Affine Modeling for theComplexity of Vector Quantizers

Estevan P. Seraco and Jose Gabriel R.C. Gomes1

We use a scalar function Θ to describe the complexity of data compression sys-tems based on vector quantizers (VQs). This function is associated with the analoghardware implementation of a VQ, as done for example in focal-plane image com-pression systems. The rate and distortion of a VQ are represented by a Lagrangiancost function J . In this work we propose an affine model for the relationship betweenJ and Θ, based on several VQ encoders performing the map R

M → {1, 2, . . . , K}. Adiscrete source is obtained by partitioning images into 4× 4 pixel blocks and extract-ing M = 4 principal components from each block. To design entropy-constrainedVQs (ECVQs), we use the Generalized Lloyd Algorithm. To design simple interpola-tive VQs (IVQs), we consider only the simplest encoder: a linear transformation,followed by a layer of M scalar quantizers in parallel – the K cells of R

M are definedby a set of thresholds {t1, . . . , tT}. The T thresholds are obtained from a non-linearunconstrained optimization method based on the Nelder-Mead algorithm.

The fundamental unit of complexity Θ is transistor: we only count the transistorsthat are used to implement the signal processing part of a VQ analog circuit: innerproducts, squares, summations, winner-takes-all, and comparators. The complexityfunctions for ECVQs and IVQs are as follows: ΘECVQ = 2KM + 9K + 3M + 4 andΘIVQ = 4Mw1+2Mw2+3Mb1+Mb2+4M +3T , where Mw1 and Mw2 are the numbersof multiplications by positive and by negative weights. The numbers of positive andnegative bias values are Mb1 and Mb2. Since ΘECVQ and ΘIVQ are scalar functionsgathering the complexities of several different operations under the same unit, theyare useful for the development of models relating rate-distortion cost to complexity.

Using a training set, we designed several ECVQs and plotted all (J , Θ) pointson a plane with axes log10(Θ) and log10(J) (J values from a test set). An affinemodel log10(Θ) = a1 log10(J) + a2 became apparent; a straightforward applicationof least squares yields the slope and offset coefficients. This procedure was repeatedfor IVQs. The error between the model and the data has a variance equal to 0.005for ECVQs and 0.02 for IVQs. To validate the ECVQ and IVQ complexity models,we repeated the design and test procedure using new training and test sets. Then,we used the previously computed complexity models to predict the Θ of the VQsdesigned independently: the error between the model and the data has a varianceequal to 0.01 for ECVQs and 0.02 for IVQs. This shows we are able to predictthe rate-distortion performance of independently designed ECVQs and IVQs. Thisresult serves as a starting point for studies on complexity gradients between J and Θ,and as a guideline for introducing complexity constraints in the traditional entropy-constrained Lagrangian cost.

1Universidade Federal do Rio de Janeiro, COPPE, Electrical Engineering Program. E-mails:{estevan, gabriel}@pads.ufrj.br. This work was supported by Brazilian research funding agencies:CNPq, FAPERJ, and FUJB/UFRJ.

2009 Data Compression Conference

1068-0314/09 $25.00 © 2009 IEEE

DOI 10.1109/DCC.2009.55

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