Often the domain and the range of an original signal xt are modeled as contin uous. The lloyd algorithm functions as follows source distribution. This results in a nonuniformquantizer and is referred to as pdf optimized quantizer. Lloydmax scalar quantizer also known as pdfoptimized quantizer given m, the optimal bi and yi that minimize mse satisfy. Chapter 5 sampling and quantization often the domain and the range of an original signal xt are modeled as contin uous. A smooth input signal can take on a stairstep shape after quantization. Channel optimized quantizer decoders, in turn, depend on the source, the quantizer encoder, the channel, and the index assignment. The optimized quantized values of a lloydmax quantizer are estimated by setting. Optimized product quantization 1 optimized product quantization tiezheng ge, kaiming hey, qifa ke, and jian sun abstractproduct quantization pq is an effective vector quantization method. Uniform quantizers are defined by quantization intervals, which have all the same step size.
Li has to be an odd number to create a zerozone for the quantizer. Nonlinear tonetransfer curve of quantizer, showing a linear region. Variable rate image compression method with deadzone. A lloydmax based quantizer of lvalues for awgn and. Leveraging the very same data structure that is used to provide nonexhaustive search, i. Optimize quantization parameters using the lloyd algorithm. Finding an optimal solution to the above problem results in a quantizer sometimes called a mmsqe minimum meansquare quantization error solution, and the resulting pdfoptimized nonuniform quantizer is referred to as a lloydmax quantizer, named after two people who independently developed iterative methods to solve the two sets of. Due to the symmetry of the pdf and the quantizer design, the mse distortion can. Significant improvement over the uniform quantizer. Joint quantizer optimization based on neural quantizer for. The essence of pq is to decompose the highdimensional vector space into the cartesian product of subspaces and then quantize these subspaces separately.
When the pdf of the analog sample is uniform, the decision intervals and output levels of the lloydmax quantizer can be computed analytically as shown below. Example of two quantizers that satisfy the optimality criteria. But this gain is applied on the input signal not on the quantizer. Learning vector quantization extensive survey about the ieeehistory. For this paper, we assume a uniform source on 0,1, a uniform quantizer encoder, a channel optimized quantizer decoder, and a binary symmetric channel with bit. Optimized product quantization 1 optimized product. We propose a variable rate image compression method with deadzone quantizer. For conventional codec, signal is decorrelated with orthonormal transformation, and uniform quantizer is introduced. The lloyd max algorithm 10 is a wellknown approach to designing nonuniform quantizers optimized according to the prevailing pdf of the input signal.
A product quantizer can generate an exponentially large codebook at very low memorytime cost. The essence of pq is to decompose the highdimensional vector space into the. Ee368b image and video compression quantization no. Optimum quantizer, obtained with the lloyd algorithm. A lloyd quantizer is the optimal quantizer minimum distortion for a given number of. Ee398a image and video compression quantization no. If the quantizer has b bits accuracy and covers the entire range 2a. Nonuniform quantizationpdfoptimized quantization problem.
The maximally decimated mband filter bank with the bank of pdfoptimized. Optimal uniform quantizer given the number of regions, n find the optimal value of. If lnfxx is a strictly concave function of x, then there is only one optimal quantizer, there is only one quantizer with m levels that satisfies and, and this is the unique best quantizer. Comparing this sil values to the table in the previous slide where pdf optimized uniform quantizer were shown, well see that these are center values are considerably improved. Optimal quantization values not necessarily the region midpoints. They are updated during the quantizerquantize method, and are reset by the resetlog function. Following a quite common search option, a coarse quantizer is used to index data by inverted lists, and residuals between data points and centroids are pqencoded. The example shows a piecewise constant pdf fu u that takes on only two positive values, say. In your example, the nonlinearity has a large gain at the origin the characteristic function is steep while it causes some sort of saturation effect at large amplitudes. Along with the preceding properties, quantizer objects have readonly states. Observe the number of output values that falls in outer levels and inner levels if they match the assumed pdf. Quantizers with uniform encoders and channel optimized. For each j 1, n, yj represents some side information, for example, previously. Note that for this to happen the actual input pdf probability density function should match the pdf thats used in designing the pdf optimized quantizer, otherwise the expected benefit might even turn into a flaw when there is a mismatch.
Point density functions depend on the quantizer decoders. In general, the expected distortion increases when a quantizer is used. An early application of gmm was found in adaptive block quantization applied to image coding 1. Pdf optimized parametric vector quantization of speech line. Product quantization pq is an effective vector quantization method.
Design scalar quantizer with 4 quantization indices with minimum expected distortion d optimum quantizer, obtained with the lloyd algorithm decision thresholds 0. Average distortion closedform solutions for pdfoptimized uniform quantizers for gaussian rv only exist for n2 and n3 optimization of is conducted numerically v u k, k k p k k0 n 1 1, k d 1 12 p k k 2 k0 n 1. Note that the rate of the quantizer is be calculated as r. Codebook is a length n vector that contains the output set. Overview of different variations of vector quantization. Wang, lab manual for multimedia lab, experiment on. Lec 43 principles of communication systemsi optimal lloyd max quantizer design iit kanpur duration. Face, a 1024 x 768 size image of a raccoon face, is used here to illustrate how kmeans is used for vector quantization.
The gain matrix l will he the key parameter in the predictive quantizer design and will be discussed in detail below. That is, the time or spatial coordinate t is allowed to take on arbitrary real values perhaps over some interval and the value xt of the signal itself is allowed to take on arbitrary real values again perhaps within some interval. Nonuniform scalar quantizer electrical engineering and. When the pdf of the input analog samples is not uniform, then the quantization steps are not constant and the optimal solutions are obtained by solving the transcendental equations 2. The lloyd quantizer that is constructed using an iterative procedure provides the minimum distortion for a given number of reconstruction levels, in other words, it generates the pdfoptimized scalar quantizer. Optimized product quantization tiezheng ge, kaiming he, qifa ke, and jian sun abstractproduct quantization pq is an effective vector quantization method. Non uniform quantization functions optimal output alphabets and levels lloyds. Due to the symmetry of the pdf and the quantizer design, the mse distortion can be written as. Assuming that you know the pdf of the samples to be quantized design the quantizers step so that it is optimal for that pdf. For uniform pdfx max x f x x 12x max want to uniformly quantize an rv x ux max,x max assume that desire m rls for r. Reconstruction levels of quantizer are uniformly spaced quantizer step size, i.
The optimized codebook has the same vector size as. The set of inputs and outputs of a quantizer can be scalars scalar quantizer. The code below shows how the quantiz function uses partition and codebook to map a real vector, samp, to a new vector, quantized, whose entries are either 1, 0. Solution depends on input pdf and can be done numerically for. This is typically achieved using a rounding operation, for example. Our solution in this work is locally optimized product quantization lopq. In this case, the decision intervals are all equal as well as the intervals between the output levels and the quantizer is called a uniform quantizer. Chapter 6 adaptive quantization without side information1. Principles of communication systemsi 7,079 views 36.
Uniform quantizer an overview sciencedirect topics. Pierre duhamel, michel kieffer, in joint sourcechannel decoding, 2010. Let quantization levels be the midpoint of the quantization regions except. Consider a symmetric scalar quantizer with 3 intervals, qx 8 a and a quantizer input with a zeromean laplace pdf, fx 1 2m e jxj m a derive the optimal reconstruction value bas a function of the decision threshold afor mse distortion. The level li of the initial quantizer is determined based on the computational cost and signaltoquantization noise sqnr of the quantizer. The optimal space decomposition is important for the pq. Instead of devoting attention to the signal being quantized, we shall consider its probability density function. The quantizer block discretizes the input signal using a quantization algorithm.
More recent examples of applications include clustering. Firstly, the autoencoder network is trained with radogaga 6 framework, which can make the latents. Under what circumstances is a nonuniform quantizer. This is to be expected because these are the optimal quantizers for these two distributions while the uniform quantizer was not optimal for the gaussian and the. The block uses a roundtonearest method to map signal values to quantized values at the output that are defined by the quantization interval. Still the result should be quantized by a uniform quantizer afterwards. A method to sparse eigen subspace and eigenportfolios. Rather than allocate a uniform step size, as would be optimum for a uniform pdf, the lloyd max approach identifies decision boundaries according to the mean values of equal area partitions of the pdf curve.
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