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コンピュータビジョン4.2節

 コンピュータビジョン4.2節

研究室のゼミで発表したRichard Szeliski 著,玉木徹ら訳の『コンピュータビジョン − アルゴリズムと応用』4.2節のスライド

Takahiro Kawashima

June 13, 2018
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  1. Τοδͷݕग़ ඍ෼͸ઢܗԋࢉͰ͋ΔͷͰଞͷϑΟϧλԋࢉͱՄ׵ Ψ΢εϑΟϧλؔ਺Λ Gσ(x) = 1 2πσ2 exp ( −

    x2 + y2 2σ2 ) ͱ͢Δ ฏ׈Խޙͷը૾ͷޯ഑Λ Jσ(x) ͱॻ͘ͱɼ Jσ(x) = ∇[Gσ(x) ∗ I(x)] = [∇Gσ(x)] ∗ I(x) (4.20) ͱͳΓɼΨ΢εϑΟϧλؔ਺ͷඍ෼ͱͷͨͨΈࠐΈͰදݱͰ͖Δ 8
  2. Τοδͷݕग़ Ψ΢εϑΟϧλؔ਺ͷඍ෼ͷධՁ ∇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
  3. Τοδͷݕग़ thinning ʮΤοδʹରͯ͠ਨ௚ͳํ޲ͷޯ഑ڧ౓͕࠷େʹͳΔ࠲ඪʯΛٻ ΊΕ͹Α͍ ˠً౓ͷ 2 ֊ඍ෼ (ϥϓϥγΞϯ) Λߟ͑Ε͹Αͦ͞͏ͩ ͜ͷ

    2 ֊ඍ෼ͷ஋ Sσ(x) ͸ɼ∇2 = ∇ · ∇(= div grad) ΑΓ Sσ(x) = ∇ · Jσ(x) = [∇2Gσ(x)] ∗ I(x) (4.22) 11
  4. Τοδͷݕग़ thinning Ψ΢εϑΟϧλͷϥϓϥγΞϯͷධՁ ∇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
  5. Τοδͷݕग़ thinning sign(Sσ(xi)) ̸= sign(Sσ(xj)) ͱͳΔྡ઀ϐΫηϧ xi, xj ͓Αͼθ ϩަࠩ఺

    xz Λ୳͢ Sσ(xi) ͱ Sσ(xj) ͱΛ݁Ϳઢ෼͕θϩͱަࠩ͢Δ఺ xz ΛٻΊΔ 15
  6. 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
  7. Τοδͷݕग़ εέʔϧબ୒ͱϘέྔਪఆ ޿͍ײ౓ͰΤοδΛͱΓ͍ͨͳΒʁ ˠεέʔϧεϖʔεͷΞϓϩʔν 1. ͍͔ͭ͘ͷ σ Λ༻ҙ 2. ͦΕͧΕͷ

    σ ʹ͍ͭͯޯ഑ ͱ 2 ֊ඍ෼Λܭࢉ 3. ҆ఆʹΤοδΛݕग़Ͱ͖Δ ࠷খͷ σ Λબ୒ɼͦΕΑΓ େ͖͍ σ Ͱݕग़͞ΕͨΤο δΛ௥Ճ 18
  8. Τοδͷ࿈݁ νΣΠϯίʔυ 8 ͭͷํ֯ (N, NE, E, SE, S, SW,

    W, NW) Λ 3bit ͰίʔυԽ (ਤ 4.34) 23
  9. Τοδͷ࿈݁ arc-length parameterization ʮހʯͷ௕͞ͱΤοδ࠲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. ࢝఺ʹ໭Δ·Ͱ܁Γฦ͢ 25
  10. 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
  11. ·ͱΊ • άϨʔεέʔϧը૾Ͱ͸ً౓ޯ഑ͰΤοδΛݕग़ ϊΠζআڈ΋ಉ࣌ʹߦ͏ͨΊʹΨ΢γΞϯϑΟϧλͷ 1 ֊ඍ ෼ͱͨͨΈࠐΉ • thinning ͍ͨ͠৔߹͸

    LoG ϑΟϧλΛ͔͚ͯθϩަࠩ఺Λٻ ΊΔ • Χϥʔը૾ͷΤοδݕग़͸໌౓ɾ৭ɾςΫενϟͳͲͷ౷ܭ ྔ͕༗ޮ • thinning ͞ΕͨΤοδͷ࿈݁͸νΣΠϯίʔυ΍ arc-length parameterization ͕༗ޮ • arc-length parameterization ޙ͸Ϛονϯά΍ϊΠζআڈΛ͠ ΍͍͢ 31
  12. References I [1] R. Rao. Image sampling, pyramids, and edge

    detection. https://courses.cs.washington.edu/courses/cse455/ 09wi/Lects/lect3.pdf, 2009.