  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.
