This thesis develops array signal processing theories for selected biomedical applications involving acoustic waves. Specifically, we consider source localization in the interior of sensor arrays for lung sound localization and efficient algorithms for photoacoustic imaging. Lung sound localization provides quantitative results to the extent and location of lung disorders. Photoacoustic imaging is important for the early detection of cancer and has numerous other biomedical applications. Previous lung sound localization methods cannot deal with multiple sources or have analytical performance measures. We propose two methods utilizing the eigen basis decomposition of the wavefield and the Minimum Variance spectrum for multiple source localization. Analytical performance measures were derived for resolution and spatial aliasing. The performance of our methods for lung sound localization together with the performance measures were proven by simulations. We consider the photoacoustic inversion problem from a frequency invariant source localization perspective. Complete series and fast photoacoustic inversion methods have not been developed for the circular and spherical sensor geometries. A new theory is developed for photoacoustic reconstruction where the source distribution is expanded with a suitable series expansion such that separating the modes in the wavefield expansion at particular frequencies, separates the information in the source expansion. This theory is applied for photoacoustic inversion using a circular acquisition geometry. The source is expanded using a Fourier Bessel series and the coefficients are estimated by processing frequencies corresponding to the Bessel zeros. The proposed method is faster than previous approaches and the derivation is valid even for finite measurement bandwidth. This new theory is flexible enough to be applied for arbitrary sensor geometries and allows the selection of a minimum number of frequency samples for reconstruction. For previous frequency domain methods, there was no way to determine the minimum number of frequency samples required. Further, numerical experiments proved the effectiveness of our approach. The extension of the proposed theory for photoacoustic inversion with a spherical array geometry was proposed. This new method expands the source distribution with a spherical Fourier Bessel series whose coefficients were now obtained by processing frequencies corresponding to the spherical Bessel zeros. Using computational order analysis and numerical experiments, this proposed method was shown to be faster than the backprojection and the Fourier series methods. To enhance the reconstruction of our proposed methods, we introduced a sub{u00AD}gradient based Total Variation (TV) minimization and an alternating projections post processing method. Both these methods were designed to handle the large data sets present in photoacoustic tomography. Applications of these two post processing ideas to previously proposed inversion methods are either difficult or impossible. The proposed inversion methods provide projection of the source distribution onto a set of basis functions. Therefore, these two post processing methods were developed to reconstruct a source distribution that preserves these projections and ensures that the source distribution is non-negative. Numerical experiments performed showed that reconstruction quality was improved by applying these two post processing methods. Further, the TV minimization method provided better reconstruction when compared to the alternating projections method
