The distributed, partial, and conditional Karhunen-Loeve transforms

The Karhunen-Loeve transform (KLT) is a key element of many signal processing tasks, including approximation, compression, and classification. Many recent applications involve distributed signal processing where it is not generally possible to apply the KLT to the signal; the KLT must be approximated in a distributed fashion. Investigations were carried out on the distributed approximations to the KLT. First, explicit solutions to special cases were presented including a partial KLT, a conditional KLT, and the combination of these two special cases. These results were used to derive an algorithm that finds the best distributed approximation to the KLT. Applications of the results from sensor networks and distributed databases were discussed.

1 Introduction

Distributed signal processing and communication is an integral part of many of the crucial contemporary applications. Consider for example a scene filmed by multiple cameras, or environmental data recorded by multiple sensors: the signals at the sensors are correlated. In the non-distributed case, one would apply the KLT, thus obtaining uncorrelated data streams which can now be compressed separately from each other. Suppose however that communication between the sensors is expensive, or that they cannot communicate at all. Then, signal processing must be done in a distributed fashion, and the full KLT cannot be applied to the data. In this paper, we show how the concept of the KLT extends to such a distributed scenario. For a state of the art of the key results on the KLT in the non-distributed case, we refer to the excellent exposition in [1]. The importance of distributed source coding is further elaborated e.g. in [2]. Figure 1: The distributed KLT problem: Distributed compression of multiple correlated vector sources