Absolute moment block truncation coding and its ...

1148

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 10, OCTOBER 1984

Absolute Moment Block Truncation Coding and Its Application to Color Images

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MAXIM0 D. LEMA AND 0. ROBERT MITCHELL

Rain Rate (rnrn/hr)

Fig. 4. Comparisonoffiveyearsofannualrainratedistributionstakenat Denver, CO, with distribution predictions made using using the year-toyear variability derived from the original Rice-Holmberg model.

Abstract-A new quantizationmethod that uses thecriterion of preserving sample absolute moments is presented. This is based on the same basic idea for block truncation coding of Delp and Mitchell but itis simpler in any practical implementation. Moreover, output equationsare thosefor a two-level nonparametric minimum mean square error quantizer when the threshold isfixed to the sample mean. The application of this method to single frame color images is developed. A color image codingsystem that usesabsolutemoment block truncationcoding of luminance and chroma information is presented. Resulting color images show reasonable performance with bit rates as low as 2.13 bitdpixel.

I. INTRODUCTION

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1 1 1 1 1 1 1 1 l 1 1 1 1 1 1 1 1 l 1 I

lob 20 40 60 80 100 120 140 160 180 X l O 1 2 k ’ 4 O Rain Rate (rnrnihr)

Fig.5.Comparisonoffive years of annual rain rate distributions taken at Denver,CO, with distributionpredictionsmadeusingtheyear-to-year variability derived from the modified Rice-Holmberg model.

SR atvarious tion using theyear-to-yearstandarddeviation exceedance percentile levels. These are labeled “ O S % , ” “5%,” “95%,” and “99.5%,” correspondingly. Also included on Figs. 4 and 5 are five years (1966-1970) of annualdistributions of rainrateobtainedfromobserved dataforDenver, CO,providedby theWeather-Environment Group of the U.S. Air ForceCommunicationsCommandat Scott Air Force Base, IL. Fig. 4 shows results using the earlier SR values given in [ 11, whereas Fig. 5 shows results using SR values corresponding to those given by (4). Clearly, the bounds of the predictions in Fig. 5 encompass the data, whereas those ofFig. 4 do not. While this may possibly be an exceptional instance,itis,however,apparentthatatleastinsomecases the S R ’ s of Figs. 1-3 will be more realistic than those given in

Thispaperpresents a newquantizationmethodthatuses the theidea of preserving momentsas adesigncriterionof quantizer. An extensive study of moment-preserving quantization was done by Delp in [ 21 and was applied t o still images byDelpandMitchell [ 11. [ 31 and to movingimageryby HealyandMitchell [ 4 1 . A majorapplicationofmomentpreserving quantizationleads t o what is knownasblock truncation coding or BTC. This coding method takes a finite number of samples and tries to preserve their first two sample moments, assigningatwo-levelquantizerdesigned forthat goal.Theequationsofthequantizerarerelativelysimple, andblocktruncationcodinghasshownrobustbehaviorin the presence of channel noise [ 1 1. It also gives good reconstructed images, since the method preserves local characteristics of spatialblocks of the image important to the human of these blocks the observer. The method computes for each sample mean and the sample standard deviation, thatis, .

m

where M is the total number of pixels in the block, and xi represents the grey value of each pixel. Bothvaluesaretransmittedalongwith a bitplanewhich contains ones in those places where x i 2 5 and zeros otherwise.At the receiver, the block is reconstructed with a twolevel quantizer that preserves the sample mean and the sample variance. The outputvalues that achieve this goal are (see [ 1] )

REFERENCES E. J. Dutton andH. T. Dougherty, “Year-to-year variability of rainfall for microwave applications in the U.S.A.,” ZEEE Trans. Commun., vol.COM-27,pp.829-832,May1979. P. L. Rice and N. R. Holmberg, “Cumulative time statisticsof surface point-rainfallrates,” ZEEE Trans. Commun., vol.COM-21,pp. 1131-1136,Oct.1973. E. J. Dutton, “Earth-space attenuation prediction procedures4atto 16 GHz,” Nat. Tech. Inform. Service [NTIS], Springfield, VA, OT Rep. 77-123,PB269-228/AS, May 1977. E. J. Dutton, H. T. Dougherty, and R. F. Martin, Jr., “Prediction of European rainfall and link performance coefficients at 8 to 30 GHz,” NTIS, Springfield, VA, APIA-000804, Aug. 1974. B.R.BeanandE. J. Dutton,“Radiometeorologicalparametersand climatology,” Telecommun. J., vol. 43, pp. 427-435, June 1976. E.J.Dutton,“Earth-spaceattenuationpredictionsforgeostationary satellite links in the U.S.A.,” NTIS, Springfield, VA,NTIA Rep. 7810, PB289-8411AS.Oct. 1978.

where q is the number of pixels greater than or equal to i j ,

Paper approved by the Editor for Signal Processing and Communication Electronics of the IEEE Communications Society for publication without oral presentation.ManuscriptreceivedNovember22,1982;revisedAugust 4, 1983. This work was supported in part by the U.S. Army Research Office. M.D.LemaiswiththeDepartmentofElectricalEngineering,Purdue University,WestLafayette,IN47907,andtheEmpresaNacionalde Telecommunicaciones (ENTEL), Argentina. 0. R. Mitchell is with the Department of Electrical Engineering, Purdue University, West Lafayette, IN 47907. ’

0090-6778/84/1000-1148$01.OO 0 1984 IEEE

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-32, NO. 10,OCTOBER 1984

Themethodpresentedinthispaperpreservesabsolute moments rather than standard moments (see [ 5 1). It will be shown that this method gives Similar pictorial results to block it gives simplerequationswhichlead truncation coding but t o fastercomputationand smaller meansquareerrorthan BTC. Therelationbetween AMBTC andthenonparametric minimummeansquareerror (MMSE)two-levelquantizeris given in Section 111. An application of this method to color images is discussed. The problem of coding color images starts with the trichromatic nature of human color vision and the correlation that generally I exists between color planes of actual pictures. It is therefore "I t important to consider some decorrelating operation. A system that uses absolute moment preservation is presented. Finally, Fig. 1. Input-outputrelationfora typical two-levelquantizer. resultsfromcodingactualcolorimageswiththeproposed system are given. Inserting into (7) CODING 11.ABSOLUTE MOMENT BLOCK TRUNCATION (AMBTC) mz= x i - 3 4 - m3+ x i +?(m - q ) Since the principal idea used in block truncation coding is forx$=r) fOIXi>?j t o achieve compression while preserving some sample moments, there exist other variants that lead to simpler results. Here, a new method of coding still images that preserves absolute sample moments is presented. Let us call it absolute moment block truncation coding orAMBTC. Letadigitizedimagebedividedintoblocksof n X n Equation (8) is a more efficient way of computing ii. Let us pixels. Each block is quantized in such a way that each resultdefine a quantity that will be useful: ingblockhasthesamesamplemeanandthesamesample xi firstabsolutecentralmomentofeachoriginalblock.Let be the grey level of a pixel in the block where 1