Background on the TIPSH Algorithm

The TIPSH algorithm was designed for the task of denoising signals and images in which the data take the form of counts (e.g., number of photons per pixel registered over a given period of observation, etc.). In general, denoising refers to an attempt to separate ``signal'' from ``noise'' in the data, and keep only the former. The TIPSH algorithm approaches this task using a relatively new tool in the signal and image processing toolbox, a wavelet transform.

Wavelet transforms take the data as input, and output a modified form of the data. In this modified form, the information in the data typically is de-correlated and compressed. In other words, the usual spatial relationships (or correlations) between data in, say, nearby pixels of an image, are broken in such a way that each piece of the modified data may be dealt with individually; treating each pixel individually in the original data ignores the fact that the structure in nearby regions of an image typically is related. The compression achieved by a wavelet transform usually means that the ``signal'' in the original data is condensed into just a small fraction of the modified data, which not surprisingly take on quite large numeric values; the rest of the modified data contains essentially just the ``noise'', and takes on much smaller values.

In this modified form, the problem of denoising then can be interpreted as one of finding a good threshold, separating the ``small'' from the ``large''. Given that the ``small'' are mainly random fluctuations due to noise, statistical techniques are needed to adjust the threshold properly, so as to choose a value for which it is highly unlikely that the ``noise'' will exceed, yet not so high so as to include appreciable amounts of ``signal''. Only that part of the modified data found to exceed the threshold value is used to reconstruct the ``signal'' underlying the data (through application of the reverse process of the wavelet transform).

Many techniques have been put forward in the last 10 years or so for dealing with the basic denoising problem using wavelets and thresholds. However, the TIPSH algorithm is one of the first to incorporate the statistical arguments necessary to account explicitly for the randomness and uncertainty in counting measurements specifically. Most other techniques are intended for data taking values along a continuous scale. When the data are observed counts, however, the statistical properties of the noise are different. TIPSH uses a particularly simple type of wavelet transform (the Haar wavelet) that essentially computes the differences of counts in nearby pixels. The statistical behavior of such differences may be studied using tools from statistics and mathematics, and thresholds derived accordingly.

Technical info and software related to TIPSH can be found here.

Questions or problems: Contact Dave Dixon