第7卷第1期 智能系统学报 Vol.7 No.1 2012年2月 CAAI Transactions on Intelligent Systems Feh.2012 D0I:10.3969/j.i8sn.1673-4785.201111001 Thermodynamics-inspired inverse halftoning via multiple halftone images SAIKA Yohei,AOKI Toshizumi? (1.Department of Information and Computer Engineering,Gunma National College of Technology,580 Toriba,Maebashi, 371-8530,Japan;2.Department of Natural Science,Gunma National College of Technology,580 Toriba,Maebashi,371- 8530,Japan) Abstract:Based on an analogy between thermodynamics and Bayesian inference,inverse halftoning was formulated using multiple halftone images based on Bayesian inference using the maximizer of the posterior marginal (MPM)estimate.Applying Monte Carlo simulation to a set of snapshots of the Q-Ising model,it was demonstrated that optimal performance is achieved around the Bayes-optimal condition within statistical uncertainty and that the performance of the Bayes-optimal solution is superior to that of the maximum-a- posteriori(MAP)estimation which is a deterministic limit of the MPM estimate.These properties were qualitatively confirmed by the mean-field theory using an infinite-range model established in statistical me- chanics.Additionally,a practical and useful method was constructed using the statistical mechanical itera- tive method via the Bethe approximation.Numerical simulations for a 256-grayscale standard image show that Bethe approximation works as good as the MPM estimation if the parameters are set appropriately. Keywords:inverse halftoning;statistical mechanics;Monte Carlo simulation;Bethe approximation; CLC Number:TP39 Document Code:A Article ID:1643-4785(2012)01-0086-09 1 Introduction of generating a halftone image comprised of a set of black and white dots that are visually similar to the o- Researchers have long used Bayesian inference to riginal image as seen by the human vision system. investigate various issues in information science and Many techniques,such as the organized dither meth- technology such as problems related to image and sig- od671,have been proposed for generating such image. nal processing3.During the last two to three dec- The inverse of digital halftoning,i.e.,"inverse ades,theoretical physicists have studied such prob- halftoning,"is used to reconstruct the original image lems by using an analogy between statistical mechanics from the halftone image,and many techniques have and the maximizer of the posterior marginal (MPM)been proposed for doing this.For an instance, estimate based on Bayesian inference.In the early Wong9 proposed statistical techniques for a halftone stage of development in this field,physicists studied image obtained using the error diffusion method.Then, image restoration and error-correcting codes!4s)to de- Stevensonapplied maximum-a-osterioi(MAP)es- termine the feasibility of applying statistical mechanics timation to inverse halftoning of halftone images conver- to these problems.Researchers have since then applied ted using either a dither method or an error diffusion statistical mechanics to various problems in information method.Recently,Saika et al.(constructed a science and technology. Bayes-optimal solution on the basis of the statistical Researchers have been investigating digital half- mechanics of the Q-Ising model for the inverse halfton- toning print technology7 for many years with the aim ing problem.It uses the MPM estimate based on Bayesian inference.Other researchers have investiga- Received Date:2011-11-01. Corresponding Author:SAIKA Yohei.E-mail:yoheisaika@gmail.com. ted super resolution,another typical problem in infor-
第1期 SAIKA Yohei,et al:Thermodynamics-inspired inverse halftoning via multiple halftone images ·87… mation science and technology.The objective is to con- ven if a uniform model prior is assumed.However,a struct a high-resolution image using multiple low-reso- fatal contour appeared in the images reconstructed u- lution images.Several researchers have triedu sing fewer than 16 halftone images.The contour can be sing Bayesian information processing from the statistical removed by the MPM estimate using fewer than 16 half- mechanical point of view. tone images if an appropriate model of the true prior In this study,a practical and useful method for which can enhance smooth structures is used.A practi- inverse halftoning is proposed.It combines a statistical cal and useful method was then developed for recon- mechanical iterative method and a framework of super structing images based on the Bethe approximation resolution to use multiple halftone images.This ap- which approximates the MPM estimate.Simulation proach was adopted with the hope that statistical me-shows that Bethe approximation reconstructs the origi- chanics of Bayesian information processing would be a-nal image almost as good as Monte Carlo simulation vailable by using an analogy between statistical me-and that only 11~15 iterations are needed to solve the chanics and the MPM estimate based on Bayesian in- self-consistent equations derived from the Bethe ap- ference.In particular,the statistical mechanical itera- proximation. tive method was used based on the Bethe approximation The content of this paper is as follows.The gener- to approximate the MPM estimate established in the al prescription of statistical mechanics and the analogy field of Bayesian information processing.Then,in or- between statistical mechanics and Bayesian inference der to realize inverse halftoning with high image quali- were briefly reviewed,describing the general formula- ty,a framework of super resolution was also applied to tion of inverse halftoning using multiple halftone images multiple halftone images for inverse halftoning.Next,based on the statistical mechanics of the Q-Ising mod- in order to clarify the performance of Bethe approxima-el.The performance of the present method was then tion from the theoretical point of view,the efficiency of examined.Monte Carlo simulation and analytical esti- the MPM estimate for inverse halftoning was examined mation were conducted based on the mean-field theo- using multiple halftone images based on Monte Carlo rys,using the infinite-range model;its statistical per- simulation for a set of snapshots of the Q-Ising model formance for a set of snapshots of the Q-Ising model and the mean-field theory using the infinite-range mod- was estimated along with its performance for a realistic el.First,Applying the Monte Carlo simulation to a set image.Finally,the performance of the presented sta- of snapshots of the 8-grayscale Q-Ising model,the sta- tistical mechanical iterative method based on the Bethe tistical performance of the MPM estimate was clarified approximation was estimated. using 4 halftone images rewritten by the organized dith- er method based on the metric using the mean square 2 General formulation error (MSE).The simulation showed that the optimal 2.1 statistical mechanics performance was achieved around the Bayes-optimal As shown in Fig.1,a principal goal of statistical condition within statistical uncertainty and that the op- mechanics is to clarify thermodynamics of many-body timal performance is superior to that of the MAP esti- systems starting with interactions between microscopic mation,which corresponds to the deterministic limit of elements.In general prescription of statistical mechan- the MPM estimate.These properties were qualitatively ics,thermal average of macroscopic physical quantity confirmed by analytical estimation based on the mean- can be estimated as an ensemble average over all possi- field theory using the infinite-range model.On the oth- ble states using a probability distribution: er hand,from the practical point of view,the perform- (1) ance was investigated based on the Monte Carlo simula- P(1s)=zpl-BIs)】 tion for realistic images,i.e.,a 256-grayscale stand-for a given Hamiltonian H(S).In this equation,a ard image“Lena”with256×256 pixels.The simula- set of the Ising spin statesS is used as a set of typi- tion showed that the MPM estimate can reconstruct the cal microscopic elements.The unit of temperature is original image using more than 36 halftone images,e- taken such that Boltzmann's constant g is unified.As
·88 智能系统学报 第7卷 a result,B =1/T.Normalization factor Z is called the the Bethe approximation.As a recent development of “partition function”: statistical mechanics,researchers have clarified that sta- Z=∑∑…∑exp[-BH({S)]. tistical mechanics serves a framework and various tech- 防 niques for probabilistic information processing based on The probability distribution in Eq.(1)is termed the the analogy (Fig.2)between statistical mechanics and "Boltzmann factor".Using the Gibbs-Boltzmann distri- Bayesian inference using the MPM estimate. bution,the thermal average of macroscopic quantity A(S)can be estimated as Statistical mechanics Bayesian inference Statistical (w=7品多系ep-1s1)1a(1s) Thermodynamics Corresponden9e。 performance Average on Correspondence Posterior Canonical ensemble Macroscopic substance probability Water Electron,Spin Correspondence Bit,Pixel Thermodynamic (Macroscopic)properties Fig.2 Analogy between statistical mechanics of phys- Ensemble average ical system and Bayesian inference of informa- ◆-Canonical ensemble tion system Time average Statistical Molecule mechanics 2.2 general formulation Microscopic elements Here,on the basis of the statistical mechanics of Electron,Spin- the Q-Ising model,the general formulation for inverse Microscopic substance halftoning is shown using multiple halftone images Fig.1 Main goal of statistical mechanics based on Bayesian inference which is obtained through the MPM estimate.As shown in Fig.3,the framework Though it is difficult to calculate this macroscopic is composed of two parts.One is the forward process quantity directly,it can be estimated by using the corresponding to digital halftoning,and the other is the approximation theories such as the mean-field theory and inverse process corresponding to inverse halftoning. Forward process(Digital halftoning) Inverse process(Inverse halftoning) t, {,] (r,} [x2,] Original image Reconstructed image r} {x2 C2. True prior Model prior Pr(,子） Likelihood Pr(z Organized dither method :Pr( Pr ( { T A set of the dithered images The set of the dithered images Fig.3 Framework of inverse halftoning via multiple halftone images
第1期 SAIKA Yohei,et al:Thermodynamics-inspired inverse halftoning via multiple halftone images ·89 tone images(m,n)using the organized dither method and a p x p Bayer threshold array M,. Here,Ty（m,n)=0,1,x,y=1,2,…,L,and m,n=1,2,.,p.Two example threshold arrays are shown in Figs.5(a)and (b).The threshold array (a)Snapshot of (b)256-grayscale (c)Halftone version Q-Ising model with "Lena"image with of (a)created using M.is given by 100x100 pixels 256×256 pixels organized dithe method and 2x2 4M2 M,=4Mo+3U。 4M2+2U2l Bayer threshold array 4M2+Un」 M=9 (3) 0328402341042 4816562450185826 (d)Halftone image (e)Grayscale image (f)Grayscale image created from (b)using obtained using 4 obtained using 64 124443614 466 38 organized dither halftone images under halftone images 60285220623054 method and8×8 Bayes-optimal without using prior Bayer threshold array condition (=1)information(//T =0. 08210 3 351143 33 9 41 MSE/O=0.131149) 124146 51195927491757 25 31119 154773913 45537 157135 6331552361295321 (a)4×4 (b)8×8 Fig.54×4and8×8 Bayer threshold arrays (g)Grayscale image (h)Grayscale image (i)Grayscale image U is an n xn matrix,the elements of which are obtained using 64 obtained using 16 obtained using 16 halftone images with halftone images halftone images and unified.Threshold M,is an integer from 0 to p2-1. JT.=1 (MSE/ without using prior appropriate model Q=0.012455) information (J/T=0,of true prior (T=I The halftone versions of the original images in Figs.4 MSE/0=0.162267)MSE/O-0.122582) (a)and (b)are shown in Figs.3(c)and (d). Fig.4 Original images,halftone versions of the origi- When the original image is converted into the nal images and reconstructed grayscale images (m,n)-th halftone image(m,n),a one-to-one In the first step of the forward process,a set of o- correspondence between each pixel of the original im- riginal grayscale images was considered,where age and the threshold of the Bayer threshold ar- 5y=0,1,…,Q-1andx,y=1,2,…,L.These ray M is first created by making use of the corre- images were generated by the assumed true prior ex- spondence: pressed as a probability distribution: 专，→M(g+m)%p,(y+n)%p, (4) P以)=zm-光： where (x+m)%p denotes a surplus that divides (x+ m)by p,and it is the same with (y+n)%p.Then, 含宫t-w户+(w-门. as shown in Fig.6,a threshold procedure is carried out for each pixel of the original image by the corre- (2) sponding thresholdM: where J,and T:are parameters for tuning "smooth- Tk,w（m,n)= ness"in the pattern of the original image.Fig.4(a) 0(5y-M(x+a%p,y+n%p·Q/p2-1/2). shows a sample of the original image,i.e.,a snapshot 0(.)is a unit-step function given by of the 8-state Q-Ising model on a square lattice.Then, as shown in Fig.4(b),a 256-grayscale standard image 8(x)=0,x0. with 256 x 256 pixels,such as "Lena",was used to The halftone images in Figs.4(c)and (d)are visual- investigate the performance for realistic images. ly similar to the original image as seen by the human In the second step of the forward process,each o- vision system although information on the original im- riginal image is rewritten into p'kinds of half- age was lost through the halftone procedure
90 智能系统学报 第7卷 In the inverse halftoning process,the original im- increase in the number of the halftone images. age is reconstructed so as to maximize the posterior r-7 marginal probability.The value of the (x,y)-th pixel 6 in the reconstructed image is given by 51 M61 Restricted range of arg max Pr() gray-level by three halfone images =4 4X4 Bayer-type threshold array 21 08210 M=2 16 124146 电京0 31119 Halfone image 3 5735 25502550 The number of the halftone images 0250255 Comparison 1151141413 25502550 Fig.7 Range of grayscale restricted by the likelihood 1201181714 025300 in Eg.(6)used for the present method 127币19019 Then,the reconstructed image is obtained by 1361289074 256-grayscale original image y=⑧(8，y）, Fig.6 Digital halftoning using 4 x 4 Bayer-type where threshold array w=名Pia1产)， The posterior probability is estimated using the Bayes formula, ()=】 叫e-k+》-0e-k-2. Pr()= To estimate the performance of the present meth- Pr({z})Pr({P2}1{) od,MSE was compared between the original and re- ∑Pr({z)Pr({x2}I{z) constructed images: as well as the assumed model of the true prior along with the likelihood.The model of the true prior ex- sE=A名，-，月 pressed by To estimate the statistical performance,MSE was a)ep averaged over the set of original images: 含-+-s) sE=名(1E)2云会，-6片 3 Performance is used so as to enhance smooth structures appearing in the patter of reconstructed images.Furthermore,a 3.1 Monte Carlo simulation model of the true prior is used,which is assumed to To quantitatively investigate the statistical per- have the same form as the true prior in Eq.(2)so that formance of the present method,a set of the snapshots an MPM estimate is constructed,including the Bayes- of the Q-Ising model was used on the square lattice.A optimal solution.The likelihood that is expressed by set of the snapshots of the 8-grayscale Q-Ising model the conditional probability representing the organized with 100x100 pixels(Fig.4(a))was used for the o- dither method via the Bayer threshold array is also riginal images.The parameters in Eq.(2)were set to used: J,=T,=1.Each original image was converted into four kinds of halftone images by the organized dither Pr({1{1)=ΠΠΠ6(rw,6(g method using the 2 x 2 Bayer-type threshold array in M(x*m%,+0p·Q/p2-1/2).(6) Egs.(3)and (4).Then,20 000 Monte Carlo steps As seen from Fig.7,the likelihood has the prop- were performed using the MPM estimate based on the erty that the possible range of the gray-level z be- Metropolis algorithm. comes narrower at each pixel with the increase in the First,it was determined how MSE depends on number of the halftone images.So,it is expected that parameter T in Eq.(5)for J=1.As shown in Fig.8, the accuracy of the performance is improved with the it is found that the performance of the present method
第1期 SAIKA Yohei,et al:Thermodynamics-inspired inverse halftoning via multiple halftone images 91 is improved by introducing the model of the true prior and that the performance is optimal under the Bayes- os2.dle-kg)》× optimal condition T=T,=1 within statistical uncer- (8) tainty. em-最-月 0.010r The summation in this equation applies to all over the 0.008 lattice sites.Then,on the basis of the saddle-point 差amd conditions on the configuration average free-energy in Eq.(8),self-consistent equations are derived on mo 0.004 and m: 0.002 1111111 0-1 m=[幻- 5exp[B,(2m5-m6)], 0 Z. 0.5 1.0 1.5 2.0 1 m=[(z〉]= Fig.8 MSE as a function of T obtained by 艺eplA,(2mt-m)】X Z, p2 Monte Carlo simulation for a set of 6(5,0(：-k9)exp[B.(2me-m)]5 snapshots of Q-Ising model D These results suggest that the MPM estimate based 85.0:-4号)ep[B.(2mu-m] Q-1 on Bayesian inference works effectively for inverse half- toning by making use of the framework of the super res- where olution with multiple halftone images while the MPM estimate is improved by introducing the model of the Z=ep[a,(2m6-m)1. true prior appropriately. Here B,=1/T,and B =1/T.Using the solution to 3.2 Mean-field theory these self-consistent equations on mo and m,MSE av- To qualitatively clarify the statistical performance eraged over the assumed true prior in Eq.(7)can be of the MPM estimate based on Bayesian inference,the estimated from 0-1 mean-field theory was tested using the infinite-range ,exp[B,(2mo5-m)](⊙()-专)2 model,which was established in terms of statistical MSE Q-1 mechanics.These models seem to be very artificial in exp[B,(2m5-m6)] the sense that all pixels neighbor each other.However, the aim here is not to establish a model of practical where usefulness but to understand generic features of macro- 直6(5,0(e-k9)esp[R.(2ms-m2)]z scopic variables,such as MSE. 2= a5.0:-号)ep[R.(2m-m] 0-1 To use the mean-field theory for estimating the performance of the MPM estimate,the infinite-range versions of the model and true priors are first intro- Using the solutions to the self-consistent equations duced: on mo and m,MSE was analytically estimated without statistical uncertainty. )=-- 。(7) First,to clarify the performance of this model,it was estimated how MSE depends on parameter T.As ,-], shown in Fig.9,it was found that the optimal perform- both of which are considered to approximate the as- ance is realized around the Bayes-optimal condition, sumed model and true priors in two dimensions.Fol- that is T=T,,and J=J,.These results indicate that lowing the strategy of the mean-field theory via the infi- the analytical estimate obtained using the infinite-range nite-range model4,the configuration average of the model qualitatively supports the results obtained using partition function of the model system is then consid- Monte Carlo simulation for the set of the snapshots of ered: the Q-Ising model.It was found that MSE monotonical- [n]=-2-最（发-× ly decreased with an increase in the number of halftone images and that perfect inverse halftoning was achieved
·92 智能系统学报 第7卷 with p2=0.These results show that the analytical esti- is found that the MPM estimate reconstructs original mate qualitatively supports the results obtained using image more accurately than the conventional Gaussian Monte Carlo simulation for a two-dimensional model. filter via the 3 x3 kerel,although the performance of 1.89565 the MPM estimate is almost the same as that of the 1.89555 MAP estimation.Also,it is found from the patterns of the reconstructed images in Figs.2(f),(i)and Figs. 1.89545 10(a),(c),that the fatal contour will not appear in 1.89535 the reconstructed image obtained by the MPM estimate via the 16 halftone images if appropriate model of the 1.895256 2 true prior is used;however,that contour appeared in the patterns of the reconstructed images obtained by the Fig.9 MSE as a function of T obtained by analytical Gaussian filter. estimate which was generated using infinite- range model 3.3 Realistic image performance To investigate the performance of the MPM esti- mate for realistic images,Monte Carlo simulation was used to derive MSE for obtaining the standard "Lena" (a)256-grayscale image obtained (b)256-grayscale image obtained by the MPM estimate via the 16 image (256-grayscale;256 x 256 pixels).Then,the by the MPM estimate via the 64 halftone images when=1,7= halftone images,when=l, performance of the MPM estimate was compared with 0.001(MSE/Q-0.131149) T0.001(MSE/Q-0.012417) those of other methods,such as the MAP estimation corresponding to the T limit of the MPM estimate and the conventional filter using a 3 x3 Gaussian ker- nel. First,the dependence of MSE on parameter Tm, was investigated when the MPM estimate was used for 64 (c)256-grayscale image obtained (d)256-grayscale image obtained by the conventional Gaussian by the conventional Gaussian filter kinds of halftone images obtained by conversion from the filter using the kernel with the using the kemel with the size 3x3 size 3x3 via the 16 halftone via the 64 halftone images "Lena"image using the organized dither method via an images (MSE/C-0.292 306) (MSE/0-0.013620 8x8 Bayer threshold array.Simulation showed that the Fig.10 Reconstructed images obtained by the MAP optimal performance was achieved in the wide range of estimation and the conventional filter using the parameter T at J=1.This result is evident in Fig. the 3 x3 Gaussian kernel for the 256- 4(f).Then,perfect inverse halftoning was achieved grayscale standard image“Lena”with256× when 256 kinds of halftone images were used,irrespec- 256 pixels tive of the settings for J and T,as shown in Fig.4(g). 3.4 Bethe approximation Next,when prior information was not used for the MPM Here a statistical mechanical iterative method is estimate,as shown in Fig.4(h),a false contour ap- used with Bethe approximation to construct a practical peared in the image reconstructed using 16 kinds of and useful method.Bethe approximation has been used halftone images.However,this contour was removed by to approximate thermodynamics of many-body physical appropriately introducing the model of the true prior,as systems by solving the self-consistent equations on a set shown in Fig.4(i). of effective fields.Here,a set of local magnetizations Next,the performance of the MPM estimate was {m(0<m<Q-1,x,y=1,2,,L)is used compared with other methods,such as the conventional on the square lattice to estimate macroscopic properties Gaussian filter using the 3 x3 kernel,the MAP estima- of the present method.Then,the set of the local tion for a set of 16(64)halftone images of the 256- magnetizationm is determined as a solution to the grayscale standard image "Lena"in Fig.2 (a).As self-consistent equations: shown in Figs.10(a),(b)and Figs.2 (f),(i),it
第1期 SAIKA Yohei,et al:Thermodynamics-inspired inverse halftoning via multiple halftone images ·93 m(s+1)=Z× approximation to carry out the Bayesian inference was estimated to obtain the MPM estimate.The results 名三idw-别x show that Bethe approximation obtains the reconstruc- ted images in Figs.12 (a)and (b)with 11 ~15 steps en-2iw:m()小} (9) respectively by choosing initial halftone images,such as Fig.4(d).As shown in the case of the Monte Carlo om{mw(s)}(0<m(s)<Q-1,x,y=1,2,…, simulation,it is found that false contour appears in the L,s=0,1,2,.),where s denotes the number of reconstructed image in Fig.12 (a)and that such con- steps.Here,the effective Hamiltonian H(; tour can be removed,as shown in Fig.12 (b),by the m.(s))in Eq.(9)is set as present method,using an appropriate model of the true H(;m.(s))=J ()2+ (x'yEDs prior.Here,the optimal value of J/T was determined by trial and error without using a technique for parame- 2（y-m(s)户，(10) ter estimation,such as the EM algorithms) and the normalization factor Z as a三三i6。- epl-之g:mo))} (11) (a)256-grayscale image recon- (b)256-grayscale image recon- Here,as shown in Fig.11,D..is a set of lattice structed using MPM estimate structed using MPM estimate and Bethe approximation for and Bethe approximation for points{(x,y),(x+1,y),(x,y+1),(x-1,y), T=0 T=1 (y-1)and msx (s)is a set of real numbers Fig.12 Reconstructed images obtained from the Bethe from 0 to Q-1 arranged on the lattice points around approximation using 16 kinds of the halftone D.The reconstructed image is obtained using the so- images lution to the self-consistent equations in Egs. From above results,it can be seen that the per- (9)~(11): formance of the Bethe approximation is similar to that 三y=8(m,y. of MPM estimation which uses Monte Carlo simulation for realistic images,such as the 256-grayscale standard image“Lena”with256×256 pixels. m-1.m M. 4 Conclusion As seen above,after outlining the statistical me- m.2 chanics which is used to describe the thermodynamics of the spin model,an analogy between statistical me- chanics and the MPM estimate was given based on Bayesian inference.Considering this analogy,the gen- eral formulation was then constructed for inverse half- toning using multiple halftone images.From the theo- retical point of view,to determine the performance of Fig.11 A set of pixels in D,(D (y), the present method,Monte Carlo simulation was car- (x+1),(x,J+1),(x-1y),(x,y-1)) ried out to obtain a set of the snapshots of the Q-Ising and a set of effective fieldsm around model.It was found that the performance was optimal D under the Bayes-optimal condition within statistical un- To investigate the performance of the presented certainty,and that the optimal performance was superi- statistical mechanical iterative method,numerical sim-or to that of the MAP estimation.Then,using analyti- ulation was conducted for the 256-grayscale "Lena" cal estimation through building a mean-field infinite- image with 256 x256 pixels.The performance of Bethe range model,the properties obtained by the Monte
94 智能系统学报 第7卷 Carlo simulation were qualitatively confirmed.Further- 1973:11-15. more,it was found that the present method is effective [8]MICELI C M,PARKER K J.Inverse halftoning[J].Jour- even for realistic images if more than 64 kinds of half- nal of Electronic Imaging,1992,1(2):143-151. tone images are used.It was also found that the fatal [9]WONG P W.Inverse halftoning and kemel estimation for error diffusion[J].IEEE Transactions on Image Process- contour will appear in the pattern conducted of the re- ing,1995,4(4):486498. constructed image if the MPM estimate is conducted u- [10]STEVENSON R L.Inverse halftoning via MAP estimation sing fewer than 16 kinds of halftone images without u- [J].IEEE Transactions on Image Processing,1997,6 sing the model of the true prior,and that this contour (4):574-583. can be removed by using the MPM estimate based on [11]SAIKA Y,INOUE J,TANAKA H,et al.Bayes-optimal an appropriate model of the true prior.Finally,from solution to inverse halftoning based on statistical mechanics the practical point of view,the performance of the sta- of the Q-Ising model [J].Central European Joumal of Physics,2009,7(3):444456. tistical mechanical iterative method was evaluated using [12]TIPPING M E,BISHOP C M.Bayesian image super-reso- the Bethe approximation.Numerical simulation using lution[M]//BECKER S,THRUN S,OBERMAYER K. the standard image "Lena"demonstrated that the Be- Advances in Neural Information Processing Systems.Cam- the approximation reconstructs the original image within bridge,USA:The MIT Press,2003:1279-1286. 11 ~15 steps and that by using Monte Carlo simula- [13 KANEMURA A,MAEDA S,ISHII S.Superresolution tion,Bethe approximation performs as good as the with compound Markov random fields via the variational MPM estimate for this problem.From these results,it EM algorithm J].Neural Networks,2009,22(7): 1025-1034. can be concluded that the statistical mechanical itera- [14]DEMPSTER A P,LAIRD N M,RUBIN D B.Maximum tive method performs more effectively than the conven- likelihood from incomplete data via the EM algorithm[J]. tional MAP estimation. Journal of the Royal Statistical Society-Series B:Statisti- Future work includes improving the performance cal Methodological,1977,39(1):1-38. of the statistical mechanical iterative method by using [15]TANAKA K,INOUE J,TITTERINGTON D M.Probabi- more accurate information on the model prior of listic image processing by means of the Bethe approxima- grayscale images and on the MTF-function of the hu- tion for the Q-Ising model [J].Journal of Physics A: Mathematical and General,2003,36(43):11023-11035. man vision system.Also,as the parameter selection is About the authors: important in these problems based on the Bayesian in- SAIKA Yohei,received a B.S.de- ferences,the authors will construct a technique of in- gree in physics from Tokyo Science Uni- verse halftoning based on the parameter estimation u- versity,Tokyo,Japan in 1989,an M.S. sing the EM algorithm. degree in physics from Tokyo Institute of Technology in 1991,and a Ph.D.degree References: in physics from Tokyo Institute of Technol- [1]BESAG J.On the statistical analysis of dirty pictures[J]. ogy in 1995.His research interests are Journal of the Royal Statistical Society-Series B:Statistical statistical mechanics,application of physics to information,and Methodological,1986,48(3):259-302. optimization by means of quantum annealing.He worked with [2]GONZALES R C,WOODS R C.Digital image processing Wakayama National College of Technology,from 1995 to 2011, [M].Reading,USA:Addison Wiley,1986. where he studied information science and technology related to [3]WINKLER G.Image analysis,random fields and dynamic image processing in the Department of Electrical and Computer Monte Carlo methods:a mathematical introduction M ] Technology.Since 2011,he is working at Gunma National Col- Berlin,Germany:Springer-Verlag,1995. lege of Technology,Maebashi,Japan. [4]NISHIMORI H.Statistical physics of spin glasses and infor mation processing:an introduction[M].Oxford,UK:Ox- AOKI Toshizumi,received a B.S. ford Press,2001. degree in physics from Nagoya University, [5]TANAKA K.Statistical-mechanical approach to image pro- Nagoya,Japan in 1976,an M.S.degree cessing[J].Journal of Physics A:Mathematical and Gener- in physics from Nagoya University in al,2002,35(37):R31-R150. 1978,and a D.Sc.in physics from Nago- [6]ULICHNEY R.Digital halftoning[M].Cambridge,USA: ya University in 1981.He has studied The MIT Press,1987. condensed matter physics,such as low-di- [7]BAYER B E.An optimum method for two-level rendition of mensional quantum systems.Since 1987,he is working at Gun- continuous tone pictures [C]//IEEE Interational Confer- ma National College of Technology,Maebashi,Japan. ence on Communications.New York,USA:IEEE Press