The International Conference on Image Processing, Santa Barbara, California, 26-29 October 1997Conventional synthesis filters in subband systems lose their optimality when additive noise due, for example, to signal quantization, disturbs the subband components. The multichannel representation of subband signal is combined with the statistical model of input signal to derive the multirate state-space model for filter bank system with additive noises. Thus the signal reconstruction problem in subband system can be formulated as the process of optimal state estimation in the equivalent multirate state-space model. With the input signal embedded in the state vector, the multirate Kalman filtering provides the minimum-variance reconstruction of input signal. Using the powerful Kronecker product notation, the results and derivations can then be extended to the 2-D cases. Incorporated with the vector dynamical model, the 2-D multirate state-space model for 2-D Kalman filtering is developed. Computer simulation with the proposed 2-D multirate Kalman filter gives favorable results.published_or_final_versio

Ho, KL

Ni, JQ

Ni, JS

Shen, MH

Tse, KW

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Title Multirate Kalman filtering approach for optimal two-dimensionalsignal reconstruction from noisy subband systemsAuthor(s) Ni, JQ; Ho, KL; Tse, KW; Ni, JS; Shen, MHCitation Ieee International Conference On Image Processing, 1997, v. 1,p. 157-160Issued Date 1997URL http://hdl.handle.net/10722/54045Rights©1997 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising orpromotional purposes or for creating new collective works forresale or redistribution to servers or lists, or to reuse anycopyrighted component of this work in other works must beobtained from the IEEE.Multirate Kalman Filtering Approach for Optimal Two-Dimensional Signal Reconstruction from Noisy Subband Systems J.Q. Ni' K.L. Ho' K.W. Tse' and J.S. Ni** M.H. Shen** 'Department of Electrical and Electronic Engineering The Uniuersin of Hong Kong, Pokficlam Roud, Hong Kong jqni@eee. hku. hk Wuhan Universi+ of Hydraulic and Electric Engineering, Wuhan, P. R. China ** Abstract - Conventional synthesis filters in subband systems lose their optimality when additive noise due, for example, to signal quantization, disturbs the subband components. The multichannel representation of subband signal is combined with the statistical model of input signal to derive the multirate state-space model for filter bank system with additive noises. Thus the signal reconstruction problem in subband system can be formulated as the process of optimal state estimation in the equivalent multirate state-space model. With the input signal embedded in the state vector, the multirate Kalman filtering provides the minimum-variance reconstruction of input signal. Using the powerful Kronecker product notation, the results and derivations can then be extended to the 2-D cases. Incorporated with the vector dynamical model, the 2-D multirate state-space model for 2-D Kalman filtering is developed. Computer simulation with the proposed 2-D multirate Kalman filter gives favorable results. I. Introduction Recently multirate signal processing ha5 become the domain of extensive research. The concept of multirate signal processing is to decompose the original signal into complementary frequency bands and then process them separately in each subband. There have been tremendous progress on the design of decimation and interpolation filters or analysishynthesis filter banks that allow perfect reconstruction[5]. The conventional approach for PR filter bank systems are based on the assumption that the subband components are free of noise. However, in practical application with filter bank systems, the subband components are always contaminated by noises due to the effect of quantization, round off and other corruption, therefore, the perfect reconstruction is no longer possible. The goal of the paper is to improve the applicability of filter bank systems by proposing the multirate Kalman synthesis filter to replace the conventional synthesis filters, with emphasis on 2-D cases. With the input signal embedded in the state vector, the multichannel representation of subband signals is combined with the statistical model of input signal to derive the multirate state-space model for the filter bank systems and the subband noises are assumed to be additive ones in this model. On the basis of this model. the multirate Kalman filter can be constructed to provide the minimum- variance estimation of the input signal based on observations of noisy subband components. The concept of multirate Kalman synthesis filtering is first given in [3]. Our works differ from those of [3] in that the philosophy to derive the state-space model is different and the multirate state-space model suitable for 2-D Kalman filtering is developed. 11. Filter bank Systems and Problem Formulation 11.1 Filter bank systems Let (HL (z ) ,GA ( 2 ) :  k = 0,1,... , M - I }  denotes the .M band filter bank systems. The bank of filtcrs {HI i:)} constitutes the analysis filters. Each filter output is down- sampled and transmitted to the receiver. where they are up-sampled and fed into thc bank of synthcsis filters { G ,  ( t  j} for signal reconstruction. 11.2 Multichannel representation of subband signals we shall consider only the M = 2 ,  i.e., two band case in the interest of convenient description, the derivations and results can be easily generalized to M > 2 case at the cost of notational complexity. Let {h, ,  g , :  i = 0.1)denotes the impulse response of 2 - band analysiskynthesis filter bank. we have the equivalent multichannel representation of subband signals y,(nj, i = 0,l Without any loss of generality, and H(k) is the multichannel impulse response matrix of the form h, , (k)  = h , ( 2 k )  andIz,,(k)= h,(2k -1)  i =0,1 The h,, ( n )  andf, ( n )  above are the so-called polyphase 157 0-8186-8183-7/97 $10.00 0 1997 IEEE components of h,(n) and f ( n )  respectively. If additive noisy corruption are included in the subband components, the received subband signal r( n )  can be expressed as follows: r(n) = y(n)+ v(n) (2-3) v(n)= [v,(n) ~ , ( n ) ] ~  in (2-3) is the additive-noise disturbance vector. 111. Derivation of Multirate State-Space Model for 1-D Signal 111.1 The Basic Signal Model Recall the multichannel representation of subband signal (2-1). Let p = A+ A ,  where 1 and A are chosen such that h, (-2A - 1) = 0 and h, (2A + 1) = 0 i = 0,l . By substituting k = k - A ,we can rewrite (2-1) as a causal form P y(n) = H(k - A)f ( n  + A  - k )  (3-1) k 0  (3- 1) can be equivalently described by a state-space model (A,B,C,D) . If we denote the state vector as x(n)=[f(rl+A--l)r q n + a - 2 1 r  ... f ( n - - ~ ) ~ ] ' ,  we then have x( n + 1) = AX( n )  + Bf (n + A) (3-2a) We further denote H , ( k )  = [h , (k )  h , , ( k ) ]  and notice that h, (-21 - 1) = 0 ,  the subband output can be written as = Cx(n) + Df[2( n + A)] (3-2b) Considering the effect of noisy disturbance vector v(n) , we can formulate the problem as the form of state- space model x ( n  + 1) = Ax(n)+ Bf ( n  +A) x(-I) = x-I r(n) = Cx(n)+Df[2(n+A)]+v(n) (3-3) 111.2 The Statistical Model characteristic of input signal should have the form[ 11 The state space model describing the statistical z ' (k+ l )=@~' (k)+Gu(k)  Z'(-l)=z'-, f ( k )  = W ( k )  (3-4) where the z'(k) and u ( k )  are the state vector and driving source, respectively. Examining the state-space models described by (3-3) and (3-4) reveals that the two systems evolve in different time scale. Alter some algebra and let z(n) = z'[2(n+A)], we have the equivalent block generation model (3-5) of thc form: z(n+1)=@*z(n)+G2u2(n) I ( n  til) = H,z(n) t Qzuz(n) f P ( n  + A l l  = W n )  (3-5)  111.3 The Multirate State-Space Model The basic model (3-3) can be augmented with the statistical model (3-5) to yield the final combined multirate state-space model. w(n + 1)  = Aw(n) +Bu,(n) r(n) = Cw(n) + v(n) +(-I) = ik, (3-6) where w(n)=[x(n)' ~ (n) ' ] ' ,  is the state vector of system model (3-6). The parameter matrices A, B and _ _  are de temned as follows (3-7) IV. Derivation of Multirate State-Space Model for 2-D Signal IV. 1 The Basic Signal Model When implementing the 2-D filter banks, the dominant approach is to apply two I-D filter banks separately. Let {hf , gf: i = 0,1 } and {h,' , gl: i = 0,1 } denote two I-D filter banks for column- and row-wise operations, respectively. The original 2-D signal f ( m ,  n )  can then be decomposed into four subband components gjj(m,n) i, j = 0,l through the 2-D separate analysis banks. We have the 2-D multichannel rcprcsentation of the form If we further define F(m, n )  = [F(m - A', n )  F(m - A' + 1, n )  ... F(m+ X, .)Ir and after some algebra (4-1) can be expressed as the more compact form y ( m , n )  = x H ' F ( m , n  -t% - k ) H ' ( k  -%)' where E'= [ H'(A') H c ( A ' - l )  ... H'(-X)].  For the sake of derivation of 2-D multirate state-space model, we express ( 4-2 ) in a more convenient form with the powerful Kronecker product notation, namely, Y(n1.n) = i ( H r ( k -  ,l')@W ) f (m ,n  + K  - k )  = tC ,? (n i . t i+K  - k )  - p' - - (4-2) 1 =o 1-0 k=0 (4-3) 158 where - Y ( ~ J Z )  = [ ~ ~ ~ , ( m , r ~ )  ~ ~ ~ ~ ( m , n )  y,, (m,n) yll(m,n)]'  and F(m,n+j l ' )= [f[2( m - A' ),2( 12 + ;1' )I.. f[2(m + 2 ) + 1,2( n + ;1' )] f[2(m- A' ) , 2 (n  + X) + 1]..f[2(m + 2') + 1,2(n + X )  + I]]' The column vector F ( m , n + X )  happens to be constituted by two adjacent column vectors f[m,Z(n+X)] and f[m,2(n+;1')+1]. As in 1-D case, we define state vector as x( m, n )  = - - - - [ G( m, ,I + x - I)' F( m, n + x - 2)' . . . F( m, ,I - *' )'I' , and consider the effect of noisy disturbance r (m,n)=y(m,n)+Iv(m,n) ,  wethenhave x(m, n + 1) = Ax(m, n )  + BF(m,n + X) ?(m,-l) = i,n F(n2,n) = Cx(m,n)+Df[m,2(n+il')]+ V(m,n )  (4-4) - IV.2 The Vector Dynamical Model Consider the wide-sense stationary random field f ( t , n ) ,  for large portion of nature images, their autocorrelation functions R( t, n )  are well characterized by ( 4-5 ) PI On the basis of (4-5) a vector model can be developed [2], and have the form in our configuration: z( m, n + 1) = @'z( m, n )  + G,u2 (m, n) F( m, n + il' ) = H2 z(m, n ) + Q,u, (m, n ) f [m,2(n+X)I= Hz(m,n) u[m,2(n+X)]  [ u[m,2(n+X)+l ]  R(r,n) = aexp(-,u,ltl-,u,inl) (4-5) - (4-6) 1, and where u 2 ( m , n )  = z(m, k) and U? (m, k )  are the 4( p ,  + 1) - dimensional state vector and driving white noise vector, respectively. IV.3 The 2 - 0  Multirate State-Space Model The 2-D multirate state-space model with consideration of the statistics of object plane and noisy disturbances in subbands can be constructed by combining the basic model (4-4) with the object plane model (4-6) to yield a augmented multirate state-space model (4-7). w ( m , n + l ) =  Aw(m,n)+Bu,(m,n) +(m,- I )=  +,,, - - (4-7) - r(m, n )  = Cw(m, n )  + S(m,n) The vector w ( m , n )  = [x(m,n)' z(m,n)']' is the state vector of system (4-7). The Kalman synthesis filter equations for 2-D subband system are given below for the sake of completeness iU(m.n) = [I-K(m,n)Clxiv(m, n-  I)+K(m,n)F(m, n) K(m,n) = P(m,n)C'[CP(m,n)Fr + R(nz,n)]-' P(m, ti + 1) = A[I - K(m, n)C]P(m. n)Ar +FQ(m, n)Br  (4-8a) (4-8b) (4-bc) - -  with +(m,-l) = G,,,, P(m,-11-1) = P, for m= 0,1;.. . R and C l  are the covariance matrices of i( n )  and u2 ( n )  , respectively. V. Numerical Results and Discussions Computer simulation is carried out to show the feasibility and effectiveness of the proposed 2-D Kalman filtering for optimal 2-D signal reconstruction from noisy subband systems. We compare the performance of the proposed 2-D multirate Kalman filter versus the conventional PR filters for 2-D signal reconstruction under different noise levels. The simulation was implemented on the SGVR4000 platform with Matlab. We adopted a set of 2-band QMF PR filter bank {h(n)}of length 8 in our simulation [4]. The 2x2 subband decomposition of test image is implemented through two I-D separate analysis filter bank{h;,hr). with h' ( n )  = hr(n)  = h,(n) ,  i = 0,l. ,=n.i Two quantitative measures, SNR,InSn and SNR, , are defined to reflect the input noise level and reconstruction performance, respectively. We would denote SNR,?" as S N R , ,  since we adopted the same SNR level in each subband in our simulation. The test image Hillside is of size 160 x 160 x 8 ( fig. 1 (a) ). The image is converted to zero mean prior to 2 x 2 subband decomposition and its autocorrelation function is found to have the form of R ( t , n )  = 1168.3exp(-23591tl-0.221nl) ( f : continuous in horizontal direction ). White additive noise at different S N R  levels was added to all the four subband images. Fig.2 demonstrates the reconstruction performance comparison with both the proposed multirate Kalman filtering and the conventional PR synthesis filters under different S N R  level. It is observed that the improvement in reconstruction SNR with the proposed 2-D multirate Kalman filtering is considerable. Fig1 .(e) shows that, even in the extremely low SNR, case, the main structure of the original image is still distinguishable on the reconstructed image with 2-D multirate Kalman filtering. VI. Conclusion In this paper, we combine the multichannel representation of subband signal with the statistical model of input signal to derive the multirate state-space model for noisy filter bank systems, with emphasis on 2- D cases. With the input signal embedded in the state vector, the issue of signal reconstruction can be formulated as the optimal state estimation with multirate Kalman filtering. Since the Kalman filtering is one- dimensional in nature, the vector dynamical model is incorporated to develop the 2-D multirate state-space model suitable for 2-D multirate Kalman filtering. Computer simulation results demonstrate significant improvement in SNR, of reconstructed 2-D signal, under different noise levels, with the proposed 2-D multirate 159 Kalman filtering approach over those with conventional synthesis filter. The nonstationarity of nature images, however, can be tackled, with adaptive Kalman filtering approach based on the image segmentation, which remains the topic for further research. References [ I ]  N.E. Nahi, Estimation Theory and Applications, New York:Wiley, 1969. [2] N.E. Nahi, “Recursive Image Enhancement-Vector Processing,” IEEE Trans. Corn., Vol. COM-21, pp.305- 31 1, Apr. 1973. [3] B. Chen, “Optimal Signal Reconstruction in Noisy Filter Bank Systems: Multirate Kalman Synthesis Filtering Approach,” IEEE Trans. Signal Processing, Vo1.43, pp.2496-2504, Nov., 1995. [4] A.Delopoulos, “Optimal Filter Banks for Signal Reconstruction from Noisy Subband Components,” IEEE Trans. signal processing, Vol. 44, pp212-224, Feb., 1996. [5] P.P. Vaidyanathan, “Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications: A Tutorial ,”Proc. of The IEEE, Vo1.78, No.!, pp.56-93, Jan. 1990. Fig. 1 .  (a) Original image Hillside of size 160 x 160 ; (b) 2 x 2 subband decomposition of Hillside; (c) noise-corrupted subband image at SNR, = 0 db ; (d) reconstructed image with conventional PR filter banks at SNR, = 0 db ; (e) reconstructed image with 2-D Kalman filtering at SNR, = 0 db; (f) reconstructed image with conventional PR filter banks at SNR, = 8 db ; (g) reconstructed image with 2-D Kalman filtering at SNR, = 8 db . SNR M (ma) Fig.2 Reconstruction S N R  versus additive noise SNR level 160 

Multirate Kalman filtering approach for optimal two-dimensional signal reconstruction from noisy subband systems

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Multirate Kalman filtering approach for optimal two-dimensional signal reconstruction from noisy subband systems

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