Theories And Matrix-Based Algorithms For The Constrained Minimax Design Of Two-Dimensional FIR Digital Filters

Two-dimensional(2-D)digital filters have found lots of applications in image processing,robotics and computer vision,seismic signal processing,radar signal processing,astronomical signal processing and so on.Finite impulse response(FIR)filters can easily attain linear-phase characteristics and are always stable,which has attracted wide attention from scholars at home and abroadThe optimization problem of 2-D filters is essentially a bivariate function approximation problem.Since the approximation theory of the bivariate function is incomplete,and the design coefficients of 2-D filters increase with the square of the filter order,the design problem of 2-D filters is comparatively complicatedThe traditional algorithms for the design problem of 2-D filters arrange the coefficients of 2-D filters into vectors,and then extend the algorithms for the one-dimensional filters into 2-D case.This results in high computational complexity and occupies lots of computer memory.Some resent algorithms have exploited the matrix nature of the coefficients of 2-D filters to solve the design problem,resulting in a significant saving in computations and memory required.While these algorithms cannot directly solve the constrained Minimax design problem of 2-D digital filters.The quadrantally symmetric 2-D FIR digital filters are the most widely used 2-D linear-phase FIR digital filters.Based on the quadrantally symmetric 2-D FIR digital filters,this thesis studies the constrained Minimax design problem of 2-D linear-phase FIR digital filters.The frequency-domain inequality constraints and the time-domain equality constraints are considered.Firstly,this thesis establishes a mathematical model of the Minimax design problem of 2-D linear-phase FIR digital filters in matrix form.The known basis functions,weighting function,desired frequency response,constraints and the unknown impulse response all appear as matrices.The elements of the unknown matrix are related to the impulse responses of the 2-D filter.Then,an efficient matrix-based algorithm is proposed to solve the Minimax design problem of 2-D FIR digital filters with frequency-domain inequality constraints.This method converts the constrained Minimax design problem into a series of unconstrained weighted Minimax design problems,where the weighting function is appropriately updated in each iteration.These unconstrained weighted Minimax design problems cannot be solved by the analytic method,thus a matrix-based iterative reweighted least-squares(IRLS)algorithm is used.Moreover,the proposed algorithm is guaranteed to converge to the optimal solution if it exists.Finally,the Minimax design problem of 2-D FIR digital filters with time-domain equality constraints is studied.By some matrix transformation and introducing some new matrix variables,the constrained Minimax design problem involving one matrix variable is converted into an unconstrained one involving three matrix variables,and a new matrix-based IRLS algorithm is proposed to solve it.The proposed IRLS algorithm includes two loops:one for updating the weighting function and the other for solving the weighted least squares(WLS)subproblems.The optimality condition of these WLS subproblems is expressed as a linear operator equation by defining a linear operator acting on the coefficient matrices of the 2-D filter.Then,a matrix-based generalized conjugate gradient algorithm is proposed to solve the linear operator equation.The convergence of the proposed algorithm can be guaranteed according to linear operator theory.Moreover,the algorithms for the design of 2-D FIR digital filters with time-domain equality constraints are extended to the design of the 2-D linear-phase FIR half-band filters.Then an efficient algorithm for the Minimax design of the 2-D linear-phase FIR half-band filters is obtained,and the algorithm can fast converge to the optimal solution.Simulation results demonstrate that the proposed matrix-based algorithms are more computationally efficient,need smaller memory space compared with the existing methods,and can accurately solve the constrained Minimax design of quadrantally symmetric 2-D FIR digital filters.In addition,these matrix-based algorithms can be further extended to the design of other 2-D linear-phase FIR digital filters.