Speeding Up of the Complex Discrete Wavelet Transform by Using Lifting Scheme

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ϦϑςΟϯάεΩʔϜʹΑΔෳૉ਺཭ࢄ΢ΣʔϒϨοτม׵ͷ

ߴ଎Խʹؔ͢Δݚڀ

ষɹ஧

_{, ౢ຤ ߉༞, ཛྷɹ༔໵, ށాɹߒ, ळ݄ɹ୓ຏ}

๛ڮٕज़Պֶେֶ ػց޻ֶܥɼѪ஌ݝ๛ڮࢢఱഢொӢ੃έٰ̍_-̍

Speeding Up of the Complex Discrete Wavelet Transform by Using Lifting

Scheme

Zhong Zhang, Kosuke Shimasue, Yuya Arashi, Hiroshi Toda and Takuma Akiduki

1-1 Hibarigaoka, Tempaku-cho, Toyohashi, Aichi 441-8580, Japan E-mail: [email protected]

Abstract It is well known that the Lifting Scheme (LS) allows us to design the fast calculation method of the Discrete Wavelet Transform (DWT). However, unfortunately, the LS can be only adopted for the wavelets having a compact support. For example, Meyer wavelet, which is a famous orthonormal wavelet basis, has no compact support. Additionally, the Complex Discrete Wavelet Transform (CDWT) is steady and useful for many signal processing applications, however, its imaginary part is constructed from the wavelets having no compact support. Therefore, we cannot adopt the LS for these analyses. In this study, we propose the design method of the LS filters for all the orthonormal wavelet bases without relation to having a compact support or not. We adopt our proposed method for the CDWTs using Meyer wavelet and Daubechies 6 wavelet, and confirm their steady analyses and fast calculation speeds.

Keywords: complex discrete wavelet transform, lifting scheme, calculation amount, Meyer wavelet, Daubechies 6 wavelet 1. ͸͡Ίʹ ΢ΣʔϒϨοτม׵_{(Wavelet Transform, WT) ʹΑ} Δप೾਺ղੳ͸ۙ೥ɼं྆ͷҟৗԻɼϊΠζআڈͳͲ ʹԠ༻͞Ε༗༻ੑ͕֬ೝ͞Ε͍ͯΔ[1]ɽ΢ΣʔϒϨο τม׵͸ہॴੑΛ࣋ͭϚβʔ΢ΣʔϒϨοτ_(Mother Wavelet, MW) Λ֦େॖখɾฏߦҠಈͤͯ͞৴߸Λղ ੳ͢Δख๏Ͱ͋Γɼ৴߸ͷҰ෦෼ͷΈʹൃੜ͍ͯ͠Δ ඇఆৗ੒෼ΛؚΉ৴߸ʹରͯ͠༗ޮͳղੳΛߦ͑Δ ಛ௃Λ͍࣋ͬͯΔ[2, 3]ɽ·ͨɼ΢ΣʔϒϨοτม׵ ͸େ͖͘࿈ଓ΢ΣʔϒϨοτม׵_{(Continuous Wavelet} Transform, CWT) ͱ཭ࢄ΢ΣʔϒϨοτม׵ (Discrete Wavelet Transform, DWT) ʹ෼͚ΒΕΔɽDWT ͸ CWT ʹൺ΂ͯॲཧ͕ߴ଎Ͱ͋Γɼ৴߸ͷϊΠζআڈ΍σʔ λѹॖͳͲʹΑ͘༻͍ΒΕΔɽ͔͠͠ɼ_{DWT ͷղੳ} Ͱ͸৴߸ͷҐ૬ʹґଘ͠ม׵݁Ռ͕มಈͯ͠͠·͏γ ϑτෆมੑͷܽ೗ͱ͍͏໰୊͕͋Δ_{[4, 5]ɽ͜ͷγϑ} τෆมੑͷܽ೗Λղܾ͢ΔͨΊߟҊ͞Εͨͷ͕ɼෳૉ ਺཭ࢄ΢ΣʔϒϨοτม׵_{(Complex Discrete Wavelet} Transform, CDWT) Ͱ͋Δ [6, 7, 8, 9]ɽCDWT ͸࣮਺ ෦ͱڏ਺෦ͷ2 ͭͷ΢ΣʔϒϨοτͰม׵ɾ࠶ߏ੒Λ ฒߦʹߦ͏ͨΊɼܭࢉྔ͕௨ৗͷ_{DWT ͱൺֱ͠ 2 ഒ} Toyohashi University of Technology,

Journal of Signal Processing, Vol.23, No.2, pp.55-74, March 2019

論　文

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Ҏ্ͱͳͬͯ͠·͏ܽ఺͕͋Δɽ

·ͨ_{CDWT Ͱ͸ɼܭࢉํ๏ʹ௨ৗɼMallat ͷଟॏ} ղ૾౓ղੳ(Multi Resolution Analysis, MRA) Λ༻͍Δ [10]ɽMRA ʹ͓͚Δม׵Ͱ͸ɼϑΟϧλॲཧޙʹμ ΢ϯαϯϓϦϯά͕࣮ߦ͞ΕΔɽҰํɼ_{MRA ΑΓ΋} ͞Βʹߴ଎ͳܭࢉख๏ͱͯ͠ɼϦϑςΟϯάεΩʔϜ (Lifting Scheme, LS) ͕ఏҊ͞Ε͍ͯΔ [11, 12]ɽLS ʹ ͓͚Δม׵Ͱ͸ɼϑΟϧλॲཧͷલʹμ΢ϯαϯϓϦ ϯά͕࣮ߦ͞ΕΔͨΊɼܭࢉྔ͸ཧ࿦తʹ_{MRA ͷ໿} ൒෼ʹͳΔͱݴΘΕ͍ͯΔɽ͔͠͠LS ͸ίϯύΫτ αϙʔτΛ༗͢Δ΢ΣʔϒϨοτʢ͢ͳΘ্ͪ࣌ؒ࣠ Ͱ༗ݶͷ௕͞Λ࣋ͭ΢ΣʔϒϨοτʣʹͷΈద༻Ͱ͖ Δͱ͍͏੍ݶ͕͋ΓɼMeyer ΢ΣʔϒϨοτͷΑ͏ͳ ίϯύΫταϙʔτΛ࣋ͨͳ͍΢ΣʔϒϨοτʹɼ_LS ͷϑΟϧλઃܭ͸ద༻Ͱ͖ͳ͍ɽ·ͨCDWT Ͱ͸γϑ τෆมੑΛ࣮ݱͤ͞ΔͨΊʹڏ਺෦ͷ΢ΣʔϒϨοτ Λࢉग़͢Δ͕ɼڏ਺෦ͷ΢ΣʔϒϨοτ͸௨ৗίϯύ ΫταϙʔτΛ࣋ͨͳ͍ɽैͬͯɼίϯύΫταϙʔ τΛ࣋ͨͳ͍΢ΣʔϒϨοτʹ΋ద༻Ͱ͖Δ_{LS ͷϑΟ} ϧλΛߏ੒͢Δख๏ͷ։ൃ͸ॏཁͳ՝୊Ͱ͋Δɽ ຊݚڀͰ͸ɼίϯύΫταϙʔτΛ࣋ͨͳ͍΢Σʔ ϒϨοτʹ΋ద༻Ͱ͖ΔLS ͷϑΟϧλઃܭํ๏Λఏ Ҋ͢Δɽͦͯ͠ɼఏҊख๏Λ֤छͷෳૉ਺΢ΣʔϒϨο τม׵΁ద༻͠ɼઃܭ๏ͷ৴པੑΛ֬ೝ͢Δɽ͞Βʹ ઃܭͨ͠LS ϑΟϧλΛ༻͍ͨ CDWT Λߏ੒͠ɼܭࢉ ྔͷ࡟ݮɼ͓Αͼܭࢉ଎౓ͷߴ଎ԽΛ֬ೝ͢Δɽ 2. ίϯύΫταϙʔτΛ࣋ͨͳ͍΢ΣʔϒϨοτʹ ΋ద༻Ͱ͖Δ_{LS ϑΟϧλઃܭ๏ͷఏҊɹɹɹɹ} 2.1 ઃܭʹඞཁͳ਺ֶͷఆٛ౳ʹ͍ͭͯ ͜͜Ͱ͸جຊతͳ਺ֶͷఆٛ౳ʹ͍ͭͯड़΂Δɽຊ ݚڀͰ͸ɼ࣮਺શମͷू߹Λ_{Rɼ੔਺શମͷू߹Λ Z} Ͱද͢ɽ࣍ʹn, j, k ౳ͷม਺͸ৗʹ੔਺Λද͢΋ͷ ͱͯ͠ɼ਺ྻ͸_{h_n_}ɼ{h_k_{} ౳Ͱද͢ɽຊݚڀͰѻ͏਺} ྻ͸ ℓ2_{(Z) ʹଐ͠ɼ∑} n∈Z|hn|2<∞ ͕੒ཱ͢Δ਺ྻ {hn} ͱ͢Δɽ࣍ʹ਺ྻ_{h_n_{} ͷ z ม׵ h(z) Λ࣍ͷΑ͏ʹఆٛ} ͢Δɽ h(z) =∑ k∈Z hkz−k (1) ϑΟϧλ܎਺_{h_n_{} Ͱܗ੒͞ΕΔϑΟϧλͰ͸ɼೖྗ৴} ߸Λ_{x_n_{}ɼग़ྗ৴߸Λ {y}_n_{} ͱͯ͠ɼ࣍ͷ৞ΈࠐΈܭࢉ} ͕ߦΘΕΔɽ yn= ∑ k∈Z hkxn−k (2) ͜ͷͱ͖͕࣍ࣜ੒ཱ͢Δɽ y(z) = h(z)x(z) (3) ࣜ(3) ʹ h(z) = zɼ͋Δ͍͸ h(z) = z−1_{Λ୅ೖ͢Δͱɼ} ग़ྗ৴߸͸ೖྗ৴߸Λ_{1 αϯϓϧਐΊΔ৴߸ɼ͋Δ͍} ͸1 αϯϓϧ஗ΒͤΔ৴߸ʹͳΔɽͦ͜ͰϒϩοΫμ ΠΞάϥϜͰ͸ɼ৴߸Λ_{1 αϯϓϧਐΊΔॲཧΛ zɼ1} αϯϓϧ஗ΒͤΔॲཧΛz−1_{Ͱද͢ɽ·ͨϒϩοΫμ} ΠΞάϥϜதͷˣ_{2 ͸ɼμ΢ϯαϯϓϦϯάΛද͠ɼ} ೖྗ৴߸_{x_n_{} ͸࣍ͷΑ͏ͳग़ྗ৴߸ {y}_n_{} ʹม׵͞ΕΔɽ} yn=x2n (4) ·ͨˢ_{2 ͸ɼΞοϓαϯϓϦϯάΛද͠ɼҎԼͷΑ͏} ͳม׵͕ߦΘΕΔɽ yn=   0,xn/2, n = 2m, m ∈ Zotherwise (5) ࣍ʹຊݚڀͰѻ͏΢ΣʔϒϨοτ͸ɼਖ਼ن௚ަ΢Σʔϒ Ϩοτجఈʹݶఆ͢Δɽͱ͜ΖͰ_{Sweldens[12] ͸ɼਖ਼} ن௚ަ΢ΣʔϒϨοτجఈɼ·ͨ͸૒௚ަ΢ΣʔϒϨο τجఈʹ͓͚Δɼ࠶ߏ੒ϩʔύεϑΟϧλ܎਺_{h_n_}ɼ͓ Αͼ࠶ߏ੒ϋΠύεϑΟϧλ܎਺_{g_n_{} Λ༻͍ͯ LS Λ} ߏ੒͢ΔϑΟϧλΛಋ͖ग़͍ͯ͠Δ͕ɼͲͪΒͷ৔߹ ΋΢ΣʔϒϨοτ͕ίϯύΫταϙʔτΛ࣋ͭ͜ͱ͕ ඞཁ৚݅ͱ͞Ε͍ͯΔɽ͔͠͠ຊݚڀͰ͸ίϯύΫτ αϙʔτΛ࣋ͭɼ࣋ͨͳ͍ʹ͔͔ΘΒͣɼ͢΂ͯͷਖ਼ ن௚ަ΢ΣʔϒϨοτجఈʹ_{LS Λద༻͢Δํ๏Λఏ} Ҋ͢Δʢͨͩ͠ίϯύΫταϙʔτΛ࣋ͨͳ͍ਖ਼ن௚ ަ΢ΣʔϒϨοτجఈʹ͓͚Δ_{LS ͸ɼݫີͳҙຯͰ} ਖ਼ن௚ަม׵ʹ͸ͳΒͳ͍͕ɼ࣮༻্͸े෼ͳਫ਼౓Λ ࣋ͭɼۙࣅతͳਖ਼ن௚ަม׵Λߏங͢Δʣɽ Ұൠతʹɼਖ਼ن௚ަ΢ΣʔϒϨοτجఈͷઃܭऀʹ ΑΓɼͦͷ΢ΣʔϒϨοτݻ༗ͷτΡʔεέʔϧ਺ྻ {pn} ͕ެද͞Ε͓ͯΓɼ࠶ߏ੒ϩʔύεϑΟϧλ܎਺ {hn}ɼ͓Αͼ࠶ߏ੒ϋΠύεϑΟϧλ܎਺ {gn} ͸ɼ࣍ ࣜΑΓٻ·Δɽ hn= √1 2pn, gn= 1 √ 2qn (6) ͨͩ͠ qn=(−1)1−np1−n (7) ίϯύΫταϙʔτΛ࣋ͭਖ਼ن௚ަ΢ΣʔϒϨοτج ఈͷ৔߹ɼτΡʔεέʔϧ਺ྻ_{p_n_{} ͸༗ݶͷθϩͰ} ͳ͍߲Ͱߏ੒͞Εɼ·ͨίϯύΫταϙʔτΛ࣋ͨͳ ͍ਖ਼ن௚ަ΢ΣʔϒϨοτجఈͰ͸ɼແݶݸͷθϩͰ

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ਤ1 z ม׵ͷׂΓࢉ Fig.1 Division for z-transforms

ͳ͍߲Λ࣋ͭɽຊݚڀͰ͸ɼ͜ΕΒ͢΂ͯͷਖ਼ن௚ަ ΢ΣʔϒϨοτجఈʹରͯ͠ɼ_{LS ͕ߏ੒Ͱ͖Δํ๏} ΛఏҊ͢ΔɽͦͷͨΊʹ͸ɼ਺ྻʹz ม׵Λࢪͨ͠ଟ ߲ࣜಉ࢜ͷಛघͳׂΓࢉΛ࣮ߦ͢Δඞཁ͕͋Δɽͦͷ ํ๏ΛҎԼʹࣔ͢ɽ 2 ͭͷ a(z)ɼb(z) ͸ɼͦΕͧΕ m ߲ɼ͓Αͼ m − 1 ߲ʢ_{m > 1, m ∈ Zʣͷ࿈ଓ߲ͨ͠Λ࣋ͭ z ͷଟ߲ࣜͱ} ͠ɼa(z) Λ b(z) ͰׂΔҎԼͷׂΓࢉΛߦ͏ɽ͢ͳΘͪ ׂΒΕΔ΄͏ͷ_{a(z) ʹ͓͚Δɼ࠷ߴ࣍਺ͱ࠷௿࣍਺͕} ফڈ͞ΕΔΑ͏ͳɼ঎u(z)ɼ৒༨ r(z) ͕ಘΒΕΔׂΓ ࢉΛߦ͏ɽ͢Δͱ_{a(z) = u(z)b(z) + r(z) ͕੒ཱ͠ɼྫ} ͑͹ a(z) = 5z2₊_6z1₊_7z0₊_8z−1₊_9z−2 b(z) = 1z2₊_2z1₊_3z0₊_4z−1 ͷ࣌ɼਤ1 ͷΑ͏ͳܭࢉʹΑΓɼ ঎: u(z) = 5z0₊_2.25z−1 ༨৒: r(z) = −6.25z1_{− 12.5z}0_{− 18.75z}−1 ͕ಘΒΕΔɽͳ͓ɼ͜ͷΑ͏ͳׂΓࢉ͸ɼa(z) ͕ m ߲ Ͱɼ_{b(z) ͕ m − 1 ߲ʢm > 1, m ∈ ZʣͰ͋Ε͹ৗʹՄ} ೳͰ͋Γɼͦͷ঎΍༨৒΋Ұҙతʹܾ·Δɽ࣍અͰఏ Ҋ͢Δɼ_{LS ͷϑΟϧλ܎਺ͷࢉग़๏Ͱ͸ɼ͜ͷׂΓ} ࢉΛ༻͍Δ͕ɼ্هͷܭࢉՄೳͳ৚͕݅ৗʹอͨΕΔ Α͏ʹɼΞϧΰϦζϜΛߏங͢Δɽ 2.2 ৽ͨͳ LS ͷϑΟϧλ܎਺ͷࢉग़๏ͷఏҊ ͜͜Ͱɼ·ͣLS ʹద༻͢Δਖ਼ن௚ަ΢ΣʔϒϨο τجఈͷɼτΡʔεέʔϧ਺ྻ_{p_n_{} ͷ߲ͷ༗ޮൣғ} Λɼద੾ͳਖ਼ͷ੔਺_{mʢm > 0, m ∈ ZʣΛ༻͍ͯɼ} −2m ≤ n ≤ 2m ͷൣғʹઃఆ͢Δɽ࣍ʹ࠶ߏ੒ϩʔ ύεϑΟϧλ܎਺_{h_n_{}ɼ͓Αͼ࠶ߏ੒ϋΠύεϑΟϧ} λ܎਺_{g_n_{} ͸ɼࣜ (6)ɼ(7) Λ༻͍ͯτΡʔεέʔϧ਺} ྻ_{p_n_{} ΑΓɼ࣍ͷ༗ޮൣғΛٻΊΔ͜ͱʹ͢Δɽͦ͠} ͯҎԼͷൣғΛɼ͜ΕΒϑΟϧλ܎਺ͷ༗ޮൣғͱఆ ΊΔɽ {hn: −2m ≤ n ≤ 2m, n ∈ Z} (8) {gn: −2m + 1 ≤ n ≤ 2m + 1, n ∈ Z} (9) ਖ਼ͷ੔਺m ͷઃఆํ๏Ͱ͋Δ͕ɼѻ͏ਖ਼ن௚ަ΢Σʔ ϒϨοτجఈʹΑΓҟͳΔɽྫ͑͹ɼ_{Meyer ΢Σʔϒ} Ϩοτ͸ίϯύΫταϙʔτΛ࣋ͨͳ͍ਖ਼ن௚ަ΢ ΣʔϒϨοτجఈͰ͋ΓɼτΡʔεέʔϧ਺ྻ_{p_n_{} ͷ} θϩͰͳ͍߲͸ແݶʹ͋Δ͕ɼҰൠతʹ_{{n : −40 ≤} n ≤ 40, n ∈ Z} ͷൣғͷ߲Λѻ͑͹ɼे෼ͳਫ਼౓Ͱ ม׵Ͱ͖Δ͜ͱ͕஌ΒΕ͍ͯΔ_{[9]ɽͦ͜Ͱਖ਼ͷ੔਺} m ͸ m = 20 ʹઃఆ͢Δɽ͢Δͱ Meyer ΢ΣʔϒϨο τͷϑΟϧλ܎਺_{h_n_}ɼ{g_n_{} ͷ༗ޮൣғ͸ɼͦΕͧΕ} {hn: −40 ≤ n ≤ 40, n ∈ Z}ɼ{gn: −39 ≤ n ≤ 41, n ∈ Z} ͱͳΔɽ·ͨΑ͘஌ΒΕͨਖ਼ن௚ަ΢ΣʔϒϨοτ جఈͷ_{Daubechies 6 ΢ΣʔϒϨοτ͸ίϯύΫτα} ϙʔτΛ࣋ͪɼτΡʔεέʔϧ਺ྻ_{p_n_{} ͷθϩͰͳ} ͍߲͸_{p_n _{: n = 0, 1, · · · , 11} ͷൣғͷ 12 ݸʹݶΒ} ΕΔɽͦ͜Ͱm = 6 ʹઃఆ͢ΔͱɼDaubechies 6 ΢ ΣʔϒϨοτͷ͜ΕΒͷ܎਺ͷ༗ޮൣғ͸ɼͦΕͧΕ {hn: −12 ≤ n ≤ 12, n ∈ Z}ɼ{gn: −11 ≤ n ≤ 13, n ∈ Z} ͱͳΔɽ͜͏͢Δͱɼ_{h_n_}ɼ{g_n_{} ͷͲͪΒʹ΋θϩͷ} ஋Λ߲͕࣋ͭͰͯ͘Δ͕ɼ͜ΕΒ͸θϩͷ஋ͷ߲ͱ͠ ͯଞͷ߲ͱಉ౳ʹѻ͏ɽͦͯ͠ޙड़͢Δɼ͜ΕΒ͕ؔ ܎ͨ͠ଟ߲ࣜͷׂΓࢉʹ͓͍ͯ͸ɼྫ֎తʹ_{0 ÷ 0 = 0}

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͕੒ཱ͢Δͱߟ͑ɼ2.1 અͷଟ߲ࣜͷׂΓࢉͷϧʔϧ Λద༻͢Δɽ࣍ʹ_{Sweldens [12] ͷख๏ʹैͬͯɼh(z)} ͷۮ਺܎਺ͷz ม׵ he(z)ɼح਺܎਺ͷ z ม׵ ho(z)ɼ͓ Αͼ_{g(z) ͷۮ਺܎਺ͷ z ม׵ g}_e_{(z)ɼح਺܎਺ͷ z ม׵} go(z) ΛҎԼͷΑ͏ʹఆٛ͢Δɽ he(z) = m ∑ k=−m h2kz−k, ho(z) = m−1 ∑ k=−m h2k+1z−k (10) ge(z) = m ∑ k=−m+1 g2kz−k, go(z) = m ∑ k=−m g2k+1z−k (11) ͢Δͱ, LS ͷϑΟϧλ܎਺͸ҎԼͷखॱͰٻΊΒΕΔɽ (1) ҎԼͷΑ͏ʹɼa0(z)ɼb0(z) Λઃఆ͢Δɽ a0(z) = he(z), b0(z) = ho(z) (12) ͦͯ͠࠷ॳ͸i = 0 ʹઃఆͯ͠ɼҎԼʹܝ͛Δ ॲཧΛଓ͚Δɽ (2) 2.1 અʹׂࣔͨ͠ΓࢉʹΑΓɼai(z) ͱ bi(z) ͔Βɼ ҎԼͷؔ܎Λຬͨ͢ɼ঎_u_i+1_{(z) ͱ৒༨ r}_i+1_{(z) Λ} ٻΊΔɽ ai(z) = ui+1(z)bi(z) + ri+1(z) (13) (3) ৽ͨͳ ai+1(z)ɼbi+1(z) Λ࣍ͷΑ͏ʹٻΊΔɽ

ai+1(z) = bi(z), bi+1(z) = ri+1(z) (14)

΋͠΋bi+1(z) = 0 Ͱ͋Ε͹ɼ͜͜ͰॲཧΛऴྃ ͢Δɽ͔͠͠_b_i+1_{(z) 0 Ͱ͋Ε͹ɼ੔਺஋ i Λ} 1 ͭ૿΍ͯ͠ɼ(2) ʹ໭ͬͯॲཧΛଓ͚Δɽ Ҏ্ͷΑ͏ʹͯ͠ಘΒΕͨɼ࠷ऴతͳ_a_i+1_{(z) ͸ఆ਺ͱ} ͳΔͷͰɼ͜ΕΛҎԼͷΑ͏ʹఆ਺_{K ͱ͓͘ɽ} K = ai+1(z) (15) ·্ͨهͷҰ࿈ͷ࡞ۀ͕ਖ਼ৗʹऴྃ͢Ε͹ɼ࠷ऴత ͳ੔਺iɼ͓Αͼਖ਼ͷ੔਺ m ͷؒʹ i + 1 = 2m ͕੒ ཱ͠ɼ_{z ͷଟ߲ࣜ u}_n_{(z) ʹؔͯ͠͸ɼશ෦Ͱ 2m ݸͷ} {un(z) : n = 1, 2, · · · , 2m} ͕ಘΒΕɼ࣍ͷ͕ࣜ੒ཱ͢Δɽ   h_he(z) o(z)    = 2m ∏ n=1   un₁(z) 1₀      K₀    (16) ͜͜Ͱ࣍ͷΑ͏ͳ_P₀_{(z) Λߟ͑Δɽ} P0(z) =   he(z) g 0 e(z) ho(z) g0o(z)    = 2m ∏ n=1   un₁(z) 1₀      K_{0 1/K}0    (17) ࣜ(17) ͷ g0 e(z) ͱ g0o(z) ͸ӈลΛܭࢉ͢Δ͜ͱͰٻΊΒ ΕΔɽଓ͍ͯ࣍ͷࣜ_{(18) Λຬͨ͢ s(z) ͷಋग़Λߦ͏ɽ} P(z) = P0(z)    1 s(z)₀ ₁    (18) ͜͜Ͱ_{P(z) ͸࣍ͷࣜ (19) ͷߦྻΛද͢ɽ} P(z) =    _hhe_o(z) g_{(z) g}e_o(z)_(z)    (19) ࣜ(19) ͷ ge(z)ɼgo(z) ͸ࣜ (11) ΑΓٻ·Δɽࣜ (17)ɼ (18)ɼ(19) ͔Β࣍ͷࣜ (20) ͕ಘΒΕΔɽ    _hhe(z) ge(z) o(z) go(z)    =    he(z) g 0 e(z) ho(z) g0o(z)       1 s(z)₀ ₁    (20) Sweldens ͷจݙ [12] ΑΓ det(P(z)) = 1 ͕੒ཱ͢Δͷ Ͱɼ࣍ͷࣜ_{(21) ͕ಘΒΕΔɽ} �� he(z) ge(z) ho(z) go(z) �� =1 (21) ࣜ_{(20)ɼ(21) Λ࿈ཱํఔࣜͱͯ͠ղ͘͜ͱͰ࣍ͷࣜ} (22) ͷΑ͏ͳ s(z) ΛٻΊΔ͜ͱ͕Ͱ͖Δɽ s(z) = g0 o(z)ge(z) − g0e(z)go(z) (22) Ҏ্ͷΑ͏ʹͯ͠ࢉग़ͨ͠_{s(z)ɼ͓Αͼ {u}_n(z) : n = 1, 2, · · · , 2m} Λ༻͍ͯɼP(z) ͸࣍ͷࣜ (23) ͷΑ͏ʹද ͤΔɽ P(z) =    1 s₀ 1₁(z)       _t 1 0 1(z) 1       1 s₀ 2₁(z)       _t1 0 2(z) 1    · · ·    1 s₀ m₁(z)       _t 1 0 m(z) 1       1 s₀ last₁(z)       K_{0 1/K}0    (23) ͜͜ʹ༻͍ͨ_s_n_(z)ɼt_n_(z)ɼs_last_{(z) ΛҎԼʹࣔ͢ɽ} sn(z) = u2n−1(z), tn(z) = u2n(z), n = 1, 2, · · · , m (24) slast(z) = K2s(z) (25) Ҏ্͕_{LS ͷϑΟϧλͷઃܭํ๏Ͱ͋Δɽ͜͜Ͱɼs}₁_(z) ʙsm(z)ɼt1(z)ʙtm(z)ɼslast(z) ͕ٻΊΒΕͨ LS ͷϑΟ ϧλ܎਺ͱͳΓɼ͜ΕΒΛ༻͍ͨม׵ɼ͓Αͼٯม׵ ͸ਤ2ɼ3 ͷΑ͏ʹͳΔɽຊݚڀͰઃܭͨ͠ LS ͷϑΟ ϧλ܎਺͸_s_last_{(z) Λআ͖͢΂ͯ 2 ߲ͱͳΔ͕ɼ஋͕θ} ϩͷ߲͸ॲཧΛলུ͔ͯ͠·Θͳ͍ɽͳ͓ίϯύΫτ αϙʔτΛ࣋ͨͳ͍΢ΣʔϒϨοτͷ৔߹ɼslast(z) ͷ ߲਺͸େ͖͘ͳΓ͕ͪͰ͋Δ͕ɼ͜ΕΒ͸ద੾ͳେ͖ ͞ʹলུԽ͢Δ͜ͱ͕Ͱ͖Δʢͦͷํ๏͸࣍અͰɼ࣮ ࡍͷઃܭྫΛܝ͛ͯઆ໌͢Δʣɽ

(5)

ਤ₂ _{LS ʹΑΔม׵} Fig.2 Forward transform by LS

ਤ3 LS ʹΑΔٯม׵ Fig.3 Inverse transform by LS

3. LS ʹΑΔ CDWT ͷߏ੒ 3.1 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ϑΟϧλͷઃܭ Meyer ΢ΣʔϒϨοτ͸Α͘஌ΒΕͨਖ਼ن௚ަ΢Σʔ ϒϨοτجఈͰ͋Δɽ͜ͷ΢ΣʔϒϨοτʹඞཁͳ਺ ྻ͸_{Toda ͷจݙ [13] ʹܝࡌ͞Ε͍ͯΔͷͰɼ2.2 અͰ} ࣔͨ͠Α͏ʹm = 20 ʹઃఆ͠ɼLS ͷϑΟϧλΛࢉग़ ͢Δ͜ͱ͕Ͱ͖Δɽ͢Δͱ_s₁_(z)ʙs₂₀_(z)ɼt₁_(z)ʙt₂₀_(z)ɼ slast(z) ͷ LS ͷϑΟϧλ͕ٻ·Δɽslast(z) Λআ͍ͯ͢ ΂ͯ_{2 ߲ͷϑΟϧλͱͳΔ͕ɼs}_last_{(z) ͚ͩ͸ 100 ߲Λ} ௒͑Δଟ߲ࣜͱͳΔɽͦ͜Ͱ_s_last_{(z) ͷ߲ͷলུԽΛߦ} ͏ඞཁ͕͋Δɽࣜ(25) ΑΓ slast(z) ͸ slast(z) = K2s(z) Ͱද͞ΕΔ͜ͱΛߟྀ͠ɼ͜͜Ͱ͸_s_{(z) ͷ߲ͷলུԽ} Λߟ࡯͢Δɽ_{s(z) Λߏ੒͢Δ਺ྻΛ {s}_n_{} Ͱද͢ͱɼਤ} 4 ʹࣔ͢Α͏ʹɼ͜ͷ਺ྻͷ֤߲ͷઈର஋͸ n = 0, 1 Ͱ࠷େ஋0.62 ΛऔΓɼࠨӈରশʹ྆αΠυʹͳͩΒ ͔ʹθϩʹ޲͔ͬͯݮਰ͍ͯ͠Δ͜ͱ͕Θ͔Δɽͦ͜ Ͱ͖͍͠஋ θʢθ >_{0ʣΛઃఆ͠ɼ਺ྻ {s}n} ͷ྆αΠυ ʹ͓͍ͯɼθ ΑΓઈର஋͕খ͍͞஋ͷ߲Λɼۙࣅతʹ θϩͱݟͳͯ͠੾ΓࣺͯΔ͜ͱʹ͢Δɽ͜͜Ͱ࢒ͬͨ ਺ྻΛ_{s′ n}ɼ੾ΓࣺͯΒΕͨ਺ྻΛ {sθn} ͱ͢Δͱɼͦ ΕΒͷ_{z ม׵ͷؒʹ͸࣍ͷ͕ࣜ੒ཱ͢Δɽ} s(z) = s′_{(z) + s}θ_(z) ₍₂₆₎ ͜͜Ͱ͖͍͠஋ θ ΛՄมͤͯ͞ɼLS ͕ग़ྗ͢Δ܎਺ ͷޡࠩΛ؍࡯͢Δɽαϯϓϧɾσδλϧ৴߸_{{ f}_n_{} ͸ɼ} N Λਖ਼ͷ੔਺ͱͯ࣍ࣜ͠Ͱද͞ΕΔ΋ͷΛ࢖༻͢Δɽ fn=sin ( πn2 2N ) , 0 ≤ n < N (27) ਤ₄ _{{S n}} Fig.4 {S n} ਤ₅ αϯϓϧ৴߸ Fig.5 Sample signal

N = 4096 ͷ࣌ɼ͜Ε͸ਤ 5 ͷΑ͏ͳ 4096 ݸͷ༗ݶͷ ߲Λ΋ͭεΟʔϓ৴߸ͱͳΔɽ͜ͷ৴߸ΛϨϕϧ0 ͷ εέʔϦϯά܎਺_{c_0,n_{} ͱͯ͠ೖྗ͠ɼैདྷͷ MRA} ͱɼ_{LS ʹΑΓಘΒΕͨɼϨϕϧ −1 ͷͦΕͧΕͷε} έʔϦϯά܎਺ɼ͓Αͼ΢ΣʔϒϨοτ܎਺Λൺֱ͠ɼ ैདྷͷMRA ʹର͢Δ LS ͷޡࠩΛࢉग़͢Δɽ ͜͜Ͱਤ_{5 ͷΑ͏ͳεΟʔϓ৴߸Λޡࠩͷࢉग़ʹ༻} ͍Δཧ༝Λड़΂Δɽ_{CDWT ΍ DWT ͸͞·͟·ͳ৔໘} ʹར༻͞ΕΔͨΊɼͦͷ͢΂ͯͷঢ়گʹ͓͚ΔޡࠩΛ ਖ਼֬ʹࢉग़͢Δ͜ͱ͸ෆՄೳͰ͋Δɽ͔͠͠ਤ_{5 ͷΑ} ͏ͳεΟʔϓ৴߸͸DC ۙล͔ΒφΠΩετप೾਺ۙ ล·Ͱɼ΄΅ຬวͳ͘ۉ౳ʹप೾਺੒෼ΛؚΜͰ͍Δ ͨΊɼσδλϧॲཧʢͱΓΘ͚΢ΣʔϒϨοτม׵౳ʣ ͷɼ͞·͟·ͳঢ়گʹ͓͚ΔޡࠩΛฏۉతʹ༧ଌ͢Δ ͷʹదͨ͠αϯϓϧ৴߸ͱͯ͠ɼޡࠩଌఆ༻ʹ༻͍Β ΕΔ͜ͱ͕ଟ͍ɽຊݚڀͰ΋ਤ5 ͷεΟʔϓ৴߸Λɼ ޡࠩͷࢉग़ʹదͨ͠৴߸ͱͯ͠࠾༻ͨ͠ɽ ࣍ʹLS ʹؔ͢Δॏཁͳੑ࣭Λઆ໌͓͔ͯ͠ͳ͚Ε ͹͍͚ͳ͍ɽ͜͜Ͱߦ͏ॲཧͷΑ͏ʹɼ༗ݶݸͷ߲ʹ ΑΓߏ੒͞ΕΔσδλϧ৴߸Λ_{LS Ͱม׵͢Δ৔߹ɼ} ର৅ͱͳΔ৴߸ͷαϯϓϧ਺N ͸ɼద੾ͳਖ਼ͷ੔਺ ℓ Λ༻͍ͯ_{N = 2}ℓ_{ͰදͤΔΑ͏ʹઃఆ͠ɼݸʑͷϑΟ} ϧλॲཧ͸ɼ८ճ৞ΈࠐΈʹΑΓ࣮ߦ͞Εͳ͚Ε͹͍ ͚ͳ͍ɽ΋ͦ͠͏͠ͳ͚Ε͹ɼਂࠁͳ࿪͕ൃੜͯ͠͠ ·͏৔߹͕͋Δɽ͜Ε͸͢΂ͯͷ_{LS ͕࣋ͭڞ௨ͷੑ} ࣭Ͱ͋ΓɼͲͷΑ͏ͳLS Ͱ͋Εɼආ͚ͯ௨Εͳ͍஫ ҙ఺Ͱ͋Δʢ΋͠ɼͲ͏ͯ͠΋८ճ৞ΈࠐΈͷӨڹΛ ආ͚͍ͨ৔߹͸ɼೖྗ৴߸ͷยํͷ୺ʹద੾ͳ߲਺ͷ θϩͷ৴߸Λૠೖ͢Ε͹Α͍ɽͨͱ͑͹_{Meyer ΢Σʔ} ϒϨοτͰϨϕϧ _{jʢ j < 0, j ∈ Zʣ·Ͱม׵͢Δ࣌ɼ} ೖྗ৴߸ͷऴΘΓʹ_{40 × 2}− j_{ݸͷθϩΛ෇Ճ͢Ε͹ɼ}

(6)

८ճ৞ΈࠐΈͷӨڹ͸΄΅ղফ͞ΕΔʣɽ͜͜Ͱ८ճ ৞ΈࠐΈʹ͍ͭͯઆ໌͢ΔɽϑΟϧλ܎਺_{h_n_{} ͱɼ༗} ݶͷN ݸʢN = 2ℓ_{, ℓ >}_{0, ℓ ∈ Zʣͷ߲Λ࣋ͭσδλ} ϧ৴߸_{{ f}_n _{: n = 0, 1, · · · , N − 1} ͱͷ८ճ৞ΈࠐΈ͸ɼ} ࣍ͷࣜ_{(28) Ͱܭࢉ͞ΕΔɽ} yn=∑ k∈Z hkf(n−k) % N, n = 0, 1, · · · , N − 1 (28) ͨͩ͠n % N ͸ n Λ N Ͱׂͬͨ৒༨Λද͢ɽຊདྷͷ८ ճ৞ΈࠐΈͷܭࢉͰ͸ɼແݶͷ߲Λ࣋ͭपظ_{N ͷ਺ྻ} Λग़ྗ͢Δ͜ͱʹͳΔ͕ɼ࣮ࡍͷܭࢉͰ͸ɼn = 0 ͔ Β_{N − 1 ·Ͱͷ߲ΛٻΊΕ͹े෼ͳͷͰʢଞͷ߲͸प} ظੑʹΑΓٻΊΒΕΔͷͰʣɼຊݚڀͰ͸্هͷΑ͏ ʹ_{n ʹ੍ݶΛઃ͚ͨܗͰ८ճ৞ΈࠐΈΛද͢ɽਤ 2ɼ3} ͷ_{LS ͷ֤ϑΟϧλ͸ɼҎ্ͷΑ͏ͳ८ճ৞ΈࠐΈʹ} ΑΓॲཧ͞Εͳ͚Ε͹͍͚ͳ͍ɽ͞ΒʹຊݚڀͰ͸ɼ ޡࠩͷ൑ఆج४ͱͳΔैདྷͷ_{MRA ͷϑΟϧλॲཧ΋ɼ} ८ճ৞ΈࠐΈʹΑΓߦ͏ɽ Ҏ্ͷ৚݅ͷ΋ͱɼैདྷͷ_{MRA ʹΑΓಘΒΕͨϨϕ} ϧ_{−1 ͷεέʔϦϯά܎਺Λ {c}O_−1,n_{: n = 0, 1, · · · , N/2−} 1}ɼ΢ΣʔϒϨοτ܎਺Λ {dO −1,n: n = 0, 1, · · · , N/2 − 1}ɼ·ͨɼ͖͍͠஋ θ Λ༻͍ͯ {sn} ͷ੾ΓࣺͯΛߦͬ ͨ_{LS ʹΑΓಘΒΕͨϨϕϧ −1 ͷεέʔϦϯά܎਺} Λ_{cθ −1,n : n = 0, 1, · · · , N/2 − 1}ɼ΢ΣʔϒϨοτ܎ ਺Λ_{dθ −1,n : n = 0, 1, · · · , N/2 − 1} ͱͯ͠ɼLS ʹ͓ ͚ΔεέʔϦϯά܎਺ͷ࣮ଌޡࠩʢ_{measured scaling} error: MSEʣ [dB]ɼ΢ΣʔϒϨοτ܎਺ͷ࣮ଌޡࠩ ʢmeasured wavelet error: MWEʣ [dB] ΛɼͦΕͧΕ࣍

ͷΑ͏ʹࢉग़͢Δɽ MSE(θ) = 10 log    ∑N/2−1 n=0 ���cO−1,n− cθ−1,n�� 2 ∑N/2−1 n=0 ���cO−1,n�� 2    [dB] (29) MWE(θ) = 10 log    ∑N/2−1 n=0 ���dO−1,n− dθ−1,n�� 2 ∑N/2−1 n=0 ���dO−1,n�� 2    [dB] (30) ͜͜Ͱ͖͍͠஋ θ Λ10−10_, ₁₀−9_, _{· · · , 10}−1_ͱՄม͞ ͤͯɼ্هͷ࣮ଌޡࠩΛࢉग़ͯ͠Έͨͱ͜Ζɼਤ6 ͷ Α͏ʹͳͬͨɽ͜ͷਤΛݟͯ΋Θ͔ΔΑ͏ʹɼ_MWE(θ) ͸ɼۃΊͯখ͍͞஋ͷ_{−249.1dB ΛอͪɼશମΛ௨͠} ͯຆͲมΘΒͳ͍ɽ͔͠͠_{MSE(θ) ͸ɼ͖͍͠஋ θ ͕} খ͍࣌͞ʹ͸े෼ʹখ͍͞Ұఆͷ஋_{−101.7dB Λอͬ} ͍ͯΔ͕ɼθ = 10−5_{͋ͨΓ͔Β࣍ୈʹେ͖͘ͳͬͯ} ͍͘ͷ͕Θ͔Δɽ͜ͷ܏޲͸ɼଞͷίϯύΫταϙʔ τΛ࣋ͨͳ͍΢ΣʔϒϨοτʹ΋ڞ௨ͯ͠ݱΕΔ͜ͱ ͕࣮ݧʹΑΓΘ͔͕ͬͨɼ͜ͷΑ͏ͳ܏޲͕ͳͥى͜ Δͷ͔Λߟ࡯͢Δ͜ͱ͸ผͷػձʹৡΓɼຊݚڀͰ͸ MSE(θ) ͕࣍ୈʹେ͖͘ͳΓ࢝ΊΔϙΠϯτΛਖ਼֬ʹ ೺Ѳ͢Δ͜ͱΛߟ͑ΔɽͳͥͳΒɼ͜ͷϙΠϯτΛج ४ʹɼ͖͍͠஋ θ Λઃఆ͢Δ͜ͱ͕ɼޡࠩΛ཈͑ͳ͕ Β_{s_n_{} ͷ߲਺Λ߹ཧతʹ࡟ݮ͢Δखஈʹܨ͕Δͱߟ} ͔͑ͨΒͩɽͦ͜Ͱ͖͍͠஋ θ ʹର͢ΔεέʔϦϯά ܎਺ͷޡࠩΛɼผͷࢹ఺͔Βߟ࡯͢Δɽ͢ͳΘͪ੾Γ ࣺͯΛߦΘͳ͍ΦϦδφϧͷ_{s_n_{} Λ༻͍ͨॲཧʹର͢} Δɼ͖͍͠஋ θ ʹΑΔ੾ΓࣺͯʹΑͬͯൃੜ͢Δޡࠩ Λɼ_{sθ n} ͷ৘ใ͔Βਪఆ͢Δํ๏Λߟ࡯͢Δɽࣜ (25)ɼ (26) ͓Αͼਤ 2 ΑΓɼs(z) Λ s′_{(z) ʹมߋ͢Δ͜ͱʹΑ} ΓɼϨϕϧ_{−1 ͷεέʔϦϯά܎਺ {c}_−1,n_{} ʹൃੜ͢Δ} ޡࠩ_{eθ n : n = 0, 1, · · · , N/2 − 1} ͸࣍ͷΑ͏ʹදͤΔɽ eθ n= ∑ k∈Z sθ kd−1,(k−n) % (N/2), n = 0, 1, · · · , N/2 − 1 (31) ͨͩ͠_{d_−1,n_{} ͸ɼ{s}_n_{} ͷ੾ΓࣺͯΛߦΘͳ͍࣌ͷϨϕ} ϧ_{−1 ͷ΢ΣʔϒϨοτ܎਺Λද͢ɽͦͯ͠ {s}n} ͷ੾Γ ࣺͯΛߦΘͳ͍_{LS ʹΑΓٻΊͨϨϕϧ −1 ͷεέʔϦ} ϯά܎਺Λ_{c_−1,n_{}ɼ͖͍͠஋ θ ʹΑΓ {s}_n_{} ͷ੾Γࣺͯ} Λߦͬͨ_{LS ʹΑΓٻΊͨεέʔϦϯά܎਺Λ {c}θ −1,n} Ͱද͢ͱɼ͕࣍ࣜ੒ཱ͢Δɽ eθ n=c−1,n− cθ−1,n, n = 0, 1, · · · , N/2 − 1 (32) ͕ͨͬͯ͠ɼ͖͍͠஋ θ ʹΑΔs(z) ͷ੾ΓࣺͯʹΑΓ

ൃੜ͢Δޡࠩ_{(estimated scaling error: ESE) [dB] ͸࣍} ͷΑ͏ʹදͤΔɽ ESE(θ) = 10 log    ∑N/2−1 n=0 ��eθn��2 ∑N/2−1 n=0 ���c−1,n��2    [dB] (33) ͱ͜ΖͰ८ճ৞ΈࠐΈͱߴ଎ϑʔϦΤม׵ʢfast Fourier transform, FFTʣ͸਌࿨ੑ͕Α͘ɼ·ͨ N = 4096 = 212 ʹઃఆͯ͋͠ΔͷͰɼม׵ʹΑΓಘΒΕΔݸʑͷ܎਺ ͷ਺ྻ΋FFT ͕ՄೳͰ͋Δɽ͜͜Ͱ FFT Λಋೖ͢Δɽ ਤ6 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔεέʔ Ϧϯά܎਺ͱ΢ΣʔϒϨοτ܎਺ͷଌఆޡࠩ

Fig.6 Measurements of scaling and wavelet coeﬃcients errors of LS using Meyer wavelet

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N = 2ℓ_{ʢℓ >} _{0, ℓ ∈ Zʣͱͯ͠ɼN ͷपظΛ࣋ͭ਺ྻ}

{ fn} ͷ FFT { ˆfk}ɼ͓Αͼͦͷٯม׵ʢinverse fast Fourier

transform, IFFTʣ͸࣍ͷΑ͏ʹఆٛ͞ΕΔɽ FFT : ˆfk= N−1 ∑ n=0 fne−i2π kn N_, IFFT : f_n₌ 1 N N−1 ∑ k=0 ˆfkei2π kn N (34) ͳ͓_{{ ˆf}_k_{} ΋पظ N} Λ࣋ͭɽ࣍ʹύʔηόϧͷఆཧʢPar-seval’s theoremʣΑΓ͕࣍ࣜ੒ཱ͢Δɽ N−1 ∑ n=0 | fn|2= 1 N N−1 ∑ k=0 | ˆfk|2 (35) ·ͨ_{N ߲ͷ਺ྻ { f}_n_{} ͱ {h}_n_{} ͷ८ճ৞ΈࠐΈ͕ {y}_n_{} Ͱ} ͋Δ࣌ɼ͢ͳΘͪࣜ(28) ͕੒ཱ͢Δ࣌ɼ͕࣍ࣜ੒ཱ ͢Δɽ ˆyk= ˆfkˆhk, k = 0, 1, · · · , N − 1 (36) FFT ͷಛ௃Ͱ͋Δࣜ (35)ɼ(36) ͷੑ࣭Λ༻͍ͯɼࣜ (33) ͸࣍ͷΑ͏ʹදͤΔɽ ESE(θ) = 10 log    ∑N/2−1 k=0 ���ˆeθk�� 2 ∑N/2−1 k=0 ���ˆc−1,n�� 2    (37) = 10 log    ∑N/2−1 k=0 ��� ˆsθkˆd−1,k�� 2 ∑N/2−1 k=0 ���ˆc−1,k��2    [dB] (38) ࣜ_{(38) ͷதͷ {ˆc}_−1,k_{}ɼ{ ˆd}_−1,k_{} ͸ɼͦΕͧΕɼ੾Γࣺͯ} ΛߦΘͳ͍_{LS ʹ͓͚ΔϨϕϧ −1 ͷεέʔϦϯά܎਺} {c−1,n}ɼ΢ΣʔϒϨοτ܎਺ {d−1,n} Λ FFT ͨ͠΋ͷͰ ͋Δɽ͜͜Ͱ࣮ࡍʹܭࢉ͞Εͨɼ͜ΕΒͷઈର஋_|ˆc_−1,k_|ɼ | ˆd−1,k| ʹ͍ͭͯৄࡉʹௐ΂ͯΈͨͱ͜Ζɼ͔ͳΓͷྖҬ ʹ͓͍ͯɼͲͪΒ΋ಉ͡Ұఆͷ஋ʹͳΔ͜ͱ͕Θ͔ͬ ͨɽͦ͜Ͱ͜ΕΒͷ஋Λۙࣅతʹఆ਺_{AʢA > 0ʣͱ} ͯ͠ѻ͏͜ͱʹ͢Δɽ͢ͳΘͪ࣍ͷۙࣅ͕ࣜ੒ཱ͢Δ ͱߟ͑Δɽ |ˆc−1,k| ≈ A, | ˆd−1,k| ≈ A (39) ͜ΕΒΛ୅ೖࣜ͠Λ੔ཧ͢Δͱࣜ(38) ͸࣍ͷΑ͏ʹද ͤɼ_{sθ n} ͷ৘ใ͔ΒޡࠩΛਪఆ͢Δ໨త͕ୡ੒͞ΕΔɽ ESE(θ) = 10 log    ∑ n∈Z �� sθ n��2    [dB] (40) ͜͜Ͱ_{(29) ͱ (40) Ͱද͞ΕΔεέʔϦϯά܎਺ͷޡ} ࠩΛਤ_{7 ʹܝ͛Δɽ͜ͷਤΛݟΔͱɼθ ͷେ͖ͳྖҬͰ} ͸ɼMSE(θ) ͱ ESE(θ) ͕Α͘੔߹͍ͯ͠Δ͜ͱ͕Θ͔ Δɽͦ͜Ͱ θ ͕े෼ʹখ͍͞ͱ͖ͷఆৗঢ়ଶͷMSE(θ) ΛE [dB] ͱ͠ɼ ES E(θ) = 10 log    ∑ n∈Z �� sθ n��2    ≈ E [dB] (41) ਤ7 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔεέʔ Ϧϯά܎਺ͷ࣮ଌޡࠩͱਪఆޡࠩ

Fig.7 Measured and estimated errors of scaling coeﬃ-cients of LS using Meyer wavelet

ද1 Meyer ΢ΣʔϒϨοτʹΑΔ ESE(θ) ͱ MSE(θ) Table.1 ESE(θ), MSE(θ) and number of terms of {sn}

us-ing Meyer wavelet

θ ESE(θ) [dB] MSE(θ) [dB] Num. of {s′n}

1 × 10−5 _−98.5 _−98.2 ₃₆ 2 × 10−5 _−94.1 _−93.6 ₃₂ 3 × 10−5 _−88.9 _−88.7 ₃₀ 4 × 10−5 _−88.9 _−88.7 ₃₀ 5 × 10−5 _−80.1 _−80.5 ₂₄ Λຬͨ͢ θ Λݟ͍ͩ͠ɼ͜ͷ෇ۙͷ θ Λ͖͍͠஋ʹ͢Ε ͹ɼ_{s_n_{} ʹର͢Δ߹ཧతͳ੾Γࣺ͕ͯߦ͑Δͱ൑அͨ͠} ʢগͳ͘ͱ΋ࣜ_{(41) Λຬͨ͢ θ ΑΓ΋খ͘͞ θ ͷ஋Λઃ} ఆͯ͠΋ɼແବʹ_{s′_n_{} ͷ߲਺͕૿͑Δ͹͔ΓͰɼޡࠩ͸} ఆৗঢ়ଶʹͳ͍ͬͯΔͱߟ͑ΒΕΔʣɽ_{Meyer ΢Σʔϒ} Ϩοτͷ৔߹ɼθ =10−10_, ₁₀−9_, ₁₀−8_, ₁₀−7_ʹ͓͍ͯɼ ࣜ_{(29) Ͱࢉग़͞ΕΔ MSE(θ) ͸͢΂ͯ −101.7dB ͱɼ΄} ΅ఆৗঢ়ଶʹ͋ͬͨɽͦ͜Ͱ_{ESE(θ) ≈ −101.7dB Λۙ} ࣅతʹຬͨ͢ θ Λௐ΂ͨͱ͜ΖɼESE (10−5_{) = −98.5dB} ͕ީิʹݟ͔ͭͬͨɽͦ͜Ͱ₁₀−5_{≤ θ ≤ 5 × 10}−5_෇ۙ ͷ_{ESE(θ)ɼMSE(θ)ɼ͓Αͼ੾ΓࣺͯΒΕͨޙͷ {s}′ n} ͷ߲਺Λௐ΂ͨͱ͜Ζɼද1 ͷΑ͏ʹͳͬͨɽ͜ͷද Λݟͯ΋ɼࣜ_{(41) Λຬͨ͢ θ ΑΓେ͖͍ྖҬͷ θ ʹ} ͓͍ͯɼESE(θ) ͱ MSE(θ) ͕Α͘੔߹͍ͯ͠Δ͜ͱ͕ Θ͔Δɽ͕ͨͬͯࣜ͠_{(40) Ͱܭࢉ͞ΕΔ ESE(θ) Λج} ४ʹͯ͠ɼॴ๬ͷਫ਼౓͕ಘΒΕΔΑ͏ʹ θ Λઃఆͯ͠ ΋Α͍ͱߟ͑ΒΕΔɽͳ͓ɼ͜͜Ͱѻ͍ͬͯΔ_Meyer ΢ΣʔϒϨοτͷ_{LS ͸ɼޙड़͢Δ CDWT Ͱ΋࢖༻͢} Δ͜ͱΛߟྀ͠ɼ_{−90dB ҎԼͷޡࠩΛ֬อ͍ͨ͠ɽͦ} ͜Ͱද_{1 ΑΓ θ = 2 × 10}−5_{ʹઃఆͨ͠ɽ͢Δͱදʹ΋}

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ࣔͨ͠Α͏ʹɼ_{32 ߲ͷ਺ྻ {s}′ n: −15 ≤ n ≤ 16, n ∈ Z} ͕ಘΒΕΔ͕ɼࣜ_{(25) ʹै͍ɼҎԼͷࣜ (42) Ͱද͞} ΕΔϑΟϧλs′ last(z) ΛٻΊɼ͜ΕΛਤ 2ɼ3 ͷ LS ͷ ϑΟϧλ_s_last_{(z) ͱͯ͠࢖༻͢Δɽ·ͨɼࢉग़ͨ͠ LS} ͷϑΟϧλ܎਺Λ෇࿥_{1 ʹܝࡌ͓ͯ͘͠ɽ} s′ last(z) = K2s′(z) (42) 3.2 LS ϑΟϧλΛ༻͍Δ DWT ͷಛੑ 3.2.1 ԋࢉྔͷൺֱͱॲཧ଎౓ͷଌఆ ͜ͷઅͰ͸࠷ॳʹ_{Meyer ΢ΣʔϒϨοτΛ༻͍ͨै} དྷͷMRA ʹΑΔ DWT ͷม׵ͱٯม׵ɼ͓Αͼ LS ʹ ΑΔ_{DWT ͷม׵ͱٯม׵ͷԋࢉྔͷൺֱΛߦ͏ɽຊ} ݚڀͰ͸ԋࢉྔΛ࣍ͷΑ͏ʹఆٛ͢Δɽ͢ͳΘͪ_{2 ͭ} ͷ࣮਺ಉ࢜ͷֻ͚ࢉͱɼ2 ͭͷ࣮਺ಉ࢜ͷ଍͠ࢉ·ͨ ͸Ҿ͖ࢉΛ૊Έ߹Θͤͯɼ_{1 ୯Ґͷܭࢉྔͱఆٛ͢Δɽ} ͦͯ͠N ߲ͷ਺ྻΛೖྗ৴߸ͱͯ͠ԋࢉྔΛ N ͷഒ ਺Ͱද͢ɽ·ͨม׵ͷॲཧ͸ɼೖྗ৴߸ΛϨϕϧ_{0 ͷ} N ߲ʢN ͸ਖ਼ͷ੔਺ʣͷεέʔϦϯά܎਺ {c0,n} ͱ͠ɼ Ϩϕϧ_{−1 ͷ N/2 ߲ͷεέʔϦϯά܎਺ {c}_−1,n_}ɼ͓Α ͼ_{N/2 ߲ͷ΢ΣʔϒϨοτ܎਺ {d}_−1,n_{} Λग़ྗ͢Δ·} Ͱͱ͢Δɽ·ͨٯม׵͸͜ͷٯͷॲཧͱ͢Δɽ͢Δͱɼ ͦΕͧΕͷԋࢉྔ͸࣍ͷΑ͏ʹදͤΔɽ ɾ_{MRA: ม׵ɼٯม׵ڞʹ: 81N,} ɾLS: ม׵ɼٯม׵ڞʹ: 56N. ͕ͨͬͯ͠ม׵ɼٯม׵ͲͪΒ΋ɼLS ʹΑΔԋࢉྔ ͸ैདྷͷ_{MRA ͷ 69.1% ͱͳΔɽ࣍ʹɼͰ͖Δ͚ͩಉ} ͡৚݅Ͱ૊·ΕͨϓϩάϥϜʹΑΔͦΕͧΕͷॲཧ࣌ ؒΛଌఆ͠ɼൺֱ͢Δɽ͜͜Ͱ༻͍ͨϓϩάϥϜπʔ ϧɼ͓Αͼ࢖༻ͨ͠ίϯϐϡʔλͷػछ͸ҎԼͷΑ͏ ͳ΋ͷͰ͋Δɽ ɾMicrosoft Visual C++ 2005ɼ

ɾ_{Intel Core2 CPU 6400, 2.13GHz, RAM 4GByte.} Ҏ্ͷ৚݅ͷ΋ͱɼͦΕͧΕͷॲཧ͸Ϩϕϧ_{−3 ·Ͱ} ͷม׵ɼٯม׵Λ࣮ߦ͠ɼͦΕͧΕ_{100 ճͷॲཧΛߦ} ͍ɼͦͷฏۉ࣌ؒΛଌఆͨ͠ͱ͜Ζɼਤ8 ͷΑ͏ʹ ͳͬͨɽͦͯ͠ೖྗ৴߸ͷ߲਺ͷ࠷େ஋_{N = 131072} ʹ͓͚ΔɼͦΕͧΕͷଌఆ݁Ռ͸ҎԼͷΑ͏ʹͳͬͨɽ

ɾMRA: ม׵: 251.7msec, ٯม׵: 266.7msec, ɾ_LS: ม׵_{: 161.1msec, ٯม׵: 160.7msec.} Ҏ্ͷ݁ՌΑΓɼ_{LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ} MRA ͷ 64.0%ɼٯม׵͸ 60.3% ͱͳͬͨɽ͜ͷΑ͏ʹ ଌఆ͔࣌ؒΒࢉग़͞Εͨ݁Ռ͸ɼԋࢉྔΑΓਪఆ͞Ε

(a) Forward MRA and forward LS

(b) Inverse MRA and inverse LS

ਤ₈ _{Meyer ΢ΣʔϒϨοτΛ༻͍ͨ MRA ͱ LS ʹ} ΑΔॲཧ࣌ؒʢαϯϓϧ਺_{N ͷೖྗ৴߸ΛϨϕϧ −1 ∼} −3 ʹม׵)

Fig.8 Processing times of MRA and LS using Meyer wavelet (an N samples input data is transformed to level −1 ∼ −3) Δ69.1% ΑΓߴ଎ʹͳΔ͕ɼ͜ͷཧ༝͸ C++ϓϩάϥ ϛϯάͷಛੑ͔ΒਪଌͰ͖Δɽ͢ͳΘͪ_{Meyer ΢Σʔ} ϒϨοτʹΑΔ_{MRA Ͱ͸ 81 ݸͷ௕େͳ਺ྻΛ༻͍ͯ} ৞ΈࠐΈܭࢉΛߦΘͳ͚Ε͹͍͚ͳ͍͕ɼ͜ͷΑ͏ͳ ৔߹ͷҰൠతͳ_{C++ϓϩάϥϛϯάͰ͸ɼ਺ྻΛ഑ྻ} ม਺ʹऩΊͯॲཧ͢ΔɽҰํɼLS Ͱ͸ຆͲͷॲཧ͕ 2 ݸͷ਺ྻʹΑΔ৞ΈࠐΈܭࢉͱͳΔͨΊɼ_{2 ݸͷ௨ৗ} ม਺Λ༻ҙ͢Δɽͱ͜ΖͰ഑ྻม਺ʹΑΔॲཧ͸௨ৗ ม਺ʹൺ΂ͯݪཧ্஗͘ͳΔ͜ͱ͕஌ΒΕ͍ͯΔʢ഑ ྻม਺Ͱ͸ϙΠϯλʔΛࢀর͠ɼͦͷϙΠϯλʔʹऩ ೲ͞Ε͍ͯΔ਺஋ΛಡΈࠐΉख͕ؒඞཁͰɼ௨ৗม਺ Λ࢖༻͢Δ৔߹ΑΓ΋஗͘ͳΔʣɽ͜ͷࠩʹΑΔॲཧ ଎౓ͷҧ͍͕ଌఆ݁ՌʹݱΕͨ΋ͷͱਪଌͰ͖Δɽ ͜͜Ͱ͞Βʹϓϩάϥϛϯάʹؔ࿈ͯ͠ɼ΋͏ͻͱ ͭॏཁͳ͜ͱΛड़΂ΔɽMRA ͷม׵ͱٯม׵ͷԋࢉ ྔ͸ͲͪΒ΋ಉ͡Ͱ͋Δ͕ɼ࣮ࡍͷॲཧ଎౓͸ɼٯม

(9)

׵ͷํ͕ɼม׵ΑΓ΍΍஗͘ͳΔͷ͕ҰൠతͰ͋Δɽ MRA ͷม׵Ͱ͸৞ΈࠐΈॲཧޙʹμ΢ϯαϯϓϦϯά Λ࣮ߦ͠ɼٯม׵Ͱ͸ΞοϓαϯϓϦϯάޙʹ৞Έࠐ ΈॲཧΛ࣮ߦ͢Δɽ͔͜͠͠ͷखॱ௨Γʹͦͷ··࣮ ߦ͞ΕΔ͜ͱ͸كͰɼܭࢉྔΛগ͠Ͱ΋ܰݮ͢ΔΑ͏ ʹϓϩάϥϛϯά͞ΕΔͷ͕ҰൠతͰ͋ΔɽຊจͰࣔ ͢ཧ࿦తͳԋࢉྔ΋ɼ͜ͷܰݮ͕ཧ૝తʹ͏·͍ͬ͘ ͨঢ়ଶΛࢉग़͍ͯ͠Δɽ͔࣮͠͠ࡍͷϓϩάϥϛϯά ʹ͓͍ͯ͸ɼ͜ͷܭࢉྔܰݮͷͨΊʹɼ͍͔ͭ͘ͷલ ॲཧ͕ඞཁʹͳΔɽ্ͦͯ͠هͷܭࢉྔܰݮ͸ٯม׵ ͷ΄͏͕೉͘͠ɼ_{MRA ͷٯม׵ͷॲཧ଎౓͸ɼม׵} ΑΓ΋΍΍ྼΔͷ͕ҰൠతͰ͋ΔɽͦͷͨΊ_{LS ʹΑ} Δม׵ɼٯม׵ͷॲཧ࣌ؒΛMRA ͱͷൺ཰Ͱදͨ͠ ৔߹ɼ_{LS ͷٯม׵ͷ΄͏͕ɼLS ͷม׵ΑΓ΋ྑ޷ͳ} ݁Ռ͕ಘΒΕ͕ͪͰ͋Δ͕ɼ͜Ε͸MRA ͷٯม׵ͷ ϓϩάϥϛϯάͷ೉͠͞ʹىҼ͍ͯ͠Δͱߟ͑ͯΑ͍ɽ Ҏ্ͷΑ͏ʹϓϩάϥϛϯάͷಛੑʹࠨӈ͞Εɼ࣮ ࡍͷॲཧ଎౓͸ɼͳ͔ͳ͔ཧ࿦௨Γʹ͸ͳΒͳ͍͕ɼ ຊݚڀͰ͸_{MRAɼLS ͷͲͪΒ΋ɼͦΕͧΕͷಛ௃Λ} ׆͔͠࠷ળͷॲཧ଎౓͕ಘΒΕΔΑ͏ʹϓϩάϥϛϯ ά͠ɼͦͷൺֱΛߦ͏ɽ 3.2.2 ม׵ͱٯม׵Λ௨ͨ͠ޡࠩͷଌఆ ࣜ_{(27) Ͱද͞ΕΔαϯϓϧ৴߸ { f}_n_{}ʢͨͩ͠ N =} 4096ʣΛೖྗ৴߸ͱ͠ɼม׵ɼٯม׵Λ௨ͯ͠ಘΒΕ ͨग़ྗ৴߸Λ_{{ f}′ n} ͱ͢Δͱɼޡࠩ͸࣍ͷΑ͏ʹܭࢉͰ ͖Δɽ error = 10 log    ∑_N−1 n=0 ��� fn− fn′��2 ∑_N−1 n=0 | fn|2    [dB] (43) ·ͨม׵͸Ϩϕϧ_{−3 ·Ͱߦ͏͜ͱʹͨ͠ɽ͢ΔͱҎ} Լͷ݁Ռ͕ಘΒΕͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −101.4 dBɼ} ɾ_{LS ʹΑΔޡࠩɿ} _{−249.5 dBɽ} ͜ͷΑ͏ʹม׵ɼٯม׵Λ௨ͨ͠ޡ͕ࠩඇৗʹখ͘͞ ͳΔͷ͸LS ͷಛ௃Ͱ͋Δɽ 3.3 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ LS Խ Meyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ֓ཁΛड़΂ Δɽ͜ͷCDWT ͷ࣮਺෦ʹ͸ɼલઅͰઆ໌ͨ͠ Meyer ΢ΣʔϒϨοτΛ༻͍Δɽͦͯ͠ڏ਺෦ʹ͸ɼ_Meyer ΢ΣʔϒϨοτͱHilbert ม׵ϖΞΛ੒͢৽ͨͳਖ਼ن௚ ަ΢ΣʔϒϨοτجఈΛઃܭͯ͠࢖༻͢Δ͜ͱʹͳΔ ͕ɼͦͷৄࡉ͸จݙ_{[13] Ͱ঺հ͞Ε͍ͯΔɽਤ 9 ʹ LS} ਤ₉ _{LS ʹΑΔ CDWT ͷม׵} Fig.9 Forward transform of CDWT by LS

ʹΑΔCDWT ͷ෼ղΞϧΰϦζϜʢٯม׵ͷ࠶ߏ੒ ΞϧΰϦζϜ͸লུʣΛࣔ͢ɽਤࣔͷΑ͏ʹɼCDWT ʹ͸ม׵લͷิ͕ؒඞཁͱͳΓɼ·ͨٯม׵ͷਤ͸ল ུͯ͋͠Δ͕ɼٯม׵ޙͷٯิؒ΋ඞཁͱͳΔɽ͜ͷ ৄࡉ͸จݙ_{[13] ʹ঺հ͞Ε͍ͯΔɽ} 3.3.1 ڏ਺෦ͷ LS Խ จݙ[13] ʹڏ਺෦ͷਖ਼ن௚ަ΢ΣʔϒϨοτجఈ ͷτΡʔεέʔϧ਺ྻ͕ܝࡌ͞Ε͍ͯΔͷͰɼ͜Ε Λجʹɼࣜ(8)ɼ(9) ʹ͓͚Δ m Λ 20 ʹઃఆͯ͠ LS Λߏ੒͢Δ͜ͱ͕Ͱ͖Δɽ·ͨ θ = 2 × 10−5_ʹઃఆ ͢Δ͜ͱʹΑΓɼ_{ESE(θ) = −92.0[dB]ɼ31 ߲ͷ਺ྻ} {s′ n: −15 ≤ n ≤ 15, n ∈ Z} ͕ಘΒΕΔɽ͜ͷΑ͏ʹ͠ ͯಘΒΕͨڏ਺෦ͷ_{LS ͷϑΟϧλ܎਺Λ෇࿥ 2 ʹܝ} ࡌ͓ͯ͘͠ɽ࣍ʹN ݸͷ߲਺Λ࣋ͭ਺ྻΛೖྗ৴߸ͱ ͢ΔͱɼͦΕͧΕͷԋࢉྔ͸࣍ͷΑ͏ʹදͤΔɽ ɾ_{MRA: ม׵ɼٯม׵ڞʹ: 81N,} ɾLS: ม׵ɼٯม׵ڞʹ: 55.5N. ͕ͨͬͯ͠ม׵ɼٯม׵ͲͪΒ΋ɼ_{LS ʹΑΔԋࢉྔ} ͸ैདྷͷMRA ͷ 68.5% ͱͳΔɽ࣍ʹ 3.2.1 અͱಉ͡ ৚݅Ͱɼڏ਺෦ͷ΢ΣʔϒϨοτʹΑΔॲཧ࣌ؒΛଌ ఆͨ͠ͱ͜Ζɼೖྗ৴߸ͷ߲਺N = 131072 ʹ͓͚Δɼ ͦΕͧΕͷଌఆ݁Ռ͸ҎԼͷΑ͏ʹͳͬͨɽ ɾ_{MRA: ม׵: 251.8msec, ٯม׵: 265.0msec,} ɾLS: ม׵: 160.3msec, ٯม׵: 159.4msec. Ҏ্ͷ݁ՌΑΓɼ_{LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ} MRA ͷ 63.7%ɼٯม׵͸ 60.2% ͱͳͬͨʢॲཧ࣌ؒ ͷάϥϑ͸ਤ_{8 ͱຆͲಉ͡ʹͳΔͷͰলུ͢Δʣɽ͜} ͜Ͱ΋3.2.1 અͷ Meyer ΢ΣʔϒϨοτʹΑΔ MRA ΍_{LS ͱಉ༷ʹɼଌఆ͔࣌ؒΒࢉग़͞Εͨ݁Ռ͸ɼԋ} ࢉྔΑΓਪఆ͞ΕΔ68.5% ΑΓߴ଎ʹͳΔ͕ɼͦͷཧ ༝͸_{3.2.1 અͰ΋ड़΂ͨΑ͏ʹɼC++ϓϩάϥϛϯά} ͷಛੑ͔Βઆ໌Ͱ͖Δ΋ͷͱࢥΘΕΔɽ

(10)

ද2 Meyer ΢ΣʔϒϨοτʹΑΔ CDWT ʹ͓͚Δॲཧ࣌ؒʢlevel −1 ʙ −3ɼN = 131072ʣ Table.2 Processing times of CDWT using Meyer wavelet, level −1 to level −3, N = 131072

CDWT (msec) ICDWT (msec)

Real part Imag. part Interpolation Real part Imag. part Inv. Interpolation MRA 254.7 251.4 301.6 263.8 265.3 295.5 LS 161.5 160.1 301.2 159.4 159.8 296.6 ·ͨ_{3.2.2 ͱಉ͡৚݅ʹΑΔม׵ɼٯม׵Λ௨ͨ͠} ޡࠩ͸ҎԼͷΑ͏ʹଌఆ͞Εͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −104.3 dBɼ} ɾ_{LS ʹΑΔޡࠩɿ} _{−265.5 dBɽ} 3.3.2 LS Խ͞Εͨશମతͳ CDWT ͷධՁ CDWT ͸ɼ࣮਺෦ͷม׵ɼڏ਺෦ͷม׵ɼͦͯ͠ ิؒͷ_{3 ͭͷॲཧͰߏ੒͞ΕΔɽ·ͨ CDWT ͷٯม} ׵ʢInverse CDWT, ICDWTʣ͸ɼ࣮਺෦ͷٯม׵ɼڏ ਺෦ͷٯม׵ɼͦͯ͠ٯิؒͷ_{3 ͭͷॲཧͰߏ੒͞Ε} ΔɽͦͷͨΊɼCDWTɼICDWT ͷԋࢉྔ͸ɼ֤ॲཧͷ ԋࢉྔͷ૯࿨Ͱࢉग़͠ͳ͚Ε͹͍͚ͳ͍ɽ_{N ߲ͷ਺ྻ} Λೖྗ৴߸ͱͯ͠ɼϨϕϧ_{−1 ͔ΒϨϕϧ −3 ·Ͱͷɼ} Meyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͓Αͼ ICDWT ͷ૯߹తͳԋࢉྔ͸࣍ͷΑ͏ʹࢉग़͞ΕΔɽ ɾ_{MRA: CDWTɼICDWT ڞʹ: 445.5N,} ɾLS: CDWTɼICDWT ڞʹ: 357.125N. ͕ͨͬͯ͠_{CDWTɼICDWT ͲͪΒ΋ɼLS ʹΑΔԋ} ࢉྔ͸ैདྷͷMRA ͷ 80.2% ͱͳΔɽ࣍ʹ 3.2.1 અͱ ಉ͡৚݅Ͱॲཧ࣌ؒΛଌఆͨ͠ͱ͜Ζਤ_{10 ͷΑ͏ʹ} ͳͬͨɽͦͯ͠ೖྗ৴߸ͷ߲਺ͷ࠷େ஋N = 131072 ʹ͓͚ΔɼͦΕͧΕͷଌఆ݁Ռ͸ҎԼͷΑ͏ʹͳͬͨɽ ɾ_{MRA: CDWT: 807.7msec, ICDWT: 824.6msec,} ɾLS: CDWT: 623.3msec, ICDWT: 615.8msec. Ҏ্ͷ݁ՌΑΓɼLS ʹΑΔ CDWT ͷॲཧ࣌ؒ͸ैདྷ ͷ_{MRA ͷ 77.2%ɼICDWT ͸ 74.7% ͱͳͬͨɽ͜ͷ} Α͏ʹଌఆ͔࣌ؒΒࢉग़͞Εͨ݁Ռ͸ɼԋࢉྔΑΓਪ ఆ͞ΕΔ_{80.2% ΑΓߴ଎ʹͳΔ͕ɼͦͷཧ༝͸ 3.2.1} અͰ΋ड़΂ͨΑ͏ʹɼC++ϓϩάϥϛϯάͷಛੑ͔Β આ໌Ͱ͖Δ΋ͷͱࢥΘΕΔɽ ࣍ʹ_{N = 131072 ʹ͓͚ΔɼCDWTɼICDWT ͷ಺෦} ͷ֤ॲཧ࣌ؒΛɼಉ͡৚݅Ͱݸผʹଌఆͯ͠Έͨͱ͜ Ζɼද_{2 ͷΑ͏ʹͳͬͨɽද 2 ͷ਺஋ΑΓɼ࣮਺෦ͷ} LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ MRA ͷ 63.4%ɼٯ ม׵͸_{60.4% ͱͳΓɼ͜Ε͸ 3.2.1 અͰࣔͨ͠ Meyer}

(a) Forward CDWTs by MRA and LS

(b) Inverse CDWTs by MRA and LS

ਤ10 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ MRA ͱ LS ʹ ΑΔ_{CDWT ͷܭࢉ࣌ؒʢαϯϓϧ਺ N ͷೖྗ৴߸Λ} Ϩϕϧ_{−1 ∼ −3 ʹม׵)}

Fig.10 Processing times of CDWTs by MRA and LS us-ing Meyer wavelet (an N samples input data is trans-formed to level −1 ∼ −3) ΢ΣʔϒϨοτ୯ମͰଌఆͨ͠਺஋ʢม׵64.0%ɼٯ ม׵_{60.3%ʣͱ΄΅ಉ݁͡Ռʹͳͬͨɽ·ͨڏ਺෦ͷ} LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ MRA ͷ 63.7%ɼٯ ม׵͸_{60.2% ͱͳΓɼ͜Ε΋ 3.3.1 અͰࣔͨ͠ڏ਺෦} ͷ΢ΣʔϒϨοτ୯ମͰଌఆͨ͠਺஋ʢม׵63.7%ɼ ٯม׵60.2%ʣͱ΄΅ಉ݁͡Ռʹͳͬͨɽ͜ͷ͜ͱ͔

(11)

ΒɼCDWTɼICDWT ʹ͓͚Δ LS ͸ਖ਼ৗʹػೳ͠ɼॲ ཧͷߴ଎Խʹد༩͍ͯ͠Δͱߟ͑ΒΕΔɽ ࣍ʹ3.2.2 ͱಉ͡৚݅ʹΑΔม׵ɼٯม׵Λ௨ͨ͠ ࿪͸ҎԼͷΑ͏ʹଌఆ͞Εͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −103.8 dBɼ} ɾLS ʹΑΔޡࠩɿ −140.3 dBɽ LS ʹΑΔޡࠩ͸ैདྷͷ MRA ΑΓ֨ஈʹΑ͘ͳͬͯ ͍Δ͕ɼ୯ମͷ΢ΣʔϒϨοτʹΑΔLS ΑΓ͸ྼΔɽ ͜Ε͸ิؒͱٯิؒʹΑΓൃੜ͢Δޡ͕ࠩӨڹ͍ͯ͠ Δ΋ͷͱߟ͑ΒΕΔɽ 3.4 Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ LS ͷઃܭ Daubechies 6 ΢ΣʔϒϨοτ͸ίϯύΫταϙʔτ Λ࣋ͭਖ਼ن௚ަ΢ΣʔϒϨοτجఈͰ͋ΓɼτΡʔε έʔϧ਺ྻ_{p_n_{} ͷθϩͰͳ͍߲͸ɼ{n = 0, 1, · · · , 11}} ͷൣғͷ12 ݸʹݶΒΕΔɽͦ͜ 2.2 અͰࣔͨ͠Α͏ ʹ_{m = 6 ʹઃఆ͠ɼLS ͷϑΟϧλΛࢉग़͢Δ͜ͱ͕} Ͱ͖Δɽ͢Δͱࢉग़͞ΕͨϑΟϧλ܎਺ͷҰ෦͸θϩ ʹͳΔ͕ɼLS ʹ͓͚Δɼ͜ΕΒͷϑΟϧλॲཧ͸ল ུ͢Δ͜ͱ͕Ͱ͖Δɽ·ͨ_s_last_{(z) ͸ 11 ߲ͷϑΟϧλ} ͱͳΔ͕ɼ͜Ε͸ҎԼͷΑ͏ʹͯ͠লུԽ͕ՄೳͰ ͋Δɽ·ͣ_{Meyer ΢ΣʔϒϨοτͷ࣌ͱಉ͡Α͏ʹɼ} MSE(θ) ͱ ESE(θ) Λࢉग़͠ɼൺֱͯ͠ΈΔͱਤ 11 ͷ Α͏ʹͳΔɽ͢ͳΘͪ͜ΕΒ_{2 ͭͷޡࠩ͸ຆͲಉ͡ۂ} ઢΛඳ͘ͷͰɼࣜ_{(40) ΑΓ ESE(θ) Λܭࢉ͠ɼޡࠩͷ} ஋͕_{−200[dB] ͔Β −40[dB] ʹٸʹ௓Ͷ্͕Δखલͷ} θ =10−3_{ʹɼ͖͍͠஋Λઃఆ͢Ε͹Α͍ɽ͜ͷ࣌ɼ}_{s′ n} ͸_{6 ߲ͷ਺ྻ {s}′ n: n = −5, −4, · · · , 0} ͱͳΔ͕ɼ͜ͷ Α͏ʹͯ͠ಘΒΕͨ_{LS ͷϑΟϧλ܎਺Λ෇࿥ 3 ʹܝ} ࡌ͓ͯ͘͠ɽͳ͓ɼ͜ͷΑ͏ͳੑ࣭͸ɼଞͷίϯύΫ ਤ11 Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹ ΑΔεέʔϦϯά܎਺ͷܭଌޡࠩͱਪఆޡࠩ

Fig.11 Measured and estimated errors of scaling coeﬃ-cients of LS using Daubechies 6 wavelet

ταϙʔτΛ࣋ͭਖ਼ن௚ަ΢ΣʔϒϨοτجఈʹ͓͍ ͯ΋؍࡯͞Ε͓ͯΓɼ্هͱಉ͡खॱʹΑΓ_{LS Λઃ} ܭ͢Δ͜ͱ͕ՄೳͰ͋Δ͜ͱ΋֬ೝ͞Εͨɽ࣍ʹ3.2.1 અͱಉ͡Α͏ʹͯ͠ԋࢉྔΛࢉग़͢Δͱɼ_{N ߲ͷ਺ྻ} Λೖྗ৴߸ͱͯ͠ҎԼͷΑ͏ʹදͤΔɽ ɾMRA: ม׵ɼٯม׵ڞʹ: 12N, ɾ_LS: ม׵ɼٯม׵ڞʹ_{: 8.5N.} ͕ͨͬͯ͠ม׵ɼٯม׵ͲͪΒ΋ɼ_{LS ʹΑΔԋࢉྔ} ͸ैདྷͷMRA ͷ 70.8% ͱͳΔɽ ࣍ʹॲཧ࣌ؒͷଌఆͰ͋Δ͕ɼ_{3.2.1 અͱಉ͡৚݅} Ͱଌఆͨ͠ͱ͜Ζɼਤ_{8 ΍ਤ 10 ͷΑ͏ͳɼϦχΞͳ} ௚ઢ͸ಘΒΕͳ͔ͬͨɽ͜ͷݪҼ͸ɼԋࢉྔͷҧ͍ʹ ͋ΔͱࢥΘΕΔɽ͢ͳΘͪ_{MRA ͷԋࢉྔͰൺֱ͢Δ} ͱɼίϯύΫταϙʔτΛ࣋ͨͳ͍Meyer ΢Σʔϒ Ϩοτ͕_{81N Ͱ͋Δͷʹର͠ɼ୹͍ίϯύΫταϙʔ} τΛ࣋ͭDaubechies 6 ΢ΣʔϒϨοτ͸ۇ͔ 12N Ͱ ͋Δɽ͕ͨͬͯ͠Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ ͨॲཧ࣌ؒ͸ɼ_{Meyer ΢ΣʔϒϨοτͷ਺෼ͷҰఔ౓} ʹͳΓɼଌఆ࣌ؒͷޡ͕ࠩग़΍͔ͬͨ͢΋ͷͱߟ͑Β ΕΔɽͦ͜Ͱ_{3.2.1 અͱಉ͡৚݅Ͱɼଌఆճ਺͚ͩΛ} 1000 ճʹҾ্͖͛ͯɼͦͷฏۉ࣌ؒΛࢉग़͢Δ͜ͱ ʹͨ͠ʢҎޙɼ_{Daubechies 6 ΢ΣʔϒϨοτʹؔ࿈͢} Δॲཧ࣌ؒͷଌఆ͸ɼ͢΂ͯ_{1000 ճɼଌఆͨ͠ฏۉ} ࣌ؒͱ͢Δʣɽ͢Δͱਤ12 ͷΑ͏ͳ௚ઢతͳάϥϑ ͕ಘΒΕɼೖྗ৴߸ͷ߲਺_{N = 131072 ʹ͓͚Δɼͦ} ΕͧΕͷଌఆ݁Ռ͸ҎԼͷΑ͏ʹͳͬͨɽ

ɾ_{MRA: ม׵: 41.8msec, ٯม׵: 45.7msec,} ɾ_LS: ม׵_{: 33.7msec, ٯม׵: 32.9msec.} Ҏ্ͷ݁ՌΑΓɼ_{LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ} MRA ͷ 80.6%ɼٯม׵͸ 72.0% ͱͳΓɼԋࢉྔ͔Β ਪఆ͞ΕΔ_{70.8% ΑΓେ͖ͳ஋ͱͳͬͨɽ} ͜ͷݱ৅ʹؔͯ͠ߟ࡯͢Δɽ্هͰ΋આ໌͕ͨ͠ɼ Meyer ΢ΣʔϒϨοτʹൺ΂ͯɼDaubechies 6 ΢Σʔ ϒϨοτͷॲཧ଎౓͸ѹ౗తʹ଎͍ɽͱ͜ΖͰਤ_{2, 3} Λݟͯ΋Θ͔ΔΑ͏ʹɼLS ʹ͓͍ͯ͸৴߸Λۮ਺߲ͱ ح਺߲ʹ෼཭ͨ͠Γ෼཭͞Εͨ৴߸ΛҰͭʹ·ͱΊΔ खֻ͕͔͍ؒͬͯΔɽॲཧ͕࣌ؒൺֱతɼ௕͍Meyer ΢ΣʔϒϨοτͰ͸ɼ͜ΕΒͷखؒ͸ຆͲແࢹͰ͖ͨ ͕ɼॲཧ࣌ؒͷ୹͍Daubechies 6 ΢ΣʔϒϨοτͰ͸ ແࢹ͢Δ͜ͱ͕Ͱ͖ͳ͘ͳΓɼॲཧ࣌ؒͷݮগ཰͕খ ͘͞ͳͬͯ͠·ͬͨ΋ͷͱߟ͑ΒΕΔɽͦ͜Ͱɼࢀߟ ·Ͱʹ্هͷखؒΛআ͍ͨLS ͷ࡞ۀ෦෼ʢ͜ΕΛ LS ͷίΞ࡞ۀ෦෼ͱݺͿ͜ͱʹ͢ΔʣͷΈͷ࡞ۀ࣌ؒΛ ܭଌ͢ΔϓϩάϥϛϯάΛ৽ͨʹ࡞੒ͯ͠ଌఆͨ͠ͱ ͜ΖɼҎԼͷΑ͏ʹͳͬͨɽ

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(a) Forward MRA and forward LS

(b) Inverse MRA and inverse LS

ਤ₁₂ _{Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ MRA} ͱLS ʹΑΔܭࢉ࣌ؒʢαϯϓϧ਺ N ͷೖྗ৴߸ΛϨ ϕϧ_{−1 ∼ −3 ʹม׵)}

Fig.12 Processing times of MRA and LS using Daubechies 6 wavelet (an N samples input data is trans-formed to level −1 ∼ −3) ɾ_{LS ίΞ: ม׵: 30.4msec, ٯม׵: 30.4msec.} ͜ΕΒ͸ɼผ్ʹ࡞੒ͨ͠ϓϩάϥϛϯάʹΑΔଌఆ ݁ՌͰ͋Δ͕ɼ্هͷ਺஋ΑΓݸʑͷॲཧ࣌ؒΛࢉग़ ͯ͠ΈΔͱɼ࣍ͷ͜ͱ͕ݴ͑Δɽ͢ͳΘͪDaubechies 6 ΢ΣʔϒϨοτʹ͓͍ͯ͸ɼίΞ࡞ۀ෦෼Ҏ֎ͷɼ৴ ߸Λ෼཭ͨ͠Γɼ·ͱΊͨΓ͢Δॲཧ͕࣌ؒɼLS શ ମͷ͓Αͦ_{1 ׂఔ౓Λ઎ΊΔͨΊɼLS ͷॲཧ଎౓͸ɼ} ཧ࿦஋ΑΓଟগɼམͪΔ͜ͱΛ༧Ίߟྀ͓͔ͯ͠ͳΕ ͹͍͚ͳ͍ɽ ࣍ʹ_{3.2.2 અͱಉ͡Α͏ʹͯ͠ɼม׵ͱٯม׵Λ௨} ͨ͠ޡࠩΛଌఆͨ͠ͱ͜ΖɼҎԼͷ݁Ռ͕ಘΒΕͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −232.4 dBɼ} ɾ_{LS ʹΑΔޡࠩɿ} _{−285.4 dBɽ} 3.5 Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ LS Խ Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ࣮ ਺෦ʹ͸ɼલઅͷ_{Daubechies 6 ΢ΣʔϒϨοτΛ༻͍} Δɽͦͯ͠ڏ਺෦ʹ͸ɼDaubechies 6 ΢ΣʔϒϨοτ ͱ_{Hilbert ม׵ϖΞΛ੒͢ਖ਼ن௚ަ΢ΣʔϒϨοτج} ఈΛ༻͍Δ͜ͱʹͳΔ͕ɼͦͷৄࡉ͸จݙ[7] Ͱ঺հ ͞Ε͍ͯΔɽ·ͨ_{LS ͷߏ੒๏΋ Meyer ΢ΣʔϒϨο} τΛ༻͍ͨCDWT ͱຆͲಉ͡ͳͷͰɼҎԼʹཁ఺ͩ ͚Λड़΂Δɽ·ͣڏ਺෦ͷਖ਼ن௚ަ΢ΣʔϒϨοτج ఈͷτΡʔεέʔϧ਺ྻ_{p_n_{} ͸จݙ [7] ͷํ๏Ͱٻ} Ίͯɼ༗ݶͷ_{23 ߲ͷ਺ྻ {p}_n _{: −6 ≤ n ≤ 16, n ∈ Z}} ʹ੾Γ٧ΊΔʢ͜ͷ࣌ɼ∑_n∈Z_|p_n_|2 = 2 ͕੒ཱ͢Δ͜ ͱΛߟྀ͢ΔͱɼτΡʔεέʔϧ਺ྻͷޡࠩΛਖ਼֬ ʹٻΊΔ͜ͱ͕Ͱ͖ɼͦΕ͸_{−102.7dB ͱܭࢉ͞Εͨ} ͷͰɼे෼ͳਫ਼౓͕อͨΕ͍ͯΔͱ൑அͨ͠ʣɽࣜ (8)ɼ(9) ʹ͓͚Δ m ͸ 8 ʹઃఆ͠ɼ·ͨࣜ (26) Ͱද ͞ΕΔ_{22 ߲ͷϑΟϧλ {s}_n_{} ͷ੾Γ٧Ίʹؔͯ͠͸ɼ} θ =8 × 10−5ʹઃఆ͠ɼ_{MSE(θ) = −79.1[dB]ɼ8 ߲ͷ} ਺ྻ_{s′ n : n = −6, −5, · · · , 1} ͕ಘΒΕͨɽ͜ͷΑ͏ʹ ͯ͠ಘΒΕͨLS ͷϑΟϧλ܎਺Λ෇࿥ 4 ʹܝࡌͯ͠ ͓͘ɽͦͯ͠_{N ߲ͷ਺ྻΛೖྗ৴߸ͱ͢ΔͱɼͦΕͧ} Εͷԋࢉྔ͸࣍ͷΑ͏ʹࢉग़͞ΕΔɽ ɾMRA: ม׵ɼٯม׵ڞʹ: 23N, ɾ_LS: ม׵ɼٯม׵ڞʹ_{: 15N.} ͕ͨͬͯ͠ม׵ɼٯม׵ͲͪΒ΋ɼ_{LS ʹΑΔԋࢉྔ} ͸ैདྷͷMRA ͷ 65.2% ͱͳΔɽ࣍ʹ 3.4 અͱಉ͡৚ ݅ͰɼN = 131072 ʹ͓͚Δॲཧ࣌ؒΛଌఆͨ͠ͱ͜ ΖҎԼͷΑ͏ʹͳͬͨɽ

ɾMRA: ม׵: 77.2msec, ٯม׵: 89.0msec, ɾ_LS: ม׵_{: 55.2msec, ٯม׵: 51.4msec.} Ҏ্ͷ݁ՌΑΓɼ_{LS ʹΑΔม׵ͷॲཧ࣌ؒ͸ैདྷͷ} MRA ͷ 71.5%ɼٯม׵͸ 57.8% ͱͳΓɼม׵ͷ΄͏ ͸ԋࢉྔ͔Βਪఆ͞ΕΔ65.2% ΑΓ΋େ͖͘ɼ·ͨ ٯม׵͸খ͘͞ͳͬͨɽ͜ͷ݁Ռ͸͜Ε·Ͱͷߟ࡯Λ େ͖͘෴͢΋ͷͰ͸ͳ͍ɽ·ͣม׵ʹؔͯ͠ݴ͑͹ɼ Meyer ΢ΣʔϒϨοτͱൺ΂ͯԋࢉྔ͕গͳ͍΢Σʔ ϒϨοτͷ৔߹ɼLS ͷίΞ࡞ۀ෦෼Ҏ֎ͷ૬ରతͳ ෛ୲͕େ͖͘ɼॲཧ଎౓ͷ޲্͕ಘΒΕʹ͔ͬͨ͘ͱ ߟ͑ΒΕΔʢ_{3.4 અࢀরʣɽ·ͨٯม׵Ͱ͸ྑ޷ͳ݁} Ռ͕ಘΒΕ͍ͯΔ͕ɼ͜Ε͸MRA ͷٯม׵ͷ஗͕͞ େ͖ͳҰҼͱݴ͑Δʢ_{3.2.1 અࢀরʣɽࢀߟ·Ͱʹ 3.4} અͰࣔͨ͠LS ͷίΞ࡞ۀ෦෼ͷΈͷ࡞ۀ࣌ؒΛɼ3.4 અͱಉ͡Α͏ʹͯ͠ଌఆͨ͠ͱ͜Ζɼ

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ද3 Daubechies 6 ΢ΣʔϒϨοτʹΑΔ CDWT ʹ͓͚Δॲཧ࣌ؒʢlevel −1 ʙ −3ɼN = 131072ʣ Table.3 Processing times of CDWT using Daubechies 6 wavelet, level −1 to level −3, N = 131072

CDWT (msec) ICDWT (msec)

Real part Imag. part Interpolation Real part Imag. part Inv. Interpolation

MRA 42.3 77.1 86.7 46.5 88.0 53.4 LS 34.1 55.0 86.6 33.0 51.2 52.6 ɾ_{LS ίΞ: ม׵: 51.7msec, ٯม׵: 48.5msec.} ͱͳΓɼίΞ࡞ۀ෦෼Ҏ֎ͷॲཧ͕࣌ؒ_{LS શମͷ਺} ύʔηϯτΛ઎ΊΔ͜ͱ͕Θ͔Δɽ࣍ʹ3.2.2 અͱಉ ͡৚݅ʹΑΔม׵ɼٯม׵Λ௨ͨ͠࿪͸ҎԼͷΑ͏ʹ ଌఆ͞Εͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −106.2 dBɼ} ɾ_{LS ʹΑΔޡࠩɿ} _{−291.0 dBɽ} ࣍ʹ_{3.3.2 અͱಉ͡Α͏ʹͯ͠ɼϨϕϧ −1 ͔ΒϨ} ϕϧ_{−3 ·ͰͷɼDaubechies 6 ΢ΣʔϒϨοτΛ༻͍} ͨ_{CDWT ͓Αͼ ICDWT ͷ૯߹తͳԋࢉྔ͸ҎԼͷ} Α͏ʹࢉग़͞ΕΔɽ ɾ_{MRA: CDWT: 113.25N,} _{ICDWT: 96.25N,} ɾ_LS: _{CDWT: 93.125N, ICDWT: 76.125N.} ͜ͷΑ͏ʹ_{MRAɼLS ͷͲͪΒʹ͓͍ͯ΋ɼCDWT ͷ} ԋࢉྔ͸ICDWT ͷԋࢉྔΑΓ΋ଟ͘ͳΔ͕ɼ͜Ε͸ Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷಛ௃Ͱ ͋Δʢৄࡉ͸จݙ[7] ʹৡΔ͕ɼDaubechies 6 ΢Σʔ ϒϨοτΛ༻͍ͨ_{CDWT ͷิؒʹ͸ɼICDWT ͷٯ} ิؒΑΓ΋ଟ͘ͷܭࢉྔ͕ඞཁͱͳΔʣɽҎ্ͷԋࢉ ྔΑΓɼCDWT ʹ͓͚Δ LS ʹΑΔԋࢉྔ͸ MRA ͷ 82.2%ɼICDWT ʹ͓͚Δ LS ʹΑΔԋࢉྔ͸ MRA ͷ 79.1% ͱͳΔɽ ࣍ʹ_{3.4 અͱಉ͡৚݅Ͱ CDWT ͷશମతͳॲཧ࣌} ؒΛଌఆͨ͠ͱ͜Ζɼೖྗ৴߸ͷ߲਺N = 131072 ʹ ͓͚ΔɼͦΕͧΕͷଌఆ݁Ռ͸ҎԼͷΑ͏ʹͳͬͨɽ ɾMRA: CDWT: 206.1msec, ICDWT: 187.9msec, ɾ_LS: _{CDWT: 175.7msec, ICDWT: 136.8msec.} Ҏ্ͷ݁ՌΑΓɼLS ʹΑΔ CDWT ͷॲཧ࣌ؒ͸ MRA ͷ_{85.2% ͱͳΓɼԋࢉྔ͔Βਪఆ͞ΕΔ 82.2% ΑΓ} ΍΍্ճͬͨɽ·ͨ_{LS ʹΑΔ ICDWT ͷॲཧ࣌ؒ͸} MRA ͷ 72.8% ͱͳΓɼԋࢉྔ͔Βਪఆ͞ΕΔ 79.1% ΑΓԼճͬͨɽ͜Ε͚ͩͰ͸ධՁ͕͠ʹ͍͘ͷͰɼ͞ ΒʹN = 131072 ʹ͓͚ΔɼCDWTɼICDWT ͷ಺෦ͷ ֤ॲཧ࣌ؒΛɼಉ͡৚݅Ͱଌఆͯ͠Έͨͱ͜Ζɼද₃ ͷΑ͏ʹͳͬͨɽද_{3 ͷ਺஋ΑΓɼ࣮਺෦ͷ LS ʹΑΔ} ม׵ͷॲཧ࣌ؒ͸MRA ͷ 80.6%ɼٯม׵͸ 71.0% ͱ ͳΓɼ͜Ε͸_{3.4 અͰࣔͨ͠ Daubechies 6 ΢ΣʔϒϨο} τ୯ମͰଌఆͨ͠਺஋ʢม׵80.6%ɼٯม׵ 72.0%ʣ ͱ΄΅ಉ݁͡Ռʹͳͬͨɽ·ͨڏ਺෦ͷ_{LS ʹΑΔม} ׵ͷॲཧ࣌ؒ͸_{MRA ͷ 71.3%ɼٯม׵͸ 58.2% ͱͳ} Γɼ͜Ε΋ຊઅͰࣔͨ͠ڏ਺෦ͷ΢ΣʔϒϨοτ୯ମ Ͱଌఆͨ͠਺஋ʢม׵_{71.5%ɼٯม׵ 57.8%ʣͱ΄΅} ಉ݁͡Ռʹͳͬͨɽ͜ͷ͜ͱ͔ΒDaubechies 6 ΢Σʔ ϒϨοτΛ༻͍ͨ_{CDWTɼICDWT ʹ͓͍ͯ΋ɼLS ͸} ਖ਼ৗʹػೳ͠ɼॲཧͷߴ଎Խʹد༩͍ͯ͠Δͱߟ͑Β ΕΔɽ ࣍ʹ_{3.2.2 અͱಉ͡৚݅ʹΑΔม׵ɼٯม׵Λ௨͠} ͨ࿪͸ҎԼͷΑ͏ʹଌఆ͞Εͨɽ ɾ_{MRA ʹΑΔޡࠩɿ −101.4 dBɼ} ɾ_{LS ʹΑΔޡࠩɿ} _{−102.1 dBɽ} ͜ΕΒͷޡࠩ͸ิؒɼٯิؒͷӨڹΛड͚ͨ݁Ռͱࢥ ΘΕΔ͕ɼͲͪΒ΋࣮༻্͸े෼ͳਫ਼౓ͱࢥΘΕΔɽ 4. LS ʹΑΔ CDWT ͷγϑτෆมੑͷ֬ೝ MRA Λ༻͍ͨ CDWT ͷɼ֤Ϩϕϧͷม׵ɼٯม ׵ʹ͓͍ͯγϑτෆมੑ͕੒ཱ͢Δ͜ͱ͸ɼจݙ[13] Ͱ͢Ͱʹ֬ೝ͞Ε͍ͯΔ͕ɼ_{LS Λ༻͍ͨ CDWT ʹ͓} ͍ͯ΋ɼಉ͡Α͏ʹγϑτෆมੑ͕੒ཱ͢Δ͜ͱΛɼ ͜ͷষͰ֬ೝ͢ΔɽͦͷͨΊʹɼ·ͣγϑτෆมੑͷ ఆٛΛ໌֬ʹ͠ɼ࣍ʹγϑτෆมੑͷ֬ೝํ๏Λݕ ౼͢Δɽͦͯ͠ਖ਼ن௚ަ΢ΣʔϒϨοτجఈͷMeyer ΢ΣʔϒϨοτΛ༻͍ͨ_{DWT Ͱ͸γϑτෆมੑ͕੒} ཱ͠ͳ͍͕ɼMeyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT Ͱ ͸γϑτෆมੑ͕੒ཱ͢Δ͜ͱΛɼ_{LS Λ༻͍ͨܭࢉ} Ͱ֬ೝ͢Δɽͦͯ͠ࢀߟ·Ͱʹ_{MRA Λ༻͍ͨܭࢉͰ} ΋ɼಉ݁͡ՌʹͳΔ͜ͱΛ֬ೝ͢Δɽ

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4.1 γϑτෆมੑͷఆٛ ࡞༻ૉ_{G ʹΑΓɼؔ਺ f (t) ∈ L}2(R) ͸ؔ਺ g(t) ∈ L2_{(R) ʹม׵͞ΕΔ΋ͷͱ͢Δɽ͢Δͱɼ͜ͷม׵͸} ࣍ͷΑ͏ʹදͤΔɽ g(t) = G f (t) (44) ࣍ʹϢχλϦ࡞༻ૉ_T_dʢͨͩ͠_{d ∈ RʣΛɼ࣍ͷΑ͏} ʹఆٛ͢Δɽ Td f (t) = f (t − d) (45) ͢ͳΘͪϢχλϦ࡞༻ૉ_T_d͸ɼؔ਺ _{f (t) ∈ L}2_{(R) Λɼ} ڑ཭_{d ∈ R ΄ͲɼฏߦҠಈͤ͞Δ΋ͷͰ͋Δɽ͜͜Ͱ} ࣍ͷ͕ࣜ੒ཱ͢Δ࣌ɼ࡞༻ૉ_{G ʹ͓͍ͯɼγϑτෆม} ੑ͕੒ཱ͢Δͱఆٛ͢Δɽ (G Td f )(t) = (TdG f )(t), ∀ f (t) ∈ L2(R), ∀d ∈ R (46) ͢ͳΘͪ೚ҙͷؔ਺_{f (t) ∈ L}2_{(R) ʹର͠ G Λ࡞༻ͤ͞} ͯ࣍ʹ೚ҙͷڑ཭d ͚ͩฏߦҠಈͨ͠΋ͷ͕ɼઌʹڑ ཭_{d ͚ͩฏߦҠಈ͔ͤͯ͞Β࣍ʹ G Λ࡞༻ͤͨ͞΋} ͷʹ౳͘͠ͳΔ࣌ɼ_{G ʹ͓͍ͯγϑτෆมੑ͕੒ཱ͢} Δͱఆٛ͢Δɽ 4.2 γϑτෆมੑͷ֬ೝํ๏ Meyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ֤Ϩϕϧ ʹ͓͍ͯγϑτෆมੑ͕੒ཱ͢Δ͜ͱ͸ɼจݙ_{[13] ౳} Ͱཧ࿦తʹূ໌͞Ε͍ͯΔ͕ɼ࣮ࡍʹͲͷఔ౓ͷਫ਼౓ Ͱγϑτෆมੑ͕੒ཱ͍ͯ͠Δ͔Λݕূ͢Δʹ͸ɼ࣮ ࡍʹܭࢉ͢Δඞཁ͕͋Δɽ͔ࣜ͠͠_{(46) ʹैͬͯɼ͋} ΒΏΔ೚ҙͷؔ਺_{f (t) ∈ L}2_{(R) ʹରͯ͠ܭࢉΛߦ͏ͷ} ͸ෆՄೳͰ͋Δɽͦ͜ͰຊষͰ͸ɼೖྗ৴߸ͱͯ࣍͠ ͷΑ͏ͳσδλϧͷΠϯύϧε৴߸_{δn} Λ༻͍Δ͜ͱ ʹ͢Δɽ δn=   1 ,0 , n = 0otherwise (47) ͜ͷΑ͏ͳΠϯύϧε৴߸͸ɼਤ_{13 ͷΑ͏ͳσδλ} ϧ৴߸ͱͯ͠ද͞ΕΔ͕ɼφΠΩετप೾਺ҎԼͷશ प೾਺੒෼ΛۉҰʹؚΜͰ͓ΓɼγϑτෆมੑΛ֬ೝ ͢Δͷʹదͨ͠৴߸ͱݴ͑Δɽͦͯ͜͠ͷΠϯύϧε ৴߸_{δ_n_{} Λɼڑ཭ d = 0, 1, · · · , 7 ΄ͲฏߦҠಈͨ͠ɼ} ҎԼͷΑ͏ͳ_{8 ݸͷ৴߸Λ༻ҙ͢Δɽ} {δn−d}, d = 0, 1, · · · , 7 (48) ਤ13 Πϯύϧε৴߸_{δ_n_} Fig.13 Impulse signal {δn}

͜ΕΒ߹ܭ_{8 ݸͷ৴߸ʹରͯ͠ɼϨϕϧ −1 ͔ΒϨϕ} ϧ_{−3 ·Ͱͷ֤Ϩϕϧͷม׵ɼ͓Αͼٯม׵Λࢪ͠ɼͦ} ͷग़ྗ৴߸ΛಘΔʢৄࡉ͸จݙ[13] ౳ʹৡΔ͕ɼҰൠ తʹϨϕϧ_{−1 ͔ΒϨϕϧ JʢJ < 0, J ∈ Zʣ·Ͱͷγ} ϑτෆมੑΛௐ΂Δʹ͸ɼ1 αϯϓϧͣͭฏߦҠಈ͠ ͨΠϯύϧε৴߸͕2−J_{ݸ΄ͲඞཁͱͳΔɽ͜͜Ͱ͸} J = −3 Ͱ͋Δ͔Βɼ2−J₌_{8 ݸͷΠϯύϧε৴߸Λ༻} ҙ͢Δඞཁ͕͋Δʣɽ͜ͷΑ͏ʹͯ͠ಘΒΕͨग़ྗ৴ ߸͸ɼ֤ϨϕϧͷΠϯύϧεԠ౴৴߸ͱݟͳ͢͜ͱ͕ Ͱ͖Δ͕ɼ֤Ϩϕϧʹ͓͍ͯɼͰ͖Δ͚ͩಉ͡ܗঢ়ͷ ΠϯύϧεԠ౴৴߸ͷग़ྗ͕ಘΒΕΔ΄͏͕ɼΑΓࣜ (46) ͕ਫ਼౓Α͘࠶ݱ͞Ε͍ͯΔͱߟ͑ΒΕɼγϑτෆ มੑ͕ΑΓਫ਼౓Α࣮͘ݱ͞Ε͍ͯΔͱ൑அͰ͖Δɽ 4.3 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ DWT ʹ͓͚Δγ ϑτෆมੑͷݕূ ·ͣ࠷ॳʹɼMeyer ΢ΣʔϒϨοτʹΑΔ LS Λ༻ ͍ͨ_{DWT ʹ͓͚ΔγϑτෆมੑΛݕূ͢ΔɽMeyer} ΢ΣʔϒϨοτʹΑΔLS ʹͯɼਤ 13 ʹࣔͨ͠ 8 ͭ ͷ৴߸ʹର͠ɼͦΕͧΕݸผʹɼϨϕϧ_{−3 ·Ͱ DWT} Λࢪ͢ɽ͜ͷ࣌ͷৄࡉͳॲཧ৚݅Λࣔ͢ɽͦΕͧΕͷ ϑΟϧλॲཧ͸ɼ_{3.1 ʹࣔͨ͠ LS ͷಛੑΛߟྀͯ͠ɼ} ८ճ৞ΈࠐΈʹΑΓߦ͏ɽͦͯ͠८ճ৞ΈࠐΈͷӨڹ Λආ͚ΔͨΊɼਤ13 ʹࣔͨ͠ 8 ͭͷ৴߸͸ɼͦΕͧΕ −1024 ≤ n ≤ 1023 ͷൣғΛ੾Γऔͬͯɼ2048 αϯϓ ϧͷ৴߸ͱͯ͠ॲཧ͢Δɽͳ͓ɼҰൠతͳDWT ʹ͓ ͍ͯ͸ิؒॲཧ͕লུ͞ΕΔ৔߹͕ଟ͍ͷͰɼ͜͜Ͱ ΋লུ͢Δɽͦͯ͠ٯม׵͸ɼͦΕͧΕͷϨϕϧʹ͓ ͍ͯݸผʹߦ͏ɽྫ͑͹Ϩϕϧ_{−1 ͷ΢ΣʔϒϨοτ} ͷٯม׵͸ɼϨϕϧ_{−1 ͷΈͷ΢ΣʔϒϨοτ܎਺Λ} ͦͷ··࢒͠ɼͦͷଞͷ΢ΣʔϒϨοτ܎਺΍εέʔ Ϧϯά܎਺Λ͢΂ͯθϩʹมߋͯ͠ɼٯม׵Λࢪ͢ɽ ಉ͡Α͏ʹͯ͠ɼϨϕϧ_{−2 ͷ΢ΣʔϒϨοτɼϨϕ} ϧ_{−3 ͷ΢ΣʔϒϨοτɼ͓ΑͼϨϕϧ −3 ͷεέʔϦ} ϯάؔ਺ͷɼͦΕͧΕͷٯม׵Λ࣮ߦ͢Δɽ͜͏͢Δ ͜ͱʹΑΓɼ֤Ϩϕϧͷม׵ɼٯม׵Λ௨ͨ͠ग़ྗ৴

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߸ΛݸผʹऔΓग़͢͜ͱ͕ՄೳͰ͋Δɽ͜͏ͯ͠ಘΒ Εͨग़ྗ৴߸Λɼ֤Ϩϕϧຖʹɼͻͱͭͷάϥϑʹฒ ΂ͯܗঢ়Λൺֱͯ͠ΈΔɽ͢ͳΘͪͦΕͧΕͷग़ྗ৴ ߸ͷࢁͷத৺Λதԝʹͯ͠ɼ_{128 αϯϓϧͷྖҬΛ੾} Γग़͠ɼ͜ΕΒͷ৴߸Λ_{128 αϯϓϧִؒͰฒ΂ͨά} ϥϑΛਤ_{14 ʹࣔ͢ɽ࣍ʹ֤ग़ྗ৴߸ {out}_n_{} ͷΤωϧ} Ϊ_{energy Λ} energy =∑ n∈Z |outn|2 (49) Ͱܭࢉ͠ɼͦͷΤωϧΪมಈͷ༷ࢠΛਤ_{15 ʹࣔ͢ɽ͜} ΕΒਤ14ɼ15 Λݟͯ΋Θ͔ΔΑ͏ʹɼೖྗ৴߸ͷΠ ϯύϧε৴߸͕_{1 αϯϓϧͣͭฏߦҠಈ͢Δͱͱ΋ʹɼ} ֤Ϩϕϧͷग़ྗ৴߸ͷ೾ܗɼ͓ΑͼΤωϧΪ͕ɼ͔ͳ Γมಈ͍ͯͯ͠ɼγϑτෆมੑ͸੒ཱ͍ͯ͠ͳ͍͜ͱ ͕Θ͔Δɽ͜͜ͰΤωϧΪมಈΛɼ਺஋Ͱද͢͜ͱΛ ߟ͑Δɽ͢ͳΘͪɼͦΕͧΕͷϨϕϧʹ͓͍ͯɼग़ྗ ৴߸ͷฏۉΤωϧΪΛಋ͖ग़͠ɼฏۉΤωϧΪͱͷࠩ ͷઈର஋͕࠷େͱͳΔग़ྗ৴߸Λݟ͚ͭग़͠ɼͦͷࠩ ͷઈର஋ΛฏۉΤωϧΪʹର͢Δൺ཰Ͱදͯ͠ΈΔͱ ҎԼͷΑ͏ʹͳͬͨɽ Wavelets of level − 1 : 0.119, Wavelets of level − 2 : 0.358, Wavelets of level − 3 : 0.358, Scaling functions of level − 3 : 0.119. ͜ͷΑ͏ʹܭࢉ͞Εͨൺ཰ΛΤωϧΪมಈ཰ͱݺͿ͜ ͱʹ͢Δ͕ɼ͜Ε͸ͻͱͭͷγϑτෆมੑͷਫ਼౓Λද ͢ࢦ਺ͱݟͳ͢͜ͱ͕Ͱ͖Δɽ౰વɼΤωϧΪมಈ཰ ͕খ͍͞΄Ͳɼਫ਼౓Α͘γϑτෆมੑ͕࣮ݱ͞ΕͯΔ ͱߟ͑ΒΕΔ͕ɼ্هͷΑ͏ʹ_{1 ׂ͔Β 3 ׂҎ্΋Τ} ωϧΪ͕มಈ͍ͯ͠Δ৔߹ʹ͸ɼγϑτෆมੑ͸࣮ݱ ͞Ε͍ͯͳ͍ͱߟ͑ͯΑ͍ɽ ࣍ʹɼMRA Λ༻͍ͨ DWT ʹ͓͍ͯɼ্هͱ·ͬͨ ͘ಉ͡Α͏ʹݕূͯ͠Έͨͱ͜ΖɼຆͲಉ݁͡Ռ͕ಘ ΒΕͨɽ͢ͳΘͪ_{LS ͱ·ͬͨ͘ಉҰͷΤωϧΪมಈ} ཰͕ಘΒΕɼਤ14ɼ15 ͱ΄΅ಉ͡άϥϑ͕ಘΒΕͨɽ 4.4 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ʹ͓͚Δ γϑτෆมੑͷ֬ೝ Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔ CDWT ʹ ͍ͭͯɼ4.3 અͱಉ͡Α͏ʹͯ͠ݕূͨ͠ͱ͜Ζɼਤ 16ɼ17 ͕ಘΒΕͨɽ·ͣਤ 16 ͔ΒΘ͔Δ͜ͱ͸ɼೖ ྗ৴߸ͷΠϯύϧε৴߸͕1 αϯϓϧͣͭฏߦҠಈ͠ ͯ΋ɼͦͷग़ྗ৴߸ͷ೾ܗ͸ຆͲมԽ͠ͳ͍ͱ͍͏͜ ͱͰ͋Δɽ·ͨਤ17 ͔Β͸ɼͦͷग़ྗ৴߸ͷΤωϧ Ϊ΋ຆͲมಈ͍ͯ͠ͳ͍͜ͱ͕Θ͔Δɽ͜ͷΤωϧΪ มಈ཰Λ4.3 અͱಉ͡Α͏ʹͯ͠ܭࢉͯ͠Έͨͱ͜Ζɼ ҎԼͷΑ͏ʹͳͬͨɽ Wavelets of level − 1 : 1.18 × 10−11_, Wavelets of level − 2 : 5.05 × 10−7_, Wavelets of level − 3 : 1.93 × 10−6_,

Scaling functions of level − 3 : 1.77 × 10−6_.

͜ͷΑ͏ʹΤωϧΪมಈ཰΋ۃΊͯඍখͰ͋Γɼߴ͍ ਫ਼౓Ͱγϑτෆมੑ͕੒ཱ͍ͯ͠Δ͜ͱ͕Θ͔Δɽ ࣍ʹɼ_{MRA Λ༻͍ͨ CDWT ʹ͓͍ͯɼ্هͱ·ͬ} ͨ͘ಉ͡Α͏ʹݕূͯ͠Έͨͱ͜ΖɼຆͲಉ݁͡Ռͱ ਤ͕ಘΒΕͨɽͨͩ͠ΤωϧΪมಈ཰ʹ͓͍ͯ͸ɼҎ ԼͷΑ͏ʹ͞ΒʹΑ͍݁Ռ͕ಘΒΕͨɽ Wavelets of level − 1 : 5.89 × 10−13_, Wavelets of level − 2 : 6.20 × 10−9_, Wavelets of level − 3 : 2.19 × 10−7_,

Scaling functions of level − 3 : 2.56 × 10−7_.

Ҏ্ͷΑ͏ʹγϑτෆมੑͷਫ਼౓ʹؔͯ͠͸ɼ_{LS Α} Γ_{MRA ͷ΄͏͕ɼ΍΍Α͍ͱ͍͏݁Ռ͕ಘΒΕ͕ͨɼ} ࣮༻্͸ɼͲͪΒ΋े෼ʹߴ͍ਫ਼౓Ͱγϑτෆมੑ͕ ࣮ݱ͞Ε͍ͯΔͱݴͬͯΑ͍ɽ 5. ·ͱΊ ैདྷͷLS ϑΟϧλͷઃܭ๏͸ɼίϯύΫταϙʔ τΛ࣋ͭ΢ΣʔϒϨοτʹͷΈʹରԠ͍ͯͨ͠ɽຊݚ ڀͰ͸͜ΕΛվળ͠ɼίϯύΫταϙʔτΛ࣋ͭɼ࣋ ͨͳ͍ʹؔΘΒͣɼ͢΂ͯͷਖ਼ن௚ަ΢ΣʔϒϨοτ جఈʹ͓͍ͯɼҰఆͷखॱΛ౿Ί͹࣮֬ʹLS ϑΟϧ λ͕ઃܭͰ͖Δख๏ΛఏҊͨ͠ɽͦͯ͜͠ΕΛ_CDWT ʹద༻ͨ͠ɽಘΒΕͨओͳ݁Ռ͸࣍ͷ௨ΓͰ͋Δɽ 1. ίϯύΫταϙʔτΛ࣋ͨͳ͍ Meyer ΢Σʔϒ Ϩοτ΍ɼίϯύΫταϙʔτΛ࣋ͭ_Daubechies 6 ΢ΣʔϒϨοτ౳ʹɼఏҊख๏ͷ LS ϑΟϧλ ͷઃܭ๏Λద༻͠ɼ͢΂ͯͷ৔߹ʹ͓͍ͯԋࢉ ྔΛɼैདྷͷ_{MRA ͷ 65% ͔Β 71% ʹݮগ͢} Δ͜ͱ͕Ͱ͖ͨɽ 2. ࣮ࡍͷϓϩάϥϛϯάʹఏҊख๏ͷ LS Λద༻͠ ॲཧ࣌ؒΛଌఆͨ͠ͱ͜ΖɼίϯύΫταϙʔ τΛ࣋ͨͳ͍_{Meyer ΢ΣʔϒϨοτ౳ͷ৔߹ɼ}

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(a) Wavelets of level −1

(b) Wavelets of level −2

(c) Wavelets of level −3

(d) Scaling functions of level −3

ਤ₁₄ _{Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔ DWT} ͷΠϯύϧεԠ౴৴߸

Fig.14 Impulse response signals of DWT by LS using Meyer wavelet

ਤ15 Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔ DWT ͷΠϯύϧεԠ౴৴߸ͷΤωϧΪมಈ

Fig.15 Fluctuation of impulse response energy of DWT by LS using Meyer wavelet

(a) Wavelets of level −1

(b) Wavelets of level −2

(c) Wavelets of level −3

(d) Scaling functions of level −3

ਤ ₁₆ _{Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔ} CDWT ͷΠϯύϧεԠ౴৴߸

Fig.16 Impulse response signals of CDWT by LS using Meyer wavelet

ਤ ₁₇ _{Meyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ʹΑΔ} CDWT ͷΠϯύϧεԠ౴৴߸ͷΤωϧΪมಈ Fig.17 Fluctuation of impulse response energy of CDWT by LS using Meyer wavelet

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ॲཧ࣌ؒΛैདྷͷMRA ͷ 60% ͔Β 64% ʹݮ গ͢Δ͜ͱ͕Ͱ͖ͨɽ 3. ͔͠͠ίϯύΫταϙʔτΛ࣋ͭ Daubechies 6 ΢ΣʔϒϨοτͷॲཧ࣌ؒ͸ɼ_{72% ͔Β 81% ͷ} ݮগʹͱͲ·ͬͨɽ͜Ε͸_{LS ಛ༗ͷॲཧɼ͢ͳ} Θͪ৴߸Λۮ਺߲ͱح਺߲ʹ෼཭ͨ͠Γɼ෼཭ ͞Εͨ৴߸ΛҰͭʹ·ͱΊΔॲཧʹӨڹ͞Εͨ ͨΊͱߟ͑ΒΕΔɽ 4. Meyer ΢ΣʔϒϨοτ΍ Daubechies 6 ΢Σʔϒ ϨοτΛ༻͍ͨɼϨϕϧ_{−1 ͔ΒϨϕϧ −3 ·} ͰͷCDWTɼICDWT ʹ͓͚Δ૯߹తͳԋࢉྔ ͸ैདྷͷ_{MRA ͷ 80% લޙʹݮগ͢Δ͜ͱ͕Ͱ} ͖ͨɽ·ͨDaubechies 6 ΢ΣʔϒϨοτΛ༻͍ ͨ_{CDWT Λআ͍ͯɼ࣮ࡍͷॲཧ࣌ؒ͸ɼैདྷ} ͷMRA ʹൺ΂ͯ 73% ͔Β 77% ʹݮগ͕ͨ͠ɼ Daubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷ Έɼ_{85% ͷݮগʹͱͲ·ͬͨɽ͜Ε͸্هͷ 3.} Ͱࣔͨ͠LS ಛ༗ͷॲཧ͕ݪҼͱߟ͑ΒΕΔɽ 5. ఏҊख๏Λ༻͍ͯɼDWT ΍ CDWT ͷγϑτෆ มੑͷݕূΛߦͬͨͱ͜Ζɼैདྷ๏ͷMRA ͱ ΄΅ಉ౳ͷɼਫ਼౓ͷߴ͍݁Ռ͕ಘΒΕͨɽ ࢀߟจݙ

[1] R. L. Allen and D. W. Mills: Signal Analysis, IEEE Press, 2004. [2] I. Daubechies: Ten Lectures on Wavelets, SIAM, Philadelphia,

1992.

[3] ࡗݪɹਐ: ΢ΣʔϒϨοτϏΪφʔζΨΠυ, ౦ژిػେֶग़ ൛ہ, 1998.

[4] S. G. Mallat: A Wavelet Tour of Signal Processing, Academic Press, New York, 1999.

[5] ށాɹߒɼষɹ஧ɼ઒ാɹ༸তɿ࠷৽΢ΣʔϒϨοτ࣮ફߨ ࠲ɿೖ໳ͱԠ༻ɿ৴߸ͷجૅ͔Β࠷৽ཧ࿦·Ͱɼιϑτόϯ ΫΫϦΤΠςΟϒɼ2005.

[6] ষɹ஧ɼށాɹߒɿγϑτෆมͳෳૉ਺཭ࢄ΢ΣʔϒϨοτ ม׵ ୈ1 ใɿෳૉ਺཭ࢄ΢ΣʔϒϨοτม׵ͷཧ࿦ͱݪཧɼ Journal of Signal Processing, Vol.11, No.5, pp.387–400, 2007.9. [7] ށాɹߒɼষɹ஧ɿγϑτෆมͳෳૉ਺཭ࢄ΢ΣʔϒϨοτ ม׵ ୈ2 ใɿ௚ަ΢ΣʔϒϨοτΛجʹͨ͠ෳૉ਺΢Σʔ ϒϨοτઃܭ๏ͷҰఏҊɼJournal of signal processing, Vol.11, No5, pp.401–412, 2007.9.

[8] ށాɹߒɼষɹ஧ɿγϑτෆมͳෳૉ਺཭ࢄ΢ΣʔϒϨοτ ม׵ ୈ3 ใɿ৽ͨͳෳૉ਺཭ࢄ΢ΣʔϒϨοτม׵ͷܭࢉ๏ɼ Journal of signal processing, Vol.11, No5, pp413–424, 2007.9. [9] ށాɹߒɼষɹ஧ɿ׬શγϑτෆมΛ࣮ݱ͢Δෳૉ਺཭ࢄ

΢ΣʔϒϨοτม׵ɼJournal of signal processing, Vol.12, No3, pp155–166, 2008.3.

[10] S. G. Mallat: A Theory for multiresolution signal decomposi-tion: The wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.11, No.7, pp.674–693, 1989.

[11] W. Sweldens: The lifting scheme: A construction of second generation wavelets, SIAM Journal on Mathematical Analysis, Vol.29, No.2, pp.511–546, 1998.

[12] I. Daubechies and W. Sweldens: Factoring wavelet transforms into lifting steps, The journal of Fourier Analysis and Applica-tions, Vol.4, Issue 3, pp.247–269, 1998.

[13] H. Toda and Z. Zhang: Perfect translation invariance with a wide range of shapes of Hilbert transform pairs of wavelet bases, In-ternational Journal of Wavelets, Multiresolution and Information Processing, Vol.8, No.4, pp.501–520, 2010.

෇࿥_{1ɿMeyer ΢ΣʔϒϨοτΛ༻͍ͨ LS ϑΟϧλ} s1(z) = +2.00247847229z0+2.00247847229z−1 t1(z) = −0.08779405478z1− 0.08779405478z0 s2(z) = −4.53805587250z0− 4.53805587250z−1 t2(z) = −3.86333490297z1− 3.86333490297z0 s3(z) = +0.03725674177z0+0.03725674177z−1 t3(z) = +6.80946801390z1+6.80946801390z0 s4(z) = +3.72181356513z0+3.72181356513z−1 t4(z) = −0.02496044226z1− 0.02496044226z0 s5(z) = −4.46467530810z0− 4.46467530810z−1 t5(z) = −0.67170824688z1− 0.67170824688z0 s6(z) = +1.31190100289z0+1.31190100289z−1 t6(z) = −0.63828208542z1− 0.63828208542z0 s7(z) = +2.91362991601z0+2.91362991601z−1 t7(z) = −0.12833350788z1− 0.12833350788z0 s8(z) = −14.48542806103z0− 14.48542806103z−1 t8(z) = +0.08129179823z1+0.08129179823z0 s9(z) = +2.64477316058z0+2.64477316058z−1 t9(z) = −0.04596808984z1− 0.04596808984z0 s10(z) = −5.78257467171z0− 5.78257467171z−1 t10(z) = −0.00798136961z1− 0.00798136961z0 s11(z) = +3.00763739414z0+3.00763739414z−1 t11(z) = +0.05844873839z1+0.05844873839z0 s12(z) = −3.48946828568z0− 3.48946828568z−1 t12(z) = +9.71773389036z1+9.71773389036z0 s13(z) = +0.00012012804z0+0.00012012804z−1

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t13(z) = −9.78037754204z1− 9.78037754204z0 s14(z) = +23.57098211188z0+23.57098211188z−1 t14(z) = +0.03574644724z1+0.03574644724z0 s15(z) = −7.25785115300z0− 7.25785115300z−1 t15(z) = +0.51053703949z1+0.51053703949z0 s16(z) = −1.02421210881z0− 1.02421210881z−1 t16(z) = +2.93591247114z1+2.93591247114z0 s17(z) = −0.06918495722z0− 0.06918495722z−1 t17(z) = −7.69684792544z1− 7.69684792544z0 s18(z) = −0.20679957193z0− 0.20679957193z−1 t18(z) = +0.92938362903z1+0.92938362903z0 s19(z) = +1.17225047445z0+1.17225047445z−1 t19(z) = +1.11356797980z1+1.11356797980z0 s20(z) = −0.35378838790z0− 0.35378838790z−1 t20(z) = +2.39426562661z1+2.39426562661z0 slast(z) = −0.00002130607z15+0.00004012566z14 −0.00003582645z13− 0.00003622490z12 +0.00005949173z11_{− 0.00013141802z}10 −0.00038083269z9+0.00015217532z8 −0.00227155702z7− 0.00275497090z6 −0.00622708354z5− 0.02963239525z4 −0.03680761783z3− 0.16933627095z2 −0.35045784078z1_{− 0.62110464843z}0 −0.62110464843z−1_{− 0.35045784078z}−2 −0.16933627095z−3_{− 0.03680761783z}−4 −0.02963239525z−5_{− 0.00622708354z}−6 −0.00275497090z−7_{− 0.00227155702z}−8 +0.00015217532z−9_{− 0.00038083269z}−10 −0.00013141802z−11₊_{0.00005949173z}−12 −0.00003622490z−13− 0.00003582645z−14 +0.00004012566z−15_{− 0.00002130607z}−16 K = 1.00000740582550530 ʢຊ _{LS ϑΟϧλ͸ Meyer ΢ΣʔϒϨοτΛ༻͍ͨ} CDWT ͷ࣮਺෦ͷ LS ϑΟϧλͱͯ͠΋࢖༻Մೳʣ ෇࿥_{2ɿMeyer ΢ΣʔϒϨοτΛ༻͍ͨ CDWT ͷڏ} ਺෦ͷ_{LS ϑΟϧλ} s1(z) = −0.25710679074z0− 1.29766364230z−1 t1(z) = +2.33574383037z1− 0.58421729627z0 s2(z) = −0.25640585088z0− 0.50431445778z−1 t2(z) = −2.84457925454z1+3.79643401033z0 s3(z) = −1.17399580040z0+0.33324315269z−1 t3(z) = +0.47006334034z1+0.84057384094z0 s4(z) = +1.65075672465z0− 1.92536744741z−1 t4(z) = −2.92238700670z1− 0.38293607267z0 s5(z) = −0.02467553907z0+0.39366032919z−1 t5(z) = −7.41921559107z1− 20.23717020390z0 s6(z) = +0.04773116791z0− 0.03529445641z−1 t6(z) = +25.49461261726z1− 67.06989055964z0 s7(z) = +0.01359865917z0+0.02155762328z−1 t7(z) = −48.78040385415z1+9.36003380121z0 s8(z) = −0.09056712577z0+0.03182350151z−1 t8(z) = −9.68231428365z1+14.90595107462z0 s9(z) = −0.04664019991z0− 0.17726437150z−1 t9(z) = +5.30975413052z1− 0.46275716954z0 s10(z) = +10.60459354726z0− 0.28639353192z−1 t10(z) = +0.01381212224z1− 0.05159830891z0 s11(z) = −0.01579589881z0+0.63777857268z−1 t11(z) = −0.01374609415z1− 0.07637878930z0 s12(z) = +28.84626727597z0+11.30022292882z−1 t12(z) = −0.09816464312z1+0.05859163785z0 s13(z) = +16.10217771738z0+3.69224972760z−1 t13(z) = +0.00720456878z1− 0.07910575545z0 s14(z) = +20.38005799247z0− 8.40043059030z−1 t14(z) = +0.10923353069z1− 0.04426219974z0 s15(z) = +11.56955109188z0− 18.68976073532z−1 t15(z) = +0.04109966416z1+1.42679139608z0 s16(z) = −0.69903849147z0+0.06252848967z−1 t16(z) = −12.87180822995z1− 24.70130189154z0 s17(z) = +0.05912367872z0− 0.00845013057z−1 t17(z) = −21.71180235612z1− 31.93125335742z0

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s18(z) = −0.02632872404z0+0.03783033697z−1 t18(z) = −93.40273410962z1+36.76521280538z0 s19(z) = +0.01057512468z0+0.01030580009z−1 t19(z) = +48.00255983869z1− 108.38776448769z0 s20(z) = +0.01712068495z0− 0.06094994132z−1 t20(z) = +16.38601960719z1+2.69321466514z0 slast(z) = −0.00001804020z15+0.00002181573z14 +0.00000158460z13_{− 0.00004754087z}12 +0.00007220613z11_{− 0.00000105758z}10 −0.00021662646z9+0.00051052970z8 −0.00058673477z7− 0.00011108927z6 +0.00229237393z5_{− 0.00648090587z}4 +0.01240972578z3− 0.01883676762z2 +0.03816188887z1_{− 0.12977677491z}0 +0.01832324917z−1₊_{0.01420366593z}−2 −0.01303383713z−3₊_{0.00722476667z}−4 −0.00267998001z−5₊_{0.00026239509z}−6 +0.00054577474z−7_{− 0.00051047701z}−8 +0.00022719602z−9_{− 0.00000945025z}−10 −0.00006436391z−11+0.00004228038z−12 +0.00000177148z−13_{− 0.00002389812z}−14 +0.00001931580z−15 K = 0.77981132999328650 ෇࿥_{3ɿDaubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ LS ϑΟ} ϧλ s1(z) = 0 (লུՄೳ) t1(z) = −0.22550617850z0 s2(z) = −0.72734207431z−1 t2(z) = −1.48306856670z0 s3(z) = −4.88162027275z−1 t3(z) = +0.34080609530z0 s4(z) = +0.00073353405z−1 t4(z) = −0.34021921990z0 s5(z) = +4.94624866289z−1 t5(z) = +1.98801890409z0 s6(z) = +2.34091715249z−1 t6(z) = +4.15443726440z0 slast(z) = +0.00012016296z5− 0.00087097335z4 +0.00303911366z3_{− 0.00786344499z}2 +0.02475081739z1_{− 0.24070648715z}0 K = 0.11154074335000000 ʢຊ_{LS ϑΟϧλ͸ Daubechies 6 ΢ΣʔϒϨοτΛ༻͍} ͨCDWT ͷ࣮਺෦ͷ LS ϑΟϧλͱͯ͠΋࢖༻Մೳʣ ෇࿥_{4ɿDaubechies 6 ΢ΣʔϒϨοτΛ༻͍ͨ CDWT} ͷڏ਺෦ͷ_{LS ϑΟϧλ} s1(z) = −0.46183155518z−1 t1(z) = +1.00500330626z0 s2(z) = +0.03278242130z−1 t2(z) = −1.21534677122z0 s3(z) = −0.21829933804z−1 t3(z) = −0.51705587237z0 s4(z) = +1.09129594421z−1 t4(z) = −0.57710977201z0 s5(z) = −0.07997480461z−1 t5(z) = +0.35683913342z0 s6(z) = −0.83732987193z0+1.06413198165z−1 t6(z) = +0.68223644021z1+0.08277305419z0 s7(z) = −0.02877340974z0− 0.83205963954z−1 t7(z) = −0.09124110438z1+3.12478895787z0 s8(z) = +0.03656378877z0− 1.01367537250z−1 t8(z) = +4.88634193788z1+26.44342151768z0 slast(z) = −0.00000223627z6+0.00003372437z5 −0.00015849363z4+0.00045217157z3 −0.00118402540z2+0.00561204977z1 −0.03780968719z0+0.00000088578z−1 K = −0.03379964546560699