C S N F'sLGORITHM FOR COMPUTING THE DISCRETE RCSDON TReNSFORM
W I T H SOME CSPPLICfATIONS
R a m w a h Q . R . .N. S r i n i v a m a K. R a j g o p a l D e p t . o f E l e c t r i c a l E n g g . , ' S c h o o l o f E l e c t r i c a l E n g g . , I n d i a n I n s t i t u t e o f S c i e n c e , P u r d u e U n i v e r s i t y ,
B a n g a l o r e
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560 012, I N D I A . West L a f a y r t t e , I n d i a n a , I N 47907-0501, USA.ABS TRAC T
The enormous g r o w t h i n t h e a p p l i c a t i o n a r e a s 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 c o m p u t a t i o n s a r e o f t e n r e q u i r e d , h a s . l e d t o t h e d e v e l o p m e n t o f t h e D i s c r e t e Radon T r a n s f o r m ( DRT 1 . T h i s p a p e r p r d p o s e s a method f o r c o m p u t i n g t h e DRT b y e x p l o i t i n g t h e c o n v e r s e 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 F o u r i e r T r a n s f o r m s , u s i n g t h e FFT a l g o r i t h m . S i m u l a t i o n o f t h e DRT a l g o r i t h m i s p r e s e n t e d and some a p p l i c a -
t i o n s a r e d i s c u s s e d . The p r o p o s e d a l g o r i t h m c a n be e x t e n d e d f o r f i l t e r i n g i n t h e Radon S p a c e .
1. I n t r o d u c t i o n
T h e Radon transform (RT), apart from its key role in the reconstruction of multidimen- sional signals from their projections (Com- puter Assisted Tomography C121), i s a power- ful tool in various image processing and machine vision applications [lo], a s it of- fers significant advantages for general image representation and manipulations. T h e Central S l i c e Theorem which relates the RT (spatial concentrations) o f a n image with i t s Fourier transform (FT)(spatial frequencies), reinforces the considerations underlying the u s e of the projection space a s a 'Recognition Domain'. Significant features in the image can be detected in a sufficiently sized projection space Cll]. Projection gen- eration i s being used to aid feature selec- tion [8,11,14-17]. The Hough Transform ( H T ) C 7 1 , which i s an algorithm for detecting straight line segments in pictures i s a spe- cial c a s e of the RT [SI.
Tile computation of the RT i s thus gain- ing importance. It is n o t possible to obtain the R T directly from discrete data. T h e R T for discrete data has been defined in a num- ber 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 inter- polation 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 D R T by Beylkin C3I based o n the formulation given by Scheibner C181. However, the slope intercept form of a straight line used here introduces complications, a s these parameters a r e not bound. Apart from requiring the storage of a large number o f huge 'transform' matrices, this algorithm appears to be fast only for rectangular arrays with o n e of the dimen- sions being considerably small and hence suitable for seismic applications. Hinkle e t a1.[101 have considered simpler schemes, however, for a special purpose architecture.
A new approach to compute the D R T of a digital image using F F T algorithms based o n the (converse of) Central Slice Theorem (CST) Cl21 i s explained in the following sections.
T h e emphasis has been to develop a method that i s appreciably fast, of moderate cost, relatively simple and amenable for parallel implementation. T h e organization of t h e rest of the paper i s a s follows. T h e CST and its converse a r e reviewed in the following Soc- tions. T h e DRT algorithm i s developed in Sec- tion 4, followed by simulation results in Section 5. Some important applications a r e discussed in Section 6. Section 7 concludes the paper.
2 . . T h e R a d o n T r a n s f o r m
Let f(x,y) represent a two dimensional (2-D) function on the plane and F(u,v), its FT. T h e Radon Tansform of f(x,y) i s the in- tegral 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)= -m-m
il
f ( x , y ) 6 ( t - x c o s € J - y s i n e ) d x d y ( 1 ) T h e straight line i s represented in i t snormal form',
t = x cost3 + y sine ( 2 ) where, ' t ' i s the radial distance and 8 , the angle made by the normal. T h e term 'projection' at an angle 8 refers to the R T integral ( 1 ) evaluated at 8. p(t,e) will be written a s p,(t) to emphasize that a 1-D
78
4.4.1
CH2766 - 4/89/0000 - 0078 0 1989 IEEE
f u n c t i o n i s c o n s i d e r e d , w i t h 8 a s a parameter.
5 . The C e n t r a l B l i c e Theormr
T h e C e n t r a l S l i c e T h e o r e m s t a t e r t h a t t h e F o u r i e r T r a n s f o r m o f t h e p r o j e c t i o n a t a n a n g l e 8 ( t h e R T a l o n g p a r a l l e l l i n e s a t a n g l e 8) d e n o t e d by P*(w), i s a c e n t r a l 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 ) , a t t h e s a m e a n g l e , 8 :
FT(p.(t)) = P e ( w ) = F ( w c o s 6 , w s i n e ) (3) C o n v e r s e l y , t h e p r o j e c t i o n p*( t) m a y b e o b t a i n e d by u s i n g t h e i n v e r s e FT:
pl(t)=F-+CPl(w))=F-+( F ( w c o s 8 , w s i n e ) 1 ( 4 ) E q u a t i o n ( 4 ) s t a t e s t h a t 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 p r o j e c t i o n 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 R a d o n S p a c e by c o n s i d e r i n g s e v e r a l e q u i - a n g u l a r s l i c e s .
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 D e f i n i t i o n : T h e D i s c r e t e R a d o n T r a n s f o r m ( D R T ) of a d i g i t a l i m a g e f ( m , n ) a t a n . a n g l e 8, i s t h e s u m of t h e v a l u e s of f ( m , n ) a l o n g t h e c o r r e s p o n d i n g o r t h o g o n a l d i r e c t i o n s .
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 t h e C e n t r a l S l i c e T h e o r e m ( e q u a t i o n ( 4 ) ) t h e D R T may b e d e f i n e d 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 ) a t 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 t h e c e n t r a l s l i c e s o f t h e 2-D DFT o f f ( m , n ) , a t t h e same a n g l e 8.
T h e a b o v e d e f i n i t i o n p r o v i d e s t h e b a s i s f o r t h e c o m p u t a t i o n of t h e D R T f r o m 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 m e t h o d i s e f f i c i e n c y , 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 c o r - r 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 t h e i m a g e o n a C a r t e s i a n g r i d , it i s r e q u i r e d t o m a p t h e s a m p l e s o n t o a p o l a r ramter. A s i m p l e m e t h o d i s i n v e r s e d i s t a n c e l i n e a r i n t e r p o l a t i o n . H o w e v e r , it i s i m p o r - t a n t t o n o t e t h a t t h e a c c u r a c y of t h e t e c h - n i q u e d e p e n d s o n t h e t y p e of i n t e r p o l a t i o n used. T h e s t e p s in t h e m e t h o d are:
1) C o m p u t e t h e 2-D D F T X(k,l) of t h e i m a g e x(m,n)r
& ~ x ( m , n ) e - ~ c 3 m / N ) c r n k - i n ) ,
k,l=1,2,..N ( 5 )
X ( k D 1 ) = msln.,
T h i s c a n b e d o n e by f i r s t 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 p a r a l l e l ) 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 r e s u l t ( in parallel u s i n g F F T a l g o r i t h m s .
2 ) C o m p u t e t h e c e n t r a l s l i c e s of t h e 2-D DFT. T h i s r e q u i r e s a m a p p i n g o f t h e s a m p l e s o f t h e F T f r o m t h e C a r t e s i a n r a s t e r o n t o a p o l a r raster. I n v e r s e d i s t a n c e l i n e a r i n t e r - p o l a t i o n i s used f o r i l l u s t r a t i o n ( f o r l i n e s p a s s i n g t h r o u g h t h e l a t t i c e p o i n t s , i n t e r -
p o l a t i o n i s u s e d f o r i l l u s t r a t i o n ( f o r l i n e s p a s s i n g t h r o u g h t h e l a t t i c e p o i n t s , inter- p o l a t i o n i s n o t r e q u i r e d and t h e DRT i s o b - t a i n e d 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+ ( X ( k + , l + ) / d A ( w , 8 ) )
i a 1
3
( 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 a n d d i s t a n c e s w ( r a d i a l f r e q u e n c y ) f r o m t h e o r i g i n . d i ( w , 8 ) d e n o t e t h e d i s t a n c e s b e t w e e n t h e p o i n t ( w , 8 ) o n t h e p o l a r g r i d a n d i t s f o u r n e a r e s t C a r t e s i a n n e i g h b o r s . Fig.
l ( a ) s h o w s t h e p a r a m e t e r s f o r t h e d e f i n i t i o n of l i n e a r i n t e r p o l a t i o n . Fig. l ( b ) s h o w s t h e p o l a r r a s t e r i n d i c a t i n g 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 n e e d b e p r e c o m p u t e d a n d s t o r e d o n l y f o r p o i n t s in t h e f i r s t q u a d r a n t , d u e t o s y m m e t r y .
XP*(W) =
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 g e t t h e DRT- t h i s c a n b e d o n e a t 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 paral- lel.
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 d e s c r i b e d a b o v e , t h e e r r o r will be i n d e p e n d e n t f r o m p o i n t t o point.
T h e a b o v e a l g o r i t h m i s e f f i c i e n t a s F F T c a n b e u s e d t o c o m p u t e t h e D F T a n d i s a l s o a m e n a b l e f o r p a r a l l e l i m p l e m e n t a t i o n . It i s i m p o r t a n t t o n o t e t h a t , 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 p a r a l l e l i m p l e m e n t a t i o n i s of- ten a d e s i r a b l e f e a t u r e , i r r e s p e c t i v e o f t h e e x a c t n u m b e r of c o m p u t a t i o n s i n v o l v e d .
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 - t i o n 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, h o w e v e r , in t h e c o n t e x t of l i n e a r fea- t u r e e n h a n c e m e n t in SFIR images. V e r y 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 h a s b e e n d e v e l o p e d 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 g e n e r a t e d t e s t i m a g e of s i z e 64x64 a r e s h o w n , i n figs. 2-4. Fig. 2 s h o w s t h e i m a g e u n d e r con- s i d e r a t i o n . F i g s . 3 ( a ) - ( d ) s h o w t h e projec- t i o n s o b t a i n e d by t h e w e i g h t e d s u m m a t i o n s c h e m e u s i n g t h e 'pixel' a s s u m p t i o n , a s out- lined by B a r r e t [2]. T h e w e i g h t s h a v e b e e n s e t e q u a l 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 a s s u m e d a r o u n d t h e C a r t e s i a n g r i d p o i n t s , 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). F i g s . 4(a)-(d) s h o w t h o s e c o m p u t e d by t h e p r o p o s e d D R T a l g o r i t h m . It may b e o b s e r v e d t h a t a t 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 a w a y f r o m t h e a x e s , 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 p r o p o s e d 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 c a n b e a t t r i b u t e d t o t h e f o l l o w i n g : ( 1 ) T h e p o i n t s w i t h i n t h e c i r c l e i n s i d e t h e C a r t e s i a n f r e q u e n c y r a s t e r a r e 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 f i g . 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 f o r s l i c e s f a r a w a y f r o m t h e axes. T h i s m a y b e m i n i m i z e d 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 s p a c e d 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 m e n t i o n e d e a r l i e r , s i m p l e l i n e a r i n t e r p o l a - t 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 RS-
t i m a t e t h e p o l a r s a m p l e s , e s p e c i a l l y i n t h e c a s e o f i m a g e s w i t h c o n s i d e r a b l e h i g h f r e q u e n c y
.
c o n t e n t . T h i s is a . l i m i t a t i o n of t h e p r o p o s e d a l g o r i t h m . H o w e v e r , a h i g h d e g r e e o f a c c u r a c y i s n o t r e q u i r e d i n m a n y a p p l i c a t i o n s .6 . a p p l i e a t i o n m
T h e 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. H o w e v e r , s t r i c t l y s p e a k i n g , 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 t h a n 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 b i n a r y i m a g e r e d u c e s t o i t s HT. T h e c o n v e n t i o n a l H o u g h t e c h n i q u e i s l i m i t e d by s l o w s p e e d a n d e x c e s s i v e m e m o r y r e q u i r e m e n t s . In a d d i t i o n , a s S u t e r C l 9 1 h a s p o i n t e d 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 i n t h e s h a p e a n d t h e l o c a t i o n o f t h e p e a k s i n t h e p a r a m e t e r ( T r a n s f o r m s p a c e , w h i c h m a y 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. T h e p r o p o s e d 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 i n a s t r a i g h t l i n e d e t e c - t i o n s c h e m e , m a k i n g i t v e r y e f f i c i e n t . R e c e n t l y , D. C a s a c e n t a n d R . K r i s h n a p u r a m 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 i n i m a g e s . 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 p r e - p r o c e s s i n g a n d h i g h d i m e n - s 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 r e q u i r e d i n t h e r e c e n t m e t h o d s i n 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 p r o p o s e d a l g o r i t h m s e r v e s 'as a f a s t a n d 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 a n d c u r v e d o b j e c t r e c o g n i t i o n .
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 m a y b e e x t e n d e d f o r f i l t e r i n g i n t h e R a d o n S p a c e , a s f o l l o w s . T h e c o n v o l u t i o n t h e o r e m i n t h e R a d o n S p a c e E61 s t a t e s t h a t t h e R T of t h e c o n v o l u - t i o n o f a f u n c t i o n f ( x , y ) w i t h a f i l t e r i n g 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 ) a n d h*(t), r e s p e c t i v e l y , f o r a l l 8:
R T ( f ( X , y ) t t h ( X , y ) ) = p , o t h a ( t ) ~ h ~ ( t ) = P ~ ' ( t ) ( 7 ) In t h e f r e q u e n c y d o m a i n :
P ~ ' ( w ) = H m ( w ) P m . ( w ) (8)
S t e p - 2 of t h e D R T a l g o r i t h m m a y b e m o d i f i e d b y i n t r o d u c i n g H,(w) a s a w e i g h t i n g f a c t o r t o o b t a i n t h e s a m p l e s o f t h e F T o f t h e f i l t e r e d f u n c t i o n o n a p o l a r grid:
H e ( w )
2,
X ( k + , l + ) / d + ( w , B )P*'(w) = ( 9 )
9
( 1 1 d + ( w , e ) 1 'i = 1
By t h e C S T , P',(w) f o r m s a c e n t r a l 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 i m a g e f'(x,y), w h i c h may b e 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 t e c h n i q u e . T h u s , 2-D l i n e a r s h i f t - i n v a r i a n t f i l t e r s c a n b e 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 f i l t e r s , by w o r k i n g in t h e R a d o n S p a c e . .The s c h e m e is i l l u s t r a t e d in fig. 5. T h i s t e c h n i q u e is p a r t i c u l a r l y u s e f u l f o r i m p 1 e m e n t ing 2- D c i r c u 1 a r 1 y s y m m e t r i c f 1 1 ters.
T h e t e c h n i q u e o u t l i n e d a b o v e s u g g e s t s a p p l i c a t i o n t o multi-directional f i l t e r i n g , a t o p i c t h a t h a s n o t r e c e i v e d m u c h a t t e n t i o n . pernark: T h e p r o p o s e d 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 c o m p u t a - t i o n o f t h e D R T h a s been presented. I t s ap- p l i c a t i o n 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 l i n e s e g m e n t s a n d e x t e n s i o n f o r c u r v e d o b j e c t r e c o g n i t i o n h a v e been discussed. The 4190- r i t h m i s a m e n a b l e f o r p a r a l l e l i m p l e m e n t a - tion, a c u r r e n t t r e n d i n s i g n a l a n d i m a g e processing. A m o d i f i c a t i o n o f t h e a l g o r i t h m f o r f i l t e r i n g i n t h e R a d o n S p a c e h a s been proposed.
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