Analogic cellular PDE machines

This paper gives an overview on analogic cellular array architectures that can also be used to approximate partial differential equations (PDEs). Cellular arrays are massively parallel computing structures composed of cells placed on a regular grid. These cells interact locally an th e array can have both local and global dynamics. The software of this architecture is an analogic algorithm that builds on analog and logical spatio-temporal instructions of the underlying hardware, that is a locally connected cellular nonlinear network (CNN). Within this framework two classes of PDEs, motivated also by image processing methodologies will be discussed: (i) reaction-diffusion (local) types and (ii) contrast modification (global) types. It will be shown that based on cellular diffusion and wave-computing formulations these classes can be approximated on existing CNN Universal Machine (CNN-UM) chips. Thus, the last generation of stored program topographic array microprocessors with integrated sensing and computing could also be viewed as the first prototypes of analogic cellular PDE machines implemented on silicon.

[1]  Valeri Mladenov,et al.  Cellular Neural Networks: Theory And Applications , 2004 .

[2]  Carlo Rovelli,et al.  Discreteness of area and volume in quantum gravity [Nucl. Phys. B 442 (1995) 593] , 1994, gr-qc/9411005.

[3]  Leon O. Chua,et al.  The CNN paradigm , 1993 .

[4]  Tamás Roska,et al.  Solving partial differential equations by CNN , 1993 .

[5]  Vadim N. Biktashev,et al.  Autowave principles for parallel image processing , 1991 .

[6]  A. Rodriguez-Vazquez,et al.  A 64/spl times/64 CNN universal chip with analog and digital I/O , 1998, 1998 IEEE International Conference on Electronics, Circuits and Systems. Surfing the Waves of Science and Technology (Cat. No.98EX196).

[7]  Leon O. Chua,et al.  Characterization and dynamics of pattern formation in cellular neural networks , 1996 .

[8]  Petros Maragos,et al.  Differential morphology and image processing , 1996, IEEE Trans. Image Process..

[9]  Csaba Rekeczky,et al.  PDE-Based Histogram Modification With Embedded Morphological Processing Of The Level-Sets , 2003, J. Circuits Syst. Comput..

[10]  L. Chua,et al.  Simulating nonlinear waves and partial differential equations via CNN. I. Basic techniques , 1995 .

[11]  A. Dawidziuk,et al.  Minimum size 0.5 micron CMOS programmable 48 by 48 CNN test chip , 1997 .

[12]  Leon O. Chua,et al.  CNN: A Vision of Complexity , 1997 .

[13]  Tamás Roska,et al.  CNN-based spatio-temporal nonlinear filtering and endocardial boundary detection in echocardiography , 1999 .

[14]  Marco Gilli,et al.  On Stability of Cellular Neural Networks , 1999, J. VLSI Signal Process..

[15]  Guillermo Sapiro,et al.  Shape preserving local histogram modification , 1999, IEEE Trans. Image Process..

[16]  T. Roska,et al.  A double time-scale CNN for solving 2-D Navier-Stokes equations , 1994, Proceedings of the Third IEEE International Workshop on Cellular Neural Networks and their Applications (CNNA-94).

[17]  S. Wolfram,et al.  Two-dimensional cellular automata , 1985 .

[18]  Leon O. Chua,et al.  The CNN is universal as the Turing machine , 1993 .

[19]  Guillermo Sapiro,et al.  Contrast Enhancement via Image Evolution Flow , 1997, CVGIP Graph. Model. Image Process..

[20]  Tamás Roska,et al.  The CNN universal machine: an analogic array computer , 1993 .

[21]  G. Sapiro,et al.  Histogram Modification via Differential Equations , 1997 .

[22]  A.G. Radvanyi.  On the rectangular grid representation of general CNN networks , 2000, Proceedings of the 2000 6th IEEE International Workshop on Cellular Neural Networks and their Applications (CNNA 2000) (Cat. No.00TH8509).

[23]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[24]  Tommaso Toffoli,et al.  Computation and Construction Universality of Reversible Cellular Automata , 1977, J. Comput. Syst. Sci..

[25]  Tamás Roska,et al.  A Nonlinear Wave Metric and its CNN Implementation for Object Classification , 1999, J. VLSI Signal Process..

[26]  James P. Keener,et al.  Propagation and its failure in coupled systems of discrete excitable cells , 1987 .

[27]  Tamás Roska,et al.  A DOUBLE TIME—SCALE CNN FOR SOLVING TWO‐DIMENSIONAL NAVIER—STOKES EQUATIONS , 1996 .

[28]  Lin-Bao Yang,et al.  Cellular neural networks: theory , 1988 .

[29]  Leon O. Chua,et al.  Pattern formation properties of autonomous Cellular Neural Networks , 1995 .

[30]  Tamás Roska,et al.  The CNN universal machine , 1993 .

[31]  Ángel Rodríguez-Vázquez,et al.  A CNN UNIVERSAL CHIP IN CMOS TECHNOLOGY , 1996 .

[32]  Leon O. Chua,et al.  The analogic cellular neural network as a bionic eye , 1995, Int. J. Circuit Theory Appl..

[33]  Tamás Roska,et al.  CNN‐based spatio‐temporal nonlinear filtering and endocardial boundary detection in echocardiography , 1999, Int. J. Circuit Theory Appl..

[34]  P. Rabinowitz.  Russian Numerical Analysis: Approximate Methods of Higher Analysis . L. V. Kantorovich and V. I. Krylov. Translated from the third Russian edition by Curtis D. Benster. Interscience, New York, 1959. xv + 681. $17. , 1961, Science.

[35]  Leon O. Chua,et al.  Computing with Front Propagation: Active Contour And Skeleton Models In Continuous-Time CNN , 1999, J. VLSI Signal Process..

[36]  L. Chua,et al.  Simulating nonlinear waves and partial differential equations via CNN. II. Typical examples , 1995 .

[37]  Tommaso Toffoli,et al.  Cellular Automata as an Alternative to (Rather than an Approximation of) Differential Equations in M , 1984 .

[38]  L. Kantorovich,et al.  Approximate methods of higher analysis , 1960 .