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Physica D 60 (1992) 259-268North-Holland

N o n l i n e a r t o t a l v a r i a t i o n b a s e d n o i s e r e m o v a l a l g o r i t h m s *L e o n i d I . R u d i n 1, S t a n l e y O s h e r a n d E m a d F a t e m i 2Cognitech Inc., 2800, 28th Street, Suite 101, Santa Monica, CA 90405, USA

A constrained optimization type of num erical algorithm for rem oving noise from im age s is presented. The totalvariation o f the im age is minimize d subject to co nstraints involving the statistics of the no ise. The c onstraints are im posedusing Lagrange multipliers. The solution is obtained using the gradient-projection m ethod. This amounts to solving a timedependent partial differential equation on a manifold determined by the constraints. As t---~0o the solution conv erges to asteady state w hich is the denoised image. The numerical algorithm is simple and relatively fast. The results appe ar to b estate-of-the-art f or very n oisy images. The method is noninvasive, yielding sharp edges in the image. The technique couldbe interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature ofthe level set divided by the magnitude of the gradient of the image, and a second step which projects the im age back ontothe constraint set.

1 . I n t r o d u c t i o n

T h e p r e s e n c e o f n o i s e in i m a g e s is u n a v o i d -a b l e . I t m a y b e i n t r o d u c e d b y t h e i m a g e f o r m a -t i o n p r o c e s s , i m a g e r e c o r d i n g , i m a g e t r a n s m i s -s i o n , e tc . T h e s e r a n d o m d i s to r t io n s m a k e i t d if -f ic u lt t o p e r f o r m a n y r e q u i r e d p i c t u r e p r o c e ss i n g .F o r e x a m p l e , t h e f e a t u r e o r i e n t e d e n h a n c e m e n ti n t r o d u c e d i n r e f s . [ 6 ,7 ] is v e r y e f f e c t i v e i n r e -s t o r i n g b l u r r y i m a g e s , b u t i t c a n b e " f r o z e n " b ya n o s c i l l a t o r y n o i s e c o m p o n e n t . E v e n a s m a l la m o u n t o f n o is e i s h a r m f u l w h e n h i g h a c c u r a c y isr e q u i r e d , e . g . a s i n s u b c e l l ( s u b p i x e l ) i m a g ea n a l y s i s .I n p r a c t i c e , t o e s t i m a t e a t r u e s i g n a l i n n o i s e ,t h e m o s t f r e q u e n t l y u s e d m e t h o d s a r e b a s e d o nt h e l e a s t s q u a r e s c r i t e r i a . T h e r a t i o n a l e c o m e sf r o m t h e s t a t i s t i c a l a r g u m e n t t h a t t h e l e a s ts q u a r e s e s t i m a t i o n i s t h e b e s t o v e r a n e n t i r e

* Research supported by DAR PA SBIR Contract#DAAH01-89-C0768 and by AFOSR Contract #F49620-90-C-0011.1 E-mail: cogni!leonid@ aerospace.aero.org.2 Current address: Institute for M athematics and its Appli-

cations, University of Minnesota, Minneapolis, MN 55455,USA.

e n s e m b l e o f a ll p o s s i b le p i c tu r e s . T h i s p r o c e d u r eis L 2 n o r m d e p e n d e n t . H o w e v e r i t h a s b e e nc o n j e c t u r e d i n r e f . [ 6 ] t h a t t h e p r o p e r n o r m f o ri m a g e s is t h e t o t a l v a r i a t i o n ( T V ) n o r m a n d n o tt h e L 2 n o r m . T V n o r m s a r e e s s e n ti a ll y L 1 n o r m so f d e r i v a ti v e s , h e n c e L 1 e s t i m a t i o n p r o c e d u r e sa r e m o r e a p p r o p r i a t e f o r th e s u b j e c t o f i m a g ee s t i m a t i o n ( r e s t o r a ti o n ) . T h e s p a c e o f f u n c t io n so f b o u n d e d t o t a l v a r i a t i o n p la y s a n i m p o r t a n tr o l e w h e n a c c u r a t e e s t i m a t i o n o f d i s c on t in u i ti e si n s o l u t i o n s i s r e q u i r e d [ 6 ,7 ] .

H i s t o r i c a ll y , t h e L ~ e s t i m a t i o n m e t h o d s g ob a c k t o G a l i l e o ( 1 6 3 2 ) a n d L a p l a c e ( 1 7 9 3 ) . I nc o m p a r i s o n t o t h e l e a s t s q u a r e m e t h o d s w h e r ec l o s e d f o r m l i n e a r s o l u t i o n s a r e w e l l u n d e r s t o o da n d e a s i ly c o m p u t e d , t h e L 1 e s t i m a t i o n is n o n -l i n e a r a n d c o m p u t a t i o n a l l y c o m p l e x . R e c e n t l yt h e s u b j e c t o f L 1 e s t i m a t i o n o f s t a ti s ti c a l d a t a h a sr e c e i v e d r e n e w e d a t t e n t i o n b y th e s t a ti st ic a lc o m m u n i t y , s e e e . g . r e f . [ 1 3 ] .

D r a w i n g o n o u r p r e v i o u s e x p e r i e n c e w i t hs h o c k r e l a t e d i m a g e e n h a n c e m e n t [ 6 , 7 ] , w e p r o -p o s e t o d e n o i s e i m a g e s b y m i n i m i z i n g t h e to t a lv a r i a t i o n n o r m o f t h e e s t i m a t e d s o l u ti o n . W ed e r i v e a c o n s t r a i n e d m i n i m i z a t i o n a l g o r i t h m a s a

0167-2789/92/$05.00 1992 - Elsevier Scien ce Publishers B.V. All rights reserved


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2 6 0 L . I . R u d i n e t a l. / N o i s e r e m o v a l a l g o r i th m s

t i m e d e p e n d e n t n o n l i n e a r P D E , w h e r e t h e c o n -s t r a i n t s a r e d e t e r m i n e d b y t h e n o i s e s t a t i s t i c s .

T r a d i t i o n a l m e t h o d s a t t e m p t t o r e d u c e /r e m o v e t h e n o i s e c o m p o n e n t p r i o r t o f u r t h e ri m a g e p r o c e s s i n g o p e r a t i o n s . T h i s is t h e a p -p r o a c h t a k e n i n t h i s p a p e r . H o w e v e r , t h e s a m eT V / L 1 p h i l o s o p h y c a n b e u s e d t o d e s i g n h y b r i da l g o r i t h m s c o m b i n i n g d e n o i s i n g w i t h o t h e r n o i s es e n s i t i v e i m a g e p r o c e s s i n g t a s k s .

2. N onlin ear partial differential equations baseddeno i s ing a lgor i thms.

L e t t h e o b s e r v e d i n t e n s i t y f u n c t i o n u 0 ( x , y )d e n o t e t h e p i x e l v a l u e s o f a n o i s y im a g e f o r x ,y ~ O . L e t u ( x , y ) d e n o t e t h e d e s i r e d c l e a ni m a g e , s oU o ( X , y ) = u ( x , y ) + n ( x , y ) , ( 2 . 1 )w h e n n i s t h e a d d i t i v e n o i s e .

W e , o f c o u r s e , w i s h t o r e c o n s t r u c t u f r o m u 0 .M o s t c o n v e n t i o n a l v a r i a ti o n a l m e t h o d s i n v o lv e al e a s t s q u a r e s L 2 f i t b e c a u s e t h i s l e a d s t o l i n e a re q u a t i o n s . T h e f ir s t a t t e m p t a l o n g t h e s e l i n es w a sm a d e b y P h i ll ip s [ 1] a n d l a t e r r ef i n e d b y T w o m e y[ 2 , 3 ] i n t h e o n e - d i m e n s i o n a l c a s e . I n o u r t w o -d i m e n s i o n a l c o n t i n u o u s f r a m e w o r k t h e i r c o n -s t r a in e d m i n i m i z a t io n p r o b l e m ism i n i m i z e f ( U x x + U y y ) 2 (2 .2a )s u b j e c t t o c o n s t r a i n t s i n v o l v i n g t h e m e a nf u = f U o ( 2 . 2 b )a n d s t a n d a r d d e v i a t i o nf ( u - u0) 2 = t r z . (2 .2c)T h e r e s u l t i n g l i n e a r s y s t e m i s n o w e a s y t o s o l v eu s i n g m o d e r n n u m e r i c a l l i n e a r a l g e b r a . H o w -e v e r , t h e r e s u l t s a r e a g a i n d i s a p p o i n t i n g ( b u t

b e t t e r t h a n t h e M E M ) w i th t h e s am ec o n s t r a i n t s ) - s e e e .g . r e f . [ 5].

T h e L 1 n o r m i s u s u a l ly a v o i d e d s in c e t h ev a r i a t i o n o f e x p r e s s i o n s l i k e Salu[ d x p r o d u c e ss i n g u l a r d i s t r i b u t i o n s a s c o e f f i c i e n t s ( e . g . 6 f u n c -t i o n s ) w h i c h c a n n o t b e h a n d l e d i n a p u r e l y a l g e -b r a i c f r a m e w o r k . H o w e v e r , i f L 2 a n d L 1 a p p r o x i -m a t i o n s a r e p u t s i d e b y s i d e o n a c o m p u t e rs c r e e n , i t is c l e a r t h a t t h e L 1 a p p r o x i m a t i o n l o o k sb e t t e r t h a n t h e " s a m e " L 2 a p p r o x i m a t io n . T h e" s a m e " m e a n s s u b j e c t t o t h e s a m e c o n s t r a i n t s .T h i s m a y b e a t l e a s t p a r t l y p s y c h o l o g i c a l ; h o w -e v e r , i t i s w e l l k n o w n i n s h o c k c a l c u l a t i o n s t h a tt h e L 1 norm o f t h e g r a d i e n t i s t h e a p p r o p r i a t es p a c e . T h i s i s b a s i c a l l y t h e s p a c e o f f u n c t i o n s o fb o u n d e d t o t a l v a r i a ti o n : B V . F o r f r e e , w e g e t t h er e m o v a l o f s p u r i o u s o s c i l l a t i o n s , w h i l e s h a r p s i g -n a l s a r e p r e s e r v e d i n t h i s s p a c e .

I n r e f . [ 6 ] t h e f i r s t a u t h o r h a s i n t r o d u c e d an o v e l i m a g e e n h a n c e m e n t t e c h n i q u e , c a l l e dS h o c k F i l t er . I t h a d a n a l o g y w i t h s h o c k w a v ec a l c u l a t i o n s i n c o m p u t a t i o n a l f l u i d m e c h a n i c s .T h e f o r m a t i o n o f d i s c o n t i n u i t i e s w i t h o u t o s c i l l a -t io n s a n d r e l e v a n c e o f t h e T V n o r m w a s ex -p l o r e d h e r e .

I n a p a p e r w r i t t e n b y t h e f ir s t t w o a u t h o r s [ 7] ,t h e c o n c e p t o f t o t a l v a r i a t i o n p r e s e r v i n g e n -h a n c e m e n t w a s f u r t h e r d e v e l o p e d . F i n i t e d i f f e r -e n c e s c h e m e s w e r e d e v e l o p e d t h e r e w h i c h w e r eu s e d t o e n h a n c e m i l d l y b l u r r e d i m a g e s s ig n if i-c a n t l y w h i l e p r e s e r v i n g t h e v a r i a t i o n o f t h e o r ig i -n a l i m a g e .

A d d i t i o n a l l y , i n [ 8 ] , A l v a r e z , L i o n s a n d M o r e ld e v i s e d a n i n t e r e s t i n g s t a b l e i m a g e r e s t o r a t i o na l g o r i t h m b a s e d o n m e a n c u r v a t u r e m o t i o n , s e ea l s o r e f . [ 9 ] . T h e m e a n c u r v a t u r e i s j u s t t h eE u l e r - L a g r a n g e d e r i v a t i v e o f t h e v a r i a t i o n .

W e t h e r e f o r e s t a t e t h a t t h e s p a c e o f B V f u n c -t i o n s i s t h e p r o p e r c l a s s f o r m a n y b a s i c i m a g ep r o c e s s i n g t a s k s .

T h u s , o u r c o n s t r a i n e d m i n i m i z a t i o n p r o b l e mis :

d x d y ( 2 .3 a )i n i m i z e ~ x 2 + Uy2as u b j e c t t o c o n s t r a i n t s i n v o l v i n g u 0 .


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L.I . Ru din et al. / Noise removal algorithms 261I n o u r w o r k s o f a r w e h a v e t a k e n t h e s a m e t w o

c o n s t r a i n t s a s a b o v e :f u d x d y = f u o d x d y . ( 2 . 3 b )n 12

T h i s c o n s t r a i n t s i g n i f i e s t h e f a c t t h a t t h e w h i t en o i s e n ( x , y ) i n ( 2 .1 ) is o f z e r o m e a n a n d

f ( u _ u 0) 2 d x d y0 = Or2 w he re o" > 0 i s giv en(2 .3c )

A s t i n c r e a s e s , w e a p p r o a c h a d e n o i s e d v e r s io no f o u r i m a g e .

W e m u s t c o m p u t e A ( t ). W e m e r e l y m u l ti p ly( 2 . 5 a ) b y ( u - u 0 ) a n d i n t e g r a t e b y p a r t s o v e r 12.I f s t e a d y s t a t e h a s b e e n r e a c h e d , t h e l e f t s i d e o f( 2 . 5 a ) v a n i s h e s . W e t h e n h a v e

A - 2o.2 + Uy( ( u ) x u x (u ) ruY ~ ] d x d y . ( 2 . 6 )2 2 q ' - 2 2 , /3

T h e s e c o n d c o n s t r a i n t u s e s a p r i o r i i n f o r m a t i o nt h a t t h e s t a n d a r d d e v i a t i o n o f th e n o i s e n ( x , y ) iso r.

T h u s w e h a v e o n e l i n e a r a n d o n e n o n l i n e a rc o n s t r a i n t . T h e m e t h o d i s t o t a l l y g e n e r a l a s r e -g a r d s n u m b e r a n d s h a p e o f c o n st ra i n ts .

W e a r r i v e a t t h e E u l e r - L a g r a n g e e q u a t i o n s

- A1 - A2(u - Uo) in 12, w i th (2.4a )OuO n 0 o n t h e b o u n d a r y o f 12 = 0 1 2 . ( 2 .4 b )

T h e s o l u t i o n p r o c e d u r e u s e s a p a r a b o l i c e q u a -t i o n w i t h t i m e a s a n e v o l u t i o n p a r a m e t e r , o re q u i v a l e n t l y , t h e g r a d i e n t d e s c e n t m e t h o d . T h i sm e a n s t h a t w e s o l v e

uy

- A ( u - u 0 ) , f o r t > O , x , y E / 2 , (2 .5a )u ( x , y , O ) g i v e n , ( 2 .5 b )O nO n 0 o n a 1 2 . ( 2 .5 c )

N o t e , t h a t w e h a v e d r o p p e d t h e f ir s t c o n s t r a i n t( 2 . 3 b ) b e c a u s e i t i s a u t o m a t i c a l ly e n f o r c e d b yo u r e v o l u t i o n p r o c e d u r e ( 2 . 5 a - c ) i f t h e m e a n o fu ( x , y , 0 ) i s t h e s a m e a s t h a t o f u 0 ( x , y ) .

T h i s g i v e s u s a d y n a m i c v a l u e A ( t ), w h i c ha p p e a r s t o c o n v e r g e a s t- -- ~o o. T h e t h e o r e t i c a lj u s t i f i c a t i o n f o r t h i s a p p r o a c h c o m e s f r o m t h ef a c t t h a t i t i s m e r e l y t h e g r a d i e n t - p r o j e c t i o nm e t h o d o f R o s e n [ 1 4 ] .

W e a g a i n r e m a r k t h a t ( 2 .5 a ) w i t h A = 0 a n dr i g h t p a r t m u l t i p l i e d b y [V u I w a s u s e d i n r e f . [ 8]a s a m o d e l f o r s m o o t h i n g a n d e d g e d e t e c t i o n .F o l l o w i n g r e f . [ 9 ] w e n o t e t h a t t h i s e q u a t i o nm o v e s e a c h l e v e l c u r v e o f u n o r m a l t o i ts e l f w i t hn o r m a l v e l o c i t y e q u a l t o t h e c u r v a t u r e o f t h el e v e l s u r f a c e d i v i d e d b y t h e m a g n i t u d e o f t h eg r a d i e n t o f u . O u r a d d i t i o n a l c o n s t r a i n ts a r en e e d e d t o p r e v e n t d i s t o r t i o n a n d t o o b t a i n an o n t r i v i a l s t e a d y s t a t e .

W e r e m a r k t h a t G e m a n a n d R e y n o l d s , i n av e r y i n t e r e s t i n g p a p e r [ 1 0 ] , p r o p o s e d m i n i m i z i n gv a r i o u s n o n l i n e a r f u n c t i o n a l s o f t h e f o r m

f q ~ ( ~ x + u 2 ) d x d y12

w i t h c o n s t r a i n t s . T h e i r o p t i m i z a t i o n i s b a s e d o ns i m u l a t e d a n n e a l i n g , w h i c h i s a c o m p u t a t i o n a l l ys lo w p r o c e d u r e u s e d t o f i n d t h e g l o b a l m i n i m u m .W e , b y c o n t r a s t , s e e k a f a s t P D E s o l v e r t h a tc o m p u t e s a " g o o d " l oc al m i n i m u m o f th e T Vf u n c t i o n a l . T h e r e i s r e a s o n t o b e l i e v e t h a t t h el o c a l e x t r e m a a p p r o a c h i s m o r e r e l e v a n t t o t h i si m a g e p r o c e s s i n g t a s k .

F i n a l l y , w e n o t e t h a t w e o r i g i n a ll y i n t r o d u c e dt h i s m e t h o d i n t w o c o n f i d e n t i a l c o n t r a c t r e p o r t s[11,12] .


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262 L.I . Rud in et a l . / Noise removal a lgorithmsT h e n u m e r i c a l m e t h o d i n t w o sp a t i al d i m e n -

s i o n s i s a s f o l l o w s . W e l e t :x ~ = i h , y / = j h , i , j = O , 1 , . . . , N , w i t h

N h = 1 , ( 2 . 7 a )t / 1 = n A t , n = 0 , 1 , . . . , ( 2 . 7 b )

/1u i j = u ( x i , y ~ , t , ) , ( 2 . 7 c )o = U o ( i h , j h ) + c r q ~ ( i h , j h ) . ( 2 . 7 d )u o

T h e m o d i f i e d i n i t i a l d a t a a r e c h o s e n s o t h a tt h e c o n s t r a i n t s a r e b o t h s a t i s f i e d i n i t i a l l y , i . e . q ~h a s m e a n z e r o a n d L 2 n o r m o n e .

T h e n u m e r i c a l a p p r o x i m a t i o n t o ( 2 . 5 ) , ( 2 . 6 ) i sn + l nUO = Ui j

+ - - , A . , 2 .uq) + (m(A+uq, Ay-un))2)l/2ij/::A + u q+ A y_ y . . . . . u n Ax u n ) ) 2 ) 1/2( A + u q + ~ ,m ~a+ q , _ _ 0 , , ,

- A t A " ( u ~ - U o ( i h , j h ) ) , ( 2 . 8 a )f o r i, j = l , . . . , N ,w i t h b o u n d a r y c o n d it i on s

n n n nU O j ~ U l j , U N j ~ i, N _ I , j )

!

( a )

/1 n nU i o ~ U i N ~- U i , N - 1 "( 2 . 8 b )

H e r eA X u i j = " T -( U i Z. 1 , j - - U i j ) ( 2 . 9 a )a n d s i m i l a r l y f o r A Y uq .r e ( a , b ) = m i n m o d ( a , b )

_ ( s g n a + s g n b ) m i n ( l a l , I b l ) ( 2 . 9 b )a n d A/1 i s d e f i n e d d i s c r e e t l y v i a

= - - - +20 .2 . . (A +uq )x 0 x /1(A+uq)(A+u,j)

x n 2 n 2V ( A + u i j ) ~ - (A Y+ u i j )y 0 y n ~ ]_ ( A + u q ) ( A + k u q ) ) J ( 2 . 9 c )

x n 2 y n 2 "V(a +u.) + (au,;)A s t e p s i z e r e s t r ic t i o n i s i m p o s e d f o r st a b i li t y :

A th-~ ~< c . ( 2 .9 d )

3 . R e s u l t sW e h a v e r u n o u r t w o - d i m e n s i o n a l d e n o i s i n g

a l g o r i t h m o n g r a p h s a n d r e a l i m a g e s .T h e g r a p h s a n d i m a g e s d i s p l a y e d t a k e o n in -

Signal

Fig. 1. (a) "B ars" . (b) Plot o f (a). (e) Plot of noisy "bar s", SNR = 1.0. (d) Noisy "ba rs", SNR = 1.0. (e) Plot of therecon structio n from (d). (e) TV reconstruction from (d). (g) Plot of the reconstruction error.


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26 4 L . I . R u d i n e t a l . / N o i s e r e m o v a l a lg o r i th m sN o i s y S i g n al

Recovered Signal( c

Error( e )

Fi g . 2 . ( a ) P l o t o f f ig . l a p l us no i s e , S N R = 0 .5 . ( b ) N o i sy fi g. l a , SN R = 0 .5 . ( c ) P l o t o f t he r econs t r uc t i on f r om ( b ) . T Vr econs t r uc t i on f r om ( b ) . ( e ) P l o t o f t he r econs t r uc t i on e r r o r .


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L . I . R u d i n e t a l. I N o i s e r e m o v a l a l g o r it h m s 26 5

t e g e r v a l u e s f ro m 0 t o 2 5 5. W h e n G a u s s i a n w h i ten o i s e i s a d d e d t h e r e s u l t i n g v a l u e s g e n e r a l l y l i eo u t s i d e t h i s r a n g e . F o r d i s p la y p u r p o s e s o n l y w et h r e s h o l d ; h o w e v e r , t h e p r o c e s s i n g t a k e s p l a c eo n a f u n c t i o n w h o s e v a l u e s g e n e r a l l y l i e a r b i t -r a r i l y f a r o u t s i d e t h e o r i g i n a l r a n g e .

S i g n a l t o n o i s e r a t i o ( S N R ) i s d e f i n e d b y :

S N R = Z a ( u u - t~)2Z n ( n u ) 2 , ( 3 . 1 )

w h e r e t i i s t h e m e a n o f t h e s i g n a l u u a n d n o i s t h en o i s e .F i g s. 1 a n d 2 c o n c e r n t h r e e p a r a l le l s te p s t a k e n

0 I- - I I I '2 ~

I I If l l : . t

I ! ! ! : il l I t l E :m3 , , , , -m4 - - -

s = i i i I I I

I l l - - :~t l I = '~

I I I ~ .~:;I I1 ~ ~ ~. _ _ 0

" " I

Fi g . 3 . ( a ) "R es o l u t i on C ha r t " . ( b ) N o i sy "R e so l u t i o n C ha r t " , S N R = 1 .0 . ( c) W i ene r f il t e r r econs t r uc t i on f r om ( b ) . ( d ) T Vr econs t r uc t i on f r om ( b ) .


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266 L . I . R u d i n e t a l. / N o i s e r e m o v a l a lg o r i th m s

from fig. 3a. This i s 38 by 38 pixels wide 256 grayleve l s b l ack and whi t e or ig ina l image . F ig . l ashow s the o r ig ina l s igna l . F ig . l b shows i ts i n t en-s i ty p lo t . F ig . l c shows the in t ens i ty of the no i sys igna l wi th addi t ive Gauss i an whi t e no i se , s igna lto no i se ra t io SNR 1 . F ig . l d shows the no i sys ig n a l. F i g . l e sh o w s a g rap h o f t h e r eco v e red

sh a rp s i g n a l an d f i g . I f sh o w s t h e r eco v e reds igna l . F ina l ly , f ig . l g shows the e r ror which i sf a i r l y " h o l l o w " . I t i s z e ro b o t h w i t h i n t h e o r i g i -n a l s t ep s an d a l so b ey o n d a f ew p ix e ls o u t s i d e o ft h em . F i g. 2 a sh o w s t h e i n t en s i ty p l o t o f a n o isys i g n al w h e n S N R = 1 , t w i ce a s m u ch G au ss i anwhi t e no i se as s igna l . F ig . 2b shows the no i sy

Fi g . 4 . ( a) "A i r p l a ne" . ( b ) N o i sy "A i r p l a ne" , SN R = 1 .0 . ( e ) W i ene r f il t er r econs t r uc t i on f r om ( b ) . ( d ) T V r eco ns t r uc t i on f r om(b).


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L . L R u d i n e t a l . / N o i s e r e m o v a l a l g o r i th m s 26 7i m ag e . F i g . 2 c sh o w s t h e i n t en s i t y p l o t o f t h ereco v e red s i g n a l an d f i g . 2 d sh o w s t h e r eco v e redi m ag e . F i n a l ly , f ig . 2 e sh o w s t h e a l m o s t " h o l l o w "e r r o r .

I t a p p e a r s t h a t o u r d e n o i s i n g p r o c e d u r e b e a t st h e cap ab i l it y o f th e h u m an ey e - s ee fig s. l b , 2 ba n d 2 c .T h e r em a i n i n g f i g u re s a r e 2 56 g ray l ev e l s t an -

d a r d b l a c k a n d w h i t e i m a g e s t a k e n f r o m t h eU S C IP I i m ag e d a t a b a se . F i g . 3 a sh o w s t h eor ig ina l 256 x 256 p ixe ls reso lu t ion ch ar t . F ig . 3bsh o w s t h e r e su l t o f ad d i n g G au ss i an w h i t e n o i se,S N R 1 . F ig . 3 c sh o w s t h e r e su l t o f o u r d en o i s i n ga lgor i thm. F ina l ly f ig . 3d shows the resu l t o fu s i n g a W e i n e r f i l t e r w h e re t h e p o w er sp ec t ru mwas es t imated f rom f ig . 3b . Not i ce f ig . 3d has a

Fig . 5 . ( a) " T an k" . (b ) No is y "T a nk " , SNR = 1 .0 . ( c ) W iene r f il t er recons t ruc t ion f rom (b ) . (d ) T V recons t ruc t ion f rom (b ) ,


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26 8 L . I . R u d i n e t a l. / N o i s e r e m o v a l a lg o r i th m sl o t o f b a c k g r o u n d n o i s e w h i c h m a k e s i t p r o b -l e m at i c for autom at i c pr oc e s s i ng . F i g . 4a shows a256 256 a i r p lane i n the de se r t ( c l e an i m age ) .F i g . 4b shows the r e su l t o f addi ng Gauss i anwh i te no i se - S N R 1 . F i g . 4c sho ws the re su l t o fa de noi s i ng v i a our a l gor i thm . F i g . 4d shows ther e su l t o f a We i ne r f i l t e r de noi s i ng wi th the tr ues p e c t r u m e s t i m a t e d f r o m t h e n o i s y i m a g e , v i a am ovi ng ave r age . F i g . 5a shows the or i g i na l512 512 p i c tur e o f a tank . F i g . 5b sho ws ther e s u l t o f a d d i n g G a u s s i a n w h i t e n o i s e S N R 4 .F i g . 5c shows a We i ne r f i l t e r de noi s i ng wi thspe c tr um e s t i m ate s f r om f i g . 5b . F i g . 5d showso u r a l g o r i t h m a p p l i e d t o t h e s a m e w i n d o w .Not i c e that the d i sc ont i nu i t i e s ar e m uc h c l e ar e ri n the l as t c ase . A l so Wi e ne r r e s tor at i on hasosci l latory artifacts.

Our r e c e nt e xpe r i m e nts i nd i c ate that the useo f m o r e c o n s t r a in t s ( i n f o r m a t i o n a b o u t t h e n o i s eand the i m age ) i n th i s m e thod wi l l y i e l d m or ede ta i l s o f the so l u t i on i n our de noi s i ng pr o-c e d u r e .

R e f e r e n c e s[1 ] B . R . Fr i eden , Res to r ing wi th max imum l ike l ihood andmax imum en t ro py , J . Op t . Soc. Am . 62 (1972) 511 .[2] D . L . Ph i l l ip s , A t echn ique fo r the nu mer ica l so lu t ion o fce r t a in in teg ra l equa t ions o f the f i r s t k ind , J . ACM 9(1962) 84.

[3] S . T wom ey , O n the num er ica l s o lu t ion o f Fredho lmin teg ra l equa t ions o f the f i rs t k ind by the inve rs ion o fthe l inea r s ys tem produced by quadra tu re , J . ACM 10(1963) 97.[4] S . T w omey , T he app l i ca t ion o f numer ica l f il t e ring to thes o lu t ion o f in teg ra l equa t ions encoun te red in ind i rec ts ens ing meas urem ents , J . F rank l in Ins t. 297 (1965) 95 .[5 ] B . R . Hu n t , T h e app l i ca t ion o f cons t ra ined l ea st s qua re se s t ima t ion to image re s to ra t ion by d ig i t a l compute r ,IE E E T rans . Comput . 22 (1973) 805 .[6 ] L . Rud in , Images , numer ica l ana lys i s o f s ingu la r i t i e sand s hock f i l t e r s , Ca l t ech , C . S . Dep t . Repor t #T R:5250:87 (1987).[7 ] S . Os he r and L . I . Rud in , Fea tu re o r i en ted image en -hanc em ent us ing s hock fi lt er s, S IA M J . N um. A na l . 27(1990) 919.[8] L . Alvarez , P.L . Lions and J .M. Morel , Image se lec t ives mooth ing a nd edge de tec t ion by non l inea r d if fus ion ,SIAM J . Num. Ana l . 29 (1992) 845 .[9] S . O s he r and J . S e th ian , Fron t s p ropaga t ing wi thcurva tu re dependen t s peed : Algor i thms bas ed on aHa mi l ton- Jaco b i fo rm ula t ion , J . Comp ut . Phys . 79(1985) 12.[10] D . Ge ma n and G . Reyn o lds , Cons t ra ined re s to ra t ionand the recove ry o f d i s con t inu i t ie s , p rep r in t (1990) .[11] E . F a temi , S . Os he r a nd L . I . Ru d in , Remov ing no i s ewi thou t exces sive b lu r r ing , Cogn i tech Repor t #5 , (12 /8 9 ) , d e li v e r ed to D A R P A U S A r m y M i ss il e C o m m a n du n d e r c o n t ra c t # D A A H 0 1 - 8 9 - C - 0 7 68 .[12] L . I . Rud in and S . Os he r , Recons t ruc t ion and enhance -me n t o f s igna l s us ing non- l ine a r non-os c i ll a to ry va r i a -t iona l me thods , C ogn i tech Repo r t #7 (3 /90) , de l ive redt o D A R P A U S A r m y M i ss il e C o m m a n d u n d e r co n t ra c t# D A A H 0 1 - 8 9 - C - 0 7 6 8 .[13] Y. D odg e, Sta t is t ica l da ta analys is based on the L j nor ma n d r e l a t e d m e t h o d s ( N o r t h - H o l l a n d , A m s t e r d a m ,1987).[14] J . G . Ros en , T he g rad ien t p ro jec t ion me thod fo r non-l inea r p rogramm ing , Pa r t I I , non l inea r cons t ra in t s, J.Soc . Indus t . Appl . Math. 9 (1961) 514.

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