Lifting construction based on Bernstein bases and application in image compression

A novel framework for the construction of biorthogonal wavelets based on Bernstein bases with an arbitrary order of vanishing moments using the lifting scheme is proposed. We explore the field of application of it in still image compression. The major contributions of this work can be summarised highlighting the following three aspects. First and foremost, we propose an algorithm that is used to increase the vanishing moments of wavelets from biorthogonal symmetrical wavelets based on Bernstein bases by the lifting scheme. An iterative algorithm for designing the lifting scheme is proposed, which is based on the relationship between the vanishing moments of the wavelet and multiples of zeros of z = 1. The authors provide formulas of the lifting scheme for the construction of wavelets with an arbitrary order of vanishing moments. In addition, the lifting scheme is the shortest among the lifting schemes with the same order of vanishing moments increasing and, more importantly, it is the only one possible. Second, to guarantee the symmetry of the lifting (dual lifting) biorthogonal filters, explicit formulas of the lifting scheme with an arbitrary order of vanishing moments are introduced, which simultaneously have the above two characteristics. With our method, a new family of the parameterisation with symmetry of filters and the related library of biorthogonal symmetric waveforms are presented. Finally, we present a new transform rule aiming at image compression and its corresponding algorithm. Applying the parameterisation of filters constructed in this paper, by adjusting their coefficients, we can realise the transform rule and obtain a new transform. We explore the possibility of applying the presented transforms in image compression at different compression rates, and the results of the experiments prove to be comparable with the CDF9/7 and several state-of-the-art wavelet transforms.