The fast discrete Radon transform. I. Theory - Image ... - CiteSeerX

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993

382

The Fast Discrete Radon Transform-I:

Theory

Brian T. Kelley and Vijay K. Madisetti, Member, IEEE

Abstract-A new inversion scheme for reconstruction of images from projections based upon the slope-intercept form of the discrete Radon transform is presented. A seminal algorithm for the forward and the inverse transforms was proposed by Beylkin in 1987. However, as proposed, the original algorithm demonstrated poor dispersion characteristics for steep slopes and could not invert transforms based upon nonlinear slope variations. By formulating the computation as a discrete computation of the continuous Radon transform formula, we explicitly derive fast new generalized inversion methods that overcome the original shortcomings. The generalized forward (FRT) and inverse algorithm (IFRT) proposed are fast, eliminate interpolation calculations, and convert directly between a raster scan grid and a rectangular/polar grid in one step. Part I1 of this paper describes the implementation of the algorithm on a massively parallel computer, and a new time-domain formulation.

I. INTRODUCTION

T

HE Radon transform and its ill-conditioned inverse were first formulated by J. Radon in 1917. However, its widespread application has suffered from the numerical intensity of inversion. The projections of a region gathered over all possible angles constitutes a Radon transform. Many classical image reconstruction techniques can be decomposed into an inversion of the classical Radon transform or generalized versions of the Radon transform [l], [2]. Image reconstruction by means of the inverse Radon transform allows the determination of a systems internal structure without physically probing the interior. For this reason, the Radon transform is used in a wide variety of applications such as tomography, ultrasound, x-ray, nuclear magnetic resonance imaging, optics, stress analysis, and geophysics, to name just a few [ 3 ] ,[4]. In d-D, the Radon transform maps a function to its integral over (d-1)-D hyperplanes at various directions in d-D space and at various distances from the zero coordinate to its transform domain. In 2-D, the continuous computation projects or maps an image plane to line integrals computed at various phase angles and intercept coordinates (T - p transform). While an exact inversion formula can be written for the continuous case [4], many methods have been proposed for implementation of the DRT and its inverse [5], [4]. The primary difficulty associated with the Radon transform stems from the requirement for an inversion procedures based upon finite (and sometimes arbitrary) number of projections. Fig. 1, for instance, illustrates the general method for collecting Manuscript received December 20, 1991; revised January 5, 1993. This work was supported in part by the National Science Foundation (NSF) under Grant MIPS-(9211725). The associate editor responsible for coordinating the review of this paper and approving it for publication was Dr. David B. Harris. The authors are with the National Center of Excellence in DSP, School of Electrical Engineering, Georgia Institute of Technology Atlanta, GA 303320250. IEEE Log Number 9208877

y

Source

Object

Projections

pv

Fig. 1. Image reconstruction via the Radon transform. The inverse Radon transform applied to the projection data reconstructs the image.

projections from source radiation (detector not shown). Knowledge of all projections of the distribution constitute a Radon transform. Image reconstruction requires the application of the inverse Radon transform to the possibly limited collection of projections. The two most popular inversion methods include back projection algorithms based upon Fourier domain interpretations and iterative reconstruction techniques [6], [7]. A distinctly different approach to these methods can be derived in the T - p [4] domain by formulating the discrete approximation to the continuous problem as a linear algebra problem. In this case, Beylkin [SI demonstrated that if discrete versions are based upon a discretization of Radon’s original formula, the inverse transform can be computed only approximately. Among the varied difficulties associated with the discrete Radon transform computation are the conversion between radial coordinates and a faster scan format, the interpolation required to compute the required line integrals approximations on a rectangular grid, and the significant computational requirements necessary for calculation of the inverse. We demonstrate a new method (FRT) [9] that, by operating in the 1.5-D frequency domain,’ successfully overcomes these problems. By reformulating the algebraic approach proposed in the seminal paper by Beylkin [8] as an approximation problem, we directly illustrate the exact relationship between the continuous Radon transform (CRT) and this discrete formulation. More importantly, by formulating the problem as an approximation to the continuous formula a generalized inversion procedure which allows for image reconstruction based upon an arbitrary collection of line integrals is fully illustrated. This new method can be viewed as passing the image through a 1.5-D filter that performs the entire DRT computation in one step. This method of implementation computes the T - p version of the CRT [4]. In this paper, we explicitly and rigorously derive a version of the discrete fast Radon transform (FRT) in a way that

’

By a 1.5 D operation, we imply that each column of the 2-D signal is filtered by a unique I-D operator.

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KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM

allows us to also postulate a generalized inverse. In Section 11, we introduce the (T - p ) DRT algorithm, directly illustrate the relationship between the discrete algorithm and the exact continuous 2-D Radon transform formula, and define the generalized DRT equation. In Section 111, we analyze the discrete KM delta function, determine the exact form of the time domain function, and proved that it is not accurate for large slopes. In Section IV, we derive a new formula for computation of the discrete KM delta function which is accurate for large slopes. In Section V, we outline the FRT algorithm and present an example. In Section we present two new inversion algorithms for the generalized FRT. we also show that the inversion identity introduced in [81 is a special case of our first inversion algorithm based upon the determination of the pseudoinverse [lo]. Then, we derive a new fast inversion procedure that can be computed in a similar fashion to the FRT. This fast discrete inverse Radon transform formula (IFRT) is generalized so as to allow for reconstruction based upon an FRT image consisting of a collection of possibly arbitrary set of line integrals. 11. THEDISCRETERADONTRANSFORM

In this paper we have adopted the following notation: CRT: Continuous Radon Transform ICRT: Inverse Continuous Radon Transform DRT: Discrete Radon Transform FRT: Discrete Fast Radon Transform IFRT: Discrete Inverse Fast Radon Transform PIFRT: Discrete Pseudo-Inverse Fast Radon Transform Let 2 and R represent the set of all integers and the set of all real numbers, respectively. A continuous 2-D Kh4 line impulse function, S(n,m ) , is defined by

S(n,m)=

{

M.

0.

if ( n - m,l2= o if ( n - m12 # o

and

2) The impulse function in (4) performs only the rotation operation while the input image is translated. In Z-D, the CRT computes line integrals along various angles and intercepts in an image plane. Discrete approximations to line integrals can be problematic since the discrete data are restricted to lie only on specific grid points. Most classical methods rely upon some form of explicit interpolation [5],[3]. A. ~i~~~~~~F

R

T

A

~

~

~

~

~

The FRT is based upon discretization of the continuous formula. This requires an approximation to continuous KM delta. Furthermore, in order to generalize the following discussion, we define two additional functions, c(.) and d(.). Definition 1: 1

if N is even

Definition 2: c(N) = c =

{ 01

if N is even if N i s odd

Definition 3: T(>

1) = aspect ratio of an image number of samples along length number of samples along width

‘

Definition 4: Let the adjacency function A(z1, 2 2 , m,, n ) be dejined as

o(I), if ( n - m12 5 &

A(z1, z2, m, n ) =

+e;

(21 - m)2

(22

- n )2

5

€ 20

O ( E ) , otherwise,

(5) where n,m,t2,e: E R, function will be given by

z1,zl

E 2. The discrete KM delta

Equation (6) will also be referred to as the discrete line The continuous l-D unit function, s ( n ) ,is given function. In Sections IV and 111, exact mathematical definitions by S(n) = 6 ( n , 0 ) . are presented. In 2-D, the cRT Of u ( t , s ) 6e.y R { u ( t , z ) }= U 8 ( 7 , P ) ) Assume that u ( t , z ) is of finite support. Let y ( n , s ) reprecan be written in T - p form as follows [4]: sent the discrete approximation to U R ( 7 , p ) , and let z ( m , l ) represent the discrete version of u ( t ,x). We can approximate R { u ( t . s ) }= ‘(‘7 z)‘(t - PZ - 7 )dz dt (2) (4) as follows: . -,

s_r,

Let t’ = t

M

- T.

/ / /m M

R{U(t,z ) } = U R ( 7 , p )=

X

u(t’ + T , z)S(t’ - p z ) dz dt’, ( 3 )

J-x

J--0c o=

J-00

./-a

1-

8-

”=-cc

hl’

~ ( t +’ T , z)S(t’,pz)d s dt’ . (4)

Replacing the unit impulse by the two-dimensional line impulse function. Note (2) and (4) differ as follows: 1) The impulse function in (2) performs both the rotation and translation operation.

-.

M

We can rewrite (2) as follows:

L’

for

n’ = -(”-1) 2

,

“-1

2

and

N ’ = 2 M ’ + 1 (8)

~

~

~

~

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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993

1

angularvaMMe unn impulse htegmli

i

Note that (11) represents the causal form of the DRT. By eliminating negative times, we simplify the time-frequency domain correspondence in the delta function approximations (see Sections 111 and IV) since the DFT can now be computed without any negative arguments. Let an odd 1-D sequence be defined as any discrete sequence possessing an odd number of samples and an even 1-D sequence be defined as any sequence containing an even number of discrete samples. More precisely,

:

?... ....,..... ..:: ; ..::.., ’.%.. ... . .. .. .., ..... .... . . .

I

x(2) =

odd even

if i = 0, 1,2, . . . , N - 1, ifi=0,1,2,...,N-1,

N = 2m + 1 N=2m m E Z (12)

I

I

I

(C)

Fig. 2. We can transform the computation to an equivalent causal form via a coordinate axis shift. The original time domain image and impulse sequence are shown in (a). The original convolution involving negative times and arguments is shown in (b). The corresponding convolution involving positive arguments is performed in a new shifted coordinate system.

Equation (7) represents the Discrete Radon Transform (DRT) formula that will serve as the basis for the fast algorithm explained later. We now wish to rewrite (7) in a manner that allows us to replace the original sequences by causal versions. 2M‘ 2L’

z’(m - 2M’

y(n, s ) =

+ n, 1

-

L’)

A d - D sequence is defined as odd if dimensions 1,2,. . . d consist solely of odd 1-D sequences. Likewise, a d - D sequence is defined as even if dimensions 1 , 2 , . . . d consists solely of even 1-D sequences. The extension of this theory to d - D cases involving a mixture of both even and odd sequences is straightforward. The remainder of this discussion assumes d = 2. If the input sequence is even, the convolution involves half sample delays. In 2-D, the variable M represents number of samples along the length of the image, while N represents the number of intercept values along the length of the output. Although theoretically N + CO, we apply a finite extent approximation. For the sake of simplicity, we will also assume that M = N for the remainder of the paper. This is achieved by zero padding M to the desired value of N. In matrix form this equation can be rewritten in the following manner: N-1

m=O 1=0

.8(m- M’, s(1 - L’)) n = 0.1. . . . .2M’. s = 0, 1, . . ,2S’

m=O

where ’ (9) ALs(m)is a matrix of support [0 : S - 1, 0 : L - 11 such Fig. 2 illustrates the coordinate axis shift represented by (9). that for a given value of m, the matrix element at index This coordinate axis shift is useful for the eventual transfor( s , I ) = S(m - ( N - 1)/2, ~ ( -l ( L - 1)/2)); mation to the equivalent frequency domain computation (see y,(n) is a row vector of support [0 : S - I] and with the Sections I11 and IV). s th element equal to y(n,s), and We now define a new causal input sequence, x(m,l),so x L ( m )is a row vector of support [0 : L - I] and with that z ( m .1 ) = d ( m - 2M’, 1 - L’). Then the l’th element equal to z(m,1). Note that (13) differs from the derivation of [SI in that 1) We eliminate times and index values < 0 in order to incorporate the causal form of the DRT equation. m = O 1=0 This greatly simplifies the eventual inclusion of FFT n = 0.1;. . ,2N’. s = O , l , . . . ,2S’ (10) computation blocks (Section 111). As written above, the output of the multichannel filter converts 2) Equation (13) is shown to be explicitly derived from an (assumed odd) [2M’ 1 x 2L’ 11 input sequence into an (11), which in turn can be simply related to (4). This link odd [2N’ 1 x 2 s ’ 11 output sequence. We generalize to is critical to the derivation of the generalized inversion the even and odd case as follows: equation (see Section VI-B). A I - 1 L-1 In this paper, we define the discrete Fourier transform of gs(n)as follows: y(n. s ) = z(m n , I)

+

+

+

+

+

m=O

/=O

.8(m- ( N - 1)/2, s(Z - ( L - l ) / 2 ) ) n = 0.1.. . . N - 1, s = 0 , 1 , . . . s - 1 (11)

I

l

l

.

._

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KELLEY AND MADISETI'I: THE FAST DISCRETE RADON TRANSFORM

and the inverse discrete Fourier transform of y s ( k )

(15)

+

If we constrain cL(n)to be periodic with period N , c L ( N m ) = g L ( m ) ;then the Discrete Fourier transform of (13) with respect to 72 yields:

(4

(b)

Fig. 3. If the slope parameter s is varied uniformly as in (a), the standard r - p transform is computed. If the parameter s is varied nonuniformly to obtain uniform radial coverage as in (b), the Radon transform computation is similar to that based upon the normal equations of a line.

where to the straightforward evaluation of ( l l ) , we instead propose the use of the generalized equation Y(n,

and

= !A.,

g(s))

M-1 L-1

F{8(m - ( N - I ) / 2 , s(l - ( L - 1)/2))}

= A,s(IC) (18)

=

z ( m + n, l)8(m- ( N - 1)/2,

N-1

8(m - ( N - 1)/2, S ( l - ( L - ])/a)) m=O

. exp( -227rmklN)

exp(-2 for0

g [ y + S ( l - ?)I 5k5

%-d

.rro [E$ A' for

+

- d+

+ cs(l -

y

1 5 IC 5 N - 1

From the above equations, we note that: 1) Hermitian symmetry is enforced in order to preserve time domain "realness;" 2) the frequency domain delta function, ALS( I C ) , represents the discrete Fourier transform of the causal, shifted version of time domain function, S(m,sl); and 3) by applying the discrete Fourier transform operator to the m index in (18), 8(m,s l ) is explicitly constrained to be periodic in m with period N . Thus the discrete Radon transform computation can be carried out by

R { z ( m .1 ) ) = F-'{ys(k)} = F-'{zL(k)A~s(k)}. (19) We present an example of the forward computation in Section V.

Equation (20) allows the DRT computation to be performed for lines oriented at an arbitrary collection of angles. We note that this rigorous inclusion of g(s) differentiates this treatment from that of [8] as follows: 1) By the proper choice of g(s) we derive algorithms for computing both the forward and inverse DRT for any arbitrary set of slopes. We present a method of inverting the (20) in Section VI. 2) We propose methods that allow us to compute 8(m (N-1)/2, g(s)(l-(L-l)/Z)) for steeply dipping slopes (i.e., large 1g(s)l as defined by lemma 3.2.) without the dispersion problems of [8] or aliasing effects. For instance, by choosing g ( s ) = B 2s

s -s + 1 -

0

< B < CO,

s = O , l , . . . , S- 1

(21)

one formulates the discrete computation of the standard I- - p method as shown in Fig. 3(a). The line integrals for this method are illustrated in Fig. 4( b) with B = 16. Any computation based upon the choice of g( .) as represented by (21) will be referred to as linear slope sampling. As can be observed, this choice has the drawback of sampling more densely (spatially) at the higher slope values. Other nonlinear choices of g( s) can remedy this inequity. For instance, by choosing g(s) = tan

2s-s+1

B. Generalized Nonlinear Sampling Strategies

As currently formulated, the FRT computation based upon the DRT formula in (11) evaluates discrete line integrals over the slopes s = 0 , 1 , 2 ; . . . N - 1. This is due to the 1-1 correspondence between the array indices of y(.,s) and the slope parameter in (11). In the discussion that follows, we will continue to assume that s = (0, I , 2.. . . , S - 1). As opposed

one can obtain the set of line integrals illustrated in Fig. 4(a). Any computation based upon the choice of g( .) as represented by (22) will be referred to as linear angular sampling. When B = tan(.ir(S - 1)/(2S)),the computation resembles the classical DRT [ 7 ] , [4] based upon the normal equation as

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386

shown in Fig. 3(b) when the angle 6' is varied uniformly.2 For clarity, we will continue to insert g ( s ) = s in most of our derivations, with the implicit understanding that g ( s ) can vary more arbitrarily. The reintroduction of g ( s ) and methods of inverting (20) for arbitrary g ( s ) functions are pursued further in Section VI-B.

5 4

3

111. THEDISCRETEKM DELTAFUNCTION(DKMD)

The Discrete KM Delta function (DKMD) function, d(m, s l ) , representing the periodic counterpart of 6(m,s l ) , is defined by taking the inverse transform of (18). In addition, a Hermitian symmetry constraint is enforced in order to insure that the corresponding time domain function is real. In order to analyze the DKMD function (S(m,s l ) ) further and to insight into the proposed method, we explore the form o (even and odd) discrete delta function further as follows Theorem I: Nj2-d

i ( m . s l )= - 1 + - l S ' c + 2 N

cos

r=l

I [

[ 21Y ~

,

.., ..................... ... . .. . ... ..

(m,- s l ) (23)

-2

-4

Proof: We note that this formulation is based upon the relation

0

2

4

2

4

(a)

H ( k ) = Fw and

H * ( k )= H * ( N - k )

(24)

Consequently, 1

h ( m )= N

_ ~y - d k=l

.H(k

+ N/2

-

d

k=l

+ e)

I .e x p ( i 2 r m ( k+ N / 2 - d + c ) / N )

+

1

+

{H(O) c H ( N / 2 ) } .

-2

-4

Fig. 4. Two reasonable methods of radial sampling are linear angular sampling as in (a) and linear slope sampling as in (b). However, any arbitrary set of slopes can be easily accommodated.

(27)

Note that H ( k + N / 2 - d + c ) = H * ( N / 2 + d - c - k ) . Then 1 h ( m )= N

{

N

0 (b)

+ .e x p ( i 2 r m ( k+ N / 2

. H * ( N / 2 d - c - k)

_ t -d

d

+

H(k)exp(i2rmk/N)

k=l

'The variable n in this instance does not directly correspond to the similar variable found in the classical DRT.

-

d

+c)/N)

1

r-

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KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM

N/2-d

N

k=l

Remark I : From the preceding derivation, we note that m = sl implies nearly constructive addition and that the terms tend to add destructively when m $ sl. The discrete formulation requires samples that are constrained to lie only at grid points. When Im - sll > IN[,the noise due to aliasing is added to the steeply dipping lines. Remark 2: To avoid costly interpolation procedures, the FRT method of computation discussed earlier computes these discrete line-integral functions in the frequency domain. Unfortunately, the 2-D periodic DKMD functions used to approximate particular line integrals can alias for steeply dipping slopes.

The explicit delta computation can be carried out using Fig. 18 or 23. Equation (23) requires more computation than (18). In 5(a), we note that the slope of the line may be a small rational number, in which case the discrete samples along the direction of the line are not guaranteed to fall on possibly sparse grid point locations. In such cases, we therefore expect to observe a function that appropriately disperses energy among neighboring grid points in such a way that the amplitudes of the discrete line function are slowly varying. This is illustrated most clearly in Fig. 5(a). In 5(b), (c), the approximation is intuitively appealing. In these cases, the amplitudes also remain uniform across the image plane. Figure 5(d) reveals the presence of aliasing line segments at the top left and bottom right of the image. In fact, for square arrays, such line aliasing occurs above any line dipping steeper than 45". Thus the frequency domain algorithm of Beylkin [8] cannot accurately compute steeply dipping line integrals for the forward transform or the reconstructed image unless N >> L. This obstacle in conjunction with the nonexistence of a generalized inverse, appear to be major hindrances to the widespread use of Beylkin's [SI method. Lemma 3.2: The DKMD function cannot tolerate steeply dipping slopes whose magnitude is greater than arctan ( r )for an image of aspect ratio r . Proof: Any set of S arbitrary slopes can be chosen for the FRT. Unless N ---f 03 as s 4 30, the DKMD approximation displays aliasing. From (23), it is observed thus

The amplitudes of the periodic delta function are nonzero at r r ~- sl = UN for v = O , & l , z t 2 , . . . Only when m sl z 0, though, are nonzero amplitudes desirable. The proper constraint is Iniax{m}l

+ Imax{s} max{l}l

5N

-1

A. Aliasing and the 2-Dim Periodic Delta Function

The FRT method of computation relies upon performing the sampling for the discrete Radon transform in the frequency domain as opposed to the time domain. Although the required convolution operation can be performed efficiently in the frequency domain via the 1-D FRT, operations involving the discrete Fourier transform impose specific periodicity constraints that can often lead to unwelcome anomalies. Aliasing and the approximation of linear convolutions by circular convolutions are two artifacts resulting from the constraints imposed by discrete periodic signals. Of these, the line function aliasing of the DKMD is the more severe problem. Figure 5(a-d) display the discrete delta function line approximation of 1 versus m for S ( m . s l ) for slopes of .Y = ( a ) / ( l O ) , s = 0.777, s = -1.0, and .Y = 1.8, respectively. The image size is L(= 41) x S(= 41) pixels with N = 41.

-.

'

n

0 In a square image array, an effective procedure that avoids line aliasing for FRT slopes above s = 1 ( = 45") can be devised by changing the aspect ratio of the array. That is, if the original input image of aspect ratio a is zero padded along the vertical axis so as to create a new image of aspect ratio r , the new image can tolerate steeply dipping slopes as high as arctan(r) without delta function line aliasing. For instance, for r = 12, a maximum acceptable slope of 85.2" can be

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388

40

35 1,

30 0.5.

25 0.

20 15

40

\ -

1 E '-

10 : 5-

> .

I

,

.

10


> r ) . For instance Fig. 6 illustrates the DKMD method of [8] corresponding to (23) for a 65 x 65 array (i.e., L = N = 65, r = 1) with s = -7. As illustrated in Fig. 6(a), the majority of energy adds noise to the discrete Radon transform computation. We now describe a new, novel KM Steep Dip (KMSD) procedure that accurately computes the KM delta function for steep dips (1 < s < CO).Figure 6(b) illustrates the robustness

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KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM

40

35 30 25

20 15 30

10 10

0-0

5 10

20

30

40

35 1.

30

0.8. 0.6.

25

0.4.

20

0.2.

15

!%!

c---

30

40

10

20 0

5

10

20

30

40 (d )

Fig. 5.

of this new method for s = -7. We will present both the even and odd case here. The full proof for the even case can be found in the Appendix. We can derive from (23)

. r

71

for N even. We note that 8(m,d) represent by (33) is symmetric about the origin and that the DKMD formula provides a valid computation for (slope) s < r .

n

(Continued).

For s > r , we interchange the 1 and m indices and compute with the variable s representing slopes of value l/s. Physically computing the entire 2-D DKMD function and rotating by 90" can entail severe communication and memory overhead [ 111. We perform the operation analytically. For the even case, this leads to the following formula:

8( m ,1

;) [; =

I+exp(-irsm) + 2

1cos [ 7 2?rr (1- sm)]]

N/2--1

r=l

(34)

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Then

9 where 1 - exp(0) g(T1

( b)

(a)

Fig. 6. Contour (a) illustrates the DKMD formula with a slope value of .$ = -7.-Y = L = 65. Note that the line integral centered at 32 is the correct function. The other unit impulses degrade the computation by adding temporally shifted noise. Contour (b) demonstrates the KMSD formula for .$ = -7. Note that the energy in (b) is properly concentrated without aliasing or dispersion.

k,

= 1 - “XP( R/N)

s2 = - 2 7 r i ( T S

-i7T

[

rs)

1

+k).

Proof: see the appendix. Remark3: Note that the computation of the KMSD function can be performed efficiently via an N length FFT. When N > L , all but the central L coefficients are discarded. Though Fig. 6(a) demonstrates the superiority of the new KMSD formula for steep dips, a second source of error We point out that (33) differs from (34) primarily by the inherent in all frequency domain computations is that created interchanging of variables 1 and m,. Figure 6( b) demonstrates by the use of cyclic signal constraints. For slopes, s > T , the computation for the slope ( l / s ) = -7 and T = 1 the reader is encouraged to convince himself that no cyclic convolution (regardless of the accuracy of the time domain (i.e., square array). The advantages of 6(b) over 6(a) are line function) can accurately model the linear convolution of clear. The KMSD function does not alias. Another subtle (11). For any computation involving slopes whose magnitude advantage of the KMSD function is its superior dispersion is greater than r , (11) must be computed in the time domain. characteristics at larger slope values. Figure 7(a) demonstrates Since the DKMD formula aliases at these large slope the time the DKMD computation for s = 8, N = 64, and L = domain KMSD formula in (60) and (69), evaluated as a time 8 and Fig. 7(b) illustrates the KMSD computation for the domain convolution, should be used for the computation when same parameters. The DKMD function gathers relatively few s > r . samples at the larger slope values, whereas the KMSD function can interpolate between the samples. Theorem 2: For an even sequence, if we define the causal V. THE FRT ALGORITHM frequency domain KMSD function as The steps for computing the FRT for g ( s ) = s are listed in Fig. 8. Note that we assume that the augmented input image z(n,,I ) is of extent [0 : N - 1, 0 : L - 11 and is transformed into a FRT output image of extent [0 : N - 1, 0 : s - 11. Then A remarkable property of the FRT is the simplicity in choosing any arbitrary set of slopes when performing the discrete Radon transform computation. Furthermore, as pointed out earlier, when sampling in the frequency domain, the need for data interpolation is completely eliminated. In 2-Dim, the where Radon transform maps line integrals at specific angles and g’(r. k . s) forO