Regularized Tight Frame Design Based On Bernstein Polynomial

The Bernstein polynomial is a useful tool in designing digital filters.In ad-dition,regularity determines the smoothness of wavelet function and affects the stability of wavelet coefficient reconstruction.We combine the regularity of filters with Bernstein polynomial,and the constructed filter can set the regular order more flexibly.First,we start from the regular definition of the filter,and give the mag-nitude square functions P(z)and P₁(z)based on Bernstein polynomial.Then the filter banks with high order are constructed by spectral decomposition.We set the relationship between the low-pass filter and the high pass filter,and set the coef-ficients of Bernstein polynomial to satisfy the perfect-reconstruction requirements.The constructed filter banks form a tight frame.Second,the coefficients of Bern-stein polynomial are optimized by solving an optimization problem.Last,based on the regular tight frame constructed in this paper,the construction method of the dual tight frame is given.We starting from the definitions of regularization and vanishing moments to give the filter high positive order and vanishing moment.The filter constructed by Bernstein Polynomials in this paper is regarded as a primal tight frame,and the expression of dual filters is further given.Then,the filters are obtained by optimizing the L²norm of the amplitude frequency response error of the primal tight frame and dual tight frame.So we give examples of 4-band regular tight frame and its dual tight frame,and apply it to image denoising.The experimental results show that:The regular tight frame constructed by our method can improve the effect of denoising;The method of optimizing the coefficients of Bernstein polynomials is effective.