Optimization design of mth-band FIR filters with application to image processing

TLDR: In this paper, a second-order cone programming (SOCP) optimization approach is proposed for the design of M th-band filters, which is more appropriate for the 2-D M thband filter design than the SOCP approach because of its efficient and simple optimization structure.

Abstract: Cone programming (CP) is a class of convex optimization technique, in which a linear objective function is minimized over the intersection of a set of affine constraints. Such constraints could be linear or convex, equalities or inequalities. Owing to its powerful optimization capability as well as flexibility in accommodating various constraints, the cone programming finds wide applications in digital filter design. In this thesis, fundamentals of linear-phase M th-band FIR filters are first introduced, which include the time-domain interpolation condition and the desired frequency specifications. The restriction of the interpolation matrix M for linear-phase two-dimensional (2-D) M th-band filters is also discussed by considering both the interpolation condition and the symmetry of the impulse response of the 2-D filter. Based on the analysis of the M th-band properties, a semidefinite programming (SOP) optimization approach is developed to design linear-phase 1-0 and 2-D M th-band filters. The 2-D SOP optimization design problem is modeled based on both the mini-max and the least-square error criteria. In contrast to the 1-D based design, the 2-D direct SDP design can offer an optimal equiripple result. A second-order cone programming (SOCP) optimization approach is then presented as an alternative for the design of M th-band filters. The performances as well as the design complexity of these two design approaches are justified through numerical design examples. Simulation results show that the performance of the SOCP approach is better than that of the SDP approach for 1-D M th-band filter design due to its reduced computational complexity for the worst-case, whereas the SDP approach is more appropriate for the 2-D M th-band filter design than the SOCP approach because of its efficient and simple optimization structure. Moreover, the designed M th-band filters are proved useful in image interpolation according to both the visual quality and the peak signal-to-noise ratio (PSNR) for the images with different levels of details.