Fast algorithms for the wavelet transform.

This research was carried out in collaboration with Dr. Umberto Amato while visiting Istituto per Applicazioni della Matematica of Naples, Italy.

The discrete wavelet transform (DWT) has proved its utility in various signal processing problems such as data compression or noise reduction. It is therefore important, both for research and for industrial applications, to dispose of fast software performing the various versions of the DWT.
Several software tools for this purpose were designed by Dr. Vuza, which are currently used in the activity of the named institute. Since Wavelet Theory is relatively new, software product adapted to specific needs are not easy available yet so that we had to develop our own.
An object-oriented approach to the DWT by developing a hierarchy of C++ classes covering the cases of periodic DWT and finite-interval DWT has been proposed . The code was presented in two versions:
The assembly version was optimized for speed by means of a non-standard use of the floating-point unit in order to achieve the following purposes:
Both 16 bit and 32 bit versions were produced.
The table below presents a comparison between the timings achieved with the C++ and assembly versions respectively. The test consisted in running the DWT for 200 times on a sequence of 4096 data and taking the average execution time, expressed below in milliseconds. The test was performed on a 90 Mhz PENTIUM computer running Windows for Workgroups 3.11 with the Win32s add-on.
16 bit code 32 bit code
C++ code 166.14 52.45
ASM code 26.36 16.75
It is expected that applications based on processors not as highly optimized as the PENTIUM (in particular, those without branch prediction) could take an even greater advantage from the proposed assembly routines.
A possible research subject could be therefore the adaption of the above methods to other types of DWT beside periodic and finite interval and to other kinds of processors.
For applications it is equally important to dispose of filter coefficients for DWT computed to a high precision. In the report mentioned below the authors proposed a program for the MATHEMATICA environment for computing the boundary filter coefficients needed by the finite interval DWT which takes advantage of the capability of MATHEMATICA of working with arbitrary high degrees of precision. Research on this topic could also be continued for other types of DWT.

Reference from list of works related to this topic: technical report [1].