On the rate-distortion performance and computational efficiency of the Karhunen-Loeve transform for lossy data compression

Abstract

We examine the rate-distortion performance and computational complexity of linear transforms for lossy data compression. The goal is to better understand the performance/complexity tradeoffs associated with using the Karhunen-Loeve transform (KLT) and its fast approximations. Since the optimal transform for transform coding is unknown in general, we investigate the performance penalties associated with using the KLT by examining cases where the KLT fails, developing a new transform that corrects the KLT's failures in those examples, and then empirically testing the performance difference between this new transform and the KLT. Experiments demonstrate that while the worst KLT can yield transform coding performance at least 3 dB worse than that of alternative block transforms, the performance penalty associated with using the KLT on real data sets seems to be significantly smaller, giving at most 0.5 dB difference in our experiments. The KLT and its fast variations studied here range in complexity requirements from O(n(2)) to O(n log n) in coding vectors of dimension n. We empirically investigate the rate-distortion performance tradeoffs associated with traversing this range of options. For example, an algorithm with complexity O(n(3/2)) and memory O(n) gives 0.4 dB performance loss relative to the full KLT in our image compression experiments.