F'sLGORITHM
FOR COMPUTING THE
RCSDON
TReNSFORM
W I T H SOME CSPPLICfATIONS
CSN
DISCRETE
Ramwah Q . R .
.N.
'School o f E l e c t r i c a l Engg.,
Purdue Uni v e r s i t y ,
West L a f a y r t t e , I n d i a n a ,
I N 47907-0501, USA.
Dept. o f E l e c t r i c a l Engg.,
I n d i a n I n s t i tut e o f Science,
560 012, I N D I A .
Bangalore
-
ABS TRAC T
The enormous g r o w t h i n
the application
areas o f
t h e Radon
T r a n s f o r m and t h e f a c t
t h a t d i g i t a l computations a r e o f t e n r e q u i r e d ,
has .led t o t h e development o f
the Discrete
Radon T r a n s f o r m ( DRT 1 .
T h i s paper prdposes
a method f o r c o m p u t i n g t h e DRT b y
exploiting
t h e converse o f
t h e C e n t r a l S l i c e Theorem
r e l s t i n g t h e Radon
and
Fourier
Transforms,
using
t h e FFT a l g o r i t h m .
Simulation o f the
DRT a l g o r i t h m i s p r e s e n t e d and some
applicat i o n s a r e discussed.
The p r o p o s e d a l g o r i t h m
in
t h e Radon
c a n be e x t e n d e d f o r f i l t e r i n g
Space.
1.
K. R a j g o p a l
Srinivama
Introduction
The Radon transform (RT), apart from its
key role in the reconstruction of multidimensional signals from their projections (Computer Assisted Tomography C121), is a powerful tool in various image processing and
machine vision applications [lo],
a s it offers
significant
advantages for general
image representation and manipulations.
The
Central Slice Theorem which relates the RT
(spatial concentrations) of an image with i t s
Fourier transform (FT)(spatial
frequencies),
reinforces the considerations underlying the
use of the
projection
space
as
a
'Recognition Domain'. Significant features in
the image can be detected in a sufficiently
sized projection space Cll].
Projection generation is being used to aid feature selection [8,11,14-17].
The Hough Transform (HT)
C 7 1 , which is an algorithm for detecting
straight line segments in pictures i s a special case of the RT [SI.
Tile computation of the RT is thus gaining importance.
It is not possible to obtain
the R T directly from discrete data. The R T
for discrete data has been defined in a number of ways, each suitable to a particular
application. Some require the interpolation
of the available data values to a dense grid
[3,18], and some other require a weighting of
the discrete data [11,14].
An approximation
to
the
RT for discrete data has been
generally referred to in the literature a s
the DRT. However, strictly speaking, the DRT
involves a summation of the discrete data
along various straight lines in the plane of
the grid. This invariably involves an interpolation o f the available data.
In many applications, there i s a need for
simpler and fast methods for computing the
RT. Recently, a -fast algorithm has been
proposed for the computation of the DRT by
Beylkin C3I based on the formulation given by
Scheibner C181. However, the slope intercept
form of a straight line used here introduces
complications, a s these parameters are not
bound. Apart from requiring the storage of a
large number of huge 'transform' matrices,
this algorithm appears to be fast only for
rectangular arrays with one of the dimensions being considerably small and hence
suitable for seismic applications. Hinkle et
a1.[101
have considered simpler schemes,
however, for a special purpose architecture.
A new approach to compute
the DRT of
a
digital image using FFT algorithms based on
the (converse of) Central Slice Theorem (CST)
Cl21 i s explained in the following sections.
The emphasis has been to develop a method
that i s appreciably fast, of moderate cost,
relatively simple and amenable for parallel
implementation. The organization of the rest
of the paper is a s follows. The CST and its
converse are reviewed in the following Soctions. The DRT algorithm i s developed in Section 4, followed by simulation results in
Section 5. Some important applications are
discussed in Section 6. Section 7 concludes
the paper.
2..
T h e Radon Transform
Let f(x,y) represent a two dimensional
(2-D) function on the plane and F(u,v),
its
FT. The Radon Tansform of f(x,y) is the integral of f(x,y) along all possible straight
lines in the plane containing the function,
provided the line integral exists:
m m
R{f(x,y)>=p(t,B)=
il
f(x,y)6(t-xcos€J-ysine)dxdy
straight line
normal form',
The
t = x cost3 + y sine
is
represented
in
its
(2)
where, ' t ' is the radial distance and 8 , the
angle
made
by
the
normal.
The term
'projection' at an angle 8 refers to the R T
p(t,e) will be
integral ( 1 ) evaluated at 8.
written a s p,(t)
to emphasize that a 1-D
4.4.1
78
(1)
-m-m
CH2766 - 4/89/0000 - 0078 0 1989 IEEE
function i s
parameter.
considered,
with
8
as
a
5 . The C e n t r a l B l i c e Theormr
T h e Central S l i c e T h e o r e m s t a t e r that
t h e F o u r i e r Transform o f t h e
projection a t
a n a n g l e 8 ( t h e R T a l o n g parallel l i n e s at
a n g l e 8) denoted by P*(w), i s a central s l i c e
o f t h e 2-D F o u r i e r T r a n s f o r m of t h e i m a g e
f ( x , y ) , at t h e s a m e a n g l e , 8 :
FT(p.(t))
= Pe(w) = F ( w c o s 6 , w s i n e )
(3)
Conversely, t h e projection p*( t) may be
obtained by using t h e i n v e r s e FT:
pl(t)=F-+CPl(w))=F-+(
F(wcos8,wsine) 1
(4)
Equation ( 4 ) s t a t e s that t h e i n v e r s e F T
of t h e 'central' s l i c e of t h e 2-D F T o f t h e
i m a g e f ( x , y ) g i v e s t h e projection o f t h e
i m a g e a t t h e s a m e angle.
It i s t h u s p o s s i b l e
t o fill t h e Radon S p a c e by
considering
several equi-angular slices.
polation i s used f o r illustration ( f o r l i n e s
passing t h r o u g h t h e l a t t i c e points, interpolation is n o t required and t h e DRT i s obtained by s u m m i n g t h e c o r r e s p o n d i n g d a t a
values):
i+
XP*(W) =
(
X
3
ia1
(
(
k+,l+)/dA(w,8)
1 / d+(w,8) 1
)
(6)
XP*(w) a r e t h e s a m p l e s o f t h e F T of x(m,n)ak
a n g l e 8 and d i s t a n c e s w
(radial frequency)
from t h e origin. d i ( w , 8 ) d e n o t e t h e d i s t a n c e s
between t h e point ( w , 8 ) o n t h e polar grid and
i t s four n e a r e s t C a r t e s i a n neighbors. Fig.
l(a) shows the parameters for the definition
of linear interpolation.
Fig. l ( b ) s h o w s t h e
polar r a s t e r indicating t h e s e t
of p o i n t s
o n t o w h i c h t h e C a r t e s i a n s a m p l e s a r e mapped.
T h e d i s t a n c e s d i s need be precomputed and
stored o n l y for p o i n t s in t h e f i r s t quadrant,
d u e t o symmetry.
3) P e r f o r m t h e i n v e r s e F F T of t h e t h e s e
s l i c e s t o get t h e DRT- t h i s can b e d o n e at
all a n g l e s i n d e p e n d e n t l y and h e n c e in parallel.
4. D i s e r m t r R a d o n T r a n s f o r m A l g o r i t h m
T h e D i s c r e t e Radon T r a n s f o r m
( D R T ) of a digital i m a g e f(m,n) at a n .angle
8, i s t h e sum of t h e v a l u e s of f(m,n) along
t h e corresponding orthogonal directions.
Definition:
CIS a c o n s e q u e n c e of t h e c o n v e r s e o f
the
Central S l i c e Theorem
( equation (4)) the
D R T may be defined in t e r m s o f t h e D F T o f t h e
image, a s follows:
The DRT o f a d i g i t a l
image
f(m,n)
at
Remark: In a local i n t e r p o l a t i o n s c h e m e s u c h
a s t h e o n e described above, t h e e r r o r will be
independent from point t o point.
The
above algorithm is efficient a s
F F T c a n be used t o c o m p u t e t h e D F T and i s
a l s o a m e n a b l e for parallel implementation. It
i s i m p o r t a n t t o n o t e that, s u i t a b i l i t y o f an
a l g o r i t h m f o r parallel i m p l e m e n t a t i o n i s often a d e s i r a b l e feature, i r r e s p e c t i v e o f t h e
exact n u m b e r of c o m p u t a t i o n s involved.
an
m g l r 9, i s t h e i n v e r s e DFT o f t h e samples on
the central slices o f
a t t h e same a n g l e 8.
t h e 2-D DFT o f
f(m,n),
The above definition provides the basis
t h e c o m p u t a t i o n of t h e D R T from t h e D F T
o f f(m,n).
T h e a d v a n t a g e of s u c h a method i s
efficiency, s i n c e F F T a l g o r i t h m s c a n b e used
t o c o m p u t e t h e DFT.
S i n c e t h e 2-D D F T corr e s p o n d s t o t h e s a m p l e s of t h e (2-D) F T of
the image on a Cartesian
grid, it i s
required t o m a p t h e s a m p l e s o n t o a polar
ramter. A s i m p l e method i s i n v e r s e d i s t a n c e
linear interpolation. However, it i s important t o n o t e that t h e a c c u r a c y of t h e technique
d e p e n d s o n t h e t y p e of interpolation
used. T h e s t e p s in t h e method are:
for
1) C o m p u t e t h e 2-D D F T
i m a g e x(m,n)r
X(k,l)
of
the
& ~ x ( m , n ) e - ~ c 3 m / N )c r n k - i n ) ,
k,l=1,2,..N
(5)
T h i s c a n be d o n e by first c o m p u t i n g t h e
t r a n s f o r m s o f t h e r o w s ( in parallel ) 81 t h e
t r a n s f o r m s of t h e c o l u m n s o f t h e result ( in
parallel
using F F T algorithms.
X ( k D 1 )=
msln.,
2 ) C o m p u t e t h e central s l i c e s of t h e
2-D
DFT.
This requires a mapping of the samples
o f t h e F T from t h e C a r t e s i a n r a s t e r o n t o a
polar raster.
I n v e r s e d i s t a n c e linear interpolation i s used f o r illustration ( f o r lines
passing through t h e lattice points, inter-
Remark:
f3 s i m i l a r a p p r o a c h f o r t h e c o m p u t a tion of t h e R T h a s been c o n s i d e r e d by M u r p h y
E131, however, in t h e c o n t e x t of linear feature enhancement
in
SFIR
images.
Very
r e c e n t l y , a t r a n s p u t e r i m p l e m e n t a t i o n has
been developed f o r t h e s a m e C91.
3. S i m u l a t i o n
S i m u l a t i o n r e s u l t s o n a c o m p u t e r generated
test i m a g e of s i z e 64x64 a r e s h o w n
,in
figs. 2-4. Fig. 2 s h o w s t h e i m a g e u n d e r consideration. Figs.
3(a)-(d) s h o w t h e project i o n s o b t a i n e d by t h e weighted s u m m a t i o n
s c h e m e using t h e 'pixel' a s s u m p t i o n , a s outlined by B a r r e t [2]. T h e w e i g h t s h a v e been
s e t equal t o t h e l e n g t h s of i n t e r s e c t i o n of
t h e l i n e s ( 2 ) w i t h t h e p i x e l s assumed around
t h e C a r t e s i a n grid points,
in o r d e r t o
c l o s e l y a p p r o x i m a t e (1). Figs. 4(a)-(d) s h o w
t h o s e c o m p u t e d by t h e proposed D R T algorithm.
It may be observed t h a t at a n g l e s c o r r e s p o n d ing t o t h e F o u r i e r s l i c e s away from t h e axes,
t h e p r o j e c t i o n s c o m p u t e d by t h e proposed D R T
a l g o r i t h m d o n o t c o m p a r e well w i t h t h o s e of
fig. 3.
T h i s can b e attributed
to the
following:
( 1 ) T h e p o i n t s within t h e c i r c l e
inside the Cartesian frequency raster are
c o n s i d e r e d , w h i c h a m o u n t s t o t r u n c a t i o n ( see
fig. l.l(b) ) . T h i s e f f e c t b e c o m e s s e v e r e for
s l i c e s f a r a w a y from t h e axes. T h i s may be
minimized by c o n s i d e r i n g all t h e u n i f o r m l y
spaced p o i n t s a l o n g t h e s l i c e s so a s t o en-
4.4.2
79
c o m p a s s t h e e n t i r e s q u a r e raster.
( 2 ) As
mentioned earlier, s i m p l e linear interpolat i o n m a y n o t b e s u f f i c i e n t t o a c c u r a t e l y RSt i m a t e t h e p o l a r samples, e s p e c i a l l y i n t h e
c a s e o f images
with
considerable
high
f r e q u e n c y content. T h i s is a . l i m i t a t i o n of
t h e proposed algorithm. H o w e v e r ,
a high
d e g r e e o f a c c u r a c y i s n o t required in many
applications.
.
6.
applieationm
The
H o u g h T r a n s f o r m ( H T ) h a s been
r e c o g n i z e d in t h e l i t e r a t u r e a s a s p e c i a l
c a s e o f t h e RT. However, s t r i c t l y speaking,
i t i s t h e D R T w h i c h i s c l o s e r t o t h e H T than
t h e RT. T h i s c a n b e a p p r e c i a t e d f r o m t h e f a c t
t h a t t h e D R T ( a n d n o t t h e R T ) o f a binary
i m a g e r e d u c e s t o i t s HT.
T h e conventional
H o u g h t e c h n i q u e i s limited by s l o w s p e e d and
e x c e s s i v e m e m o r y requirements.
In a d d i t i o n ,
a s S u t e r C l 9 1 h a s pointed o u t , o n e o f t h e
d e f i c i e n c i e s o f t h e H T i s a d i s t o r t i o n in t h e
s h a p e a n d t h e location o f t h e p e a k s in t h e
parameter
( T r a n s f o r m s p a c e , w h i c h may b e
e l i m i n a t e d by t h e u s e . o f t h e DRT.
The
proposed D R T a l g o r i t h m c a n r e p l a c e t h e H o u g h
T r a n s f o r m a t i o n s t e p in a s t r a i g h t line detection
scheme,
making i t very efficient.
Recently,
D. C a s a c e n t and R .
Krishnapuram
C41 h a v e d e v e l o p e d a t e c h n i q u e w h i c h i s a n
e x t e n s i o n o f t h e H T , t o d e t e c t and l o c a t e
c u r v e s in images. T h i s a l g o r i t h m a v o i d s t h e
e x t e n s i v e pre-processing
and h i g h
dimens i o n a l i t y o f t h e H o u g h S p a c e t h a t i s required
in t h e r e c e n t m e t h o d s in g e n e r a l i z i n g t h e
H o u g h T r a n s f o r m Cl].
T h e proposed a l g o r i t h m
s e r v e s 'as a f a s t and e f f i c i e n t ( a n d p a r a l l e l )
i m p l e m e n t a t i o n o f t h e H o u g h s c h e m e and c u r v e d
o b j e c t recognition.
T h e D R T a l g o r i t h m d e s c r i b e d a b o v e may b e
extended f o r filtering in t h e R a d o n S p a c e , a s
follows. T h e c o n v o l u t i o n theorem in t h e R a d o n
S p a c e E61 s t a t e s that t h e R T of t h e c o n v o l u tion o f a f u n c t i o n f(x,y) w i t h a filtering
f u n c t i o n h ( x , y ) is t h e c o n v o l u t i o n of t h e i r
p r o j e c t i o n s p m ( t ) and h*(t),
respectively,
f o r all 8:
RT(f(X,y)tth(X,y))=p,otha(t)~h~(t)=P~'(t) (7)
In t h e f r e q u e n c y domain:
P ~ ' ( w ) = Hm(w)Pm.(w)
(8)
Step-2 of t h e D R T
a l g o r i t h m may b e
a s a weighting
m o d i f i e d by i n t r o d u c i n g H,(w)
factor t o obtain the samples of the F T of the
filtered f u n c t i o n o n a p o l a r grid:
He(w)
P*'(w)
=
2,
9(
i = 1
X(k+,l+)/d+(w,B)
(9)
1 1 d+(w,e) 1'
By t h e C S T , P',(w)
f o r m s a central s l i c e
of t h e (2-D) F T o f t h e f i l t e r e d
image
f'(x,y),
w h i c h may be o b t a i n e d by d i r e c t
F o u r i e r i n v e r s i o n technique. T h u s , 2-D linear
shift-invariant f i l t e r s c a n be r e a l i z e d by a
s e t of d e c o u p l e d 1 - D filters, by working in
t h e R a d o n Space. .The s c h e m e is illustrated in
fig. 5. T h i s t e c h n i q u e is particularly useful
for imp 1 emen t ing 2-D c i rcu 1 a r 1 y s y m m e t r i c
f 1 1 ters.
T h e technique outlined above suggests
a p p l i c a t i o n t o multi-directional filtering, a
topic t h a t h a s n o t received m u c h attention.
pernark: T h e proposed a l g o r i t h m i s a m e n a b l e t o
t r a n s p u t e r i m p l e m e n t a t i o n C91.
7 . Summary
O n e f f i c i e n t a l g o r i t h m f o r t h e computat i o n o f t h e D R T h a s been presented.
I t s application f o r t h e d e t e c t i o n o f s t r a i g h t line
segments and
extension for curved object
recognition
h a v e been discussed.
The 4190rithm i s a m e n a b l e f o r parallel implementation,
a c u r r e n t trend in signal and i m a g e
processing.
A modification o f t h e algorithm
f o r f i l t e r i n g in t h e Radon S p a c e h a s been
proposed.
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80
POLAR SnflPLE
Fig.
d
Fig.
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-- --
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2:
The T e s t Image
60
7
NXN
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Transform W i t h Some A p p l i c a t i o n s t o V e l o c i t y
Estimation",
EE Tech.
R e p o r t 8301,
March
1981, R i c e U n i v e r s i t y , Texas.
Cl91 B.W.
Suter,
"6 P a r a l l e l Algorithm f o r
t h e D i s c r e t e Radon T r a n s f o r m " , P r o c . Computer
A r h i t n c t u r e f o r P a t t e r n A n a l y s i s and Image
Data Base Management", pp: 394-398, 1985.
loo
30
20
0
40
20
50
Modi t iod
1ntorp1n.-
.
Polar-Rect.
I nt o r p l n
-
2-D
IFFT
,
f'(m),n)
Fig. 5
4.4.4
81