《A Translation-Invariant Wavelet》.pdf

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 2, FEBRUARY 1996 225 A Translation-Invariant Wavelet Representation Algorithm with Applications Jie Liang and Thomas W. Parks, Fellow, IEEE Abstract- We address the time-varying problem of wavelet to recover the signal from the local extrema representation of transforms, and a new translation-invariantwavelet representa- its wavelet transform. tion algorithm is proposed. Using the algorithm introduced by In this paper, we propose a new approach to the translation- Beylkin, we compute the wavelet transform for all the circular time shifts of a length-N signal in O(N log N ) operations. The varying problem of wavelet transforms. Using the algorithm wavelet coefficients of the time shift with minimal cost are selected introduced by Beylkin [l],we compute the wavelet transform as the best representation of the signal using a binary tree for all the circular time shifts of a length-N discrete-time search algorithm with an appropriate cost function.We apply the signal in O ( N logN ) operations. With a carefully chosen translation-invariantrepresentation algorithm to a geoacoustic cost function, the wavelet coefficients of the time shift with data compression application. The results show that the new algorithm can reduce the distortion (the squared error in our minimal cost are selected as the best representation of the case) substantially, if the input signals are transients that are signal using a binary tree search algorithm similar to that used sensitive to time shifts.