New lattice vector quantizer design procedures for nonuniform sources that yield excellent performance while retaining the structure required for fast quantization are described. Analytical methods for truncating and scaling lattices to be used in vector quantization are given, and an analytical technique for piecewise-linear multidimensional companding is presented. The uniform and piecewise-uniform lattice vector quantizers are then used to quantize the discrete cosine transform coefficients of images, and their objective and subjective performance and complexity are contrasted with other lattice vector quantizers and with LBG training-mode designs.
TREE SUMMARYThe Linde, Buzo, Gray (LBG) algorithm is the most common approach to designing vector quantizers (VQs) [I]. Two disadvantages to LBG vector quantizers are their quantization/encoding complexity and the fact that only a locally optimal design is guaranteed. Lattice-based vector quantizers can greatly simplify the quantization operation, but lattice VQs can only be shown to be optimal for uniformly distributed sources or to be asymptotically optimal as the number of output points becomes large 12, 31. The design of a uniform lattice VQ for a specified lattice involves truncating the lattice to achieve the desired rate and scaling the lattice to obtain the minimum distortion for a certain source input probability density function (pdf). The number of points within a chosen normed radius from the origin in an N dimensional lattice can be calculated using the Jacobi theta functions for the I2 norm [2] or some other functions for the 11 norm [4]. To date, however, the scaling of the lattice has only been accomplished experimentally [5, 61. That is, for a particular truncated lattice and the given source pdf, a scale factor is selected and the resulting distortion is computed based upon simulations. A new scale factor is chosen and the distortion for the new scaling is also found via simulations. This procedure is repeated until a relative minimum distortion is obtained.We present here the first completely analytical method for scaling the truncated cubic lattice to minimize the mean squared error (MSE) distortion criterion and an approximate analytical technique for truncating the cubic lattice according to an 11 or 12 norm radius. The scaling and truncation expressions are given in terms of the VQ rate and dimension and the pdf of the radial norm. The efficacy of this new analytical design procedure is demonstrated by comparing the results obtained for scalar quantization of memoryless Gaussian and Laplacian sources with well-known optimal designs and by comparing the resulting 8 and 16 dimensional VQs with those obtained using the experimental, simulation-based approach. Results at rates 2-8 bits/sample are in excellent agreement, but at 1 bit/sample the new analytical expressions are somewhat in error.The possibility of a multidimensional companding approach to improving VQ performance over that produced by uniform VQs has been suggested, but no design procedures have been forthcoming. We describe in this paper a method for multidimensional piecewise linear companding which partitions the N-dimensional space into radial bands about the origin, where the radial parameter is measured in terms of the I 1 or I2 norm. The spacing of the VQ lattice points in the region immediately surrounding the origin is some value CO, the spacing in the first concentric region is s c~, in the next concentric region szco, and so on, where s > 1. For the cubic lattice, an expression for the radius rc, where the region with lattice (0.80) (1.68) (4.18) (16.19) 24.11 26.31 26.43 E6 77 25.83 (0.77) (1.62) (3.49) (15.92) Table 1....
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