On Properties of Locally Optimal Multiple Description Scalar Quantizers with Convex Cells

It is known that the generalized Lloyd method is applicable to locally optimal multiple description scalar quantizer (MDSQ) design. However, it remains unsettled when the resulting MDSQ is also globally optimal. We partially answer the above question by proving that for a fixed index assignment there is a unique locally optimal fixed-rate MDSQ of convex cells under Trushkin's sufficient conditions for the uniqueness of locally optimal fixed-rate single description scalar quantizer. This result holds for fixed-rate multiresolution scalar quantizer (MRSQ) of convex cells as well. Thus, the well-known log-concave probability density function (pdf) condition can be extended to the multiple description and multiresolution cases. Moreover, we solve the difficult problem of optimal index assignment for fixed-rate MRSQ and symmetric MDSQ, when cell convexity is assumed. In both cases we prove that at optimality the number of cells in the central partition has to be maximal, as allowed by the side quantizer rates. As long as this condition is satisfied, any index assignment is optimal for MRSQ, while for symmetric MDSQ an optimal index assignment is proposed. The condition of convex cells is also discussed. It is proved that cell convexity is asymptotically optimal for high resolution MRSQ, under the $r$th power distortion measure.