Constrained and hierarchical density estimation for image reconstruction and sensor networks

In this paper, we propose constrained and hierarchical density estimations for image reconstruction and sensor networks. The estimation result of the conventional integrated squared-error (ISE) based solution has problem such as degeneracy, therefore the necessity of maximum entropy constraint arises. The cost function with maximum entropy penalty can be solved the problem using Newton's method and the augmented cost function with maximum entropy constraint can have a closed form solution for each step with hierarchical approach by binary tree. Since K-mean algorithm in ISE based algorithm is hard-limiting case of expectation-maximization (EM) algorithm, we propose a maximum-entropy expectation-maximization (MEEM) algorithm to fully utilize the whole sample information. However the MEEM algorithm only considers over-fitting problem. Therefore we also propose an attraction-repulsion expectation-maximization (AREM) algorithm that seeks an equilibrium between over-fitting and over-smoothing in density estimation by incorporating attraction and repulsion forces among the Gaussian functions and determining the optimal balance between the competing forces experimentally. For the reduction of the computational burden of the EM algorithm, we expand the conventional EM algorithm into a kernel based EM algorithm. The proposed algorithm can use the samples represented by kernel functions as well as the samples represented by the conventional delta functions together.;Moreover, we propose to extend the use of the classical EM algorithm for image recovery from randomly sampled data and sensor field estimation from randomly scattered sensor networks. We further propose to use our approach in density estimation, image recovery and sensor field estimation. Computer simulations are given to demonstrate the effectiveness of the proposed algorithms.