Compressing 3D Measurement Data Under Interval Uncertainty

The existing image and data compression techniques try to minimize the mean square deviation between the original data f(x, y, z) and the compressed-decompressed data f̃(x, y, z). In many practical situations, reconstruction that only guaranteed mean square error over the data set is unacceptable. For example, if we use the meteorological data to plan a best trajectory for a plane, then what we really want to know are the meteorological parameters such as wind, temperature, and pressure along the trajectory. If along this line, the values are not reconstructed accurately enough, the plane may crash – and the fact that on average, we get a good reconstruction, does not help. In general, what we need is a compression that guarantees that for each (x, y), the difference |f(x, y, z) − f̃(x, y, z)| is bounded by a given value ∆ – i.e., that the actual value f(x, y, z) belongs to the interval [f̃(x, y, z)−∆, f̃(x, y, z) + ∆]. In this paper, we describe new efficient techniques for data compression under such interval uncertainty. 1 Formulation of the Problem Compression is important. At present, so much data is coming from measuring instruments that it is necessary to compress this data before storing and processing. We can gain some storage space by using lossless compression, but often, this gain is not sufficient, so we must use lossy compression as well. Successes of 2-D image compression. In the last decades, there has been a great progress in image and data compression. In particular, the JPEG2000 standard (see, e.g., [8]) uses the wavelet transform methods together with other efficient compression techniques to provide a very efficient compression of 2D images. Within this standard, we can select different bitrates (i.e., number of bits per pixel that is required, on average, for the compressed image), and depending on the bitrate, get different degrees of compression. 2 Olga Kosheleva et al. When we select the highest possible bitrate, we get the lossless compressions that enables us to reconstruct the original image precisely. When we decrease the bitrate, we get a lossy compression; the smaller the bitrate, the more the compressed/decompressed image will differ from the original image. 2-D data compression. In principle, it is possible to use these compression techniques to compress 2D measurement data as well. Compressing 3-D data: layer-by-layer approach. It is also possible to compress 3D measurement data f(x, y, z) – e.g., meteorological measurements taken in different places (x, y) at different heights z. One possibility is simply to apply the 2D JPEG2000 compression to each horizontal layer f(x, y, z0). Compressing 3-D data – an approach that uses KLT transform: general idea. Another possibility, in accordance with Part 2 of JPEG2000 standard, is to first apply the KLT transform to each vertical line. Specifically, we: – compute the average value f̄(z) = N−1 ·∑ x,y f(x, y, z) of the analyzed quantity at a given height z, where N is the overall number of horizontal points (x, y); – compute the covariances between different heights: V (z1, z2) = N−1 · ∑ x,y (f(x, y, z1)− f̄(z1)) · (f(x, y, z2)− f̄(z2)); – find the eigenvalues λk and the eigenvectors ek(z) of the covariance matrix V (z1, z2); we sort these eigenvectors into a sequence e1(z), e2(z), . . . so that |λ1| ≥ |λ2| ≥ . . .; – finally, we represent the original 3D data values f(x, y, z) as a linear combination f(x, y, z) = f̄(z) + ∑