Frame Bounds Of The Laplacian Pyramid:Computation And Optimization

Laplacian pyramid （LP） has been proved to be an effective tool in multi-resolutionanalysis,which has been widely used in image processing and communication. LP can be viewed asa special oversampled filter bank frame that provides a redundant signal representation. The upper(lower) bound of a frame is an important index in the analysis and design of filter bank frames.With state space methods in control theory, there have been some new results for one-dimensionaland two-dimensional separable filter bank frame, successfully implemented in the analysis ofwavelet frames. However, there is a lack of the effectively numerical methods to compute the framebounds of the LP. This paper investigates the frame bounds of the LP, mainly including thefollowing two ingredients:(1) For one-dimensional case, this paper studies the computation and optimization of framebounds for the LP frame. For any given N-level LP, an algorithm is developed to compute itspolyphase matrix, based on which the linear matrix inequality (LMI) conditions are provided tocompute the frame bounds. We show that the frame bound ratio can be decreased by adjusting thegain of each sub-channel without changing frequency selective property. The minimal ratio as wellas the corresponding optimal gain factors has been obtained by solving some LMIs, which can beeasily solved by existing handy software. Various examples are given to show the effectiveness ofthe proposed methods.(2) For two-dimensional case, this paper investigates the bounds computation problems for LPand provides a frequency-independent solution. Firstly, the state space realization of2-D discreteFBs is given in the form of Roesser model. By using the generalized KYP lemma, a directcomputation method is presented. Finally, extensive examples are given on wavelet and LP framesto demonstrate the effectiveness of the proposed method for2-D frames.