情報処理学会第 78 回全国大会

3Q-06ίʔυਐߦͱଟॏԻεϖΫτϧͷ֊૚ϕΠζϞσϧʹ‫ͮ͘ج‬ԻָԻ‫ڹ‬৴߸ͷԻߴਪఆඌౡ ༏ଠ †தଜ ӫଠ ‡† ‫౎ژ‬େֶ ޻ֶ෦৘ใֶՊ1.

ࢳࢁ ࠀण ‡‡ ‫౎ژ‬େֶ େֶӃ৘ใֶ‫ڀݚ‬Պ ஌ೳ৘ใֶઐ߈͸͡Ίʹࣗಈ࠾ේͱ͸ɼԻָԻ‫ڹ‬৴߸தʹ‫·ؚ‬ΕΔԻߴͱԻՁΛਪఆ͢Δ໰୊Ͱ͋ΓɼಛʹԻߴਪఆʹ͍ͭͯ͸ଟ͘ͷ‫͞ͳ͕ڀݚ‬Ε͍ͯΔɽैདྷɼԻߴਪఆΛ࣮‫͢ݱ‬Δʹ͸ɼࠞ߹ԻͰ͋ΔԻָԻ‫ڹ‬৴߸ͷԻ‫ݯ‬෼཭͕ඞཁͰ͋Δͱߟ͑ΒΕ͖ͯͨɽྫ͑͹ɼඇෛ஋ߦྻҼࢠ෼ղ (NMF) Λ༻͍Δͱɼࠞ߹ԻͷεϖΫτϩάϥϜ͔ΒɼҟͳΔԻߴ
ʹରԠ͢Δ‫ج‬ఈεϖΫτϧͱ֤‫ج‬ఈͷΞΫςΟϕʔγϣϯ (Իྔ) ΛਪఆͰ͖Δ [1,2]ɽ͜͜ͰɼΞΫςΟϕʔγϣϯ͸࿈ଓྔͰ͋ΔͷͰɼ֤࣌ࠁʹ͓͚Δ֤Իߴͷ on/oﬀΛ൑ఆ (ԻߴਪఆʹϐΞϊϩʔϧͷग़ྗ) ͢ΔͨΊʹ͸ɼରԠ͢Δ‫ج‬ఈεϖΫτϧͷΞΫςΟϕʔγϣϯʹର͢Δೋ஋ԽΛߦ͏ඞཁ͕͋ͬͨɽ͔͠͠ɼᮢ஋ઃఆ͕ࠔ೉Ͱ͋Δ͚ͩͰ͸ͳ͘ɼߴਫ਼౓ͳԻ‫ݯ‬෼཭͕Ͱ͖Ε͹Իߴਪఆ΋༰қʹͳΔͷͱಉ༷ɼԻߴ͕‫ط‬஌Ͱ͋Ε͹Ի‫ݯ‬෼཭͸༰қʹͳΔͱ͍͏ܲͱཛͷؔ܎͕ଘࡏ͢ΔͨΊɼ͜ͷΑ͏ͳॎྻॲཧ͸ຊདྷద੾Ͱ͸ͳ͔ͬͨɽຊߘͰ͸ɼԻ‫ݯ‬෼཭ͱԻߴਪఆΛҰ‫ͨ͏ߦʹڍ‬Ίɼࠞ߹Իͷੜ੒աఔΛ֬཰తʹఆࣜԽ͠ɼͦͷ‫ٯ‬໰୊Λղ͘ͱ͍͏ΞϓϩʔνΛͱΔ [1, 3] (ਤ 1)ɽ·ͣɼϕʔλաఔ NMF (BP-NMF) [4] ͱಉ༷ɼ֤࣌ࠁʹ͓͍֤ͯ‫ج‬ఈͷ on/oﬀ Λ੍‫͢ޚ‬Δೋ஋ม਺Λಋೖ͢Δɽ͜ΕʹΑΓɼΞΫςΟϕʔγϣϯͷ஋ʹؔΘΒͣɼೋ஋ม਺͕ oﬀ ΛऔΔ৔߹ʹ͸ɼରԠ͢ΔԻߴ͸ࠞ߹Իͷੜ੒ʹ͸‫د‬༩͠ͳ͘ͳΔɽ͞Βʹɼ֤࣌ࠁʹ͓͚ΔԻߴͷ૊Έ߹Θͤ͸ίʔυʹґଘ͢Δ͜ͱ͔ΒɼίʔυભҠΛજࡏม਺‫ྻܥ‬ͱ͠ɼԻߴ‫܈‬ͷ on/oﬀ (ϐΞϊϩʔϧ) Λग़ྗ͢ΔӅΕϚϧίϑϞσϧ (HMM) Λߏ੒͢Δɽ͢ͳΘͪɼHMMΛԻߴ‫ʹ܈‬ର͢Δࣄલ෼෍ɼNMF Λࠞ߹Իʹର͢ΔԻߴ‫܈‬ͷ໬౓ؔ਺ͱͨ͠౷Ұతͳ֊૚ϕΠζϞσϧ͕ఆࣜԽ͞ΕΔɽ͜ͷछͷ‫ޠݴ‬ϞσϧͱԻ‫ڹ‬Ϟσϧͷ౷߹͸ɼҰൠతͳԻ੠ೝࣝγεςϜͱಉ༷Ͱ͋Δ͕ɼΪϒεαϯϓϦϯάΛ༻͍ͯԻߴਪఆΛߦ͏ͱಉ࣌ʹɼ྆ϞσϧΛҰ‫ֶ͠ͳࢣڭʹڍ‬श͢Δ఺͕ҟͳΔɽ

2.

٢Ҫ ࿨Ղ ‡ఏҊ๏ఏҊϞσϧ͸ɼԻ‫ڹ‬ϞσϧͰ͋Δ NMF ͱ‫ޠݴ‬Ϟσϧ
Ͱ͋Δ HMM ͔ΒͳΔɽҎԼʹ֬཰ϞσϧͷఆࣜԽͱύϥϝʔλͷࣄ‫ޙ‬෼෍ͷਪఆํ๏ʹ͍ͭͯઆ໌͢Δɽ

2.1 Ի‫ڹ‬ϞσϧͷఆࣜԽೋ஋ม਺͕ಋೖ͞Εͨ NMF Ͱ͸ɼࠞ߹Իͷৼ෯εϖ×TΫτϩάϥϜ X ∈ RFʹରͯ͠ɼX ≈ W (H ⊙ S) ͱ+×K
͍͏௿ϥϯΫۙࣅΛߦ͏ɽ͜͜ͰɼW ∈ RF
͸K ‫ݸ‬+K×Tͷ‫ج‬ఈεϖΫτϧ‫܈‬ɼH ∈ RK×T͓ΑͼS∈{0,1}+͸ରԠ͢ΔΞΫςΟϕʔγϣϯϕΫτϧ‫͓܈‬Αͼೋ஋ม਺ϕΫτϧ‫͋Ͱ܈‬ΓɼF ͸प೾਺Ϗϯ਺ɼT ͸ϑϨʔϜ਺Λද͢ɽ·ͨɼप೾਺ϏϯΛ f (1 ≤ f ≤ F )ɼϑϨʔ
ϜΛ t (1 ≤ t ≤ T )ɼ‫ج‬ఈΛ k (1 ≤ k ≤ K) Ͱද͢ɽNMF ͸‫؍‬ଌߦྻ X ͱ࠶ߏ੒ߦྻ W (H ⊙ S) ͱͷۙࣅ‫ࠩޡ‬Λ࠷খԽ͢Δ໰୊Ͱ͋Δ͕ɼ࣍ࣜͰఆٛ͞ΕΔ໬
Pitch Estimation for Music Audio Signals based on a Hierarchical
Bayesian Model of Chords and Spectrograms: Yuta Ojima, EitaNakamura, Katsutoshi Itoyama, and Kazuyoshi Yoshii (Kyoto Univ.)ਤ 1: Ϟσϧͷશମ૾౓ؔ਺࠷େԽͱͯ͠ͷղऍ͕ՄೳͰ͋Δɽ!
Xf t | Wf k Hkt Skt ∼ Poisson ( k Wf k Hkt Skt )͜͜Ͱɼ໬౓ؔ਺ͱͯ͠ϙΞιϯ෼෍Λ༻͍ͨ [1]ɽຊ
‫Ͱڀݚ‬͸͞Βʹɼࣄલ෼෍Λಋೖ͢Δ͜ͱʹΑΓɼNMFͷϕΠζϞσϧΛߏ੒͢Δɽ

Wf k ∼ Gamma (a, b) , Hkt ∼ InverseGamma (c, d)͜͜ͰɼW ʹ͸ΨϯϚࣄલ෼෍Λ༩͑Δ͜ͱͰ [5]ɼௐ
೾ߏ଄ͷΑ͏ͳεύʔεͳ‫ج‬ఈεϖΫτϧʹ༠ಋͨ͠ɽ
ҰํɼH ʹ͸‫ٯ‬ΨϯϚ෼෍Λ༩͑Δ͜ͱͰɼඇεύʔεͳΞΫςΟϕʔγϣϯʹ༠ಋͨ͠ɽHkt ͕΄΅ 0 Ͱ͋Δ৔߹ʹ͸ɼೋ஋ม਺ Skt ͷ஋ʹؔΘΒͣͦͷ‫ج‬ఈ k ͸໬౓ؔ਺ʹ‫د‬༩͠ͳ͍ͨΊɼSkt ͷਪఆ͕ద੾ʹߦ͑ͳ͘ͳͬͯ͠·͏ɽҰํɼHkt ͕ৗʹ͋Δఔ౓ͷେ͖͞Λ΋ͭΑ͏Ͱ͋Ε͹ɼ࣮ࡍʹ͸ͦͷԻߴ͕ଘࡏ͠ͳ͚Ε͹ɼSkt ͸ 0 ΛऔΒ͟ΔΛಘͣɼద੾ʹϚεΫͱͯ͠ͷ‫ػ‬ೳΛՌͨ͢ɽ࣮ࡍʹ͸ɼ࣌ؒํ޲ͷ‫׈‬Β͔͞Λߟྀ͢ΔͨΊɼ‫ٯ‬ΨϯϚ࿈࠯ࣄલ෼෍Λ༻͍ͨ [5]ɽ2.2 ‫ޠݴ‬ϞσϧͷఆࣜԽίʔυભҠΛજࡏม਺‫ ྻܥ‬Z = {z1 , · · · , zT } (zt ∈{1, · · · , I}) ʹ࣋ͪɼ‫؍‬ଌม਺‫ ྻܥ‬S = {s1 , · · · , sT } (ఏҊϞσϧશମ͔Β‫ݟ‬Ε͹જࡏม਺) Λग़ྗ͢ΔϕΠζ
HMM ΛఆࣜԽ͢Δɽ͜͜ͰɼI ͸ίʔυͷछྨ਺ (ঢ়ଶ਺) ͱͨ͠ɽ·ͣɼঢ়ଶભҠϞσϧ͸࣍ࣜͰ༩͑ΒΕΔɽ
zt | zt−1 , ψzt−1 ∼ Categorical(ψzt−1 ), ψi ∼ Dirichlet(1I )͜͜Ͱɼψi ͸ίʔυ i ʹ͓͚ΔભҠ֬཰Ͱ͋Γɼແ৘ใσΟϦΫϨࣄલ෼෍Λ༩͑ͨɽҰํɼԻߴ‫܈‬ͷग़ྗϞσϧ͸࣍ࣜͰ༩͑ΒΕΔɽ

Skt | zt , πzt k ∼ Bernoulli(πzt k ), πzt k ∼ Beta(e, f )͜͜Ͱɼ֤࣌ࠁ t ʹ͓͚ΔԻߴ k ͷ on/oﬀ ͸ද͕ग़Δ֬཰ πzt k ͷίΠϯτεͰܾఆ͞ΕΔɽ͢ͳΘͪɼSkt = 1
ͱͳΔ֬཰͸ɼͦͷ࣌ࠁʹ͓͚ΔίʔυʹΑͬͯҟͳΔɽ࣮ࡍʹ͸ɼߏ੒ԻͷԻߴͷ૬ରతͳؔ܎͕ಉ͡ʹͳΔίʔυಉ࢜͸ɼԻߴͷग़‫཰֬ݱ‬Λ‫ڞ‬༗͢ΔΑ͏ʹͨ͠ɽ

2.3 ࣄ‫ޙ‬෼෍ͷਪ࿦ࣄ‫ޙ‬෼෍ p(W , H, S, Z, π, ψ|X) Λۙࣅ‫͢ࢉܭ‬ΔͨΊɼΪϒεαϯϓϦϯάΛ༻͍Δɽ·ͣɼNMF ͷύϥϝʔλ W , H ͱ HMM ͷύϥϝʔλ π, ψ Λద౰ʹॳ‫ظ‬Խ͠ɼp(S|W , H, π, ψ, X) ͔ΒԻߴ‫ ܈‬S Λαϯϓϧ͢

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情報処理学会第 78 回全国大会

ਤ 4: "MUS-scn15_12_ENSTDkCl"ʹ͓͚Δίʔυߏ଄ਪఆ݁Ռ
ද 1: Իߴਪఆ݁Ռͷ f ஋Ϟσϧ‫ ޠݴ‬+ Ի‫ڹ‬ϞσϧԻ‫ڹ‬ϞσϧͷΈ

ਤ 2: "MUS-scn15_12_ENSTDkCl"ʹର͢ΔԻߴਪఆ݁
ՌɽԻߴਪఆ݁Ռͷഎ‫ܠ‬৭͸ίʔυਪఆ݁ՌΛද͢ɽਤ 3: "MUS-scn16_3_ENSTDkCl"ʹର͢ΔԻߴਪఆ݁Ռ

3.

࣮‫ݧ‬

ຊख๏ͷ༗ޮੑΛ֬ೝ͢ΔͨΊɼMAPS σʔλϕʔ
ε [8] ͷ͏ͪ"ENSTDkCl"ͷϥϕϧ͕෇͞ΕͨϐΞϊ‫ۂ‬30 ‫ۂ‬ͷ๯಄ 30 ඵؒʹର͠ɼԻߴਪఆΛߦ͍ɼ‫ج‬ఈͷ੍‫ݶ‬ͷΈΛՃ͑ͨ BP-NMF ͱൺֱͨ͠ɽೖྗԻ‫ڹ‬৴߸ͱͯ͠༧Ί variable-Q ม‫[ ׵‬9] ͓ΑͼϦαϯϓϦϯάʹΑΓɼप೾਺Ϗϯ 926ɼ࣌ؒϑϨʔϜ 3000 ͷεϖΫτϩάϥϜ΁ͱม‫ޙͨ͠׵‬ɼௐ೾ଧָ‫ث‬Ի෼཭ [10] Λߦͬͨ΋ͷΛ༻͍ͨɽͳ͓ɼϋΠύʔύϥϝʔλ͸࣮‫ݧ‬తʹܾఆͨ͠ɽ·ͨɼHMM ͷॳ‫཰֬ظ‬ͷࣄલ෼෍͸Ұ༷Ͱ͋Δͱ͠ɼભҠ֬཰͸ɼϑϨʔϜ୯ҐͰͷίʔυͷભҠ͸‫ى‬͜Γʹ͍ͨ͘Ίࣗ‫ݾ‬ભҠΛ 1−5.0×10−8 ͱ͠ɼଞঢ়ଶ΁ͷભҠ͕σΟϦΫϨ෼෍ʹै͏ͱԾఆͨ͠ɽίʔυ͸ 12छྨͷϧʔτԻʹର͠ɼMajor ͱ Minor ͷ 2 ͭͷίʔυ‫ֶ͕ܗ‬श͞ΕΔ͜ͱΛҙਤͯ͠‫ ܭ‬24 छྨΛ༻ҙͨ͠ɽ‫ج‬ఈ͸ௐ೾ߏ଄Λද͢‫ج‬ఈͱඇௐ೾ߏ଄Ͱ͋ΔࡶԻΛද‫͢ݱ‬Δ‫ج‬ఈΛ 1 ͭͣͭ༻ҙͨ͠ɽௐ೾ߏ଄‫ج‬ఈΛγϑτ͢Δ͜ͱͰԻߴʹରԠͤ͞Δɽ·ͨɼࡶԻද‫جݱ‬ఈͷεύʔεੑ͸‫ޠݴ‬Ϟσϧʹґଘͤͣ͞ɼैདྷͷ BP-NMFͱಉ༷ʹܾఆͨ͠ɽධՁ͸Իߴਪఆ݁Ռͱਖ਼ղσʔλΛ‫ࢉʹج‬ग़ͨ͠ f ஋ʹΑΓߦ͏ɽͳ͓ɼධՁʹ͋ͨͬͯ‫ۂ‬શମͷΦΫλʔϒͷ‫ޡ‬Γ͸‫ڐ‬༰ͨ͠ɽධՁ࣮‫ݧ‬ͷ݁ՌΛද 1 ʹࣔ͢ɽද 1 ΑΓɼ‫ޠݴ‬ϞσϧΛՃ͑ͯ΋Իߴਪఆਫ਼౓ͷฏ‫ۉ‬஋͸޲্͠ͳ͔ͬͨɽ͔͠͠ɼ‫݁ʹͱ͝ۂ‬ՌΛൺֱ͢Δͱ 30 ‫ۂ‬த 18 ‫ʹۂ‬ର͢ΔԻߴਪఆਫ਼౓͕޲্͍ͯͨ͠ɽ͜ͷ͜ͱ͔ΒɼҰ෦ͷ‫ۂ‬࠶‫཰ݱ‬
70.02%
73.82%

ద߹཰
58.00%
56.56%

ʹରͯ͠େ͖͘ਫ਼౓͕མ͍ͪͯΔ͜ͱ͕Θ͔Δɽ‫ޠݴ‬Ϟσϧͷ౷߹ʹΑΓԻߴਪఆਫ਼౓͕ 3.3 ϙΠϯτ޲্ͨ͠‫ۂ‬Λਤ 2 ʹɼ7 ϙΠϯτѱԽͨ͠‫ۂ‬Λਤ 3 ʹࣔ͢ɽ·ͨɼલऀʹର͢Δίʔυߏ଄ਪఆ݁ՌΛਤ 4 ʹࣔ͢ɽਤ 2,3 ΑΓɼಉ͡Իͷ‫͖ڹ‬Λ࣋ͭ‫܁‬Γฦ۠ؒ͠͸ಉҰͷঢ়ଶׂ͕Γ౰ͯΒΕ͓ͯΓɼ‫ޠݴ‬Ϟσϧ͕ਖ਼ֶ͘͠
श͞Ε͍ͯΔ͜ͱ͕෼͔Δɽ·ͨɼਤ 4 ΑΓɼࣄલ৘ใ
͕ͳ͘ͱ΋ԻָԻ‫ڹ‬৴߸͔Βίʔυͷߏ଄͕֓Ͷਖ਼֫͘͠ಘͰ͖͍ͯΔ͜ͱ͕෼͔Δɽਤ 2 Ͱ͸࢒‫ڹ‬ԻΛर͏‫ޡ‬Γ͕ɼਤ 3 Ͱ͸සग़͢ΔԻ͕ίʔυߏ଄ͱͯ͠ਖ਼ֶ͘͠
शͰ͖ͳ͍͜ͱʹΑΔ‫ޡ‬Γ͕‫ݟ‬ΒΕΔɽ

4.

Δɽ࣍ʹɼp(W , H|S, X) ͔Β W , H Λɼp(π, ψ, Z|S)͔Β π, ψ, Z ΛͦΕͧΕαϯϓϧ͢Δɽ͜ͷͱ͖ɼNMFͱ HMM ͸จ‫[ ݙ‬4, 6] Λࢀߟʹಠཱʹߋ৽Ͱ͖Δɽ͜ΕΛ൓෮͢Δ͜ͱͰࣄ‫ޙ‬෼෍ʹऩଋ͢Δɽ͜ͷΑ͏ʹɼԻ‫ڹ‬Ϟσϧͱ‫ޠݴ‬ϞσϧΛҰ‫࠷ʹڍ‬దԽͯ͠ԻߴਪఆΛߦ͏Ξϓϩʔν͸ɼจ‫[ ݙ‬7] Ͱ΋‫ݟ‬ΒΕΔɽf஋
63.44%
64.05%

͓ΘΓʹຊߘͰ͸ HMM ʹΑΔ‫ޠݴ‬Ϟσϧͱ BP-NMF ʹΑΔԻ‫ڹ‬ϞσϧΛ౷߹͠ɼԻߴਪఆΛߦ͏ͨΊͷख๏ΛఏҊͨ͠ɽίʔυߏ଄Λߟྀͯ͠΋Իߴਪఆ݁Ռ͸޲্͠ͳ͔͕ͬͨɼ͜Ε͸‫ʹۂ‬ΑΓίʔυߏ଄Λ֫ಘͰ͖ͳ͔ͬͨ͜ͱ͕‫ݪ‬ҼͰ͋Δͱߟ͑ΒΕΔɽͦͷͨΊ‫ޠݴ‬ϞσϧʹվળΛՃ͑Δ͜ͱͰίʔυߏ଄Λਖ਼֫͘͠ಘͰ͖Ε͹ɼਫ਼౓ͷ޲্͕‫ظ‬଴Ͱ͖Δɽ͜ΕΛ౿·͑ɼࠓ‫ޙ‬͸ӅΕηϛϚϧίϑϞσϧΛಋೖ͢Δ͜ͱͰίʔυਐߦΛϑϨʔϜ୯ҐͰ͸ͳ͘ഥ୯ҐͰଊ͑ɼίʔυߏ଄ਪఆ݁Ռͷਫ਼౓޲্ΛਤΔ༧ఆͰ͋Δɽँࣙ ຊ‫ڀݚ‬ͷҰ෦͸ JSPS Պ‫ݚ‬අ 24220006, 26700020, 26280089,

JST CREST ͷࢧԉΛड͚ͨɽࢀߟจ‫ݙ‬[1] P. Smaragdis et al. Non-negative matrix factorization for
polyphonic music transcription. IEEE WASPAA, 177–180,2003.
[2] N. Bertin et al. Enforcing harmonicity and smoothness inBayesian non-negative matrix factorization applied to polyphonic music transcription. IEEE TASLP, 18(3):538–549,
2010.
[3] K. Yoshii et al. Infinite composite autoregressive models formusic signal analysis. ISMIR, 79–84, 2012.
[4] D. Liang et al. Beta process non-negative matrix factorization with stochastic structured mean-field variational inference. arXiv, 1411.1804, 2014.
[5] A T. Cemgil et al. Conjugate gamma markov random
fields for modelling nonstationary sources. ICA, 697–705.
Springer, 2007.
[6] S. L. Scott. Bayesian methods for hidden Markov models.
JASA, 97(457), 2002.
[7] H. Kameoka et al. Context-free 2D tree structure model ofmusical notes for Bayesian modeling of polyphonic spectrograms. ISMIR, 307–312, 2012.
[8] V. Emiya et al. Multipitch estimation of piano sounds using
a new probabilistic spectral smoothness principle. IEEETASLP, 18(6):1643–1654, 2010.
[9] C. Schörkhuber et al. A Matlab toolbox for eﬃcientperfect reconstruction time-frequency transforms with logfrequency resolution. AES Conf. SA, 2014.
[10] D. Fitzgerald. Harmonic/percussive separation using median filtering. DAFX, 1–4, 2010.

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