On theMeasurement ofCurvature ina Quantized Environment

Some aspects oftheapplication ofthemathematical concept ofcurvature as a practical descriptor ofshapeforpattern recognition andimageprocessing applications are investigated. Theaccuracytowhichthecurvature oftheboundary ofa binary (silhouette) shapecan be estimated froma givenquantized version ofthatshapedepends upon twofactors intheestimation process:thecontour-tracing algorithm bywhichboundary points on thequantized shape are defined andthemethodwhereby the curvature function estimate issmoothed topartially remove the quantization noiseresulting fromthedigitization oftheoriginal shape. Inthis paper,sixcontour-tracing algorithms aredescribed and usedtoextract curvature functions fromquantized testshapes in theabsence ofsmoothing ofanykind.Modelsoftheaveragequan- tization noise characteristics inthefrequency domain arethende- veloped forthecurvature functions corresponding toeachofthe sixcontour-tracing algorithms. Themodels areusedtocomparethe performance ofthesixalgorithms on thebasisofthequantization noise characteristics ofeach. Itisfoundthattheuseful bandwidth ofcurvature functions obtained fromquantized shapes issomewhat less thanthetheoretical limit imposed bytheNyquist sampling theorem. Themodelsde- veloped servetodemonstrate thevariation intheuseful bandwidth ofthecurvature functions corresponding toeachcontour-tracing algorithm. Thevariation inbandwidth withquantizing arrayresolu- tion canalso bepredicted withthemodels. Therelationship between thebandwidth ofa curvature function andtheamountofdetail representable inthecorresponding curveintheplane isthenex- plored. Finally, some characteristics oftheprocessofsmoothing curvature functions toeliminate quantization noise arepresented. IndexTerms-Contour coding, contour tracing, curvature, image processing, patterm recognition, quantization noise, two-dimensional.