Quantizers with Parameterized Distortion Measures

Abstract

In many quantization problems, the distortion function is given by the Euclidean metric to measure the distance of a source sample to any given reproduction point of the quantizer. We will in this work regard distortion functions, which are additively and multiplicatively weighted for each reproduction point resulting in a heterogeneous quantization problem, as used for example in deployment problems of sensor networks. Whereas, normally in such problems, the average distortion is minimized for given weights (parameters), we will optimize the quantization problem over all weights, i.e., we tune or control the distortion functions in our favor. For a uniform source distribution in one-dimension, we derive the unique minimizer, given as the uniform scalar quantizer with an optimal common weight. By numerical simulations, we demonstrate that this result extends to two-dimensions where asymptotically the parameter optimized quantizer is the hexagonal lattice with common weights. As an application, we will determine the optimal deployment of unmanned aerial vehicles (UAVs) to provide a wireless communication to ground terminals under a minimal communication power cost. Here, the optimal weights relate to the optimal flight heights of the UAVs.

Energy-Efficient Node Deployment in Heterogeneous Two-Tier Wireless Sensor Networks With Limited Communication Range

“…For the synthetic data, we provide the experimental results in two heterogeneous two-tier WSNs: (i) WSN1: A heterogeneous WSN including 1 FC and 20 APs; (ii) WSN2: A heterogeneous WSN including 4 FCs and 20 APs. We consider the same target domain Ω as in [3], [8], i.e., Ω = [0, 10] 2 .…”

“…i.e, every node location is generated with uniform distribution on Ω. In order to make a fair comparison to prior work, similar to the experimental setting in [3], [8], the maximum number of iterations is set to 100, FCs and APs are denoted, respectively, by black and red circles. Other parameters are provided in Table I.…”

“…Note that unlike the regular Voronoi diagrams, the Voronoi cells defined in ( 7) may be non-convex, not star-shaped or even disconnected. The following proposition establishes that the generalized Voronoi diagram in (8) provides the optimal cell partitions, i.e., R * (P, Q, T ) = V(P, Q, T ) for a given P, Q, T .…”

“…Proof of Lemma 2: Using Lemma 3 in [8], it can be easily shown that the optimal quantization regions are two closed intervals. Without loss of generality, let R = {R 1 , R 2 }, where R 1 = [0, r] and R 2 = [r, 1] be the optimal partitioning.…”

Using Quantization to Deploy Heterogeneous Nodes in Two-Tier Wireless Sensor Networks

“…We provide the experimental results in two heterogeneous two-tier WSNs: (i) WSN1: A heterogeneous WSN including 1 FC and 20 APs; (ii) WSN2: A heterogeneous WSN including 4 FCs and 20 APs. We consider the same target domain Ω as in [1], [6], i.e., Ω = [0, 10] 2 . The data rate density function is set to a uniform function, f (ω) = 1 Ω dA = 0.01.…”