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4.2 અ Τοδ ౡوେ June 11, 2018 ిؾ௨৴େֶ ঙݚڀࣨ B4
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࣍ 1. Τοδͷݕग़ 2. Τοδͷ࿈݁ 2
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Τοδͷݕग़
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Τοδͷݕग़ ྠֲઢͳͲͷΤοδ͖ΘΊͯଟ͘ͷใΛؚΉ ਓखʹΑΔΤοδݕग़ (ਤ 4.31) ˠ͜ΕΛύιίϯ༷ʹΒ͍ͤͨ 3
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Τοδͷݕग़ ୯७ͳΤοδͷݕग़ํ๏ɿΤοδΛٸܹͳًมԽͱͯ͠ѻ͏ ˠًͷޯΛߟ͑Δ I(x) ΛϐΫηϧ x = (x, y)⊤ ্ͷًͱ͢Δͱɼًޯ J(x) J(x) = ∇I(x) = ( ∂I ∂x , ∂I ∂y ) (x) (4.19) 4
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Τοδͷݕग़ ϕΫτϧ J(x) ͷ • ͖ɿًؔͷ࠷ٸޯํ • େ͖͞ɿًؔͷมԽ߹͍ 5
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Τοδͷݕग़ ߴपʹϊΠζ͕ଟ͍ ˠϩʔύεϑΟϧλͰฏԽ͔ͯ͠ΒޯΛܭࢉ ローパス フィルタ 6
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Τοδͷݕग़ ϑΟϧλద༻ޙޯͷ͖͕ਖ਼͘͠อଘ͞Ε͍ͯͯ΄͍͠ ˠԁܗͷϑΟϧλ ՄೳͳԁܗϑΟϧλΨεϑΟϧλͷΈ (3.2 અɼਤ 3.14) ˠΤοδݕग़ͷͨΊͷϩʔύεϑΟϧλΨγΞϯ͕ఆ൪ 7
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Τοδͷݕग़ ඍઢܗԋࢉͰ͋ΔͷͰଞͷϑΟϧλԋࢉͱՄ ΨεϑΟϧλؔΛ Gσ(x) = 1 2πσ2 exp ( − x2 + y2 2σ2 ) ͱ͢Δ ฏԽޙͷը૾ͷޯΛ Jσ(x) ͱॻ͘ͱɼ Jσ(x) = ∇[Gσ(x) ∗ I(x)] = [∇Gσ(x)] ∗ I(x) (4.20) ͱͳΓɼΨεϑΟϧλؔͷඍͱͷͨͨΈࠐΈͰදݱͰ͖Δ 8
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Τοδͷݕग़ ΨεϑΟϧλؔͷඍͷධՁ ∇Gσ(x) = ( ∂ ∂x , ∂ ∂y )⊤ Gσ(x) = ( ∂ ∂x , ∂ ∂y )⊤ 1 2πσ2 exp ( − x2 + y2 2σ2 ) = 1 σ2 (−x, − y)⊤ 1 2πσ2 exp ( − x2 + y2 2σ2 ) ((4.21) ࣜͱ߹Θͳ͍͕ͨͿΜ͜ΕͰ͍͋ͬͯΔ) 9
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Τοδͷݕग़ thinning ΤοδΛ 1 ըૉͷଠ͞Ͱදݱ͍ͨ͠߹͕ଟ͍ (ࡉઢԽ; thinning) (ը૾ [1] ΑΓ) 10
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Τοδͷݕग़ thinning ʮΤοδʹରͯ͠ਨͳํͷޯڧ͕࠷େʹͳΔ࠲ඪʯΛٻ ΊΕΑ͍ ˠًͷ 2 ֊ඍ (ϥϓϥγΞϯ) Λߟ͑ΕΑͦ͞͏ͩ ͜ͷ 2 ֊ඍͷ Sσ(x) ɼ∇2 = ∇ · ∇(= div grad) ΑΓ Sσ(x) = ∇ · Jσ(x) = [∇2Gσ(x)] ∗ I(x) (4.22) 11
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Τοδͷݕग़ thinning ΨεϑΟϧλͷϥϓϥγΞϯͷධՁ ∇2Gσ(x) = ∇ · [ 1 σ2 (−x, − y)⊤ 1 2πσ2 exp ( − x2 + y2 2σ2 )] = ∂ ∂x [ − x 2πσ4 exp ( − x2 + y2 2σ2 )] + ∂ ∂y [ − y 2πσ4 exp ( − x2 + y2 2σ2 )] = 1 2πσ2 ( x2 + y2 − 2σ2 σ4 ) exp ( − x2 + y2 2σ2 ) 12
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Τοδͷݕग़ thinning ∇2Gσ(x) ͷΛແࢹˠ LoG(Laplacian of Gaussian) ϑΟϧλ LoG(x) = ( x2 + y2 − 2σ2 σ4 ) exp ( − x2 + y2 2σ2 ) 13
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Τοδͷݕग़ thinning Sσ(x) ͷූ߸͕มԽ ˠ૬ରతͳ໌Δ͕͞มԽ Sσ(x) ͷθϩަࠩΛ୳ͤ Α͍ 14
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Τοδͷݕग़ thinning sign(Sσ(xi)) ̸= sign(Sσ(xj)) ͱͳΔྡϐΫηϧ xi, xj ͓Αͼθ ϩަࠩ xz Λ୳͢ Sσ(xi) ͱ Sσ(xj) ͱΛ݁Ϳઢ͕θϩͱަࠩ͢Δ xz ΛٻΊΔ 15
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Τοδͷݕग़ thinning Sσ(xj) − Sσ(xi) xj − xi (xz − xi) + Sσ(xi) = 0 ∴ xz = xiSσ(xj) + xjSσ(xi) Sσ(xj) + Sσ(xi) ͕ಘΒΕΔɽ3 ࣍ݩҎ্ͷ߹ಉ༷ʹ xz = xiSσ(xj) + xjSσ(xi) Sσ(xj) + Sσ(xi) (4.25) Ͱ͋Δ 16
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Τοδͷݕग़ εέʔϧબͱϘέྔਪఆ LoG ʹదͳ σ ΛઃఆˠӶ͍/ಷ͍ΤοδΛநग़ (ਤ 4.32, (b), (c)) 17
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Τοδͷݕग़ εέʔϧબͱϘέྔਪఆ ͍ײͰΤοδΛͱΓ͍ͨͳΒʁ ˠεέʔϧεϖʔεͷΞϓϩʔν 1. ͍͔ͭ͘ͷ σ Λ༻ҙ 2. ͦΕͧΕͷ σ ʹ͍ͭͯޯ ͱ 2 ֊ඍΛܭࢉ 3. ҆ఆʹΤοδΛݕग़Ͱ͖Δ ࠷খͷ σ ΛબɼͦΕΑΓ େ͖͍ σ Ͱݕग़͞ΕͨΤο δΛՃ 18
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Τοδͷݕग़ εέʔϧબͱϘέྔਪఆ ͍ σ ͰΤοδΛநग़ (ਤ 4.32, (f)) 19
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Τοδͷݕग़ Χϥʔը૾ͰͷΤοδݕग़ Χϥʔը૾ͰΤοδݕग़Λ͍ͨ͠ ୯७ʹًޯΛݟΔͱɼً৭ؒͷΤοδΛݕग़Ͱ͖ͳ͍ ղܾҊ 1ɿRGB ֤͝ͱʹًޯΛܭࢉ͢Δ • ֤৭Ͱූ߸ͷҟͳΔޯ͕ग़Δͱɼ୯७ͳ͠߹ΘͤͰ૬ ࡴ͕ى͜Δ ղܾҊ 2ɿ֤ըૉͷपลͰہॴతͳ౷ܭྔΛ͍Ζ͍ΖௐΔ • ୯७ͳًɾ໌ɾ৭͚ͩͰͳ͘ɼςΫενϟͷมԽͳͲ ଊ͑ΒΕΔ 20
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Τοδͷݕग़ ਤ 4.33ɽBGɿ໌ɼCGɿ৭ɼTGɿςΫενϟ 21
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Τοδͷ࿈݁
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Τοδͷ࿈݁ நग़͞ΕͨΤοδΛ࿈݁ͯ͠Ұܨ͗ʹ͍ͨ͠ thinning ͞ΕͨΤοδͷըૉใΛ͍࣋ͬͯΔͱָ ˠ͍ۙΛ୳ࡧͯ͠ܨ͛Α͍ ΤοδΛ࿈݁͢ΔͱΑΓѹॖͨ͠දݱ͕ՄೳʹͳΔ 22
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Τοδͷ࿈݁ νΣΠϯίʔυ 8 ͭͷํ֯ (N, NE, E, SE, S, SW, W, NW) Λ 3bit ͰίʔυԽ (ਤ 4.34) 23
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Τοδͷ࿈݁ νΣΠϯίʔυ νΣΠϯίʔυͰͷΤϯίʔυޙɼϥϯϨϯάεූ߸Ͱ͞Βʹѹ ॖͰ͖Δ ϥϯϨϯάεූ߸ ܁Γฦ͠ͷจࣈΛͦͷճͰදݱ AAAABBBCCCCC ˠ A4B3C5 24
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Τοδͷ࿈݁ arc-length parameterization ʮހʯͷ͞ͱΤοδ࠲ඪΛ༻͍ͯදݱ (ਤ 4.35) 1. x0 = (1, 0.5)⊤ ͔Βελʔτ 2. s = 0 ʹ x0 ͷ࠲ඪΛͦΕͧΕϓϩοτ 3. x1 = (2, 0.5)⊤ 4. s = ∥x1 − x0∥ = 1 ʹ x1 ͷ࠲ඪΛͦΕͧΕϓϩοτ 5. ࢝ʹΔ·Ͱ܁Γฦ͢ 25
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Τοδͷ࿈݁ arc-length parameterization Q. Կ͕͏Ε͍͠ͷ͔ʁ A. ϚονϯάฏԽͳͲͷॲཧ͕༰қʹͳΔ ܗঢ়ͷࣅͨΤοδΛߟ͑Δ (ਤ 4.36) 26
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Τοδͷ࿈݁ arc-length parameterization 1. Τοδͷ࠲ඪͷฏۉ ¯ x0 = ∫ S x(s)ds Λݮࢉ 2. s Λ 0 ∼ S ͔Β 0 ∼ 1 ʹਖ਼نԽ 3. ͦΕͧΕʹ͍ͭͯϑʔϦΤม 27
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Τοδͷ࿈݁ arc-length parameterization ͱͷΤοδಉ͕࢜εέʔϦϯάͱճసͷҧ͍͔͠ͳ͍ ˠϑʔϦΤมͷ݁ՌڧͱҐ૬ͷζϨ͔͠ҟͳΒͳ͍ͣ (։͕࢝ҟͳΔͱઢܗͷҐ૬ͷζϨग़Δ) 28
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Τοδͷ࿈݁ arc-length parameterization ࢄԽ࣌ʹੜ͡ΔϊΠζͷฏԽʹ༗ޮ ͔͠͠ී௨ʹฏԽϑΟϧλΛ͔͚Δͱॖখͯ͠ฏԽ͞ΕΔ ਤ 4.37(a), ԁͷܘ͕ॖখ͍ͯ͠Δ 29
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Τοδͷ࿈݁ arc-length parameterization 2 ֊ඍʹجͮ͘Φϑηοτ߲Λ͔͢ɼΑΓେ͖ͳ (ͦ͢ͷ ͍ʁ) ฏԽϑΟϧλΛ༻͍Δ ਤ 4.37(b) 30
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·ͱΊ • άϨʔεέʔϧը૾ͰًޯͰΤοδΛݕग़ ϊΠζআڈಉ࣌ʹߦ͏ͨΊʹΨγΞϯϑΟϧλͷ 1 ֊ඍ ͱͨͨΈࠐΉ • thinning ͍ͨ͠߹ LoG ϑΟϧλΛ͔͚ͯθϩަࠩΛٻ ΊΔ • Χϥʔը૾ͷΤοδݕग़໌ɾ৭ɾςΫενϟͳͲͷ౷ܭ ྔ͕༗ޮ • thinning ͞ΕͨΤοδͷ࿈݁νΣΠϯίʔυ arc-length parameterization ͕༗ޮ • arc-length parameterization ޙϚονϯάϊΠζআڈΛ͠ ͍͢ 31
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References I [1] R. Rao. Image sampling, pyramids, and edge detection. https://courses.cs.washington.edu/courses/cse455/ 09wi/Lects/lect3.pdf, 2009.