Thee-dimensional Quantum Cellula Neual Netwok and Its Application to Image Pocessing * Sen Wang, Li Cai, Huanqing Cui, Chaowen Feng, Xiaokuo Yang Science College, Ai Foce Engineeing Univesity Xi an 701, China wangsen1998-0@163.com Abstact A thee-dimensional quantum cellula neual netwok is poposed by using the quantum cellula automata as neuon cells. The thee-dimensional quantum cellula neual netwok consists of two layes of quantum cellula automata aay and possesses the A cloning template, B cloning template, and theshold. The image pocessing functions such as hole filling and cone detecting wee pefomed by using the polaization of quantum cellula automata as pixel value and selecting diffeent cloning templates and thesholds. The SIMULINK model is employed to simulate image pocessing functions and the simulation esults demonstate the effectiveness of the poposed quantum cellula neual netwoks. Keywods image pocessing; quantum cellula neual netwok; cellula neual netwok; quantum cellula automata I. INTRODUCTION The eal-time image pocessing of lage size images on a geneal-pupose compute is a time and esouce consuming task. It would be of geat pactical benefit to develop highpaallel data pocessing hadwae[1]. Cellula neual netwoks (CNN) have attacted plenty of inteest as a pomising achitectue fo paallel data pocessing[-3]. Howeve, hadwae implementation of CNN has always been challenging in tems of powe consumption and integation density. The neuon cells in CNN ae mainly constucted by lage numbe of tansistos in pio wok, and esult in high powe consumption and low packing density. A quantum cellula neual netwok (QCNN) model[4-] has been poposed, in which neuon cells ae constucted by quantum cellula automata (QCA)[6]. QCNNs epesent innovative achitectues in the field of CNNs, and possess the advantages of the QCAs and CNNs, whose peculiaities lie in the extemely high packing density, ulta low powe consumption, and simplified inteconnection. Recently, QCNNs ae used to ealize ulta-small chaotic geneatos fo contol and secue communication[7-]. Additionally, in [11], two-dimensional QCNNs ae poposed and applied to image pocessing. Howeve, the image pocessing functions of the two-dimensional QCNNs ae moe simple, because it can only ealize the A cloning template of CNNs. In this pape, A thee-dimensional QCNN is poposed. The new stuctue of the QCNN contains two layes of QCA aay and possesses the A cloning template, B cloning template, and theshold. Hole filling opeation and cone detecting This wok was suppoted by the National Natual Science Foundation of China (gant nos. 61401498), and the Pogam of Shaanxi Povincial Natual Science fo Basic Reseach (gant no. 14JQ8343) opeation ae pefomed by selecting diffeent cloning templates and thesholds. This study could be vey impotant fo futue ulta-high density ealization and applications of CNNs. II. THREE-DIMENSIONAL QUANTUM CELLULAR NEURAL NETWORKS The neuon cells in QCNNs ae constucted by QCAs. A QCA consists of an aay of quantum-dots and tunnel unctions. Quantum-dots ae connected locally by the inteactions of the electons contained in Quantum-dot[6]. Electons can tunnel between the quantum dots in the same cell though tunnel unctions. The electons tend to occupy antipodal sites within the cell due to thei mutual electostatic epulsion, thus an isolated cell possesses two complete polaized states as shown in Fig. 1. The achitectue of QCNN is simila to that of CNN, and the inteaction between neighboing cells is not by connecting wies but by Coulomb foce. Fig. shows the stuctue of thee-dimensional QCNN. In Fig., blocks epesent QCA cells, top QCA aay is input laye, and bottom QCA aay is output laye. The input laye accepts data input and the value of cells ae fixed when QCNNs evolve. The value of output laye cells will vay based on the input and output laye cells. Fig. 1 The two complete polaized states of QCA 978-1-090-07-4/17/$31.00 17 IEEE ICIS 17, May 4-6, 17, Wuhan, China 411
whee is a positive intege denoting the adius of the neighbohood. W ( i, ; k, l ) is the weight of output laye cell C( k, output laye cell C( i, ), (, ;, ) laye cell C( k, l ) to output laye cell (, ) to U i k l is the weight of input C i. Moeove, all cells on the same laye ae assumed to be identical, hence, W ( i, ; k, and U( i, ; k, ae independent of i and. Consequently, the W ( i, ; k, l ) and U( i, ; k, can be expessed as QCNNs can be descibed by a set of state vaiables which contains the classical cell polaization P and a quantum phase angle ϕ. The dynamics of the output laye cell k can be descibed as follows [1] dpk = γ 1 Pk sinϕk dϕk P = EP+ γ cosϕ 1 k k k k Pk whee is the Planck constant, γ is the inte-dot tunneling enegy inside each cell, E k is the electostatic enegy cost of two adacent fully polaized cells having opposite polaization, and P k is the weighted sum of the neighboing cells polaization. Conside an M N thee-dimensional QCNN, having M N cells aanged in M ows and N columns of evey laye. We denote the cell on the ith ow and th column by C( i, ). In thee-dimensional QCNN, the effect of input laye to output laye need to be taken into account, and the state equation of in output laye can be descibed as follows the cell ( ) dp dϕ = a 1 P sinϕ i, i, i, (1) W( i, kl ;, ) Pkl, U( i, klp ;, ) kl, () = i, input whee a Ckl (, ) N( i, ) Ckl (, ) N( i, ) P + ϕ + b i, a cos 1 Pi, = γ, input kl, i, P is the polaization of input laye cell C( k, l ), and b is the theshold of input laye to output laye. N ( i, ) is the neighbohood of cell C( i, ) and is defined as Fig. The stuctue of the thee-dimensional QCNN ( ) max i k, l N ( i, ) = C( k, 1 k M 1 l N (3) (, ;, ) (, ) (, ;, ) (, ) W i k l A k i l U i k l B k i l When =1, the matix A and B ae 3 3 squae aays as shown in Fig. 3, and ae nomally called as A and B cloning templates, espectively. Moeove, the Coulomb inteaction between output laye, is ecipocal, and is weighted by cell C( k and ( ) W ( i, ; k, l ), hence, we have (, ;, ) W ( k, l; i, ) W i k l Fom (4) and (), it can be deduced that (4) = () (, ) (, ) A k i l = A i k l (6) Fom (6), it can be deived that the A cloning template is centosymmetic. III. APPLICATION OF QUANTUM CELLULAR NEURAL NETWORKS TO IMAGE PROCESSING In ode to apply QCNNs to image pocessing, it is needed to implement the following pocedue in advance: a) The pixels ae constucted by cells in the same laye, and the pixel value is epesented by the polaization of the cell. b) Assume that the pixel values -1 and 1 coespond to black and white, espectively, and the pixel values between - 1 and 1 coespond to diffeent gay-scale level. B(-1,-1) B(-1,0) B(-1,1) B(0,-1) B(1,-1) B(0,0) B(1,0) B(0,1) B(1,1) Fig. 3 Cloning templates A cloning template; B cloning template 41
Diffeent image pocessing function can be acquied by chosen diffeent A, B cloning templates and theshold b. In the following numeical calculation, the aay of evey laye is, and othe system paametes ae chosen as: a = 0.0 = 1. A. Simulation of Quantum Cellula Neual Netwoks The output laye cells dynamics ae descibed as (), and is shown in Fig.4. the SIMULINK model of cell ( ) In Fig.4, fom In1 to In8 ae polaizations of neighbohood of output laye cell C( i, ), and fom In9 to In17 ae polaizations of input laye cells. Out1 is the, and Out is the polaization of the output laye cell ( ) quantum phase angle of the output laye cell C( i, ). Theshold is the theshold b, Akl ( k, l = 1,,3) and Bkl ( k, l = 1,,3) ae the A and B cloning templates, espectively. The dynamics behavio of QCNNs can be acquied by connecting the SIMULINK model based on the stuctue of QCNNs. B. Hole Filling Opeation When the A and B cloning templates ae chosen as Fig., and theshold b =, hole filling opeation can be achieved as shown in Fig. 6. The oiginal image is tansmitted to input laye, and the final image is acquied fom output laye. Fom the Fig.6, it can be seen that the inne of 8 was filled by white. C. Cone Detecting Opeation When the theshold b = 60,and the A and B cloning templates ae chosen as Fig. 7, cone detecting opeation can be achieved as shown in Fig. 8 and Fig.9. Fig. Hole filling opeation template A cloning template; B cloning template Fig. 6 Hole filling opeation initial image of output laye; input image of input laye; tansient image of output laye; (d) steady image of output laye. (d) Fig. 4 The SIMULINK model of the output laye cell C( i, ) 413
Fig. 7 Cone detecting opeation template A cloning template; B cloning template Fig. 8 Cone detecting opeation of squae input image of input laye; tansient image of output laye; steady image of output laye. Fig. 9 Cone detecting opeation of diamond input image of input laye; tansient image of output laye; steady image of output laye. In ode to implement cone detecting opeation, the oiginal image is tansmitted to input laye and output laye simultaneously, and the final image is acquied fom output laye. Fom Fig. 8 and Fig. 9, it can be seen that fou cone of the squae and diamond ae detected, espectively. In [11], we poposed a two-dimensional QCNN which only has the A cloning template. In this pape, the poposed theedimensional QCNN possesses A cloning template, B cloning template, and theshold. Compaed with two-dimensional QCNNs, thee-dimensional QCNNs can pefom moe image pocessing functions because of the divesity in choosing cloning templates and thesholds. Moeove, when the B cloning template and theshold ae set to zeo, theedimensional QCNN is tansfomed into a two-dimensional QCNN. IV. CONCLUSIONS This pape poposed a thee-dimensional QCNN by using QCAs as neuon cells. The poposed QCNN contains two layes of QCA aay and intoduces the concepts of A cloning template, B cloning template, and theshold. The functions of the hole filling and cone detecting opeation wee pefomed by using the polaization of QCA cell as pixel value and selecting diffeent cloning templates and thesholds. The SIMULINK model is employed to simulate the image pocessing functions and the simulation esults demonstate the effectiveness of the poposed QCNNs. Additionally, the QCNNs possesses the advantages of the simplified inteconnection, the extemely high packing densities and low powe consumption, and will become a vey impotant egime fo futue evolutions in the field of CNN. The study in this pape povides valuable infomation about QCNNs fo futue application in high-paallel signal pocessing such as image pocessing and patten ecognition. REFERENCES [1] A. Khitun, M. Q. Bao, and K. L. Wang, Magnetic cellula nonlinea netwok with spin wave busfo image pocessing, Supelattices and Micostuctues, vol. 47, pp. 464-483, Novembe. [] L. O. Chua, and L. Yang, Cellula Neual Netwoks: Theoy, IEEE Tansactions on Cicuits and Syestems, vol. 3, pp. 17-173, Octobe 1988. [3] W. Shimoda, and K. Yanai, CNN-based food image segmentation without pixel-wise annotation, Lectue Notes in Compute Science, Vol. 981, pp. 449-47, Januay. [4] G. Tóth, C. S. Lent, P. D. Tougaw, Y. Bazhnik, W. Weng, W. Pood, R. W. Liu, and Y.-F. Huang, Quantum cellula neual netwoks, Supelattices Micostuctues, vol., pp. 473 478, Apil 1996. [] L. Fotuna, M. L. Rosa, D. Nicolosi, and D. Poto, Nanoscale system dynamical behavios: fom quantum-dot-based cell to 1-D aays, IEEE Tansactions on Vey Lage Scale Integation Syetems, vol. 1, pp. 1167-1173, Novembe 04. [6] C. S. Lent, P. D. Tougaw, and G. H. Benstein, Quantum cellula automata, Nanotechnology, vol. 4, pp. 49-7, Januay 1993. [7] S. Wang, L. Cai, Q. Kang, G. Wu, and Q. Li, The chaacteistics of nonlinea chaotic dynamics in quantum cellula neual netwoks, Chinese Physics B, vol. 17, pp. 837-07, August 08. [8] S. Wang, L. Cai, B. Zhang, and H. Y. Zhao, Chaotic function poective synchonization of quantum cellula neual netwok, Miconanoelectonic Technology, vol. 3, pp. 7-1, Januay 16( in Chinese) 414
[9] C. H. Yang, Z. M. Ge, C. M. Chang, and S. Y. Li, Chaos synchonization and chaos contol of quantum-cnn chaotic system by vaiable stuctue contol and impulse contol, Nonlinea Analysis: Real Wold Applications, vol. 11, pp. 1977-198, Novembe. [] S. Wang, L. Cai, N. Zhang, C. W. Feng, and X. K. Yang, Synchonization and multichannel secue communicaiton of quantum cellula neual netwok, Poceedings of the 11th Wold Congess on Intelligent Contol and Automation, Shengyang, China, 14, pp. 4040-4044. [11] S. Wang, L. Cai, Q. Kang, Q. Li, and G. Wu, The two-demensional quantum cellula neual netwok and its applications to image pocessing, Reseach and Pogess of solid State Electonics, vol. 8, pp. 340-34, Septembe 08( in Chinese). [1] Á. I. Csugay, W. Pood, and C. S. Lent, Signal pocessing with neaneighbo-coupled time-vaying quantum-dot aays, IEEE Tansactions on Cicuits and Systems-I: Fundamental Theoy and Applications, vol. 47, pp. 11-13, August 00. 4