Mathematical analysis of the energy compaction affected by the dimensionality of Karhunen-Lòeve transform

Transform coding is a essential tool for picture coding applications, and coding schemes that can achieve high energy compaction are essential. In terms of energy compaction, Karhunen-Loeve transform (KLT) is known to be optimal. The energy compaction provided by KLT depends on the statistical property of the input signal and the dimensionality of KLT. However, the quantitative effect of KLT dimensionality on energy compaction has not been clarified. This paper establishes a mathematical model of the relationship among the dimensionality of KLT and the energy compaction of the transform coefficients, using mathematical tools in quantum information theory.