Vol.2014-MUS-104 No.14
2014/8/26

৘ใॲཧֶձ‫ڀݚ‬ใࠂ
IPSJ SIG Technical Report

MachineDancing: μϯεಈ࡞σʔλͷࣗಈ෼ੳʹ‫ͮ͘ج‬Իָʹ࿈ಈͨ͠μϯεੜ੒ख๏ਂࢁ ֮1‫ޙ‬౻ ਅ޹1֓ཁɿμϯεͱԻָ͕࿈ಈͨ͠σʔλ͔Β‫ػ‬ցֶशΛߦ͍ɺ৽͘͠ೖྗ͢Δָ‫ʹۂ‬ରͯ͠ 3 ࣍‫ݩ‬ίϯϐϡʔλάϥϑΟοΫεͷΩϟϥΫλͷμϯεΛࣗಈੜ੒Ͱ͖Δख๏ MachineDancing ΛఏҊ͢ΔɻैདྷɺμϯεͷஅยΛ४උͦ͠ΕΛ֬཰ϞσϧͳͲΛ༻͍ͯԻָʹ߹Θͤͯ઀ଓ͢Δ͜ͱͰμϯεࣗಈੜ੒͕࣮‫͞ݱ‬Ε͖ͯͨɻ͔͠͠μϯεͷஅยΛ੾ΓషΓ͢ΔͷΈͰɺμϯεಈ࡞ࣗମͷֶशɾੜ੒ख๏ͱ͸ͳ͓ͬͯΒͣɺੜ੒݁ՌͷόϦΤʔγϣϯʹ‫ݶ‬ք͕͋ͬͨɻຊ‫Ͱڀݚ‬͸μϯεಈ࡞ͷ֬཰Ϟσϧͱͯ͠Ψ΢γΞϯϓ
ϩηεʢGPʣΛ༻͍ɺμϯεͱԻָͷରԠؔ܎ͷΈͰͳ͘ɺμϯεಈ࡞ࣗମΛ΋ֶश͢Δ͜ͱͰɺ৽ͨͳ
ಈ࡞Λָ‫ʹۂ‬࿈ಈͯࣗ͠ಈੜ੒Ͱ͖Δख๏ΛఏҊ͢Δɻ

1. ͸͡Ίʹ
3 ࣍‫ݩ‬ʢ3DʣίϯϐϡʔλάϥϑΟοΫεʢCGʣͷΩϟϥΫλ͸޿͘ී‫͕ͨ͠ٴ‬ɺͦͷಈ࡞ͷ੍࡞͸ਓखʹΑΔ΋ͷ͕ଟ͍ɻΩϟϥΫλͷঢ়‫ʹگ‬Ԡͯ͡ଟ༷ͳಈ࡞Λͭ͘Δ
ʹ͸ਓखͰ͸‫ݶ‬ք͕͋Γɺಈతʹࣗಈੜ੒͢Δඞཁ͕͋Δɻ
ͦ͜Ͱຊ‫Ͱڀݚ‬͸ɺԻָʹ࿈ಈͨ͠μϯεΛର৅ʹɺ‫ػ‬ցֶशʹ‫ͳͨ৽͍ͯͮج‬ಈ࡞Λࣗಈੜ੒Ͱ͖Δख๏Λ࣮‫͢ݱ‬Δ͜ͱΛ໨తͱ͢ΔɻԻָʹ࿈ಈͨ͠ 3D ͷ CG ΩϟϥΫλͷಈ࡞ΛฤूͰ͖Διϑτ΢ΣΞ MikuMikuDance (MMD) *1 ʹΑΓɺμϯεಈը͸ MMD ͷొ৔ҎલΑΓखܰʹ࡞੒Ͱ͖ΔΑ͏ʹͳͬͨɻMMD Ͱ༻͍Δ͜ͱ͕Ͱ͖ΔΩϟϥΫλͷϞσϧ

(࢟‫ܗ‬΍ؔઅͷߏ଄ͳͲ) ͸Πϯλʔωοτ্ʹଟ͘ެ։͞

ਤ 1

MachineDancing ͷ֬཰Ϟσϧ: μϯε‫ޠ‬ኮ͔ΒΨ΢γΞϯ
ϓϩηεʹΑͬͯμϯεಈ࡞ͱԻ‫ڹ‬ಛ௃ྔ͕ग़ྗ͞ΕΔɻμϯεͱָ‫ۂ‬ͷରԠ෇͍ͨσʔλϕʔε͔Β‫ػ‬ցֶशΛߦ͍ɺ೚ҙΕ͓ͯΓɺࣗ෼ࣗ਎ͰϞσϧΛσβΠϯͰ͖ͳ͍ਓͰ΋Ϟͷָ‫ʹۂ‬ରͯ͠μϯεΛࣗಈੜ੒͢Δ͜ͱ͕Ͱ͖Δɻ

σϧΛެ։͢Δਓͷ͓͔͛ͰΞχϝʔγϣϯΛ࡞੒Ͱ͖Δɻ·ͨΩʔϑϨʔϜͱ‫ݺ‬͹ΕΔಛఆͷ࣌ࠁͰͷΩϟϥΫ
ΔΑ͏഑੍ྀͯ͠࡞͢Δඞཁ͕͋ΔɻୈೋʹɺμϯεΛೖλͷ࢟੎ (ϙʔζ) Λฤू͠ɺͦΕΒΩʔϑϨʔϜͷϙʔζ

ྗ͢Δ͜ͱʹ͕͔͔࣌ؒΔͨΊͰ͋Δɻࣗવͳಈ͖ʹ͢Δ

ؒͷิؒํ๏Λઃఆ͢Δ͚ͩͰɺҰ࿈ͷΩϟϥΫλͷಈ࡞ͨΊʹ͸ɺҰ‫ۂ‬Λ௨ͯ͡ଟ਺ͷΩʔϑϨʔϜΛઃఆ͢Δඞ
Λ࡞੒͢Δ͜ͱ͕Ͱ͖Δɻཁ͕͋Γɺ͔͠΋ͦΕΒͷิؒ‫ۂ‬ઢΛࢼߦࡨ‫͕ͳ͠ޡ‬Βܾ͔͠͠μϯεΛ੍࡞͢Δ͜ͱ͸ґવେมͳ࡞‫͋Ͱۀ‬Δɻ
Ίͳ͚Ε͹ͳΒͳ͍ɻগͳ͍ૢ࡞Ͱࣗવͳಈ࡞ͱ͢ΔͨΊɺୈҰʹԻָʹରԠͨ͠μϯεಈ࡞Λ੍࡞͢Δͷ͕೉͍ͨ͠

ηϯασόΠε Kinect ͳͲʹΑΔਓؒͷ࢟੎৘ใͷऔಘΊͰ͋Δɻಈ͖͕୯ௐʹͳΒͳ͍Α͏े෼ͳόϦΤʔγϣ(ϞʔγϣϯΩϟϓνϟ) Λར༻͢Δ͜ͱ΋ՄೳͰ͋Δ͕ɺϯͷಈ࡞Λߟ͑Δඞཁ͕͋Δɻ·ͨ 3D ΩϟϥΫλʹର͠μϯε͕གྷΕͳ͚Ε͹༻͍Δ͜ͱ͕Ͱ͖ͳ͍ɻ

ͯෆࣗવͳಈ࡞͕༰қʹઃఆͰ͖ͯ͠·͏ͨΊɺࣗવʹͳຊ‫Ͱڀݚ‬͸͜ΕΒͷࠔ೉Λࠀ෰Ͱ͖ΔɺԻָʹ࿈ಈ͠1*1࢈‫ٕۀ‬ज़૯߹‫ॴڀݚ‬
National Institute of Advanced Industrial Science and Technology (AIST)
http://www.geocities.jp/higuchuu4

c 2014 Information Processing Society of JapanͨμϯεΛࣗಈੜ੒Ͱ͖Δख๏ MachineDancing ΛఏҊ͢Δɻָ‫ͱۂ‬μϯε͕ରԠ෇ֶ͍ͨशσʔλΛ༻͍ͯɺμϯεಈ࡞ͱͯࣗ͠વͳಈ࡞ͷ‫ػ‬ցֶशΛߦ͏͜ͱͰɺ೚ҙͷ

1Vol.2014-MUS-104 No.14
2014/8/26

৘ใॲཧֶձ‫ڀݚ‬ใࠂ
IPSJ SIG Technical Reportָ‫ʹۂ‬ରͯ͠μϯεΛࣗಈੜ੒Ͱ͖Δɻͳ͓μϯεಈ࡞Λ੾ΓషΓ͢Δ‫ط‬ଘͷख๏ͱ͸ҟͳΓɺֶशσʔλͷμϯε
Λ୯ʹग़ྗ͢ΔͷͰ͸ͳ͘ɺָ‫ʹۂ‬Ԡͨ͡৽ͨͳಈ࡞Λੜ
੒͢Δ͜ͱ͕Ͱ͖Δɻ·ֶͨशσʔλΛେ͖͘ม͑Ε͹ɺͦΕʹ൐ͬͯେ͖͘ҟͳͬͨμϯεΛੜ੒Ͱ͖ΔɻԻָʹ࿈ಈͨ͠μϯεΛࣗಈੜ੒͢Δैདྷͷ‫Ͱڀݚ‬͸ɺԻָͷςϯϙ [1][13]ɺϏʔτ‫ݕ‬ग़ͷͨΊͷಛ௃ྔ [5][14]ɺ
Իߴͱ࿨Ի [11]ɺϝϩσΟͷ֓‫[ ܗ‬12] ͳͲ͕μϯεͱରԠ͚ͮͯ෼ੳ͞Ε༻͍ΒΕ͍ͯΔɻ·ͨ 2 ͭҎ্ͷԻ‫ڹ‬ಛ௃ྔΛ૊Έ߹Θͤͯ༻͍Δख๏ [3] ΍ɺ1 ‫ۂ‬தͷԻ‫ڹ‬ಛ௃ྔͷྨࣅ౓ߦྻΛ༻͍Δख๏ [8] ͕͋ΔɻӅΕϚϧίϑϞσ
ϧʢHMMʣΛ༻͍ͯμϯεͱԻָͷରԠؔ܎ΛϞσϧԽ͠μϯεੜ੒Λߦ͏ख๏΋ఏҊ͞Ε͍ͯΔ [11]ɻ
ҰํͰμϯεಈ࡞ͷ࣌‫ྻܥ‬Λ෼ੳ͠ɺ࣌ؒతʹ཭ΕͨΩʔϑϨʔϜؒͷࣗવͳิؒ΍ɺਓؒΒ͍͠ಈ͖Λ࣮‫͢ݱ‬Δ‫ݚ‬‫ڀ‬΋ߦΘΕ͍ͯΔɻHMM [16]ɺಈతϕΠδΞϯωοτϫʔΫ (DBN) [7]ɺ֊૚తσΟϦΫϨաఔӅΕϚϧίϑϞσϧ

ਤ 2 ຊख๏ͰμϯεΛ෼ੳɾੜ੒͢Δࡍʹ༻͍Δ෦ҐʢϘʔϯʣɻ

(HDP-HMM) [17]ɺKernel Canonical Correlation Analysisମͷத৺ΛදΘ͢Ϙʔϯ͸ϫʔϧυ࠲ඪʹΑͬͯදΘ͠ɺଞͷ

(KCCA) [4]ɺΨ΢γΞϯϓϩηε (GP) Λ࢖͏ํ๏ [9]ɺજϘʔϯ͸઀ଓ͍ͯ͠ΔϘʔϯͱͷ઀఺Λத৺ͱ͢ΔճసΛද‫͢ݱ‬ΔΫΥʔλχΦϯʹΑͬͯද‫͞ݱ‬ΕΔɻਤதͷΩϟϥΫλ

ࡏม਺ͷ‫੻ي‬ͷτϙϩδʔΛߟྀͨ͠ Topological Gestureͷ 3D Ϟσϧ͸ koron ࢯʹΑ੍ͬͯ࡞͞Εͨɻ

Analysis (TGA) [10][15] ͕ఏҊ͞Ε͍ͯΔɻ·ͨμϯεಈ࡞Ͱ͸ͳ͘ਓؒͷಈ࡞Ұൠͷ‫ͯ͠ͱڀݚ‬ɺGaussian Pro-‫͠ܭ‬ɺͦΕΛ࢖ָͬͯ‫ʹۂ‬࿈ಈͨ͠μϯεಈ࡞Λࣗಈੜ੒cess Dynamical Models (GPDM) [18]ɺMulti-layer Joint

͢Δํ๏Λड़΂Δɻ͸͡ΊʹԻָͱμϯεΛͦΕͧΕѻ͏

Gait-Pose Manifolds (multi-layer JGPM) [2] ͳͲΛ༻͍ͨΊͷಛ௃ྔʹ͍ͭͯड़΂Δɻ࣍ʹμϯεಈ࡞ͷ֬཰తੜͯɺਓؒͷಈ࡞Λ௿࣍‫ݩ‬ͷۭؒ΁ඇઢ‫ʹ૾ࣸܗ‬Αͬͯॖୀ੒ϞσϧΛઃ‫͠ܭ‬ɺσʔλʹ‫ͦͮ͘ج‬ͷϞσϧͷֶशΞϧͤͯ͞෼ੳɾิؒ͢Δख๏͕ఏҊ͞Ε͍ͯΔɻ

ΰϦζϜΛड़΂Δɻ͞ΒʹԻָΛμϯεಈ࡞ͷ֬཰తੜ੒

͜ͷΑ͏ʹैདྷͷํ๏Ͱ͸ָ‫ʹۂ‬࿈ಈͨ͠μϯε͕ੜ੒ϞσϧͱରԠ෇͚ΔͨΊͷํ๏Λٞ࿦͠ɺ࠷‫ͦʹޙ‬ͷϞσ͞ΕΔ΋ͷͷɺμϯεಈ࡞ͷ࣌‫ྻܥ‬Λ෼ੳֶ͠शσʔλʹϧΛ༻͍ͯ৽ͨͳָ‫ʹۂ‬ରͯ͠μϯεΛࣗಈੜ੒͢Δํ๏

ͳ͍৽ͨͳμϯεಈ࡞Λࣗಈੜ੒͢Δʹ͸ࢸ͍ͬͯͳ͍ɻ

Λड़΂Δɻͳ͓ຊ‫Ͱڀݚ‬͸ָ‫ۂ‬ͷϏʔτͱখઅઢ͕ɺ֊૚

·ͨμϯε΍ਓؒͷಈ࡞ͷࣗવͳิؒΛߦ͏ͨΊʹ GP ΛతͳϏʔτߏ଄ͷਪఆ [19] ౳ʹΑͬͯద੾ʹ༩͑ΒΕΔ΋࢖͏ख๏͕ఏҊ͞Ε͍ͯΔ͕ɺͦΕΛ༻ָ͍ͯ‫ʹۂ‬ରͯ͠ͷͱ͢ΔɻμϯεΛࣗಈੜ੒Ͱ͖ͳ͔ͬͨɻͦ͜Ͱ MachineDancing
Ͱ͸

2.1 μϯεੜ੒ͷͨΊͷԻ‫ڹ‬ಛ௃ྔͱμϯεͷσʔλߏ଄

1. HMM ʹΑΔμϯεͱԻָͷରԠؔ܎ͷϞσϧԽ

2.1.1 μϯεੜ੒ͷͨΊͷԻ‫ڹ‬ಛ௃ྔ

2. GP ʹΑΔμϯεಈ࡞ͱԻָಛ௃ྔͷϞσϧԽԻָԻ‫ڹ‬৴߸Λ෼ੳ͢ΔͨΊʹ޿͘༻͍ΒΕ͍ͯΔϝϧʹΑΓɺָ‫ͱۂ‬μϯε͕ରԠ෇ֶ͍ͨशσʔλΛ༻͍ͯ‫ػ‬प೾਺έϓετϥϜ܎਺ ʢMFCCʣʢ16 ࣍‫ݩ‬ʣͱɺ௚લͷցֶशΛߦָ͍‫ʹۂ‬ରͯ͠৽͍͠μϯεಈ࡞Λੜ੒͢Δɻ

෼ੳ۠ؒͷ MFCC ͱͷࠩ෼Ͱ͋Δ ∆MFCCʢ16 ࣍‫ݩ‬ʣΛख๏ͷུ֓Λਤ 1 ʹࣔ͢ɻMachineDancing Ͱ༻͍ΒΕ࿈Ͷͨ 32 ࣍‫ݩ‬ͷϕΫτϧΛԻ‫ڹ‬ಛ௃ྔͱͯ͠༻͍Δɻ͜

Δ֬཰Ϟσϧ͸ɺԻָͷԻ‫ڹ‬ಛ௃ྔΛ‫؍‬ଌ‫͠ͱྻܥ‬ɺྨࣅΕʹΑΓ‫ۂ‬தͷ੝Γ্͕Γɺྨࣅ෦෼ʢ‫܁‬Γฦ͞ΕΔαϏͨ͠μϯεಉ࢜Λ·ͱΊͨʮμϯε‫ޠ‬ኮʯ͕ӅΕঢ়ଶͱͳͳͲʣɺԻ৭ͷ൑ผΛ͋Δఔ౓ߦ͏͜ͱ͕Ͱ͖Δɻ͔͠͠Δ HMM Ͱ͋ΔɻԻָͷಛ௃ྔͷग़ྗ֬཰͸Ψ΢γΞϯϓ

͜ΕΒ͸ද૚తͳಛ௃ྔͰ͋ΔͨΊɺμϯεͷಈ࡞ͱΑΓϩηεʹΑܾͬͯఆ͞Εɺ·ͨμϯεಈ࡞΋Ψ΢γΞϯϓ

ີ઀ʹؔ܎͢Δಛ௃ྔ͸ɺࠓ‫ݕޙ‬౼ͷ༨஍͕͋ΔɻϩηεʹΑ֤ͬͯӅΕঢ়ଶ͔Β֬཰తʹੜ੒͞ΕΔɻμϯ

2.1.2 μϯεಈ࡞ͷσʔλߏ଄εͷࣗಈੜ੒͸ɺָ‫ۂ‬ͷԻ‫ڹ‬ಛ௃ྔΛ‫؍‬ଌͱͨ͠৔߹ͷ໬

3 ࣍‫ݩ‬ΩϟϥΫλͷϞσϧʹ͸Ϙʔϯͱ‫ݺ‬͹ΕΔਓؒͷ౓࠷େͷμϯεಈ࡞Λ୳ࡧ͢Δ໰୊ͱͯ͠ఆࣜԽͰ͖Δɻࠎ֨ʹ૬౰͢Δߏ଄͕͋Γɺ͜ͷϘʔϯΛฏߦҠಈ΋͘͠

2. Իָʹ࿈ಈ͢Δμϯεͷ֬཰తੜ੒Ϟσϧ͸ճసͤ͞Δ͜ͱͰΩϟϥΫλΛಈ͔͢͜ͱ͕Ͱ͖Δɻମͷத৺ΛදΘ͢ϘʔϯͷҐஔ͸ಈ͖ʹґଘͤͣ‫ݻ‬ఆ͞ΕͨຊઅͰ͸Իָʹ࿈ಈ͢Δμϯεͷ֬཰తੜ੒ϞσϧΛઃc 2014 Information Processing Society of JapanσΧϧτ࠲ඪʹ‫ͮ͘ج‬Ґஔ࠲ඪ (x1 , x2 , x3 ) ͱද‫͖Ͱݱ‬Δɻ

2

Vol.2014-MUS-104 No.14
2014/8/26

৘ใॲཧֶձ‫ڀݚ‬ใࠂ
IPSJ SIG Technical Report͔͠͠࿹΍଍ͱ͍ͬͨϘʔϯ͸ɺଞͷϘʔϯͱ઀ଓ͞Εͯ
͍ΔͨΊɺҐஔ࠲ඪͰද͢ͱ࿹ͱ಑ମ͕཭ΕΔͳͲ࢟੎ͱ
ͯ͠ෆՄೳͳ΋ͷ͕ද‫͞ݱ‬Εͯ͠·͍ɺ;͞Θ͘͠ͳ͍ɻ
͜ΕΒͷϘʔϯʹ͍ͭͯ͸ɺମͷத৺ʹ͍ۙํ΁઀ଓ͞Ε͍ͯΔϘʔϯʢ਌Ϙʔϯʣͱͷ઀఺Λத৺ͱͯ͠ɺ‫ج‬ຊ࢟
੎ʢਤ 2 ͷ࢟੎ʣ͔ΒͲΕ͚ͩ 3 ࣍‫ݩ‬తʹճస͞Ε͔ͨͰද‫͖Ͱݱ‬Δɻ

3 ࣍‫ݩ‬ͷճస͸ճసߦྻʹΑΔද‫ݱ‬ɺΦΠϥʔ֯ʹΑΔද‫ݱ‬ɺΫΥʔλχΦϯʹΑΔද‫͕͋ݱ‬Δ͕ɺຊ‫Ͱڀݚ‬͸ΫΥʔλχΦϯΛ༻͍Δɻճసߦྻ͸ 3 × 3 ͷߦྻͰ͋Γɺ୯ҐϕΫτϧͷ޲͖ͷมԽΛઢ‫ܗ‬ม‫Ͱ׵‬දͨ͠ͱ͖ͷม‫͋Ͱྻߦ׵‬ΔɻΦΠϥʔ֯͸ɺ3 ͭͷ࣠ΛࣄલʹܾΊͨॱ൪Ͱத৺࣠ͱͯ͠ճసͤ͞Δ֯౓Ͱ 3 ࣍‫ݩ‬ճసΛද‫͢ݱ‬

ਤ 3μϯεಈ࡞ͷ෼ੳ۠ؒ΁ͷ੾Γग़͠ํ: 0.5 ഥͷΦʔόʔϥο

Δํ๏Ͱ (θx , θy , θz ) ͱදͤΔɻΫΥʔλχΦϯ Q ͸ճసϓΛͤ͞ͳ͕Β 1 খઅִؒͰಈ࡞Λ෼ੳ͢Δɻςϯϙͷҟͳલ‫ޙ‬ͷϕΫτϧͷ֎ੵ (q1 , q2 , q3 ) ͱ಺ੵ q0 ͔ΒͳΔ࢛‫ݩ‬Δָ‫ۂ‬ͷμϯε͔Β੾Γग़ͨ͠ಈ࡞ಉ࢜ͰൺֱΛ͢ΔͨΊɺ1਺Ͱ͋ΓɺQ = q0 + q1 i + q2 j + q3 k ͱදΘͤΔɻ͜͜ͰখઅΛ 1.0 ͱ͢ΔΑ͏ਖ਼‫ن‬Խ͢Δɻ

i2 = j2 = k2 = ijk = −1ɺij = −ji = kɺjk = −kj = iɺki = −ik = j Ͱ͋ΔɻΫΥʔλχΦϯͰ͸ 4 ͭͷύϥϝʔλʹΑͬͯ࢟੎ͷճసΛදΘ͢͜ͱ͕Ͱ͖Δɻ੒͞Ε͍ͯΔͱଊ͑Δ͜ͱ͕Ͱ͖Δɻຊ‫Ͱڀݚ‬͸ָ‫ʹۂ‬࿈ಈͨ͠μϯεಈ࡞Λ෼ੳ۠ؒ΁ͱ෼ΫΥʔλχΦϯ͸ճసߦྻΑΓύϥϝʔλ਺͕গͳ͘ɺ

ׂͯ͠ଊ͑ɺ෼ੳ۠ؒ͝ͱʹμϯεಈ࡞͕μϯε‫ޠ‬ኮ͔Β

·ͨΦΠϥʔ֯ͷද‫ݱ‬ͷ໰୊Ͱ͋Δɺճస͢Δ࣠ͷॱ൪ͱ֬཰తʹੜ੒͞Ε͍ͯΔɺͱ͍͏ൃ૝Ͱ֬཰తੜ੒Ϟσϧ֯౓ʹΑͬͯ͸໨తͷ࢟੎ʹಋ͘ද‫͕ݱ‬ଘࡏ͠ͳ͍৔߹

Λઃ‫͢ܭ‬Δɻ

΍ɺ࢟੎ؒΛิؒ͢ΔࡍʹେճΓͱͳΔ‫੻ي‬Λඳ͍ͯ͠·μϯε‫ޠ‬ኮ͸ͲͷΑ͏ͳ࣌ؒత௕͞Ͱఆٛ͞ΕΔ΂͖ͩ͏δϯόϧϩοΫ‫ݱ‬৅Λආ͚Δ͜ͱ͕Ͱ͖Δɻ͞Βʹ‫ٿ‬໘

Ζ͏͔ɻԻָʹ࿈ಈͨ͠μϯε͸ԻָͷഥͷҐஔʹେ͖͘ઢ‫͕ؒิܗ‬؆୯ʹߦ͑ΔͳͲͷศརͳੑ࣭͕͋Δɻ

ؔ࿈͓ͯ͠Γɺ࣌ؒͷ୯Ґͱͯ͠ɺ1 ഥͷ௕͞Λ‫ن‬४ʹ͢ຊ‫Ͱڀݚ‬͸ମͷಈ͖Λද‫͢ݱ‬ΔͨΊʹඞཁͳ࿹ɺ଍ͳͲ

Δͷ͕;͞Θ͍͠ɻҰൠʹ෼ੳ͕۠ؒ௕͍΄Ͳ௕͍࣌ؒత

ͷ 20 ຊͷϘʔϯΛ༻͍ͨɻ༻͍ͨϘʔϯΛਤ 2 ʹࣔ͢ɻߏ଄Λ࣋ͬͨμϯε‫ޠ‬ኮΛ෼ੳͰ͖Δ͕ɺ‫ޠ‬ኮ਺͕๲େʹશ 20 ຊͷϘʔϯͷ͏ͪɺମͷத৺ΛදΘ͢ 3 ࣍‫࠲ݩ‬ඪͷͳͬͯ͠·͏ܽ఺͕͋Δɻຊ‫Ͱڀݚ‬͸ 4 ෼ͷ 4 ഥࢠΛԾఆ

3 ͭͷύϥϝʔλͱɺ࢒Γ 19 ຊͷϘʔϯͷճసΛදΘ͢

͠ɺ1 খઅ 4 ഥͷதͰͷ‫ڧ‬ഥͱऑഥͰͲͷΑ͏ʹμϯεಈΫΥʔλχΦϯͷ 4 ͭͷύϥϝʔλΛ 1 ͭͷϕΫτϧ΁ͱ࡞͕ҧ͏ͷ͔ʹ͍ͭͯ‫͖Ͱ࡯؍‬Δ۠෼ͷ௕͞ͱ͢ΔͨΊɺ1

ଋͶɺ79 ࣍‫ݩ‬ͷϕΫτϧͱ֤ͯ࣌͠ࠁͰͷΩϟϥΫλͷ࢟খઅͷμϯεΛ‫ج‬ຊ୯Ґͱߟ͑Δɻͦͷ্Ͱ৽ͨͳఏҊͱ੎Λද‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δɻ

ͯ͠ɺখઅͷઌ಄ͷಈ࡞΁લͷখઅ͔Βଓ͘ಈ࡞΋߹Θͤͯ෼ੳ͢ΔͨΊʹɺલͷখઅͷ 3 ഥ໨͔Β࢝·Γɺͦͷখ

2.2 μϯεಈ࡞ͷ֬཰తੜ੒Ϟσϧઅͷ 4 ഥΊ࠷‫Ͱޙ‬ऴΔ 1.5 খઅͷ෼ੳ۠ؒΛ༻͍ͯμϯε

2.2.1 μϯε‫ޠ‬ኮΛ෼ੳ͢Δɻμϯεಈ࡞ͷ෼ੳ۠ؒͷ༷ࢠΛਤ 3 ʹࣔ͢ɻָ‫ʹۂ‬࿈ಈͨ͠μϯεಈ࡞͸෦෼ʹ੾Γ෼͚ͯଊ͑Δ͜͜ͷΑ͏ʹμϯεಈ࡞͕ 0.5 খઅ෼Φʔόʔϥοϓ͠ͳ͕

ͱ͕Ͱ͖Δɻ࣮ࡍɺαϏ΍ A ϝϩͷ‫܁‬Γฦ͠΍Ի‫ڹ‬ಛ௃͕Β੾Γग़͞Ε෼ੳ͢Δํ๏ΛఏҊ͢Δɻͳ͓ɺ෼ੳ͢Δμࣅ͍ͯΔ෦෼Ͱ͸ྨࣅͨ͠ಈ࡞͕‫ݟ‬ΒΕΔ͜ͱ͕ଟ͍ɻ·ϯεʹରԠָͨ͠‫ʹۂ‬Αͬͯςϯϙ͕ҟͳΔͨΊɺ੾Γग़ͨΑΓখ͞ͳ࣌ؒ୯Ґʹ͓͍ͯ΋ɺ2 ͭͷ࢟੎͕ަ‫ݱʹޓ‬͞ΕΔμϯεͷ࣌ؒ௕͕ҟͳΓɺ‫ʹ͍ޓ‬ൺֱ͢Δͷ͕ෆศΘΕΔͳͲ‫܁‬Γฦ͠ߏ଄͕μϯεಈ࡞ͷதʹ‫ݟ‬ΒΕΔɻ‫܁‬Ͱ͋Δɻͦ͜Ͱ 1 খઅ͕ 1.0 ͱͳΔΑ͏ʹ࣌ؒͷਖ਼‫ن‬ԽΛΓฦ͞Ε͍ͯΔ෦෼͸ɺͦͷ৔ॴͷԻָ΍ɺͦΕ͕ԿճΊߦ͏ɻͷ‫܁‬Γฦ͔͠ʹԠͯ͡ɺมԽͯ͠‫ݱ‬ΘΕΔ͜ͱ͕ଟ͍͕ɺ

2.2.2 μϯε‫ޠ‬ኮʹ‫ͮ͘ج‬μϯεಈ࡞ͷ֬཰తੜ੒ϞσϧͦΕΒ͕ྨࣅ͍ͯ͠ΔͨΊɺಉ͡ಈ࡞͔Β೿ੜͯ͠ੜ·Εઌ಄͔Β n ൪໨ͷ෼ੳ۠ؒͷμϯε Dn ͕ɺμϯε‫ޠ‬ͨಈ࡞ͱͯ͠ଊ͑Δ͜ͱ͕Ͱ͖Δɻ

ኮ vn ͱͯ͠෼ੳ͢Δ͜ͱ͕Ͱ͖Δͱ͢Δɻ͜͜Ͱ Dn͜ͷΑ͏ͳྨࣅͨ͠μϯεಈ࡞Λੜ੒͢Δ‫ج‬ຊ‫ͳͱܗ‬Δ

͸෼ੳ۠ؒதͷ࢟੎ΛදΘ͢ϕΫτϧ d ΛଋͶͨߦྻμϯεಈ࡞ͷ͜ͱΛɺຊ‫Ͱڀݚ‬͸ʮμϯε‫ޠ‬ኮʯͱ‫Ϳݺ‬ɻ[d1 , · · · , ds , · · · , dS ] Ͱ͋Δɻds ͸‫ࡏݱ‬ͷ࣮૷Ͱ͸ 79 ָ࣍‫ۂ‬தͷ A ϝϩͷ‫܁‬Γฦ͠ʹ͓͍ͯɺྨࣅͨ͠μϯεಈ࡞‫ݩ‬ϕΫτϧͰ͋ΔɻS ͸෼ੳ۠ؒதͰ‫؍‬ଌ͞ΕΔ࢟੎ͷ‫ݸ‬͕͋Δ৔߹ɺͦΕΒ͸ྨࣅͨ͠‫ڞ‬௨ͨ͠μϯε‫ޠ‬ኮ͔Βੜ਺Ͱ͋Δɻμϯεಈ࡞ͷσʔλʹ͸ҰఆͷϑϨʔϜϨʔτT

c 2014 Information Processing Society of Japan3

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IPSJ SIG Technical ReportͰ࣌ࠁ͝ͱͷ࢟੎͕‫ه‬ड़͞ΕΔ͔ɺ࣌ؒతʹͱͼͱͼͷͱද‫͞ݱ‬Εͨͱ͖ɺॏΈ܎਺ {aj }Jj=1 ͕ฏ‫ ۉ‬0ɺ෼ࢄ‫ڞ‬෼ΩʔϑϨʔϜͷ࣌ࠁ͝ͱͷ࢟੎͕‫ه‬ड़͞Ε͍ͯΔɻͨ͠ࢄߦྻ Σp ͷΨ΢ε෼෍ʹै͍ɺ͕ͬͯɺ෼ੳ۠ؒதʹ‫·ؚ‬ΕΔ࢟੎σʔλͷ‫ ਺ݸ‬S ͸ɺָ

a = (a1 , · · · , aJ ) ∼ N (0, Σp )(4)‫ۂ‬ͷςϯϙ΍ΩʔϑϨʔϜͷৄࡉͳઃఆ͞Ε౓߹͍ʹԠ͡
ͱͳΔͱ͢Δɻ͜ͷͱ͖ Representer ఆཧʹΑΓɺ‫ج‬ఈؔͯҟͳ͍ͬͯΔՄೳੑ͕͋Δɻ
෼ੳ۠ؒ಺ͷμϯεಈ࡞ Dn ͕μϯε‫ޠ‬ኮ vn Λ‫֬ʹݩ‬਺ φj ͕ਖ਼ఆ஋ΧʔωϧͰ͋Δͱ͖ʹ͸ɺୈҰͷԾఆͷ΋཰తͳ༳Β͗Λ΋ͬͯੜ੒͞ΕΔͱͯ͠ɺ֬཰ P (Dn |vn )ͱͰਪఆ͞ΕΔ f ͕ࣜʢ3ʣͷ‫ܗ‬Λ࣋ͪɺJ ͸ D ʹ‫·ؚ‬ΕΛಋೖ͢Δɻ·ͨμϯε‫ޠ‬ኮ͸ྡ߹͏෼ੳ۠ؒʹϥϯμϜ

Δ࢟੎ͷ਺ʢߦྻͷߦ਺ʣͱͳΔ͜ͱ͕஌ΒΕ͍ͯΔɻຊʹ࿈ͳΔΘ͚Ͱ͸ͳ͘ɺ͋Δ‫ޠ‬ኮͷ‫ʹޙ‬͸ͲͷΑ͏ͳ‫ޠ‬ኮ͕ଓ͖ͦ͏͔ͱ͍͏ภΓ͕͋Δͱߟ͑ΒΕΔɻͦ͜Ͱ‫ޠ‬ኮ͔Β‫ޠ‬ኮ΁ͱભҠ͢Δ֬཰ P (vn |vn−1 ) Λಋೖ͢Δɻ͜ͷͱ͖ N ‫ݸ‬ͷ෼ੳ۠ؒͷμϯεಈ࡞ {Dn }N
n=1 ͕μϯ
ε‫ޠ‬ኮ {vn }N
n=1 ͔Βੜ੒͞ΕΔ֬཰͸ҎԼͷΑ͏ʹ‫Ͱࢉܭ‬‫Ͱڀݚ‬͸ਖ਼ఆ஋Χʔωϧͱͯ͠ RBF Χʔωϧλ2
φtj (t) = k (t, tj ) = exp − |t − tj |
2

(5)

Λ༻͍ͨɻ
͜ΕΒ 2 ͭͷԾఆͷ΋ͱ P (D, a) = P (D|a) P (a) Λ‫ܭ‬ࢉ͠ɺ͜ͷಉ࣌෼෍தͷ a Λੵ෼ফ‫͢ڈ‬Δͱɺ෼ੳ۠ؒͷ

͖Δɻ
N
!
 Y
N
P {Dn }NP (Dn |vn ) P (vn |vn−1 )(1)n=1 , {vn }n=1 =
n=1ͨͩ͠ɺද‫ه‬ͷศ্ٓ P (v1 |v0 ) = P (v1 ) ͱͨ͠ɻҎ‫ޙ‬ͷࣜͰ΋ΠϯσοΫε͕ 0 ͱͳΔ৔߹͸ಉ༷ʹѻ͏ɻ͜Ε͸

HMM Ͱ͋Γɺ‫؍‬ଌ͕μϯεಈ࡞ɺӅΕঢ়ଶ (જࡏม਺) ͕μϯεಈ࡞ͷ֬཰తੜ੒Ϟσϧ͸P (D|t; σ, λ)



1 !
1
exp − tr DT K−1 D
= q
2
|t|α
(2π)
|K|α

(6)

ͱ‫·ٻ‬Δɻ͜Ε͸Ψ΢γΞϯϓϩηεʹΑΔϞσϧԽͰμϯε‫ޠ‬ኮͰ͋ΔɻP (Dn |vn ) Λग़ྗ֬཰ɺP (vn |vn−1 ) ͸ભҠ֬཰ɺP (v1 ) Λॳ‫Ϳݺͱ཰֬ظ‬ɻ͜͜Ͱ P (Dn |vn ) ͸͋Δɻͨͩ͠ t ͸෼ੳ۠ؒதͰ࢟੎͕‫؍‬ଌ͞ΕΔ࣌ࠁΛ
TଋͶͨϕΫτϧ t = (t1 , · · · , ts , · · · , tS ) ɺα ͸ϕΫτϧ dμϯε‫ޠ‬ኮ͕༩͑ΒΕͨͱ͖ʹͲͷΑ͏ͳμϯεಈ࡞͕ग़ͷ࣍‫਺ݩ‬ɺK ͸ RBF Χʔωϧʹ‫ͮ͘ج‬ΧʔωϧߦྻͰྗ͞ΕΔ͔ʹ͍ͭͯͷ֬཰ϞσϧͰ͋Γɺ͜ΕΛʮμϯε(K)ij = k (ti , tj ) + σ −2 δti tj Ͱ͋Γɺδ ͸ΫϩωοΧʔͷσ‫ޠ‬ኮ͔Βͷ֬཰తੜ੒Ϟσϧʯͱ‫Ϳݺ‬ɻ߲࣍Ͱઃ‫͞ܭ‬ΕΔ
ϧλͰ͋Δɻ
͜ͷ֬཰తੜ੒ϞσϧΛ্‫ه‬ͷϞσϧͱ౷߹͢Δ͜ͱͰɺμϯεಈ࡞ͷ֬཰తੜ੒ϞσϧΛͭ͘Δ͜ͱ͕Ͱ͖Δɻ

2.2.3 μϯε‫ޠ‬ኮ͔Βͷ֬཰తੜ੒Ϟσϧ2.3 μϯεಈ࡞ͷ֬཰తੜ੒ϞσϧͷֶशΞϧΰϦζϜ
લઅͰઃ‫͞ܭ‬Εͨμϯεಈ࡞ͷ֬཰తੜ੒ϞσϧΛɺσʔμϯεಈ࡞ʹ͸प‫ظ‬ੑ΍ഥ͝ͱͷಈ࡞ʹରԠؔ܎͕͋Δλʹ‫ֶ͍ͯͮج‬श͢Δํ๏Λٞ࿦͢Δɻ͜ͱ͔Βɺ͜ΕΒΛ֬཰తʹϞσϧԽ͍ͨ͠ɻಉ͡μϯε‫ޠ‬ኮͱͯ͠ଊ͑Δ͜ͱ͕Ͱ͖Δ 2 ͭͷࣅͨμϯεಈ࡞ D1
ͱ D2 ͕͋Δͱ͖ɺ͜ΕΒ 2 ͭͷߦྻʹ‫·ؚ‬ΕΔ࢟੎Λද

2.3.1 μϯε‫ޠ‬ኮͷֶशμϯε‫ޠ‬ኮ {vk }K
k=1 Λֶश͢Δํ๏Λٞ࿦͢Δɻμϯεͷ‫ޠ‬ኮʹ͸ࣗવ‫ޠݴ‬ͷΑ͏ͳ͋Β͔͡Ίܾ·ͬͨ‫ޠ‬ኮͷΘ͢ϕΫτϧͷ‫਺ݸ‬͸ಉ͡ͱ͸‫ݶ‬Βͳ͍ɻͦ͜ͰɺD ʹ‫ؚ‬ηοτ͸ͳ͍ͷͰɺμϯεಈ࡞ͷྨࣅ౓ʹ‫͍ͯͮج‬Ϋϥε·ΕΔ࢟੎ [d1 , · · · , dS ] Λɺ1 খઅͷ௕͕͞ 1.0 ͱͳΔΑ͏λϦϯάΛߦֶ͍श͢Δɻਖ਼‫ن‬Խ͞Εͨ࣌ࠁΛม਺ͱͯ͠ɺds = f (ts ) ͱදͤΔ࿈ଓ࠷΋ద੾ʹμϯε‫ޠ‬ኮͱμϯεಈ࡞ͷରԠؔ܎͕औΕͯؔ਺ f Λਪఆ͢Δඞཁ͕͋Δɻ͜͜Ͱ ts ͸࢟੎ ds ͕෼ੳ۠ؒ಺Ͱ‫؍‬ଌ͞ΕΔ࣌ࠁ (−0.5 < ts < 1.0) Ͱ͋Δɻf Λਪఆ͢Δʹ͋ͨͬͯɺ࣍ͷ 2 ఺ΛԾఆ͢Δɻ͍Δঢ়ଶ͸ɺμϯε‫ޠ‬ኮׂ͕Γ౰ͯΒΕͨͱ͖ͷֶशσʔPNλͷର਺໬౓ L = n=1 log P (Dn |tn , v ∗ ; σvn , λvn ) ͕࠷େ

ͱͳΔͱ͖ͱ͍͑Δɻ͜ͷঢ়ଶ͸ɺద౰ͳμϯε‫ޠ‬ኮͱμୈҰͷԾఆͱͯ͠ɺ֤࣌ࠁͰͷ࢟੎ ds ͸ɺf (ts ) ͷ஋͔ΒΨ΢εϊΠζ͕ՃΘͬͯ‫؍‬ଌ͞ΕΔͱ͢Δɻ͢ͳΘͪ!
|ds − f (ts ) |2 ∼ N 0, σ 2 I(2)!
ͱॻ͚Δɻ͜͜Ͱ N 0, σ 2 I ͸ฏ‫ྵ͕ۉ‬ϕΫτϧͰɺ෼ࢄ‫ڞ‬෼ࢄߦྻ͕ σ 2 I Ͱ͋Δଟ࣍‫ݩ‬Ψ΢ε෼෍Ͱ͋ΔɻୈೋͷԾఆͱͯ͠ɺؔ਺ f (t) ͕ɺద੾ʹબ͹Εͨʢඇઢ‫ܗ‬ʣ‫ج‬ఈؔ਺ φj (t) ͷॏΈ෇͚࿨Ͱf (ts ) =JX

aj φj (ts )j=1

c 2014 Information Processing Society of Japanϯεಈ࡞ͷରԠؔ܎ͷॳ‫ظ‬ঢ়ଶ͔ΒɺEM ΞϧΰϦζϜʹ
‫ͮ͘ج‬ΫϥελϦϯάͰ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ

• E-step: ‫ޠ‬ኮͱμϯεಈ࡞ͷରԠؔ܎Λ΋ͱʹɺ֤‫ޠ‬ኮ v ͝ͱʹ σv ͱ λv Λ‫ٻ‬ΊΔ• M-step: σʔλશମͷ໬౓͕࠷େͱͳΔΑ͏ɺ‫ޠ‬ኮͱμϯεಈ࡞ͷରԠؔ܎ΛܾΊΔ
ͷ 2 ͭͷεςοϓͷ൓෮ʹΑͬͯߦ͑ΔɻҎԼɺE-step ͱM-step ͷͦΕͧΕͷৄࡉΛड़΂Δɻ
(3)

2.3.1.1 E-stepM ‫ݸ‬ͷ෼ੳ۠ؒͷμϯεಈ࡞ {D1 , · · · , DM } ͕μϯε4

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IPSJ SIG Technical Report‫ޠ‬ኮ v ∗ Ͱ͋ΔͱΘ͔ͬͨ৔߹ɺμϯε‫ޠ‬ኮ v ∗ ͕༩͑ΒΕͨͱ͖ͷμϯεಈ࡞ͷ֬཰తੜ੒ϞσϧΛࣜʢ6ʣʹ‫ج‬T
͍ͮͯ࡞Δ͜ͱ͕Ͱ͖ΔɻD ʹ D∗ = D1 , · · · , DM

Λ ୅ ೖ ͠ ɺt ΁ ֤ ෼ ੳ ۠ ؒ ʹ ͓ ͍ ͯ ࢟ ੎ ͕ ‫ ؍‬ଌ ͞ Ε Δ
TΛ ୅ ೖ ͠ ͨ ͱ ͖ ͷ ɺσ࣌ ࠁ ͷ ߦ ྻ t∗ = t1 , · · · , tMΕͨԻ‫ڹ‬ಛ௃ྔͷྻΛ {Mn }N
n=1 ͱ͢Δɻ
μϯεಈ࡞ͷٞ࿦ͱಉ༷ʹɺԻ‫ڹ‬ಛ௃ྔʹ΋࣌ؒతͳߏ
଄͕͋Γɺখઅதͷಛ௃ྔϕΫτϧಉ࢜ͷ૬͕ؔμϯεಈ࡞ʹؔ࿈͍ͯ͠Δͱߟ͑ΒΕΔɻͦ͜Ͱ P (M|v) ΛΨ΢γΞϯϓϩηεʹΑͬͯμϯεಈ࡞ͷ৔߹ͱಉ༷ʹϞσϧԽͱ λ Λม਺ͱ‫͖ͱͨݟ‬ͷ஋ɺ͢ͳΘͪ໬౓Λ࠷େԽ͍ͨ͢͠ɻ͔͠͠ɺಉ͡μϯεʹରͯ͠ରԠ͢ΔԻ‫ڹ‬ಛ௃ྔΔΑ͏ʹ σ ͱ λ Λ‫ٻ‬ΊΕ͹ྑ͍ɻ۩ମతʹ͸ର਺໬౓ͷྻ͸ଟ༷Ͱ͋Γɺ୯Ұͷ֬཰෼෍Ͱද‫͢ݱ‬Δ͜ͱ͸Ͱ͖

∗

∗

∗

∗

LL (σ, λ, v ) = log P (D |t , v ; σ, λ) Λ࠷େԽ͢Δ σv∗ ͱͳ͍ͱߟ͑ΒΕΔɻͦ͜Ͱෳ਺ͷ֬཰෼෍ͷॏΈ෇͖࿨ʹλv∗ ΛɺScaled Conjugate Gradient [7] ͳͲͷ࠷దԽ๏ʹΑͬͯ֬཰Λ‫͢ࢉܭ‬Δ͜ͱΛߟ͑ΔɻΑͬͯ‫ٻ‬ΊΔɻ͜ͷ໬౓͸‫ޠ‬ኮ v ∗ ͷԼͰूΊΒΕͨμϯ͋Β͔͡Ί {Mn }N
n=1 Λɺμϯεಈ࡞ΛΫϥελϦϯά

εಈ࡞͔Βֶशͯ͠‫͞ࢉܭ‬ΕΔ΋ͷͰ͋ΔͷͰɺ৚݅ʹ v ∗ͨ͠৔߹ͱಉ༷ͷૢ࡞ͰɺK ‫ݸ‬ͷΫϥελ {zk }K
k=1 ʹ෼ׂ

͕ೖ͍ͬͯΔ͜ͱʹ஫ҙ͞Ε͍ͨɻ

2.3.1.2 M-stepE-step ͰಘΒΕΔ‫ޠ‬ኮ {vk }K
k=1 ʹରԠ͢Δ֬཰తੜ͓ͯ͘͠ɻ͜ͷͱ͖ P (Mn |v) ͸ z Λม਺ʹՃ͑ͯɺX
P (Mn , z|vn )P (Mn |vn ) =z੒ Ϟ σ ϧ ͕ ͋ Ε ͹ ɺ͋ Δ μ ϯ ε ಈ ࡞ D̂ ͕ Ͳ ͷ ‫ ޠ‬ኮ ͔=Δɻ͢ͳΘͪɺ࠷΋໬΋Β͍͠D̂ Λੜ੒͢Δ‫ޠ‬ኮ v̂ ͸v̂ = argmaxv log P D̂|t̂, v; σv , λv ʹΑͬͯ‫ٻ‬ΊΔ͜ͱ͕P (Mn |z) P (z|vn )

(11)

z

Βੜ੒͞ΕΔ΋ͷ͔Λɺର਺໬౓࠷େͱͳΔ‫ޠ‬ኮͷ֬཰తੜ੒ϞσϧΛ୳ࡧ͢Δ͜ͱͰௐ΂Δ͜ͱ͕Ͱ͖

X

ͱॻ͚ΔɻP (Mn |z) ͸Ի‫ڹ‬ಛ௃ͷΫϥελ z ͕༩͑ΒΕͨͱ͖ͷԻ‫ڹ‬ಛ௃ྔͷྻͷ֬཰Ͱ͋Γɺ͜Ε͸ΫϥελϦ
ϯάʹΑͬͯ͢Ͱʹ‫͍ͯͬ·ٻ‬ΔΨ΢γΞϯϓϩηεʹ‫ج‬Ͱ͖Δɻ͍ܾͮͯఆ͞ΕΔɻP (z|vn ) ͸ɺμϯεͷ‫ޠ‬ኮ͕༩͑ΒΕ

2.3.2 ग़ྗ֬཰ɾભҠ֬཰ɾॳ‫཰֬ظ‬ͷֶशͨͱ͖ͷԻ‫ڹ‬ಛ௃ྔͷΫϥελ͕ग़‫͢ݱ‬Δ֬཰Ͱ͋Γɺֶग़ྗ֬཰ɺ͢ͳΘͪμϯεಈ࡞ D ͕μϯε‫ޠ‬ኮ v ͔Βशσʔλʹ͓͍ͯɺμϯε‫ޠ‬ኮ͝ͱʹɺͦͷμϯε‫ޠ‬ኮͱੜ੒͞ΕΔ֬཰͸ɺ࠷దԽ͞ΕͨϞσϧύϥϝʔλ σv ͱಉ࣌ʹͲͷΑ͏ͳԻ‫ڹ‬ಛ௃ྔͷΫϥελ͕‫؍‬ଌ͞Ε͔ͨΛλv Λ༻͍ͯɺֶशσʔλ͕༩͑ΒΕͨ΋ͱͰͷ৚݅෇͖֬

Χ΢ϯτ͢Δ͜ͱͰ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ
ࣜʢ11ʣʹ͓͍ͯɺP (Mn |z) ͸Ψ΢γΞϯϓϩηεͰ͋P
ΓɺΨ΢γΞϯϓϩηεʹॏΈ P (z|vn )ʢ z P (z|vn ) = 1ʣ཰ΛҎԼͷΑ͏ʹ‫͢ࢉܭ‬Ε͹Α͍ɻP (D|v, D∗ , t∗ , t)


1
1 
exp − tr ZT K̂−1 Z
= q
2
|t|α
(2π)
|K|α

͕͔͚ΒΕ͍ͯΔͨΊɺ͜Ε͸ࠞ߹Ψ΢γΞϯϓϩηεͱ(7)

2.4.2 μϯεಈ࡞ͷ֬཰తੜ੒Ϟσϧͱͷ౷߹͋Δμϯε‫ޠ‬ኮ v ͷ΋ͱͰɺͲͷΑ͏ͳ MFCC+∆MFCC

ͨͩ͠Z = D − AK−1 D∗ , K̂ = B − AT K−1 A
λv 2 ∗|ti − tj |2 + σv 2 δt∗i tj(A)ij = exp −
2


λv 2
|ti − tj |2 + σv 2 δti tj(B)ij = exp −2

ͳ͍ͬͯΔɻ

(8)͕ੜ੒͞ΕΔ͔ͷ֬཰ʹ‫͍ͯͮج‬ɺμϯεಈ࡞ͷ֬཰తੜ੒Ϟσϧͱ౷߹͠ɺԻָʹ࿈ಈ͢Δμϯεಈ࡞ͷ֬཰తੜ

(9)
(10)

Ͱ͋Δɻ͜͜Ͱࣜ ʢ7ʣͷ஋͸‫ޠ‬ኮ v ͷԼͰूΊΒΕͨμϯεಈ࡞͔Βֶशͯ͠‫͞ࢉܭ‬ΕΔ΋ͷͰ͋ΔͷͰɺ৚݅ʹ੒ϞσϧΛͭ͘Δ͜ͱ͕Ͱ͖ΔɻHMM Ͱ͋Δࣜ (1) ʹμϯε‫ޠ‬ኮ {vn }N
n=1 Λಋೖͯ͠ม‫͢ܗ‬Δͱɺ!
N
P {Mn }N
n=1 , {Dn }n=1
X

!
N
N
=
P {Mn }N
n=1 , {Dn }n=1 , {vn }n=1{vn }Nn=1v ͕ೖ͍ͬͯΔɻॳ‫ͱ཰֬ظ‬ભҠ֬཰͸ɺֶशσʔλͷ͢΂ͯͷμϯεಈ࡞ʹ͍ͭͯ‫ޠ‬ኮΛਪఆ্ͨ͠Ͱɺ‫ޠ‬ኮͷग़‫ݱ‬ճ਺ɺྡΓ߹͏‫ޠ‬ኮͷભҠճ਺Λ΋ͱʹ‫͢ࢉܭ‬Ε͹Α͍ɻ

=

X

N
Y

P (Mn |vn ) P (Dn |vn ) P (vn |vn−1 ) (12)n=1{vn }Nn=1

Λಘͯɺμϯεಈ࡞ͷΨ΢γΞϯϓϩηεʹ‫ͮ͘ج‬ϞσϧͱԻ‫ڹ‬ಛ௃ྔͱμϯε‫ޠ‬ኮͷؔ܎ͷϞσϧΛ༻͍ͯ֬཰͕

2.4 Իָʹ࿈ಈ͢Δμϯεಈ࡞ͷϞσϧԽ‫͖Ͱࢉܭ‬Δ͜ͱ͕Θ͔Δɻ

2.4.1 ԻָͱμϯεͷରԠؔ܎ͷ֬཰Ϟσϧμϯεಈ࡞ͷ‫ޠ‬ኮ v ͷ΋ͱͰͷԻָԻ‫ڹ‬৴߸Λ෼ੳ͢Δεಈ࡞ͷ෼ੳ۠ؒͱಉ͡Α͏ʹɺҰͭલͷখઅͷ 3 ഥ໨͔

2.5 Իָʹ࿈ಈͨ͠μϯεͷࣗಈੜ੒!

N͜ͷΑ͏ʹಘΒΕͨϞσϧ P {Mn }N
n=1 , {Dn }n=1 ΛΒࠓͷখઅͷ 4 ഥΊ·ͰΛ੾Γग़ͯ͠෼ੳ͢Δɻ੾Γग़͞μϯεಈ࡞‫ྻܥ‬Λੜ੒͢Δɻμϯεͷࣗಈੜ੒͸༩͑ΒΕͨΊɺԻ‫ڹ‬ಛ௃ྔ MFCC+∆MFCC ͷྻʹ͍ͭͯ΋ɺμϯc 2014 Information Processing Society of Japan΋ͱʹɺϏʔτͱখઅઢ͕ਪఆࡁΈͷ৽ָ͍͠‫ʹۂ‬ରͯ͠

5

Vol.2014-MUS-104 No.14
2014/8/26

৘ใॲཧֶձ‫ڀݚ‬ใࠂ
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3. μϯεͷࣗಈੜ੒࣮‫ݧ‬ຊख๏ʹΑ࣮ͬͯࡍʹμϯεಈ࡞ָ͕‫͔ۂ‬Βੜ੒Ͱ͖Δ͔Λ͔֬Ίͨɻ·ͨࣗಈੜ੒݁Ռͱֶशσʔλʹ‫·ؚ‬ΕΔμϯεಈ࡞Λൺֱ͠ɺֶशσʔλͷμϯεʹ‫͍͠৽͍ͯͮج‬ಈ࡞Λࣗಈੜ੒Ͱ͖͍ͯΔ͜ͱΛ֬ೝͰ͖ͨɻੜ੒݁Ռ͸ϗʔϜϖʔδ https://staff.aist.go.jp/s.fukayama/

MachineDancing/index-j.html ͔Β֬ೝͰ͖Δɻ
60 ‫ۂ‬ͷμϯεͱָ‫͕ۂ‬ରԠ෇͍ͨσʔλΛ༻͍ϞσϧͷֶशΛߦͬͨɻμϯεͷσʔλ͸ MMD ͷϞʔγϣϯσʔλϑΥʔϚοτ vmd ʹΑͬͯ‫ه‬ड़͞Ε͍ͯΔɻָ‫ۂ‬ͷϏʔτͱখઅઢ͸ɺ֊૚తͳϏʔτߏ଄ͷਪఆ [19] ʹΑͬͯ‫ٻ‬Ίͨ‫ʹޙ‬ɺͦͷਪఆ‫ޡ‬ΓΛख࡞‫Ͱۀ‬मਖ਼ͨ͠ɻ0.5
খઅͣͭΦʔόʔϥοϓΛͤ͞ͳ͕Βɺ1.5 খઅͷ෼ੳ۠ؒ௕Ͱμϯεಈ࡞Λ੾Γग़͠ɺϞσϧͷֶशΛߦͬͨɻੜ
੒͞Εͨμϯε͸ɺ3D ͷ༷ʑͳ CG ΩϟϥΫλʢྫ͑͹ɺ
ਤ 2 ͷ koron ࢯʹΑͬͯ࡞੒͞ΕͨॳԻϛΫͷ 3D Ωϟϥਤ 4μϯεಈ࡞ͷࣗಈੜ੒: ֤μϯε‫ޠ‬ኮ͔ΒΨ΢γΞϯϓϩηε

ΫλʣΛ༻͍ϨϯμϦϯάͨ͠ɻʹΑͬͯੜ੒͞ΕΔ 1.5 খઅ୯Ґͷμϯεಈ࡞͸ɺ3 ഥ໨ 4 ഥ໨ʹ͓͍ͯϑΣʔυΠϯɾϑΣʔυΞ΢τ͢ΔΑ͏ʹՃࢉ͞

4. ͓ΘΓʹ

Εɺָ‫ۂ‬Λ௨ͨ͡μϯεಈ࡞͕ੜ੒͞ΕΔɻ্ͷਤதͰ͸ΩϟຊߘͰ͸ָ‫ͱۂ‬μϯε͕ରԠ෇ֶ͍ͨशσʔλΛ༻͍ͯϥΫλͷ͕࢟ϑΣʔυΠϯɾϑΣʔυΞ΢τ͍ͯ͠Δ͕ɺ࣮ࡍʹ͸ΩϟϥΫλͷϘʔϯͷҐஔɾճస͕ઢ‫ؒ͞ิܗ‬ΕΔɻ

‫ػ‬ցֶशΛߦ͍ɺ೚ҙͷ৽ָ͍͠‫ʹۂ‬࿈ಈͨ͠μϯεಈ࡞Λࣗಈੜ੒͢Δख๏ MachineDancing ΛఏҊͨ͠ɻֶश

ͨԻָಛ௃ྔͷ΋ͱͰ໬౓࠷େͷμϯεΛ୳ࡧ͢Δ໰୊ͱσʔλதͷμϯεಈ࡞ͷஅยΛ੾ΓషΓ͢ΔͷͰ͸ͳ͘ɺͯ͠‫ؼ‬ணͰ͖Δɻ͔֤࣌͠͠ࠁͰͷμϯεಈ࡞Λ࠷దԽ͢Δ͜ͱ͸ࠔ೉Ͱ͋ΔͷͰɺμϯε‫ޠ‬ኮͷྻ {vn }N
n=1 Λ࠷ద

ԻָͱμϯεͷରԠؔ܎ʹՃ͑ͯɺμϯεಈ࡞ࣗମΛ΋ֶ
शͰ͖Δ֬཰ϞσϧΛઃ‫͠ܭ‬ɺͦͷֶशΞϧΰϦζϜΛಋ

Խ͢Δ໰୊ͱͯ͠ղ͖ɺಘΒΕͨμϯε‫ޠ‬ኮͷྻ͔Βμϯεಈ࡞Λੜ੒͢Δ 2 εςοϓʹΑͬͯࣗಈੜ੒Λߦ͏ɻग़ͨ͠ɻ·ͨμϯε‫ޠ‬ኮͱ͍͏֓೦Λಋೖ͠ɺ࣌ؒΛม਺
ͱ͢Δ࿈ଓؔ਺ʹΑͬͯμϯεಈ࡞ΛϞσϧԽͨ͜͠ͱ

·ͣ͸͡Ίʹɺ෼ੳ۠ؒ͝ͱʹ༩͑ΒΕͨԻ‫ڹ‬ಛ௃ྔ∗ N{Mn }Nn=1 ͷ΋ͱͰɺ໬౓࠷େͷμϯε‫ޠ‬ኮͷ‫{ ྻܥ‬vn }n=1ͰɺҟͳΔςϯϙͷָ‫Ͳͳۂ‬Λ‫ؚ‬Ήଟ༷ͳֶशσʔλΛ༻͍ͯ֬཰ϞσϧΛֶशͰ͖ΔΑ͏ʹͳͬͨɻΛ‫ٻ‬ΊΔɻ{vn∗ }Nn=1 = argmax

NY‫ࡏݱ‬ͷ MachineDancing Ͱ͸ɺָ‫ۂ‬Λ༩͑Δͱ‫׬‬શʹࣗ

P (Mn |vn ) P (vn |vn−1 )(13){vn }Nn=1 n=1͜Ε͸Ի‫ڹ‬ಛ௃ྔ͕‫؍‬ଌɺμϯε‫ޠ‬ኮ͕ӅΕঢ়ଶͰ͋Δ

ಈͰμϯεΛੜ੒͢Δख๏͕ͩɺࠓ‫ޙ‬ຊख๏ΛԠ༻ͯ͠Ϣʔβ͕ΠϯλϥΫςΟϒʹμϯεΛ੍࡞Ͱ͖ΔΠϯλʔϑΣʔεΛߏங͢Δ༧ఆͰ͋Δɻ

HMM Ͱ͋ΓɺViterbi ΞϧΰϦζϜΛ༻͍ͯղ͘͜ͱ͕Ͱ
͖Δɻँࣙຊ࿦จͷਤதʹొ৔͢Δ 3D ΩϟϥΫλ͸ɺϐΞ࣍ʹಘΒΕͨμϯε‫ޠ‬ኮͷྻ {vn∗ }Nn=1 Λ༻͍ͯɺ෼ੳ

ϓϩɾΩϟϥΫλʔɾϥΠηϯεʹ‫͍ͯͮج‬ΫϦϓτϯɾ۠ؒ͝ͱʹμϯεಈ࡞ Dnew
Λੜ੒͢Δɻμϯε‫ޠ‬ኮ vnnϑϡʔνϟʔɾϝσΟΞ‫ࣜג‬ձࣾͷΩϟϥΫλʔʮॳԻͱͯ͠ΫϥελϦϯά͞Εͨμϯεಈ࡞ΛूΊͨߦྻTD∗ = D1 , · · · , DM Λ༻͍ɺΨ΢γΞϯϓϩηεͷ֤࣌ࠁͰͷฏ‫ۉ‬Λ‫ٻ‬ΊΔ͜ͱͰੜ੒Ͱ͖Δɻ·ͨɺμϯεಈ࡞ϛΫʯΛ࢖༻͠·ͨ͠ɻ·ͨΩϟϥΫλͷ 3D Ϟσϧ͸koron ࢯʹΑ੍ͬͯ࡞͞Εͨ΋ͷͰ͢ɻຊ‫ڀݚ‬ͷҰ෦͸JST CRESTʮOngaCRESTʯϓϩδΣΫτͷࢧԉΛड͚͕֬཰ϞσϧʹΑͬͯϞσϧԽ͞Ε͍ͯΔ͜ͱͰɺֶशͰ
·ͨ͠ɻಘΒΕͨ෼ࢄʹ‫͍ͯͮج‬ฏ‫ۉ‬஋ʹ৙ཚΛ༩͑ͯɺੜ੒͢Δ౓ʹҟͳͬͨμϯεಈ࡞Λੜ੒͢Δ͜ͱ΋ՄೳͰ͋Δɻ࠷ࢀߟจ‫ݙ‬‫ʹޙ‬ɺ֤෼ੳ۠ؒͰμϯε‫ޠ‬ኮ͔Βੜ੒͞Εͨμϯεಈ࡞[1]͸ɺ֤খઅͷ 3 ഥΊ 4 ഥ໨Ͱલ‫ޙ‬ͷಈ͖͕ϑΣʔυΞ΢τɾΠϯ͢ΔΑ͏ʹɺઢ‫ؒิܗ‬Λߦ͍ɺָ‫ۂ‬શମʹ࿈ಈ͢ΔμϯεΛੜ੒͢Δɻิؒͷ༷ࢠΛਤ 4 ʹࣔ͢ɻc 2014 Information Processing Society of Japan[2]

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