BIOLOGICALLY INSPIRED FILTERS UTILIZING SPECTRAL PROPERTIES OF TOEPLITZ-BLOCK-TOEPLITZ MATRICES

Authors

  • Dragan Vidacic
  • Richard A. Messner

DOI:

https://doi.org/10.47839/ijc.14.4.820

Keywords:

Lateral Inhibition Excitation, Recurrent Neural Networks, Toeplitz-Block-Toeplitz Matrices.

Abstract

The construction of filters arising from linear neural networks with feed-backward excitatory-inhibitory connections is presented. Spatially invariant coupling between neurons and the distribution of neuron-receptor units in the form of a uniform square grid yield the TBT (Toeplitz-Block-Toeplitz) connection matrix. Utilizing the relationship between spectral properties of such matrices and their generating functions, the method for construction of recurrent linear networks is addressed. By appropriately bounding the generating function, the connection matrix eigenvalues are kept in the desired range allowing for large matrix inverse to be approximated by a convergent power series. Instead of matrix inversion, the single pass convolution with the filter obtained from the network connection weights is applied when solving the network. For the case of inter-neuron coupling in the form of a function that is expandable in a Fourier series in polar angle, the network response filter is shown to be steerable.

References

H.K. Hartline, F. Ratliff, Inhibitory interaction of receptor units in the eye of Limulus, J. Gen. Physiol., (40) 3 (1957), pp. 357-376.

H.K. Hartline, F. Ratliff, Spatial summation of inhibitory influences in the eye of Limulus, and the mutual interaction of receptor units, J. Gen. Physiol., (41) 5 (1958), pp. 1049-1066.

B.S. Gutkin, C.E. Smith, Conditions for noise reduction and stable encoding of spatial structure by cortical neural networks, Biol. Cybern., (82) 6 (2000), pp. 469-475.

G.G. Furman, Comparison of models for subtractive and shunting lateral-inhibition in receptor-neuron fields, Biol. Cybern., (2) 6 (1965), pp. 257-274.

R.A. Messner, Smart Visual Sensors for Real-Time Image Processing and Pattern Recognition Based Upon Human Visual System Characteristics, PhD Dissertation, Clarkson University, 1984.

H.H. Szu, R.A. Messner, Adaptive invariant novelty filters, Proc. IEEE, (74) 3 (1986), pp. 518-519.

S.E. Palmer, Vision Science Photons to Phenomenology, A Bradford Book, The MIT Press, Cambridge, Massachusetts, London, England, 1999.

M.G. Luniewicz, R.A. Messner, Effects of lateral subtractive inhibition within the context of a polar-log spatial coordinate mapping, in: D.P. Casasent (Ed.), Proc. SPIE Intelligent Robots and Computer Vision VII, 1988, vol. 1002, pp. 58-65.

Y. Yu, T. Yamauchi, Y. Choe, Explaining low-level brightness-contrast illusions using disinhibition, in A.J. Ijspeert, M. Murata, N. Wakamiya (Eds.), Biologically Inspired Approaches to Advanced Information Technology: First International Workshop, BioADIT 2004, Lausanne, Switzerland, 2004, Revised Selected Papers, Lecture Notes in Computer Science, 2004, vol. 3141, pp. 166-175.

Y. Yu, Y. Choe, Angular disinhibition effect in a modified Poggendorff illusion, in K.D. Forbus, D. Gentner, T. Regier (Eds.), Proceedings of the 26th Annual Conference of the Cognitive Science Society, Mahwah, NJ, USA, (2004), pp. 1500-1505.

M. Wax, T. Kailath, Efficient inversion of Toeplitz-block Toeplitz matrix, IEEE Trans. Acoust, Speech Signal Process., (31) 5 (1983), pp. 1218-1221.

N. Kalouptsidis, G. Carayannis, and D. Manolakis, On block matrices with elements of special structure, Proc. IEEE Int. Conf. on Acoustics, Speech and Signal Processing, ICASSP’82, Paris, France, (May 1982), vol. 7, pp. 1744-1747.

A.E. Yagle, A fast algorithm for Toeplitz-block-Toeplitz linear systems, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP’01, Salt Lake City, USA, (2001), vol. 3, pp. 1929-1932.

S.J. Reeves, Fast algorithm for solving block banded Toeplitz systems with banded Toeplitz blocks, Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, ICASSP’02, Orlando, USA, (2002), vol. 4, pp. 3325-3328.

D.A. Bini, B. Meini, Solving block banded block Toeplitz systems with banded Toeplitz blocks, in F.T. Luk (Ed.), Proc. SPIE, Advanced Signal Processing Algorithms, Architectures, and Implementations IX, 1999, vol. 3807 pp. 300-311.

U. Grenander, G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea Publishing, New York, 1984.

S. Serra, Preconditioning strategies for asymptotically ill conditioned block Toeplitz systems, BIT Num. Math., (34) (1994), pp. 579-594.

P. Tilli, On the asymptotic spectrum of Hermitian block Toeplitz matrices with Toeplitz blocks, Math. Comp., (66) 219 (1997), pp. 1147-1159.

W.T. Freeman, H.H. Adelson, The design and use of steerable filters, IEEE Trans. Pattern Anal. Mach. Intell., (13) 9 (1991), pp. 891-906.

S. Barnet, Matrices: Methods and Applications, Oxford University Press, New York, 1990.

R.A. Young, L.M. Lesperance, W.W. Meyer, The Gaussian derivative model for spatial-temporal vision: I. Cortical model, Spat. Vis., (14) 3,4 (2001), pp. 261-319.

E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, New Jersey, 1971.

J.B. Price, M.H. Hayes, Steerable filter cascades, Proc. IEEE Intentional Conference on Image Processing ICIP’99, Kobe, Japan, (October 1999), vol. 2, pp. 880-884.

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Published

2015-12-28

How to Cite

Vidacic, D., & Messner, R. A. (2015). BIOLOGICALLY INSPIRED FILTERS UTILIZING SPECTRAL PROPERTIES OF TOEPLITZ-BLOCK-TOEPLITZ MATRICES. International Journal of Computing, 14(4), 198-207. https://doi.org/10.47839/ijc.14.4.820

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