Gaussian Golomb CodesSeishi TAKAMURA and Yoshiyuki YASHIMA

NTT Cyber Space Laboratories, NTT CorporationY-517A 1-1 Hikarino-oka, Yokosuka, Kanagawa, 239-0847 Japan

Email: {takamura.seishi, yashima.yoshiyuki}@lab.ntt.co.jp

Gaussian sources are observed in vast wide areas. Contrarily to what happens for thegeometric and double-sided geometric distributions, there is no simple, instantaneous codefor the normal distribution.

This paper tackles this problem by mapping the normal distribution into the geometricdistribution before applying Golomb codes, which is optimal for geometric distributions.In our mapping, a pair of normally-distributed i.i.d. integers (say (x, y)) is concatenated andthen mapped to one natural number z(x, y). The conditions that z shall satisfy are:minx,y z(x, y) = 0, l(x, y) < l(a, b)⇒ z(x, y) < z(a, b) and z(x, y) = z(a, b)⇔ (x, y) = (a, b),where l(x, y) is an arbitrary distance measure between the origin and the grid point (x, y),such as the Euclidean norm. Fig. 1 is an example of such a mapping. When the signal iscorrelated, l(x, y) = x2 − 2ρxy+ y2, where ρ is the autocorrelation of the source, is suitable.The mapping can be easily obtained using a computer program. In addition, if the upper-and lower- bounds of the source is known, pre-calculated mapping table can be stored in thememory because it is independent of source statistics. Of course, this table is not needed tobe downloaded / transmitted. After this mapping, z is made geometrically-distributed andconventional Golomb codes can be efficiently applied.

Coding efficiency comparison for our codes (Gaussian Golomb) and ordinary Golombcodes for quantized Gaussian sources is shown in Fig. 2. The unit-variance Gaussian dis-tribution is quantized with step size of d. The quantizer maps input values within eachbin into integers. Our codes constantly yield better coding efficiency, which is higher than98%. In particular, the coding efficiency for d < 0.1 area is stably higher than 99.5%.

We also conducted the experiment on actual pixel value residual data of decoded video,which was observed to be quasi-Gaussian in Fig. 3. Our coding efficiency is no worse than90%, in average 94.1%. Conventional Golomb codes yield 87.4% in average, which isabout 7 points worse.

In addition, we propose an estimation method of optimal Golomb parameter usingsource variance to prevent exhaustive parameter search, which is verified to be practicallyefficient enough.

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Figure 2: Coding efficiencyfor quantized Gaussian source

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Figure 3: Coding effi-ciency for actual data sets

2007 Data Compression Conference (DCC'07)0-7695-2791-4/07 $20.00 © 2007