Discrete Wavelet Transform Comes Of Age
by Paul McGoldrick
I have been writing in support of image compression using wavelets for at least six years - as an admirer of the perfectly beautiful mathematical models represented by wavelets and the efficiency it allows. There has been only one commercial semiconductor adaptation of wavelets and that has been from Analog Devices with its ADV601 and ADV611/612 products. They offer unique solutions for lossless compression over an enormous range and are ideal for all sorts of uses, both lossless and lossy.
Compared to DCT (discrete cosine transform) compression and decompression, wavelets represent a much purer solution using functions that iteratively decompose an image in frequency bands. Although physicists and engineers have been trying to decompose waveforms for a century - the most noticeable being the success of Fourier - the theory behind wavelets emanated from a paper by Grossman and Morlet ("Decomposition of Hardy functions into square integrable wavelets of constant shape," SIAM J. Math. Anal., 15(4), pp 723-736, July 1984.) and it took until 1991/92 before practical solutions were envisaged.
Although wavelets missed out in MPEG-2 - and they should not have if purely technical issues had really been of account, and that has been accepted with the implementation of wavelets in static texture encoding in the MPEG-4 content-based video coding standard - another opportunity has arisen for them in the more venerable standard of JPEG.
JPEG2000 is the latest ITU image compression standard from the JPEG (Joint Photographic Experts' Group) committee and uses the discrete wavelet transform for compression and decompression with an extremely efficient coding scheme. Discrete wavelets are used, rather than continuous (which are commonly more applicable to derive range in a radar, or some kind of moving image recognition system) and standard FIR filters are likely to be used by most systems for the construction of the required filterbanks. A full system will require the transform of the image, quantization, and entropy encoding with the receive channel exactly reversing those processes.
JPEG2000 should find uses in Internet imaging, medical imaging, digital photography, remote surveillance and sensing, color fax, printing, and mobile imaging. The PSNR (peak-signal-to-noise ratio) of a JPEG2000 signal is 42 dB or better at a compression ratio of 1.8 bit/pixel and remains above 34 dB all the way down to a compression rate of 0.4 bit/pixel - opening the markets for many new sorts of imaging products.
Manufacturer ADI took early advantage of this standard - which is Part 1 of the standard - and I hope it and other companies will take note, and advantage, of the subsequent Parts: Part 2 will allow for different quantization approaches and custom macro-like wavelets, while Part 3 will focus on situations where the same piece of hardware (like a digital camera) is being used for both motion and still image capture.
Analog Devices has stuck with wavelets and the ADV-JP2000 - reviewed in analogZONE on June 11, 2002 - is the first
and, so far as I know, still the only semiconductor solution available anywhere
for JPEG2000. I applaud the company's continued belief in a technology for
which the rewards really are now on the way as wavelets become big, very
big.