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A multigrid approach to the scalar quantization problem

机译：标量量化问题的多网格方法

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摘要

A multigrid framework for the one-dimensional (scalar) fixed-rate quantization problem is presented. The framework is based on the Lloyd-Max iterative process, which is a central building block in many quantization algorithms. This process iteratively improves a given initial solution and generally converges to a local minimum of the quantization distortion. Contrary to the classical Lloyd-Max process, the convergence of the multigrid algorithm is practically independent of the number of representation levels sought. Using this approach, a local minimum is reached at the cost of just a few Lloyd-Max iterations. The complexity of the proposed method is O(n) operations for the continuous case and 3N+O(n) for the discrete case, where n is the number of representation levels sought and N is the size of the discrete probability density function. In addition to its independent attributes, this work is a precursor to the more important vector quantization problem, for which a multiscale framework is also being developed.
机译：提出了用于一维（标量）固定速率量化问题的多网格框架。该框架基于Lloyd-Max迭代过程，这是许多量化算法中的核心组成部分。该过程迭代地改进了给定的初始解，并且通常收敛到量化失真的局部最小值。与经典的Lloyd-Max过程相反，多重网格算法的收敛性实际上与所寻求的表示级别数目无关。使用这种方法，只需几次Lloyd-Max迭代即可达到局部最小值。所提出方法的复杂性是连续情况下的O（n）个操作，离散情况下的3N + O（n），其中n是寻求的表示级别数，N是离散概率密度函数的大小。除了其独立的属性外，这项工作是更重要的矢量量化问题的前兆，为此也正在开发多尺度框架。

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