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Joint binary image deconvolution and blur identification in the context of two-dimensional storage channels
Signal Processing 91 (2011) 2426–2431
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Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Joint binary image deconvolution and blur identification in the context of two-dimensional storage channels Frederic Lehmann INSTITUT TELECOM, TELECOM SudParis, departement CITI, UMR-CNRS 5157, 91011 Evry Cedex, France
a r t i c l e in f o
abstract
Article history: Received 17 January 2011 Received in revised form 22 April 2011 Accepted 23 April 2011 Available online 7 May 2011
We consider blurring of binary images and corruption by ambient noise occuring on two-dimensional storage channels. Since coding is generally used in such systems, the deconvolution problem can be treated jointly with decoding. Several methods have been proposed in the literature under the name of turbo equalization to mitigate the degradation introduced by such channels. However, the problem of blur identification has rarely been addressed previously. In this paper, we propose a technique for estimating the 2D channel coefficients, along with the variance of the ambient noise. The proposed estimation algorithm is adaptive and performed jointly with turbo equalization, so as to limit the number of known pilot symbols needed to bootstrap the channel estimator. Interestingly, we found that the computational complexity of the proposed joint channel estimation and turbo equalization method depends heavily on the sensitivity of existing turbo equalization methods to 2D channel parameter mismatch. & 2011 Elsevier B.V. All rights reserved.
Keywords: Image deconvolution Blur identification Turbo equalization Two-dimensional storage channels Channel parameter mismatch
1. Introduction Linear image restoration is a well-known problem in the field of image processing, pattern recognition and computer vision, which consists in restoring an image degraded by blur and noise [1–3]. Recent developments in the area of optical and magnetic storage systems sparked renewed interest in this subject. Since the materials used in today’s recording technology are expected to reach their physical limits in the near future, new systems using multi-track optical storage [4] and holographic storage [5] have been proposed. By increasing the data density, these new recording media have the potential to meet the growing storage demand. However, unlike conventional storage technologies, two-dimensional intersymbol interference (2D ISI) appears during the readback process. Therefore, the output of two-dimensional storage channels can be regarded as binary images, corrupted by blur and ambient noise.
Tel.: þ33160764633; fax:33160764433.
E-mail address: [email protected] 0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.04.027
Various deconvolution techniques have been proposed in the literature. Among the simplest techniques, the minimum mean-square error (MMSE) equalizer [6], the decision feedback equalizer (DFE) [7] and the Viterbi algorithm [8], have been extended from 1D to 2D channels. It is well known that soft-input soft-output (SISO) equalization combined with channel decoding, also known as turbo equalization [9,10], can achieve close to optimal performances. Several SISO equalizers suitable for 2D channels are available in the literature. The iterative multistrip (IMS) algorithm [11,12] processes each row of observations independently with a BCJR equalizer [13]. A similar technique, obtained by replacing BCJR equalization by Gaussian belief propagation (GaBP) [14], has the advantage that the complexity is not exponential in the memory of the 2D channel. Another SISO technique based on generalized belief propagation (GBP) was proposed in [15], but the complexity of this method grows rapidly with the size of the 2D binary images to be stored. In this paper, we propose an iterative semi-blind solution to the problem of blur identification caused by the 2D channel, based on a maximum-likelihood framework. The critical first iteration relies on a number of
F. Lehmann / Signal Processing 91 (2011) 2426–2431
known pilot symbols [16] needed to boostrap the system. After a number of rounds of turbo equalization, symbol decisions which are reliable enough are used as tentative decisions to re-estimate the 2D channel parameters. By exploiting the channel decoder output to refine the channel estimates, we obtain an iterative process [17], which allows to limit the number of pilot symbols to a small fraction of each stored binary image. Interestingly, we found that the computational complexity of the proposed joint channel estimation and turbo equalization method depends heavily on the sensitivity of existing turbo equalization methods to 2D channel parameter mismatch. This paper is organized as follows. Section 2 introduces the considered system model for 2D storage channels. Section 3 presents the proposed joint channel estimation and turbo equalization approach. Finally, experimental results are given in Section 4 to compare the performances of the proposed channel estimation technique for several existing turbo equalization methods.
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2D storage channel Error Correcting Encoder
Pilot Symbol Insertion
{ bi,j}
{ yi,j}
{ hm,n} { ni,j}
Fig. 2. System model for 2D storage channels.
This matrix of data is fed to a 2D ISI channel modeling the 2D storage channel. Let fhm,n ,0 rm r M1,0 rn r N1g be the 2D ISI coefficients. As suggested in [12], by introducing a suitable delay, non-causal 2D channels can always be converted to causal 2D channels, therefore, the channel output can be written as yi,j ¼
M1 X N1 X
hm,n bim,jn þ ni,j ,
0 r i rI1,
0 rj r J1
m¼0n¼0
ð1Þ
2. System model In 2D storage systems, the data are written on the medium in the form of 2D binary images. Each binary image is encoded with an error correcting code, in order to correct the errors introduced by the storage material and the readback process. Let O ¼ fs ¼ ði,jÞ,0 r i r I1,0 r j r J1g be the 2D lattice, on which the data symbols are defined. Each binary image is a matrix of binary data of the form fbs ,s 2 Og, with bs 2 f1, þ1g. It is well known that the variations of the 2D storage channel in the context of magnetic and holographic storage can be significant [18,19]. For simplicity, we will assume that the 2D storage channel parameters are constant on a binary image and we insert a set of Np Np pilot symbols in each binary image for the purpose of channel estimation. A binary image can, therefore, we described as the matrix of binary data fbi,j ,0 ri rI1,0 r j r J1g, illustrated in Fig. 1.
0
Np-1
j
J-1
0 Pilot symbols Np-1
where ni,j N ð0, s2 Þ is an additive white Gaussian noise (AWGN) term. The complete system model is illustrated by Fig. 2. Let us define the vector of 2D ISI coefficients of size MN 1 as 2 3 hM1,N1 6 7 ^ 6 7 6 7 6 h0,N1 7 6 7 6 7 ^ h¼6 ð2Þ 7, 6 7 6 hM1,0 7 6 7 6 7 ^ 4 5 h0,0 then we can collect the parameters of the 2D channel in k ¼ ðh, s2 Þ. 3. Joint channel estimation and turbo equalization In this section, we first describe the general structure of the iterative detection algorithm used for joint channel estimation and turbo equalization in Section 3.1. Then we introduce the proposed channel estimation technique in Section 3.2. 3.1. Iterative detection algorithm
i
bi,j
I-1
Fig. 1. Binary image example with pilot symbols inserted in the upper left corner.
The proposed algorithm for joint channel estimation, equalization and decoding in the context of 2D storage channels is depicted in Fig. 3. The inner loop performs joint equalization and decoding in a typical turbo equalization fashion [9,10]. Namely, a SISO 2D equalizer processes the received observation matrix {yi,j} and delivers soft reliabilities on the binary image {bi,j} in the form of extrinsic log-likelihood ratios [9,10]. At the initialization, we assume no prior information on the binary image (except for the pilot symbols, which are perfectly known). The soft output of the equalization stage is used as prior information on the
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{ yi,j}
SISO 2D Equalization
Pilot Symbol Removal
Channel Estimation
Pilot Symbol Insertion
SISO Decoder
Fig. 3. Iterative detection system with joint 2D channel estimation.
binary image by a SISO decoder. The SISO decoder exploits the parity-check constraints of the error correcting code to deliver refined extrinsic log-likelihood ratios on the binary image. Then, the soft output of the decoding stage is used as prior information on the binary image {bi,j} to start a new round of 2D equalization, and so on. Nt iterations of this process are performed until convergence is reached, where one iteration corresponds to one round of 2D equalization, followed by one round of decoding. The outer loop updates the 2D channel parameter estimates by using the known pilot symbols and tentative decisions at the output of the SISO decoder, with the method introduced in Section 3.2. This channel re-estimation process is repeated Nc times, until no further improvement on the bit-error rate (BER) is achieved.
0
Np-1
j
J-1
0 Pilot symbols Np-1 {bi,j} N
i
bi-2,j-2 bi-2,j-1 bi-2,j s=(i,j),xs= bi-1,j-2 bi-1,j-1 bi-1,j bi,j-2 bi,j-1 bi,j
M
I-1 3.2. Channel estimation For each s 2 O, let xs be the M N column vector containing the data symbols, on which observation ys depends. Fig. 4 illustrates the portion of the binary image, which must be raster scanned columnwise to obtain xs , for a given value of s ¼(i,j). Let S be any subset of O, such that fxs ,s 2 Sg is formed of either pilot symbols or tentative symbol decisions. In Appendix A, we show that the data-aided maximumlikelihood estimate of k, given the observations fys ,s 2 Sg is expressed as 8 " #1 " # > P P > > T ^ > xs xs ys xs h¼ > > < s2S s2S ð3Þ T P > > ðys h^ xs Þ2 > > > > : s^ 2 ¼ s2S : jSj We now define the subset S O, on which the channel estimator should operate. The hard decision on data symbol bs is said to be reliable if the log-likelihood ratio of all the elements of xs , after the latest round of decoding, has an absolute value larger than a threshold, say 415. Then s 2 S if and only if bs is reliable or bs is a pilot symbol. The algorithm of Table 1 summarizes the proposed joint channel estimation, equalization and decoding procedure.
Fig. 4. Illustration of the portion of the binary image corresponding to xs, where s ¼ (i,j) (here M ¼ 3, N ¼3).
Table 1 Joint channel estimation, equalization and decoding procedure. 1. Initialize the set S: pilot symbols only 2. Initialize the 2D channel parameters according to (3) 3. Repeat Nc times Initialize all the turbo equalization messages (equiprobable data symbols) Perform Nt rounds of turbo equalization, given the current 2D channel parameter estimates Update S: pilot symbolsþ reliable hard decisions on the data symbols Re-estimate the 2D channel parameters according to (3)
It is expected that when Nc grows, the subset of reliable tentative decisions S becomes larger, so that the channel estimates also become more reliable. 3.3. Complexity evaluation The computational complexity per data symbol of the IMS equalizer (resp. the GaBP equalizer) is Oð2MN Þ (resp. OððMNÞ3 Þ) [11,14]. Since equalization dominates the
F. Lehmann / Signal Processing 91 (2011) 2426–2431
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computational load of the algorithm in Table 1, its computational complexity can be evaluated as OðNc Nt IJ2MN Þ (resp. OðNc Nt IJðMNÞ3 Þ) if IMS equalization (resp. GaBP equalization) is used. 4. Experimental results The performance of the proposed scheme is assessed by computer simulations in terms of bit-error rate (BER) versus the signal-to-noise ratio (SNR) defined as PM1 PN1 2 h SNR ¼ m ¼ 0 n2 ¼ 0 m,n , Rs where R denotes the coding rate times the pilot insertion rate. The output of a regular low-density parity-check (LDPC) encoder is modulated with binary shift keying (BPSK) to obtain the input of the 2D ISI channel of size I¼J¼ 30. We consider 2D equalizers based on the IMS algorithm [11] and the GaBP algorithm [14], which are SISO equalizers processing each row of observations one by one. Guard bands, similar to those described in [20], are used to initialize and terminate the equalizer processing each row of observations in a known state. In particular for the GaBP equalizer in [14], the prior distribution of the initial state, is chosen as a Gaussian distribution whose mean is the binary data vector contained in the guard band and whose covariance matrix is set to 103 IMN . The weight factor appearing in this method was set empirically to w¼0.7. Unless otherwise stated, turbo equalization is simulated for the following channels with M ¼N ¼3. The first channel is representative of moderate ISI 2 3 0:0993 0:352 0:0993 6 7 1 0:352 5 Channel A : ½hm,n ¼ 4 0:352 0:0993 0:352 0:0993 and is associated with a rate-4/5 (3,15) regular LDPC code. The second channel, representative of severe ISI, is a linear approximation of the Philips two-dimensional optical storage channel [11] 2 3 1 1 0 6 7 Channel B : ½hm,n ¼ 4 1 2 1 5 0 1 1 and is associated with a rate-1/2 (3,6) regular LDPC code. 4.1. Sensitivity to imperfect channel and SNR estimation We assume that the channel parameters are not known a priori and must be estimated by exploiting a set of 8 8 pilot symbols, arranged as in Fig. 1. Simultaneous channel parameter estimation and turbo equalization is performed using the algorithm of Table 1, with Nc outer iterations for the purpose of channel re-estimation and Nt inner turbo equalization iterations per outer iteration. Fig. 5 shows the results obtained for channel A and B when IMS-based turbo equalization is employed with Nt ¼6 iterations. For the sake of comparison, the BER obtained with perfect knowledge of the channel is also shown. It can be noticed that for Nc ¼1 (pilot-only channel
Fig. 5. IMS-based turbo equalization for channel A and B with channel estimation (Nc ¼1,5, Nt ¼ 6 iterations).
Fig. 6. GaBP-based turbo equalization for channel A and B with channel estimation (Nc ¼1, Nt ¼ 6 iterations).
estimation), the BER performances of joint channel estimation and turbo equalization are far from the knownchannel case. The difference can partly be gained back by augmenting the number of outer iterations, but at the expense of multiplying the computational complexity by Nc. For both channel A and B, convergence is reached for Nc ¼5. The same experiments were carried out for GaBPbased turbo equalization and the results are reported in Fig. 6, with Nc ¼1 and Nt ¼6. These results show that performing the outer loop only once is sufficient. This means that performing GaBP turbo equalization based on the rough initial channel parameter estimates (i.e. based solely on the pilot symbols) instead of the true values of the parameters, has no noticeable impact on the performances. The small 0.3 dB difference observed between the BER curves for perfect channel knowledge and channel estimation, corresponds exactly to the decrease in effective information rate due to the pilot insertion.
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By comparing Figs. 5 and 6, we observe that for channel B, GaBP turbo equalization is not necessarily worse than IMS turbo equalization, when the 2D channel must be estimated. This is somehow counter-intuitive, since GaBP turbo equalization is a rough approximation of message passing based on a Gaussian approximation [14], while IMS turbo equalization implements exact message passing, when the channel is perfectly known. We explain the observed behavior by the fact that the IMS turbo equalizer is very sensitive to channel and SNR mismatch, while the GaBP turbo equalizer is not. 4.2. Long memory optical storage channel In order to demonstrate the effectiveness of the proposed joint channel estimation and turbo equalization method for 2D storage channels with long memory, we adopt the page-oriented optical memory (POM) coherent channel model of [6] (for a value of blur W¼1.3), truncated to M¼N ¼5: Channel C : ½hm,n 2 0:0269 0:0478 6 0:0478 0:0851 6 6 ¼6 6 0:1513 0:2693 6 4 0:0478 0:0851 0:0269 0:0478
0:1513
0:0478
0:2693 0:8522
0:0851 0:2693
0:2693
0:0851
0:1513
0:0478
0:0269
3
0:0478 7 7 7 0:1513 7 7 7 0:0478 5 0:0269
A rate-4/5 (3,15) regular LDPC code is used. In this case, IMS turbo equalization would be a formidable task, since each BCJR equalizer would have to work on a trellis with 220 states (i.e. more than one million possible discrete states). However, GaBP turbo equalization, whose complexity does not grow exponentially in the memory of the 2D channel, is still manageable. Fig. 7 compares turbo equalization with simultaneous channel estimation (Nc ¼1, Nt ¼ 6 iterations), known-channel turbo equalization (Nt ¼6 iterations) and knownchannel MMSE equalization followed by 10 rounds of LDPC decoding. Here, turbo equalization is based on GaBP
equalization. The MMSE method is not satisfactory. Again, the 0.3 dB power efficiency difference between turbo equalization with and without perfect channel knowledge, is due solely to the rate loss induced by pilot insertion. 5. Conclusion In this paper, we introduced a joint deconvolution and blur identification method, which is suitable for binary images. The targeted application is the restoration of binary images on 2D storage channels. The proposed method, which uses the turbo equalization framework, inserts a few pilot symbols in each binary image for the purpose of channel estimation. Channel estimation is performed iteratively, based on the known pilot symbols and tentative decisions, obtained from the most reliable symbols at the output of the decoder. Our simulations showed that the IMS-based turbo equalizer, though quasioptimal for perfect knowledge of the 2D channel, is very sensitive to channel estimation and SNR mismatch. Therefore, the 2D channel must be re-estimated several times before convergence is reached, which augments the computational complexity considerably. The GaBP-based turbo equalizer, though sub-optimal for perfect knowledge of the 2D channel, does not suffer from this shortcoming. Thus, in the presence of channel uncertainty, the GaBP-based turbo equalizer provides a good complexity/ performance tradeoff, even for 2D channels with long memory. Appendix A. Data-aided maximum-likelihood 2D channel estimation Let Y ¼ fys ,s 2 Sg and B ¼ fbs ,s 2 Sg, where S O. We remind that B contains known data symbols. Our purpose is to calculate
k^ ¼ arg maxpðYjB; kÞ,
ð4Þ
where according to the channel model of Section 2 Y pðys jxs ; kÞ: pðYjB; kÞ ¼
ð5Þ
k
100
s2S
Differentiating (5) with respect to k, we obtain X @logpðys jxs ; kÞ @pðYjB; kÞ ¼ pðYjB; kÞ @k @k s2S
BER
10−1
ð6Þ
or equivalently
10−2
@logpðYjB; kÞ X @logpðys jxs ; kÞ ¼ : @k @k s2S known channel turbo equalizer
10−3
Using (7), the solution of (4) verifies X @logpðys jxs ; kÞ ¼ 0: ^ @k
estimated channel turbo equalizer− Nc=1 known channel MMSE equalizer
s2S
10−4 0
2
4
6
8 10 12 SNR (dB)
14
16
Fig. 7. GaBP-based turbo equalization for channel C.
18
ð7Þ
ð8Þ
k¼k
Again from Section 2, we have 1 1 T pðys jxs ; kÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2 ½ys h xs 2 , 2 2s 2ps
ð9Þ
F. Lehmann / Signal Processing 91 (2011) 2426–2431
therefore 8 T > @logpðys jxs ; kÞ ys h xs > > ¼ xs < 2 @h s > @logpðy jx ; k Þ 1 s s T > > ¼ ½s2 ðys h xs Þ2 : : 2 @s2
ð10Þ
^ s^ 2 Þ Injecting (10) into (8), the parameter estimates k^ ¼ ðh, must satisfy 8P ^T > > < ðys h xs Þxs ¼ 0 s2S ð11Þ T P 2 P > > : s^ ¼ ðys h^ xs Þ2 : s2S
s2S
We readily recognize in the first equation of (11), the set of orthogonality conditions characteristic of a least squares problem [21]. Thus, the desired result (3) follows. References [1] M. Sugiyama, H. Ogawa, A unified method for optimizing linear image restoration filters, Signal Processing 82 (2002) 1773–1781. [2] S. Ibrahim Sadhar, A.N. Rajagopalan, Restoration of scanned photographic images, Signal Processing 86 (2006) 1035–1048. [3] J.P. Oliveira, J.M. Bioucas-Dias, M.A.T. Figueiredo, Adaptive total variation image deblurring: a majorization–minimization approach, Signal Processing 89 (2009) 1683–1693. [4] W. Coene, Two-dimensional optical storage, in: Technical Digest of Optical Data Storage (ODS) Conference, Vancouver, Canada, 2003. [5] J. Ashley, M.-P. Bernal, G.W. Burr, H. Coufal, H. Guenther, J.A. Hoffnagle, C.M. Jefferson, B. Marcus, R.M. Macfarlane, R.M. Shelby, G.T. Sincerbox, Holographic data storage, IBM J. Res. Develop. 44 (3) (2000) 341–368. [6] K.M. Chugg, X. Chen, M.A. Neifeld, Two-dimensional equalization in coherent and incoherent page-oriented optical memory, J. Opt. Soc. Am. A 46 (3) (1999) 549–562. [7] J.K. Nelson, A.C. Singer, U. Madhow, Multi-directional decision feedback for 2D equalization, in: Proceedings of the ICASSP 04, Montreal, Canada, 2004.
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[8] J.F. Heanue, K. Gurkan, L. Hesselink, Signal detection for page-access optical memories with intersymbol interference, Applied Optics 35 (14) (1996) 2431–2438. [9] D. Reynolds, X. Wang, Low-complexity turbo-equalization for diversity channels, Signal Processing 81 (2001) 989–995. [10] L. Zhang, Y. Shang, Y. Cheng, H. Xiang, Improved turbo equalization based on soft ISI cancellation, Signal Processing 89 (2009) 1812–1820. [11] M. Marrow, J.K. Wolf, Detection of 2-dimensional signals in the presence of ISI and noise, in: Proceedings of the ISITA 04, Parma, Italy, 2004. [12] J.B. Soriaga, P.H. Siegel, J.K. Wolf, M. Marrow, On achievable rates of multistage decoding on two-dimensional ISI channels, in: Proceedings of the ISIT 05, Adelaide, Australia, 2005. [13] L.R. Bahl, J. Cocke, F. Jelinek, J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Transactions on Information Theory 20 (1974) 284–287. [14] F. Lehmann, Turbo equalization of two-dimensional intersymbol interference channels using Gaussian belief propagation, in: Proceedings of the ICASSP 10, Dallas, TX, USA, 2010. [15] N. Shental, S. Shamai, I. Kanter, A.J. Weiss, Y. Weiss, Discrete-input two-dimensional Gaussian channels with memory: estimation and information rates via graphical models and statistical mechanics, IEEE Transactions on Information Theory 54 (4) (2008) 1500–1513. [16] P. Salvo Rossi, R. Muller, O. Edfors, Linear MMSE estimation of time–frequency variant channels for MIMO–OFDM systems, Signal Processing 91 (2011) 1157–1167. [17] J. Guo, Y. Shang, S. Ren, H. Xiang, A turbo receiver combined with CFO compensation in frequency-domain for OFDMA uplink, Signal Processing 90 (2010) 2253–2264. [18] J. Cioffi, W. Abbott, K.D. Fisher, A survey of adaptive equalization for magnetic-disk storage channels, in: Proceedings of the Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, 1998. [19] D. Park, J. Lee, Holographic data storage channel model with intensity factor, IEEE Transactions on Magnetics 45 (5) (2009) 2268–2271. [20] A.H.J. Immink, W.M.J. Coene, A.M. van der Lee, C. Busch, A.P. Hekstra, J.W.M. Bergmans, J. Riani, S.J.L.V. Beneden, T. Conway, Signal processing and coding for two-dimensional optical storage, in: Proceedings of the Globecom 03, San Francisco, USA, 2003. [21] S. Haykin, Adaptive filter theory, Prentice Hall, Upper Saddle River, NJ, 2002.
Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Fast communication
Joint binary image deconvolution and blur identification in the context of two-dimensional storage channels Frederic Lehmann INSTITUT TELECOM, TELECOM SudParis, departement CITI, UMR-CNRS 5157, 91011 Evry Cedex, France
a r t i c l e in f o
abstract
Article history: Received 17 January 2011 Received in revised form 22 April 2011 Accepted 23 April 2011 Available online 7 May 2011
We consider blurring of binary images and corruption by ambient noise occuring on two-dimensional storage channels. Since coding is generally used in such systems, the deconvolution problem can be treated jointly with decoding. Several methods have been proposed in the literature under the name of turbo equalization to mitigate the degradation introduced by such channels. However, the problem of blur identification has rarely been addressed previously. In this paper, we propose a technique for estimating the 2D channel coefficients, along with the variance of the ambient noise. The proposed estimation algorithm is adaptive and performed jointly with turbo equalization, so as to limit the number of known pilot symbols needed to bootstrap the channel estimator. Interestingly, we found that the computational complexity of the proposed joint channel estimation and turbo equalization method depends heavily on the sensitivity of existing turbo equalization methods to 2D channel parameter mismatch. & 2011 Elsevier B.V. All rights reserved.
Keywords: Image deconvolution Blur identification Turbo equalization Two-dimensional storage channels Channel parameter mismatch
1. Introduction Linear image restoration is a well-known problem in the field of image processing, pattern recognition and computer vision, which consists in restoring an image degraded by blur and noise [1–3]. Recent developments in the area of optical and magnetic storage systems sparked renewed interest in this subject. Since the materials used in today’s recording technology are expected to reach their physical limits in the near future, new systems using multi-track optical storage [4] and holographic storage [5] have been proposed. By increasing the data density, these new recording media have the potential to meet the growing storage demand. However, unlike conventional storage technologies, two-dimensional intersymbol interference (2D ISI) appears during the readback process. Therefore, the output of two-dimensional storage channels can be regarded as binary images, corrupted by blur and ambient noise.
Tel.: þ33160764633; fax:33160764433.
E-mail address: [email protected] 0165-1684/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2011.04.027
Various deconvolution techniques have been proposed in the literature. Among the simplest techniques, the minimum mean-square error (MMSE) equalizer [6], the decision feedback equalizer (DFE) [7] and the Viterbi algorithm [8], have been extended from 1D to 2D channels. It is well known that soft-input soft-output (SISO) equalization combined with channel decoding, also known as turbo equalization [9,10], can achieve close to optimal performances. Several SISO equalizers suitable for 2D channels are available in the literature. The iterative multistrip (IMS) algorithm [11,12] processes each row of observations independently with a BCJR equalizer [13]. A similar technique, obtained by replacing BCJR equalization by Gaussian belief propagation (GaBP) [14], has the advantage that the complexity is not exponential in the memory of the 2D channel. Another SISO technique based on generalized belief propagation (GBP) was proposed in [15], but the complexity of this method grows rapidly with the size of the 2D binary images to be stored. In this paper, we propose an iterative semi-blind solution to the problem of blur identification caused by the 2D channel, based on a maximum-likelihood framework. The critical first iteration relies on a number of
F. Lehmann / Signal Processing 91 (2011) 2426–2431
known pilot symbols [16] needed to boostrap the system. After a number of rounds of turbo equalization, symbol decisions which are reliable enough are used as tentative decisions to re-estimate the 2D channel parameters. By exploiting the channel decoder output to refine the channel estimates, we obtain an iterative process [17], which allows to limit the number of pilot symbols to a small fraction of each stored binary image. Interestingly, we found that the computational complexity of the proposed joint channel estimation and turbo equalization method depends heavily on the sensitivity of existing turbo equalization methods to 2D channel parameter mismatch. This paper is organized as follows. Section 2 introduces the considered system model for 2D storage channels. Section 3 presents the proposed joint channel estimation and turbo equalization approach. Finally, experimental results are given in Section 4 to compare the performances of the proposed channel estimation technique for several existing turbo equalization methods.
2427
2D storage channel Error Correcting Encoder
Pilot Symbol Insertion
{ bi,j}
{ yi,j}
{ hm,n} { ni,j}
Fig. 2. System model for 2D storage channels.
This matrix of data is fed to a 2D ISI channel modeling the 2D storage channel. Let fhm,n ,0 rm r M1,0 rn r N1g be the 2D ISI coefficients. As suggested in [12], by introducing a suitable delay, non-causal 2D channels can always be converted to causal 2D channels, therefore, the channel output can be written as yi,j ¼
M1 X N1 X
hm,n bim,jn þ ni,j ,
0 r i rI1,
0 rj r J1
m¼0n¼0
ð1Þ
2. System model In 2D storage systems, the data are written on the medium in the form of 2D binary images. Each binary image is encoded with an error correcting code, in order to correct the errors introduced by the storage material and the readback process. Let O ¼ fs ¼ ði,jÞ,0 r i r I1,0 r j r J1g be the 2D lattice, on which the data symbols are defined. Each binary image is a matrix of binary data of the form fbs ,s 2 Og, with bs 2 f1, þ1g. It is well known that the variations of the 2D storage channel in the context of magnetic and holographic storage can be significant [18,19]. For simplicity, we will assume that the 2D storage channel parameters are constant on a binary image and we insert a set of Np Np pilot symbols in each binary image for the purpose of channel estimation. A binary image can, therefore, we described as the matrix of binary data fbi,j ,0 ri rI1,0 r j r J1g, illustrated in Fig. 1.
0
Np-1
j
J-1
0 Pilot symbols Np-1
where ni,j N ð0, s2 Þ is an additive white Gaussian noise (AWGN) term. The complete system model is illustrated by Fig. 2. Let us define the vector of 2D ISI coefficients of size MN 1 as 2 3 hM1,N1 6 7 ^ 6 7 6 7 6 h0,N1 7 6 7 6 7 ^ h¼6 ð2Þ 7, 6 7 6 hM1,0 7 6 7 6 7 ^ 4 5 h0,0 then we can collect the parameters of the 2D channel in k ¼ ðh, s2 Þ. 3. Joint channel estimation and turbo equalization In this section, we first describe the general structure of the iterative detection algorithm used for joint channel estimation and turbo equalization in Section 3.1. Then we introduce the proposed channel estimation technique in Section 3.2. 3.1. Iterative detection algorithm
i
bi,j
I-1
Fig. 1. Binary image example with pilot symbols inserted in the upper left corner.
The proposed algorithm for joint channel estimation, equalization and decoding in the context of 2D storage channels is depicted in Fig. 3. The inner loop performs joint equalization and decoding in a typical turbo equalization fashion [9,10]. Namely, a SISO 2D equalizer processes the received observation matrix {yi,j} and delivers soft reliabilities on the binary image {bi,j} in the form of extrinsic log-likelihood ratios [9,10]. At the initialization, we assume no prior information on the binary image (except for the pilot symbols, which are perfectly known). The soft output of the equalization stage is used as prior information on the
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{ yi,j}
SISO 2D Equalization
Pilot Symbol Removal
Channel Estimation
Pilot Symbol Insertion
SISO Decoder
Fig. 3. Iterative detection system with joint 2D channel estimation.
binary image by a SISO decoder. The SISO decoder exploits the parity-check constraints of the error correcting code to deliver refined extrinsic log-likelihood ratios on the binary image. Then, the soft output of the decoding stage is used as prior information on the binary image {bi,j} to start a new round of 2D equalization, and so on. Nt iterations of this process are performed until convergence is reached, where one iteration corresponds to one round of 2D equalization, followed by one round of decoding. The outer loop updates the 2D channel parameter estimates by using the known pilot symbols and tentative decisions at the output of the SISO decoder, with the method introduced in Section 3.2. This channel re-estimation process is repeated Nc times, until no further improvement on the bit-error rate (BER) is achieved.
0
Np-1
j
J-1
0 Pilot symbols Np-1 {bi,j} N
i
bi-2,j-2 bi-2,j-1 bi-2,j s=(i,j),xs= bi-1,j-2 bi-1,j-1 bi-1,j bi,j-2 bi,j-1 bi,j
M
I-1 3.2. Channel estimation For each s 2 O, let xs be the M N column vector containing the data symbols, on which observation ys depends. Fig. 4 illustrates the portion of the binary image, which must be raster scanned columnwise to obtain xs , for a given value of s ¼(i,j). Let S be any subset of O, such that fxs ,s 2 Sg is formed of either pilot symbols or tentative symbol decisions. In Appendix A, we show that the data-aided maximumlikelihood estimate of k, given the observations fys ,s 2 Sg is expressed as 8 " #1 " # > P P > > T ^ > xs xs ys xs h¼ > > < s2S s2S ð3Þ T P > > ðys h^ xs Þ2 > > > > : s^ 2 ¼ s2S : jSj We now define the subset S O, on which the channel estimator should operate. The hard decision on data symbol bs is said to be reliable if the log-likelihood ratio of all the elements of xs , after the latest round of decoding, has an absolute value larger than a threshold, say 415. Then s 2 S if and only if bs is reliable or bs is a pilot symbol. The algorithm of Table 1 summarizes the proposed joint channel estimation, equalization and decoding procedure.
Fig. 4. Illustration of the portion of the binary image corresponding to xs, where s ¼ (i,j) (here M ¼ 3, N ¼3).
Table 1 Joint channel estimation, equalization and decoding procedure. 1. Initialize the set S: pilot symbols only 2. Initialize the 2D channel parameters according to (3) 3. Repeat Nc times Initialize all the turbo equalization messages (equiprobable data symbols) Perform Nt rounds of turbo equalization, given the current 2D channel parameter estimates Update S: pilot symbolsþ reliable hard decisions on the data symbols Re-estimate the 2D channel parameters according to (3)
It is expected that when Nc grows, the subset of reliable tentative decisions S becomes larger, so that the channel estimates also become more reliable. 3.3. Complexity evaluation The computational complexity per data symbol of the IMS equalizer (resp. the GaBP equalizer) is Oð2MN Þ (resp. OððMNÞ3 Þ) [11,14]. Since equalization dominates the
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computational load of the algorithm in Table 1, its computational complexity can be evaluated as OðNc Nt IJ2MN Þ (resp. OðNc Nt IJðMNÞ3 Þ) if IMS equalization (resp. GaBP equalization) is used. 4. Experimental results The performance of the proposed scheme is assessed by computer simulations in terms of bit-error rate (BER) versus the signal-to-noise ratio (SNR) defined as PM1 PN1 2 h SNR ¼ m ¼ 0 n2 ¼ 0 m,n , Rs where R denotes the coding rate times the pilot insertion rate. The output of a regular low-density parity-check (LDPC) encoder is modulated with binary shift keying (BPSK) to obtain the input of the 2D ISI channel of size I¼J¼ 30. We consider 2D equalizers based on the IMS algorithm [11] and the GaBP algorithm [14], which are SISO equalizers processing each row of observations one by one. Guard bands, similar to those described in [20], are used to initialize and terminate the equalizer processing each row of observations in a known state. In particular for the GaBP equalizer in [14], the prior distribution of the initial state, is chosen as a Gaussian distribution whose mean is the binary data vector contained in the guard band and whose covariance matrix is set to 103 IMN . The weight factor appearing in this method was set empirically to w¼0.7. Unless otherwise stated, turbo equalization is simulated for the following channels with M ¼N ¼3. The first channel is representative of moderate ISI 2 3 0:0993 0:352 0:0993 6 7 1 0:352 5 Channel A : ½hm,n ¼ 4 0:352 0:0993 0:352 0:0993 and is associated with a rate-4/5 (3,15) regular LDPC code. The second channel, representative of severe ISI, is a linear approximation of the Philips two-dimensional optical storage channel [11] 2 3 1 1 0 6 7 Channel B : ½hm,n ¼ 4 1 2 1 5 0 1 1 and is associated with a rate-1/2 (3,6) regular LDPC code. 4.1. Sensitivity to imperfect channel and SNR estimation We assume that the channel parameters are not known a priori and must be estimated by exploiting a set of 8 8 pilot symbols, arranged as in Fig. 1. Simultaneous channel parameter estimation and turbo equalization is performed using the algorithm of Table 1, with Nc outer iterations for the purpose of channel re-estimation and Nt inner turbo equalization iterations per outer iteration. Fig. 5 shows the results obtained for channel A and B when IMS-based turbo equalization is employed with Nt ¼6 iterations. For the sake of comparison, the BER obtained with perfect knowledge of the channel is also shown. It can be noticed that for Nc ¼1 (pilot-only channel
Fig. 5. IMS-based turbo equalization for channel A and B with channel estimation (Nc ¼1,5, Nt ¼ 6 iterations).
Fig. 6. GaBP-based turbo equalization for channel A and B with channel estimation (Nc ¼1, Nt ¼ 6 iterations).
estimation), the BER performances of joint channel estimation and turbo equalization are far from the knownchannel case. The difference can partly be gained back by augmenting the number of outer iterations, but at the expense of multiplying the computational complexity by Nc. For both channel A and B, convergence is reached for Nc ¼5. The same experiments were carried out for GaBPbased turbo equalization and the results are reported in Fig. 6, with Nc ¼1 and Nt ¼6. These results show that performing the outer loop only once is sufficient. This means that performing GaBP turbo equalization based on the rough initial channel parameter estimates (i.e. based solely on the pilot symbols) instead of the true values of the parameters, has no noticeable impact on the performances. The small 0.3 dB difference observed between the BER curves for perfect channel knowledge and channel estimation, corresponds exactly to the decrease in effective information rate due to the pilot insertion.
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By comparing Figs. 5 and 6, we observe that for channel B, GaBP turbo equalization is not necessarily worse than IMS turbo equalization, when the 2D channel must be estimated. This is somehow counter-intuitive, since GaBP turbo equalization is a rough approximation of message passing based on a Gaussian approximation [14], while IMS turbo equalization implements exact message passing, when the channel is perfectly known. We explain the observed behavior by the fact that the IMS turbo equalizer is very sensitive to channel and SNR mismatch, while the GaBP turbo equalizer is not. 4.2. Long memory optical storage channel In order to demonstrate the effectiveness of the proposed joint channel estimation and turbo equalization method for 2D storage channels with long memory, we adopt the page-oriented optical memory (POM) coherent channel model of [6] (for a value of blur W¼1.3), truncated to M¼N ¼5: Channel C : ½hm,n 2 0:0269 0:0478 6 0:0478 0:0851 6 6 ¼6 6 0:1513 0:2693 6 4 0:0478 0:0851 0:0269 0:0478
0:1513
0:0478
0:2693 0:8522
0:0851 0:2693
0:2693
0:0851
0:1513
0:0478
0:0269
3
0:0478 7 7 7 0:1513 7 7 7 0:0478 5 0:0269
A rate-4/5 (3,15) regular LDPC code is used. In this case, IMS turbo equalization would be a formidable task, since each BCJR equalizer would have to work on a trellis with 220 states (i.e. more than one million possible discrete states). However, GaBP turbo equalization, whose complexity does not grow exponentially in the memory of the 2D channel, is still manageable. Fig. 7 compares turbo equalization with simultaneous channel estimation (Nc ¼1, Nt ¼ 6 iterations), known-channel turbo equalization (Nt ¼6 iterations) and knownchannel MMSE equalization followed by 10 rounds of LDPC decoding. Here, turbo equalization is based on GaBP
equalization. The MMSE method is not satisfactory. Again, the 0.3 dB power efficiency difference between turbo equalization with and without perfect channel knowledge, is due solely to the rate loss induced by pilot insertion. 5. Conclusion In this paper, we introduced a joint deconvolution and blur identification method, which is suitable for binary images. The targeted application is the restoration of binary images on 2D storage channels. The proposed method, which uses the turbo equalization framework, inserts a few pilot symbols in each binary image for the purpose of channel estimation. Channel estimation is performed iteratively, based on the known pilot symbols and tentative decisions, obtained from the most reliable symbols at the output of the decoder. Our simulations showed that the IMS-based turbo equalizer, though quasioptimal for perfect knowledge of the 2D channel, is very sensitive to channel estimation and SNR mismatch. Therefore, the 2D channel must be re-estimated several times before convergence is reached, which augments the computational complexity considerably. The GaBP-based turbo equalizer, though sub-optimal for perfect knowledge of the 2D channel, does not suffer from this shortcoming. Thus, in the presence of channel uncertainty, the GaBP-based turbo equalizer provides a good complexity/ performance tradeoff, even for 2D channels with long memory. Appendix A. Data-aided maximum-likelihood 2D channel estimation Let Y ¼ fys ,s 2 Sg and B ¼ fbs ,s 2 Sg, where S O. We remind that B contains known data symbols. Our purpose is to calculate
k^ ¼ arg maxpðYjB; kÞ,
ð4Þ
where according to the channel model of Section 2 Y pðys jxs ; kÞ: pðYjB; kÞ ¼
ð5Þ
k
100
s2S
Differentiating (5) with respect to k, we obtain X @logpðys jxs ; kÞ @pðYjB; kÞ ¼ pðYjB; kÞ @k @k s2S
BER
10−1
ð6Þ
or equivalently
10−2
@logpðYjB; kÞ X @logpðys jxs ; kÞ ¼ : @k @k s2S known channel turbo equalizer
10−3
Using (7), the solution of (4) verifies X @logpðys jxs ; kÞ ¼ 0: ^ @k
estimated channel turbo equalizer− Nc=1 known channel MMSE equalizer
s2S
10−4 0
2
4
6
8 10 12 SNR (dB)
14
16
Fig. 7. GaBP-based turbo equalization for channel C.
18
ð7Þ
ð8Þ
k¼k
Again from Section 2, we have 1 1 T pðys jxs ; kÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2 ½ys h xs 2 , 2 2s 2ps
ð9Þ
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therefore 8 T > @logpðys jxs ; kÞ ys h xs > > ¼ xs < 2 @h s > @logpðy jx ; k Þ 1 s s T > > ¼ ½s2 ðys h xs Þ2 : : 2 @s2
ð10Þ
^ s^ 2 Þ Injecting (10) into (8), the parameter estimates k^ ¼ ðh, must satisfy 8P ^T > > < ðys h xs Þxs ¼ 0 s2S ð11Þ T P 2 P > > : s^ ¼ ðys h^ xs Þ2 : s2S
s2S
We readily recognize in the first equation of (11), the set of orthogonality conditions characteristic of a least squares problem [21]. Thus, the desired result (3) follows. References [1] M. Sugiyama, H. Ogawa, A unified method for optimizing linear image restoration filters, Signal Processing 82 (2002) 1773–1781. [2] S. Ibrahim Sadhar, A.N. Rajagopalan, Restoration of scanned photographic images, Signal Processing 86 (2006) 1035–1048. [3] J.P. Oliveira, J.M. Bioucas-Dias, M.A.T. Figueiredo, Adaptive total variation image deblurring: a majorization–minimization approach, Signal Processing 89 (2009) 1683–1693. [4] W. Coene, Two-dimensional optical storage, in: Technical Digest of Optical Data Storage (ODS) Conference, Vancouver, Canada, 2003. [5] J. Ashley, M.-P. Bernal, G.W. Burr, H. Coufal, H. Guenther, J.A. Hoffnagle, C.M. Jefferson, B. Marcus, R.M. Macfarlane, R.M. Shelby, G.T. Sincerbox, Holographic data storage, IBM J. Res. Develop. 44 (3) (2000) 341–368. [6] K.M. Chugg, X. Chen, M.A. Neifeld, Two-dimensional equalization in coherent and incoherent page-oriented optical memory, J. Opt. Soc. Am. A 46 (3) (1999) 549–562. [7] J.K. Nelson, A.C. Singer, U. Madhow, Multi-directional decision feedback for 2D equalization, in: Proceedings of the ICASSP 04, Montreal, Canada, 2004.
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[8] J.F. Heanue, K. Gurkan, L. Hesselink, Signal detection for page-access optical memories with intersymbol interference, Applied Optics 35 (14) (1996) 2431–2438. [9] D. Reynolds, X. Wang, Low-complexity turbo-equalization for diversity channels, Signal Processing 81 (2001) 989–995. [10] L. Zhang, Y. Shang, Y. Cheng, H. Xiang, Improved turbo equalization based on soft ISI cancellation, Signal Processing 89 (2009) 1812–1820. [11] M. Marrow, J.K. Wolf, Detection of 2-dimensional signals in the presence of ISI and noise, in: Proceedings of the ISITA 04, Parma, Italy, 2004. [12] J.B. Soriaga, P.H. Siegel, J.K. Wolf, M. Marrow, On achievable rates of multistage decoding on two-dimensional ISI channels, in: Proceedings of the ISIT 05, Adelaide, Australia, 2005. [13] L.R. Bahl, J. Cocke, F. Jelinek, J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Transactions on Information Theory 20 (1974) 284–287. [14] F. Lehmann, Turbo equalization of two-dimensional intersymbol interference channels using Gaussian belief propagation, in: Proceedings of the ICASSP 10, Dallas, TX, USA, 2010. [15] N. Shental, S. Shamai, I. Kanter, A.J. Weiss, Y. Weiss, Discrete-input two-dimensional Gaussian channels with memory: estimation and information rates via graphical models and statistical mechanics, IEEE Transactions on Information Theory 54 (4) (2008) 1500–1513. [16] P. Salvo Rossi, R. Muller, O. Edfors, Linear MMSE estimation of time–frequency variant channels for MIMO–OFDM systems, Signal Processing 91 (2011) 1157–1167. [17] J. Guo, Y. Shang, S. Ren, H. Xiang, A turbo receiver combined with CFO compensation in frequency-domain for OFDMA uplink, Signal Processing 90 (2010) 2253–2264. [18] J. Cioffi, W. Abbott, K.D. Fisher, A survey of adaptive equalization for magnetic-disk storage channels, in: Proceedings of the Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, 1998. [19] D. Park, J. Lee, Holographic data storage channel model with intensity factor, IEEE Transactions on Magnetics 45 (5) (2009) 2268–2271. [20] A.H.J. Immink, W.M.J. Coene, A.M. van der Lee, C. Busch, A.P. Hekstra, J.W.M. Bergmans, J. Riani, S.J.L.V. Beneden, T. Conway, Signal processing and coding for two-dimensional optical storage, in: Proceedings of the Globecom 03, San Francisco, USA, 2003. [21] S. Haykin, Adaptive filter theory, Prentice Hall, Upper Saddle River, NJ, 2002.