Sparsity Motivated Auditory Wavelet Representation and Blind Deconvolution
Aniruddha Adiga · 2018
In many scenarios, events such as singularities and transients that carry important information about a signal undergo spreading during acquisition or transmission and it is important to localize the events. For example, edges in an image, point sources in a microscopy or astronomical image are blurred by the point-spread function (PSF) of the acquisition system, while in a speech signal, the epochs corresponding to glottal closure instants are shaped by the vocal tract response. Such events can be extracted with the help of techniques that promote sparsity, which enables separation of the smooth components from the transient ones. In this thesis, we consider development of such sparsity promoting techniques. The contributions of the thesis are three-fold: (i) an auditory-motivated continuous wavelet design and representation, which helps identify singularities; (ii) a sparsity-driven deconvolution technique; and (iii) a sparsity-driven deconvolution technique for reconstruction of nite-rate-of-innovation (FRI) signals. We use the speech signal for illustrating the performance of the techniques in the first two parts and super-resolution microscopy (2-D) for the third part. In the rst part, we develop a continuous wavelet transform (CWT) starting from an auditory motivation. Wavelet analysis provides good time and frequency localization, which has made it a popular tool for time-frequency analysis of signals. The CWT is a multiresolution analysis tool that involves decomposition of a signal using a constant-Q wavelet filterbank, akin to the time-frequency analysis performed by basilar membrane in the peripheral human auditory system. This connection motivated us to develop wavelets that possess auditory localization capabilities. Gammatone functions are extensively used in the modeling of the basilar membrane, but the non-zero average of the functions poses a hurdle. We construct bona de wavelets from the Gammatone function called Gammatone wavelets and analyze their properties such as admissibility, time-bandwidth product, vanishing moments, etc.. Of particular interest is the vanishing moments property, which enables the wavelet to suppress smooth regions in a signal leading to sparsi cation. We show how this property of the Gammatone wavelets coupled with multiresolution analysis could be employed for singularity and transient detection. Using these wavelets, we also construct equivalent lterbank models and obtain cepstral feature vectors out of such a representation. We show that the Gammatone wavelet cepstral coefficients (GWCC) are effective for robust speech recognition compared with mel-frequency cepstral coefficients (MFCC). In the second part, we consider the problem of sparse blind deconvolution (SBD) starting from a signal obtained as the convolution of an unknown PSF and a sparse excitation. The BD problem is ill-posed and the goal is to employ sparsity to come up with an accurate solution. We formulate the SBD problem within a Bayesian framework. The estimation of lter and excitation involves…