This Article 
 Bibliographic References 
 Add to: 
Concurrent Error Detection in Wavelet Lifting Transforms
October 2004 (vol. 53 no. 10)
pp. 1291-1302
Wavelet transforms, central to multiresolution signal analysis and important in the JPEG2000 image compression standard, are quite susceptible to computer-induced errors because of their pipelined structure and multirate processing requirements. Such errors emanate from computer hardware, software bugs, or radiation effects from the surrounding environment. Implementations use lifting schemes, which employ update and prediction estimation stages, and can spread a single numerical error caused by failures to many output transform coefficients without any features to warn data users. This paper proposes an efficient method to detect the arithmetic errors using weighted sums of the wavelet coefficients at the output compared with an equivalent parity value derived from the input data. Two parity values may straddle a complete multistage transform or several values may be used, each pair covering a single stage. There is greater error-detecting capability at only a slight increase in complexity when parity pairs are interspersed between stages. With the parity weighting design scheme, a single error introduced at a lifting section can be detected. The parity computation operation is properly viewed as an inner product between weighting values and the data, motivating the use of dual space functionals related to the error gain matrices. The parity weighting values are generated by a combination of dual space functionals. An iterative procedure for evaluating the design of the parity weights has been incorporated in Matlab code and simulation results are presented.

[1] D.S. Taubman and M.W. Marcellin;, JPEG2000 Image Compression Fundamentals, Standards and Practice. Boston: Kluwer Academic, 2002.
[2] Int'l Org. for Standardization (ISO) and Int'l Electrotechnical Commission (IEC), ISO/IEC 15444-1 International Standard Information Technology-JPEG2000-Image Coding System. Geneva: ISO Copyright Office, 2000.
[3] C.S. Burrus, R.A. Gopinath, and H. Guo, Introduction to Wavelets and Wavelet Transforms, A Primer. Upper Saddle River, N.J.: Prentice Hall, 1998.
[4] C. Constantinescu, Trends and Challenges in VLSI Circuit Reliability IEEE Micro, vol. 23, pp.14-19, July/Aug. 2003.
[5] P. Shivakumar et al., "Modeling the Effect of Technology Trends on the Soft Error Rate of Combinatorial Logic," Proc. Int'l Conf. Dependable Systems and Networks, IEEE CS Press, 2000, pp. 389-398.
[6] N. Seifert, Z. Xiaowei, and L. W. Massengill, Impact of Scaling on Soft-Error Rates in Commercial Microprocessors IEEE Trans. Nuclear Science, vol. 49, no. 6,pt. 1, pp. 3100-3106, Dec. 2002.
[7] J.Y. Jou and J.A. Abraham, "Fault Tolerant FFT Networks," IEEE Trans. Computers, Vol. 37, May 1988, pp. 548-561.
[8] M. Tsunoyama and S. Naito, A Fault-Tolerant FFT Processor Digest of Papers, 21st Int'l Symp. Fault-Tolerant Computing (FTCS-21), pp. 128-135, June 1991.
[9] D.L. Tao, C.R.P. Hartmann, and Y.S. Chen, A Novel Concurrent Error Detection Scheme for FFT Networks IEEE Trans. Parallel and Distributed Systems, vol. 4, no. 2, pp. 198-221, Feb. 1993.
[10] F. Lombardi and J.C. Muzio, Concurrent Error Detection and Fault Location in an FFT Architecture IEEE J. Solid-State Circuits, vol. 27, pp. 728-736, May 1992.
[11] C.G. Oh and H.Y. Youn, “On Concurrent Error Detection, Location, and Correction of FFT Network,” Proc. 23rd IEEE Fault-Tolerant Computing Symp. (FTCS-23), pp. 596-605, June 1993.
[12] G.G. Oh, H.Y. Youn, and V.K. Raj, An Efficient Algorithm-Based Concurrent Error Detection for FFT Networks IEEE Trans. Computers, vol. 44, no. 9, pp. 1157-1162, Sept. 1995.
[13] S.J. Wang and N.K. Jha, Algorithm-Based Fault Tolerance for FFT Networks IEEE Trans. Computers, vol. 43, no. 7, pp. 849-854, July 1994.
[14] G.R. Redinbo, Concurrent Error Detection in Fast Unitary Transform Algorithms Proc. Int'l Conf. Dependable Systems and Networks, pp. 37-46, June 2001.
[15] J. Daubechies and W. Sweldens, Factoring Wavelet Transforms into Lifting Steps J. Fourier Analysis and Applications, vol. 4, pp. 247-269, 1998.
[16] A. Jensen, A. la Cour-Harbo, Ripples in Mathematics The Discrete Wavelet Transform. Berlin: Springer-Verlag, 2001.
[17] G.R. Redinbo, “Generalized Algorithm-Based Fault Tolerance: Error Correction via Kalman Estimation,” IEEE Trans. Computers, vol. 47, no. 6, pp. 639-655, June 1998.
[18] W.F. Chang and C.W. Wu, Low-Cost Modular Totally Self-Checking Checker Design for m-out-of-n Code IEEE Trans. Computers, vol. 48, no. 8, pp. 815-826, Aug. 1999.
[19] I.N. Herstein, Topics in Algebra, second ed. New York: John Wiley&Sons, 1975.
[20] J.L. Goldberg, Matrix Theory with Applications. New York: McGraw-Hill, 1991.

Index Terms:
Wavelet transforms, concurrent error detection, weighted sum parity, real number codes, algorithm-based fault tolerance, error gain matrices, dual space functionals, biorthogonality.
G. Robert Redinbo, Cung Nguyen, "Concurrent Error Detection in Wavelet Lifting Transforms," IEEE Transactions on Computers, vol. 53, no. 10, pp. 1291-1302, Oct. 2004, doi:10.1109/TC.2004.74
Usage of this product signifies your acceptance of the Terms of Use.