法線分布関数フィルタリングの実装と評価

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(1)Vol.2016-CG-164 No.2 2016/9/21. ৘ใॲཧֶձ‫ڀݚ‬ใࠂ IPSJ SIG Technical Report. ๏ઢ෼෍ؔ਺ϑΟϧλϦϯάͷ࣮૷ͱධՁ ಙ٢ ༤հ1,a). ֓ཁɿຊߘ͸ 2016 ೥ʹൃද͞Εͨ Kaplanyan et al. ͷ‫ڸ‬໘൓ࣹͷΞϯνΤΠϦΞγϯάख๏ͷ࣮૷ͱධ Ձʹ͍ͭͯใࠂ͢Δɽ͜ͷख๏͸ࢹઢํ޲ͱޫ‫޲ํݯ‬ͷதؒϕΫτϧͷඍ෼Λ༻͍ͯ๏ઢ෼෍ؔ਺ΛϑΟ ϧλϦϯά͢Δͱ͍͏΋ͷͰ͋Δɽ͔࣮͠͠ࡍʹ͸ඍ෼ͷਪఆ‫͕ࠩޡ‬େ͖͍ͨΊɼ๬·͍݁͠ՌΛಘΒΕ ͳ͍͜ͱ͕ଟ͍ɽ͜ͷ͜ͱ͸ Kaplanyan et al. ΋࿦จதͰࢦఠ͓ͯ͠Γɼ࣮༻ʹ͸ϑΟϧλϦϯάʹόΠ ΞεΛՃ͑ͯਖ਼֬͞ΑΓ҆ఆੑΛ༏ઌ͢Δ͜ͱΛਪ঑͍ͯ͠Δɽͦ͜Ͱզʑ͸ͲͷΑ͏ͳόΠΞεΛՃ͑ Δͷ͕࣮༻తͳͷ͔ௐ΂ΔͨΊɼ࣮‫ݧ‬Λߦͬͨɽͦͷ݁ՌɼσΟϑΝʔυϨϯμϦϯά޲͚ͷۙࣅख๏͕ ࠷΋҆ఆ͍ͯ͠Δ͜ͱ͕෼͔ͬͨɽ. Evaluation of Normal Distribution Function Filtering for Specular Anti-Aliasing Tokuyoshi Yusuke1,a). 1. ๏ઢ෼෍ؔ਺ͷϑΟϧλϦϯά. Σ=σ. 2. ϩʔϓεϖʔεʹ͓͚Δඍ෼Λ༻͍ͯ๏ઢ෼෍ؔ਺ΛϑΟ ϧλϦϯά͢Δख๏ΛఏҊͨ͠ɽ͔͠͠ͳ͕Βɼ࣮ࡍʹ͸ ඍ෼ͷਪఆ͸ෆ҆ఆͰ‫͕ࠩޡ‬େ͖͘ͳΓ΍͍͢ɽͦͷͨ Ίɼ࣮༻ʹ͸ϑΟϧλϦϯάʹόΠΞεΛՃ͑Δ͜ͱ͕ਪ ঑͞Ε͍ͯΔɽͦ͜ͰຊߘͰ͸ͲͷΑ͏ͳόΠΞεΛՃ͑ Δͷ͕࣮༻తͳͷ͔Λ‫͢ূݕ‬Δɽ. ΕΔɽਪఆ͞ΕͨεϩʔϓεϖʔεͰͷதؒϕΫτϧͷ ඍ෼ʢϐΫηϧؒͷࠩ෼ʣΛ Δhu , Δhv ͱ͠ɼϐΫηϧͷ ϑΟϧλϦϯάΧʔωϧΛ෼ࢄ σ 2 = 0.25 ͷΨ΢ε෼෍ͱ Ծఆ͢ΔͱɼεϩʔϓεϖʔεͰͷதؒϕΫτϧͷ‫ڞ‬෼ࢄ ߦྻ͸ҎԼͷࣜͰ༩͑ΒΕΔɽ 1 a). ‫ࣜג‬ձࣾεΫ΢ΣΞɾΤχοΫε [email protected]. ⓒ 2016 Information Processing Society of Japan. . Δhv. .. (1). ͢ΔɽBeckmann ෼෍͸εϩʔϓεϖʔεͰ͸Ψ΢ε෼෍ Ͱ͋Γɼ‫ڸ‬໘ͷૈ͞ύϥϝʔλͷೋ৐ α2 ͸๏ઢ෼෍ͷ෼ ࢄΛೋഒͨ͠΋ͷͰ͋ΔɽதؒϕΫτϧͷ෼෍͸ϚΠΫϩ ϑΝηοτͷ๏ઢ෼෍ͳͷͰɼϑΟϧλϦϯάΛղੳతͳ ৞ΈࠐΈʹΑͬͯද͢͜ͱ͕Ͱ͖Δɽͦͷ݁ՌɼϑΟϧλ Ϧϯά‫ޙ‬ͷ‫ڸ‬໘ͷૈ͞͸ҎԼͷߦྻͱͳΔɽ. A=. α2. 0. 0. α2. ͷਪఆ͸ɼϐΫηϧγΣʔμʔͰྡ઀ϐΫηϧͱͷࠩ෼Λ R ‫ٻ‬ΊΔ໋ྩʢe.g., DirectX Ͱ͸ ddx/ddyʣʹΑͬͯߦΘ. Δhu. ࣍ʹ෺ମද໘ͷ๏ઢ෼෍ؔ਺Λ Beckmann ෼෍ [1] ͱԾఆ. 1.1 ख๏ ϦΞϧλΠϜϨϯμϦϯάʹ͓͚ΔதؒϕΫτϧͷඍ෼. T . Δhv. Kaplanyan et al. [4] ͸‫ڸ‬໘൓ࣹͷΞϯνΤΠϦΞγϯ άΛߦ͏ͨΊʹɼࢹઢํ޲ͱޫ‫޲ํݯ‬ͷதؒϕΫτϧͷε. Δhu. + 2Σ.. (2). ۙ೥өը΍ήʔϜ౳ͰҰൠతʹ࢖༻͞Ε͍ͯΔ GGX ෼ ෍ [7], [8] ͷૈ͞ύϥϝʔλ͸ɼBeckmann ෼෍ͷૈ͞ύϥ ϝʔλͰۙࣅՄೳͰ͋ΔɽͦͷͨΊຊߘͰ͸͜ͷૈ͞ߦྻ. A Λ GGX ෼෍ʹద༻͢Δɽ ঘɼૈ͞ߦྻΛ࢖ͬͯҰൠԽͨ͠ GGX ૒ํ޲൓ࣹ཰ ෼෍ؔ਺ʢbidirectional reflectance distribution function,. BRDFʣʹ͍ͭͯ͸‫ݩ‬࿦จʹ‫͞ࡌه‬Ε͍ͯͳ͍ɽͦͷͨΊɼ ใࠂऀ͕ಋग़ΛߦͬͨࣜΛ §A.1 ʹ‫ه‬ड़͢Δɽ. 1.

(2) Vol.2016-CG-164 No.2 2016/9/21. ৘ใॲཧֶձ‫ڀݚ‬ใࠂ IPSJ SIG Technical Report. 1.2 ό΢ϯσΟϯάϘοΫεΛ࢖ͬͨϑΟϧλϦϯά લड़ͷख๏͸ཧ࿦తʹ͸ਖ਼͍͕͠ɼ࣮ࡍʹ͸‫ࠩޡ‬ͷӨ‫ڹ‬ Λ‫͘ڧ‬ड͚ΔͷͰ๬·͍݁͠ՌΛಘΒΕͳ͍͜ͱ͕͋Δ. }. ͨͩ͠ɼ๏ઢ෼෍ؔ਺ϑΟϧλϦϯάͰ͸ॎԣͷϐΫη ϧͷมԽྔΛ‫ݟ‬ΔͷͰɼର֯ํ޲ͷϐΫηϧͷ๏ઢ͸ແ͘ ͯ΋ྑ͍ɽͦͷͨΊຊߘͰ͸ҎԼͷΑ͏ʹলུͨ͠ɽ. ͱ‫ݩ‬࿦จͰड़΂ΒΕ͍ͯΔɽͦ͜ͰϑΟϧλϦϯάΛ҆ఆ. float3 SumNeighborPixels( float2 pixelID, float3 normal ). ͤ͞ΔͨΊʹɼKaplanyan et al. ͸όΠΞεͷ͋Δա৒ͳ. { float2 dir = 1.0 - 4.0 * frac( pixelID * 0.5 );. ϑΟϧλϦϯάΛ࢖͏͜ͱΛਪ঑ͨ͠ɽ͜Ε͸ Δhu , Δhv. float3 horz = normal + ddx_fine( normal ) * dir.x;. Ͱ࡞ΒΕΔฏߦ࢛ล‫ܗ‬ͷό΢ϯσΟϯάϘοΫεͷॎԣͷ. float3 vert = normal + ddy_fine( normal ) * dir.y;. ෯Λҟํੑ BRDF ͷૈ͞ύϥϝʔλͱͯ͠࢖͏ͱ͍͏΋ ͷͰ͋Δɽ. return normal + horz + vert; }. 2.1.2 HLSL Shader Model 6.0 1.3 ࣮‫݁ݧ‬Ռ ࣮‫݁ݧ‬ՌΛਤ 1 ʹࣔ͢ɽ͜ͷϑΟϧλϦϯάख๏ʹΑͬ ͯΤΠϦΞγϯάΛ཈͑Δ͜ͱ͕Ͱ͖ͨҰํͰɼೖࣹ͕֯ ઙ͍৔߹ʹա৒͔ͭҟํੑͷ‫͍ڧ‬ϑΟϧλϦϯά͕ߦΘΕ ͓ͯΓɼ͜ΕʹΑͬͯ‫ʹٯ‬ΞʔςΟϑΝΫτ͕ൃੜͯ͠͠. DirectX 12 ͷ HLSL Shader Model 6.0 Ͱ͸ྡ઀ϐΫη ϧͷ஋Λ௚઀ࢀর͢Δ໋ྩ͕௥Ճ͞Εͨɽ͜ΕΛ༻͍Δ͜ ͱͰ্ड़ͷ pixel quad message passing ๏ΑΓ΋୯७͔ͭ ਫ਼౓ͷߴ͍ฏ‫ۉ‬஋Λ‫ٻ‬ΊΔ͜ͱ͕ՄೳʹͳΔͱߟ͑ΒΕΔɽ ࠓ‫ޙ‬͸͜ͷ৽໋ྩΛ࢖࣮ͬͨ૷Λߦ͍͍ͨͱߟ͍͑ͯΔɽ. ·͍ͬͯΔɽ͜ͷ‫ݪ‬Ҽ͸εϩʔϓεϖʔεʹ͓͚Δதؒϕ ΫτϧͷมԽྔ͕ೖࣹ֯ͷਖ਼઀ʹൺྫ͢ΔͨΊͩͱߟ͑Β ΕΔɽͦͷͨΊɼೖࣹ͕֯ઙ͘‫ͭ׌‬๏ઢ͕ϐΫηϧؒͰม Խ͢Δ৔߹ɼ͜ͷதؒϕΫτϧͷมԽྔͷ‫ۃ͕ࠩޡ‬Ίͯେ ͖͘ͳͬͯ͠·͏Α͏Ͱ͋Δɽ͜͏ͨ͠ΞʔςΟϑΝΫτ ͸ Kaplanyan et al. ͕ओு͢ΔΑ͏ʹό΢ϯσΟϯάϘο ΫεΛ࢖ͬͨϑΟϧλϦϯάͰܰ‫ݮ‬ՄೳͰ͋Δɽ͠ͳ͠ͳ ͕ΒɼͦΕͰ΋ґવͱͯ͠໨ཱͭΞʔςΟϑΝΫτ͕ൃੜ ͯ͠͠·͍ͬͯΔɽ. 2. ฏ‫ۉ‬๏ઢϕΫτϧΛ༻͍ͨۙࣅ தؒϕΫτϧΛ༻͍ͨϑΟϧλϦϯά͸ޫ‫ݯ‬ຖʹ‫͢ࢉܭ‬ Δඞཁ͕͋Γɼޫ‫͕਺ݯ‬ଟ͍৔߹ʹ‫͕ྔࢉܭ‬ଟ͘ͳΔɽ· ͨɼσΟϑΝʔυϨϯμϦϯάʹ༻͍Δ͜ͱ͕Ͱ͖ͳ͍ͱ ͍͏໰୊͕͋Δɽͦ͜ͰσΟϑΝʔυϨϯμϦϯάʹର͠ ͯ͸ɼதؒϕΫτϧͷ୅ΘΓʹۙ๣ 4 ϐΫηϧͷ๏ઢͷฏ ‫ۉ‬Λ༻͍Δۙࣅ͕ఏҊ͞Ε͍ͯΔɽͦ͜Ͱ͜ͷۙࣅ๏ʹͭ ͍ͯ඼࣭ͷධՁΛߦͬͯΈΔ͜ͱʹͨ͠ɽۙ๣ 4 ϐΫηϧ ͷ๏ઢϕΫτϧͷฏ‫ۉ‬ͷ‫ٻ‬Ίํ͸‫ݩ‬࿦จʹ‫ه‬ड़͞Ε͍ͯͳ ͍ͷͰɼຊߘͰղઆ͢Δɽ. 2.2 ࣮‫݁ݧ‬Ռ ࣮‫݁ݧ‬ՌΛਤ 2 ʹࣔ͢ɽ‫ʹͱ͖͜΂͘ڻ‬ೖࣹ͕֯ઙ͍৔ ߹ͰͷΞʔςΟϑΝΫτͷൃੜΛ཈͑Δ͜ͱ͕Ͱ͖ͨɽ͜ ͷཧ༝ͱͯ͠͸ɼதؒϕΫτϧͱҟͳΓฏ‫ۉ‬๏ઢ͸෺ମද ໘ʹରͯ͠ઙ͘ͳΓʹ͍ͨ͘Ίͱߟ͑ΒΕΔɽͦͷͨΊɼ ཧ࿦্͸தؒϕΫτϧΛ༻͍Δͷ͕ਖ਼͍͠ͱߟ͑ΒΕΔ͕ɼ ฏ‫ۉ‬๏ઢϕΫτϧΛ༻͍ͨํ͕࣮༻ੑ͸ߴ͍ͱ‫͑ݴ‬Δɽ. 3. ౳ํੑۙࣅ σΟϑΝʔυϨϯμϦϯάͰ͸ɼG-buffer [6] ͷϝϞϦʔ ࢖༻ྔΛ཈͑ΔͨΊ౳ํੑͷૈ͞ύϥϝʔλʹͨ͠ํ͕๬ ·͍͠ɽͦ͜Ͱ Kaplanyan et al. ͕ओு͢ΔΑ͏ʹɼό΢ ϯσΟϯάϘοΫεΛ࢖ͬͨϑΟϧλϦϯάʹ͓͍ͯૈ͞ ͷ࠷େ஋Λ༻͍ͨ౳ํੑϑΟϧλϦϯάΛߦͬͬͯΈΔ͜ ͱʹͨ͠ɽ·ͨ௥Ճ࣮‫ͯ͠ͱݧ‬ɼ‫ڞ‬෼ࢄߦྻͷ࠷େͷ‫ݻ‬༗ ஋Λ༻͍ͯૈ͞ͷਪఆΛߦͬͯΈͨɽ. 3.1 ࣮‫݁ݧ‬Ռ ࣮‫݁ݧ‬ՌΛਤ 3 ʹࣔ͢ɽҟํੑϑΟϧλϦϯάͱҟͳ Γɼ྆ऀͱ΋ࢹ֮తʹෆշͳΞʔςΟϑΝΫτ͕ൃੜ͢Δ. 2.1 ࣮૷ 2.1.1 Pixel quad message passing ϐΫηϧγΣʔμʔʹ͓͍ͯɼྡ઀ϐΫηϧͷ஋Λಘ Δํ๏ͱͯ͠ pixel quad message passing ๏ [5] ͕͋Δɽ DirectX Ͱ͸ྡ઀ϐΫηϧͷ஋ͱͷਖ਼֬ͳࠩ෼ΛಘΔ໋ྩ ͱͯ͠ɼddx fine/ddy fine ໋ྩ͕͋Δɽ͜ΕΛ༻͍Δ͜ͱ ͰɼҎԼͷΑ͏ʹ 4 ϐΫηϧͷ๏ઢͷ૯࿨Λ‫ٻ‬ΊΔ͜ͱ͕ ՄೳͱͳΔɽ float3 SumNeighborPixels( float2 pixelID, float3 normal ) {. ͜ͱ͸ͳ͘ͳͬͨɽҰํͰɼશମతʹա৒ͳϑΟϧλϦϯ ά͕ߦΘΕͯ͠·͍ͬͯΔɽ͔͠͠ͳ͕ΒதؒϕΫτϧΛ ༻͍ͨҟํੑͷϑΟϧλϦϯάͱҧ͍ɼߴप೾ͷΞʔςΟ ϑΝΫτͰ͸ͳ͍ͷͰࢹ֮తʹ‫ڐ‬༰͠қ͍ͱߟ͑ΒΕΔɽ. Kaplanyan et al. ͕ఏҊ͍ͯ͠Δૈ͞ͷ࠷େ஋Λ༻͍ΔΑ Γ΋ɼ࠷େͷ‫ݻ‬༗஋Λ༻͍ͯૈ͞Λਪఆͨ͠ํ͕ա৒ͳ ϑΟϧλϦϯάΛए‫ׯ‬཈͑Δ͜ͱ͕Ͱ͖ͨɽ. 4. ·ͱΊ. float2 dir = 1.0 - 4.0 * frac( pixelID * 0.5 ); float3 horz = normal + ddx_fine( normal ) * dir.x;. ຊ‫ڀݚ‬ใࠂͰ͸ Kaplanyan et al. ͷ๏ઢ෼෍ؔ਺ϑΟϧ. float3 vert = normal + ddy_fine( normal ) * dir.y;. λϦϯάͷ࣮૷ͱධՁΛߦͬͨɽ࣮‫ݧ‬ͷ݁Ռɼཧ࿦௨Γͷ. float3 diag = horz + ddy_fine( horz ) * dir.y; return normal + horz + vert + diag;. ⓒ 2016 Information Processing Society of Japan. ࣮૷Ͱ͸ߴप೾ͷΞʔςΟϑΝΫτ͕ൃੜ͢Δ͜ͱ͕֬. 2.

(3) Vol.2016-CG-164 No.2 2016/9/21. ৘ใॲཧֶձ‫ڀݚ‬ใࠂ IPSJ SIG Technical Report. (a) ϑΟϧλϦϯάແ͠. (b) ‫ڞ‬෼ࢄߦྻΛ࢖ͬͨϑΟϧλϦϯά. (c) ό΢ϯσΟϯάϘοΫεΛ࢖ͬͨϑΟϧλ Ϧϯά. ਤ 1: தؒϕΫτϧͷඍ෼Λ༻͍ͨҟํੑϑΟϧλϦϯάɽೖࣹ͕֯ઙ͍৔߹ʹෆշͳΞʔςΟϑΝΫτ͕ൃੜ͍ͯ͠Δɽ ೝ͞Εͨɽ͔͠͠ͳ͕Βɼۙࣅख๏Ͱ͋ΔσΟϑΝʔυϨ ϯμϦϯά޲͚ͷ౳ํੑϑΟϧλϦϯάͰ͸ɼ෭࡞༻ͱ͠ ͯ͜ΕΒͷ໰୊ΛճආՄೳͰ͋Δ͜ͱ͕෼͔ͬͨɽͦͷͨ ΊɼϑΥϫʔυϨϯμϦϯάͰ͋ͬͯ΋ɼ͜ͷσΟϑΝʔ υϨϯμϦϯά޲͚ͷۙࣅΛߦͬͨํ͕ྑ͍Մೳੑ͕͋ Δɽ͜ͷۙࣅख๏͸ա৒ͳϑΟϧλϦϯάʹΑͬͯେ͖ͳ όΠΞεΛੜΈग़͕͢ɼ͜ͷόΠΞε͸௿प೾Ͱ͋Δɽ͜ ͷΑ͏ͳ௿प೾ͷόΠΞε͸ࢹ֮తʹෆշʹͳΓʹ͍͘ͷ ͰɼήʔϜ౳ͷ‫ָޘ‬Λ໨తͱͨ͠ΞϓϦέʔγϣϯʹର͠ (a) ‫ڞ‬෼ࢄߦྻΛ࢖ͬͨϑΟϧλ (b) ό΢ϯσΟϯάϘοΫεΛ Ϧϯά. ࢖ͬͨϑΟϧλϦϯά. ͯ͸ద͍ͯ͠ΔՄೳੑ͕͋Δɽ ँࣙ. ຊ࿦จͷϙϦΰϯϞσϧʹ͸ҎԼͷ URL Ͱެ։. ਤ 2: ฏ‫ۉ‬๏ઢͷඍ෼Λ༻͍ͨҟํੑϑΟϧλϦϯάɽೖ. ͞Ε͍ͯΔσʔλΛ࢖༻ͤͯ͞௖͍ͨɽσʔλͷఏ‫ਂʹڙ‬. ࣹ͕֯ઙ͍৔߹ͷΞʔςΟϑΝΫτ͕தؒϕΫτϧΛ࢖ͬ. ँ͢Δɽ. ͨ৔߹ʹൺ΂ͯେ͖͘‫ݮ‬গ͍ͯ͠Δɽ. • F. Meinl and M. Dabrovic, URL: http://www.crytek.com/cryengine/cryengine3/downloads. ෇. ࿥. A.1 ‫ڞ‬෼ࢄߦྻϕʔεͷ GGX BRDF ϚΠΫϩϑΝηοτ BRDF[2] ͸ҎԼͷࣜͰ༩͑ΒΕΔɽ. ρ(ω i , ω o ) =. (a) ‫ڞ‬෼ࢄߦྻͷ࠷େͷ‫ݻ‬༗஋Λ (b) ό΢ϯσΟϯάϘοΫεͷ࠷ ࢖ͬͨϑΟϧλϦϯά. େ෯Λ࢖ͬͨϑΟϧλϦϯά. ਤ 3: ฏ‫ۉ‬๏ઢͷඍ෼Λ༻͍ͨ౳ํੑϑΟϧλϦϯάɽ྆ ऀͱ΋ෆշͳߴप೾ΞʔςΟϑΝΫτΛ΄΅‫׬‬શʹআ‫Ͱڈ‬ ͖Δ͕ɼա৒ͳϑΟϧλϦϯάʹΑͬͯ‫͍ڧ‬௿प೾ͷόΠ. G2 (ω i , ω o , ω m )D(ω m )F (ω m · ω o ) . 4|ω i · n||ω o · n|. ͜͜Ͱ n ͸෺ମද໘ͷ๏ઢɼω m =. ͸தؒϕ. Ϋ τ ϧ ɼD(ω m ) ͸ Ϛ Π Ϋ ϩ ϑ Ν η ο τ ͷ ๏ ઢ ෼ ෍ ؔ ਺ɼG2 (ω i , ω o , ω m ) ͸ masking-shadowing ؔ਺ɼͦͯ͠. F (ω m · ω o ) ͸ϑϨωϧ൓ࣹ߲Ͱ͋ΔɽSmith ͷ masking ؔ਺Λ G1 (ω o ) =. 1 1+Λ(ω o ). ͱ͢Δͱɼheight correlated. masking-shadowing ؔ਺͸ҎԼͷࣜͰ༩͑ΒΕΔɽ. Ξε͕ൃੜ͢Δɽ࠷େͷ‫ݻ‬༗஋Λ༻͍ͨํ͕ա৒ͳϑΟϧ λϦϯάΛ཈͑Δ͜ͱ͕ՄೳͰ͋Δɽ. ω i +ω o ||ω i +ω o ||. G2 (ω i , ω o , ω m ) =. 1 . 1 + Λ(ω i ) + Λ(ω o ). GGX ෼෍Ͱද͞ΕΔϚΠΫϩαʔϑΣε͸‫ܗ‬ঢ়ʗ৳ॖෆ. ⓒ 2016 Information Processing Society of Japan. 3.

(4) ৘ใॲཧֶձ‫ڀݚ‬ใࠂ IPSJ SIG Technical Report. Vol.2016-CG-164 No.2 2016/9/21. มʢshape/stretch invarianceʣͰ͋Γɼૈ͞ύϥϝʔλ α ͸ϚΠΫϩαʔϑΣεͷ৳ॖͷεέʔϧͰ͋Δɽͦͷͨ Ί๏ઢ෼෍ؔ਺ͱ Smith ͷ masking-shadowing ؔ਺͸෺ ମද໘ͱதؒϕΫτϧͷ৳ॖ͔Β‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δ [3]ɽ ‫ط‬ଘͷ anisotropic GGX BRDF ͸઀ઢۭؒͰͷ࣠ํ޲ʹ ϚΠΫϩαʔϑΣεΛ৳ॖͤ͞ΔϞσϧͰ͋ΔɽͦΕʹର ͠ɼ‫ڞ‬෼ࢄߦྻϕʔεͷ BRDF Ͱ͸ϚΠΫϩαʔϑΣε ͷ৳ॖํ޲͸ૈ͞ߦྻ A ͷ‫ݻ‬༗ϕΫτϧͰ͋Γɼ৳ॖͷ εέʔϧͷೋ৐͕ A ͷ‫ݻ‬༗஋Ͱ͋Δɽ͜ͷ͜ͱ͔Βɼ઀ ઢۭؒʹ͓͍ͯ ω m = [xm , ym , zm ] ͱ͢Δͱ‫ܗ‬ঢ়ෆมΑΓ ๏ઢ෼෍ؔ਺͸ҎԼͷࣜͱͳΔɽ. D(ω m ) =. 1 . 2 )2 π det(A) ([xm , ym ]A−1 [xm , ym ]T + zm. ·ͨ઀ઢۭؒʹ͓͍ͯ ω o = [xo , yo , zo ] ͱ͢Δͱɼ৳ॖෆ มΑΓɼSmith ͷ masking-shadowing ؔ਺ʹରͯ͠ҎԼͷ ͕ࣜಋग़͞ΕΔɽ. [xo , yo ]A[xo , yo ]T + zo2 Λ(ω o ) = −0.5 + . 2|zo | ࢀߟจ‫ݙ‬ [1]. [2]. [3]. [4]. [5] [6]. [7]. [8]. Beckmann, P. and Spizzichino, A.: Scattering of Electromagnetic Waves from Rough Surfaces, MacMillan (1963). Cook, R. L. and Torrance, K. E.: A Reflectance Model for Computer Graphics, ACM Trans. Graph., Vol. 1, No. 1, pp. 7–24 (1982). Heitz, E.: Understanding the Masking-Shadowing Function in Microfacet-Based BRDFs, J. Comput. Graph. Tech., Vol. 3, No. 2, pp. 48–107 (2014). Kaplanyan, A. S., Hill, S., Patney, A. and Lefohn, A.: Filtering Distributions of Normals for Shading Antialiasing, HPG’16, pp. 151–162 (2016). Penner, E.: Shader Amortization using Pixel Quad Message Passing, GPU Pro 2, A K Peters, pp. 349–367 (2011). Saito, T. and Takahashi, T.: Comprehensible Rendering of 3-D Shapes, SIGGRAPH Comput. Graph., Vol. 24, No. 4, pp. 197–206 (1990). Trowbridge, T. S. and Reitz, K. P.: Average Irregularity Representation of a Rough Surface for Ray Reflection, J. Opt. Soc. Am, Vol. 65, No. 5, pp. 531–536 (1975). Walter, B., Marschner, S. R., Li, H. and Torrance, K. E.: Microfacet Models for Refraction Through Rough Surfaces, EGSR’07, pp. 195–206 (2007).. ⓒ 2016 Information Processing Society of Japan. 4.

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