部分空間拘束とエピポーラ拘束を利用した2組の時系列画像における画像間対応推定

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(1)2006−CVIM−156（20） 2006／11／10. 社団法人情報処理学会研究報告 IPSJ SIG Technical Report. ෦෼ۭؒ߆ଋͱΤϐϙʔϥ߆ଋΛར༻ͨ͠ 2 ૊ͷ࣌‫ྻܥ‬ը૾ʹ͓͚Δը૾ؒରԠਪఆ ߴ‫ڮ‬. ल࿨†. ਿຊ. ໜथ. Ԟ෋. ਖ਼හ. ౦‫ۀ޻ژ‬େֶେֶӃཧ޻ֶ‫ڀݚ‬Պ‫ػ‬ց੍‫ޚ‬γεςϜઐ߈ ˟ 152–8550 ౦‫౎ژ‬໨ࠇ۠େԬࢁ 2–12–1–S5–22 E-mail: {hidekazu,shige,mxo}@ok.ctrl.titech.ac.jp ͋Β·͠ ຊ࿦จͰ͸ɼ2 ૊ͷ࣌‫ྻܥ‬ը૾Λར༻ͨ͠ը૾ؒͷରԠਪఆख๏ΛఏҊ͢ΔɽఏҊख๏Ͱ͸ɼ2 ૊ͷ࣌‫ྻܥ‬ ը૾தͷશಛ௃఺ΛͦΕͧΕ௥੻͠ɼը૾ؒʹଘࡏ͢ΔΤϐϙʔϥ߆ଋΛར༻ͯ͠ɼҰํͷ࣌‫ྻܥ‬ը૾ͷͦΕͧΕͷ ಛ௃఺‫੻ي‬Λଞํͷ࣌‫ྻܥ‬ը૾ͷશಛ௃఺‫͕੻ي‬ଐ͢Δ෦෼ۭؒʹ͋ͯ͸ΊΔɽ͜ΕʹΑΓɼҰํͷ࣌‫ྻܥ‬ը૾͔Β ಘΒΕͨಛ௃఺‫ʹ੻ي‬ର͢Δଞํͷ࣌‫ྻܥ‬ը૾தͷಛ௃఺‫੻ي‬Λਪఆ͢Δ͜ͱ͕Ͱ͖ɼը૾ؒͷରԠ͕ಘΒΕΔɽ͜ ͷํ๏͸ɼըૉ஋Λར༻ͨ͠εςϨΦը૾ରԠ఺୳ࡧͱ͸ҟͳΓɼ྆ํͷը૾Ͱಉ࣌ʹ‫؍‬ଌ͞Ε͍ͯͳ͍఺ʹରͯ͠ ΋ରԠ͕ಘΒΕΔͱ͍͏ಛ௃Λ࣋ͭɽ͞ΒʹɼಘΒΕͨରԠΛ΋ͱʹɼର৅ͷ 3 ࣍‫ܗݩ‬ঢ়Λ෮‫͢ݩ‬Δ͜ͱ΋Ͱ͖Δɽ ߹੒ը૾͓Αͼ࣮ը૾Λ༻͍࣮ͨ‫ݧ‬Λ௨ͯ͡ɼఏҊख๏ͷ༗ޮੑΛࣔ͢ɽ Ωʔϫʔυ. ࣌‫ྻܥ‬ը૾ɼը૾ؒରԠਪఆɼ෦෼ۭؒ߆ଋɼΤϐϙʔϥ߆ଋ. Image Correspondence Estimation from Subspace Constraint and Epipolar Constraint on a Pair of Image Sequences Hidekazu TAKAHASHI† , Shigeki SUGIMOTO, and Masatoshi OKUTOMI Department of Mechanical and Control Engineering, Graduate School of Sceience and Engineering, Tokyo Institute of Technology, 2–12–1 Ookayama, Meguro-ku, Tokyo, 152–8550 Japan E-mail: {hidekazu,shige,mxo}@ok.ctrl.titech.ac.jp Abstract In this paper, we propose a novel approach for image correspondence estimation using a pair of synchronized image sequences. In the proposed approach, after tracking the feature points in each image sequence over several frames, we utilize the consistent epipolar constraints on the image pairs for fitting each trajectory in one image sequence to the motion subspace derived from all trajectories in the other sequence. Then the stereo correspondence of each trajectory is obtained. Dissimilarly to the conventional stereo correspondence estimation based on matching using pixel values, the proposed approach enables us to obtain the image correspondences even though the trajectories are observed in only one image sequence. The validity of the proposed approache is shown by the experiments using synthetic and real images. Key words spacio-temporal imagesɼimage correspondence estimation, subspace constraints, epipolar constraints ͜ͷΑ͏ͳը૾ؒͷରԠ͚͔ͮΒ 3 ࣍‫ݩ‬৘ใΛऔಘ͢Δ. 1. ͸ ͡ Ί ʹ. ୅දతͳٕज़Ͱ͋Δɽͨͩ͠ɼ͜ͷ 2 ख๏ʹ͓͚ΔରԠ. ը૾͔Βγʔϯͷ 3 ࣍‫ݩ‬৘ใΛऔಘ͢ΔͨΊʹ͸ɼෳ ਺ͷը૾ؒͰͷ࠲ඪͷରԠ͚ͮΛߦ͏͜ͱ͕ෆՄܽͰ͋ Γɼ͜ͷରԠ͚ͮ͸ɼҰൠʹըૉ஋ͷ࣋ͭ৘ใΛར༻͠ ͯߦΘΕΔɽSFM(shape from motion) ͱεςϨΦ๏͸ɼ ——————————————————. ͚ͮͷ೉қ౓͸ͦΕͧΕҟͳΔɽ. SFM(ྫ͑͹ [1], [2]) Ͱ͸ɼྡ઀͢Δը૾ϑϨʔϜ͸ඍ খʹมԽ͢Δࢹ఺ҠಈʹΑͬͯಘΒΕɼ2 ϑϨʔϜؒͰ ͷըૉ஋͸΄ͱΜͲมԽ͠ͳ͍ͨΊɼը૾ؒͷରԠ͚ͮ ͸ൺֱత༰қͰ͋Δɽͨͩ͠ɼΧϝϥӡಈΛಉ࣌ʹਪఆ ͢Δࠔ೉͞΍ɼઈରతͳ 3 ࣍‫ݩ‬৘ใΛऔಘ͢Δ͜ͱ͸Ͱ. † ‫ࡏݱ‬͸ (‫ )ג‬σϯιʔ (˟ 448-8661 Ѫ஌‫מݝ‬୩ࢢত࿨ொ 1-1) ʹ‫ۈ‬຿ɽ. −163− 1.

(2) ͖ͳ͍͜ͱͳͲ͕ܽ఺ͱͯ͠‫͛ڍ‬ΒΕΔɽҰํɼεςϨ. ख๏ΛఏҊ͍ͯ͠Δɽ͜ͷํ๏͸ɼҼࢠ෼ղ๏͕ಛ௃఺. Φ๏Ͱ͸ɼΧϝϥҐஔ͸ࣄલʹΩϟϦϒϨʔγϣϯͰ͖. ‫੻ي‬ΛΞϑΟϯ෦෼ۭؒʹ͋ͯ͸ΊΔ͜ͱ͔Β [7]ɼఏҊ. ΔͨΊ‫ط‬஌Ͱ͋Γɼը૾ؒͷରԠ෇͚ͷΈ͔Βγʔϯͷ. ख๏ͱྨࣅͨ͠ํ๏ͱ‫͑ݴ‬ΔɽHo Β͸ɼ2 ͭͷ࣌‫ྻܥ‬ը. 3 ࣍‫ݩ‬৘ใ͕ಘΒΕΔɽ͔͠͠ɼSFM ͱൺֱ͢Δͱɼେ. ૾ͷ‫ͱ੻ي‬Χϝϥ֎෦ύϥϝʔλ͔Βಋग़͞ΕΔߦྻ͕. ͖͘ҟͳΔࢹ఺Ґஔ͔ΒࡱӨ͞Εͨը૾Λར༻͢ΔͨΊɼ. 4 ࣍‫ݩ‬෦෼ۭؒΛுΔ͜ͱΛར༻͠ɼͦͷ෦෼ۭؒͷ‫ج‬ ఈΛ‫ٻ‬ΊΔ͜ͱʹΑΓεςϨΦը૾ؒͷରԠΛ‫ٻ‬Ί͍ͯ Δɽͨͩ͠ɼ͜ͷख๏Ͱ͸ɼ෦෼ۭؒͷ‫ج‬ఈΛ‫ٻ‬ΊΔࡍ ʹɼεςϨΦը૾ؒͷ 4 ఺Ҏ্ͷରԠΛಘΔ͜ͱ͕ෆՄ ܽͰ͋Δͷʹର͠ɼఏҊख๏͸ɼεςϨΦը૾ؒͷରԠ ͸શ͘ඞཁͱ͠ͳ͍ɽ͢ͳΘͪɼఏҊख๏Ͱ͸ɼεςϨ ΦΧϝϥ͕෺ମͷදཪΛͦΕͧΕ‫؍‬ଌ͠ɼ2 ͭͷΧϝϥ ʹಉ࣌ʹ‫؍‬ଌ͞Εͨಛ௃఺͕શ͘ଘࡏ͠ͳ͍‫ͳ୺ۃ‬৔߹ Ͱ΋ɼରԠΛ‫ٻ‬ΊΔ͜ͱ͕ՄೳͰ͋Δɽ ຊ࿦จͷߏ੒͸ҎԼͷͱ͓ΓͰ͋Δɼ·ͣɼ2 ষͰ͸ɼ ࣌‫ྻܥ‬ը૾தͷ‫͕੻ي‬༗͢Δ෦෼ۭؒ߆ଋʹ͍ͭͯ֓આ ͢Δɽ3 ষͰ͸ɼҰํͷ࣌‫ྻܥ‬ը૾͔ΒಘΒΕͨ‫੻ي‬Λɼ εςϨΦը૾ؒͷΤϐϙʔϥ߆ଋར༻ͯ͠෦෼ۭؒʹ͋ ͯ͸ΊΔఏҊख๏ʹ͍ͭͯઆ໌͢Δɽ4 ষͰ͸ɼ෦෼ۭ ؒͷ࣍‫ݩ‬Λద੾ʹਪఆ͢ΔͨΊͷ 1 ͭͷΞϓϩʔνͱ͠ ͯɼ3 ࣍‫ܭݩ‬ଌ݁ՌΛը૾্ʹ౤Өͨ݁͠Ռͱɼը૾͔ ΒಘΒΕͨಛ௃఺‫ͱ੻ي‬Λൺֱ͢Δํ๏ʹ͍ͭͯ঺հ͢ Δɽ5 ষͰ͸ɼ߹੒ը૾ͱ࣮ը૾Λ༻͍࣮ͨ‫ݧ‬Λߦ͍ɼఏ Ҋख๏ͷ༗ޮੑΛࣔ͢ɽ࠷‫ʹޙ‬ຊ࿦จΛ·ͱΊΔɽ. ΦΫϧʔδϣϯ΍౤Ө࿪ΈͳͲͷӨ‫ʹڹ‬Αͬͯը૾ؒͷ ਖ਼͍͠ରԠ͚ͮ͸ࠔ೉ͱͳΔɽ ͦ͜Ͱɼຊ࿦จͰ͸ɼSFM ͷରԠ෇͚ͷ༰қ͞Λར༻ ͯ͠ɼϕʔεϥΠϯ௕ͷେ͖ͳ 2 ͭͷεςϨΦը૾ؒͷ ରԠΛ‫ٻ‬ΊΔख๏ΛఏҊ͢ΔɽఏҊख๏Ͱ͸ɼ‫ྻߦૅج‬ ͕ΩϟϦϒϨʔγϣϯ͞ΕͨεςϨΦΧϝϥ͔ΒಘΒΕ Δ࣌‫ྻܥ‬ը૾Λར༻͢Δɽ͜͜Ͱ͸ɼ࣌‫ྻܥ‬ը૾தͷ֤ ϑϨʔϜؒͷରԠ͸ըૉ஋Λར༻ͯ͠ಘΔ΋ͷͷɼες ϨΦը૾ؒͷରԠ͸ըૉ஋Λར༻ͤͣɼಛ௃఺‫͕੻ي‬༗ ͢Δ෦෼ۭؒ߆ଋ [2] ͱɼεςϨΦը૾ؒʹ੒ཱ͢ΔΤϐ ϙʔϥ߆ଋͱΛར༻ͯ͠୅਺తʹ‫ٻ‬ΊΔɽ ఏҊख๏Ͱ͸ɼ·ͣɼ‫ج‬४ը૾ྻ͔Βಛ௃఺Λ௥੻͠ɼ ‫ج‬४ը૾ྻͷશಛ௃఺‫ʹ੻ي‬ΑͬͯுΒΕΔ෦෼ۭؒΛ ਪఆ͢Δ (෦෼ۭؒΛ‫ٻ‬ΊͨҰํͷը૾ྻΛ‫ج‬४ը૾ྻ ͱ͠ɼଞํΛࢀরը૾ྻͱ͢Δ)ɽ͜͜Ͱ͸ɼΞϑΟϯ ΧϝϥϞσϧʹΑͬͯಘΒΕΔ 3 ࣍‫ݩ‬෦෼ۭؒΛ֦ு͠ ͯɼҰൠతͳΧϝϥϞσϧΛ૝ఆͨ͠ΑΓߴ࣍‫ݩ‬ͷ෦෼ ۭؒΛ૝ఆ͢Δɽͦͯ͠ɼεςϨΦը૾ؒͷΤϐϙʔϥ ߆ଋΛར༻ͯ͠ɼࢀরը૾ྻͷಛ௃఺‫੻ي‬Λͦͷ෦෼ۭ. 2. ෦෼ۭؒ߆ଋ. ؒʹ͋ͯ͸ΊΔɽ͜Ε͸ɼࢀরը૾ྻ͔ΒಘΒΕ֤ͨಛ. ·ͣɼ࣌‫ྻܥ‬ը૾தͷ‫͕੻ي‬༗͢ΔΞϑΟϯ෦෼ۭؒ [4]. ௃఺‫ʹ੻ي‬ରԠͨ͠‫ج‬४ը૾ྻͷಛ௃఺‫੻ي‬Λ‫ٻ‬ΊΔ͜ ͱΛҙຯ͠ɼࢀরը૾ྻͷಛ௃఺ʹର͢Δશ࣌ࠁͷες. ʹ͍ͭͯ֓આ͢Δɽ͜ͷ෦෼ۭؒ͸ɼSFM ͷ୅දతͳख. ϨΦରԠ͕ಘΒΕΔɽ͜ͷํ๏Λར༻͢Δͱɼ‫ج‬४ը૾. ๏Ͱ͋ΔҼࢠ෼ղ๏ [2], [3] ΍ɼϞʔγϣϯηάϝϯςʔ. ྻͰ෦෼ۭؒΛுΔ͜ͱ͕Ͱ͖Ε͹ɼࢀরը૾ྻʹରԠ. γϣϯ [5], [6], [8] ͓͍ͯ޿͘ར༻͞Ε͍ͯΔɽ. ͢Δಛ௃఺‫ج͕੻ي‬४ը૾ྻʹଘࡏ͢Δඞཁ͸ͳ͘ɼ‫ج‬. ੩ࢭͨ͠ΧϝϥΛ༻͍ͯಈ෺ମΛࡱӨ͠ɼM ຕͷը. ४ը૾Ͱ͸‫؍‬ଌ͞Ε͍ͯͳ͍ಛ௃఺Ͱ͋ͬͯ΋‫ج‬४ը૾. ૾্ͷ N ‫ݸ‬ͷಛ௃఺ Pj , (j = 1, · · · , N ) Λ௥੻͢Δɽ. ্ͷը૾࠲ඪ͕ಘΒΕΔɽ‫ج‬४ը૾ྻͱࢀরը૾ྻͷ໾. i(i = 1, · · · , M ) ຕ໨ͷը૾ʹ͓͚Δಛ௃఺ Pj ͷը૾࠲ ඪΛ pij = (uij , vij )T ͱ͢Δͱɼj ൪໨ͷಛ௃఺ͷ‫੻ي‬ ͸ɼ2M ࣍‫ۭؒݩ‬தͷ 1 ఺ͱͯ͠ҎԼͷΑ͏ʹද͞ΕΔɽ. ׂΛೖΕସ͑ͯಉ༷ͷॲཧΛߦ͑͹ɼҰํͷը૾ʹ͓͍ ͯಘΒΕͨಛ௃఺ͷɼ΋͏Ұํͷը૾্ͷରԠ࠲ඪ͕ಘ ΒΕΔɽ. T. pj = [u1j , v1j , u2j , v2j , . . . , uM j , vM j ]. ఏҊख๏ͱಉ༷ʹɼ࣌‫ྻܥ‬εςϨΦը૾Λར༻ͯ͠ɼε. (1). ςϨΦը૾ؒͷً౓Λൺֱ͢Δ͜ͱͳ͘ରԠ఺Λਪఆ͢. ෺ମ্ʹ೚ҙͷ෺ମ࠲ඪ‫ܥ‬Λ‫ݻ‬ఆ͠ɼϫʔϧυ࠲ඪ‫ܥ‬. Δ࿮૊Έ͸ɼ͍͔ͭ͘ఏҊ͞Ε͍ͯΔɽDornaika Β [11]. Λ೚ҙʹఆΊΔɽi ຕ໨ͷը૾ʹ͓͚Δ෺ମ࠲ඪ‫ܥ‬ͷ‫ݪ‬. ͸ɼ‫ڧ‬ΩϟϦϒϨʔγϣϯࡁΈͷ 2 ୆ͷεςϨΦΧϝϥؒ. ఺Λ τi ͱ͠ɼ֤࣠ͷ‫ج‬ఈϕΫτϧΛɼͦΕͧΕ ii , ji , ki. ͷճసͱฒਐͱɼಛ௃఺ରԠ͔Βਪఆ͞ΕΔ࣌‫ྻܥ‬ը૾. ͱ͢Δɽ·ͨɼಛ௃఺ Pj ͷ෺ମ࠲ඪ‫͚͓ʹܥ‬Δ 3 ࣍‫࠲ݩ‬. ؒͷ ego-motion Λར༻͢Ε͹ɼεςϨΦը૾ؒͷରԠ͸. ඪΛ (aj , bj , cj )T ͱ͢Δɽ͜ͷͱ͖ɼi ຕ໨ͷը૾্ͷಛ. 2 ͭͷΤϐϙʔϥઢͷަ఺ͱͯ͠ಘΒΕΔ͜ͱΛࣔͨ͠ɽ ͔͠͠ɼ͜ͷ৔߹ɼ࣌‫ྻܥ‬ը૾ؒͷ ego-motion ͸ɼඍখ ͳΧϝϥӡಈ͔Βਪఆ͢Δඞཁ͕͋Γɼͦͷ ego-motion ਪఆ͸τϥοΩϯάΤϥʔʹରͯ͠‫ۃ‬Ίͯ੬ऑͰ͋Δ͜ ͱ͕஌ΒΕ͍ͯΔͨΊ [12]ɼ͜ͷํ๏Λ༻͍ͨεςϨΦ ը૾ؒͷରԠਪఆͷਫ਼౓͸௿͍ɽ ҰํɼHo Β [9], [10] ͸ɼ2 ୆ͷΞϑΟϯΧϝϥ͔Βಘ ΒΕΔεςϨΦ࣌‫ྻܥ‬ը૾ʹҼࢠ෼ղ๏ [2] Λద༻ͨ͠. ௃఺ Pj ͷ 3 ࣍‫࠲ݩ‬ඪ rij ͸ɼ࣍ࣜͰද͞ΕΔɽ. rij = τi + aj ii + bj ji + cj ki. (2). ΞϑΟϯΧϝϥϞσϧΛԾఆ͢Δͱɼrij ͕ը૾ʹ౤Ө ͞ΕΔ࠲ඪ͸ɼ࣍ࣜͰද͞ΕΔɽ Pa1 rij + q pij rij = Pa = 1 1 1 ͨͩ͠ɼPa ͸ΧϝϥͷࣹӨߦྻͰ͋Γɼ. −164− 2. (3).

(3) Pa =. Pa1 0T. q 1. . Λ‫ٻ‬ΊΔɽͨͩ͠ɼΧϝϥ͸ಉ‫͍ͯ͠ظ‬Δ΋ͷͱ͠ɼΧ. (4). ϝϥؒͷ‫( ྻߦૅج‬Fundamental Matrix) ͸‫ط‬஌ͱ͢Δɽ. ͱ͢Δɽ͢ͳΘͪɼPa1 ͱ q ͸ɼͦΕͧΕ 2×2 ͱ 2×1 ͷ ߦྻͱϕΫτϧͰ͋Γɼϫʔϧυ࠲ඪ‫ʹܥ‬ର͢ΔΧϝϥ ͷҐஔ΍Χϝϥͷ಺෦ύϥϝʔλʹΑͬͯఆ·Δɽ ࣜ (2) ͱࣜ (3) ΑΓɼ͕࣍ࣜಘΒΕΔɽ uij = Pa1 rij + q pij = vij. = m0i + aj m1i + bj m2i + cj m3i. m2i = Pa1 ji ,. m1i = Pa1 ii ,. m3i = Pa1 ki. ‫ج‬४ը૾ྻ͔ΒಘΒΕͨ‫ط‬஌ͷ‫੻ي‬ϕΫτϧΛ T pj = u1j , v1j , u2j , v2j , . . . , uM j , vM j (11) ͱ͢Δɽj = 1, 2, . . . , N1 ͷશ‫੻ي‬ϕΫτϧΛฒ΂ͯɼ࣍ ͷΑ͏ͳߦྻΛ࡞Δɽ W = p1 , p2 , . . . , pN1. (5). (6) (7). ͱ͢Δɽͦͯ͠ɼࣜ (5) Λɼi ʹ͍ͭͯ 1 ͔Β M ·Ͱॎ. ͨͩ͠ɼ. ʹฒ΂Δͱɼ͕࣍ࣜಘΒΕΔɽ. pj = m0 + aj m1 + bj m2 + cj m3 (8) T ͨͩ͠ɼmh = mTh1 , mTh2 , · · · , mThM , (h = 0, 1, 2, 3) Ͱ͋Δɽ ࣜ (8) ͸ɼ2M ࣍‫ۭؒݩ‬தͷશͯͷಛ௃఺‫੻ي‬ϕΫτϧ pj ͕ m0 Λ௨Γɼm1 ɼm2 ɼm3 ͰுΒΕΔ෦෼ۭؒʹ ‫·ؚ‬ΕΔ͜ͱ͕Θ͔Δɽ͢ͳΘͪɼΞϑΟϯΧϝϥͰࡱ Ө͞Εͨಈ෺ମ্ͷશͯͷಛ௃఺ͷ‫੻ي‬ϕΫτϧ pj ͸ 2M ࣍‫( ۭؒݩ‬ҎԼɼཤྺ࠲ඪۭؒͱ‫ )Ϳݺ‬தͷ 3 ࣍‫ݩ‬෦ ෼ۭؒʹ‫·ؚ‬ΕΔͱ͍͏ੑ࣭͕͋Δ [4]ɽ. p¯j = pj − pG ,. pG =. N1 1 pj N1 j=1. (14). ¯ Λओ੒෼෼ੳ͢Δ͜ͱͰ‫ج‬ఈ Ͱ ͋ Δ ɽͦ ͠ ͯ ɼW e1 , . . . , e2M ΛಘΔʢ஫ 1ʣɽ Χϝϥ͕ΞϑΟϯΧϝϥϞσϧΛຬͨ͢৔߹ɼpj ͸ 3 ࣍‫ݩ‬෦෼ۭؒʹ‫·ؚ‬ΕΔͷͰɼpG ͱ 3 ͭͷ‫ج‬ఈ e1 , e2 , e3 Λ༻͍ͯ࣍ࣜͷΑ͏ʹද͞ΕΔɽ pj = pG + α1j e1 + α2j e2 + α3j e3. (15). ͨͩ͠ɼα1j , α2j , α3j ͸‫ج‬ఈͷ܎਺Ͱ͋Δɽ Χϝϥ͕ΞϑΟϯΧϝϥϞσϧͷ৔߹ɼલड़ͷͱ͓Γ. 3. 2 ૊ͷ࣌‫ྻܥ‬ը૾ؒͷରԠҐஔਪఆ Ҡಈ͢Δ෺ମΛ 2 ͭͷΧϝϥͰࡱӨ͠ɼ‫ج‬४Χϝϥ ͱࢀরΧϝϥ͔ΒಘΒΕΔ 2 ૊ͷ࣌‫ྻܥ‬ը૾ͷͦΕ ͧ Ε ʹ ͓ ͍ ͯ Ҡ ಈ ෺ ମ ্ ͷ ಛ ௃ ఺ Λ τ ϥοΩ ϯ ά ͢ Δɽ͜ͷͱ͖‫ج‬४Χϝϥ͔ΒಘΒΕͨը૾ (ҎԼɼ‫ج‬ ४ը૾ͱ͢Δ) ্ͷಛ௃఺ͷཤྺ࠲ඪΛ (uij , vij )T , (i =. 1, · · · , M ), (j = 1, · · · , N1 ) ͱ͠ɼࢀরΧϝϥ͔ΒಘΒΕ ͨը૾ (ҎԼɼࢀরը૾ͱ͢Δ) ্ͷಛ௃఺ͷཤྺ࠲ඪΛ T (uik , vik ) , (i = 1, · · · , M ), (k = 1, · · · , N2 ) ͱ͢Δɽͨ ͩ͠ɼN1 ͱ N2 ͸ɼ‫ج‬४ը૾ྻͱࢀরը૾ྻʹ͓͍ͯτ ϥοΩϯά͞Εͨಛ௃఺ͷ਺ΛͦΕͧΕද͢ɽ ҎԼͰ͸ɼࢀরը૾্ͷ k ൪໨ͷಛ௃఺ͷཤྺ࠲ඪ T (uik , vik ) , (i = 1, · · · , M ) ʹରԠ͢Δɼ‫ج‬४ը૾্ͷಛ T ௃఺ͷཤྺ࠲ඪ (uik , vik ) , (i = 1, · · · , M ) Λ‫ٻ‬ΊΔ໰ ୊Λߟ͑Δɽ͢ͳΘͪɼࢀরը૾্ͷ k ൪໨ͷಛ௃఺ͷ ཤྺ࠲ඪΛฒ΂Δ͜ͱʹΑͬͯද͞ΕΔ‫ط‬஌ͷ‫੻ي‬ϕΫ τϧ T pk = u1k , v1k (9) , u2k , v2k , . . . , uM k , vM k ʹରԠͨ͠ɼ‫ج‬४ը૾ଆͷະ஌ͷ‫੻ي‬ϕΫτϧ T pk = u1k , v1k , u2k , v2k , . . . , uM k , vM k. (12). W ͷ֤ྻ͔Βશͯͷ‫੻ي‬ϕΫτϧͷॏ৺ϕΫτϧΛҾ͍ ¯ ͱ͠ɼ࣍ͷΑ͏ʹද͢ɽ ͨߦྻΛ W ¯ = p¯1 , p¯2 , . . . , p¯N W (13) 1. ͨͩ͠ɼ. m0i = Pa1 τi + q,. 3. 1 ෦෼ۭؒੜ੒. (10). ‫ج‬ఈ਺͸ 3 Ͱे෼Ͱ͋Δ͕ɼ࣮ࡍͷΧϝϥ͸ΞϑΟϯΧ ϝϥϞσϧͱ͸‫ʹີݫ‬͸ҟͳΔɽͦͷͨΊɼ࣮ࡍͷσʔ λ͸ 3 ࣍‫ݩ‬ΑΓߴ͍࣍‫ݩ‬ͷ෦෼ۭؒʹ΋‫·ؚ‬ΕΔͱߟ͑ ΒΕΔɽͦ͜ͰɼҎԼͰ͸ɼΧϝϥϞσϧΛΑΓҰൠԽ ࣍͠ͷΑ͏ʹɼΑΓߴ࣍ͷ෦෼ۭؒΛར༻͢Δɽ. pj = pG + α1j e1 + α2j e2 + . . . + ανj eν , (3 < = ν) (16) 3. 2 ରԠ఺ਪఆ. ࢀরը૾্ͷ k ൪໨ͷಛ௃఺‫੻ي‬ϕΫτϧ pk ͱɼͦΕ. ʹରԠ͢Δ‫ج‬४ը૾্ͷ‫੻ي‬ϕΫτϧ pk ͸ɼը૾ؒͷ Τϐϙʔϥ߆ଋʹΑͬͯؔ܎͚ͮΔ͜ͱ͕Ͱ͖Δɽ͢ͳ Θͪɼi ൪໨ͷϑϨʔϜͰ͸ɼ͕࣍ࣜ੒ཱ͢Δɽ ⎤ ⎡ uik ⎢ ⎥ (17) 1 F ⎣ vik uik vik ⎦ = 0,. ⎡. f11 ⎢ where F = ⎣ f21 f31. 1 f12 f22 f32. ⎤ f13 ⎥ f23 ⎦ f33. (18). ¯ W ¯ T Λ࡞੒͠ɼϞʔϝϯτߦྻͷ‫ݻ‬༗ϕΫτ ʢ஫ 1ʣɿϞʔϝϯτߦྻ M = W ϧΛ‫ٻ‬ΊΔͱɼͦͷ‫ݻ‬༗ϕΫτϧ͕‫ٻ‬ΊΔ‫ج‬ఈ e1 , . . . , e2M ʹରԠ͢Δ. 3 −165−.

(4) ·ͨɼࢀরը૾্ͷ k ൪໨ͷಛ௃఺‫੻ي‬ϕΫτϧ pk. ʹରԠ͢Δ‫ج‬४ը૾্ͷ‫ ੻ي‬pk ͸ɼ‫ج‬४ը૾্ͷ‫੻ي‬. pj , (j = 1, · · · , N1 ) ͕நग़͞ΕͨಉҰ෺ମ্ʹ͋ΔͷͰɼ ࣜ (16) Ͱද͞Εͨ෦෼ۭؒʹଐ͢Δ͸ͣͰ͋Δɽ͢ͳΘ ͪɼ͕࣍ࣜ੒ཱ͢Δɽ pk = pG + β1k e1 + β2k e2 + . . . + βνk eν , (3 < = ν) (19). ͱ͢Δɽ ࣜ (24) ͸ɼν ‫ݸ‬ͷະ஌਺ βnk (n = 1, 2, . . . , ν) ʹରͯ͠ ࣜ਺͕ M ‫ͳݸ‬ͷͰɼν Λ (3 < =ν< = M ) ͷൣғͰબ୒͢ Ε͹ɼະ஌਺ βnk Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖ɼβnk ͕ಘΒΕΕ ͹ɼࣜ (16) ͔Β‫ج‬४ը૾্ͷ‫͖Ͱࢉܭ͕੻ي‬Δ͜ͱ͕Θ ͔Δɽ ࣜ (24) Λ࣍ͷΑ͏ʹॻ͘ɽ. ͨͩ͠ɼβnk , (n = 1, 2, · · · , ν) ͸ɼn ൪໨ͷ‫ج‬ఈͷ܎਺Ͱ. Ak xk = bk. ͋Δɽ ·ͣɼࣜ (19) ͔Βɼi ൪໨ͷϑϨʔϜʹؔ͢ΔࣜΛந ग़͢Δɽ uik vik. (i). (i). ͨͩ͠ɼ. (28) Ak = MkT F1 e1 . . . eκ ⎡ ⎤ β1k ⎢ . ⎥ T T ⎢ xk = ⎣ .. ⎥ ⎦ , bk = −Mk F1 pG − Mk F2 (29) βνk. (i). = pG + β1k e1 + β2k e2 + . . . + βνk e(i) ν (20). (i). (i). (i). ͨͩ͠ɼpG , e1 , . . . , eν ͸ɼpG , e1 , . . . , eν ͷͦΕͧΕ. ͷϕΫτϧ͔Β 2i − 1 ͱ 2i ൪໨ͷߦΛऔΓग़ͨ͠ϕΫτ. Ͱ͋Δɽࣜ (27) Λ. ϧΛද͢ɽͦͯ͠ɼࣜ (17) ʹࣜ (20) Λ୅ೖ͢Δͱɼi ൪ ໨ͷϑϨʔϜʹ͍ͭͯɼ͕࣍ࣜಘΒΕΔɽ ⎧ ⎡ 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎢ β ⎢ 1k (i) (i) ⎢ . ˜ T u f1 p(i) e1 . . . eν ik G ⎢ . ⎪ ⎪ ⎣ . ⎪ ⎪ ⎩ βνk. ⎤. ⎥ ⎥ ⎥ + f2 ⎥ ⎪ ⎪ ⎦ ⎪ ⎪ ⎭. =0 ͨͩ͠ɼ. (21). xk = (ATk Ak )−1 ATk bk. (30). ͷΑ͏ʹղ͖ɼβnk (n = 1, 2, . . . , ν) Λ‫ٻ‬ΊΔɽ͜ΕʹΑ Γɼࢀরը૾্ͷ k ൪໨ͷ‫੻ي‬ϕΫτϧ pk ʹର͢Δ‫ج‬. ४ը૾্ͷ‫੻ي‬ϕΫτϧ pk ͕ಘΒΕΔɽ. ͜ͷํ๏Λɼશͯͷ k ʹ͍ͭͯͦΕͧΕ‫ٻ‬ΊΔ͜ͱʹ. ΑΓɼࢀরը૾ʹ͔͍ࣸͬͯ͠ͳ͍ಛ௃఺Ͱ΋ɼͦͷಛ ௃఺ʹରԠ͢Δ‫ج‬४ը૾্ͷಛ௃఺ͷҐஔΛશϑϨʔϜ ʹΘͨͬͯਪఆ͢Δ͜ͱ͕Ͱ͖Δɽ‫ج‬४ը૾ͱࢀরը૾. 1 uik vik ⎤ ⎡ f11 f12 ⎥ ⎢ = ⎣ f21 f22 ⎦ , f31 f32. ˜ T u ik = f1. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬. (27). . ͷ໾ׂΛೖΕସ͑ͯಉ༷ͷ‫ࢉܭ‬Λߦ͏͜ͱʹΑΓɼҰํ. (22). ͷΧϝϥʹ͔͍ࣸͬͯ͠ͳ͍ಛ௃఺Ͱ΋ɼ྆ํͷΧϝϥ. f13 ⎥ ⎢ = ⎣ f23 ⎦ (23) f33. ʹରԠ͚ͮΛߦͳ͏͜ͱ͕Ͱ͖Δɽ͢ͳΘͪɼ2 ୆Ͱಘ. ⎤. ⎡. f2. ΒΕͨը૾্ͷશͯͷಛ௃఺ͷରԠ͚ͮΛߦͳ͏͜ͱ͕ ՄೳͰ͋Δɽ. ͱ͢Δɽ͞ΒʹɼશϑϨʔϜ (i = 1, · · · , M ) ʹ͍ͭͯࣜ. 4. 3 ࣍‫ߏ࠶ݩ‬੒ʹΑΔ෦෼ۭؒͷ‫ج‬ఈ਺ͷܾఆ. (21) Λॎʹฒ΂Δͱɼ͕࣍ࣜಘΒΕΔɽ ⎧ ⎡ 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎢ ⎢ β1k MkT F1 pG e1 . . . eν ⎢ ⎢ .. ⎪ ⎪ ⎣ . ⎪ ⎪ ⎩ βνk = 0. ΞϑΟϯΧϝϥϞσϧΛԾఆͨ͠৔߹ɼ෦෼ۭؒͷ‫ج‬. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬. ⎤. ఈ਺Λ ν = 3 ͱ͢Δ͜ͱ͕Ͱ͖Δ͕ɼఏҊख๏Ͱ͸ɼΑ Γ൚༻తͳΧϝϥϞσϧΛ૝ఆ͠ɼν > = 3 ͱ͍ͯ͠Δɽ͜. ⎥ ⎥ ⎥ + F2 ⎥ ⎪ ⎪ ⎦ ⎪ ⎪ ⎭. ͷ ν ͷ஋Λܾఆ͢Δ͜ͱ͸༰қͰ͸ͳ͍͕ɼຊষͰ͸ɼ ‫ج‬ఈ਺ ν ΛదԠతʹܾఆ͢ΔͨΊͷ 1 ͭͷํ๏ͱͯ͠ɼ ରԠ͔ΒಘΒΕΔର৅෺ମͷ 3 ࣍‫ߏ࠶ݩ‬੒݁ՌΛར༻͠. (24). Ͱ͸ɼ‫ʹྻߦૅج‬Ճ͑ɼεςϨΦΧϝϥͷ಺෦ύϥϝʔ. ͨͩ͠ɼ. ⎡ MkT. ͨख๏ʹ͍ͭͯड़΂Δɽͨͩ͠ɼҎԼʹड़΂Δ ν ͷܾఆ. ⎢ ⎢ =⎢ ⎢ ⎣ ⎡. ⎢ F1 = ⎢ ⎣. ˜ T u 2k. .. ⎤. f1 ... λΛ‫ط‬஌ͱ͢Δɽ. ⎤. ˜ T u 1k. . ˜ T u Mk. ⎥ ⎥ , F2 ⎦. . f1. ຊষͰड़΂Δํ๏Ͱ͸ɼશϑϨʔϜʹ౉ͬͯશಛ௃఺. ⎥ ⎥ ⎥, ⎥ ⎦ ⎡. (25). Λ 3 ࣍‫ߏ࠶ݩ‬੒ͨ݁͠Ռ͕ɼ3 ࣍‫ݩ‬෦෼ۭؒʹ‫·ؚ‬ΕΔ ͜ͱΛར༻͠ɼͦͷ෦෼ۭؒʹ͋ͯ͸Ίͨ࣌ͷ࢒͕ࠩখ ͘͞ͳΔΑ͏ʹɼν Λܾఆ͢Δɽ͢ͳΘͪɼϑϨʔϜ͝. ⎤. f2 ⎢ . ⎥ ⎢ = ⎣ .. ⎥ ⎦ f2. ͱʹಘΒΕΔ 3 ࣍‫ܗݩ‬ঢ়͕ɼશϑϨʔϜʹ౉ͬͯҰ؏ੑ ͕͋Δ͜ͱΛධՁ‫ج‬४ͱ͢ΔɽҎԼͰ͸ɼ·ͣɼ͜ͷ 3. (26). ࣍‫ݩ‬෦෼ۭؒʹ͍ͭͯઆ໌͠ɼ࣍ʹɼͦͷ෦෼ۭؒ΁ͷ ͋ͯ͸ΊΛར༻ͨ͠ ν ͷܾఆํ๏Λઆ໌͢Δɽ. 4 −166−.

(5) 3 ষʹड़΂ͨํ๏ʹΑͬͯಘΒΕͨୈ i ൪ϑϨʔϜʹ. Camera02. ͓͚Δ k ൪໨ͷಛ௃఺ͷରԠ͔ΒɼΧϝϥͷ಺෦ɾ֎෦ ύϥϝʔλΛར༻ͯ͠ɼͦͷಛ௃఺ͷ 3 ࣍‫࠲ݩ‬ඪΛ‫ࢉܭ‬. 0.3[m]. T. ͢ΔɽಘΒΕͨ 3 ࣍‫࠲ݩ‬ඪΛ Xik = (xij , yij , zij ) ͱ͠ɼ. Object. શϑϨʔϜʹ͍ͭͯฒ΂ͨϕΫτϧ Xk Λ࣍ࣜͰද͢ɽ. Xk =. . X1k. X2k. . . . XM k. T. (31). Z. i ൪໨ͷϑϨʔϜͷϫʔϧυ࠲ඪ‫ʹܥ‬ର͢Δ෺ମ࠲ඪ ‫ܥ‬ͷ‫ج‬ఈϕΫτϧΛɼͦΕͧΕ ii ɼji ɼki ͱ͠ɼ෺ମ࠲ ඪ‫ܥ‬ͷ‫఺ݪ‬Λ τi ͱ͢Δɽk ൪໨ͷಛ௃఺ͷ 3 ࣍‫࠲ݩ‬ඪΛ ෺ମ࠲ඪ‫Ͱܥ‬දͨ͠ͱ͖ͷ࠲ඪΛɼ(aik , bik , cik )T ͱ͢Δ ͱɼXik ͸࣍ࣜͰද͞ΕΔɽ ⎤ ⎡ xik ⎥ ⎢ Xik = ⎣ yik ⎦ = τi + aik ii + bik ji + cik ki (32). 5.0[m] Y ਤ1. Αͬͯɼ্ࣜΛશϑϨʔϜʹΘͨͬͯॎʹฒ΂Δ͜ͱͰ. 5200. Xk ΛಘΔɽ. 5000. ⎢ ⎢. =⎢ ⎢. ⎣. 1 2. .. .. 4600. 1. M. -800 1200. 100th frame. 800. 50th frame. 400. 0. -400. Y. 1st frame trajectory. Z. ⎡. ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ 2 ⎥ ⎢ ⎥, = ⎢ . ⎥, = ⎢ . ⎥, = ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎦ ⎣ . ⎦ ⎣ . ⎦ ⎣. M. 0. X. 4800. (33). 1. γϛϡϨʔγϣϯ࣮‫͚͓ʹݧ‬ΔΧϝϥͱ෺ମͷ഑ஔ. 800 5400. ͨͩ͠ɼτ , i, j, k ͸ɼ࣍ࣜͰද͞ΕΔɽ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤. X. Camera01. zik. Xk = τ + ak i + bk j + ck k. 120[deg]. M. ⎤ 1. ਤ2. ⎥ ⎥ ⎥ ⎥ ⎦. 2. .. .. ‫ٿ‬ͷ‫֤࣠( ੻ي‬ͷ୯Ґ͸ mm). ¯ ͷ֤ߦϕΫτϧ͕ɼ3 ݁ՌΛࣔ͢ 3M × N2 ͷߦྻ X ࣍‫ݩ‬෦෼ۭؒʹ‫·ؚ‬ΕΔ͜ͱΛҙຯ͍ͯ͠Δɽͦ͜Ͱɼ 3< = min(M, N2 − 1) ͷൣғͰ ν ΛมԽͤ͞ɼͦΕͧ =ν< ¯ ΛಘΔɽͦͯ͠ɼX ¯ Λओ੒෼෼ੳ͠ɼ Εͷ ν ʹ͓͚Δ X. M. (34). ୈ 4 ൪໨ҎԼͷ‫د‬༩཰Λ༻͍ͯσʔλΛ 3 ࣍‫ݩ‬෦෼ۭؒ ͞Βʹɼશಛ௃఺ΛͳΒ΂Δ͜ͱͰ࣍ࣜΛಘΔɽ X = X1 X2 . . . XN2 ⎤ ⎡ 1 1 ... 1 ⎥ ⎢ a a ... a ⎢ 1 2 N2 ⎥ = τ i j k ⎢ ⎥ (35) ⎣ b1 b2 . . . bN2 ⎦. c1. c2. .... ʹ౰ͯ͸Ίͨ৔߹ͷ࢒ࠩΛ‫͠ࢉܭ‬ɼͦͷ࢒͕ࠩ࠷খʹͳ ΔΑ͏ʹ‫ج‬ఈ਺ ν Λܾఆ͢Δɽ ·ͨɼ͜ͷΑ͏ʹͯ͠ಘΒΕͨ ν Λ༻͍ͯಘΒΕΔ 3 ࣍‫ݩ‬෮‫݁ݩ‬Ռ͸ɼτϥοΩϯάΤϥʔʹΑΔ‫ࠩޡ‬Λ‫ؚ‬Μ ¯ Λ 3 ࣍‫ݩ‬෦෼ۭؒʹ͋ Ͱ͍Δͱߟ͑ΒΕΔɽͦ͜ͰɼX ͯ͸Ίͨͱ͖ͷ 3 ࣍‫ܗݩ‬ঢ়Λਪఆ݁Ռͱ͢Δ͜ͱʹΑΓɼ. cN2. τϥοΩϯάΤϥʔʹΑΔ‫ࠩޡ‬Λ௿‫͢ݮ‬Δɽ. ෺ମ࠲ඪ‫ܥ‬ͷ‫఺ݪ‬͸Ͳ͜ʹઃఆͯ͠΋Α͍ͷͰɼಛ௃ ఺ͷ 3 ࣍‫࠲ݩ‬ඪͷॏ৺Λ‫ʹ఺ݪ‬ઃఆ͢Δͱɼ࣍ࣜΛಘΔɽ ¯ = X ¯1 X ¯2 . . . X ¯N X 2 ⎤ ⎡ a1 a2 . . . aN2 ⎥ ⎢ = i j k ⎣ b1 b2 . . . bN2 ⎦ (36). c1. c2. .... cN2. ఏҊख๏ͷ༗ޮੑΛௐ΂ΔͨΊʹɼγϛϡϨʔγϣϯ ࣮‫࣮ͱݧ‬ը૾࣮‫ݧ‬ΛߦͳͬͨɽຊষͰ͸ɼ͜ΕΒͷ݁Ռ ʹ͍ͭͯड़΂Δɽ 5. 1 γϛϡϨʔγϣϯ࣮‫ݧ‬. γϛϡϨʔγϣϯ࣮‫Ͱݧ‬͸ɼਤ 1 ʹࣔ͢Α͏ʹɼΧϝ ϥͱର৅෺ମΛ഑ஔͨ͠ɽର৅෺ମ͸‫͋Ͱٿ‬Γɼ2 ୆ͷ. ͨͩ͠ɼ. ¯ k = Xk − τ , X. 5. ࣮ ‫ ݁ ݧ‬Ռ. Χϝϥ͸‫ٿ‬ͷத৺఺͔Β 5.0[m] ཭Ε͍ͯΔɽ2 ୆ͷΧϝ. τ =. N2 1 Xk N2. ϥ͸ͱ΋ʹը֯͸ 7.8[deg]ɼয఺‫཭ڑ‬͸ 35[mm] Ͱ͋Δɽ. (37). ‫ٿ‬͸શͯͷϑϨʔϜʹΘͨͬͯ 1 ฏ໘্ΛϥϯμϜʹҠ. k=1. ಈ͓ͯ͠Γɼ͜ͷͱ͖ͷ‫ٿ‬ͷҠಈ‫੻ي‬Λਤ 2 ʹࣔ͢ɽ ‫ ʹ্ٿ‬200 ఺ͷಛ௃఺Λ഑ஔ͠ɼ2 ୆ͷΧϝϥͦΕͧ. Ͱ͋Δɽ ͢ͳΘͪɼࣜ (36) ͸ɼશϑϨʔϜͷશಛ௃఺ͷ෮‫ݩ‬. ΕͰಛ௃఺ΛࡱӨ͢ΔɽΧϝϥ 1 Ͱ͸ 71 ఺ɼΧϝϥ 2 Ͱ. −167− 5.

(6) ਤ3. RMSE Error of Inferred Feature Position [pix]. Camera01. Camera02. શϑϨʔϜͰ‫؍‬ଌ͞Εͨಛ௃఺ (ࠨ:Χϝϥ 1, ӈ:Χϝϥ 2). ͸ 77 ఺ͷಛ௃఺Λ 100 ϑϨʔϜʹΘͨͬͯ௥੻ͨ͠ɽ͜. 1.2 Camera01 1 0.8 0.6 0.4 0.2 0. 0. ͷͱ͖τϥοΩϯάͨ͠ಛ௃఺Λਤ 3 ʹࣔ͢ɽ֤ϑϨʔ. ਤ4 RMSE Error of Inferred Feature Position [pix]. ϜʹͯಘΒΕͨ 2 ຕͷը૾ͷ֤ಛ௃఺ʹରͯ͠ɼը૾࠲ ඪ্Ͱฏ‫ ۉ‬0ɼඪ४ภࠩ σ[pix] ͷਖ਼‫ن‬ཚ਺ʹΑΔτϥο Ωϯά‫ࠩޡ‬ΛՃ͑ɼ3 ষʹͯड़΂ͨख๏ͱɼ4 ষʹͯड़΂ ͨ෦෼ۭؒ࣍‫ݩ‬ͷܾఆ๏Λ૊Έ߹ΘͤͨରԠҐஔਪఆΛ ߦͬͨɽ ఏҊख๏ʹΑͬͯಘΒΕͨରԠ఺Ґஔͷਪఆਫ਼౓Λਤ. 4ɼ5 ʹࣔ͢ɽਤ 4 ͸Χϝϥ 2 Λ‫ج‬४ը૾ͱͨ͠৔߹ͷਪ ఆ݁ՌΛࣔ͠ɼτϥοΩϯά‫͚͓ʹࠩޡ‬Δ σ(ԣ࣠) ͱɼΧ ϝϥ 1 ʹ͍ࣸͬͯΔಛ௃఺ΛΧϝϥ 2 ͷը૾্ʹ౤Ө͠ ͨࡍͷɼશϑϨʔϜʹ౉Δશಛ௃఺ͷਅͷ࠲ඪͱਪఆ͠ ͨ࠲ඪͱͷ RMSE(ॎ࣠) Λ͍ࣔͯ͠Δɽಉਤͷ݁Ռ͸ɼ 1 ͭͷ σ ʹ͍ͭͯ 20 ճରԠ఺ਪఆΛߦ͍ɼ֤ σ ʹ͓͚Δ ฏ‫ࠩޡۉ‬Λ‫ͨ͠ࢉܭ‬΋ͷͰ͋Δɽຊ࣮‫Ͱݧ‬͸ɼ෦෼ۭؒ ͷ‫ج‬ఈ਺͸ɼ͍ͣΕͷ σ ʹ͓͍ͯ΋ ν = 3 ͕બ୒͞Εͯ ͍Δɽ·ͨɼਤ 5 ͸ɼΧϝϥ 1 Λ‫ج‬४ը૾ͱͨ͠৔߹ͷ ਪఆ݁ՌΛࣔ͠ɼਤ 4 ͱಉ༷ʹɼΧϝϥ 2 ʹ͍ࣸͬͯΔ ಛ௃఺ΛΧϝϥ 1 ͷը૾্ʹ౤Өͨ͠ࡍͷɼਅͷ࠲ඪͱ ਪఆͨ͠࠲ඪͱͷ RMSE(ॎ࣠) Λ͍ࣔͯ͠Δɽ͍ͣΕʹ ͓͍ͯ΋ɼσ = 0 ͷͱ͖͸ɼରԠҐஔਪఆͷ RMSE ͸θ ϩͰ͋Γɼྑ޷ͳରԠҐஔਪఆ͕Ͱ͖͍ͯΔɽ·ͨɼτ ϥοΩϯά‫͕͋ࠩޡ‬Δ৔߹͸ɼඪ४ภࠩ σ ΑΓ΋ɼਪఆ ͨ͠ରԠͷ RMSE ͷํ͕௿͘ɼ4 ষʹͯड़΂ͨ 3 ࣍‫࠶ݩ‬ ߏ੒݁ՌΛར༻͢Δ͜ͱʹΑΓɼτϥοΩϯά‫ࠩޡ‬Λ௿ ‫ͨ͠ݮ‬ਪఆ͕Ͱ͖͍ͯΔ͜ͱ͕Θ͔Δɽ ࣍ʹɼΞϑΟϯΧϝϥޮՌͱબ୒͞Εͨ‫ج‬ఈ਺ͷؔ܎ Λௐ΂ͨɽը૾্Ͱͷ෺ମͷେ͖͞ɼҠಈྔ͕΄΅ಉ͡ ʹͳΔΑ͏ʹɼ3 ࣍‫ۭؒݩ‬தͰͷ෺ମͷҐஔͱҠಈྔɼ͓ ΑͼΧϝϥͷয఺‫( ཭ڑ‬9.0[mm]ʙ49.0[mm]) ΛมԽͤ͞ɼ ΞϑΟϯΧϝϥϞσϧͷޮՌ (ॳ‫ظ‬ϑϨʔϜʹ͓͚Δ෺ ମͱΧϝϥͱͷ‫཭ڑ‬Λɼ෺ମͷްΈͰׂͬͨ஋) ΛมԽ ͤ͞ͳ͕ΒɼରԠҐஔਪఆΛߦͬͨɽਤ 6ɼ7 ͸ɼΧϝϥ 2 ͷը૾ͱΧϝϥ 1 ͷը૾ΛͦΕͧΕ‫ج‬४ը૾ͱͨ͠ͱ ͖ɼΞϑΟϯΧϝϥϞσϧͷޮՌ (ԣ࣠) ͱɼબ୒͞Εͨ ‫ج‬ఈ਺ ν(ॎ࣠) ͱͷؔ܎Λ͍ࣔͯ͠Δɽಉਤͷ֤άϥϑ ͸ɼτϥοΩϯά‫ࠩޡ‬ͷඪ४ภࠩ σ Λ 0.2 ∼ 2.0[pix] ͷ ؒͰมԽͤͨ͞৔߹ʹ͓͚Δબ୒͞Εͨ‫ج‬ఈ਺Λࣔͯ͠ ͍Δɽͦͷ݁Ռ͸ɼ1 ͭͷ σ ʹ͍ͭͯ 20 ճࢼߦΛ͘Γ͔ ͑͠ɼબ୒͞Εͨ‫ج‬ఈ਺ͷฏ‫ۉ‬Λ‫ͨ͠ࢉܭ‬΋ͷͰ͋Δɽ ਤ 6ɼ7 ʹΑΓɼΞϑΟϯϞσϧͱͷဃ཭͕େ͖͘ͳΔ ΄Ͳ (ԣ࣠ͷ஋͕খ͘͞ͳΔ΄Ͳ)ɼ·ͨτϥοΩϯά‫ޡ‬. 0.5. 1 1.5 2 Tracking Error (standard dev)[pix]. 2.5. 3. ରԠ఺ਪఆਫ਼౓ (Χϝϥ 1). 1.2 Camera02 1 0.8 0.6 0.4 0.2 0. 0. 0.5. ਤ5. 1 1.5 2 Tracking Error (standard dev)[pix]. 2.5. 3. ରԠ఺ਪఆਫ਼౓ (Χϝϥ 2). ͕ࠩখ͘͞ͳΔ΄ͲɼΑΓେ͖ͳ‫ج‬ఈ਺͕બ୒͞ΕΔ܏ ޲͕͋Δ͜ͱ͕෼͔ΔɽΞϑΟϯΧϝϥ͔Βͷဃ཭͕େ ͖͍৔߹ɼν = 3 ͷ෦෼ۭؒʹ͸͋ͯ͸·Βͳ͍ͨΊɼ େ͖ͳ࣍‫ݩ‬ͷ෦෼ۭ͕ؒඞཁͱͳΔɽ͔͠͠ɼτϥοΩ ϯά‫͕ࠩޡ‬େ͖͘ͳΔͱɼߴ࣍‫ݩ‬ͷ෦෼ۭ͕ؒτϥοΩ ϯά‫ࠩޡ‬Λද‫͢ݱ‬Δۭؒʹͳͬͯ͠·͍ɼߴ࣍‫Ͱ·ݩ‬ར ༻ͯ͠ 3 ࣍‫ܗݩ‬ঢ়෮‫ݩ‬Λߦͬͨ৔߹ʹ͸ɼϑϨʔϜ͝ͱ. 3 ࣍‫ܗݩ‬ঢ়ʹ͹Β͖͕ͭੜ͡Δɽ͢ͳΘͪɼτϥοΩϯ ά‫͕ࠩޡ‬େ͖͍ͱ͖͸ɼΑΓ‫ج‬ఈ਺͕গͳ͍΄͏͕֤ϑ ϨʔϜʹ͓͚Δ 3 ࣍‫ܗݩ‬ঢ়ͷ͹Β͖͕ͭখ͍ͨ͞Ίɼখ ͞ͳ ν ͕બ୒͞ΕΔ΋ͷͱࢥΘΕΔɽ τϥοΩϯάΤϥʔ͕ฏ‫ ۉ‬0ɼ෼ࢄ 1.0[pix] ͷͱ͖ʹ͓ ͚ΔఏҊख๏ʹΑΔ 3 ࣍‫ߏ࠶ݩ‬੒݁ՌΛਤ 8 ʹࣔ͢ɽਤ. 8 ͸෮‫݁ݩ‬Ռʹਅ஋ΛॏͶͨ΋ͷͰ͋Δɽਤ 8 Λ‫ݟ‬Δͱ ਅ஋ͷ˘ͷதʹਪఆͨ͠෮‫݁ݩ‬Ռͷಛ௃఺͕‫·ؚ‬Ε͍ͯ Δɽ3 ࣍‫ܗݩ‬ঢ়ʹ͓͚Δਅ஋ͱਪఆҐஔͱͷ RMSE ͸. 1.5[mm] Ͱ͋ͬͨɽ͜ͷ͜ͱ͔Β΋ରԠ఺͕ਫ਼౓ྑ͘ਪ ఆͰ͖͓ͯΓɼఏҊख๏ͷ༗ޮੑ͕֬ೝͰ͖Δɽ 5. 2 ࣮ը૾࣮‫ݧ‬ ఏҊख๏͸ը૾্Λ௚઀୳ࡧ͢Δ͜ͱͳ͘ɼରԠ఺Λ ਪఆͰ͖ΔͨΊ 2 ୆ͷΧϝϥͰ‫ڞ‬௨Ͱ͍ࣸͬͯͳ͍෦෼ ͷରԠҐஔ΋ਪఆͰ͖ΔɽରԠҐஔਪఆͷਫ਼౓͕ྑ͚Ε ͹ɼରԠؔ܎ʹ‫͍ͨͮج‬εςϨΦ 3 ࣍‫ܗݩ‬ঢ়෮‫ݩ‬ͷ݁Ռ ΋ྑ͍͸ͣͰ͋Δɽͦ͜Ͱɼ࣮ը૾࣮‫Ͱݧ‬͸‫ڞ‬௨෦෼Λ ΄ͱΜͲઃ͚ͣʹ෺ମΛࡱӨ͠ɼ෺ମͷ 3 ࣍‫ܗݩ‬ঢ়෮‫ݩ‬ ͕ਫ਼౓ྑ͘ߦ͑Δ͜ͱΛ͔֬ΊΔ͜ͱͰɼఏҊख๏ͷ༗ ޮੑΛ֬ೝ͢Δɽ ࣮‫ࡱ͍ͨ༻ʹݧ‬Өର৅͸ਤ 9 Ͱ͋Γɼ͜ͷ෺ମʹ͸ಛ. 6 −168−.

(7) 7 σ =0.2 σ =0.6 σ =1.0 σ =1.4 σ =1.8 σ =2.0. Selected Number of Basis. 6 5 4 3 2. ࡑ࡯ࠞ࡯. 1 0 4. 6. 8. ਤ6. 10 12 14 16 18 20 Depth in 1st Frame/Object Depth. 22. 24. ࡱӨ৚݅ͱ‫ج‬ఈ਺ (Χϝϥ 1). 7 σ =0.2 σ =0.6 σ =1.0 σ =1.4 σ =1.8 σ =2.0. Selected Number of Basis. 6 5. ਤ9. ࡱӨର৅. 4 3 2 1 0 4. 6. ਤ7. 8. 10 12 14 16 18 20 Depth in 1st Frame/Object Depth. 22. 24. ਤ 10. ࡱӨ৚݅ͱ‫ج‬ఈ਺ (Χϝϥ 2). τϥοΩϯά݁Ռ (ࠨɿΧϝϥ 1ɼӈɿΧϝϥ 2). Camera01 Camera02. Camera01 Camera02 True. ਤ 11 3 ࣍‫ܗݩ‬ঢ়෮‫݁ݩ‬Ռ (ϫΠϠʔϑϨʔϜ). trajectory ਤ 8 3 ࣍‫ݩ‬෮‫݁ݩ‬Ռ. 1st frame. ௃఺௥੻༻ʹશ෦Ͱ 83 ‫ݸ‬ͷϚʔΧʔ͕͍͍ͭͯΔɽಉਤ. 64th frame. 100th frame. ਤ 12 3 ࣍‫੻يݩ‬෮‫݁ݩ‬Ռ (Side View). ͷ෺ମΛɼฏ໘্Λδάβάʹӡಈͤ͞ϚʔΧΛτϥο Ωϯάͨ͠ɽਤ 10 ͸τϥοΩϯά݁ՌΛද͓ͯ͠Γɼਤ தͷઢ͸ɼ100 ϑϨʔϜʹΘͨͬͯτϥοΩϯά͞Εͨ. ͢ɽಉਤΑΓ 2 ͭͷΧϝϥͰɼ‫ڞ‬௨Ͱ‫͍ͯ͑ݟ‬Δಛ௃఺. ಛ௃఺ͷ‫͋Ͱ੻ي‬ΔɽΧϝϥ 1 ͰࡱӨͨ͠ը૾্Ͱ͸ 42. (2 ఺) ͕ॏͳ͍ͬͯΔ͜ͱ͕෼͔Δɽ͜ΕʹΑΓɼରԠ ఺ͷਪఆਫ਼౓͕Α͘ɼྑ޷ͳ 3 ࣍‫ܗݩ‬ঢ়͕ಘΒΕ͍ͯΔ ͜ͱ͕Θ͔Δɽ ·ͨɼਤ 12 ͸ɼୈ 1 ϑϨʔϜɼୈ 64 ϑϨʔϜɼ͓Αͼ ୈ 100 ϑϨʔϜͷ෮‫݁ݩ‬Ռͱɼ෺ମͷӡಈͷ‫ͱ੻ي‬Λࣔ ͨ͠΋ͷͰ͋Δɽಉਤ͸ɼ෺ମΛਅԣ͔Β‫ͨݟ‬΋ͷΛࣔ ͓ͯ͠Γɼ෺ମͷӡಈ‫΅΄͕੻ي‬Ұ௚ઢ্ʹͳ͍ͬͯͯɼ ෺ମ͕Ұฏ໘্Λӡಈ͍ͯ͠Δ͜ͱΛΑ͘ද͍ͯ͠Δɽ ϫΠϠʔϑϨʔϜʹΑΔ෮‫݁ݩ‬Ռʹɼࡾ֯ύονΛ͋ ͯɼCG දࣔͨ͠΋ͷ͕ਤ 13(্ஈ) Ͱ͋Δɽਤ 13(Լஈ) ͸ࡱӨର৅ΛͦΕͧΕͷ CG ͷ݁Ռͷࢹ఺ʹ͋͏Α͏ʹɼ. ‫ݸ‬ͷϚʔΧʔ͕௥੻Ͱ͖ɼΧϝϥ 2 ͰࡱӨͨ͠ը૾্Ͱ ͸ 43 ‫ݸ‬ͷϚʔΧʔ͕௥੻Ͱ͖ͨɽ ఏҊख๏ʹΑΓରԠҐஔਪఆΛߦ͍ɼର৅෺ମΛ 3 ࣍ ‫ܗݩ‬ঢ়෮‫ͨ͠ݩ‬ɽຊ࣮‫ͯʹݧ‬બ୒͞Εͨ෦෼ۭؒͷ࣍‫ݩ‬ ͸ ν = 3 Ͱ͋ͬͨɽ͜Ε͸ɼτϥοΩϯά‫͕ࠩޡ‬େ͖͍ ͨΊͱߟ͑ΒΕΔɽ3 ࣍‫ܗݩ‬ঢ়෮‫݁ͨ͠ݩ‬ՌΛϫΠϠʔ ϑϨʔϜͰࣔͨ͠΋ͷ͕ɼਤ 11 Ͱ͋Δɽਤதӈ൒෼͕ Χϝϥ 1 Ͱ‫͍ͯ͑ݟ‬Δ෦෼ͷ෮‫݁ݩ‬Ռ (Χϝϥ 2 Λ‫ج‬४ ը૾ͱͨ݁͠Ռ) Λࣔ͠ɼࠨ෦෼͕Χϝϥ 2 Ͱ‫͍ͯ͑ݟ‬ Δ෦෼ͷ෮‫݁ݩ‬Ռ (Χϝϥ 1 Λ‫ج‬४ը૾ͱͨ݁͠Ռ) Λࣔ. −169− 7.

(8) ਤ 13. ্ஈɿCG ෮‫݁ݩ‬ՌɼԼஈɿਅͷ‫ܗ‬ঢ় (ࠨ͔Β SideɼFrontɼTop View). จ. ࣮෺ମΛࡱӨͨ͠΋ͷͰ͋Δɽ͍ͣΕͷ෮‫݁ݩ‬Ռ΋ਅͷ ‫ܗ‬ঢ়Λྑ͘ද͍ͯ͠Δɽ͜ͷ͜ͱ͔ΒɼఏҊख๏ʹΑΓɼ. ‫ݙ‬. [1] ଠా௚࠸: ৴པੑධՁΛ΋ͬͨΦϓςΟΧϧϑϩʔ͔Βͷ‫ܗ‬ ঢ়෮‫ͦͱݩ‬ͷҠಈମ‫ݕ‬ग़΁ͷԠ༻, ిࢠ৘ใ௨৴ֶձ࿦จࢽ,. ରԠ఺ͷҐஔਪఆ͕ਫ਼౓ྑ͘Ͱ͖͍ͯΔͱ͍͑Δɽ. Vol.J76-D-II, No.8, pp.1562-1571 (1993) [2] C.Tomasi and T.Kanade:. 6. ͓ Θ Γ ʹ. Shape and Motion from Im-. age Streams under Orthography: A Factorization Method, IJCV, vol.9, no.2, pp.137-154 (1992) [3] C.J.Poelman and T.Kanade: A Paraperspective Factoriza-. ຊ࿦จͰ͸ɼ࣌‫ྻܥ‬εςϨΦը૾Λ࢖ͬͨ৽͍͠ες. tion Method for Shape and Motion Recovery, PAMI, vol.19, no.3, pp.206-218 (1997). ϨΦରԠҐஔਪఆͷํ๏ΛఏҊͨ͠ɽఏҊख๏Ͱ͸ɼ‫ج‬ ४ը૾ྻ͔ΒಘΒΕͨಛ௃఺‫੻ي‬Λ༻͍ͯ෦෼ۭؒΛߏ. [4] ࠇᖒయٛ, ۚ୩݈Ұ: ෦෼ۭؒ෼཭๏ͱϞσϧબ୒ʹΑΔӡಈ෺ ମͷ෼཭, ৘ใॲཧֶձίϯϐϡʔλϏδϣϯͱΠϝʔδϝσΟ. ੒͠ɼͦͷ෦෼ۭؒ߆ଋͱɼΤϐϙʔϥ߆ଋͱΛ༻͍ͯɼ ࢀরը૾ྻͷ‫ʹ੻ي‬ରԠ͢Δ‫ج‬४ը૾ྻͷ‫੻ي‬Λਪఆ͠ ͨɽ·ͨɼಘΒΕͨରԠ఺͔Β 3 ࣍‫ܗݩ‬ঢ়෮‫ݩ‬Λߦ͍ɼ ֤ϑϨʔϜʹ͓͚Δ 3 ࣍‫ܗݩ‬ঢ়͕ 1 ͭͷ 3 ࣍‫ݩ‬෦෼ۭؒ. Ξ‫ڀݚ‬ձ, 2000-CVIM-124-4 (2000). [5] ࠇᖒయٛ, ۚ୩݈Ұ: ΞϑΟϯۭؒ෼཭๏ʹΑΔӡಈ෺ମͷ෼཭, ৘ใॲཧֶձ‫ڀݚ‬ใࠂ, 2001-CVIM-125-3 (2001). [6] K.Kanatani: Motion Segmentation by Subspace Separation. ʹଐ͢Δ͜ͱΛར༻ͯ͠ɼಛ௃఺‫͕੻ي‬ுΔ෦෼ۭؒͷ ࣍‫ݩ‬Λਪఆͨ͠ɽ. and Model Selection, ICCV, vol.2, pp.301-306 (2001) [7] ۚ୩݈Ұ, ੁ୩อ೭: Ҽࢠ෼ղ๏ͷ‫׬‬શϨγϐ, ৴ֶٕใ, PRMU2003-118, (2001). ఏҊख๏ͷ࠷΋େ͖ͳಛ௃͸ɼً౓৘ใΛ༻͍ͨը૾ ؒͷରԠ఺Λ௚઀୳ࡧ͢Δํ๏ͱ͸ҟͳΓɼΧϝϥؒͷ. [8] J.P.Costeria and T.Kanade: A Multibody Factorization Method for Independently Moving Objects, IJCV, vol.29,. Τϐϙʔϥ߆ଋͱɼಉҰ෺ମ্ͷಛ௃఺ͷ‫੻ي‬ϕΫτϧ ͕΋ͭ߆ଋ৚݅ͱΛར༻ͯ͠ɼ‫ز‬Կֶత߆ଋ৚݅ͷΈ͔. no.3, pp.159-179 (1998) [9] P.-K.Ho and R.Chung: Stereo-Motion with Stereo and Mo-. Βը૾ؒͷಛ௃఺ͷରԠ఺Λਪఆ͢Δ͜ͱͰ͋Δɽ͜Ε. tion in Complement, PAMI, vol.22, no.2, pp.215-220 (2000). ʹΑΓɼ྆ํͷը૾͔Βಉ͡ಛ௃఺͕‫؍‬ଌ͞Ε͍ͯͳ͘. [10] P.-K.Ho and R.Chung: Use of Affine Camera Model and All. ͯ΋ɼҰํͷը૾Ͱ‫؍‬ଌ͞Εͨಛ௃఺‫੻ي‬͸ɼ΋͏Ұํ. Stereo Pairs in Stereo-Motion, ICIV, pp.323-328 (1998) [11] F.Dornaika and R.Chung: Stereo Correspondence from Mo-. ͷը૾্ͷରԠҐஔΛ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖ͨɽ. tion Correspondence, CVPR, vol.1, pp.70-75 (1999). ࠓ‫ޙ‬͸ɼ࣮ը૾Λར༻ͨ͠ख๏ʹ͓͍ͯɼϚʔΧʔΛ ࢖༻ͤͣɼը૾্ͷಛ௃఺ͷΈΛར༻ͯ͠ఏҊख๏ͷ༗ ޮੑΛࣔ͢ํ๏ʹ͍ͭͯ‫ݕ‬౼͢Δ༧ఆͰ͋Δɽ·ͨɼ࣮. [12] Z. Zhang: Determining the Epipolar Geometry and its Uncertainty: A Review, IJCV, vol.27, no.2, pp.161-198 (1998) [13] M.A.Fischer: Random Sample consensus: A paradigm for. ‫ڥ؀‬ԼͰ͸ɼγϛϡϨʔγϣϯ࣮‫ͱݧ‬͸ҟͳΓɼτϥο. model fitting with applications to image analysis and auto-. Ωϯά‫͕ࠩޡ‬େ͖͘ͳΔ܏޲͕͋ΔͨΊɼΞ΢τϥΠΞ. mated cartography, Comm. ACM, vol.24, no.6, pp.381-395. ͷআ‫͕ڈ‬ඞཁͱͳΔͱߟ͍͑ͯΔɽͦͷํ๏ͱͯ͠ɼ‫ج‬. (1981). ఈ਺Λܾఆ͢Δࡍʹ RANSAC [13] Λಋೖ͢Δํ๏ͳͲ Λ‫ݕ‬౼͢Δ༧ఆͰ͋Δɽ 8 −170−.

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