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.
1057-7149/93$03.00 0 1993 IEEE
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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
.
._
385
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
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
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
I
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
388
40
35
1,
30
0.5.
25
0.
20
15
40
\
-
1 E
'-
10 :
5-
>
.
I
,
.
10
<
r
<
J
20
30
1
40
40
35
1,
30
25
0.5.
20
0.
15
10
0
5
10
20
30
0
40
( b)
Fig. 5. Illustration of the b ( I J I . .A\) approximation. Contours (left) and graphs (right) of 1 vs. 7n are shown for angles of 8.05"' (i.e., s = 0.1414) in (a),
37.85' (b), -45' (c), and 60.95' (d). At small angles the amplitude variation is readily apparent. At steep angles, the function wraps around (aliases).
tolerated. Zero padding has the added benefit of increasing the
total number of intercept values computed. This anti-aliasing
procedure requires 0 (r ) times more computation. A more
sophisticated method that completely circumvents the aliasing
of the DKh4D function for large slopes is presented in the
next section.
IV. THE KM STEEPDIP FORMULA
(KMSD)
As currently formulated, the DKMD formula aliases for
slopes above arctan(r) for 2-D arrays possessing an aspect
I
"
ratio of r . In addition, the aliasing for slopes above r quickly
becomes catastrophic as slope values increase (s >> 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)
I
IEEE TRANSACIIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
390
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<r<
spatial positions to points in Radon transform space. In the
g ( r . k , s) =
continuous coordinate system, the T - p transform is specified
g’(T - N , k , s ) for -$< r < N - 1
in terms of slopes and intercepts. A uniform variation of the
slope, p , generates the standard T - p Radon transform. By
and
varying the slope parameter uniformly, a nonuniform variation
1 - exp(R)
is created in the angular sampling (i.e., e). Often, it is desirable
g’(r. k . s ) =
1 - exp(R/N)
to select line integrals spaced at uniform angular distances. To
R = -27ri(rs k )
induce a uniform angular sampling, the slope parameter must
be varied nonuniformly.
Proof: The entire proof is given in the appendix.
The parameter s displayed in (18) selects the slopes of
Theorem 3: For an odd sequence let us define the causal the particular line integrals in the computation. By the proper
frequency domain KMSD function as
choice of s, uniform angular sampling of the Radon transform
can be carried out. This slope parameter in the discrete version
of the T - p Radon transform, if uniformly varied, samples
very finely near the vertices of the square grid, as compared
+
+
39 1
KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM
4.4
4.4
!
,
50
70
40’
”
30
,o
of
0
( b)
(a)
Fig. 7 . (a) illustrates the DKMD formula with a slope value of ,\ = 8, L = 8 , and S z 64. (b) demonstrates the superior dispersion characteristics
of the KMSD formula computed with the same parameters.
l<s<r
1
c
Compute the FFT of the columns
(in parallel) to form
Compute the FFT of the columns
(in parallel) to form
A L$ (k)
compute the matrix vector
gk)
.c
L
g(r,k,s) to form
compute the matrix vector
convolution E(m) *A L$ (m)
I
~(k)
compute the matrix vector
-
product
x(k) ALS (k)
YE)
Compute the IFFT of the columns
(in parailel) to form
W=Y(~*S)
The FRT is of size N x S.
I
I
I
Y(n4
Fig. 8. The FRT algorithm.
to the horizon. These two methods of sampling
- are illustrated
in Fig. 4.
= L = N = 5 (i.e., T = 1). Then a
Example: Let
5 x 5 matrix is to be discretely Radon transformed into another
392
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
0.000
0.000
0.000
0.000
1.000
0.000
0.200
0.000
-0.2472
0.000
0.000 0.000
0.000 0.6472
1.000 1.000
0.000 0.6472
0.000 0.000
0.000
0.000
0.000
0.000
0.000
0.000 1.000
0.2472 0.000
0.000 0.000
0.200 0.000
0.000 0.000
1.000 0.000
1.000 0.6472
1.000 1.000
1.000 0.6472
1.000 0.000
1.000
0.000
0.0000
0.000
0.000
,
0.000
0.000
1.000
0.000
0.000
0.000
-0.2472
0.000
0.2000
0.000
0.000
0.000
0.000
0.000
0.000
4
+
ALs(m,)&:,(m 3 )
I
0.000
1.000
0.000
0.000
0.000
matrix of size 5 x 5. Using (13) we can compute the discrete
Radon transform from an approximation to the continuous integral. Note that the N in this instance, A ~ s ( m
= 0 , l . 2,3,4)
with S = [-1. - ( 1 / 2 ) , 0 . ( 1 / 2 ) , 11 and L = [O. l , 2 . 3 , 4 ] , are
determined from (23) to be those depicted by (35) and (37).
For instance,
Ys(3) =
0.000
0.000
0.000
1.000
0.000
(36)
m=O
where, due to the periodicity assumption for the input function,
m S 3 can be restricted to the range (0, 1, 2, 3,4) by evaluating
the matrix equation modulo 5. This straightforward evaluation
of (13) as a circular convolution would yield the same result as
the linear convolution, but is less efficient than the frequency
domain method.
Since the maximum slope is equal to T ( = l),the frequency
domain DKMD algorithm could be applied. The FRT method
of computation is based upon an evaluation of (19). Since
A ~ s ( l cis) known explicitly (see (18)), no excessive coefficient
storage, or time domain computation such as that shown in
(23) is involved. From (18), A:,s(k = 0 . 1 , 2 , 3 , 4 ) are:
The FRT transform computation is carried out by
0.000
0.000
0.000
0.000
0.000
0.000
-0.2472
0.000
0.6472
1.000
1.000
0.6472
0.000
-0.2472
0.000
0.000
0.200
0.000
-0.2472
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
1.000
0.6472
0.000
-0.2472
0.000
0.000
-0.2472
0.000
0.6472
1.000
I
I
0.000
1.000
0.000
0.000
0.000
0.000
0.000
0.000
1.000
0.000
I
,
,
(35)
extremely fast computation results by exploiting this natural
concurrency.
VI.
INVERSION OF THE
FRT
We propose two new techniques for the inversion. They are
the discrete pseudo inverse fast radon transform (PIFRT) and
a direct inversion method (IFRT). The first uses a least squares
minimization to invert data. The latter is a very fast, efficient
approach based upon the forward FRT method.
A. Method I : PIFRT
Though the FRT algorithm is an efficient method for computation of the DRT, inversion of the Radon transform has
traditionally proven to be a more difficult. Since the FRT
algorithm is based upon frequency domain matrix operators,
one approach for inversion follows naturally from linear
algebra. Clearly, ifand only ifA,s(IC) is nonsingular for every
value of IC, does
&(IC)
[ALS(IC)l-lYS(k)
yield a suitable discrete inverse Radon transform formula.
Unfortunately, exact invertibility is precluded as A ~ s ( k )is
l 4
R { z ( m .I ) } =
[ALS(k)X,(k)]
-227r
generally not nonsingular for each value of IC.
n=O
Figure 9 illustrates the ill-conditioned nature of the trans(38) form matrices for specific values of IC derived using (18). The
where & ( k ) represents the FFT of the input signal along graph of Fig. 9 represents a plot of the condition number of
only the columns of the array. The IFFT of the above the each of the frequency domain transform matrices in (18) versus
matrix-vector product along the columns of the output yields the frequency index IC for the frequency domain transform
the discrete Radon transform output.
matrix. In this instance, the parameters L = S = N = 121
The primary advantages of the FRT are its simplicity and were selected. Even in the special, contrived example of a
speed. Multiple I-D FFT’s and matrix multiplications form the square matrix, the singular nature of some of the matrices is
major computational steps. Moreover, within each of these still evident near dc at the high and low-frequency indices.
steps, completely independent operations are involved. A n This suggests that the dc level of the input signal is the
most difficult part of the frequency spectrum to recover. As
matrix inversion is limited to only square singular matrices,
an alternative method to the direct inverse is the application
‘These matrices are described in the S x L plane. However, they are more
identifiable in the .U x L plane.
of the PIFRT.
,(
?)
393
KELLEY AND MADISElTI: THE FAST DISCRETE RADON TRANSFORM
find an alternative quantity (call this vector & ( k ) )
minimizes the absolute difference between & ( k )
that
and
where l(a(lpdenotes the p-norm of the vector a. For p = 2,
(39) is minimized in a least-square sense. More importantly,
the optimal value of x L ( k ) in this case can be shown to be
[lo1
1,4
3
f
0 8 -
o.6-
I
'i
1.6.(
'
04-
02-
140
0
where A,,(k)
= Q,CQF represents the singular value
decomposition (SVD) of ATs(k); and E+ is the inverse
diagonal matrix of C. We observe that
I = Q2C+QTA,s(k)
#
( [ ~ L ( k , ]* A L S ( W ) -l
[AES(k)l * a L s ( k )
unless rank { A ~ s ( k ) =
} min(L, S ) Consequently,
1) The identity introduced in Beylkin [8] is invalid when
rank {A,s(k)} < min(L,S); and,
2) the LMS formulation simplifies into the less general
identity introduced in [8] when rank {A,s(k)} =
min(L. S)
+
+
0.3090 0.95112
-0.8090
0.58782
-0.8090 - 0.58782
0.3090 - 0.95112
1.oooo
I
+
+
--0.8090
0.58782
0.3090 - 0.95112
0.3090 0.95112
-0.8090 - 0.5878i
1.0000
-0.8090 - 0.58782
0.3090 0.95112
0.3090 - 0.9511i
0.58782
-0.8090
1.0000 0.00002
I
!
+
+
+
1
1.0000
1.0000
1.oooo
1.0000
1.0000
I
"
Figure of frequency index k versus the condition number for a square
ALS(k).
Therefore, (40) represents the generalized inversion procedure.
Let the vectors & ( k ) represent the FRT transform of the data
y L ( k )according to (19). The PIFRT is expressed as
x'L(k) =
[a'Ls(k)l-'Ys(k)
(41)
where [Ais(k)]-l = Q,C+Qy. Geometrically, the pseudoinverse operation [A;,( k ) ] projects the transform vector
Y,(k) onto the column space of A ~ s ( k such
)
that the error
vector, E = Y - Ax, is orthogonal to the column space.
Since an exact inverse transformation is usually precluded, this
1.0000
1.0000
1.0000
1.oooo
1.0000
-'
1.0000
1.0000
1.0000
1.oooo
1.0000
1.00001.0000
1.oooo
1.oooo
1.0000,
,
+
-0.8090
0.58782 -0.8090 - 0.58782 0.3090 - 0.95112
1.0000
-1.0000 - 0.00002 -0.8090 - 0.58782 -0.3090 - 0.95112 0.3090 - 0.95112
-0.8090 - 0.58782 -0.8090 - 0.58782 -0.8090 - 0.58782 -0.8090 - 0.58782
-0.3090 - 0.95112 -0.8090 - 0.58782 -1.0000 - 0.00002 -0.8090
0.58782
0.3090 - 0.95112 -0.8090 - 0.58782 -0.8090 + 0.58782 0.3090 0.95112
+
+
+ 0.9511i
+ 0.95112
+ 0.95112
+ 0.95112
+ 0.95112
-0.8090 - 0.58782
-0.8090
0.58782
0.3090 0.95112
1.0000 0.00002
0.3090 - 0.95112
0.3090 0.95112 0.3090 - 0.95112
1.0000 0.00002 0.3090 - 0.95112
0.3090 - 0.95112 0.3090 - 0.95112
-0.8090 - 0.58782 0.3090 - 0.95112
-0.8090
0.58782 0.3090 - 0.95112
-0.8090
0.58782
-0.8090 - 0.58782
0.3090 - 0.95112
1.0000 0.00002
0.3090 0.95112
1.0000
-0.8090
0.58782
0.3090 - 0.9511.i
0.3090 0.95112
-0.8090 - 0.58782
-1.0000 - 0.00002
-0.8090 - 0.58782
1.0000
0.3090 0.95112
-0.8090
0.58782
-0.8090 - 0.58782
0.3090 - 0.95112
0.3090 - 0.95112
1.0000 + 0.00002
0.3090 0.95112
-0.8090
0.58782
-0.8090 - 0.58782
+
+
0.3090
0.3090
0.3090
0.3090
0.3090
+
+
+
0.3090 - 0.95112 -0.8090 - 0.58782
-0.8090 - 0.58782 -1.0000 - 0.00002
-0.8090
0.58782 -0.8090
0.58782
0.3090 0.95112 -0.3090
0.95112
1.0000 0.00002
0.3090 0.95112
+
+
+
1.0000
1.0000
1.0000
1.oooo
1.0000
Fig. 9.
+
+
+
-0.8090
-0.8090
-0.8090
-0.8090
-0.8090
+ 0.58782
+ 0.58782
+ 0.58782
+ 0.58782
+ 0.58782
+
+
+
+
+
+
0.3090 + 0.95112
-0.3090 + 0.95112
-0.8090 + 0.58782
1.0000
-0.8090 - 0.58782
0.3090 0.95112
0.3090 - 0.95112
-0.8090
0.58782
+
+
+
+
+
+
,
,
1
1
]
,
,
(37)
1
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
394
yields the smallest possible mean square error of the inverse
transformation.
equation in ( r , p ) space:
(46)
B. Method 2: Direct Inversion of the CRT
The least-squares inversion procedure introduced in
Section VI-A is based upon a linear algebraic solution; and as
such, it is most efficient in inverting problems that have been
forward transformed by the same matrix operators. Although
(41) can be applied as a general inversion procedure, our
empirical results were generally unsatisfactory. The discrete
Radon transformed data can be directly inverted with the
PIFRT. However, if the DRT data is operated upon in the DRT
domain, general inversion procedures based upon continuous
domain formulations are required for inversion. Radon
transform inversion methods are based upon approximations of
continuous domain equations. Such methods are intrinsically
based upon the relationship between the Radon transform and
Fourier transform as described by the projection slice theorem
[7], [4]. The robustness of inversion methods based upon the
projection-slice theorem increases as the number of projections
increases. An adversity associated with such methods is the
explicit conversion from polar to rectangular coordinates, as
with the back-projection operation. As an efficient counterpart
to filtered back-projection method^,^ we propose in the r - p
(or slope intercept) domain, a new discrete inversion algorithm
(IFRT) that is closely related to the forward FRT algorithm.
The relationship between the 2-D Fourier and Radon transform has been well established [7], (41. Let the continuous
Fourier and Radon transform, respectively, of u ( t ,x ) be written as follows:
32{?L(t.
x)} = U y K ,w)
u ( t ,x ) exp[ j K x
+ .7wt] dx dt
(42)
(47)
We note that the derivative of the Hilbert transform of U*(r,p)
corresponds to passing every row of the Radon transform into
a filter of frequency response J w J .Let us denote the output of
this computation as v ( t ’ , p ) . Then
+ IwIF1 { R { u ( t ,
= IwIUF2(wp,
w)*
(48)
One can rewrite the inversion formula in (47) as follows:
v(t’,p)S(t - t ’ , p z ) d t ’ d p .
(49)
27r
Equation (49) is the inversion counterpart to the forward CRT
solution.
A property associated with the above formulation is that
neither linear slope sampling nor nonuniform angular sampling
is implied. It is sometimes desirable to choose nonuniform
values for the continuous slope parameter, p , that, for instance,
might allow for uniform angular coverage of the forward
Radon transform. By replacing p with G(p),in (49), we can
introduce the generalized inversion equation:
(50)
The parameter G(p) represents the slope function. Its incorporation into (50) allows for the unification of many possible
sampling algorithms under one single framework. For instance,
by choosing G(p) = tan(0) we can write
R{?L(t.Z)}= UR(7.p)
~ ( .x)S(t
t , - p~
- 7) dx d t
.
. 6 ( t - t’,tan(0)x)sec2(O) dt’d01
(43)
In 2-D, we write the connection between the continuous
r - p transform and the original signal as follows:
or
U F 2 ( w p ,w)=
.I
U R ( r , p )e x p ( - i w r ) d r .
(44)
Define the Hilbert transform of U R ( r , p ) as follows:
(51)
as the continuous inversion equation upon which the FRT
derivation for uniform angular coverage is based. The angle corresponding to the slope is B = arctan(G(p)). Thus
what is often desired in the discrete Radon transform is not
U R ( r , G ( p ) = p ) evaluated at uniform values of p , but
U R ( r ,tan(0)) evaluated at uniform values of B. We can now
account for this and other slope functions in the generalized
inverse formula by the proper selection of G(p) in (50). We
note that generalized inversion procedures for both linear and
nonlinear G(p) functions were not considered in [8].
C. The Discrete Inverse Fast Radon Transform (IFRT)
After considerable manipulation [4], (44) can be reformulated
into the well known [12], [4] Radon transform inversion
A classical back-projection operation is based upon the normal equations
and is, of course, not applicable in the r - p domain (see Part 11).
I
7 -
Let x ( m ,I) and 7 1 4 7 2 , s) represent the discrete counterparts
,
respectively. In this framework,
of u ( t ,x ) and ~ ( t ’G(p)),
~ ( 7 2 s)
, represents the discrete value of the derivative of the
Hilbert transform of the DRT function, ~ ( ng (,s ) ) .
395
KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM
Although the vector sequence q,( 7 2 ) represents the derivative of the Hilbert transform of y (n,), this additional step is
trivially computed in the frequeGly domain [7]. In addition,
in order to suppress the amplification of the high frequency
noise caused by the Hilbert operation, we incorporated a
Shepp-Logan filter [3], (51 in all of the reconstruction examples. By discretizing (SO) in a manner similar to (4) and
following the same procedure as (4)-(11), we define the
generalized discrete inversion formula as follows:
and
F{h(s)i(m- ( N - 1)/2, g ( s ) ( l - ( L - 1)/2))}
N-1
I) =
m=O
.(I - ( L - 1)/2)) exp(-iZ"/N)
h ( s )e x d - i 2
[
g(.s)(l-
+
UJ(7L, S)
N=O
n=o
.~
( V-L(
N - 1)/2
h ( s ) i ( m- ( N - 1)/2, d . 7 )
ASL(k)=
s-1 N-1
X(711.
foro 5 k
-
n,,g(s)
. (1 - ( L - 11/21] h ( s )
(52)
= As,(k)
(58)
I
(
5
h(S)exp 4 2
-
F) ] )
d
[
+ g ( s ) (1 - y )])
for+--d+15k<N-1
where the discrete function h ( s ) represents the discrete approximation to the negative derivative of G. If g ( s ) is varied to
achieve uniform slope sampling or uniform angular sampling,
as in (21) and (22), respectively, one can express h ( s ) as
follows:
-1.
uniform slope sampling (i.e., T - p method)
h ( s )=
Comparing this formula to (41), the most notable difference
is the complete circumvention of both a matrix inversion
and/or SVD least squares inversion. We note that unlike the
FRT, both the Fourier transform computation of the A matrix
and q,(n,) signal for the IFRT use the same negative sign in
the exponential. The reason for this is simple enough-the
FRT algorithm decomposes into a correlation computation
sec2
whereas the IFRT reduces to a convolution. The computational
( for' uniform angular cocerage.
steps of the IFRT algorithm are shown in Fig. 10, and a
(53) flow diagram of the FRT and IFRT are displayed in Fig. 11.
When an arbitrary collection of slopes is described by g ( s ) , The symmetry between the forward and inverse algorithm
a suitable choice of h can be determined. The values shown in Fig. 11 are readily apparent. However, l l ( b ) requires an
in (53) represent choices of h for typical sampling techniques. additional operation involving the derivative of the Hilbert
Other choices of h ( s ) are necessary for other sampling vari- transform [3], [5], [7]. When performed in the frequency
domain, the computation required for this step is relatively
ations.
minor compared to the matrix multiply operations. Two other
The convolutional matrix equation is
more subtle differences in Figs. l l ( a ) and l l ( b ) are the sign
"1
.171(m)=
~ , ( n ) A s L ( m-, n,)
(54) change in the 1-D column F I T in the IFRT and FRT algorithm
) As,(k).
and the different composition of A ~ s ( k and
n=O
Applications of the FRT computation are demonstrated
where6
A s ~ ( 7 n is
) a matrix of support [0 : S - 1 , 0 : L - 11 such in Figs. (12) and (13). Each image consists of a grayscale
that for a given value of m, the matrix element at index 256 x 256 image. In the unshifted coordinate axis, the origin
of each input image is the coordinate ( ( L- 1)/2, ( N - 1)/2).
(s, 1 ) = h(s)S(m- ( N - 1)/2, g(.s)(l - ( L - 1)/2));
g L ( m )is a row vector of support [0 : L - 11 and with Therefore, all slopes and intercepts are computed with respect
to a (0,O) coordinate which is placed at the center of each
the l'th element equal to z ( m ,l ) , and;
u,(n) is a row vector of support [0 : S - 11 and with the image. We selected the parameters T = 4, S = L = 256,
and B = 4. In addition g ( s ) was chosen according to
s'th element equal to w(n,,s).
(22). In Fig. 12(a) we display a 256 x 256 tool image. The
In the frequency domain, the IFRT computation is
central 512 x 256 Dortion of the FRT transformed image
X l ( k )= W S ( k ) A S L ( k )
(55) is displayed in Fig. 12(b). Note that a symmetric, nonlinear
selection of slopes from [-B,B] via (22) was chosen. In
where
addition, the intercept values represent linear gradations from
S-1
, (56) [ - ( ( L - 1)/2), - ( ( L - 1)/2) 1,.. . , ( L - 1)/2 - 1,( L F{w,(n,)) = W s ( k )=
q,(n)exp
1)/2]. Each of the four bright spots in this instance corresponds
n=O
to one of the individual tool icons. The reconstruction by
means of the IFRT is displayed in Fig. 12(c). Figure 13
N-l
1
illustrates the results when the identical procedure is applied
=N n = O %(') .p' ( i27r
3
(57) to a ball image.
The computational aspects of the DRT are summarized
in Table I. Since the Hilbert transform operation is only
hATI,Orl) # A L , , ( J t z ) . Only in the special instance of h ( s ) = 1 is
O ( N 21% N ) , the total computation of the IFRT is still in
2,iL ( J J J ) the transpose of A r 5 ( TI7 ).
the same order as the FRT.
I
[
arc tan[^]),
-
+
$)
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
396
Add zeros to the columns of
the image so that the augmented
image posesses an aspect ratio
of r. The resulting matrix is of
size N x L.
Compute the FFT of the columns of
y(n,s) to form _Y(k)
Filter with the derivative of the Hilbert
transform to form VJ(k).
l c s g
l -
Compute the IFFT of
W(k) to form _w(m)
s51
compute the 1D FFT of
g(r,k,s) to form
'LL
I
compute the matrix vector
(k)
Compute the IFFT of the columns
in parallel to form Ll(n)=x(n,l)
The IFRT is of size N x L.
Truncate to the original image
, size.
+
x(nA
Fig. 10. A flowgraph of the IFRT algorithm.
VII. SUMMARY
The FRT is a very efficient, parallel, and flexible means of
computing the Radon transform. This is due to the ability to
compute any arbitrary set of projections angles with the FRT
computations. We assumed either uniformly spaced angular
projections or uniformly varying slopes for the forward FRT
transform. This represented a desired choice as opposed to
an implementation constraint. In addition, a fast computation
can be performed concurrently in each frequency index. The
DRT algorithm is based upon the discretization of a line
)
We have presented a new parameterization
in ( 7 , ~ space.
method that leads to a discrete Radon transform computation
which avoids many of the difficulties associated with previous
classical methods. This new IFRT computation introduced represents a natural extension to the FRT based upon the inverse
formula for ( T . p ) Radon transform. It represents an efficient,
theoretically sound method of inversion that circumvents least
squares and matrix inversion approaches, avoids the need for
data interpolation, and requires no classical back-projection
operation. It is computationally as efficient as the forward
FRT. Presently we are investigating an inverse formulation for
the 3-D FRT and the performance on a number of industrial
applications (e.g., in machine vision and seismic migration,
especially in conjunction with a new architecture developed
for computing the Fourier transform [13]).
APPENDIX
Proof of Theorem 2: For an even sequence the causal
frequency domain KMSD function is defined as
A(
h ( m ) = S m--
N
1
y,
[ I - q , )
+ exp(-i7rsm) + 2
N/2--1
r=l
391
KELLEY AND MADISETTI: THE FAST DISCRETE RADON TRANSFORM
+
( $ + ))
1 - exp(-i27r(
+ +))
1 - exp(-i27rN
N
1
N/2-1
r=l
+
5 +
+
1
1 - exp(-i27r(r.s
k))
1- ( 3 x 4 4 2 ( r s k ) )
.
1
N/2-1
r=l
(W
--
1-D IFFTI
Col.
+ + $))
exp(-i27r( + + + ))
1- e x p ( - i 2 ~ N (
Fig. 11. A flow diagram of the FRT and IFRT algorithm.
IN/2-1
N/2-1
1
N
1
+ exp(-ixsm) + 2
’
r=l
m=O
++
1 - exp(-i27r(rs
k ) ) exp( i
1 - exp(-i2 % ( r s k ) )
N-l
+ -1
‘.p[’
$mors)
. 2 ~ (rN)
N
r=(N/2+1)
’
1
+
+
1 - exp(-i27r((r - N ) s k ) )
1 - exp(-i2 5 ( ( r - N ) s k ) )
(21:
. exp i - m o ( r - N ) s
H(k)=
x [N S ( k ) +
N-1
(65)
m=O
N/2-1 N - l
r=l
1 - exp(-i27rN(
1 - exp(-i27r(
N / 2 - 1 .V-1
r=l
1
m=O
$- + $ ) )
+ + $ ))
m=O
1
’
+
+
I - exp(-i27r((r - N ) s k ) )
1 - exp(-i2 5 ( ( r - N ) S k ) )
. exp
(21:
i - [mo(r - N ) s - ( r - N ) l o ] ) (66)
In converting from r to 1, we are interested in using the FFT.
,
-
I
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
398
Fig. 12. (a) displays the original 2 j G x 256 tool image. The FRT of the image is illustrated in (b) and the IFRT of (b) is shown in (c).
We should compute using L = N , and this implies mo = 10.
1
H(k)= N
.Y/ 2
(1)
N
r=O
1+
1 - exp(-227r(rs
1 - exd-22
(TS
frequency domain KMSD function is defined as
A(
"2
h(m,)= 6 m, - 1, [lk))
+ +k))
( N - l)r7r
-2
[1 - .I)
N
1
+N
*T-l
r= ( ,Y/2
exp
+1 )
h(7Tb) =
[.2r
]
+
(.( N
.exp
2
-
1):
-
N
[+
1
(N-1)/2
2
r=l
2-(1)
(69)
The reader is encouraged to finish the remainder of the proof
which is similar to the even case.
1 - exp(-i27r((r - N ) . k ) )
1 - exp(-i2
((7- - N ) "
k))
+
y].)
+
- N)7r
[l - s])
0
(67)
ACKNOWLEDGMENT
Proof of Theorem 3: For an odd sequence the causal
1
T -
The authors thank Prof. Jim McClellan for his technical
comments that improved upon the presentation. They grate-
r
399
KELLEY AND MADISEITI. THE FAST DISCRETE RADON TRANSFORM
(c)
Fig. 13. (a) displays the original 2 j G x 2jG ball image. The FRT of the image is illustrated in (b) and the lFRT of (b) is shown in (c).
TABLE I
THERADONTRANSFORM
Transform
Definition
fully thank the three anonymous reviewers and Dr. David
B. Harris for detailed, critical, and insightful comments that
greatly sharpened the technical content of the paper. Thanks
are also due to Dr. John Cozzens of NSF for his support of
this project.
Complexity
Comments
REFERENCES
[ 11 V. K. Madisetti and D. G. Messerschmitt, “Seismic migration algorithms
on parallel computers,” IEEE Trans. Acoust., Speech, Signal Processing,
vol. 39, pp,
.. 1642-,654, July 1991,
[2] D. Miller, M. Oristaglio, and G. Beylkin, “A new slant on seismic
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 2, NO. 3, JULY 1993
imaging: Migration and integral geometry,” Geophysics, pp. 943-964,
1987.
A. K. Jain, Fundamentals of Digital Image Processing. Englewood
Cliffs, NJ: Prentice-Hall, 1988.
Stanley R. Deans, The Radon Transform and Some of its Applications.
New York: John Wiley and Sons, 1983.
J. L. Sanz, E. B. Hinkle, and A. K. Jain, Radon and Projection TransformBased Computer Vision. Berlin: Springer-Verlag, 1987.
Agi Iskender, P. J. Hurst, and W. K. Current, VLSI Signal Processing
IV: A Pipelined Architecture for Radon Transform Computation in a
Multiprocessor Array. New York IEEE Press, 1990.
D. E. Dudgeon and R. M. Mersereau, Multidimensional Digital Signal
Processing. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1984.
Gregory Beylkin, “Discrete radon transform,” IEEE Trans. Acoust.,
Speech, Signal Processing, vol. 35, pp. 162-172, Feb. 1987.
B. T. Kelley and V. K. Madisetti, “The fast discrete radon transform.”
Proc. Int. Cont Acoust., Speech, Signal Processing, IEEE-ICASSP’92,
1992, vol. Ill, pp. 409-412.
G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore: The
Johns Hopkins University Press, 198.5.
B. T. Kelley and V. K. Madisetti, “Optimal concurrent VLSI architectures for 2-d transposition,” in Advanced Research in VLSI, Carlo
Sequin, Ed. Cambridge, MA: MIT Press, Mar. 1991.
Enders A. Robinson, Migration of Geophysical Data. Boston, MA:
International Human Resources Development Corp., 1983.
B. Kelley and V. Madisetti, “Efficient VLSI architectures for the 1-d and
2-d arithmetic fourier transform (aft),” IEEE Trans. Signal Processing,
vol. 41, pp. 36.5-384, Jan. 1993.
Brian T. Kelley received the B.S degree in electrical engineering in 1987 from the Cornell University
and the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology in
1989 and 1992, respectively.
He is presently employed by Motorola, Inc in
Austin, TX, in the general area of VLSI for signal
processing. During the summer of 1988 he held
a research position at Naval Research Laboratory
in Washington, DC. During the summer of 1989
he worked on CMOS VLSI Design for IBM at
T J. Watson Research Center. His interests include real time digital signal
processing, VLSI signal processing, multidimensional signal processing with
applications to imaging, and in digital communication systems.
From 1987 to 1990 he held both an Office of Naval Research Fellowship
and a Presidential Fellowship at Georgia Tech. Dr Kelley is a member of
Tau Beta Pi and Eta kappa Nu He received the Thurgood Marshall Doctoral
Dissertation Fellowship awarded by Dartmouth College for 1992.
I
T---
Vijay K. Madisetti (S’82-M’89) received his B
Tech (Honors) in electronics and electrical communications engineering at the Indian Institute of
Technology (IIT), Kharagpur, India, in 1984, and
the Ph.D. in electrical engineering and computer SCIences at the University of California (UC), Berkeley,
in 1989 Dr. Madisetti was a Jagadis Bose National
Science Talent Search Fellow at IIT Kharagpur from
1979-84.
He has been an Assistant Professor of digital
signal processing in the School of Electrical Engineering, and an Adjunct Professor in the School of Earth and Atmospheric
Sciences, at the Georgia Institute of Technology, Atlanta, GA, since 1989 He
directs the industry-funded National Center of Excellence in DSP (NCEDSP),
in the School of Electrical Engineering at the same institute. His research
interests are in digital signal processing, image processing, modeling and
simulation of multiprocessor computer systems, seismic signal processing in
oil exploration, VLSI signal processing, and digital communication systems
He has published and consulted frequently in these areas. He serves on the
Academic Faculty Senate at Georgia Tech, and as an Associate Editor of
the INTERNATIONAL JOURNAL IN COMPUTER SIMULATION. He. serves as the
Technical Program Chairman of the IEEEiACM International Workshop on
Modeling, Analysis, dnd Simulation of Computer and Telecommunication
Systems-MASCOTs’94.
Dr Madisetti received a General Proficiency Prize at IIT Kharagpur in
1983, and the Demetri Angelakos Achievement Award at UC Berkeley in
1989 He was also a recipient of the Ira Kay Memorial Best Paper Prize at the
22cnd ACMiIEEE Annual Simulation Symposium in 1989, the IBM Faculty
Development Award 1990-92, and the NSF Research Initiation Award 1992