By: Preethi Ramkumar
Meaning of DWT – “Discrete Wavelet Transform”, transforms discrete signal from time domain into time-frequency domain. The transformation product is set of coefficients organized in the way that enables not only spectrum analyses of the signal, but also spectral behavior of the signal in time. This is achieved by decomposing signal, breaking it into two components, each caring information about source signal. Filters from the filter bank used for decomposition come in pairs: low pass and high pass. The filtering is succeeded by down sampling (obtained filtering result is "re-sampled" so that every second coefficient is kept). Low pass filtered signal contains information about slow changing component of the signal, looking very similar to the original signal, only two times shorter in term of number of samples. High pass filtered signal contains information about fast changing component of the signal. In most cases high pass component is not so rich with data offering good property for compression. In some cases, such as audio or video signal, it is possible to discard some of the samples of the high pass component without noticing any significant changes in signal. Filters from the filter bank are called "wavelets".
The other perspective to the same theory is based on the fact that some signals, such as audio or video signals often carry redundant information. For instance, looking at the digital picture reveals that neighboring pixels often differ very slightly. The idea is to find a mathematical relation that connects neighboring data samples (pixels) and reduces their number. Of course, inverse process is needed to reconstruct the original (picture in this case).
“…The Discrete Wavelet Transform (DWT) is a special case of the WT that provides a compact
representation of a signal in time and frequency that can be computed efficiently.” [Computer Science Department * also Music Department
Princeton]
“…The BA113FDWT IP core performs hardware 2D multi-level Discrete Wavelet Transform in either lossless or lossy mode with full compatiblity with JPEG2000 Standard (ISO/IEC 15444-1). The core is optimized for speed and is suited for high-end applications involving real-time video (e.g. SDTV or more).
It features a 16-bit datapath and is able to decompose tiles of size up to 128 by 128 pixels into up to 5 levels of decomposition.” [Barco Silex]
“…JPEG 2000's wavelet compression (known as DWT - Discrete Wavelet Transform) treats the image as a signal or wave, rather than small sets of discrete pixel values. The transform organises the image information into a continuous wave - typically with many peaks and dips - and centres it on zero. Actually, it regards the image as a series of waves, one for each colour channel (e.g. Red, Green, Blue), and it does sometimes break up big images into large tiles for ease of processing, but it's simpler to talk about one wave here - the process is exactly the same for each.”
[Technical Advisory Service for Images]
“…A growing number of performance-critical DSP application use
the discrete wavelet transform (DWT), thus prompting the need for
highly efficient DWT software implementations. Unfortunately,
the rapid evolution of computing platforms and compiler technology
makes carefully hand-tuned code obsolete almost as fast as it
is written. In this paper we describe our work on the automatic
generation of DWT implementations that are tuned to a given platform.
Our approach captures the various DWT algorithms in a
concise mathematical framework that enables the integration of
DWTs into the SPIRAL code generation system. Experiments
show the quality of our automatically generated code and provide
interesting insights; for example, the fastest code differs between
platforms and is usually based on a non-obvious combination of
DWT algorithms.” [Department of Electrical and Computer Engineering
Carnegie Mellon University,
Pittsburgh, U.S.A.]
“…The Discrete Wavelet Transform (DWT) is an efficient and useful tool for signal and
image processing applications and will be adopted in many emerging standards, starting
with the new compression standard JPEG2000 [1]. This growing “success” is due to the
achievements reached in the field of mathematics, to its multiresolution processing
capabilities, and also to the wide range of filters that can be provided. These features
allow the DWT to be tailored to suit a wide range of applications.” [School of Computer Science, The Queen's University of Belfast.]
