Many problems in computer graphics deal with complex 3D scenes where visibility, proximity, collision detection queries have to be answered. Due to the complexity of these queries and the one of the models they are applied to, data structures most often based on hierarchical decompositions have been proposed to solve them. As a result of these involved algorithms/data structures, most of the analysis have been carried out in the worst case and fail to report good average case performances in a vast majority of cases. The goal of this work is therefore to investigate geometric probability tools to characterize average case properties of standard scenes such as architectural scenes, natural models, etc under some standard visibility and proximity requests. In the first part we recall some fundamentals of integral geometry and discuss the classical assumption of measures invariant under the group of motions in the context of non uniform models. In the second one we present simple generali...

Data structures based on uniform subdivisions of the space ---also known as bucketing--- have the nice properties that they can be walked through very easily and can provide neighborhood relations at low cost. For data sets which are uniformly scattered in 2D or 3D space, this makes the implementation of algorithms such as ray tracing, nearest neighbors computation or Delaunay triangulation almost trivial. But should the processed data set admit dense clusters, the spatial partitioning does not result in data partitioning so that the performances are collapsing.

they can be walked through veryeasily and can provide neighborhoodrelationsatlowcost.Fordatasetswhichare uniformlyscatteredin2Dor3Dspace,thismakestheimplementationofalgorithmssuchasraytracing,nearestneighborscomputationorDelaunaytriangulationalmosttrivial. Butshouldtheprocesseddatasetadmitdenseclusters,the spatialpartitioningdoesnotresultindatapartitioningso thattheperformancesarecollapsing. Althoughithasbeenknownforalongtimeindimension onethatrecursivebucket-sortadmitsalinearcomplexity forawiderangeofprobabilitydensities,recursivebucketlikedatastructureshavenotreceivedanyattentioninthe computationalgeometrycommunity. Ithasbeenobserved incomputergraphicsthatthesewerethefastesttoray-trace, butthequestionofunderstandingwhytheyarenotjust anotherspacepartitioningdatastructurebutrathertheonly datastructurethatsucceedsincapturingtheprobabilistic propertiesofdatadistributionremainsopen. Thispaperisafirststepinthisdirectionandinvestigates hierarchical recursiveandnonrecursivedatastructuresforray-tracing. First,weshowthatpreciselyanalyzinganoptimized ray-traceris adifficulttaskduetothe contextsensitivityofthecallscostsofthefunctionscalled mostoften. Second, weexhibitstatisticsshowingthatif uniformgridsaredefinitelynottherightdatastructureto usefornon-uniformdistributions,recursivegridsarevery goodathandlingsuchdistributions.Third,wepresentseveralimprovementsoftheHierarchyofUniformGridsdata structure,whichresultforthebestcasesinrunningtimes improvedbyuptoafactorthreewithreferencetothepreviouslybestknownsolution.