BINARY IMAGE COMPRESSION USING THE RADON TRANSFORM
G.R. Ramesh and K. Rajgopal,
Department of Electrical Engineering,
Indian Institute of Science,
Bangalore 560 012
RT. S i m u l a t i o n r e s u l t s are p r e s e n t e d i n S c c t i o n 4 . S e c t i o n 5 c o n c l u d e s t h e paper.
ABSTRACT: A new technique for compression of
binary images is proposed. The Radon transform is used to oonvert a binary image to a
set
of
1-dimensional
(1-D)
non-binary
sequences, which are coded using 1-D teohniques. A binary image can be reconstructed
from a very small number of projections and
this leads to significant compression. The
compression ratio for a binary i n w e of size
N x N is inversely proportional to N.
2. Beconstruction of
The Radon t r a n s f o r m of an image f ( x , y )
is t h e i n t e g r a l of f ( x , y ) a l o n g a l l p o s s i b l e
l i n e s i n t h e p l a n e c o n t a i n i n g t h e image:
m m
p(e,t)=llf(x.y)*(t-x.cos(e)-y.sin(e))dxdy
. . . . . . . .(1)
-m
1.
-m
where
0 and t are t h e parameters of t h e
s t r a i g h t l i n e i n t h e n o r m a l form:
Introduction
The need f o r e f f i c i e n t s t o r a g e
and
transmission
of
graphics
and
two-tone
( b i n a r y ) images h a s i n c r e a s e d e n o r m o u s l y . As
a
result,
s e v e r a l t e c h n i q u e s have been
d e v e l o p e d f o r b i n a r y image c o m p r e s s i o n . Imp o r t a n t methods i n c l u d e r u n - l e n g t h c o d i n g ,
white block skipping, p r e d i c t i o n d i f f e r e n t i a l
q u a n t i z a t i o n (PDQ), r e l a t i v e a d d r e s s c o d i n g
and p r e d i c t i v e c o d i n g [ 1,2]. I n g e n e r a l , a l l
t h e s e t e c h n i q u e s . ( e x c e p t PDQ) d o n o t e x p l o i t
e f f i c i e n t l y , the inter-dimensional correlat i o n i n t h e 2 - d i m e n s i o n a l (2-D) d a t a . While
run l e n g h t c o d i n g and i t s e x t e n s i o n s are
s i m p l e and e f f i c i e n t , t h e y a r e v e r y s e n s i t i v e
t o channel e r r o r s .
I n 2-D l i n e a r p r e d i c t i v e
c o d i n g , t h e c h o i c e of t h e p r e d i o t i o n mask and
t h e c o m p l e x i t y of t h e p r e d i c t o r are major
i s s u e s [2,3!.
Further,
t h e e x i s t i n g techn i q u e s f o r b i n a r y image o o m p r e s s i o n d i f f e r
s i g n i f i c a n t l y from t h o s e u s e d f o r compressSon
of g r a y l e v e l images and h e n c e , a separate
c o d i n g s y s t e m is r e q u i r e d f o r b i n a r y images.
t = x.cos(8) + y.sin(8)
(2)
The term ' p r o j e c t i o n '
refers t o the
Radon t r a n s f orn i n t e g r a l ( 1) e v a l u a t e d a t
a n g l e 8,
d e n o t e d as p e ( t ) t o e m p h a s i z e t h e
f a c t t h a t i t is a 1-D f u n c t i o n . Thus, t h e
Radon t r a n s f o r m is u s e d t o p r o d u c e a s e t of
1-D s i g n a l s ( p r o j e c t i o n s ) from a 2-D s i g n a l
f(x,y).
T h i s d i m e n s i o n r e d u c i n g p r o p e r t y of
t h e Radon t r a n s f o r m a l l o w s t h e c o n v e r s i o n of
a 2-D p r o c e s s i n g t a s k to a s e t of s i m p l e r 1-D
p r o c e s s i n g t a s k s i n d e p e n d e n t l y on e a c h of t h e
projections.
The i n v e r s e Radon t r a n s f o r m mathematic a l l y corresponds t o f i n d i n g t h e function
There
are
f ( x , y ) from i t s p r o j e c t i o n s .
s e v e r a l algorithms t h a t approximate t h e inv e r s e Radon t r a n s f o r m ,
t o r e c o n s t r u c t an
image from a s e t of its p r o j e c t i o n s , s u c h as
t h e c o n v o l u t i o n b a c k p r o j e c t i c n (CBP) o r t h e
F o u r i e r r e c o n s t r u c t i o n t e c h n i q u e . However,
b i n a r y image r e c o n s t r u c t i o n a l g o r i t h m s are
b a s i c a l l y i t e r a t i v e i n n a t u r e and r e s e m b l e
the algebraic techniques f o r reconstructing
g r a y l e v e l images. A s t r i k i n g f e a t u r e of s u o h
a l g o r i t h m s is t h a t t h e y r e q u i r e v e r y f e u
projections f o r a satisfactory reconstruction
I n t h i s paper, w e p r e s e n t a new scheme
f o r c o m p r e s s i o n of b i n a r y images u s i n g t h e
The RT is u s e d as a
Radon t r a n s f o r m ( R T ) .
t o o l t o r e d u c e t h e p r o b l e m of c o d i n g 2-D b i n a r y s e q u e n c e t o t h a t of c o d i n g a s e t of 1-D,
n o n - b i n a r y s e q u e n c e s i . e . , p r o j e c t i o n s of t h e
image. The p r o c e d u r e f o r r e c o n s t r u c t i o n consists of d e c o d i n g t h e p r o j e c t i o n s , f o l l o w e d
by r e c o n s t r u c t i o n of t h e b i n a r y
image from
t h e s e p r o j e c t i o n s . A m a j o r a d v a n t a g e of t h i s
a p p r o a c h arises from t h e f a c t t h a t a b i n a r y
image c a n b e r e c o n s t r u c t e d from v e r y f e u
p r o j e c t i o n s [ 5 , 6 ] , which r e s u l t s i n s i g n i f i c a n t compression. I t is p o s s i b l e t o apply
b o t h p r e d i c t i v e and t r a n s f o r m t e c h n i q u e s i n
t h e p r o p o s e d scheme. I n t h i s paper, w e cons i d e r t h e a p p l i c a t i o n of linear p r e d i c t i o n
and t h e d i s c r e t e c o s i n e t r a n s f o r m (DCT).
[5,81-
I n t h i s paper, w e u s e t h e a l g o r i t h m
p r o p o s e d i n [SI, t o r e c o n s t r u c t a d i g i t a l b i n a r y image from its ( c o d e d ) p r o j e c t i o n s ,
at
8 = 0 , r/4, r/2 and 3r/4. The p r o j e c t i o n s of
an image of s i z e NxN, where N=2H+1, are comp u t e d as f o l l o w s :
p o ( t ) = 5: f ( t . m )
The o r g a n i z a t i o n of t h e paper is as f o l lows. S e c t i o n 2 b r i e f l y o u t l i n e s r e c o n s t r u c t i o n of b i n a r y images from t h e i r p r o j e c t i o n s .
S e c t i o n 3 d e s c r i b e s p r e d i c t i v e and t r a n s f o r m
t e c h n i q u e s f o r c o d i n g b i n a r y images u s i n g t h e
p2(t)
= a E f(m,t-m),
p,.Jt)
z
= E f(n,t),
( t ) =42 E f ( m , t + m ) ,
p3E
-H
5
-2n
t < H
s t <
-n s
-2n
( 3 a)
t <
( 3 b)
n
(3
0 )
s t < 2 1 (3 d )
The summation is p e r f o r m e d o v e r t h e NxN image
178
t h e p r e d i c t i o n mask (memory s e t ) , c o m p l e x i t y
of t h e p r e d i c t o r , and s t a b i l i t y , r e m a i n as
m a j o r i s s u e s [ 2 , 3 . 4 ] . We propose an a l g o r i t h m
which u s e s simpler 1-D LP t e c h n i q u e s on t h e
s e t of p r o j e c t i o n s ( i e . , t h e Radon t r a n s f o r m )
o f t h e b i n a r y image. S t a b i l i t y is g u a r a n t e e d
b y u s i n g t h e so c a l l e d a u t o c o r r e l a t i o n method
or e s t i m a t i n g t h e 1-D LP parameters.
s u p p o r t . Note t h a t , w h i l e t h e p r o j e c t i o n s a t
= 0 and 4 2 are .paced s t u n i t d i m t a n c e s ,
t h o s e a t 0 = r/4 and 3r/4 are s p a c e d a t 1/42
units.
8
The p r o b l e m o f r e c o n s t r u c t i n g f ( m , n )
from t h e s e t o f 4 p r o j e c t i o n s ,
is f o r m u l a t e d
as t h a t o f s o l v i n g t h e s e t of l i n e a r equa( 3 ) . The a l g o r i t h m is a n errortions,
r e d u c t i o n scheme. and b e g i n s by a s s u m i n g an
i n i t i a l estimate f o r f ( m , n ) .
I n each iterat i o n , f ( m , n ) is f i r s t e s t i m a t e d b a s e d on
p r o j e c t i o n s a t one o f t h e f o u r a n g l e s , 8 = 8 % ,
which f o r m s t h e i n i t i a l estimate f o r estimat i o n b a s e d on t h e p r o j e c t i o n a t a d i f f e r e n t
a n g l e , 8 i + 1 , as f o l l o w s :
Each o f t h e d a t a s e q u e n c e s d e ( t ) , is i n dependently
modeled as a n a u t o r e g r e s s i v e
( A R ) s e q u e n c e o f o r d e r me:
d e ( t ) = & ( t >+ e e ( t )
*
where, d e ( t )
(7)
= - E ae(k)de(t-k)
(8)
at
and $ e i t f t ) is t h e p r o j e c t i o n o f f e i ( m , n )
0=8i+i.
L e i b r ( t ) is t h e number o f p i x e l s
a l o n g t h e l i n e ( 2 ) at i n d e x p e r p e n d i c u l a r
d i s t a n c e t and 8 = 0 i + 1 . € e i + , ( t ) is t h e est i m a t i o n e r r o r i n terms of
projections.
t(m,n) denotes t h e index t corresponding t o
t h e l i n e ( Z ) , passing through t h e point
a t 0 = 8 i + i . The a b o v e s t e p is r e p e a t e d
(m,n),
to u t i l i z e the projections at a l l the angles.
is a n estimate o f d e ( t ) ;
ae(k), k=1.2,. . m e ,
are t h e AR model (LP) parameters and e e ( t ) ,
t h e e r r o r s i g n a l . The AR parameters { a e ( k ) )
are computed from t h e d a t a d e ( t ) ,
using t h e
constrained minimization procedure due t o
Burg. T h i s method g u a r a n t e e s t h e s t a b i l i t y o f
t h e i n v e r s e p r e d i c t i o n - e r r o r f i l t e r . These
parameters are u s e d t o compute % ( t ) and
h e n c e , t h e e r r o r s i g n a l e e ( t ) . The e r r o r s i g n a l s e q u e n c e s e e ( t ) are q u a n t i z e d and s t o r e d
o r coded f o r t r a n s m i s s i o n . a l o n g w i t h { a e ( k ) }
and pe. Due t o its i n h e r e n t low v a r i a n c e ,
e e ( t ) r e q u i r e s few b i t s t o r e p r e s e n t and t h i s
is t h e e s s e n c e of l i n e a r p r e d i c t i v e d a t a conpression.
The f i n a l s t e p i n e a c h i t e r a t i o n consists o f a p p l y i n g t h e c o n s t r a i n t t h a t , 0 S f
5 1. The c o n s t r a i n e d v a l u e s o f 'f(m,n)
form
t h e i n i t i a l estimate f o r t h e n e x t i t e r a t i o n .
The l a s t s t e p o f t h e r e c o n s t r u c t i o n a l g o r i t h m
involves thresholding ie., s e t t i n g t h e values
of f t o 1 i f f20.5, and zero o t h e r w i s e .
A t t h e r e c e i v e r , t h e data s e q u e n c e s
d e ( t ) are r e t r i e v e d by d r i v i n g t h e i n v e r s e
p e d i c t i o n - e r r o r f i l t e r by e e ( t ) ( e q u a t i o n 7 ) .
The p r o j e c t i o n s p e ( t ) are o b t a i n e d by a d d i n g
pe t o d e ( t ) ,
from which, t h e b i n a r y iaage is
reconstructed using t h e algorithm described
i n t h e previous section.
3. B i n a r v I m a a e i
The
follows :
(4)
where,
(5)
h
The b a s i c i d e a h e r e is t o u s e t h e Radon
t r a n s f o r m t o r e d u c e t h e b i n a r y image t o a s e t
of 1-D s e q u e n c e s i e . , p r o j e c t i o n s a t v a r i o u s
a n g l e s . The p r o j e c t i o n s o f a b i n a r y image are
1 - D n o n - b i n a r y s e q u e n c e s , which are coded by
u s i n g w e l l known 1-D t e c h n i q u e s .
pe
summarized
as
3. R e c o n s t r u c t i o n : R e t r i e v e p e ( t ) by d r i v i n g
t h e i n v e r s e p r e d i c t i o n - e r r o r f i l t e r by t h e
( q u a n t i z e d ) e r r o r s i g n a l e e ( t ) . The r e t r i e v e d
P r o j e c t i o n s are t h e n u s e d t o r e c o n s t r u c t t h e
image f ( m , n ) u s i n g a b i n a r y image r e c o n s t r u c tion algorithm.
The p r o c e d u r e o u t l i n e d above s u g g e s t s
t h e implementation of a n i n t r a - f r a m e a d a p t i v e
p r e d i c t i v e image c o d i n g scheme ( a d a p t i v e d i f f e r e n t i a l p u l s e c o d e m o d u l a t i o n o r ADPCH).
The scheme is i l l u s t r a t e d i n f i g . 1 . The b l o c k
'RBC'
i n d i c a t e s image r e c o n s t r u c t i o n f r o m
p r o j e c t i o n s and ' V A R ' c o r r e s p o n d s t o computat i o n of
variance.
C a l c u l a t i o n of
the
v a r i a n c e and t r a n s m i s s i o n of t h e q u a n t i z e r
s t e p - s i z e i n f o r m a t i o n c a n b e a v o i d e d by u s i n g
J a y a n t ' s 1-word memory a d a p t i v e q u a n t i z e r
171. A major a d v a n t a g e of t h i s q u a n t i z e r consists i n i n c r e a s i n g t h e dynamic r a n g e of t h e
quantizer.
(6)
I n t h i s paper, w e c o n s i d e r t h e a p p l i c a t i o n o f LPC and t h e DCT f o r c o d i n g t h e d a t a
s e q u e n c e s , d e ( t ) . The a l g o r i t h m is d e s c r i b e d
i n t h e following section.
3.1
be
2 . Code t h e p r o j e c t i o n s p e ( t ) u s i n g 1-D LPC.
Me.
-
may
1. Compute t h e p r o j e c t i o n s p e ( t ) of t h e d i g i t a l b i n a r y image f ( m , n ) .
The f i r s t s t e p is t h e r e f o r e , t o compute
t h e Radon t r a n s f o r m ( t h e s e t o f p r o j e c t i o n s )
using
o f t h e d i g i t a l b i n a r y image f ( m , n ) ,
( 3 ) . Each o f t h e p r o j e c t i o n s p e ( t ) ,
is made
z e r o mean by s u b t r a c t i n g from i t , its mean
de(t) = pe(t)
procedure
. .
L i n e a c v -e c
The p r i n c i p l e s o f 2-D l i n e a r p r e d i c t i v e
c o d i n g have been a p p l i e d t o b i n a r y images
[ l - 4 3 . The d i f f e r e n c e is t h a t , i n t h e case of
b i n a r y images, t h e error s i g n a l is a l s o a b i n a r y s e q u e n c e and t h e r e f o r e , n o q u a n t i z e r is
r e q u i r e d . Redundancy i n t h e b i n a r y image w i l l
be r e f l e c t e d i n l o n g r u n s o f z e r o s i n t h e err o r s i g n a l , which may b e t h e n coded u s i n g
r u n - l e n g t h t e c h n i q u e s . However, d i r e c t 2-D LP
t e c h n i q u e s c o n s i d e r e d so f a r , d o n o t e x p l o i t
efficiently,
t h e inter-dimensional
correlat i o n i n t h e 2-D d a t a . F u r t h e r , t h e c h o i c e of
3.1.1
L e t Np b e t h e number o f p r o j e c t i o n s
required t o s a t i s f a c t o r i l y reconstruct t h e
b i n a r y image, me t h e o r d e r of t h e p r e d i c t o r
and B. t h e number of b i t s r e q u i r e d t o quant i z e t h e e r r o r s i g n a l e e ( t ) . Then, i n g e n e r a l
179
Parmeters
Fig. Ua) ADKX Coder
Fig. l(b) Decoder
(assuming f o r s i m p l i c i t y , t h a t t h e same o r d e r
is u s e d f o r a l l t h e p r o j e c t i o n s ) t h e compreas i o n r a t i o , d e f i n e d as t h e r a t i o of t h e numb e r of b i t s r e p r e s e n t i n g t h e coded image t o
t h a t of t h e o r i g i n a l image, is g i v e n by:
N p ( mex8
CR =
+
a c h i e v e d i n t h e case of l a r g e images, by an
a p p r o p r i a t e c h o i c e of t h e parameters. For example, i f N=1024, Ne=16, t h e n , f o r LPC w i t h
B = 3 and 2 , CRO is a p p r o x i m a t e l y 1/14 and 1/22
respectively.
N42.B )
3.2
(9)
NxN
Transform t e c h n i q u e s have n o t been app l i e d f o r b i n a r y image c o m p r e s s i o n , s i n c e
t h e y d o n o t o f f e r s i g n i f i c a n t Compression.
However,
it i s p o s s i b l e t o u s e e f f i c i e n t
t r a n s f o r m s s u c h as t h e DCT t o compress t h e
(non-binary)
p r o j e c t i o n s of t h e image. Comp r e s s i o n is a c h i e v e d by a p p l y i n g t h e DCT to
t h e s i g n a l ( p r o j e c t i o n , i n t h e p r e s e n t case)
and r e t a i n i n g and e n c o d i n g o n l y a f r a c t i o n of
t h e DCT c o e f f i c i e n t s .
The d i s c a i d i n g of t h e
rest of t h e c o e f f i c i e n t s d o e s n o t s e r i o u s l y
d e g r a d e t h e s i g n a l . T h i s is due t o t h e excell e n t e n e r g y compaction p r o p e r t y of t h e DCT.
Advantages of t r a n s f o r m t e c h n i q u e s f o r conpression include
higher
efficiency
and
r o b u s t n e s s t o t r a n s m i s s i o n d e g r a d a t i o n s . The
a l g o r i t h m is summarized below:
8 b i t s have been assumed f o r
the prediction
c o e f f i c i e n t s . The c o n t r i b u t i o n of t h e quant i z e r s t e p - s i z e i n f o r m a t i o n and t h e mean are
n e g l e c t e d . N42 c o r r e s p o n d s t o t h e l e n g t h ( i n
number of p o i n t s ) of t h e p r o j e c t i o n d a t a and
hence t h a t of t h e e r r o r s i g n a l . I f Ne < < N .
which is normal€y t h e case, t h e n t h e compress i o n r a t i o reduces t o :
CR rs
42 N p B /
N
(10)
Thus, t h e compression r a t i o depends upon
t h e number of p r o j e o t i o n s r e q u i r e d f o r satisf a c t o r y r e c o n s t r u c t i o n of t h e image, and B is
t h e number of b i t s r e q u i r e d t o q u a n t i z e t h e
error signal.
The f a c t o r Np i n g e n e r a l
depends upon t h e o o m p l e x i t y of t h e image und e r c o n s i d e r a t i o n . However. as mentioned earl i e r , a b i n a r y image c a n b e r e c o n s t r u c t e d
from v e r y few p r o j e c t i o n s and t h i s l e a d s t o a
s i g n i f i c a n t compression. I n t h e p a r t i c u l a r
c a s e c o n s i d e r e d i n t h i s paper, w i t h 4 p r o j e c t i o n s ( N p o i n t s f o r h o r i z o n t a l and v e r t i c a l
p r o j e c t i o n s , and 2N-1 p o i n t s f o r t h o s e a l o n g
t h e two d i a g o n a l s ), t h e CR is g i v e n by:
CR=6
B / N
( i ) Compute t h e p r o j e c t i o n s p e ( t ) of t h e
image f ( m , n ) .
( i i ) Code t h e 1-D.
non-binary sequences
pe(t),
u s i n g DCT. Note t h a t t h e DCT can b e
computed v i a an FFT a l g o r i t h m .
(iii) A t t h e r e c e i v e r ,
the projections pe(t>
are r e t r i e v e d by IDCT ( v i a IFFT) and u s e d t o
r e c o n s t r u c t t h e image f ( m , n ) .
The scheme is i l l u s t r a t e d i n
(11)
3.2.1
Thus, t h e compression r a t i o is seen t o
be i n v e r s e l y p r o p o r t i o n a l t o N . T h a t is, au
t h e s i z e of t h e image i n c r e a s e s ,
t h e achieva b l e compression a l s o i n c r e a s e s l i n e a r l y , and
t h i s looks very impressive.
I n practice
however, when N is v e r y l a r g e a n d / o r t h e
image h a s f i n e r d e t a i l s , t h e image w i l l have
t o b e d i v i d e d i n t o smaller b l o c k s ( s u b i m a g e s )
and e a c h of them coded s e a p a r a t e l y . T h i s a l s o
s e r v e s t o make t h e c o d e r a d a p t t o t h e l o c a l
image c h a r a c t e r i s t i c s .
I f an NxN image is
divided uniformly i n t o NB blocks, t h e o v e r a l l
compression r a t i o is:
CRo = CRJNB
Fig. 2.
L e t C b e t h e f r a c t i o n of t h e DCT c o e f f i cients
(corresponding
to
each
of t h e
projections) retained.
Assuming
that
4
p r o j e c t i o n s r e p r e s e n t t h e image (as i n t h e
c a s e of LPC), t h e compression r a t i o is:
CRDCT = 6 BDCTC / N
(13)
BDCT is t h e number of b i t s r e q u i r e d t o r e p r e s e n t e a c h of t h e DCT c o e f f i c i e n t s (assuming a
u n i f o r m b i t a l l o c a t i o n ) . B e t t e r Compression
c a n b e a c h i e v e d by e f f i c i e n t b i t a l l o c a t i o n
of t h e DCT c o e f f i e c e n t s . A s d i s c u s s e d i n t h e
case of LPC, large images have t o b e d i v i d e d
i n t o smaller b l o c k s and e a c h of them coded
s e p a r a t e l y . A p a r t from making t h e c o d e r a d a p t
t o l o c a l image c h a r a c t e r i s t i c s ,
t h i s also
s e r v e s t o r e d u c e s t o r a g e and c o m p u t a t i o n a l
(12)
where, CR is t h e compression r a t i o w i t h t h e
NxN image c o n s i d e r e d as a whole. Thus, comp r e s s i o n is s c a l e d down by t h e f a c t o r TNB.
However, s i g n i f i c a n t compression can s t i l l b e
180
Pip. 2: Transform comprsasion scheme
r e q u i r e m e n t . I f t h e image is d i v i d e d u n i f o r t h e o v e r a l l compression
r a t i o CRo w i l l b e CRfNe, where CR is t h e comp r e s s i o n r a t i o w i t h t h e NxN image c o n s i d e r e d
as a w h o l e . With N=1024 and Na=16, t h e v a l u e s
o f CRo w i t h B D C T = ~ and C=O.4 and 0.3, are
1/18 and 1/24 r e s p e c t i v e l y .
A s nentioned earlier, a major d i s a d v a n t age of t h e p r o p o s e d t e c h n i q u e is t h e error
resulting
from
the
process
of
image
r e c o n s t r u c t i o n from p r o j e c t i o n s ; d e g r a d a t i o n
c a n b e r e l a t i v e l y s e v e r e on documents w i t h
finer details.
mly i n t o N B b l o c k s ,
5 . Canclusions
3 . 3 Bemarks:
A new t e c h n i q u e f o r b i n a r y image
comp r e s s i o n h a s been p r e s e n t e d , where t h e Radon
t r a n s f o r m is made u s e o f as a t o o l t o r e d u c e
t h e 2-D b i n a r y image c o d i n g problem t o a s e t
of 1-D
problems
of
coding
non-binary
sequences.
B o t h p r e d i c t i v e and t r a n s f o r m
t e c h n i q u e s have been d i s c u s s e d . The compress i o n r a t i o f o r a n NxN image h a s been shown t o
be i n v e r s e l y p r o p o r t i o n a l t o N .
Simulation
r e s u l t s showing t h e p e r f o r m a n c e of t h e algor i t h m have been p r e s e n t e d . V a r i o u s o t h e r feat u r e s o f t h e t e c h n i q u e have been h i g h l i g h t e d .
A p p l i c a t i o n of d i f f e r e n t r e c o n s t r u c t i o n a l g o r i t h m s are b e i n g i n v e s t i g a t e d . The e f f e c t s
d u e t o q u a n t i z a t i o n and c h a n n e l e r r o r s on t h e
q u a l i t y o f t h e r e c o n s t r u c t e d image are b e i n g
studied.
E r r o r s d u e t o q u a n t i z a t i o n and c h a n n e l
degradations tend t o get d i s t r i b u t e d over t h e
e n t i r e image d u r i n g t h e p r o c e s s o f r e c o n s t r u c t i o n from p r o j e c t i o n s .
2) A s t h e RT r e d u c e s a b i n a r y image t o a s e t
of non-binary
s e q u e n c e s , b i n a r y as well as
g r a y l e v e l images c a n b e coded u s i n g t h e
p r o p o s e d scheme.
3 ) Coding o f t h e p r o j e c t i o n s can b e p e r f o r m e d
in parallel.
4 ) The p r o j e c t i o n p r o c e s s t e d d s t o h a v e a n
a v e r a g i n g e f f e c t on background n o i s e , r e d u c ing its e f f e c t .
5 ) Major d i s a d v a n t a g e s o f t h e p r o p o s e d s c h e n e
I)
are :
( i ) e r r o r s d u e t o r e c o n s t r u c t i o n from p r o j e c -
tions.
( i i ) u n s u i t a b i l i t y for continuous on-line
image t r a n s m i s s i o n , s i n c e t h e a l g o r i t h m s f o r
t h e r e c o n s t r u c t i o n o f a b i n a r y image f r o m its
p r o j e c t i o n s are i t e r a t i v e and hence time consuming.
4.
Acknowledgment
The a u t h o r s w i s h t o t h a n k Ganesh U u r t h y
f o r u s e f u l s u g g e s t i o n s , encouragement
and s u p p o r t .
C.N.S.
Sinulation
S i m u l a t i o n s t u d i e s were c a r r i e d o u t on
s e v e r a l computer g e n e r a t e d t e s t images. One
example is p r e s e n t e d i n F i g s . 3-5.
Fig. 3
shows t h e o r i g i n a l image o f s i z e 129x129. The
s i z e o f e a c h of t h e c h a r a c t e r is 23x15. T h i s
corresponds t o a printed c h a r a c t e r
of size
3nmX2mm sampled a t 200 p o i n t s p e r i n c h . Note
however, t h a t t h e e n t i r e image d o e s n o t c o r r e s p o n d t o a s t a n d a r d document. F o u r p r o j e c t i o n s of t h e image were coded b y LPC u s i n g a
s i m p l e u n i f o r m q u a n t i z e r w i t h B L P C = ~ and 2.
The c o n p r e s s i o n r a t i o s o b t a i n e d were 1 1 7 . 2
and 1 / 1 0 . 8 r e s p e c t i v e l y . The r e s u l t s a r e d i s p l a y e d i n F i g . 4 . The c o r r e s p o n d i n g r e s u l t s
w i t h DCT c o d i n g o f t h e p r o j e c t i o n s w i t h
B D C T = ~ , C=O.4 and c=O.3, are d i s p l a y e d i n
F i g . 5 . The c o m p r e s s i o n r a t i o s a r e 1 / 6 . 7 and
1/9 r e s p e c t i v e l y . I t is p o s s i b l e t o a c h i e v e
b e t t e r p e r f o r m a n c e by a p r o p e r b i t a l l o c a t i o n
o f t h e DCT c o e f f i c i e n t s .
RBFERBNCB8
[ l ] R . B . A r p s , " B i b l i o g r a p h y on B i n a r y Image
C o m p r e s s i o n " , P r o c . I E E E , Vo1.68, No.7, J u l y
1980, pp:922-924.
[2] A . K . J a i n , Fundamentals o f D i g i t a l Image
Processing, P r e n t i c e H a l l , 1989.
[3] H .
R o b a y a s h i and L . R .
B a h l . "Image Data
Compression by P r e d i c t i v e Coding I : P r e d i c t i o n A l g o r i t h m s " , IBU J1. of Rea. and Dev.,
March 1974, pp:184-171.
[4 J T .S. Huang,
"Coding o f Two-Tone Images",
IBEE T r a n s . Communications, COH-25, V o l . 1 1 ,
November 1977.
[ 5 ] G.T.
Herman,
" R e c o n s t r u c t i o n of B i n a r y
P a t t e r n s From a Few P t o j e c t i o n s " ,
i n A GunI n t e r n a t i o n a l Computing
t h e r e t al. ( e d s ) ,
Symposium 1973, North-Holland P u b l i s h i n g C o . ,
1974.
[SI H . Soumekh, " B i n a r y Image R e c o n s t r u o t i o n
from Four P r o j e c t i o n s " , P r o c .
ICASSP'68,
pp : 1280- 1283.
[7] N.S. J a y a n t , " A d a p t i v e Q u a n t i z a t i o n With
a One-Word Memory", The B e l l System t e c h n i c a l
J o u r n a l , Vo1.52, No.?, S e p t . 1973, p e : l l l 6 1144.
I n t h e p r e s e n t case, w e have c o n s i d e r e d
p r o j e c t i o n s t o ,feeresent
the binary
image.
However, i t is p o s s i b l e t o c o n s i d e r a
four
g r e a t e r number o f p r o j p t i o n s b y u s i n g a d i s c r e t i z a t i o n o f t h e Radon t r a n s f o r m , f o l l o w e d
by a s u i t a b l e r e c o n s t r u c t i o n a l g o r i t h m .
181
F i g . 3: T e s t image
(a)
(b)
Fig. 4: Rbcautructiai of the test 4 0 fraP its projectitma codal by Ipc:
(a) B=3 ((XsV7.2) a d (b) B=2 (cIbrV10.8)
FU. 5: Rsronstnacticn of the test image frm ita pmjsotions coded
DX, with bcr=8: (a) C 4 . 4 (cEbrV8.7) and (b) 0 . 3 (cEbrl/B)
182
wing the