AB HELSINKI UNIVERSITY OF TECHNOLOGY Department of Computer Science and Engineering Telecommunications Software and Multimedia Laboratory Sami Kiminki Sound Propagation Theory for Linear Ray Acoustic Modelling Supervisor: ProfessorLauriSavioja Instructor: TapioLokki,D.Sc. (Tech.) HELSINKI UNIVERSITY ABSTRACT OF THE OF TECHNOLOGY MASTER’S THESIS Author: SamiKiminki Nameofthethesis: SoundPropagationTheoryforLinearRayAcousticModelling Date: March7,2005 Numberofpages: 100+14 Department: ComputerScienceandEngineering Professorship: T-111 Supervisor: ProfessorLauriSavioja Instructor: TapioLokki,D.Sc. (Tech.) Inthiswork,alinearrayacousticmodellingtheoryisconstructed. Thetheoryformsabase for linear ray acoustic modellingmethods. As such, the theory can be used to derive and analyseraymethods. Threeexistingraymodellingmethods(theimagesourcemethod,the radiosity method, and the ray tracing method) are shown to be derivable from the theory. Itisalsosuggestedthatthetheorycanbeusedtoderiveacousticcharacteristicsestimators such as the average reverberation time of a room. To the author’s knowledge, this is the (cid:2)rstattempttocreate atheoryfor acousticraymodelling. The theory is divided into two parts: general and acoustic. The general theory consists of general de(cid:2)nitions, time-dependent energy propagationequations, and detection equa- tions. The general part yieldstime-independentray modellingtheoryby eliminatingtime dependency, thus linking the acoustic and the graphic ray modelling. The acoustic part speci(cid:2)esthegeneralde(cid:2)nitionsasacousticde(cid:2)nitions. Thetheorylackssub-surfacescat- tering re(cid:3)ection and edge diffraction. A well-de(cid:2)ned extension path for the inclusion is considered,however. The general de(cid:2)nitions consist of mathematical and physical de(cid:2)nitions. Energy propa- gationequationsare constructedindetail, resultingin the re(cid:3)ection-iterativeconstruction and the acoustic rendering equation. The (cid:2)rst is a straightforward construction, and the second is a balance equation (cid:151) extension of the Kajiya’s rendering equation. The equa- tionsevaluateimpulseenergyresponsesandareshowntobeequivalentusinglinearoper- atoranalysis. Anexamplede(cid:2)nitionforauralizationofenergyresponsesisconstructed. Keywords: generalmodellingtheory,imagesourcemethod(ISM),radiosity,raytracing,rayacous- ticmodelling ii TEKNILLINEN KORKEAKOULU DIPLOMITY(cid:214)N TIIVISTELM˜ Tekij(cid:228): SamiKiminki Ty(cid:246)nnimi: ˜(cid:228)nenetenemisteoria lineaarisessa s(cid:228)deakustiikassa P(cid:228)iv(cid:228)m(cid:228)(cid:228)r(cid:228): 7.3.2005 Sivuja: 100+14 Osasto: Tietotekniikan osasto Professuuri: T-111 Ty(cid:246)nvalvoja: ProfessoriLauriSavioja Ty(cid:246)nohjaaja: TkTTapioLokki Ty(cid:246)ss(cid:228) rakennetaan pohjateoria lineaariselle s(cid:228)deakustiselle mallinnukselle. Teoriaa voi- daank(cid:228)ytt(cid:228)(cid:228)s(cid:228)demenetelmienjohtoonjaanalyysiin.Kolmeolemassaolevaas(cid:228)deakustista mallinnusmenetelm(cid:228)(cid:228) osoitetaan olevan johdettavissa teoriasta (kuval(cid:228)hde-, radiositeetti- jas(cid:228)teenseurantamenetelm(cid:228)).Lis(cid:228)ksiehdotetaan,ett(cid:228)teoriaavoitaisiink(cid:228)ytt(cid:228)(cid:228)my(cid:246)sakus- tisten tunnuslukujen estimointiin, esimerkkin(cid:228) j(cid:228)lkikaiunta-aika. T(cid:228)m(cid:228) on tekij(cid:228)n tiet(cid:228)- myksenmukaanensimm(cid:228)inenyritysluodakattavas(cid:228)deakustisenmallinnuksenteoria. Teoriajaetaankahteenosaan,yleiseenjaakustiseen.Yleinenosak(cid:228)sitt(cid:228)(cid:228)yleisetm(cid:228)(cid:228)ritel- m(cid:228)t,aikariippuvatenergiankulkuyht(cid:228)l(cid:246)tsek(cid:228)havainnointiyht(cid:228)l(cid:246)t.Teorianyleisest(cid:228)osasta saadaan lis(cid:228)ksi teoria s(cid:228)degra(cid:2)ikalle, kun eliminoidaan aikariippuvuudet. Akustinen osa spesi(cid:2)oiyleisetm(cid:228)(cid:228)ritelm(cid:228)takustisiksim(cid:228)(cid:228)ritelmiksi.Teoriastapuuttuupinnanalaissiron- taheijastuksissasek(cid:228)reunadiffraktio.Teorianlaajennettavuusn(cid:228)idenpuutteidenosaltaon otettuhuomioon. Yleiset m(cid:228)(cid:228)ritelm(cid:228)t koostuvat matemaattisista ja fysikaalisista m(cid:228)(cid:228)ritelmist(cid:228). Energian- kulkuyht(cid:228)l(cid:246)t konstruoidaan yksityiskohtaisesti. T(cid:228)m(cid:228) johtaa heijastusiteratiiviseen kons- truktioonsek(cid:228)akustiseenmallinnusyht(cid:228)l(cid:246)(cid:246)n.Ensimm(cid:228)inenonsuoraviivainenkonstruktio. J(cid:228)lkimm(cid:228)inen on tasapainoyht(cid:228)l(cid:246), joka on Kajiyan mallinnusyht(cid:228)l(cid:246)n laajennus. Yht(cid:228)l(cid:246)t tuottavat energiaimpulssivasteita ja konstruktiot osoitetaan yht(cid:228)l(cid:228)isiksi lineaarioperaatto- rianalyysilla. Ty(cid:246)ss(cid:228) rakennetaan esimerkinomainenm(cid:228)(cid:228)ritys energiavasteiden auralisaa- tioon. Avainsanat:yleinenmallinnusteoria,kuval(cid:228)hdemenetelm(cid:228),radiositeetti,s(cid:228)teenseuranta,akustiikan s(cid:228)demallinnus iii Acknowledgments FirstofallIthankmysupervisingprofessorLauriSaviojaandmyinstructorTapio Lokki. Theymadeitpossibleformetoconductthisworkandprovidedinvaluable insightsintothevast(cid:2)eldofacousticmodelling. Ialsothankmyclosestassociate Heli Nironen for providing lots of background information and reference mate- rial,andespeciallyforthetirelessconversationaleffortsonvarioussound-related topics. I thank professor Timo Eirola for very useful mathematical discussions, and for allowing me to present the work in his seminar in computational science and en- gineering. I also thank Jaakko Lehtinen for helping me in radiance calculus, and JanneKontkanenforvaluableinformationoncomputergraphics. Additional thanks should go to Vesa Hirvisalo, Jan Jukka Kainulainen, and Juha Tukkinen for support and otherwise important in(cid:3)uence on my work. Finally, I thank Maarit Tirri for numerous comments on the textual layout. Any remaining linguisticanomaliesare duetomypersonalstubbornnessandobsessions. iv Contents Preface xiii 1 Introduction 1 1.1 OrganizationoftheThesis . . . . . . . . . . . . . . . . . . . . . 5 2 Background 7 2.1 SoundandAcoustics . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 PhysicsofSound . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 EnergyofSound . . . . . . . . . . . . . . . . . . . . . . 12 2.2 AcousticModelling . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 ImpulseResponses . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 RayMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 ImageSourceMethod . . . . . . . . . . . . . . . . . . . 17 2.4.2 RadiosityMethod . . . . . . . . . . . . . . . . . . . . . . 19 2.4.3 RayTracingMethods . . . . . . . . . . . . . . . . . . . . 21 3 Simpli(cid:2)ed GeneralEnergy PropagationTheory 24 3.1 Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 FundamentalDe(cid:2)nitions . . . . . . . . . . . . . . . . . . 25 3.1.2 PolyhedralEnvironments . . . . . . . . . . . . . . . . . . 29 3.1.3 DiscreteEnvironments . . . . . . . . . . . . . . . . . . . 29 3.1.4 VisibilityComputation . . . . . . . . . . . . . . . . . . . 30 3.2 RadiationandRe(cid:3)ection . . . . . . . . . . . . . . . . . . . . . . 30 3.2.1 EnergyFlow . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.2 TheEnergySource . . . . . . . . . . . . . . . . . . . . . 37 3.2.3 TheObserver . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.4 Radiance . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.5 Re(cid:3)ection . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.6 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 EnergyPropagationEquations . . . . . . . . . . . . . . . . . . . 48 3.3.1 TemporalIntensityAlgebra . . . . . . . . . . . . . . . . 48 3.3.2 GeometryTermandRe(cid:3)ectionKernel . . . . . . . . . . . 52 v 3.3.3 Re(cid:3)ection-iterativeConstruction . . . . . . . . . . . . . . 55 3.3.4 AcousticRenderingEquation . . . . . . . . . . . . . . . 60 3.3.5 EquivalencyofRe(cid:3)ection-iterativeConstructionandARE 62 3.3.6 RemarkonDetection . . . . . . . . . . . . . . . . . . . . 64 3.4 RadiationatVariousFrequencies . . . . . . . . . . . . . . . . . . 64 3.4.1 MathematicalDiscussion . . . . . . . . . . . . . . . . . . 65 3.5 ConsiderationsonExtensions . . . . . . . . . . . . . . . . . . . . 67 3.5.1 ExtendingforEdgeDiffraction . . . . . . . . . . . . . . 67 3.5.2 ExtendingforSub-surface Scattering . . . . . . . . . . . 68 4 AcousticEnergy PropagationTheory 70 4.1 AdaptationoftheGeneralEnergyPropagationTheory . . . . . . 71 4.2 AuralizationofEnergyResponse . . . . . . . . . . . . . . . . . . 72 4.3 SpecializationsoftheTheory . . . . . . . . . . . . . . . . . . . . 73 4.3.1 ImageSourceMethod . . . . . . . . . . . . . . . . . . . 74 4.3.2 RadiosityMethod . . . . . . . . . . . . . . . . . . . . . . 77 4.3.3 RayTracingMethod . . . . . . . . . . . . . . . . . . . . 78 4.4 ConsiderationsonUsingtheTheory . . . . . . . . . . . . . . . . 79 5 Conclusion 81 5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2 FurtherWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 A SomeEssentialMathematics 86 A.1 EuclideanSpace . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 A.2 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 A.3 LinearOperatorAnalysis . . . . . . . . . . . . . . . . . . . . . . 93 B ABriefNoteon BDRFsand Lambertian DiffuseRe(cid:3)ections 96 Bibliography 98 vi List of Figures 2.1 A2-dimensionalspring-masssystemwithdisplacement . . . . . . 11 2.2 Direct,(cid:2)rst,andsecondorderimagesources . . . . . . . . . . . . 17 3.1 Possibleraypathsinspecularre(cid:3)ectingenvironment . . . . . . . 32 3.2 Possibleraypathsindiffusere(cid:3)ectingenvironment . . . . . . . . 33 3.3 Incidentenergy fromapointsourcetoa smallsurface patch . . . 34 3.4 Radiantincidentenergytoa smallsurface patch . . . . . . . . . . 35 3.5 Energy(cid:3)owofincidentradiation . . . . . . . . . . . . . . . . . . 40 3.6 Re(cid:3)ectionofplanarbeam . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Intensitymeasurementofplanarpropagatingwavefront . . . . . . 49 3.8 Parametersofthere(cid:3)ectionkernel . . . . . . . . . . . . . . . . . 54 3.9 Diffractionalbendingofrays . . . . . . . . . . . . . . . . . . . . 68 4.1 Re(cid:3)ectingbeam . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 A.1 Mirrorre(cid:3)ection. . . . . . . . . . . . . . . . . . . . . . . . . . . 88 vii Notations Latin alphabets A variableusedinderivationsandproofs a m3 pointinsurface geometry area m2 surfacearea operator(p.77) (cid:2) (cid:3) B re(cid:3)ectionoperator(p.62) (cid:2) (cid:3) B allre(cid:3)ectionsoperator(p.66) c m speedofwavefront(p.36) b s card cardinality,sizeofset(p.21) (cid:2) (cid:3) D delaypattern(p.72) D linearsystemresponseoperator(p.14) d totaldetection(p.48,64) b representation function of linear system response op- erator(p.14) d detectionofdirectsourcetoobserverradiation(p.55) 0 d detectionofradiance,detectionoftotalre(cid:3)ectedradi- r ationviaanynumberofre(cid:3)ections(p.57) det matrixdeterminant dtf directionaltransferfunctionforobservation(p.39,47) exitant(pre(cid:2)x)(p.37) e F patchemissionvector(p.20,78) f 1 frequency(p.42) s f generalfunction (cid:2) (cid:3) viii f 1 bidirectional re(cid:3)ection distribution function (BRDF) r sr (p.41) (cid:2) (cid:3) f BRDF ofanidealdiffusere(cid:3)ection(p.45) r;d f BRDF ofanidealspecularre(cid:3)ection (p.47) r;s G environmentalsurface geometry(p.27) g Kajiyangeometrytermwithpropagationdelay(p.53) g(cid:136) Kajiyan geometry term without propagation delay (p.53) Ha mediumabsorptionoperator(p.51) I W intensity, irradiance, energy (cid:3)ow per surface area m2 h i (p.36) I identitymatrix,identityoperator incident(pre(cid:2)x) (p.37) i (cid:17)(cid:136)(t) impulseresponse(p.72) L(W) W radiance(p.40,60) m2 L (W) hWi primaryradiance (p.60) 0 m2 ‘(t) h i time-dependent radiance in non-absorptive medium (p.52) totalpropagatedradiance(p.60,61) ‘ primaryradiance (p.54,57(cid:150)58) 0 ‘ primaryradiance afterk re(cid:3)ections(p.57,60) k ‘(cid:136) time-dependent intensity in non-absorptive medium (p.50) M mirrorre(cid:3)ection operator(p.45,87) n surfacenormal(p.27) O() asymptoticcomplexityclass P vectorofenergy(cid:3)owsofpatches(p.20,77) P patch,smallbutnotin(cid:2)nitesimalsurfacearea (p.19) k p patternofemittance(p.38) e R re(cid:3)ectionkernel(p.53) r distance,radiusofa sphere ix S losslesspropagationoperatorfor distancer (p.50) r S(cid:136) propagationoperatorwithmediumabsorptionfordis- r tancer. UsedintheconstructionofTIA(p.49) s noisesignalwithaverageunitintensity(p.72) T triangle(p.89) t time v m 3 velocity(vectorquantity) s 3 x [m] pointinspace (cid:2) (cid:3) x locationoftheobserver(p.39,55) o x raypath(p.10) p x locationoftheenergy source(p.37,55) s Lower-case Greekalphabets b re(cid:3)ectance factor,totalre(cid:3)ectance (p.42) g sensitivitypattern(p.72) d Diracdeltafunctional(p.15,45,91) q elevation angle (cid:151) angle between W and surface nor- mal(p.26) n visibilityfunction(p.27) n 1 visiblegeometry(p.28) (cid:0) n inverseprojection(p.28) p r biconicalre(cid:3)ectance factor (p.42) probabilitydensity(p.92) s energypro(cid:2)leofsub-band(cid:2)lteredimpulse(p.66,72) signal(p.14) t generalscalarorvector f azimuthangleofW(p.26) j angle y waverepresentationofafundamentalparticle(p.13) w [sr] solidangle,setofdirections(p.26) x
Description: