Enhanced Gram-Schmidt Process for Improving the Stability in Signal and Image Processing
The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors) into an orthonormal basis (a set of orthogonal, unit-length vectors). The process consists of taking each vector and then subtracting the
elements in common with the previous vectors. This paper introduces an Enhanced version of the Gram-Schmidt Process (EGSP) with inverse, which is useful for signal and image processing applications.
<p>[1] S. Haykin, Neural Networks: A Comprehensive Foundation. New York:
Macmillan, 1994.
[2] S. Haykin, Modern Filters. New York: Macmillan, 1990.
[3] S. Haykin, Adaptive Filter Theory. New Jersey: Prentice-Hall, 1991.
[4] H Lodhi and Y. Guo, “Gram-Schmidt kernels applied to structure
activity analysis for drug design”, in Proc. of 2nd. Workshop on
Datamining in Bioinformatics, Edmonton, Alberta, Canada, July 23rd.
2002, pp. 37-42.
[5] R. A. Mozingo and T. W. Miller, Introduction to Adaptive Arrays. New
York: Wiley, 1980.
[6] R. T. Compton, Adaptive Antennas: Concepts and Performance. New
Jersey: Prentice-Hall, 1988.
[7] G. W. Stewart, Introduction to Matrix Computations. Orlando:
Academic Press, 1973.
[8] J. Makhoul, “A class of all-zero lattice digital filters: properties and
applications,” IEEE Trans. Acoust. Speech and Signal Process., vol.
ASSP-26, pp.304-314, 1978.
[9] N. Ahmed and D. H. Youn, “On a realization and related algorithm for
adaptive prediction,” IEEE Trans. Acoust. Speech and Signal Process.,
vol. ASSP-28, pp.493-4974, 1980.
[10] K. A. Gallivan and C. E. Leiserson, “High-performance architectures for
adaptive filtering based on the Gram-Schmidt algorithm,” SPIE, Real
Time Signal Process. VII, vol.495, pp.30-36, 1984.
[11] M. Mastriani, “Self-Restorable Stochastic Neuro-Estimation using
Forward-Propagation Training Algorithm,” in Proc. of INNS, Portland,
OR, July 1993, vol. 1, pp. 404-411.
[12] M. Mastriani, “Self-Restorable Stochastic Neurocontrol using Back-
Propagation and Forward-Propagation Training Algorithms,” in Proc. of
ICSPAT, Santa Clara, CA, Sept. 1993, vol. 2, pp.1058-1071.
[13] M. Mastriani, “Pattern Recognition Using a Faster New Algorithm for
Training Feed-Forward Neural Networks,” in Proc. of INNS, San Diego,
CA, June 1994, vol. 3, pp. 601-606.
[14] M. Mastriani, “Predictor of Linear Output,” in Proc. IEEE Int.
Symposium on Industrial Electronics, Santiago, Chile, May 1994,
pp.269-274.
[15] M. Mastriani, “Predictor-Corrector Neural Network for doing Technical
Analysis in the Capital Market,” in Proc. AAAI International
Symposium on Artificial Intelligence, Monterrey, México, October 1994,
pp. 49-58.
[16] N. Al-Dhahir and J. M. Cioffi, "Efficiently computed reduced-parameter
input-aided MMSE equalizers for ML detection: A unified approach,"
IEEE Trans. on Info. Theory, vol. 42, pp. 903-915, May 1996.
[17] P. J. W. Melsa, R. C. Younce, and C. E. Rhors, "Impulse response
shortening for discrete multitone transceivers," IEEE Trans. on
Communications, vol. 44, pp. 1662-1672, Dec. 1996.
[18] N. Al-Dhahir and J. M. Cioffi, "Optimum finite-length equalization for
multicarrier transceivers," IEEE Trans. on Communications, vol. 44, pp.
56-63, Jan. 1996.
[19] B. Lu , L. D. Clark, G. Arslan, and B. L. Evans, "Divide-and-conquer
and matrix pencil methods for discrete multitone equalization," IEEE
Trans. on Signal Processing, in revision.
[20] G. Arslan , B. L. Evans, and S. Kiaei, "Equalization for Discrete
Multitone Receivers To Maximize Bit Rate," IEEE Trans. on Signal
Processing, vol. 49, no. 12, pp. 3123-3135, Dec. 2001.
[21] B. Farhang-Boroujeny and Ming Ding, "Design Methods for Time
Domain Equalizer in DMT Transceivers", IEEE Trans. on
Communications, vol. 49, pp. 554-562, March 2001.
[22] J. M. Cioffi, A Multicarrier Primer. Amati Communication Corporation
and Stanford University, T1E1.4/97-157, Nov. 1991.
[23] M. Ding, A. J. Redfern, and B. L. Evans, "A Dual-path TEQ Structure
For DMT-ADSL Systems", Proc. IEEE Int. Conf. on Acoustics, Speech,
and Signal Proc., May 13-17, 2002, Orlando, FL, accepted for
publication.
[24] K. V. Acker, G. Leus, M. Moonen, O. van de Wiel, and T. Pollet, "Per
tone equalization for DMT-based systems," IEEE Trans. on
Communications, vol. 49, no. 1, pp. 109-119, Jan 2001.
[25] S. Trautmann, N.J. Fliege, “A new equalizer for multitone systems
without guard time” IEEE Communications Letters, Vol.6, No. 1 pp. 34
-36, Jan 2002.
[26] M. Tuma and M. Benzi, “A robust preconditioning technique for large
sparse least squares problems”, in Proc. of XV. Householder
Symposium, Peebles, Scotland, June 17-21, 2002. pp.8
[27] H. J. Pedersen, “Random Quantum Billiards”, Master dissertation,
K0benhavns Universitet, Copenhagen, Denmark, july 1997, pp.63
[28] G. H. Golub and C. F. van Loan, Matrix Computations, (3rd Edition)
Johns Hopkins University Press, Baltimore, 1996.
[29] G. Hadley, Linear Algebra. Mass.: Wesley, 1961.
[30] G. Strang, Linear Algebra and Its Applications. New York: Academic
Press, 1980.
[31] S. J. Leon, “Linear Algebra with Applications” Maxwell MacMillan,
New York, 1990.</p>
<p>[1] S. Haykin, Neural Networks: A Comprehensive Foundation. New York:
Macmillan, 1994.
[2] S. Haykin, Modern Filters. New York: Macmillan, 1990.
[3] S. Haykin, Adaptive Filter Theory. New Jersey: Prentice-Hall, 1991.
[4] H Lodhi and Y. Guo, “Gram-Schmidt kernels applied to structure
activity analysis for drug design”, in Proc. of 2nd. Workshop on
Datamining in Bioinformatics, Edmonton, Alberta, Canada, July 23rd.
2002, pp. 37-42.
[5] R. A. Mozingo and T. W. Miller, Introduction to Adaptive Arrays. New
York: Wiley, 1980.
[6] R. T. Compton, Adaptive Antennas: Concepts and Performance. New
Jersey: Prentice-Hall, 1988.
[7] G. W. Stewart, Introduction to Matrix Computations. Orlando:
Academic Press, 1973.
[8] J. Makhoul, “A class of all-zero lattice digital filters: properties and
applications,” IEEE Trans. Acoust. Speech and Signal Process., vol.
ASSP-26, pp.304-314, 1978.
[9] N. Ahmed and D. H. Youn, “On a realization and related algorithm for
adaptive prediction,” IEEE Trans. Acoust. Speech and Signal Process.,
vol. ASSP-28, pp.493-4974, 1980.
[10] K. A. Gallivan and C. E. Leiserson, “High-performance architectures for
adaptive filtering based on the Gram-Schmidt algorithm,” SPIE, Real
Time Signal Process. VII, vol.495, pp.30-36, 1984.
[11] M. Mastriani, “Self-Restorable Stochastic Neuro-Estimation using
Forward-Propagation Training Algorithm,” in Proc. of INNS, Portland,
OR, July 1993, vol. 1, pp. 404-411.
[12] M. Mastriani, “Self-Restorable Stochastic Neurocontrol using Back-
Propagation and Forward-Propagation Training Algorithms,” in Proc. of
ICSPAT, Santa Clara, CA, Sept. 1993, vol. 2, pp.1058-1071.
[13] M. Mastriani, “Pattern Recognition Using a Faster New Algorithm for
Training Feed-Forward Neural Networks,” in Proc. of INNS, San Diego,
CA, June 1994, vol. 3, pp. 601-606.
[14] M. Mastriani, “Predictor of Linear Output,” in Proc. IEEE Int.
Symposium on Industrial Electronics, Santiago, Chile, May 1994,
pp.269-274.
[15] M. Mastriani, “Predictor-Corrector Neural Network for doing Technical
Analysis in the Capital Market,” in Proc. AAAI International
Symposium on Artificial Intelligence, Monterrey, México, October 1994,
pp. 49-58.
[16] N. Al-Dhahir and J. M. Cioffi, "Efficiently computed reduced-parameter
input-aided MMSE equalizers for ML detection: A unified approach,"
IEEE Trans. on Info. Theory, vol. 42, pp. 903-915, May 1996.
[17] P. J. W. Melsa, R. C. Younce, and C. E. Rhors, "Impulse response
shortening for discrete multitone transceivers," IEEE Trans. on
Communications, vol. 44, pp. 1662-1672, Dec. 1996.
[18] N. Al-Dhahir and J. M. Cioffi, "Optimum finite-length equalization for
multicarrier transceivers," IEEE Trans. on Communications, vol. 44, pp.
56-63, Jan. 1996.
[19] B. Lu , L. D. Clark, G. Arslan, and B. L. Evans, "Divide-and-conquer
and matrix pencil methods for discrete multitone equalization," IEEE
Trans. on Signal Processing, in revision.
[20] G. Arslan , B. L. Evans, and S. Kiaei, "Equalization for Discrete
Multitone Receivers To Maximize Bit Rate," IEEE Trans. on Signal
Processing, vol. 49, no. 12, pp. 3123-3135, Dec. 2001.
[21] B. Farhang-Boroujeny and Ming Ding, "Design Methods for Time
Domain Equalizer in DMT Transceivers", IEEE Trans. on
Communications, vol. 49, pp. 554-562, March 2001.
[22] J. M. Cioffi, A Multicarrier Primer. Amati Communication Corporation
and Stanford University, T1E1.4/97-157, Nov. 1991.
[23] M. Ding, A. J. Redfern, and B. L. Evans, "A Dual-path TEQ Structure
For DMT-ADSL Systems", Proc. IEEE Int. Conf. on Acoustics, Speech,
and Signal Proc., May 13-17, 2002, Orlando, FL, accepted for
publication.
[24] K. V. Acker, G. Leus, M. Moonen, O. van de Wiel, and T. Pollet, "Per
tone equalization for DMT-based systems," IEEE Trans. on
Communications, vol. 49, no. 1, pp. 109-119, Jan 2001.
[25] S. Trautmann, N.J. Fliege, “A new equalizer for multitone systems
without guard time” IEEE Communications Letters, Vol.6, No. 1 pp. 34
-36, Jan 2002.
[26] M. Tuma and M. Benzi, “A robust preconditioning technique for large
sparse least squares problems”, in Proc. of XV. Householder
Symposium, Peebles, Scotland, June 17-21, 2002. pp.8
[27] H. J. Pedersen, “Random Quantum Billiards”, Master dissertation,
K0benhavns Universitet, Copenhagen, Denmark, july 1997, pp.63
[28] G. H. Golub and C. F. van Loan, Matrix Computations, (3rd Edition)
Johns Hopkins University Press, Baltimore, 1996.
[29] G. Hadley, Linear Algebra. Mass.: Wesley, 1961.
[30] G. Strang, Linear Algebra and Its Applications. New York: Academic
Press, 1980.
[31] S. J. Leon, “Linear Algebra with Applications” Maxwell MacMillan,
New York, 1990.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65317", author = "Mario Mastriani and Marcelo Naiouf", title = "Enhanced Gram-Schmidt Process for Improving the Stability in Signal and Image Processing", abstract = "The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors) into an orthonormal basis (a set of orthogonal, unit-length vectors). The process consists of taking each vector and then subtracting the
elements in common with the previous vectors. This paper introduces an Enhanced version of the Gram-Schmidt Process (EGSP) with inverse, which is useful for signal and image processing applications.
", keywords = "Digital filters, digital signal and image processing,
Gram-Schmidt Process, orthonormalization. ", volume = "7", number = "3", pages = "506-5", }
