# FPGA-BASED NORMALIZATION FOR MODIFIED GRAM-SCHMIDT ORTHOGONALIZATION

### I. Sajid, Sotirios G. Ziavras, M. M. Ahmed

#### Abstract

Eigen values evaluation is an integral but computation-intensive part for many image and signal processing applications. Modified Gram-Schmidt Orthogonalization (MGSO) is an efficient method for evaluating the Eigen values in face recognition algorithms. MGSO applies normalization of vectors in its iterative orthogonal process and its accuracy depends on the accuracy of normalization. Using software, floating-point data types and floating-point operations are applied to minimize rounding and truncation effects. Hardware support for floating-point operations may be very costly in execution time per operation and also may increase power consumption. In contrast, lower-cost fixed-point arithmetic reduces execution times and lowers the power consumption but reduces slightly the precision. Normalization involves square root operations in addition to other arithmetic operations. Hardware realization of the floating-point square root operation may be prohibitively expensive because of its complexity. This paper presents three architectures, namely ppc405, ppc_ip and pc_pci, that employ fixed-point hardware for the efficient implementation of normalization on an FPGA. We evaluate the suitability of these architectures based on the needed frequency of normalization. The proposed architectures produce a less than 10-3 error rate compared with their software-driven counterpart for implementing floating-point operations. Furthermore, four popular databases of faces are used to benchmark the proposed architectures.

#### References

- Benkrid, K., Crookes, D., Bouridane, A., Con, P., and Alotaibi, K., 1999, A High Level Software Environment for FPGA Based Image Processing, Image Proc. And Its Applications. Seventh Inter. Conf., Manchester, U.K.
- Burden, R. L., and Faires, J. D., 2005. Numerical Analysis, seventh ed., Thomson.
- Chin-Chin, H., Shin-Ichi, Y., 1996. Hideji Fujika, W., and Koichiro, S., A Fuzzy Self-Tuning Parallel Genetic Algorithm For Optimization, Computers ind. Engng vol. 30, no. 4.
- Gavish, B. and Sridhar,S., 2006. “Computing the 2- median on tree networks in O (n lg n) time, Inter. Jour. of networks, vol. 26, issue 4, Wiley InterScience.
- Giraud, L., Langou, J. and Rozloznik, M., 2003. On the loss of orthogonality in the Gram-Schmidt orthogonalization process, Technical Report TR/PA/03/25, CERFACS.
- He, X., Yan, S., Hu, Y., Niyogi, P., and Zhang, H., 2005. Face Recognition Using Laplacian faces”, IEEE Trans. on Pat. Anal and Machine Intelligence, vol. 27, no. 3.
- IBM 64-bit processor local bus architecture specification version 3.5, patent no. SA-14-2534-01, May, 2001.
- Liao, J.R., 2000. Real-Time Image Reconstruction for Spiral MRI Using Fixed-Point Calculation, IEEE Trans. on Medical Imaging, vol. 19, no.7.
- Niklas, P., Franz-Erich, W. and Martin, R., 2007. Laplace spectra as fingerprints for image recognition, Computer Aided Design, vol. 39.
- Oberstar, E. L., 2007. Fixed-point representation & fractional math, Report Oberstar Consulting.
- Ortega, J.M., 1963. An Error Analysis of Householder's Method for the Symmetric Eigenvalue Problem”, Numerische Mathematik, 1-225.
- Peter, S. and Mirian, L., 1996. Area and Performance Tradeoffs in Floating-point Divide and Square-Root Implementations, ACM Comp. Sur., vol. 28, no. 3.
- Piromsopa, K., Aporntewan, C., and Chongsatitvatana, P., 2009. An FPGA Implementation of a Fixed-point Square Root Operation, ISCIT, 2001.
- Phillips, P. J., Moon, H., Rizvi, S. A., and Rauss, P. J, 2000. The Feret Evaluation Methodology for FaceRecognition Algorithms”, IEEE Trans. Pat. Anal. Mach. Intell., vol. 22, no. 10.
- Sajid, I., Ahmed, M. M. and Sageer, M., 2010 PGA based optimized architecture for face recognition using fixed point Householder algorithm, Acceptance for publication, VI South. Prog. Logic Conf., Brazil.
- Sajid, I., Ahmed, M. M., and Taj, I., 2009. Time Efficient Face Recognition Using Stable Gram-Schmidt Orthonormalization, Inter. Jour of Signal and Image Proc. and Pat. vol. 1, no.2.
- Sajid, I., Ahmed, M. M., Taj, I., Humayun, M., 2008. Design of High Performance FPGA based Face Recognition System, Proc. of Prog. In Electro.
- Stavros, P, Peter, L, and Miroslaw, B, 2003. An FPGA System for the High Speed Extraction, Normalization and Classification of Moment Descriptors, Lecture notes in computer science, 2003 - Springer.
- Sharma, A., and Paliwal, K. K., 2007. Fast principal component analysis using fixed-point algorithm, Patt. Recog. Letters, vol. 28, no. 10.
- The Database of Faces at a glance, 2009.
- Wilkinson, J. H., 1962. Error Analysis of Eigenvalue Techniques Based on Orthogonal Transformations, Journal of the Society for Industrial and Applied Mathematics, vol. 10, no. 1.
- Wen,G., Bo, C., Shiguang, S., Xilin, C., Delong, Z., Xiaohua, Z., and Debin, Z., 2008. The CAS-PEAL Large-Scale Chinese Face Database and Baseline Evaluations, IEEE Trans. Systems, man, and cybernetics-part a: systems and humans, vol. 38, no. 1, JANUARY http://www.jdl.ac.cn/peal/index.html
- Yamin, L., and Wanming, C., 1997. Implementation of Single Precision Floating Point Square Root on FPGAs, Proceeding of the 5th IEEE symposium on FPGA-based custom computing Machines, April http://www.superkits.net/whitepapers.htm.
- Yale Univ. Face Database, 2002

#### Paper Citation

#### in Harvard Style

Sajid I., G. Ziavras S. and M. Ahmed M. (2010). **FPGA-BASED NORMALIZATION FOR MODIFIED GRAM-SCHMIDT ORTHOGONALIZATION** . In *Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)* ISBN 978-989-674-029-0, pages 227-232. DOI: 10.5220/0002848702270232

#### in Bibtex Style

@conference{visapp10,

author={I. Sajid and Sotirios G. Ziavras and M. M. Ahmed},

title={FPGA-BASED NORMALIZATION FOR MODIFIED GRAM-SCHMIDT ORTHOGONALIZATION},

booktitle={Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)},

year={2010},

pages={227-232},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0002848702270232},

isbn={978-989-674-029-0},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the International Conference on Computer Vision Theory and Applications - Volume 2: VISAPP, (VISIGRAPP 2010)

TI - FPGA-BASED NORMALIZATION FOR MODIFIED GRAM-SCHMIDT ORTHOGONALIZATION

SN - 978-989-674-029-0

AU - Sajid I.

AU - G. Ziavras S.

AU - M. Ahmed M.

PY - 2010

SP - 227

EP - 232

DO - 10.5220/0002848702270232