Abstract-A new neural network-based approach is introduced for recursive computation of the principal components of a stationary vector stochastic process. The neurons of a single layer network are sequentially trained using a recursive least squares squares (RLS) type algorithm to extract the principal components of the input process. The optimality criterion is based on retaining the maximum information contained in the input sequence so as to be able to reconstruct the network inputs from the corresponding outputs with minimum mean squared error. The proof of the convergence of the weight vectors to the principal eigenvectors is also established. A simulation example is given to show the accuracy and speed advantages of this algorithm in comparison with the existing methods. Finally, the application of this learning algorithm to image data reduction and filtering of images degraded by additive and/or multiplicative noise is considered.
Abstract-Speckle effects are commonly observed in synthetic aperture radar (SAR) imagery. In airborne SAR systems the effect of this degradation reduces the accuracy of detection substantially. Thus, the elimination of this noise is an important task in SAR imaging systems. In this paper a new method for speckle noise removal is mtroduced using 2-D adaptive block Kalman filtering (ABKF). The image process is represented by an autoregressive (AR) model with nonsymmetric half-plane (NSHP) region of support. New 2-D Kalman filtering equations are derived which take into account not only the effect of speckles as a multiplicative noise but also those of the additive receiver thermal noise and the blur. This method assumes local stationarity within a processing window, whereas the image can be assumed to be globally nonstationary. A recursive identification process using the stochastic Newton approach is also proposed which can be used on-line to estimate the filter parameters based upon the information within each new block of the image. Simulation results on several images are provided to indicate the effectiveness of the proposed method when used to remove the effects of speckle noise as well as that of the additive noise.
Abstract-This paper is concerned with the development of a 2-D adaptive Kalman filtering by recursive adjustment of the parameters of an autoregressive (AR) image model with non symmetric half-plane (NSHP) region of support. The image and degradation models are formulated in a 2-D state-space model, for which the relevant 2-D Kalman filtering equations are given. The recursive parameter identification is achieved using the extension of the stochastic Newton approach to the 2-D case. This process can be implemented on-line to estimate the image model parameters based upon the local statistics in every processing window. Simulation results for removing an additive noise from a degraded image are also presented. I. INTRODUCfIONThe problem of adaptive Kalman filtering of nonhomogeneous images has attracted considerable attention during the recent years [1]- [6]. Space-invariant Kalman filters assume wide sense stationarity of the image field, which is not a satisfactory assumption for real world images. As a result, they are insensitive to abrupt changes and tend to smear the edges and reduce the contrast, resulting in an image with poor visual appearance. Kalman filters that use space-varying image models take into account the local statistical information within a processing window to adjust the filter parameters. One such method was introduced by Kaufman, Woods, and Tekalp [1], [2], which proposes an identification and estimation procedure for nonsymmetric half-plane (NSHP) image model that can be used on-line to evaluate the covariance matrix of the plant noise and the parameters of the AR model at each stage of the algorithm. Another approach proposed by Rajala et at. [3] is based upon partitioning an image into disjoint regions according to the local spatial activities determined by the directional derivative information. This method uses the nearest neighbor algorithm to determine the best previous state and 2-D interpolation scheme to improve the estimates of the initial states in each region. In a recent paper by Tekalp, Kaufman, and Woods [4], an edgeadaptive Kalman filter is derived that uses multiple image models to reduce the ringing artifacts that are caused by spaceinvariant filters. The selection of the appropriate model is done using the maximum a posteriori (MAP) method. Azimi-Sadjadi [5], [6] introduced a 2-D adaptive block Kalman filtering, which is used to remove the effects of speckle noise in synthetic aperture radar (SAR) imagery. Other important work in this area is in [7]-[9].In this paper, a 2-D recursive parameter identification process is derived using the 2-D extension of the stochastic Newton approach [10] which can be used to estimate the parameters of an AR model with NSHP region of support (ROS). In Section II, the 2-D image and degradation models with NSHP regions of support are arranged in a state-space form, in which the state propagates in two dimensions. For this dynamical model the space-varying 2-D Kalman filtering equations are given in Section III, which take into account t...
A new 2-D a d a p t i v e b l o c k Kalman f i l t e r i n g i s i n t r o d u c e d f o r p r o c e s s i n g s y n t h e t i c a p e r t u r e r a d a r (SAR) imagery. The image p r o c e s s i s r e p r e s e n t e d by an a u t o r e g r e s s i v e (AR) model with nonsyrrrmetric h a l f p l a n e (NSHP) r e g i o n of s u p p o r t a r r a n g e d i n t h e b l o c k d i a g o n a l form. New 2-D Kalman f i l t e r i n g e q u a t i o n s a r e d e r i v e d which t a k e i n t o account n o t o n l y t h e e f f e c t s of t h e m u l t i p l i c a t i v e s p e c k l e n o i s e b u t a l s o t h e a d d i t i v e r e c e i v e r t h e r m a l n o i s e . T h i s method assumes l o c a l s t a t i o n a r i t y w i t h i n one block, whereas t h e image c a n b e assumed t o b e g l o b a l l y n o n s t a t i o n a r y . A r e c u r s i v e i d e n t i f i c a t i o n p r o c e s s u s i n g t h e s t o c h a s t i c Newton approach i s a l s o p r e s e n t e d which can be used o n -l i n e t o e s t i m a t e t h e f i l t e r p a r a m e t e r s b a s e d upon t h e i n f o r m a t i o n w i t h i n e a c h new b l o c k o f d a t a r e c e i v e d . An image p r o c e s s i n g example i s p r o v i d e d t o examine t h e e f f e c t i v e n e s s of t h e proposed method when used t o remove t h e e f f e c t s of s p e c k l e n o i s e i n a SAR image. The s p e c k l e e f f e c t s a r e commonly o b s e r v e d i n images g e n e r a t e d with h i g h l y c o h e r e n t p r o p a g a t i o n a s m u l t i p l e s o f t i n y s p o t s o f v a r y i n g i n t e n s i t y s u p e r i m p o s e d on t h e o r i g i n a l image. I n a i r b o r n e s y n t h e t i c a p e r t u r e r a d a r ( S A R ) systems t h e e f f e c t of t h i s d e g r a d a t i o n reduces t h e accuracy of d e t e c t i n g a t a r g e t . Speckles have t h e c h a r a c t e r i s t i c s of a random m u l t i p l i c a t i v e n o i s e i n t h e s e n s e t h a t t h e t e r r a i n b a c k s c a t t e r ( t h e d e s i r e d image) i s m u l t i p l i e d by a s t a t i o n a r y random p r o c e s s which r e p r e s e n t s t h e e f f e c t s of c o h e r e n t f a d i n g [ l l . Another c o n s t r a i n t which makes t h e r e s t o r a t i o n p r o c e s s i n p r e s e n c e of s p e c k l e n o i s e d i f f e r e n t t h a n t h e s t a n d a r d minimum mean s q u a r e e r r o r (MMSE) f i l t e r i n g i s t h a t f o r r a d a r t h e s i g n a l i s n o n s t a t i o n a r y s i n c e t h e mean b a c k s c a t t e r changes w i t h t h e type of t a r g e t b e i n g sensed, d e s p i t e of t h e f a c t t h a t t h e n o i s e can be modelled a s being s t a t i o n a r y . S e v e r a l image r e s t o r a t i o n t e c h n i q u e s f o r SAR imagery have been proposed o v e r t h e p a s t few y e a r s [ 2 ] -[ 5 1 . L e e [2] p r o p o s e d a n approach f o r s p e c k l e n o i s e smoothing u s i n g Sigma f i l t e r s which i s b a s e d upon continuous a d a p t a t i o n of f i l t e...
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