Signal Reconstruction From Wavelet Maxima From Wavelet Modulus Maxima Using Quadratic Programming

Wavelet transform is another efficient time-frequency analysis tool after Fourieranalysis. Wavelet transforms represent signals on the time-scale plane and have themultiresolution analysis structure and good time-frequency localization. Wavelettheory has been successfully applied in many fields, such as signal and imageprocessing, data compression, quantum physics, seismology, etc. This thesis is focusedon the investigation of the algorithms to reconstruct signals from the modulus maximaof wavelet transforms.Singularity carries most information of signals. The modulus maxima of wavelettransforms can characterize the singularity in signals well. Therefore, the modulusmaxima of the wavelet transforms of signals carry most information of the signals andreconstructing the signals from the modulus maxima of the wavelet transforms isalways a hot spot in wavelet applications. A new method of signal reconstruction usingwavelet modulus maxima is presented in this thesis. Signal reconstruction using themodulus maxima of wavelet transforms boils down to a quadratic programmingproblem, which can quickly and accurately reconstruct signals from the modulusmaxima of their wavelet transforms.Quadratic programming is the simplest special optimization problem amongnonlinear programming and has been used in many fields, such as management,economics, operation research, system analysis and combinatorial optimization. Whena practical problem is transferred to a quadratic programming, it is easier to be solvedfast. In our problem, the basic assumption that the reconstructed signal has the samewavelet modulus maxima with the original signal is utilized. Though this assumptioncannot assure that the reconstructed signal strictly equals to the original signal, thereconstructed signal can approximate to the original signal with very high precision.Through solving the proposed quadratic programming, the original signal can bereconstructed from the modulus maxima of its wavelet transform.The proposed algorithm includes the following several steps. First, a one-dimensional signal is transformed into the wavelet domain. Second, the modulusmaxima of the wavelet transform are found from wavelet transform coefficients. Third,signal reconstruction is carried out by solving a quadratic programming withconstraints. Moreover, the proposed algorithm is also extended to the two-dimensionalcase and is tried to apply to data compression. The experimental results to realone-dimensional signals show that the reconstruction error are at very low level when asmall number of the modulus maxima are used. In particular, the proposed algorithmhas much smaller computational complexity than the commonly-used iterativereconstruction methods.