On Image Compression by Biorthogonal Wavelet Transforms Based on Discrete Splines

We present a new family of biorthogonal wavelet transforms and a related library of biorthogonal symmetric wavelets. We employed the interpolatory continuous and discrete splines as a tool for devising a discrete biorthogonal wavelet scheme. The construction is based on the \lifting scheme" presented by Sweldens in [1]. The lifting scheme allows custom design and fast implementation of the transforms. Brie°y, the idea of the computation is that values of the signal located at odd positions are predicted by values in the midpoints of the spline that interpolates even values of the signal. Then, the odd subarray is replaced by the di®erence between the current and the predicted subarrays.Then we employ the new odd subarray for updating the existing even subarray in order to smooth the even subarray and thus reduce the aliasing which is a consequence of the decimation. We devised a diversity of transforms using interpolatory splines of various orders as the predicting and updating aggregates. Our investigation revealed an interesting relation between the discrete splines and the popular Butterworth ̄lters [2]. The constructed ̄lter banks comprise ̄lters which act as a bi-directional (causal and anti-causal) half-band Butterworth ̄lters. The frequency response of Butterworth ̄lters are maximally °at and we succeeded in construction of the dual ̄lters with similar property. The corresponding wavelets we name the Butterworth wavelets. We established explicit formulas for the construction of wavelets with arbitrary number of vanishing moments. Although devised ̄lters have in ̄nite impulse response, the computations in our scheme are conducted in the time domain using recursive ̄ltering. The ̄lters which we use are symmetric and allow fast cascade or parallel implementation. We conducted a wide series of experiments on compression of images and seismic data using the above transforms. The transforms proved to be highly e±cient for this application. In most experiments they outperform the popular 9/7 biorthogonal ̄lter while having lower computational complexity.