方向性ウェーブレッ卜変換の提案と医用画像処理応用

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(1)情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. ํ޲ੑ΢ΣʔϒϨοτม‫׵‬ͷఏҊͱҩ༻ը૾ॲཧԠ༻ Ճ౻ɹ‫ؽ‬1,a). ষɹ஧1. ‫ాށ‬ɹߒ1. ࠓଜɹ޹2. ࡾ୐ɹ఩෉1. ੴ઒ɹ߁޺3. ֓ཁɿຊ࿦จͰ͸ɼෳૉ਺཭ࢄ΢ΣʔϒϨοτม‫( ׵‬CDWT) Λ‫ʹج‬ɼ৽‫ن‬ͷํ޲ੑ΢ΣʔϒϨοτม‫׵‬Λ ఏҊ͢Δɽ΢ΣʔϒϨοτม‫׵‬ͷҰͭͰ͋Δ CDWT ͸ɼը૾ͷۭؒप೾਺ղੳ͕ՄೳͰ͋Γɼը૾ॲཧ ख๏ͱͯ͠޿͘༻͍ΒΕ͖ͯͨɽCDWT ͸ը૾Λํ޲੒෼ʹ෼ղ͠ɼ‫ڥ‬քઢ΍ෆ࿈ଓͳઢΛํ޲ผʹଊ͑ Δࣄ͕ՄೳͰ͋Δɽ͜ͷ‫ػ‬ೳ͸ํ޲બ୒ੑͱ‫ݺ‬͹Εɼ‫ܗ‬ঢ়৘ใ΍‫ز‬Կֶಛ௃ͷநग़ʹ‫ظ‬଴ग़དྷΔɽ͔͠͠ɼ CDWT ͷํ޲બ୒ੑͰಘΒΕΔํ޲੒෼͸ 6 ํ޲ʹ‫ݶ‬ఆ͞Ε͓Γɼे෼ͳը૾ಛ௃Λఏ‫͢ڙ‬Δͱ͸‫ͳ͑ݴ‬ ͍ɽͦ͜Ͱɼଟ͘ͷํ޲ੑಛ௃Λநग़Մೳͳํ޲ੑϑΟϧλΛઃ‫͠ܭ‬ɼैདྷͷ CDWT ͱ૊Έ߹Θͤͨํ ޲ੑ΢ΣʔϒϨοτม‫׵‬ΛఏҊͨ͠ɽͦͯ͠ɼఏҊख๏Λҩ༻ը૾ॲཧʹԠ༻͠ɼͦͷ༗ޮੑΛ֬ೝͨ͠ɽ Ωʔϫʔυɿෳૉ਺཭ࢄ΢ΣʔϒϨοτม‫׵‬ɼํ޲બ୒ੑɼۭؒप೾਺ղੳɼҩ༻ը૾ॲཧ. The Directional Wavelet Transform and its Medical Image Processing Application Kato Takeshi1,a). Zhong Zhang1. Toda Hiroshi1 Imamura Takashi2 Ishikawa Yasuhiro3. Miyake Tetsuo1. Abstract: This paper proposes the novel directional wavelet transform based on the complex discrete wavelet transform(CDWT). The complex discrete wavelet transform is one of the wavelet transform and the CDWT is known as the space-frequency analysis method and image processing method. The CDWT decomposes an image into directional components and discontinuity lines and singularities are detected. This property is called Directional Selection and expected as geometrical feature and shape information extraction method. However, directional selection offers only 6 directional components and we can not obtain the enough image directional features. Thus, we design the directional filters that can detect various directional features and we combine the previous CDWT and designed filters. Finally, we apply the proposed method to medical image processing and we consider the validation of proposed method. Keywords: Complex Discrete Wavelet Transform, Directional Selection, Space-Frequency Analysis, Medical Image Processing.. 1. ͸͡Ίʹ. ख๏ͱͯ͠޿͘༻͍ΒΕ͖ͯͨɽ৴߸ɾը૾ʹ DWT Λద ༻͢Δ৔߹ɼMallat[1] ͕ఏҊͨ͠ଟॏղ૾౓ղੳ (Multi. ैདྷɼ2 ࣍‫ࢄ཭ݩ‬΢ΣʔϒϨοτม‫( ׵‬2-Dimensional. Resolution Analysis, MRA) ͷߴ଎ΞϧΰϦζϜΛ༻͍. Discrete Wavelet Transform, 2D-DWT) ͸৴߸ɾը૾ॲཧ. ͨ཭ࢄ΢ΣʔϒϨοτม‫( ׵‬Discrete Wavelet Transform,. 1. 2 3 a). DWT) ͕ଟ͘༻͍ΒΕΔɽ͜Ε͸ɼϑΟϧλϦϯάͱμ΢ ๛‫ٕڮ‬ज़Պֶେֶ 1-1 Hibarigaoka, Tempaku, Toyohashi, Aichi, 441-8156, Japan ৽ׁେֶ 8050, Ikarashi 2-no-cho, Nishi, Niigata, 950-2181, Japan ੴ઒ҩӃ 1-3-16, Honchonishi, Chuo, Saitama, Saitama, Japan [email protected]. ⓒ 2014 Information Processing Society of Japan. ϯαϯϓϦϯάΛซ༻͢ΔͨΊɼߴ଎ॲཧ͕ՄೳͰ͋Γɼ ҰൠతͳϑΟϧλόϯΫͱൺ΂ɼ‫͕ྔࢉܭ‬গͳ͍ͱ͍͏ར ఺Λ࣋ͭɽϑΟϧλϦϯάʹΑΓɼप೾਺੒෼ʹ૬౰͢Δ ΢ΣʔϒϨοτ܎਺͕ಘΒΕΔɽ2D-DWT Ͱ͸ϑΟϧλ ॲཧ͔Βɼ1 ͭͷ௿प೾੒෼ͱ 3 ͭͷߴप೾੒෼Λ‫͢ࢉܭ‬. 1.

(2) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. Δɽ௿प೾੒෼͸ɼฏ‫׈‬Խͨ͠ը૾͕ಘΒΕɼҰํͰ 3 ͭ. Ԡ༻͢Δɽ‫ڳ‬෦ CT ը૾ʹ͸ɼഏ಺෦ʹ͋Δഏ͕Μ౳ͷज. ͷߴप೾੒෼͸ɼํ޲ੑΤοδΛ‫ݕ‬ग़ͨ͠ը૾͕ಘΒΕΔɽ. ᚅੑපม͕өΓࠐΈɼҩࢣ͕਍அʹॏཁͳ৘ใΛಘΔɽ·. ͜͜Ͱํ޲ੑΤοδͱ͸ը૾ͷ 2 ࣍‫ݩ‬ฏ໘಺Ͱɼಛఆํ޲. ͨۙ೥ɼϚϧνεϥΠε CT(Computed Tomography) ͷ. ͷ‫ڥ‬քઢ΍ෆ࿈ଓͳઢͰ͋Δɽ2D-DWT Ͱ͸ɼਫฏɼਨ. ී‫ٴ‬΍༻్ͷ֦େ͔Βɼҩࢣ͕େྔͷ CT ը૾ΛಡӨ͠ͳ. ௚ɼର֯ํ޲ͷํ޲ੑΤοδ͕ಘΒΕΔͨΊɼैདྷ͸ը૾. ͚Ε͹ͳΒͣɼෛ୲૿΍‫ޡ‬਍ͷ૿Ճ͕‫ة‬ዧ͞Ε͍ͯΔɽͦ. ͷಛ௃நग़ख๏ͱͯ͠ར༻͞Ε͖ͯͨɽ. ͜Ͱɼजᚅ෦ҐΛը૾ॲཧʹΑΓɼࣗಈͰೝࣝ͢Δը૾਍. ͔͠͠ɼैདྷͷ MRA Λ༻͍ͨ 2D-DWT Ͱ͸ɼ‫ݕ‬ग़ͨ͠. அ͕‫ظ‬଴͞Ε͍ͯΔɽຊ‫Ͱڀݚ‬͸ɼํ޲ੑ΢ΣʔϒϨοτ. ը૾ͷಛ௃ͷҐஔʹΑͬͯղੳ݁Ռʹ͹Β͖͕ͭੜ͡ɼ࣌ෆ. ม‫͔׵‬Βಘͨಛ௃Λ‫ʹج‬ɼजᚅ෦ҐͷೝࣝՄೳ͔Λ‫ݕ‬౼. มੑΛ࣋ͨͳ͍ɽͦͷͨΊɼ‫ͳ݈ؤ‬ը૾ॲཧ͕ࠔ೉Ͱ͋ͬ. ͢Δɽ. ͨɽ͜ͷ໰୊ʹରͯ͠ɼ‫ాށ‬Β [2] ͸‫׬‬શγϑτෆมෳૉ ਺཭ࢄ΢ΣʔϒϨοτม‫( ׵‬Perfect Translation Invariance. Complex Discrete Wavelet Transform, CDWT) ΛఏҊ͠ ͍ͯΔɽCDWT ͸ଞʹ΋ Selesnick Β [5] ͕ఏҊ͢Δ΋ͷ. 2. 2 ࣍‫ݩ‬ෳૉ਺཭ࢄ΢ΣʔϒϨοτม‫׵‬ͷۛຯ 2.1 2 ࣍‫ݩ‬ෳૉ਺཭ࢄ΢ΣʔϒϨοτม‫׵‬ͷ‫ࢉܭ‬ ैདྷͷ DWT ΍ 2D-DWT ʹ͸ɼγϑτෆมੑͷܽ೗ͱ. ΋͋Δ͕ɼຊ࿦จͰ͸͜ΕΒΛ૯শͯ͠ɼCDWT ͱ͢Δɽ. ‫ݺ‬͹ΕΔऑ఺ɼ͢ͳΘͪը૾ͷಛ௃ͷҐஔΑͬͯม‫݁׵‬. CDWT ͸৴߸ͷҐ૬ʹΑΒͣɼ‫ͳ݈ؤ‬ղੳ͕ՄೳͰ͋Δ. Ռ͕ҟͳΓɼ‫ͳ݈ؤ‬ը૾ॲཧ͕ࠔ೉ͱͳΔऑ఺͕͋ͬͨɽ. ͱ͍͏େ͖ͳར఺Λ࣋ͭɽ. ‫ాށ‬Β [2] ͷఏҊ͢Δ CDWT ͸ɼMeyer ͷ௚ަ΢Σʔϒ. ͞ΒʹɼKingsbury Β͸ɼCDWT Λ 2 ࣍‫֦ʹݩ‬ுͨ͠ 2. ϨοτΛ‫ʹج‬ઃ‫͞ܭ‬Ε͓ͯΓɼ‫׬‬શγϑτෆมੑΛ࣮‫͠ݱ‬. ࣍‫ݩ‬ෳૉ਺཭ࢄ΢ΣʔϒϨοτม‫( ׵‬2-Dimensional Com-. ͍ͯΔɽCDWT ͸࣮਺෦ͱ‫਺ڏ‬෦ʹ෼͔Εͨ 2 ͭͷ௚ަ. plex Discrete Wavelet Transform, 2D-CDWT) ʹΑͬͯಘ. ΢ΣʔϒϨοτʹΑΓߏ੒͞ΕΔͷ͕ಛ௃Ͱ͋Δɽ͢ͳ. ΒΕ࣮ͨ਺෦ͱ‫਺ڏ‬෦ͷ΢ΣʔϒϨοτ܎਺ʹ࿨ͱࠩͷ‫ܭ‬. ΘͪεέʔϦϯάؔ਺͸ɼ࣮਺෦ͱ‫਺ڏ‬෦ͷεέʔϦϯ. ࢉΛద༻͠ɼը૾ͷํ޲੒෼Λ‫͢ࢉܭ‬Δख๏ΛఏҊͨ͠ɽ. άؔ਺ φR (x)ɼφI (x) ͕͋Γɼ·ͨϚβʔ΢ΣʔϒϨοτ. ͜Ε͸ CDWT ͷํ޲બ୒ੑͱ‫ݺ‬͹ΕΔɽ͜Ε͸࣮਺෦ͱ. (Mother WaveletɼMW) ΋ಉ͘͡ɼ࣮਺෦ɼ‫਺ڏ‬෦ͷ MW. ‫਺ڏ‬෦ͷҐ૬ࠩʹΑͬͯɼ࣮਺෦ͱ‫਺ڏ‬෦ͷ΢ΣʔϒϨο. ψ R (x)ɼψ I (x) ͕͋Δɽͳ͓ MRA ͷߴ଎ΞϧΰϦζϜʹ. τ܎਺͕‫ׯ‬ব͍͋͠ɼಛఆํ޲ͷ೾‫ܗ‬ͷΈ͕‫ڧ‬ௐ͞ΕΔͨ. ༻͍Δ࣮਺෦ͷϩʔύεɾϋΠύεϑΟϧλͷϑΟϧλ܎. ΊͰ͋Δ [3]ɽ. R I I ਺Λ {aR k }ɼ{bk }ɼ·ͨ‫਺ڏ‬෦ͷͦΕΒΛ {ak }ɼ{bk } ͱ͢. ͜ΕΒͷํ޲੒෼͸ෳૉ਺ͷϑΟϧλ͕͔͚ΒΕΔͨΊɼ. ΔɽҎ্ͷΑ͏ͳ CDWT Λ 2 ࣍‫֦ʹݩ‬ுͨ͠ 2 ࣍‫ݩ‬ෳૉ. ෳૉ਺஋Ͱ͋Γɼͦͷઈର஋Λ‫͖Ͱࢉܭ‬Δɽํ޲੒෼ͷઈର. ਺཭ࢄ΢ΣʔϒϨοτม‫׵‬ʢ2D Complex Discrete Wavelet. ஋Λ AVDC(Absolute Values of Directional Components). Transform, 2D-CDWTʣʹ͍ͭͯड़΂Δɽ. ͱ‫͢ͱͱ͜Ϳݺ‬Δ. AVDC ͸ɼߴप೾੒෼͔Β‫͞ࢉܭ‬Εɼ ͦΕΒ͸ํ޲ੑΤοδΛ‫ݕ‬஌͢ΔͨΊɼAVDC ΋ಉ༷ʹը. 2D-CDWT Ͱ͸ɼ࣮਺෦ɾ‫਺ڏ‬෦ͷεέʔϦϯάؔ਺ͱ MW ͕ɼ2 ࣍‫ݩ‬ͷ৴߸ f (x, y) Λల։͢Δɽ. ૾ͷํ޲ੑΤοδΛநग़ՄೳͰ͋Δɽ2D-CDWT Ͱ͸ɼ6. ͦΕʹ͸ɼ·ͣ࠷ॳʹεέʔϦϯάؔ਺Λ༻͍ͯิؒ. ํ޲ͷ AVDC Λ‫ࢉܭ‬ՄೳͰ͋Δɽ͜Ε͸‫ޙ‬ड़͢Δ͕ɼํ޲. Λߦ͏ඞཁ͕͋Γɼ͜ͷ࡞‫ۀ‬͸ࣜ (1) Ͱද͞Εɼྫ͑͹. બ୒ੑ͕ 3 ͭͷߴप೾੒෼͔ΒͦΕͧΕ 2 छྨͷํ޲੒෼. RI(x, y) ͷ߲͸ԣํ޲ʢx ࣠ʣʹ࣮਺෦ɼॎํ޲ʢy ࣠ʣʹ. Λ‫ࢉܭ‬ՄೳͳͨΊͰ͋Δɽ͔͠͠ɼը૾ॲཧ΍ಛ௃நग़ɼ. ‫਺ڏ‬෦ͷॲཧΛ͢Δؔ਺Ͱɼࣜ (2) ͷதͷؔ਺ RR(x, y) ͷ. ը૾ೝࣝ౳ͷଟ͘ͷԠ༻ʹର͠ɼCDWT Λߟྀͨ͠৔߹ɼ. Α͏ʹද͞ΕΔɽͳ͓ؔ਺ RI(x, y), IR(x, y), II(x, y) ʹ. 6 ํ޲ͷํ޲ੑΤοδ͸े෼͸ը૾ͷ‫ز‬Կֶಛ௃Λఏ‫͠ڙ‬. ؔͯ͠΋ಉ༷ʹද͞ΕΔɽ. ͍ͯΔͱ͸‫͍ͳ͑ݴ‬ɽ ؔ࿈ख๏ͱͯ͠ɼ఻౷తͳΨϘʔϧϑΟϧλ΍ํ޲ੑ. f (x, y) = RR(x, y)+RI(x, y)+IR(x, y)+II(x, y), (1). ϑΟϧλόϯΫΛ༻͍ͨख๏͕͋ΔɽΨϘʔϧϑΟϧλ͸ɼ. 2D-CDWT ΑΓ΋ଟ͘ͷํ޲ੑಛ௃ΛಘΒΕΔ͕ɼμ΢ϯ αϯϓϦϯά͸ద༻ग़དྷͳ͍ͨΊ‫͕ྔࢉܭ‬ଟ͍ɽ·ͨํ޲. RI(x, y) =. . R I cRI 0,kx ,ky φ (x − kx )φ (y − ky ).. (2). kx ,ky. ੑϑΟϧλόϯΫ͸‫ྔࢉܭ‬͸গͳ͍͕ɼγϑτෆมੑΛ࣋. ࣜ (2) ͷதͷ܎਺ cRI 0,kx ,ky Ͱ͋Δ͕ɼ͜ΕΒ͸ࣜ (1) ͷ. ͨͳ͍ɽ·ͨۙ೥Ͱ͸ΧϯλʔϨοτม‫׵‬΍ΧʔϒϨοτ. f (x, y) ͕ɼղੳͷର৅ͱͳΔ 2 ࣍‫ݩ‬ͷ཭ࢄ৴߸ fnx ,ny Λ. ม‫׵‬౳΋ఏҊ͞Ε͍ͯΔ͕ɼγϑτෆมੑΛ࣋ͪͭͭɼଟ. ิؒ͢ΔΑ͏ʹઃఆ͢Δඞཁ͕͋Δɽ͜ΕΒͷ܎਺͸ɼղ. ͘ͷํ޲ੑಛ௃ΛಘΒΕΔख๏͸ະͩগͳ͍ɽຊ‫Ͱڀݚ‬͸ɼ. ੳͷର৅ͱͳΔ 2 ࣍‫ݩ‬ͷ཭ࢄ৴߸ fnx ,ny Λ༻͍ͯҎԼͷ. ৽ͨͳํ޲ੑϑΟϧλͷઃ‫ํܭ‬๏ΛఏҊ͠ɼ2D-CDWT ͱ. ࣜ (3) ΑΓ‫ٻ‬ΊΒΕΔ͜ͱ͕ɼจ‫[ ݙ‬2] ΑΓಋ͔ΕΔ ʢจ. ૊Έ߹Θͤ৽ͨͳํ޲ੑ΢ΣʔϒϨοτม‫׵‬ΛఏҊ͢Δɽ. ‫[ ݙ‬2] Ͱ͸ɼCDWT ʹΑΔ 1 ࣍‫ݩ‬৴߸ͷิؒ๏͕ఏҊ͞Ε. ·ͨɼఏҊख๏ͷ༗ޮੑΛ‫ݕ‬౼͢ΔͨΊɼ‫ڳ‬෦ CT ը૾΁. ͍ͯΔ͕ɼ͜ΕΛ 2 ࣍‫֦ʹݩ‬ு͢Δͱࣜ (3) ͕ಋ͔ΕΔʣɽ. ⓒ 2014 Information Processing Society of Japan. 2.

(3) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. 1 fnx −kx ,ny −ky φR (−kx )φI (−ky ). 4. cRI 0,nx ,ny =. (3). kx ,ky. ଞͷؔ਺ RR(x, y)ɼIR(x, y), II(x, y) ͷ܎਺ cRR 0,kx ,ky ,. II cIR 0,kx ,ky ɼc0,kx ,ky ʹؔͯ͠΋ಉ༷ʹ‫ٻ‬ΊΒΕΔɽͦͯ͠ɼ͜. ΕΒͷ܎਺ʹ͸ MW ͱεέʔϦϯάؔ਺͔Βܾఆ͞ΕΔ ϋΠύεɾϩʔύεϑΟϧλ͕ద༻͞Εɼ֤प೾਺੒෼΁ ෼ղ͞ΕΔɽྫͱͯ͠ cRI j+1,kx ,ky ͷ෼ղΛࣜ (4)-(7) ʹࣔ͢ ʢj ͸෼ղϨϕϧͰ͋Γɼ੔਺Ͱද͞ΕΔɽྫ͑͹ j = −1 ͱ͢Ε͹ cRI ɽ 0,kx ,ky ͔Β෼ղϨϕϧ −1 ΁ͷ෼ղͱͳΔʣ RI R I RI cj,nx ,ny = a2nx −kx a2ny −ky cj+1,kx ,ky , (4) kx ,ky. dRI,LH j,nx ,ny dRI,HL j,nx ,ny. . =. I RI aR 2nx −kx b2ny −ky cj+1,kx ,ky ,. (5). =. I RI bR 2nx −kx a2ny −ky cj+1,kx ,ky ,. (6). dRI,HH j,nx ,ny =. 0,LH R0,LH I0,LH Dj,nx ,ny = (Dj,nx ,ny )2 + (Dj,nx ,ny )2 , 1,LH R1,LH I1,LH Dj,nx ,ny = (Dj,nx ,ny )2 + (Dj,nx ,ny )2 .. (11). (12). (13). I RI bR 2nx −kx b2ny −ky cj+1,kx ,ky .. (7). ਺੒෼ʹؔ͢Δ‫ࢉܭ‬΋ಉ༷ʹͯ͠ߦ͏ɽྫͱͯ͠ਤ 1 ͷೖ ྗը૾ʹରͯ͠ 2D-CDWT Λద༻͠ɼಘΒΕͨ dRI,LH j,kx,ky ౳. kx ,ky. . 1 RR,LH + dII,LH (d j,nx ,ny ), 2 j,nx ,ny 1 I0,LH Dj,n = (dIR,LH − dRI,LH j,nx ,ny ), x ,ny 2 j,nx ,ny 1 R1,LH Dj,n = (dRR,LH − dII,LH j,nx ,ny ), x ,ny 2 j,nx ,ny 1 I1,LH Dj,n = (dIR,LH + dRI,LH j,nx ,ny ). x ,ny 2 j,nx ,ny R0,LH Dj,n = x ,ny. ͨͩࣜ͠ (11)ʙ(13) ͸ɼLH ੒෼ʹؔ͢ΔࣜͰɼଞͷप೾. kx ,ky. . Λద༻͠ɼը૾ͷํ޲੒෼ΛಘΔख๏ΛఏҊ͍ͯ͠Δɽ. kx ,ky. 0,LH. ʹରͯ͠ɼࣜ (11)ʙ(13) ͷ‫ࢉܭ‬Λߦ͍ɼಘΒΕͨ |Dj,nx ,ny | ౳Λਤ 2. 0,LH ʹࣔ͢ɽͳ͓ɼ|Dj,n | x ,ny. ౳͕ AVDC ͱͳΔɽ·. ౳͸εέʔϦϯά܎਺ɼ·ͨ dRI,LH j,nx ,ny. ͨɼਤ 2 ͷ݁Ռ͸ɼਤ 3 ʹैͬͯ഑ஔͨ͠΋ͷͰ͋Δɽਤ. ౳͸΢ΣʔϒϨοτ܎਺Λࣔ͢ɽࣜ (4) ͔ΒಘΒΕͨ௿प. 2 ͔Β΋෼͔ΔΑ͏ʹɼ2D-CDWT ͔ΒಘΒΕͨߴप೾੒. ࣜ (4)-(7) ͷ cRI j,nx ,ny ೾੒෼ͷ. cRI j,nx ,ny. Λɼ࠶ͼࣜ (4)-(7) ͷ. cRI j+1,kx ,ky. ʹ୅ೖ͠ɼ. ෼ͷ΢ΣʔϒϨοτ܎਺Λ༻͍ͯɼೖྗը૾ͷന͍ԁͷྠ. ࠶‫ؼ‬తʹϑΟϧλΛద༻͢ΔɽҎ্ͷΑ͏ͳϑΟϧλϦϯ. ֲΛ 6 ํ޲ʹ෼͚ͯ‫ݕ‬ग़Ͱ͖Δɽ·ͨɼಉ༷ͷ‫ʹࢉܭ‬Πϯ. άʹΑΓ֤प೾਺੒෼ͷ΢ΣʔϒϨοτ܎਺Λ‫͢ࢉܭ‬Δ. ύϧε৴߸Λೖྗͨ݁͠ՌΛਤ 4 ʹࣔ͢ɽਤ 4 ͔Β΋ํ޲. ͱɼRI(x, y) ͸ࣜ (8) ʹల։͞ΕΔɽ R I RI(x, y) = cRI J,kx ,ky φJ,kx (x)φJ,ky (y). ੑΛ࣋ͬͨ 6 ํ޲ͷ೾‫ܗ‬ͷߏ੒͕֬ೝͰ͖Δɽ. 3. ৽ͨͳํ޲ੑϑΟϧλͷઃ‫ܭ‬. kx ,ky. +. −1 j=J kx ,ky. +. −1 . . j=J kx ,ky. +. −1 j=J kx ,ky. લઅͰ͸ɼ2D-CDWT ͷ‫ͱࢉܭ‬ɼͦͷํ޲બ୒ੑʹ͍ͭ. R I dLH,RI j,kx ,ky φj,kx (x)ψj,ky (y). ͯड़΂ͨɽຊઅ͓Αͼ࣍અʹͯɼΑΓଟ͘ͷํ޲બ୒ੑΛ ಘΔख๏ΛఏҊ͢ΔɽॳΊʹຊઅͰ͸ɼଟ͘ͷ AVDC ΍ํ. R I dHL,RI j,kx ,ky ψj,kx (x)φj,ky (y). R I dHH,RI j,kx ,ky ψj,kx (x)ψj,ky (y).. ޲ੑΤοδΛநग़͢ΔͨΊͷํ޲ੑϑΟϧλΛઃ‫͢ܭ‬Δɽ. (8). ͨͩࣜ͠ (8) ͸෼ղϨϕϧ −1 ͔Β J ʢJ ͸ෛͷ੔਺ʣ· Ͱͷม‫͋Ͱ׵‬Γɼ֤ϨϕϧͷεέʔϦϯάؔ਺ɼ΢Σʔϒ Ϩοτ͸ҎԼͷΑ͏ʹද͞ΕΔɽ √ j R j φR 2 φ (2 x − k). j,k (x) = √ j R ψj,k (x) = 2 ψ R (2j x − k).. (9) (10). ࣜ (9), (10) ͸࣮਺෦ͷΈΛ͍ࣔͯ͠Δ͕‫਺ڏ‬෦΋ಉ༷ͷ ؔ܎Λ࣋ͭɽͳ͓ࣜ (1) ͷதͷؔ਺ RR(x, y), IR(x, y),. II(x, y) ʹؔͯ͠΋ɼࣜ (8) ͷ RI(x, y) ͱಉ͡Α͏ʹ෼ղ. 3.1 ํ޲બ୒ੑͱͦͷप೾਺ಛੑ ํ޲ੑϑΟϧλͷઃ‫ܭ‬ͷ‫ݕૅج‬౼ͱͯ͠ɼํ޲બ୒ੑʹ ͓͚Δ MW ͱͦͷप೾਺ಛੑʹ͍ͭͯ‫ݕ‬౼͢Δɽͳ͓ɼຊ ‫Ͱڀݚ‬͸ը૾ΛϑʔϦΤม‫͠׵‬ɼಘΒΕͨৼ෯Λप೾਺ಛ ੑͱ‫͢ͱͱ͜Ϳݺ‬Δɽࣜ (11)ɼ(12) ͷதͷ RRʙII ͷఴ͑ ࣈ͕෇͍ͨ܎਺ dRR,HH j,nx ,ny ౳͸΢ΣʔϒϨοτ܎਺Ͱ͋Δɽ. 2D-CDWT ʹ͓͍ͯߴ଎ΞϧΰϦζϜΛ༻͍Δ৔߹ɼ͜Ε Βͷ܎਺͸ࣜ (4)-(7) ʹࣔ͢Α͏ͳɼμ΢ϯαϯϓϦϯά Λ൐͏ϑΟϧλϦϯάʹΑͬͯ‫ٻ‬ΊΒΕΔ͕ɼࣜ (14) ʹࣔ ͢Α͏ʹɼೖྗը૾ͱ MW ͱͷ಺ੵͷԋࢉʹΑͬͯ΋‫ܭ‬ ࢉՄೳͰ͋Δɽ. ͞Εɼ࠷ऴతʹࣜ (1) ͷ f (x, y) ͸֤߲ͷ֤प೾਺੒෼ͷ࿨ RR,LH dRR,LH j,nx ,ny = f, ψj,nx ,ny ,. Ͱද͞ΕΔɽ. RR,LH ψj,n (x, y) x ,ny. 2.2 2D-CDWT ͷํ޲બ୒ੑ Kingsbury Β [4] ͸ϑΟϧλॲཧʹΑΓಘΒΕͨߴप೾ ੒෼ͷ΢ΣʔϒϨοτ܎਺ʹɼҎԼͷࣜ (11)ʙ(13) ͷ‫ࢉܭ‬. ⓒ 2014 Information Processing Society of Japan. =. (14). R φR j,nx (x)ψj,ny (y). RR,LH ͨͩ͠ < f, ψj,n > ౳͸ 2 ࣍ฏ໘্ʹఆٛ͞Εͨؔ x ,ny. RR,LH ਺ f (x, y)ɼψj,n (x, y) ͷ಺ੵΛද͠ɼ࣍ͷΑ͏ʹ‫͞ࢉܭ‬ x ,ny. ΕΔɽ. 3.

(4) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. ߱௬. , . ௬ ௫. ߱௫. , ਤ 1. ೖྗը૾. Fig. 1 Input image. level -1. level -2 … LL … level -2. level -1. (a). -75. 75. (b). 5 MW ͷ प ೾ ਺ ಛ ੑ (ϑ ʔ Ϧ Τ ม ‫ ͯ ͠ ׵‬ಘ ͨ ৼ ෯).(a)ψˆR0,HH (ωx , ωy )(b)ψˆR0,LH (ωx , ωy ) .. ਤ. Fig.. 5 The frequency characteristics of MWs.(a)ψˆR0,HH (ωx , ωy )(b)ψˆR0,LH (ωx , ωy )(Amplitude of the Fourier transform result).. 45. 15. -15. -45. ߱. ਤ 2 ํ޲બ୒ੑͷద༻݁Ռ (AVDC). Fig. 2 The result of directional selection(AVDC). , | |. , | |. , , | | | |. 㽢. , | |. , | |. ߱. െ߱ (b). , , | | | |. (a) ਤ 6. , | |. ߱. ௫ , | |. 㽢. ߠ. െ߱. 0 , | |. ߱. , | |. प೾਺ಛੑͱ೾‫( ܗ‬ϑΟϧλ) ͷํ޲ͷؔ܎ɽ(a)‘x’ ʹͷΈৼ ෯Λ࣋ͭप೾਺ಛੑɽ(b)(a) ͷ‫ٯ‬ϑʔϦΤม‫Ͱ׵‬ಘΒΕΔ೾‫ܗ‬. Fig. 6 The relationship between frequency characteristic in ௬. ਤ 3. the frequency domain and directional filter in space domain((a)The frequency characteristic that has spec-. ֤प೾਺੒෼ͷଳҬ. Fig. 3 The location of each frequency component.. trum only ‘x’ points. (b) The corresponding directional filter is the inverse Fourier transform of (a).. ࣜ (16) ͔Βɼํ޲੒෼ dR0,LH j,nx ,ny ౳͸ೖྗը૾ͱɼ2 ͭͷ 2. RR II ࣍‫ݩ‬ͷ MWψj,n ɼψj,n ΍εέʔϦϯάؔ਺ φII j,nx ,ny x ,ny x ,ny. ౳ͷ࿨΋͘͠͸ࠩͱͷ಺ੵʹΑͬͯಘΒΕΔ͜ͱ͕Θ͔Δɽ R0,LH R0,HH ͜͜Ͱɼψj,n ͓Αͼɼψj,n ͷप೾਺ಛੑΛߟ͑Δɽ x ,ny x ,ny. R0,LH R0,HH ਤ 5 ʹ ψj,n ͓Αͼɼψj,n ͷप೾਺ಛੑΛࣔ͢ɽ x ,ny x ,ny. ਤ4. Πϯύϧε৴߸Λೖྗͨ͠৔߹ʹ͓͚Δํ޲બ୒ੑͷద༻݁Ռ. Fig. 4 The result of directional selection using impulse signal. RR,LH f, ψj,n x ,ny. ͍ͯ఺ରশʹεϖΫτϧ͕഑ஔ͞ΕɼۭؒྖҬͰ͸ಛఆํ ޲ʹৼ෯Λ࣋ͭࣄ͕Θ͔ΔɽͦͷͨΊɼಛఆํ޲ͷৼ෯͕. =. ਤ 5 ͔Βɼํ޲੒෼Λ‫͢ࢉܭ‬Δ MW ͸ɼप೾਺ྖҬʹ͓. f (x, y). RR,LH ψj,n (x, y)dxdy. x ,ny. (15). ‫ڧ‬ௐ͞Εɼํ޲੒෼͔Βํ޲ੑΤοδ΍ըૉ஋ͷෆ࿈ଓઢ ͕ಘΒΕΔɽ͞Βʹਤ 6(a) ͷΑ͏ͳप೾਺ྖҬͰɼಛఆͷ. ·ͨࣜ (14) ͸ RR ͷྫͰ͋Δ͕ɼͦΕҎ֎ͷ RI ɼIRɼ. ఺ରশͷ఺ʹɼಉ͡ৼ෯Λ࣋ͭप೾਺ಛੑΛߟ͑Δɽ͜ͷ. II Ͱ΋ಉ༷ʹ੒ཱ͢Δɽࣜ (14) Λࣜ (11) ʹ୅ೖ͢Δͱɼ. प೾਺ಛੑΛ‫ٯ‬ϑʔϦΤม‫͢׵‬Δͱɼਤ 6(b) ͕ಘΒΕΔɽ. ࣜ (16) ͕ಘΒΕΔɽ. ਤ 5 ΍ਤ 6 ౳͔Βɼप೾਺ྖҬͷεϖΫτϧͷҐஔ (ࣼઢ ෦) ͱۭؒྖҬͷ೾‫ܗ‬ͷํ޲͸ࣜ (17) ͷΑ͏ͳ௚ަؔ܎ʹ. dR0,LH j,nx ,ny = =< f, ∴. <. RR,LH f, ψj,n x ,ny. >+<. II,LH f, ψj,n x ,ny. ͋Δ͜ͱ͕Θ͔Δɽ. >. 2. θ = tan−1 (ωy1 /ωx1 ) .. RR,LH II,LH + ψj,n ψj,n x ,ny x ,ny. dR0,LH j,nx ,ny. >. 2 R0,LH =< f, ψj,nx ,ny >,. R0,LH (x, y) = ψj,n x ,ny. (17). 3.2 ํ޲ੑϑΟϧλͷઃ‫ܭ‬ (16). RR,LH II,LH (x, y) + ψj,n (x, y) ψj,n x ,ny x ,ny. 2. ⓒ 2014 Information Processing Society of Japan. ਤ 5, 6 ͔ΒɼMW ΍೾‫ܗ‬ʢϑΟϧλʣͷप೾਺ಛੑʹ Αͬͯɼͦͷ೾‫ܗ‬ͷํ޲͕ҟͳΔࣄ͕֬ೝ͞Εͨɽͦͷͨ. .. ΊຊઅͰ͸ɼϑΟϧλͷप೾਺ಛੑΛઃ‫͠ܭ‬ɼ೚ҙํ޲ͷ. 4.

(5) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. ˆ x , ωy ) Λ ‫׵‬Λ༻͍ΔͨΊͰ͋Δɽͦͯ͠ɼप೾਺ಛੑ ψ(ω. ߱௬ ߱௬. ߴ଎‫ٯ‬ϑʔϦΤม‫ͨ͠׵‬΋ͷΛํ޲ੑϑΟϧλͱ͢Δɽ ߱௫. ਤ 7(b) தʹ͓͍ͯɼઃ‫౓֯ͨ͠ܭ‬ൣғҎ֎ͷྖҬͷৼ෯. ߠଵ ߠଶ. ͸ 0 Ͱ͋ΔɽͦͷͨΊɼઃ‫޲ํͨ͠ܭ‬ੑϑΟϧλ͸ɼॴ๬ ߱௫. (a) ਤ 7. (b). ํ޲੒෼Λ‫͢ࢉܭ‬ΔϑΟϧλͷྫ (a)θ1 ͔Β θ2 Λ‫ݕ‬ग़͢Δ ϑΟϧλ.(b)(a) ʹରԠ͢Δप೾਺ಛੑ.. ͷ֯౓ൣғͷΈΛநग़͢Δɽ. 3.3 2D-CDWT ͱํ޲ੑϑΟϧλΛ༻͍ͨ৽ͨͳํ޲ બ୒ੑͷఏҊ. Fig. 7 The example of the filter that calculate directional com-. લઅͰ͸ɼ೚ҙͷ֯౓ൣғΛநग़Մೳͳํ޲ੑϑΟϧλ. ponents(a)The filter that detects angular range from θ1. Λઃ‫ͨ͠ܭ‬ɽ͜ͷϑΟϧλͷ֯౓ൣғΛ 10[deg] ΍ 20[deg]. to θ2 .(b)The frequency characteristic of designed filter. ͱࡉ͔֯͘౓ൣғʹઃఆ͠ɼଟ͘ͷํ޲੒෼ʹ෼ղՄೳͰ. corresponding to (a).. ͋ΓΔɽͦͯ͠ଟ͘ͷํ޲ੑಛ௃ΛಘΔࣄ͕ಘΒΕΔɽ. Τοδ΍ಛ௃Λ‫ݕ‬ग़͢ΔϑΟϧλΛઃ‫͢ܭ‬Δɽࣜ (17) ͔ Βɼ೚ҙͷํ޲Λ‫ݕ‬ग़͢ΔϑΟϧλͷप೾਺ಛੑ͸ɼप೾ ਺ྖҬͰಛఆͷ֯౓ൣғʹεϖΫτϧΛ࣋ͭϑΟϧλͱͳ Δɽྫ͑͹ਤ 7 Ͱ͸ɼθ1 ͔Β θ2 ·Ͱͷಛఆͷ֯౓ൣғΛ நग़͢ΔϑΟϧλͱͳΔɽ ਤ 7 ʹࣔ͢प೾਺ಛੑ͸ɼࣼઢ෦ʹৼ෯Λ࣋ͪɼ֯౓ʹ Ԡͯ͡ৼ෯͕มԽ͢Δؔ਺Ͱ͋Δɽͦ͜Ͱɼਤ 7 ͷΑ͏ͳ प೾਺ಛੑΛࣜ (18) ͷ֯౓ θ ͷؔ਺͔Β࡞੒͢Δɽ ⎧ 0, θ<a ⎪ ⎪. ⎪ ⎪ |θ−θshif t1 | π 1−Δ ⎪ ⎪ cos[ 2 ν( 2Δ ⎪ (1−Δ)π − 1 )], a ≤ θ < b ⎨ ˆ ψ(θ) = 1, b≤θ<c. ⎪ ⎪ |θ+θshif t2 | ⎪ π 1−Δ ⎪ cos[ 2 ν( 2Δ ⎪ (1−Δ)π − 1 )], c ≤ θ < d ⎪ ⎪ ⎩ 0, d≤θ (18) ࣜ (18) ͷ cos ‫ۂ‬ઢͷա౉ྖҬʹ͸ɼ2D-CDWT Ͱ༻͍ ΔεέʔϦϯάؔ਺ͷա౉ྖҬͷ‫ۂ‬ઢΛ֯౓ θ ͷؔ਺ͱ͠ ͯɼར༻ͨ͠ [6]ɽ·ͨɼࣜ (18) தͷ ν ͸ҎԼͷࣜͰ༩͑ ΒΕΔɽ. ν(x) = x4 (35 − 84x + 70x2 − 20x3 ), 0 ≤ x ≤ 1.. ैདྷͷ 2D-CDWT ͸֤෼ղϨϕϧͷߴप೾੒෼͔Βํ ޲੒෼͕‫͞ࢉܭ‬ΕΔͨΊɼ֤෼ղϨϕϧ͸प೾਺ಛੑ͕ҟ ͳΓɼଟॏղ૾౓ͷํ޲੒෼͕‫ࢉܭ‬ՄೳͰ͋Δɽͦ͜Ͱํ ޲ੑϑΟϧλͱ 2D-CDWT Λ૊Έ߹Θͤɼଟ͘ͷํ޲੒෼ Λଟॏղ૾Ͱ‫ࢉܭ‬Մೳʹ͢Δɽ͜ͷఏҊख๏͸ҎԼͷ (1) ͔Β (5) ͷॲཧʹΑ࣮ͬͯ‫͞ݱ‬ΕΔɽ·ͨ͜ͷॲཧ͸ਤ 8 ʹରԠ͍ͯ͠Δɽ(1) ೖྗը૾ʹ CDWT ͷิؒॲཧΛద ༻͠ɼεέʔϦϯά܎਺ΛಘΔɽ(2) εέʔϦϯά܎਺ʹ ର͠ϩʔύεϑΟϧλΛ x,y ྆࣠ʹద༻͢Δɽ͜ͷ࣌ɼ௨ ৗͷ CDWT ͷΑ͏ͳμ΢ϯαϯϓϦϯά͸ద༻͠ͳ͍ɽ. (3) ϩʔύεϑΟϧλ͔Βಘͨ௿प೾੒෼ͱ‫ݩ‬ͷεέʔϦ ϯά܎਺ͷࠩ෼Λ‫͠ࢉܭ‬ɼߴप೾੒෼Λ‫͢ࢉܭ‬Δɽ͜ͷॲ ཧͷ‫ޙ‬ɼ௿प೾੒෼ʹ͸μ΢ϯαϯϓϦϯάΛద༻͢Δɽ. (4)(3) Ͱ‫ߴͨ͠ࢉܭ‬प೾੒෼ʹର͠ɼํ޲ੑϑΟϧλΛద ༻͢Δɽ(5) ࣍ͷϨϕϧͰ͸ɼ(2) ͔Β (3) ͷॲཧΛ࠶‫ؼ‬త ʹ‫܁‬Γฦ͢ɽ࣍ͷϨϕϧʹ͸μ΢ϯαϯϓϦϯάͨ͠௿प ೾੒෼Λೖྗͱ͢Δɽ্‫ه‬ͷϓϩηεʹΑΓɼ֤Ϩϕϧͷ ํ޲੒෼ͱ࠷΋௿͍௿प೾੒෼͕ಘΒΕΔɽํ޲੒෼͸ ֤ʑ࣮਺෦ͱ‫਺ڏ‬෦Λ࣋ͭɽ·ͨɼCDWT ͱಉ༷ʹํ޲ ੒෼ͷઈର஋ AVDC Λ‫ࢉܭ‬ग़དྷΔɽఏҊख๏Ͱ͸ɼAVDC. ࣜ (18) ʹ͸ɼθshif t1 , θshif t2 , a, b, c and d ͷύϥϝʔλ͕. ͸ RR, RI , IR ͓Αͼ II ͷฏํࣗ৐࿨Ͱ‫͞ࢉܭ‬ΕΔɽ·. ͋Δɽθshif t1 ͓Αͼ θshif t2 ͷύϥϝʔλ͸ cos ‫ۂ‬ઢΛฏߦ. ͨɼਤ 8 ͸ RR ͷΈΛ͍ࣔͯ͠Δ͕ɼଞͷ RI , IRɼII ʹ͓. Ҡಈ͢ΔͨΊͷύλϝʔλͰ͋Δɽύϥϝʔλ a, b, c ͓Αͼ. ͍ͯ΋ಉ༷ͷॲཧͰ͋Δɽ࢖༻͢Δํ޲ੑϑΟϧλ΋ಉ͡. d ͸ cos ‫ۂ‬ઢͷ୺఺Λ͍ࣔͯ͠Δɽྫ͑͹ɼ40 ͔Β 50[deg]. ΋ͷΛ࢖༻͢ΔɽఏҊख๏͸ը૾Λਤ 9 ͷΑ͏ʹɼ֤ํ޲. ͷ֯౓ൣғΛಘ͍ͨ৔߹ɼ[(a + b)/2 = 40 ∗ π/180(rad) Ͱ. ੒෼΁෼ղ͢Δɽਤ 9 ͸ɼ֤ํ޲੒෼ͷ֯౓ൣғΛ 10[deg]. ͋Γ, (c + d)/2 = 50 ∗ π/180 (rad) ͱͳΔɽͦͷ࣌ɼθshif t1. ͮͭʹઃఆͨ͠৔߹Ͱ͋Δɽ͜ͷ৔߹͸ 18 ‫ݸ‬ͷํ޲੒෼. ͸ɼ−π ͔Β 40 ∗ π/180(rad) ·ͰͷҠಈྔ (40 ∗ π/180 + π ). ΛಘΔ͜ͱ͕Ͱ͖Δɽ. ʹઃఆ͢ΔɽҰํͰ θshif t2 ͸ π ͔Β 50 ∗ π/180(rad) ·Ͱ. ఏҊख๏ͱैདྷͷ 2D-CDWT ͷҧ͍͸ɼߴप೾੒෼Ͱ. ͷҠಈྔ (π − 50 ∗ π/180) ʹઃఆ͢Δɽ͜Ε͸ɼεέʔϦ. ͋Δɽैདྷͷ 2D-CDWT ͕ɼx, y ʹ෼཭ͨ͠ϑΟϧλΛ. ϯάؔ਺͸ΧοτΦϑप೾਺͕ਖ਼‫ن‬Խप೾਺Ͱ π ͓Αͼ. ར༻͍ͯ͠Δͷʹର͠ɼఏҊख๏͸ 2 ࣍‫ݩ‬ඇ෼཭ͷํ޲. −π ʹઃఆ͞Ε͍ͯΔͨΊͰ͋Δɽ·ͨɼύϥϝʔλ a, b,. ੑϑΟϧλΛར༻͍ͯ͠Δɽ͜ͷϑΟϧλΛ࠾༻͢Δ͜ͱ. c ͓Αͼ d ͸ɼθshif t1 ɼθshif t2 ͱ Δ ΑΓܾఆ͞ΕΔɽ. Ͱɼ೚ҙͷ֯౓ൣғͷํ޲੒෼ͷநग़Λ࣮‫ݱ‬ՄೳͰ͋Δɽ. ਤ 7(b) ͸ 40[deg] ͔Β 50[deg] ͷ֯౓ൣғʹৼ෯Λ࣋ͭ. ͔͠͠ɼඇ෼཭ͷϑΟϧλͰ͋ΔͨΊɼैདྷͷ 2D-CDWT. प೾਺ಛੑͷྫͰ͋Δɽ͜͜Ͱɼप೾਺ಛੑ͸ ωx , ωy ͱ. ΑΓ΋‫͕ྔࢉܭ‬ଟ͍ɽ͞ΒʹɼఏҊख๏͸ CDWT Λ‫ʹج‬. ΋ʹɼ−π ͔Β π ͷൣғͰఆٛͨ͠ɽ͜Ε͸ɼઃ‫ͨ͠ܭ‬प. ͍ͯ͠ΔͨΊγϑτෆมੑΛ࣋ͭɽ. ೾਺ಛੑΛϑΟϧλͱͯ͠ར༻͢Δࡍʹɼߴ଎ϑʔϦΤม. ⓒ 2014 Information Processing Society of Japan. 5.

(6) 情報処理学会研究報告 IPSJ SIG Technical Report /ŶƉƵƚ /ŵĂŐĞ.

(7). ோோ. Vol.2014-CVIM-193 No.13 2014/9/1. &ĂƐƚ &ŽƵƌŝĞƌ dƌĂŶƐĨŽƌŵ. /ŶƚĞƌƉŽͲ ͲůĂƚŝŽŶ. Ͳ. /ŶǀĞƌƐĞ &ŽƵƌŝĞƌ dƌĂŶƐĨŽƌŵ. &ĂƐƚ &ŽƵƌŝĞƌ dƌĂŶƐĨŽƌŵ. >ŽǁWĂƐƐ &ŝůƚĞƌ, . ĞƐŝŐŶĞĚ ŝƌĞĐƚŝŽŶĂů &ŝůƚĞƌƐ. ŝƌĞĐƚŝŽŶĂůĐŽŵƉŽŶĞŶƚ ;ůĞǀĞů͗ͲϭͿǁŝƚŚĚĞƐŝŐŶĞĚ ĂŶŐƵůĂƌƌĂŶŐĞ. ZĞƉĞĂƚŽŶĞůĞǀĞůƉƌŽĐĞƐƐ;ĚĂƐŚůŝŶĞͿ ƵŶƚŝůƐƉĞĐŝĨŝĞĚĚĞĐŽŵƉŽƐŝƚŝŽŶůĞǀĞů. 2 ↓, 2 ↓. ਤ 8. ఏҊख๏. Fig. 8 The proposed directional selection based on directional. (b). (a) ਤ 11. ഏ໺಺ʹजᚅΛ࣋ͭ CT ը૾ྫ.. Fig. 11 The example of medical images that have the tumor in the lung area.. filters and the CDWT. ߱௬. ߱௫. ਤ 9. ఏҊख๏ͰಘΒΕΔํ޲੒෼ͷ֤प೾਺ಛੑ.. Fig. 9 The frequency characteristics of directional components. ਤ 12. ఏҊख๏ͷҩ༻ը૾΁ͷద༻݁Ռ.. Fig. 12 The processing result by using proposed method.. by using the proposed method.. 4. ҩ༻ը૾ॲཧ΁ͷԠ༻ ఏҊख๏ͷ༗ޮੑΛ‫ݕ‬౼͢ΔͨΊɼҩ༻ը૾ॲཧʹԠ༻ ͢Δɽຊ࿦จͰ͸ഏ಺෦ʹजᚅ෦ҐΛ࣋ͭ CT ը૾ͷපม ෦ҐೝࣝΛ‫ݕ‬౼͢Δɽࠓճ͸जᚅΛ࣋ͭ‫ ऀױ‬6 ໊͔Βɼज ᚅ෦Ґ͕͋Δ CT ը૾ 6 ຕɼजᚅ෦Ґ͕ແ͍ը૾Λ 6 ຕબ ୒͠ɼ‫ ܭ‬12 ຕͷը૾Λ࣮‫ʹݧ‬ར༻͢Δɽ (b). (a). 4.1 ‫ڳ‬෦ CT ը૾΁ͷఏҊख๏΁ͷద༻ ਤɽ11(a),(b) ʹजᚅ෦Ґ͕͋Δ‫ڳ‬෦ CT ը૾ͷྫΛࣔ ͢ɽ͜ͷը૾ͷഏ໺಺ʹϚʔΫͨ͠ന͘ΪβΪβͷลԑ‫ܗ‬ ঢ়Λ࣋ͭ෺ମ͕जᚅͰ͋Δɽ ਤɽ12 ʹɼਤɽ11 ͷͦΕͧΕͷը૾ΛఏҊख๏ʹͯม (d). (c) ਤ 10. ఏҊख๏ͷద༻݁Ռ.. Fig. 10 The result of the proposed method.. ‫݁ͨ͠׵‬ՌΛࣔ͢ɽਤɽ12 ӈ͸ɼ෼ղϨϕϧ-2 ͷఏҊख ๏Λద༻͠ɼ160[deg] ͔Β 180[deg] ͷ AVDC Λ‫ͨ͠ࢉܭ‬ ݁ՌΛ͍ࣔͯ͠Δɽಉਤࠨ͸ɼ෼ղϨϕϧ-2ɼ90[deg] ͔Β. 110[deg] ͷ AVDC Λ‫݁ͨ͠ࢉܭ‬ՌͰ͋ΔɽͦΕͧΕʹ༻ ఏҊख๏ͷద༻ྫΛਤ 10(a)ʙ(d) ʹࣔ͢ɽਤ 10 Ͱ͸ɼ. ͍Δํ޲ੑϑΟϧλͷλοϓ਺͸ 16 ʷ 16(ॎʷԣ) ఺ͱ͠. ೖྗը૾ͱͯ͠ɼਤ 1 Λ༻͍ɼ֤Ϩϕϧͷํ޲੒෼Λ‫ࢉܭ‬. ͨɽಉਤ͔Βɼजᚅ෦Ґʹͯɼߴ͍ೱ୶஋͕ಘΒΕ͓ͯΓɼ. ͨ͠ɽࠓճɼ෼ղϨϕϧ͸ −2 ͱͨ͠ɽਤ 10(a)ʙ(d) ͸͍. जᚅ෦Ґͷ‫ݕ‬ग़͕֬ೝͰ͖Δ. ͣΕ΋Ϩϕϧ −2 ͷ AVDC Λ͍ࣔͯ͠Δɽਤ 10 ͷͦΕͧ Εͷը૾͔ΒɼఏҊख๏͕ɼํ޲ੑϑΟϧλͰઃ‫֯ͨ͠ܭ‬. 4.2 ಛ௃ϕΫτϧͷ‫ͱࢉܭ‬जᚅ෦Ґೝࣝ. ౓ൣғʹै͍ɼ֤ํ޲ͷΤοδΛ‫ݕ‬ग़͍ͯ͠Δ͜ͱ͕֬ೝ. ਤɽ12(a) ͔ΒɼఏҊख๏Λ༻͍ͯजᚅ෦ҐͷΤοδ౳Λ. Ͱ͖Δɽ·ͨɼਤ 2 ͔Βɼैདྷͷ 2D-CDWT ͸ 6 ํ޲ͷ. ‫ݕ‬ग़ՄೳͰ͋Δ͜ͱΛ֬ೝͨ͠ɽ࣍ʹɼఏҊख๏ͷద༻݁. AVDC Λ‫͖Ͱࢉܭ‬ɼ֯౓ൣғ͕޿͍΋ͷ΋͕͋ͬͨɼҰํ. Ռ͔Βɼजᚅ෦ҐΛೝࣝ͢ΔͨΊʹɼͦΕͧΕͷ AVDC ͔. ͰఏҊख๏͸ɼࡉ͔͍֯౓ൣғͷ AVDC Λ‫ݕ‬ग़Ͱ͖ɼଟ. Βಛ௃ϕΫτϧʢಛ௃ྔʣΛ࡞੒͢Δɽ࡞੒ͨ͠ಛ௃ϕΫ. ͘ͷํ޲ͷ AVDC Λ‫ݕ‬ग़ՄೳͰ͋ΔɽͦͷͨΊɼఏҊख. τϧΛαϙʔτϕΫλʔϚγϯ (Support Vector Machine,. ๏͸ɼैདྷΑΓ΋ࡉ͔ͳํ޲ੑಛ௃Λը૾͔Β‫ݕ‬ग़Մೳͳ. SVM) ʹೖྗ͠ɼजᚅ෦Ґͷೝࣝ݁ՌΛ݁ՌΛಘΔɽຊ‫ݚ‬. ख๏ͩͱߟ͑ΒΕΔɽ. ‫Ͱڀ‬͸ҎԼͷํ๏Ͱಛ௃ϕΫτϧΛ‫͢ࢉܭ‬Δɽ. ( 1 ) ೖྗ͞Εͨը૾ʹఏҊख๏Λద༻͢Δɽಛ௃ϕΫτϧ ⓒ 2014 Information Processing Society of Japan. 6.

(8) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. Λߏ੒͢Δࡍʹ͸ɼϨϕϧ-4 ·Ͱͷม‫׵‬Λߦ͍ɼ֤Ϩ ϕϧͰ 9 ํ޲ (֤ํ޲੒෼͕ 20[deg]) ͷ AVDC Λ‫ࢉܭ‬ ͢Δɽ. ( 2 ) ֤ํ޲੒෼ͷը૾Λɼ༧Ίઃఆͨ͠ϒϩοΫʹ෼ׂ͢ Δɽࠓճ͸ɼ16 ʷ 16[pix] ͷϒϩοΫʹ෼ׂ͢Δɽ͜ ͜Ͱɼ֤Ϩϕϧͷ AVDC ͸μ΢ϯαϯϓϦϯά͞Εͯ ͍ΔͨΊɼ֤ը૾αΠζ͸ҟͳΓɼϨϕϧຖʹϒϩο. ਤ 13. ೝࣝϓϩηε (4) ·Ͱͷॲཧ݁Ռ.. Fig. 13 The results of from (1) to (4) in recognition process.. Ϋ͕୲౰͢Δ‫ݩ‬ը૾্Ͱͷେ͖͞͸ҟͳΔɽ. ( 3 ) ֤ϒϩοΫͷதͰɼAVDC ͷ਺஋͕େ͖͍ॱʹ 3 ఺औ. (6) ͷॲཧʹ͸ਐ·ͣɼ(7) ͷॲཧʹਐΉ (ਤɽ13(b))ɽ. Γग़͠ɼಛ௃఺ͱ͢Δɽ͜͜Ͱɼಛ௃఺ͱಉҐஔͷଞ. ( 6 ) जᚅ͕ଘࡏ͢Δͱ൑ఆ͞Εͨ৔߹ɼजᚅͱ͞ΕͨͦΕ. ͷํ޲੒෼ͷ਺஋΋औΓग़͢ɽɹ 36 ຕͷը૾͕͋Δ. ͧΕͷಛ௃ϕΫτϧ͕ ROI ಺ʹೖ͍ͬͯΔ͔Λ‫ࢉܭ‬. ͨΊɼҰͭͷಛ௃఺ʹ͖ͭɼ36 ‫ݸ‬ͷ਺஋ΛऔΓग़͢͜. ͠ɼೝࣝҐஔ͕ਖ਼͍͔͠Λ൑ఆ͢Δɽजᚅͱ͞Εͨಛ. ͱͱͳΔɽ͜ΕΛ 36 ࣍‫ݩ‬ͷಛ௃ϕΫτϧͱ͢Δɽ. ௃ϕΫτϧͷ಺ɼROI ಺ʹೖ͍ͬͯΔ਺ͱɼROI ಺ʹ. ( 4 ) શͯͷ AVDC ͷશͯͷϒϩοΫͰɼಛ௃఺ΛऔΓग़. ೖ͍ͬͯͳ͍਺Λൺֱ͠ɼROI ಺ʹೖ͍ͬͯΔ਺ͷํ. ͠ɼಛ௃ϕΫτϧΛ‫͢ࢉܭ‬Δɽ. (2) ͷॲཧͰ͸ɼ༷ʑͳେ͖͞ͷϒϩοΫ͔Βಛ௃఺Λ ‫͍ͯ͠ࢉܭ‬Δࣄʹ૬౰͢ΔͨΊɼ֦େॖখͷมԽʹରԠ͠ ͨಛ௃఺Λ‫ࢉܭ‬ՄೳͰ͋Δͱߟ͑ΒΕΔɽ. ͕ଟ͚Ε͹ɼजᚅҐஔΛਖ਼ৗʹೝࣝՄೳͰ͋ͬͨͱ͢ Δਤɽ13(c)ɽҰํͰ ROI ಺ʹೖ͍ͬͯͳ͍਺ͷํ͕ ଟ͍৔߹͸ɼҐஔͷ‫ݕ‬ग़͸ෆՄͱͨ͠ɽ. ( 7 ) ผͷ‫ࠪݕ‬ը૾Λબ୒͠ɼ(1) ͔Β (6) ͷॲཧΛ‫܁‬Γฦ͢ɽ. ࣍ʹɼ‫ͨ͠ࢉܭ‬ಛ௃ϕΫτϧΛ‫ʹج‬जᚅ෦ҐೝࣝͷͨΊ. ਤɽ13 ʹ্‫ه‬ೝ࣮ࣝ‫ݧ‬ͷ్த݁ՌΛࣔ͢ɽਤɽ13 ͸ಛ. ͷɼϥϕϧΛ෇༩͢Δɽϥϕϧ͸ɼजᚅϥϕϧͱͦΕҎ֎. ௃ϕΫτϧΛ෼ྨͨ͠ (4) ͷ݁ՌͰ͋Δɽಉਤதͷ੺ࣔ͘. ͷਖ਼ৗϥϕϧΛ༻ҙ͠ɼSVM ͷֶशʹར༻͢Δɽजᚅϥϕ. ͞Εͨ఺͕जᚅͱೝࣝ͞Εͨಛ௃ϕΫτϧ (ಛ௃఺) Ͱ͋. ϧͷ෇༩ʹ͸ɼҩࢣʹೖྗը૾Λఏࣔ͠ɼजᚅ෦ҐͷྖҬ. Δɽ͔͜͜Βɼजᚅ෦Ґ෇ۙͷಛ௃ϕΫτϧ͕ೝࣝ͞Εͯ. (Region of Interest, ROI) Λճ౴Λґཔͨ͠ɽͦͯ͠ɼ্. ͍Δࣄ͕֬ೝग़དྷΔɽ(5) ͷॲཧ͸ɼਤɽ13 ʹࣔ͢੺఺ͷ. ‫ํه‬๏Ͱ‫ͨ͠ࢉܭ‬ಛ௃ϕΫτϧͷ಺ɼҩࢣ͕ࣔͨ͠ ROI ಺. ਺͕ 100 ఺Ҏ্ͳΒ͹ɼजᚅ͋Γͱ൑ఆ͢Δ (Detect)ɽ(6). ʹ͋Δಛ௃ϕΫτϧʹɼजᚅϥϕϧΛ෇༩ͨ͠ɽROI ֎ͷ. ͷॲཧ͸ɼਤɽ13 ʹࣔ͢੺఺͕ҩࢣͷࢦఆͨ͠ ROI ಺ʹ. ಛ௃ϕΫτϧʹ͸ɼਖ਼ৗϥϕϧΛ෇༩ͨ͠ɽजᚅ͕ແ͍ը. ൒෼Ҏ্ೖ͍ͬͯΔͳΒ͹ɼजᚅҐஔΛਖ਼͘͠൑ఆͨ͠ͱ. ૾ͷ৔߹͸ಘΒΕͨಛ௃఺શͯʹਖ਼ৗϥϕϧΛ෇༩ͨ͠ɽ. ͨ͠ (Position Correct)ɽ. ࣍ʹɼಛ௃ϕΫτϧͱͦͷϥϕϧΛ SVM ʹೖྗ͠ɼֶशɾ. ·ͨɼजᚅͳ͠ͱ൑ఆ͞Εͨ΋ͷΛ Not Detect ͱ͠ɼDe-. ෼ྨʹΑΔը૾ೝ࣮ࣝ‫ݧ‬Λߦ͏ɽը૾ೝ࣮ࣝ‫ݧ‬͸ɼҎԼͷ. tect ͷ಺ɼजᚅҐஔ͕ਖ਼͔ͬͨ͠΋ͷΛ Position Correct,. खॱͰߦ͏ɽ. ༗Γͱ൑அ͞Ε͕ͨҐஔ͕ਖ਼͘͠ͳ͔ͬͨ΋ͷΛ Position. ( 1 ) 12 ຕͷ CT ը૾ͷ಺ɼ1 ຕΛ‫ࠪݕ‬ը૾ͱ͢ΔɽͦΕҎ. Wrong ͱͨ͠ɽ. ֎͸ɼSVM ͷֶश༻ը૾ͱ͢Δɽ. ( 2 ) ‫ࠪݕ‬ը૾͓Αͼֶश༻ը૾͔Βɼલड़ͷํ๏Ͱಛ௃ϕ ΫτϧΛ‫͢ࢉܭ‬Δɽ. ( 3 ) ֶश༻ը૾ͷ֤ʑ͔ΒಘΒΕͨಛ௃ϕΫτϧͱҩࢣ͕ ࢦఆͨ͠ϥϕϧΛ SVM ʹೖྗɾֶश͠ɼࣝผϞσϧ Λ‫͢ࢉܭ‬ΔɽSVM Ͱ͸ɼRBF Χʔωϧʢγ = 1ʣΛ༻ ͍ɼίετύϥϝʔλ͸ 1 ͱͨ͠ɽ·ͨɼSVM ͷલ ॲཧͱͯ͠ɼಛ௃ϕΫτϧͷ਺஋͸֤࣍‫࠷Ͱݩ‬େ஋͕. 1ɼ࠷খ஋͕ 0 ͱͳΔΑ͏ʹௐઅͨ͠ɽ ( 4 ) ‫ࠪݕ‬ը૾ͷಛ௃ϕΫτϧΛɼ(3) Ͱֶशͨࣝ͠ผϞσ. ͦͷ݁ՌఏҊख๏ʹΑΓɼजᚅͷ༗ແΛ 5/6 ͷׂ߹Ͱೝ ͍ࣝͯ͠Δ͜ͱ͕֬ೝग़དྷΔɽ·ͨɼजᚅͷҐஔʹ͍ͭͯ ͸ɼ4/6 ͷׂ߹ͰೝࣝՄೳͰ͋ͬͨɽҰํͰɼजᚅͷແ͍ ਖ਼ৗը૾ʹରͯ͠͸ɼ‫ޡ‬ೝ͕ࣝͳ͘ɼߴਫ਼౓ʹೝࣝՄೳͰ ͋Δ͜ͱ͕֬ೝ͞Εͨɽ. 5. ·ͱΊ ຊ‫Ͱڀݚ‬͸ɼैདྷͷ 2D-CDWT ΍ͦͷํ޲બ୒ੑʹ͍ͭ ͯड़΂ͨɽैདྷͷ 2D-CDWT Ͱ͸ɼ෼ղ݁Ռ͔Β 6 ํ޲ ͷ AVDCʢը૾தͷํ޲ੑΤοδ΍‫ز‬Կֶಛ௃ʣΛಘΒΕ. ϧʹೖྗ͠ɼ֤ಛ௃ϕΫτϧͷೝࣝ݁ՌΛ‫͢ࢉܭ‬Δɽ. ΔͷΈͰ͕͋ͬͨɼप೾਺ಛੑͱ AVDC ͷؔ܎Λ‫ݕ‬౼͠ɼ. ೝࣝ݁Ռͱͯ͠ɼ֤ಛ௃ϕΫτϧ͕जᚅ෦Ґ͔ਖ਼ৗϕ. ৽ͨͳํ޲ੑϑΟϧλΛઃ‫ͨ͠ܭ‬ɽͦͯ͠ɼઃ‫޲ํͨ͠ܭ‬. ΫτϧͷϥϕϧͰ͋Δ͔͕‫͞ࢉܭ‬ΕΔ (ਤɽ13(a))ɽ. ੑϑΟϧλͱैདྷͷ 2D-CDWT Λ૊Έ߹Θͤͨ৽ͨͳํ. ( 5 ) (4) ͷ݁Ռ͔Β‫ࠪݕ‬ը૾಺ͷजᚅ෦Ґͷ༗ແΛ൑ఆ͢. ޲બ୒ੑΛಘΔख๏ΛఏҊͨ͠ɽఏҊͨ͠ख๏Λҩ༻ը૾. Δɽࠓճ͸ɼजᚅ෦Ґͱ͞Εͨಛ௃ϕΫτϧ͕ 100 ‫ݸ‬. ॲཧͷजᚅ෦ҐೝࣝʹԠ༻ͨ͠ɽఏҊख๏Λར༻ͯ͠ɼज. Ҏ্͋Δ৔߹͸ɼ‫ࠪݕ‬ը૾ʹजᚅ෦Ґ͕ଘࡏ͢Δ΋ͷ. ᚅ෦ҐͷྠֲΛํ޲ผʹ‫ݕ‬஌͍ͯ͠Δ͜ͱΛ֬ೝͨ͠ɽ࣍. ͱͨ͠ɽ͜͜Ͱɼजᚅ෦Ґ͕ແ͍ͱ൑ఆ͞Εͨ৔߹͸ɼ. ʹɼఏҊख๏ͷద༻݁ՌΛར༻͠ɼಛ௃ϕΫτϧΛߏ੒͠. ⓒ 2014 Information Processing Society of Japan. 7.

(9) 情報処理学会研究報告 IPSJ SIG Technical Report. Vol.2014-CVIM-193 No.13 2014/9/1. ͨɽߏ੒ͨ͠ಛ௃ϕΫτϧͱ SVM Λར༻͠ɼजᚅͷೝࣝ ͕ՄೳͰ͋ͬͨ͜ͱΛ֬ೝͨ͠ɽजᚅͷ͋Δ 6 αϯϓϧͷ ಺ɼजᚅͱೝࣝαϯϓϧ͕ 5 ͭɼҐஔ΋ਖ਼͘͠൑ఆͨ͠΋ ͷ͕ 4 ͭͰ͋ͬͨɽҰํͰɼजᚅͷແ͍ 6 αϯϓϧͷ಺ɼ ‫ͯͬޡ‬जᚅ͋Γͱ൑ఆͨ͠΋ͷ͸֬ೝ͞Εͳ͔ͬͨ͜ͷ݁ Ռ͔ΒఏҊख๏ͷ༗༻ੑΛ֬ೝͨ͠ɽࠓ‫ޙ‬͸ɼςεταϯ ϓϧͷ֦େͱಛ௃ϕΫτϧͷߏ੒ํ๏ͷ࠶‫ݕ‬౼͕ࠓ‫ޙ‬ͷ՝ ୊ͱͳΔɽ ࢀߟจ‫ݙ‬ [1]. [2]. [3]. [4] [5]. [6]. S.G. Mallat, “A Theory for Multiresolution Signal Decomposition The Wavelet Representation”, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol.11, No.7, pp.674-693, 1989. ‫ ߒ ా ށ‬ɼষ ஧ ɼ“‫ ׬‬શ γ ϑ τ ෆ ม ੑ Λ ࣮ ‫͢ ݱ‬ Δ ෳ ૉ ਺ ཭ ࢄ ΢ Σ ʔ ϒ Ϩ ο τ ม ‫ ׵‬ʡ, ৴ ߸ ॲ ཧ, Vol. 12, No. 2 (2008), pp.156-166. T. Kato, et al, “Directional Selection property of 2Dimensional Complex Discrete Wavelet Transform and its application on defect inspection of semiconductor wafer circuits”, Transactions of the JSME, C, Vol. 79, No. 808, pp. 4901-4916, 2013. N. G. KingsburyɼImage Processing with Complex Wavelet, Phil Trans, Royal Society London A, 1999. Selesnick W. I., “The Design of Approximate Hilbert Transform Pairs of Wavelet Basesʡ, IEEE Transactions on Signal Processing , Vol. 50, No. 5 (2002), pp.11441152. ‫ ߒ ా ށ‬ɼ ষ ஧ ɼ “‫ ׬‬શ γ ϑ τ ෆ ม ੑ Λ ࣮ ‫ ͢ ݱ‬Δ ෳ ૉ ਺ ཭ ࢄ ΢ Σ ʔ ϒ Ϩ ο τɾύ έ ο τ ม ‫ ׵‬ʡ, ৴ ߸ ॲ ཧ, Vol. 14, No. 2 (2008), pp.139-152.. ⓒ 2014 Information Processing Society of Japan. 8.

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