Design &implementation of complex-valued FIR digital filters with application to migration of seismic data

One-dimensional (I-D) and two-dimensional (2-D) frequency-space seismic migration FIR digital filter coefficients are of complex values when such filters require special space domain as well as wavenumber domain characteristics. In this thesis, such FIR digital filters are designed using Vector Space Projection Methods (VSPMs), which can satisfy the desired predefined filters' properties, for 2-D and three-dimensional (3-D) seismic data sets, respectively. More precisely, the pure and the relaxed projection algorithms, which are part of the VSPM theory, are derived. Simulation results show that the relaxed version of the pure algorithm can introduce significant savings in terms of the number of iterations required. Also, due to some undesirable background artifacts on migrated sections, a modified version of the pure algorithm was used to eliminate such effects. This modification has also led to a significant reduction in the number of computations when compared to both the pure and relaxed algorithms. We further propose a generalization of the l-D (real/complex-valued) pure algorithm to multi-dimensional (m-D) complex-valued FIR digital filters, where the resulting frequency responses possess an approximate equiripple nature. Superior designs are obtained when compared with other previously reported methods. In addition, we also propose a new scheme for implementing the predesigned 2-D migration FIR filters. This realization is based on Singular Value Decomposition (SVD). Unlike the existing realization methods which are used for this geophysical application, this cheap realization via SVD, compared with the true 2-D convolution, results in satisfactory wavenumber responses. Finally, an application to seismic migration of 2-D and 3-D synthetic sections is shown to confirm our theoretical conclusions. The proposed resulting migration FIR filters are applied also to the challenging SEGIEAGE Salt model data. The migrated section (image) outperformed images obtained using other FIR filters and with other standard migration techniques where difficult structures contained in such a challenging model are imaged clearly

Deformable kernels for early vision

Early vision algorithms often have a first stage of linear-filtering that `extracts' from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs. A technique is presented that allows: 1) computing the best approximation of a given family using linear combinations of a small number of `basis' functions; and 2) describing all finite-dimensional families, i.e., the families of filters for which a finite dimensional representation is possible with no error. The technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations. The relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated. Experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi-scale 2D edge-detection kernels. The implementation issues are also discussed

MVDR broadband beamforming using polynomial matrix techniques

This thesis addresses the formulation of and solution to broadband minimum variance distortionless response (MVDR) beamforming. Two approaches to this problem are considered, namely, generalised sidelobe canceller (GSC) and Capon beamformers. These are examined based on a novel technique which relies on polynomial matrix formulations. The new scheme is based on the second order statistics of the array sensor measurements in order to estimate a space-time covariance matrix. The beamforming problem can be formulated based on this space-time covariance matrix. Akin to the narrowband problem, where an optimum solution can be derived from the eigenvalue decomposition (EVD) of a constant covariance matrix, this utility is here extended to the broadband case. The decoupling of the space-time covariance matrix in this case is provided by means of a polynomial matrix EVD. The proposed approach is initially exploited to design a GSC beamformer for a uniform linear array, and then extended to the constrained MVDR, or Capon, beamformer and also the GSC with an arbitrary array structure. The uniqueness of the designed GSC comes from utilising the polynomial matrix technique, and its ability to steer the array beam towards an off-broadside direction without the pre-steering stage that is associated with conventional approaches to broadband beamformers. To solve the broadband beamforming problem, this thesis addresses a number of additional tools. A first one is the accurate construction of both the steering vectors based on fractional delay filters, which are required for the broadband constraint formulation of a beamformer, as for the construction of the quiescent beamformer. In the GSC case, we also discuss how a block matrix can be obtained, and introduce a novel paraunitary matrix completion algorithm. For the Capon beamformer, the polynomial extension requires the inversion of a polynomial matrix, for which a residue-based method is proposed that offers better accuracy compared to previously utilised approaches. These proposed polynomial matrix techniques are evaluated in a number of simulations. The results show that the polynomial broadband beamformer (PBBF) steersthe main beam towards the direction of the signal of interest (SoI) and protects the signal over the specified bandwidth, and at the same time suppresses unwanted signals by placing nulls in their directions. In addition to that, the PBBF is compared to the standard time domain broadband beamformer in terms of their mean square error performance, beam-pattern, and computation complexity. This comparison shows that the PBBF can offer a significant reduction in computation complexity compared to its standard counterpart. Overall, the main benefits of this approach include beam steering towards an arbitrary look direction with no need for pre-steering step, and a potentially significant reduction in computational complexity due to the decoupling of dependencies of the quiescent beamformer, blocking matrix, and the adaptive filter compared to a standard broadband beamformer implementation.This thesis addresses the formulation of and solution to broadband minimum variance distortionless response (MVDR) beamforming. Two approaches to this problem are considered, namely, generalised sidelobe canceller (GSC) and Capon beamformers. These are examined based on a novel technique which relies on polynomial matrix formulations. The new scheme is based on the second order statistics of the array sensor measurements in order to estimate a space-time covariance matrix. The beamforming problem can be formulated based on this space-time covariance matrix. Akin to the narrowband problem, where an optimum solution can be derived from the eigenvalue decomposition (EVD) of a constant covariance matrix, this utility is here extended to the broadband case. The decoupling of the space-time covariance matrix in this case is provided by means of a polynomial matrix EVD. The proposed approach is initially exploited to design a GSC beamformer for a uniform linear array, and then extended to the constrained MVDR, or Capon, beamformer and also the GSC with an arbitrary array structure. The uniqueness of the designed GSC comes from utilising the polynomial matrix technique, and its ability to steer the array beam towards an off-broadside direction without the pre-steering stage that is associated with conventional approaches to broadband beamformers. To solve the broadband beamforming problem, this thesis addresses a number of additional tools. A first one is the accurate construction of both the steering vectors based on fractional delay filters, which are required for the broadband constraint formulation of a beamformer, as for the construction of the quiescent beamformer. In the GSC case, we also discuss how a block matrix can be obtained, and introduce a novel paraunitary matrix completion algorithm. For the Capon beamformer, the polynomial extension requires the inversion of a polynomial matrix, for which a residue-based method is proposed that offers better accuracy compared to previously utilised approaches. These proposed polynomial matrix techniques are evaluated in a number of simulations. The results show that the polynomial broadband beamformer (PBBF) steersthe main beam towards the direction of the signal of interest (SoI) and protects the signal over the specified bandwidth, and at the same time suppresses unwanted signals by placing nulls in their directions. In addition to that, the PBBF is compared to the standard time domain broadband beamformer in terms of their mean square error performance, beam-pattern, and computation complexity. This comparison shows that the PBBF can offer a significant reduction in computation complexity compared to its standard counterpart. Overall, the main benefits of this approach include beam steering towards an arbitrary look direction with no need for pre-steering step, and a potentially significant reduction in computational complexity due to the decoupling of dependencies of the quiescent beamformer, blocking matrix, and the adaptive filter compared to a standard broadband beamformer implementation

Recovery under Side Constraints

This paper addresses sparse signal reconstruction under various types of structural side constraints with applications in multi-antenna systems. Side constraints may result from prior information on the measurement system and the sparse signal structure. They may involve the structure of the sensing matrix, the structure of the non-zero support values, the temporal structure of the sparse representationvector, and the nonlinear measurement structure. First, we demonstrate how a priori information in form of structural side constraints influence recovery guarantees (null space properties) using L1-minimization. Furthermore, for constant modulus signals, signals with row-, block- and rank-sparsity, as well as non-circular signals, we illustrate how structural prior information can be used to devise efficient algorithms with improved recovery performance and reduced computational complexity. Finally, we address the measurement system design for linear and nonlinear measurements of sparse signals. Moreover, we discuss the linear mixing matrix design based on coherence minimization. Then we extend our focus to nonlinear measurement systems where we design parallel optimization algorithms to efficiently compute stationary points in the sparse phase retrieval problem with and without dictionary learning