s, 0) = g1 (s)8(0 - 01) + g1 (s )8(0 - 02)

that is, if there are only two projections, then (see Fig. 10.5)

bP (r, <!>) = g1 (s1) + g1 (s2)

where s1 = r cos(01 - <!> ), s2 = r cos(02 - <!> ). In general, for a fixed point (x, y) or (r, <!>), the value of back projection 9Jg is evaluated by integrating g(s, 0) over 0 for all lines that pass through that point. In view of (10.8) and (10.17), the back­ projection at (r, <!>) is also the integration of g(s, 0) along the sinusoid

s = r cos(0 - <!>) in the (s, 0) plane (Fig. 10.3b ). Remarks

The back-projection operator 91 maps a function of (s, 0) coordinates into a func­ tion of spatial coordinates (x, y) or ( r, <!> ). The back-projection b (x, y) at any pixel (x, y) requires projections from all directions. This is evident from (10.16).

Figure 10.S Back-projection of g, (s) and g2 (s) at ( r, <I>).

440 Image Reconstruction from Projections Chap. 1 0

It can be shown that the back-projected Radon transform

(10.18) is an image of f(x, y) blurred by the PSF 1 / (x2 + y2)112, that is,

j(x, y) � aJg = aJ 91,f

(10.19) where ® denotes the two-dimensional convolution in the Cartesian coordinates. In

f (x, y) = f(x, y) ® (x2 + y2)-112

polar coordinates (10.20) where ® now denotes the convolution expressed in polar coordinates (Problem

10.6). Thus, the operator aJ is not the inverse of In fact, 91, . Y3 is the adjoint of Yr [Problem 10.7]. Suppose the objectf(x, y) and its projections g(s, 0), for all 0, are discretized and mapped into vectors f and g and are related by a matrix trans­ formation g = Rf. The matrix R then is a finite difference approximation of the operator

The matrix RT would represent the approximation of the back­ projection operator aJ .

The operation J = aJ[ 91,f] gives the summation algorithm (Fig. 10.6). For a

set of isolated small objects with a small number of projections, this method gives a star pattern artifact (Fig.

The objectf(x, y) can be restored from f(x, y) by a two-dimensional (inverse) filter whose frequency responset is l�I = �� + �� , that is,

Back-projected projections f( x, y) Figure

Object f(x, y)

10.6 Summation algorithm for image reconstruction, J � .'ii g.

t Note that the Fourier transform of (.t2 + y2r112 is W + m-112• Sec. 1 0.3

The Back-Projection Operator 441

1 0.3 Filter Functions for Convolution/Filter Back-Projection Algorithms, d � 1/2 �o

TABLE

Discrete impulse Frequency response

response Filter

Impulse response

dh (md) Ram-Lak

H(�)

h(s)

h(m) �

HRL m � l�I rect(�d)

hRL (s) = re •

m=O

[2 sinc(2�os) - sinc2 (�0s)]

hRL (m) = 4d ' -sin2( 'ITm/2)

m �o

2(1 + sin 2'1T�

0s)

Shepp-Logan l�I sinc(�d) rect(�d)

'IT2 d(1 - 4m2) Low-pass

'IT2 (d2 - 4s2)

�[ �) �)] HhRdm -n + hRL (m + m

l�I cos( 'IT�d) rect( �d)

hRL(s -

+ hRL(s +

cosine

Generalized

ahRL (m) + (1 - a) · Hamming

l�I [ a + (1 - a ) cos2'1T�d] ·

1-a

rect(�d), 0 sas 1

·[hRL (m - 1) + hRL (m + 1)] Stochastic

[hRL (s - d) + hRL (s + d)]

See eq. (10.70) and Example 10.6

where :72 denotes the two-dimensional Fourier transform operator. In practice the filter l�I is replaced by a physically realizable approximation (see Table 10.3). This method [16] is appealing because the filtering operations can be implemented approximately via the FFT. However, it has two major difficulties. First, the Fourier domain computation of l�IF;,(�, 0) gives F(O, 0) = 0, which yields the total density fff(x, y) dx dy = 0. Second, since the support of YJg is unbounded, f (x, y) has to be

computed over a region much larger than the region of support off (x, y ) . A better algorithm, which follows from the projection theorem discussed next, reverses the order of filtering and back-projection operations and is more attractive for practical implementations.

10.4 THE PROJECTION THEOREM [5-7, 12, 13] There is a fundamental relationship between the two-dimensional Fourier trans­

form of a function and the one-dimensional Fourier transform of its Radon trans­ form. This relationship provides the theoretical basis for several image recon­

struction algorithms. The result is summarized by the following theorem.

Projection Theorem. The one-dimensional Fourier transform with respect to s of the projection g (s, 0) is equal to the central slice, at angle 0, of the two­ dimensional Fourier transform of the object f(x, y), that is, if

G (£, 0) = F;,(£, 0) � F(£ cos 0, £ sin 0)

Image Reconstruction from Projections Chap. 10

Figure 10.7 shows the meaning of this result. This theorem is also called the

projection-slice theorem. Proof. Using (10.6) in the definition of G (I;, e), we can write

G (l;, e) � f,,g(s, e)e-i2ir� ds

= Jft(s cos e - u sin e , s sin e + u cos e)e -i2irE-< ds du Performing the coordinate transformation from (s, u) to (x, y), [see (10.5)], this

becomes

G (l;, e) = Jf t (x , y) exp[-j2ir(xl; cas e + yl; sin e)] dxdy

= F(I; cos e, I; sin e) which proves (10.22).

Remarks From the symmetry property of Table 10. 1 , we find that the Fourier transform slice

also satisfies a similar property

(10.24) Iff(x, y) is bandlimited, then so are the projections. This follows immediately

G (-1;, e + ir) =

G (I;, e)

from the projection theorem. An important consequence of the projection theorem is the following result.

f(x, vl

f(x, y)

10.7 The projection theorem, G (�, 0) = F,, (�, 0). Sec. 1 0.4

Figure

The Projection Theorem 443

Convolution-Projection Theorem. The Radon transform of the two-dimen­ sional convolution of two functions f,(x, y) and fz(x, y) is equal to the one­ dimensional convolution of their Radon transforms, that is, if gk � £/?fb k = 1 , 2, then

t1(x - x', y - y ')f2(x ', y ') dx' dy ' = f� g,(s - s ', e)g2(s', 6) ds ' {fj (10.25) }

"' 9l

The proof is developed in Problem 10.9. This theorem is useful in the implementation of two-dimensional linear filters by one-dimensional filters. (See Fig. 10.9 and the accompanying discussion in Section 10.5.)

0) of Example The two-dimensional Fourier transform of f(x, y) is F( � i, � 2) = 8( � 1 - 4)8( � 10.1. 2 - 3) =

Example 10.3 We will use the projection theorem to obtain the g (s,

8(� cos 0 - 4)8(� sin 0 - 3). From (10.22) this gives G(�, 0) = 8(� cos 0 - 4) 8( � sin 0 - 3). Taking the one-dimensional inverse Fourier transform with respect to � and using the identity (10.9), we get the desired result

r x 8(� cos 0 - 4)8(� sin 0 - 3)ej2T<S�d�

g (s,

( ) ( j87TS )

G)e110"'s 8(0 - <j>)

= lcos0I exp cos0 8(4 tan 0 - 3) =

1 0.5 THE INVERSE RADON TRANSFORM [6, 12, 13, 17] The image reconstruction problem defined in Section 10. 1 is theoretically equiv­

alent to finding the inverse Radon transform of g (s, 6). The projection theorem is useful in obtaining this inverse. The result is summarized by the following theorem.

Inverse Radon Transform Theorem. Given g(s, 6) � Yrf, -x< s < oo,

0 :s; 0 < 1T , its inverse Radon transform is

f(x ) 1 'y = ( - ) f"fx

21T2 0 - x x cos e + y sm e - s

e)] dsd6 (10.26)

In polar coordinates

- ( 1 fp(r, <!>) =f(r cos <j>, r sm <!>) - - [(aglas)(s, e)] 2

2 ) f"fx

Ll

1T 0 r cos e ( - -x -s

ds d6 (10.27)

Proof. The inverse Fourier transform f(x, y) =

fr

+ by) di; I ds 2 when written in polar coordinates in the frequency plane, gives f(x, y) =

F(s " s2) exp[j21T(S , x

f" r 0 0

F;,(s, 6) exp[j21Ti;(x cos e +y sin e)]sdsde (10.28)

Image Reconstruction from Projections Chap. 1 0

Allowing � to be negative and 0 :::; 0 < 11', we can change the limits of integration and use (10.22) to obtain (show!)

f(x, y) = f' f., l�IF;,(�, 0)exp[j211'�(x cos 0 + y sin 0)] d� d0 = f' f l�IG (�, 0) exp[j211'�(x cos 0 + y sin 0)] d� de (10.29)

= r g (x cos 0 + y sin 0, 0) de

where (10.30) Writing l�IG as �Gsgn(�) and applying the convolution theorem, we obtain

g (s, 0) = [ .7[1{�G (�, 0)}] ® [ .7[1{sgn(�)}]

[ �� ] ® c�� )

(s, 0)

(-1z) J"' 211' a t s- t

where (1/j21r)[ag (s, 0)/as] and (- 1/j211's) are the Fourier inverses of �G (�, 0) and sgn(�), respectively. Combining (10.29) and (10.31), we obtain the desired result of (10.26). Equation (10.27) is arrived at by the change of coordinates x = r cos <I> and

y= r sin <I>· Remarks The inverse Radon transform is obtained in two steps (Fig. 10.8a). First, each

projection g (s, 0) is filtered by a one-dimensional filter whose frequency response is \�\. The result, g (s, 0), is then back-projected to yieldf(x, y). The filtering operation can be performed either in the s domain or in the � domain. This process yields two different methods of finding ur -1, which are discussed shortly.

The integrands in (10.26), (10.27), and (10.31) have singularities. Therefore, the Cauchy principal value should be taken (via contour integration) in evaluating the integrals.

Definition. The Hilbert transform of a function <l>(t) is defined as

(.!.) J'°

ljl(s) � .91<1> � <f>(s) ® ( _!_ ) = 11' -oo S d t

<f>(t)

1l'S

The symbol .sf- represents the Hilbert transform operator. From this definition it follows that g (s, 0) is the Hilbert transform of (11211')ag (s, 0)/as for each 0.

Because the back-projection operation is required for finding ur -1 , the recon­ structed image pixel at (x, y) requires projections from all directions.

Sec. 1 0.5 The Inverse Radon Transform 445 Sec. 1 0.5 The Inverse Radon Transform 445

g(s, II)

1-D filter

f(x, y)

I� I (a) Inverse radon transform

Convolution

I g(s, II) I Differentiate Hilbert

g(s, Ill

Back-project f(x, y)

(b) Convolution back-projection method

Filter

Inverse

g(s, II) Fourier transform G(E, Ill

Fourier

g(s,

Back-project f(x, y) II )

(c) Filter back-projection method

Figure 10.8 Inverse radon transform methods.

Convolution Back-Projection Method Defining a derivative operator as

(10.33) The inverse Radon transform can be written as

(10.34) Thus the inverse Radon transform operator is VP -1 = (1/27r) £8 -�''"0 . This means

f(x, y) = (l/27r)W _!k·0g

,q-1 can also be implemented by convolving the differentiated projections with 1/27rs and back-projecting the result (Figure 10.8b) .

Filter Back-Projection Method From (10.29) and (10.30), we can also write

(10.35) where Sf"is a one-dimensional filter whose frequency response is l�I, that is,

f(x, y) = W Sfg

f., l�IG(�. e)ei2"� ds

g � srg �

= .711 {l�I[ .71 g]}

Image Reconstruction from Projections Chap. 1 0

This gives (10.37) which can be implemented by filtering the projections in the Fourier domain and

back-projecting the inverse Fourier transform of the result (Fig. IO.Sc).

Example 10.4 We will find the inverse Radon transform of g(s, 0)

= G)ei10"s - <!>).

Convolution back-projection method. 'Trei10"

Using ag/as = j2

s8(0 - ti>) in (10.26)

f(X, = 2'Tr l"f� . eJIO"s (X

y) jZ'Tr - 2 COS 0+ Y Sin 0 - st1 8(0 - tj>)

0 -�

ds

1 = (�) J� ei10"s[s -(x ds

y sin <I> )f

cos <I> +

]'Tr -�

l/(�

Since the Fourier inverse of - a ) is j'Trei2"a'sgn(t), the preceding integral becomes

= lO'Tr(x cos ti> + y sin <I>) J. Filter back-projection method.

f (x, y) = exp[j2'Tr(x cos ti> + y sin ti> )t] sgn(t)I,

exp[j

G(�, 0) = G)8(� - 5)8(0 - ti>) ::? t (s, 0) =

G) f� 1�18(� - 5)8(0 - ti>) exp(j2'Trs�) d� 8(0 = ei10"s - <1>)

=?f(x,y)= f

lO x

p[j (

cos 0 + y sin 0)]8(0 - <l>) d0

ex

'Tr

= lO'Tr(x

exp[j

cos <I> + y sin <I>)]

a),f(x, y)

For <I> = tan1

will be the same as in Example 10.1.

Two-Dimensional Filtering via the Radon Transform

A useful application of the convolution-projection theorem is in the implementa­ tion of two-dimensional filters. Let

b) represent the frequency response of a two-dimensional filter. Referring to Fig. 10.9 and eq. (10.25), this filter can be implemented by first filtering for each

A (� 1,

6, the one-dimensional projection g(s, 6) by

f(x, y)

q(s, O) f(x, y) A!�, . �zl

2·D filter

f(x, y)

f(x, y)

g(s, 0)

1-D filters

Ap!t 01

Domain

f(x, y)

1-D filters

g(s, Ol f(x, y)

I� I Ap(t Ol

Figure 10.9 Generalized filter back projection algorithm for two-dimensional fil­ ter implementation.

Sec. 1 0.5 The Inverse Radon Transform 447 Sec. 1 0.5 The Inverse Radon Transform 447

becomes l�IAP (�, 0). Hence, the two-dimensional filter A (� 1 , �1) can be imple­ mented as

a (x, y) © f(x, y) = q; SfBg

wh�re %e represents a one-dimensional filter with frequency response Ap (�, 0) �

l�IAp (�, 0). 1 0.6 CONVOLUTION/FILTER BACK-PROJECTION ALGORITHMS :

DIGITAL IMPLEMENTATION [18-21] The foregoing results are useful for developing practical image reconstruction

algorithms. We now discuss various considerations for digital implementation of these algorithms.

Sampling Considerations In practice, the projections are available only on a finite grid, that is, we have

available

gn (m) � g ( s m , 0n) � [ 9cf](sm , 0n),

(10.39) where, typically, Sm = md, an = n!l., /:l. = TrlN. Thus we have N projections taken at

equally spaced angles, each sampled uniformly with sampling interval d. If � 0 is the highest spatial frequency of interest in the given object, then d should not exceed the corresponding Nyquist interval, that is,

d :::::; 1/2�

If the object is space limited,

that is, fp (r, <!>)

0, lrl > D/2, then D Md , and the number of samples should satisfy

(10.40) Choice of Filters

The filter function l�I required for the inverse Radon transform emphasizes the high-spatial frequencies. Since most practical images have a low SNR at high fre­

quencies, the use of this filter results in noise amplification. To limit the unbounded nature of the frequency response, a bandlimited filter, called the Ram-Lak filter

has been proposed. In practice, most objects are space-limited and a bandlimiting filter with a sharp cutoff frequency � 0 is not very suitable, especially in the presence

448 Image Reconstruction from Projections Chap. 1 0 448 Image Reconstruction from Projections Chap. 1 0

(10.42) Here W(�) is a bandlimiting window function that is chosen to give a more­ moderate high-frequency resppnse in order to achieve a better trade-off between

H(�) = l�IW(�)

the filter bandwidth (that is, high-frequency response) and noise suppression. Table

10.3 lists several commonly used filters. Figure

10.10 shows the frequency and

Frequency response HW Impulse response h (s)

a. E 0.2 <( -0.1

-0.2 0 0.2 0.6 0 2 3 4 5 6 Frequency, �

D istance, s

(a) RAM-LAK 0.2 0.6 0.1

a. > E 0.2 -0.1

Distance, s

(b) Shepp- Logan

-0.6 -0.2 0 0.2 0.6 0 2 3 4 5 6 Frequency, �

Distance, s

(c)

Lowpass cosine

Distance, s

(d) Generalized hamming

Figure IO. IO Reconstruction filters. Left column: Frequency response; right col- umn: Impulse response; dotted lines show linearly interpolated response.

Sec. 1 0.6 Convolution/Filter Back-Projection Algorithms 449 Sec. 1 0.6 Convolution/Filter Back-Projection Algorithms 449

interpolation 9.<si back-projection

"" f( x, y) convolution

Discrete

Discrete

h(m)

ffi N

(a) Convolution back-projection algorithm: Digital implementation ;

f-gn(ml0-j

9. (s) Discrete

Linear

"" f(x, y)

f-9.(m)c� �9.lm)�

0 K-1

(b) Filter back-projection algorithm: Digital implementation.

Figure 10. 11 Implementation of convolution/filter back projection algorithms.

the impulse responses of these filters for d = 1. Since these functions are real and even, the impulse responses are displayed on the positive real line only. For low

levels of observation noise, the Shepp-Logan filter is preferred over the Ram-Lak filter. The generalized low-pass Hamming window, with the value of a optimized for the noise level, is used when the noise is significant. In the presence of noise a better approach is to use the optimum mean square reconstruction filter also called the stochastic filter, (see Section 10.8).

Once the filter has been selected, a practical reconstruction algorithm has two major steps:

1. For each 0, filter the projections g(s, 0) by a one-dimensional filter whose frequency response is

H ( �) or impulse response is h (s ).

2. Back-project the filtered projections, g(s, 0). Depending on the implementation method of the filter, we obtain two distinct

algorithms (Fig. 10.11). In both cases the back-projection integral [see eq. (10. 17)] is implemented by a suitable finite-difference approximation. The steps required in the two algorithms are summarized next.

Convolution Back-Projection Algorithm The equations implemented in this algorithm are (Fig. 10. lla)

Convolution:

(10.43a) Back projection:

g (s, 0) = g (s, 0) © h (s)

(10.43b) 450

f(x, y) = sBg

Image Reconstruction from Projections Chap. 1 0

The filtering operation is implemented by a direct convolution in the s do­ main. The steps involved in the digital implementation are as follows:

1. Perform the following discrete convolution as an approximate realization of sampled values of the filtered projections, that is,

M g(md, nil) = gn(m) � L 8n(k)k (m - k), - :5 m :5- - l

M/2- 1

-M

2 2 where h(m) � dh(md) is obtained by sampling and scaling h(s). Table 10.3

k = -M/2

lists h (m) for the various filters. The preceding convolution can be imple­ mented either directly or via the FFT as discussed in Section 5.4.

2. Linearly interpolate gn(m) to obtain a piecewise continuous approximation of g(s, nil) as

g(s, nil) = frn(m) + (� -m [gn(m + 1) - gn(m)], )

(10.45) md :ss < (m + l)d

3. Approximate the back-projection integral by the following operation to give

N- 1

f(x, y) =f(x, y) � {l]Ng � Ll L g(x cosntl + y sin n tl, ntl)

n=O

where {lJ N is called the discrete back-projection operator. Because of the back­ projection operation, it is necessary to interpolate the filtered projections

8n(m). This is required even if the reconstructed image is evaluated on a sampled grid. For example, to evaluate

N-1

f(i Llx.j fly) = fl L g (iflx cos n fl + j fly sin n fl, nil)

n=O

on a grid with spacing (Ax, Ay), i, j = 0, ± 1, ±2, . . . , we still need to evaluate g(s, n

= though higher-order interpolation via the Lagrange functions (see Chapter -M/2, . . . ,M/2 - 1. Al­

A) at locations in between the points md, m

4) is possible, the linear interpolation of (10.45) has been found to give a good

trade-off between resolution and smoothing [18]. A zero-order hold is some­ times used to speed up the back-projection operation for hardware imple­ mentation.

Filter Back-Projection Algorithm In Fig. 10.llb, the filtering operation is performed in the frequency domain accord­

ing to the equation

g(s, 0) .9""!1[G (�, 0)H(�)]

Given H(�), the filter frequency response, this filter is implemented approximately by using a sampled approximation of G (�, 0) and substituting a suitable FFT for the

Sec. 1 0.6 Convolution/Filter Back-Projection Algorithms 451 Sec. 1 0.6 Convolution/Filter Back-Projection Algorithms 451

1. Extend the sequence gn (m ), -M 12 :s: m :s: (M /2) - 1 by padding zeros and periodic repetition to obtain the sequence gn (m)0 0 :s: m :s: K - 1 . Take its FFT to obtain G n (k) , 0 :s: k :s: K - 1 . The choice of K determines the sampling resolution in the frequency domain. Typically K = 2M if M is large; for exam­ ple, K = 512 if M = 256.

2. Sample H(�) to obtain H (k) � H(kb.�), H (K - k) � H*(k), 0 :s: k < K/2,

where * denotes the complex conjugate.

3. Multiply the sequences Gn (k) and H (k), 0 :s: k :s: K - 1 , and take the inverse

FFT of the product. A periodic extension of the result gives 8n (m), -K/2 :s: m :s: (K/2) - 1 . The reconstructed image is obtained via (10.45) and

(10.46) as before. Example 10.5

Figure 10.12b shows a typical projection of an object digitized on a 128 x 128 grid (Fig. 10.12a). Reconstructions obtained from 90 such projections, each with 256 samples per line, using the convolution back-projection algorithm with Ram-Lak and Shepp-Logan

filters, are shown in Fig. 10.12c and d, respectively. Intensity plots of the object and its reconstructions along a horizontal line through its center are shown in Fig. 10.12f through h. The two reconstructions are almost identical in this (noiseless) case. The background noise that appears is due to the high-frequency response of the recon­ struction filter and is typical of inverse (or pseudoinverse) filtering. The stochastic filter outputs shown in Fig. 10.12e and i show an improvement over this result. This filter is discussed in Section 10.8.

Reconstruction Using a Parallel Pipeline Processor Recently, a powerful hardware architecture has been developed [11] that en­

ables the high speed computation of digital approximations to the Radon transform and the back-projection operators. This allows the rapid implementation of convolution/filter back-projection algorithms as well as a large number of other image processing operations in the Radon space. Figure 10.13 shows some results of reconstruction using this processor architecture.

10.7 RADON TRANSFORM OF RANDOM FIELDS [22, 23]

So far we have considered /(x, y) to be a deterministic function. In many problems, such as data compression and filtering of noise, it is useful to consider the input

f(x, y) to be a random field. Therefore, it becomes necessary to study the properties of the Radon transform of random fields, that is, projections of random fields.

A Unitary Transform � Radon transform theory for random fields can be understood more easily by consid­

ering the operator 452

Image Reconstruction from Projections Chap. 10 Image Reconstruction from Projections Chap. 10

(a) Original object

(bl

A typical projection

(c) Ram-Lak filter (d) Shepp-Logan filter

Figure

10.12 Image reconstruction

(e) Stochastic filter

example.

Sec. 1 0.7 Radon Transform of Random Fields 453

0 16 32 48 64 (f) Object l ine

0 16 32 48 64 -64 -48 -32 - 1 6

(g) Reconstruction via RAM-LAK filter

-64 -48 -32 -16 0 16 32 48 64 -64 -48 -32 - 1 6 0 16 32 48 64 (h) Reconstruction via

(i) Reconstruction via Shepp-Logan filter

stochastic filter. Also see Example 1 0.6

where 51[112 represents a one-dimensional filter whose frequency response is lsl112• The operation

is equivalent to filtering the projections by 9'("112 (Fig. 10.14). This operation can also

be realized by a two-dimensional filter with frequency response (s� + s�)114 followed by the Radon transform.

Theorem 10.1. Let a + denote the adjoint operation of 9c . The operator 9c is unitary, that is,

(10.51) This means the inverse of a is equal to its adjoint and the a transform preserves energy, that is,

a -1 = a + = aJ :7(-112

454 Image Reconstruction from Projections Chap. 1 0

(a) Original phantom image (b) reconstruction via convolution back-projection,

abcdefg l1ijl{lmn

l1ijlclmn

opqrstu opqrst11 vw_xyz .,

vwxyz

(c) original binary image (d) reconstruction using fully constrained ART algorithm

Figure 10.13 Reconstruction examples using parallel pipeline processor.

ff rr

(10.52) This theorem is useful for developing the properties of the Radon transform for

1t(x, y)l2dxdy =

jg (s, 0)12dsd0

0 -�

random fields. For proofs of this and the following theorems, see Problem 10.13. Sec. 1 0.7

Radon Transform of Random Fields 455 Radon Transform of Random Fields 455

f Filter I I� 1112

2-D filter

I I (H + ��1114 I I

10.14 The .Cf-transform.

Radon Transform Properties for Random Fields

Definitions. Let f(x, y) be a stationary random field with power spectrum density S(£ i. £2) and autocorrelation function r(Ti.T2). Then S(£ i. £2) and r(Ti,T2) form a two-dimensional Fourier transform pair. Let SP (£, 6) denote the polar­

coordinate representation of S(£ i, £2), that is,

(10.53) Also, let rP (s, 6) be the one-dimensional inverse Fourier transform of SP (£, 6), that

SP (�, 6) � S(� cos 6, � sin 6)

is, (10.54) Applying the projection theorem to the two-dimensional function r(Ti. T2), we ob­

serve the relation

(10.55) Theorem 10.2. The operator ?f is a whitening transform in 6 for stationary

rP (s, 6) = U?r

random fields, and the autocorrelation function of g (s, 6) is given by r88 (s, 6; s ', 6') � E [g (s, 6)g (s ', 6')] = r8 (s - s ', 6)8(6 - 6') (10.56a) where

(10.56b) This means the random field g (s, 6) defined via (10.50) is stationary in s and

r8 (s, 6) = rP (s, 6)

uncorrelated in 6. Since g (s, 6) can be obtained by passing g (s,

6) through .W-112, which is independent of 6, g(s, 6) itself must be also uncorrelated in 6. Thus, the

Radon transform is also a whitening transform in 6 for stationary random fields and the autocorrelation function of g (s, 6) must be of the form

(10.57) where rg (s, 6) is yet to be specified. Now, for any given 6, we define the power

rgg (s, 6;s '; 6') � E[g(s, 6)g(s ', 6')] = rg(s - s ', 6)8(6 - 6')

spectrum density of g (s, 6) as the one-dimensional Fourier transform of its auto­ correlation function with respect to s, that is,

456 Image Reconstruction from Projections Chap. 1 0

From Fig. 10. 14 we can write (10.59)

These results lead to the following useful theorem. Projection Theorem for Random Fields

Theorem 10.3. The one-dimensional power spectrum density S8 (s, 0) of the 9l

transform of a stationary random field f(x, y) is the central slice at angle 0 of its two-dimensional power spectrum density S(s 1, s2), that is,

(10.60) This theorem is noteworthy because it states that the central slice of the two­

dimensional power spectrum density S (s 1 ,s 2 ) is equal to the one-dimensional power spectrum of g (s, 0) and not of g(s, 0). On the other hand, the projection theorem states that the central slice of a two-dimensional amplitude spectrum den­

sity (that is, the Fourier transform) F (s 1 , s 2) is equal to the one-dimensional ampli­ tude spectrum density (that is, the Fourier transform) of g(s, 0) and not of g (s, 0). Combining (10.59) and (10.60), we get

(10.61) which gives, formally,

(10.62) and

(10.63) Theorem 10.3 is useful for finding the power spectrum density of noise in the

reconstructed image due to noise in the observed projections. For example, suppose v(s, 0) is a zero mean random field, given to be stationary in s and uncorrelated in 0, with

E[v(s, 0)v(s ', 0')] = rv (s - s ', 0)8(0 - 0')

(10.64a)

(10.64b) If v(s, 0) represents the additive observation noise in the projections, then the noise

Sv (s, .71 0)

rv (s, 0)

component in the reconstructed image will be

(10.65) where

TJ(X, y) � aJ Sf'v = aJ %112ii = £7l -l ii

ii � �12v. Rewriting (10.65) as

ii = m 'TJ

Sec. 1 0.7 Radon Transform of Random Fields 457 Sec. 1 0.7 Radon Transform of Random Fields 457

(10.67) This means the observation noise power spectrum density is amplified by (�� + �1)112

by the reconstruction process (that is, by 01-1). The power spectrum STI is bounded only if l�IS" (�, 0) remains finite as �� oo. For example, if the random field v(s, 0) is

bandlimited, then TJ(X, y) will also be pandlimited and STI will remain bounded.

1 0.8 RECONSTRUCTION FROM BLURRED NOISY PROJECTIONS [22-25) Measurement Model In the presence of noise, the reconstruction filters listed in Table 10.3 are not

optimal in any sense. Suppose the projections are observed as

w (s, 0) = f., hp (s - s ', 0)g(s ', 0) ds ' + v(s, 0),

< s < oo, 0 < 0 ::5 1T The function hP (s, 0) represents a shift invariant blur (with respect to s), which may

occur due to the projectiG>n-gathering instrumentation, and v(s, 0) is additive, zero mean noise independent of f(x, y) and uncorrelated in 0 [see (10.64a)]. The opti­ mum linear mean square reconstruction filter can be determined by applying the

Wiener filtering ideas that were discussed in Chapter 8. The Optimum Mean Square Filter The optimum linear mean square estimate of f(x, y), denoted by J (x, y), can be

reconstructed from w (s, 0), by the filter/convolution back-projection algorithm (Problem 10.14)

g (s, 0) = f,, ap (s - s', e)w(s ', e) ds'

The foregoing optimum reconstruction filter can be implemented as a generalized filter/convolution back-projection algorithm using the techniques of Section 10.6. A

458 Image Reconstruction from Projections Chap. 1 0 458 Image Reconstruction from Projections Chap. 1 0

aP ( s, 0), which can change with 0.

Reconstruction from noisy projections. In the absence of blur we have hp ( s, 0) = o(s) and

(10.71) The reconstruction filter is then given by

w (s, 0) = g(s, 0) + v(s, 0)

!�ISP (�, AP (�, 0) 0)

- [Sp(�, (10.72) 0)

+ 1�1s. (�, 0)]

Note that if there is no noise, that is, s.� 0, then AP(�, 0 ) � � 1 , which is, of course,

the filter required for the inverse Radon transform.

Using (10.61) in (10.70) we can write

(10.73) where

AP(�, 0) = l�IAP (�, 0)

Note that AP (�, 0) is the one-dimensional Weiner filter for g (s, 0) given w(s, 0).

This means the overall optimum filter AP is the cascade of l�I, the filter required for the inverse Radon transform, and a window function Ay (�, 0), representing the

locally optimum filter for each projection. In practice, AP(�, 0) can be estimated

adaptively for each 0 by estimating Sw (�, 0), the power spectrum density of the observed projection w (s, 0).

Example 10.6 Reconstruction from noisy projections Suppose the covariance function of the object is modeled by the isotropic function

r(x, y) = a 2 + y 2). The corresponding power spectrum is then S (� i. b) = 2imcr 2[a2 + 4ir2 (�; + ��)t3ri or Sp (�, 0) = 2iraa 2[a2 + 4ir2 er3n. Assume there is no

blur and let r. (s, 0) = a� . Then the frequency response of the optimum reconstruction filter, henceforth called the stochastic filter, is given by

2iro:cr 2 + l�ld;. ( o:2 + 4ir2 � 2)3!2

l�l2iro:(SNR)

2 SNR � a -2

This filter is independent of 0 and has a frequency response much like that of a band-pass filter (Fig. 10.15a). Figure 10.15b shows the impulse response of the sto­ chastic filter used for reconstruction from noisy projections with a� = 5, a2= 0.0102, and a= 0.266. Results of reconstruction are shown in Fig. 10.15c through i. Com­ parisons with the Shepp-Logan filter indicate significant improvement results from the use of the stochastic filter. In terms of mean square error, the stochastic filter performs

13.5 dB better than the Shepp-Logan filter in the case of a� = 5. Even in the noiseless case (Fig. 10.12) the stochastic filter designed with a high value of SNR (such as 100), provides a better reconstruction. This is because the stochastic filter tends to moderate the high-frequency components of the noise that arise from errors in computation.

Sec. 1 0.8 Reconstruction from Blurred Noisy Projections 459

Ap (t B) 0.40

A typical frequency response of a stochastic tilter AP (t Bl = AP (-�. e)

0.6 h(s)

-------1- s

(b) I mpulse response of the stochastic filter used

Figure 10.15 Reconstruction from noisy projections.

460 Image Reconstruction from Projections Chap. 1 0

(c) Typical noisy projection, a� = 5 (d) Reconstruction via Shepp-Logan filter

(e) Reconstruction via the stochastic filter

(f) Shepp-Logan filter, a� = 1; (g) Stochastic filter, a� = 1;

Figure 10. 15 Cont'd

Sec. 1 0.8 Reconstruction from Blurred Noisy Projections 461

(h) Shepp-Logan filter, er� = 5; (i) Stochastic filter, er� = 5.

Figure 10.15 Cont'd

1 0.9 FOURIER RECONSTRUCTION METHOD [ 2 6- 29 ]

A conceptually simple method of reconstruction that follows from the projection theorem is to fill the two-dimensional Fourier space by the one-dimensional Fourier transforms of the projections and then take the two-dimensional inverse Fourier

transform (Fig. 10. 16a), that is,

(10.75) Algorithm There are three stages of this algorithm (Fig. 10. 16b). First we obtain

f(x, y) = .-721 [ c7"1g]

Gn (k) = G (kl1E, n fl0), -K/2 :::; k ::5 K/2 - l , 0 ::5 n ::5 N - l , as in Fig. (1 0 . ll b ) . Next, the Fourier domain samples available on a polar raster are interpolated to yield estimates on a rectangular raster (see Section 8. 16). In the final stage of the

algorithm, the two-dimensional inverse Fourier transform is approximated by a suitable-size inverse FFf . Usually, the size of the inverse FIT is taken to be two to three times that of each dimension of the image. Further, an appropriate window is used before inverse transforming in order to minimize the effects of Fourier domain truncation and sampling.

Although there are many examples of successful implementation of this algorithm [ 2 9], it has not been as popular as the convolution back-projection algo­ rithm. The primary reason is that the interpolation from polar to raster grid in the frequency plane is prone to aliasing effects that could yield an inferior reconstructed image.

462 Image Reconstruction from Projections Chap. 10 462 Image Reconstruction from Projections Chap. 10

G(t lll

F(t1 • t2)

f(x, y)

F i l l Fourier

space (a) The concept

Yn (m)c Gn (k)

Interpolate

F(kA�1 • /A�2l Window

2-D

f(mAx, nAy)

inverse rectangular raster

from polar to

and/or

pad zeros

F FT

(b) A practical fourier reconstruction algorithm

Figure

10.16 Fourier reconstruction method.

Reconstruction of Magnetic Resonance Images (Fig. 1 0. 17) In magnetic resonance imaging there are two distinct scanning modalities, the

projection geometry and the Fourier geometry [30]. In the projection geometry mode, the observed signal is G (�, 0), sampled at � = kA�, -K/2 ::s k ::s K/2 - l,

0 = n A0, 0 ::s n ::s N - 1, A0 = 'ITIN. Reconstruction from such data necessitates the availability of an FFf processor, regardless of which algorithm is used. For exam­ ple, the filter back-projection algorithm would require inverse Fourier transform of

(a) MRI data; (b) Reconstructed image; Figure

10.17 Magnetic resonance image reconstruction.

Sec. 1 0.9 Fourier Reconstruction Method 463

G(�, e)H(�). Alternatively, the Fourier reconstruction algorithm just described is also suitable, especially since an FFf processor is already available.

In the Fourier geometry mode, which is becoming increasingly popular, we directly obtain samples on a rectangular raster in the Fourier domain. The recon­ struction algorithm then simply requires a two-dimensional inverse FFf after win­ dowing and zero-padding the data.

Figure 10. 17a shows a 512 x 128 MRI image acquired in Fourier geometry mode. A 512 x 256 image is reconstructed (Fig. 10. 17b) by a 512 x 256 inverse FFf of the raw data windowed by a two-dimensional Gaussian function and padded by zeros.

1 0. 1 0 FAN-BEAM RECONSTRUCTION Often the projection data is collected using fan-beams rather than parallel beams

(Fig. 10. 18). This is a more practical method because it allows rapid collection of projections C9mpared to parallel beam scanning. Referring to Fig. 10.18b, the source S emits a thin divergent beam of X-rays, and a detector receives the beam after attenuation by the object. The source position is characterized by the angle 13,

and each projection ray is represented by the coordinates (CT, 13), -TI/2 :s CT < TI/2,

13 < 2TI. The coordinates of the (CT, 13) ray are related to the parallel beam coordinates (s, e) as (Fig. 10.18c)

where R is the distance of the source from the origin of the object. For a space­ limited object with maximum radius D 12, the angle CT lies in the interval

[--y, -y], -y � sin-1 (D/2R). Since a ray in the fan-beam geometry is also some ray in the parallel beam geometry, we can relate their respective projection functions

b (CT, 13) and g(s, e) .as

(10.77) If b (CT, 13) is given on a grid (<Tm' 13n), then this relation gives g (s, e) on a grid

'b (CT, 13) = g(s, e) = g(R sin CT, CT + 13)