| Popis: |
We propose and justify a new approach to constructing optimal nonlinear transforms of random vectors. We show that the proposed transform improves such characteristics of {rank-reduced} transforms as compression ratio, accuracy of decompression and reduces required computational work. The proposed transform ${mathcal T}_p$ is presented in the form of a sum with $p$ terms where each term is interpreted as a particular rank-reduced transform. Moreover, terms in ${mathcal T}_p$ are represented as a combination of three operations ${mathcal F}_k$, ${mathcal Q}_k$ and ${oldsymbol{varphi}}_k$ with $k=1,ldots,p$. The prime idea is to determine ${mathcal F}_k$ separately, for each $k=1,ldots,p$, from an associated rank-constrained minimization problem similar to that used in the Karhunen--Lo`{e}ve transform. The operations ${mathcal Q}_k$ and ${oldsymbol{varphi}}_k$ are auxiliary for f/inding ${mathcal F}_k$. The contribution of each term in ${mathcal T}_p$ improves the entire transform performance. A corresponding unconstrained nonlinear optimal transform is also considered. Such a transform is important in its own right because it is treated as an optimal filter without signal compression. A rigorous analysis of errors associated with the proposed transforms is given. |
