ランク2制約を考慮した基礎行列・エピポール推定の精度評価

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(1)2006−CVIM−156（19） 2006／11／10. 社団法人情報処理学会研究報告 IPSJ SIG Technical Report. ͛ͯ˧ 2 եᏚʾᓏૂɈə‫ݙ‬ኁᝑ՛Ͷ˙̰͂͹͟஝ࡱɶᏁঋ៼њ ‫د‬ᅏ յ‫خ‬. ࣃᬌ Ҩ. ࣦࣟ޿࡚޿࡚ᭌ ᕳဵኽ࡚ቆዲኽ Ȕ 700-8530 ࣦࣟॎ༂ँι 3-1-1. {migita,shaku}@chino.it.okayama-u.ac.jp ʲΈͥ˲: ೞൈ๪ᅖӆࢧΨɶࢧਚဠঌฉȱʭ‫ݙ‬ኁᝑ՛Ȟʱȣɸ˙̰͂͹͟ʾ஝ࡱɍʱ‫ڦ‬ᰊɰȮȣɧᷳ஝ࡱ Ꮑঋʾ޿ȴȸी‫د‬ɍʱា‫܊‬ȳ 2 ɢȞʱ᷷ॽи࡚ᇒ᠞ॄ (Sampson ᠞ॄ) ʾᅋȣʱȱЎతᇒ᠞ॄʾᅋȣʱȱᷳ Ȯʫʁᷳ‫ݙ‬ኁᝑ՛ɶஙᏫዴᬗʾ͛ͯ˧ 2 ɶᝑ՛ɰᭆࡱɍʱȱ͛ͯ˧ 3 ɶᝑ՛ʟ៭ɍȱɪȣȦ 2 ဠɨȞʱ᷷᦯ ॡᷳ͛ͯ˧ 2 եᏚʾဳឿɈəЎతᇒ᠞ॄɶೋࢵ‫( ׈‬ᐸৣ 8 ဠˌ͟˰͝˼͉) ɨᦌЭ។ʾਁəৼɰᷳ͛ͯ˧ 2 եᏚʾᓏૂɈəॽи࡚᠞ॄɶೋࢵ‫׈‬ɰʫɠɧೋ᧜។ɰѶ๪ɍʱ᷷ʗəᷳ͛ͯ˧ 2 եᏚʾᓏૂɈəЎతᇒ᠞ ॄɶೋࢵ‫׈‬ɰ‫ݙ‬ɤȸᷳʫʮᲛᏁঋɶᦌЭ។ᅇ૾ଚ໩ʟ஭൙Ʉʲɧȣʱ᷷ೠዠɨɸᷳ͛ͯ˧եᏚɶ೎ဳȮʫʁ ᇼᇒᬝతɶ᧤଼ɰʫɠɧਁʭʲʱ 4 ɢɶ஝ࡱଚ໩ɰɢȣɧᷳ់ས᠞ॄȳ஝ࡱᏁঋɰؓʒɍ৬ᯮɶԇՒడᝑ ՛ʾࢴȴ஝ࡱᏁঋʾຘ᥏ɍʱ᷷ɕɈɧᷳ͛ͯ˧եᏚȳᇼᇒᬝతɶ᧤଼ʫʮʟᨿាɨȞʱɀɪʾჼᠳȮʫʁ ࡵᱸɰʫɠɧኊɍ᷷ʗəᷳᏁঋɶ᧐ȣȳᅇɋʱ‫܊؂‬ɰɢȣɧᓏ࢕ɍʱ᷷. Evaluation of the effect of rank-2 constraint on the accuracy of epipole estimation Tsuyoshi MIGITA and Takeshi SHAKUNAGA Department of Information Technology, Faculty of Engineering, Okayama University 3-1-1 Tsushima-naka, Okayama, 700-8530, JAPAN {migita, shaku}@chino.it.okayama-u.ac.jp Abstract: The fundamental matrix or epipole can be estimated from a set of point correspondences in two uncalibrated views, and there are two important factors which greatly affect the estimation accuracy: Selection of the cost function, geometric or algebraic, and the rank-2 constraint, which is, rank-3 matrices are allowed or not. Usually, an initial estimation is obtained by an ‘algebraic without rank-2’ method, aka linear 8-point method, which is refined by a ‘geometric with rank-2’ method. In addition, we have proposed an ‘algebraic with rank-2’ method for a better initialization. This paper shows that the rank-2 constraint is more important than the cost function selection, by comparing the covariance matrices of the estimation error caused by an observation noise.. 1. ̹˴̬ͯ. ɶᬝѢ়ɰᬝɍʱ๼ॄɶᨿʙЇȴ 2 ϋ٫ʾೋࢵ‫׈‬ ɍʱɀɪɨᷳᅖӆȱʭ‫ݙ‬ኁᝑ՛ʾ஝ࡱɍʱɀɪȳ. ‫ػ‬΢˶͹ͯʾ௠৬Ɉə΢ࢧɶᦦឿଶ৬ᅖӆɰ‫ݙ‬. ɨȴʱ᷷ɀɶɪȴᷳ஝ࡱᐁഔɸ͛ͯ˧ 2 ɶᝑ՛ɰ. ɤȣɧ˶͹ͯɶॽи࡚ᇒ។ഐʾᝑȦ‫طݪ‬ᷳ‫ݙ‬ኁᝑ. ᭆࡱɄʲɵɺɮʭɮȣ᷷਀೽ᷳᇼᇒᬝత᧤଼ɶᨿ. ՛ȳᨿាɮ৯տʾഔəɍ [2]᷷‫ݙ‬ኁᝑ՛ɪ‫ض‬ᅖӆΨ. ាਵɸ৑ᠪɄʲɧȴəȳᷳ͛ͯ˧եᏚɶᨿាਵɰ. ɶࢧਚဠɶঌฉɸ˙̰͂͹͛ୖ೷ʾྀəɍɶɨᷳɀ. ᬝɍʱ។ഐɸ‫כ‬Ւɨɸɮȱɠə᷷ೠዠɨɸ͛ͯ˧. 1 −155−.

(2) 2 ɶեᏚʾ،ࢉɰᓏૂɍʱɀɪȳᷳЎతᇒ᠞ॄ/ ॽ и࡚ᇒ᠞ॄɶ᧤଼ʫʮʟᨿាɨȞʱɀɪʾ។ഐɪ ࡵᱸɶεᯃȱʭኊɍ᷷ ᲛᏁঋɶ஝ࡱɶəʝɰɸᷳೋࢵ‫׈‬ɶᇼᇒᬝతȳ. F ɰɸ 9 ҂ɶាᏩȳȞʱȳᷳᕳᅐঋɸ 7 Ɉȱɮ ȣ᷷ɮɓɮʭᷳF ɸ det(F ) = 0 ʾྀəɍ͛ͯ˧ 2 ɰᝑ՛ɰᭆࡱɄʲᷳೀɰ F ɶ˺˪͹͟ɰɸચٟȳ ɮȣəʝᷳ|F |2F = 1 ɪɈɧᖧȣȱʭɨȞʱ᷷. ᅖӆιɨɶ̰˧˿͟తʾ᝝ɍʫȦɰ᧜ՓɰᨿʙЇ. ᕳᅐঋɶత (7 ᏹ) АΨɶ (lp , rp ) ȳΫȫʭʱɪ. ȽɍʱਓាȳȞʱ᷷ɀɶʫȦɮᇼᇒᬝతʾॽи࡚. F ʾ஝ࡱɍʱɀɪȳɨȴʱ᷷ɀɶəʝɰɸᷳ๓় ȳላɶ F ɰࢧɈɧೋࢵҐ 0 ɪɮʱϜʾՠᅋɍʱ᷷. ᇒ᠞ॄɪȣȦ᷷ɈȱɈᷳ᧜Փɮᨿʙʾៜ፡ɍʱɰ ɸ‫ݙ‬ኁᝑ՛ʾϙʝሻɠɧȣʱਓាȳȞʮᷳ᦯ॡɸ. Sampson ᠞ॄɶೋࢵ‫[ ׈‬3](ᯀᐸৣೋ᧜‫ )׈‬ʾᝑȦ ȱᷳɕʲɪ‫ػ‬ፅɶચٟʾ୨ɢ‫ؖ‬ਈଚ໩ (ȸʮɀʙ໩ [6]) ʾᝑȦ᷷ȣɏʲɰɈɧʟᷳೋ՜ɶៜ፡ɨɸԀɧ ɶᨿʙʾ 1 ɪɈɧ஝ࡱʾᝑȦɶȳ಑᦯ɨȞʱ᷷ɀ ɶ‫طݪ‬ɶᇼᇒᬝతɸЎతᇒ᠞ॄɪ٤ɺʲʱ᷷ʗəᷳ ஙᏫዴᬗʾ͛ͯ˧ 2 ɶᝑ՛ɰᭆࡱɍʱɀɪʟᷳ஝ ࡱᏁঋɰɪɠɧᨿាɨȞʱȳᷳ஝ࡱʾ‫܍‬ᮎɰɍʱ. E(F ) :=. P −1 . 2 wp · lTp F rp ,. (2). p=0. ɀɀɨ wp ɸᨿʙɨȞʱ᷷᠋ᏰɸАΩɶ፯ɨᦗʍʱ᷷ ។ഐɶѡࡳΨᷳF ɶាᏩʾԀɧζʍə 9 ๓Ӭ̻˧ ̘͟ʾ f ɪೃȸ᷷ាᏩɶ᯳঄ɸ๓়ɶ᦯ʮɪɍʱ᷷. f T := [ aT bT cT ] , əɚɈ F T = [ a b c ] . (3). ា‫܊‬ɨʟȞʱ᷷᦯ॡᅋȣʭʲʱᐸৣ 8 ဠˌ͟˰͝ ˼͉ɨɸᷳɀɶեᏚʾဳឿɈᷳЎతᇒ᠞ॄʾೋࢵ. ɀʲʾᅋȣɧ়ᷳ (2) ɰჹʲə lTp F rp ɸ (lp ⊗. ‫׈‬ɍʱ᷷΢ూᷳ͛ͯ˧ 2 ɶեᏚʾ،ࢉɰᓏૂɈɧ. rp )T f ɪ ɮ ʱ ᷷ə ɚ Ɉ ᷳ(a, b, c) ⊗ (x, y, z) = (ax, ay, az, bx, by, bz, cx, cy, cz) ɪɍʱ᷷ɀʲʾ P ҂ᮀʝə๼ॄ̻˧̘͟ r ʾ๓ɶʫȦɰࡱᒳɍʱ᷷ ⎤ ⎡ .. . ⎥ ⎢ ⎥ ⎢ (4) r = Jf := ⎢ (lp ⊗ rp )T ⎥ f ⎦ ⎣ .. .. Ўతᇒ᠞ॄʾೋࢵ‫׈‬ɍʱᯀ‫ؖ‬ਈᇒଚ໩ɶ‫ث‬ᔥਵʟ ኊɄʲɧȣʱ [4]᷷ೠዠɨɸ͛ͯ˧ 2 եᏚʾᓏૂɍ ʱɀɪȳ஝ࡱᏁঋɰΫȫʱ৬ᯮʾ᝝ɍత়ʾࢴՌ Ɉᷳɕɶ৬ᯮɶᅇɋʱჼᅐɰɢȣɧᓏ࢕ɍʱ᷷ʗ əᷳЎతᇒ/ॽи࡚ᇒ᠞ॄɶ᧤଼ɰʫʱ৬ᯮʟ‫ػ‬ෲ ɰ៼њɈᷳ͛ͯ˧եᏚɰʫʱ৬ᯮɪຘ᥏ɍʱ᷷Ԋ зᇒɰɸᷳ់ས᠞ॄɰШɠɧჹʲʱ஝ࡱҐɶ 1 ๓ ɶ‫֧ޱ‬ʾັʝᷳɕɶԇՒడᝑ՛ʾຘ᥏ɍʱ᷷ೋᖧ ɶ‫طݪ‬ɰɸ KCR Ωᅝ [5] ʾ᧏૾ɍʱȳᷳЅɶ‫طݪ‬ ɰɸɀɀȱʭɶɏʲȳᅇɋʱ᷷͛ͯ˧ 2 եᏚɶဳ ឿɰʫʱ֑‫׈‬ɸ 1 ๓ɶᩁɪɈɧჹʲʱȳᷳЎతᇒ. ɀɀɨ J ɸ f ȱʭ r ʋɶ Jacobi ᝑ՛ɨȞʮᷳ޿ȴ Ʉɸ P × 9 ɨȞʱ᷷ೀɰᷳwp ʾាᏩɪɍʱࢧ័ᝑ ՛ W ʾᓏȫᷳH = J T W J ɪࡱᒳɍʱɪ়ᷳ (2) ɸ๓ɶʫȦɰɮʱ᷷. 2 ᠞ॄɶщᅋɰʫʱ֑‫׈‬ɸȈO(Dw ) ɶᩁȊɰᅢʗʱ᷷. E(f ) = f T Hf .. ɀɶ Dw ɪɸᷳॽи࡚ᇒ᠞ॄʾࡵჹɍʱᨿʙɶฉ. (5). ྅Ңॄ (əɚɈᷳᨿʙɶॳ‫ܬ‬ɸ 1 ɰ๪ួ‫׈‬Ʉʲɧȣ. ɀɶᨿʙᝑ՛ W ɸᷳЗચɶ๪Ґࢧዉᝑ՛ɨʟᖧȣ᷷. ʱɪɍʱ) ɨȞʮᷳɀɶҐȳࡵ᭩ɰࢵɄȣɀɪɸࡵ ᱸɰʫɠɧቭ᠔ɨȴʱ᷷. ɪɀʴɨᷳࡵ᭩ɰᅖӆȱʭਁʭʲə႒਍ဠঌฉɸᷳ ˜r ) ላҐɰ᠞ॄȳ֐ʸɠəʟɶɨȞʱ᷷ɀʲʾ (˜lp , r. 2. ɪೃȸ᷷‫ݙ‬ኁᝑ՛ʾ஝ࡱɍʱɪȴᷳላҐɸೞሻɨ. ࢝Ꮕᢕ‫̵ڟ‬ೡ঵ိ 2 ɢɶ˞͋͛ɪ P ҂ɶ႒਍ဠȳȞʮᷳጽ p ဠɶ. ‫͛͋˞ض‬ɰʫʱӆʾɕʲɗʲ‫ػ‬๓̻˧̘͟ lp , rp ɨ᝝ɍɪᷳ๓়ɶ˙̰͂͹͛ୖ೷ȳྀəɄʲʱ [2]᷷. lTp F rp. =0. Ȟʱȱʭᷳ់སᩁʾᅋȣɧ J ʣ H ʾ෬૾ɍʱਓា ȳȞʱȳᷳ។ഐɶ‫طݪ‬ɰɸላҐɸ౓ሻɨȞʮᷳላ ɶ J, H, F ʾᅋȣʱ᷷. 2.1. (1). ɀɶ F ɸ͛ͯ˧ 2 ɶ 3 × 3 ᝑ՛ɨȞʮᷳ࢝Ꮕᢕ‫ڟ‬. F ̵ೝᔯᐸᱛ̵‫ږ‬ᤘ. Аৼɶ។ഐɶѡࡳɶəʝ F ɶ႒ᅲҐՒ។ʾࢴӿ ɍʱ᷷‫ض‬៥‫ر‬ɸ๓়ɰʫɠɧࡱᒳɍʱ᷷. ɪ٤ɺʲʱ᷷ঌฉ̻˧̘͟ɰᅋȣə l, r ɶ៥‫ر‬ɸ ɀɶ়ɨɶी‫( د‬left/right) ʾ᝝ɍʟɶɨᷳႏჼᇒ вᒛᬝѢɪɸᬝѢɮȣ᷷. 2 −156−. ⎡ ⎢ F = [u0 u1 u2 ] ⎣. ⎤⎡. ε σ1. ⎤ v T0 ⎥⎢ T ⎥ ⎦ ⎣ v1 ⎦ . σ2 v T2. (6).

(3) 3 ɢɶ႒ᅲҐɶɮȱɨೋࢵɶႏʾ ε ɪɍʱ (޿ࢵຘ ᥏ɸᐐࢧҐɰ‫ݙ‬ɤȸ)᷷់ས᠞ॄɶɮȣላɶ F ɰ ࢧɈɧɸ ε = 0 ɨȞʱ᷷ʗəᷳ2 ɢɶ˞͋͛ɶဴ √ ဠ᣿ᮍȳፅɈȣ‫طݪ‬ፅɸ σ1 = σ2 = 1/ 2 ɪɮʱ (|F |2F = 1 ɰ໳ચ)᷷v 0 ɸ‫د‬ᅖӆɶ˙̰෎ɨȞʮ e ɪʟೃȸ᷷‫ػ‬ෲɰ u0 ɸीᅖӆɶ˙̰෎ɨȞʮ e ɪʟೃȸ᷷ɀʲʭɶ̻˧̘͟ɸ ε = 0 ɨɸ‫ڦ‬ᰊɮ ȸ˙̰෎ɨȞʱȳᷳАৼ ε = 0 ɨʟ˙̰෎ɪᓏȫ ʱ᷷ɀʲʭɶ̻˧̘͟ɰɢȣɧᷳ๓ɶਵᣏȳȞʱ᷷. Fe = T. F e. . =. εe. εe .. (13). ɀɶ়ɨ ν → ∞ ɶ෎ᭆʾᓏȫʱɀɪɰʫʮᷳ† ྯ. H†. ui ⊗ v j .. := H − −. H −hhT H − . hT H − h. (14). ɀʲɸȈ͛ͯ˧եᏚʾᓏૂɈə΢ᖑᦢᝑ՛ȊɨȞʱ᷷. (8) (9). əɚɈ (i, j) = {0, 1, 2}2 ɨȞʱ᷷. hk ɸ f ɶஙᏫዴᬗɶ๪ួᇾϬ‫ݙ‬অʾ૾ɈɧȮ ʮᷳf = σ1 h4 + σ2 h8 (ε = 0 ɶ‫ )طݪ‬ɨȞʱ᷷ʗ ə h0 = e ⊗ e = Be ɸ޺ᅋɍʱɶɨᷳཎࡐʾሂ ᅪɈɧ h ɪೃȸ‫طݪ‬ʟȞʱ᷷. 2.2. = H − H = I − ff T .. (7). e :=. Morrison ɶᦢᝑ՛Ԅ়ɰ‫ݙ‬ɤȸ๓ɶᬝѢ়ȳᨿ ាɪɮʱ (H − ɶ‫د‬ᔉཎࡐɸ Moore-Penrose ɶ΢ ᖑᦢᝑ՛ʾኊɍ)᷷

(4). T − − hh H H H + νhhT H− − T ν −1 + h H − h. ፡ࡋʾ๓ɶʫȦɰࡱᒳɍʱ᷷. . ʗəᷳ๓ɶᝑ՛Ͷ̻˧̘͟ʾࡱᒳɍʱ᷷ ⎡ ⎤ e ⎢ ⎥ B := ⎣ e ⎦ ,. h3i+j. ɀɶ។ʣɕɶ௉֧ʾ។ഐɍʱΨɨᷳSharman-. ҌҪϟ‫ک‬ᔞ. ɀɶᝑ՛ɰᬝɈɧ H † h = 0, H † f = 0ᷳH † HH † =. H † ȳ૾ጐɍʱ᷷. 2.3. ᮃ̵ͩᬨಀ. ᇼᇒᬝతɶࡱᒳɰᅋȣʱᨿʙɪɈɧᓏȫʭʲʱ ႏʾॽɢȱ୯Ⱦʱ᷷ ⎧ ⎪ wp = (|ΠF T l|2 + |ΠF r|2 )−1 ⎨ Wopt : ˜ : ˜ |2 )−1 (15) W wp = (|ΠF T ˜l|2 + |ΠF r ⎪ ⎩ W1 : wp = 1 1 0 0 əɚɈᷳ Π = (16) 0 1 0. ஙᏫዴᬗʾ͛ͯ˧ 2 ɰեᭆɍʱɰɸ̼̝ᷳ̕͟ˏ ໩ɰʫɠɧ় (5) ʾ๓ɶʫȦɰଡ଼৐ɍʱɪᖧȣ᷷. ɮȮᷳԀɧɶ wp ɰԇ᦯ɶࡱతʾகȽɧʟ஝ࡱᐁഔ ɸ‫ޱ‬ʸʭɮȣɶɨᷳАΩɨɸ΢ᖑਵʾ߆ȦϜɮȸ. Eν (f ). =. f T (H + νBB T )f .. (10). ় (7) ɰʫʮᷳB T f = εe ɨȞʱȱʭᷳBBT f =. εh Ȯʫʁ ε = eT B T f = hT f ɨȞʱ᷷਀ɠɧᷳ Ψɶ়ɸ๓ɶʫȦɰʟೃȸɀɪȳՌ೽ʱ᷷. Eν (f ). = f T (H + νhhT )f .. (11). |F |F = 1 ɶ೹ЕΩɨɀɶᬝతʾೋࢵ‫׈‬ɍʱɰɸᷳ ๓ɶܑ೎Ґ‫ڦ‬ᰊɨ ν → ∞ ɶ෎ᭆʾᓏȫʲɺᖧȣ᷷ H + νhhT − λI f = 0 (12). wp−1 ɶॳ‫ܬ‬ɸ 1 ɨȞʱɪɍʱ᷷ Wopt ȳೋᖧɶᨿʙɨȞʮᷳɀɶ౾ɶᇼᇒᬝతɸ ॽи࡚ᇒ᠞ॄɨȞʱ᷷ ȈೋᖧȊɶࡱᒳʣೋ᧜ਵɶ៴ ౦ɸৼᦗɍʱ᷷ࡵ᭩ɰɸላҐ lp , rp ɸೞሻɨȞʱȱ ˜ ɪᦌЭɈə Sampson ᠞ॄȳᅋȣ ʭᷳ់སᩁɨ W ʭʲʱ᷷ɀʲʭɶᨿʙɶ়ɸ஝ࡱɍʍȴ F ʾ‫ه‬ʚ ɶɨᷳɀʲʭɶೋࢵ‫׈‬ɸᯀᐸৣ‫ڦ‬ᰊɨȞʮᷳиʭȱ ɶ‫ؖ‬ਈଚ໩ʾាɍʱ᷷ɕɶೋ՜ɶຊ᭢ɨɸᷳW1 ɶ ʫȦɰԀɧɶᨿʙʾ 1 ɪɍʱɶȳ಑᦯ɨȞʱ᷷ɀ ɶ౾ɶᇼᇒᬝతɸЎతᇒ᠞ॄɨȞʱ᷷. W1 ɶʫȦɮ΢ូϒಫɮᦌЭȳ૾ጐɍʱɶɸᷳ޺. [4] ɨɸᷳɀʲʾ 1 ‫ޱ‬తɶᲛ๓ూው়ɰढ़ሕɈᷳ՜ ೚஝ࡱɮɈɨ។ȸూ໩ʾᦗʍɧȣʱ᷷ೠዠɨɸᷳɀ ɶూው় (ɪᰓЭɶଚ໩) ɶ។ɶ୯֧ʾ។ഐɍʱȳᷳ ɀɶ។ഐɸ។໩ɰɸјࡑɈɮȣ᷷Მ๓ూው়ʾᅋ. ȸɶ‫ࡵطݪ‬᭩ɰ wp ȳ 1 ЇᦌɰᮀιɍʱȱʭɨȞ. 1 ɸᷳৼᦗɍʱ 37 ഓɶᅖӆ՛ȱʭ 2 ഓɶᏹʾ 397. ȣɧʟᷳᯀᐸৣೋ᧜‫׈‬ʾᅋȣɧʟᷳf ʾɭɶʫȦ. ᦯ʮ᧤ʁᷳwp−1 ɶฉ྅Ңॄ Dw ɶՒॏʾኊɈə̭. ɮ̫͛͋͹̇ɨ᝝ჹɈɧʟ។ɶ୯֧ɸత࡚ᇒɰɸ. ˺̘˨͉͛ɨȞʱ᷷܏ȱʭᷳ‫ע‬Ւʾᣰȫʱ 200 ᏹ. ፅњɨȞʱ᷷. лɶᅖӆࢧɨฉ྅Ңॄ Dw ȳ 0.1 ʾΩ‫܉‬ɠɧȣʱ. ʱ᷷wp−1 ɶՒॏʾ় (15) ɶࡱᒳȱʭ។ഐᇒɰᠪʍ ʱϜɸࢂ౨ɨɸɮȣɶɨᷳࡵᱸᇒɮ៼њʾኊɍ᷷܏. 3 −157−.

(5) ‫ ܬ‬0ᷳՒడ 2 ɪɍʱ (‫׵‬ɝ E[ddT ] = 2 I)᷷. Frequency (out of 397) 250. 3.1. 200. ᤘ‫ޟ‬̵̡᫈ಳ۫ 1. ɀɀɨɸᷳΔf ʾࢴȸəʝᷳʗɏᷳΔr ʾ๓ɶʫ Ȧɰࣖᬓɍʱ᷷. 150. Δr. 100 50 0. 0. ܏ 1:. 0.5. wp−1. 1. 1.5. аɈᷳM T. 2 Dw. ɶฉ྅Ңॄ Dw ɶ̭˺̘˨͉͛. =. ΔJ f + JΔf. (17). =. M T d + JΔf ⎡. (18) ⎤. ⎢ := ⎣. .. ⎥ . ⎦(19). . (ΠF r p )T (ΠF T lp )T .. .. .. ɀɀɨ M T ɸ d ȱʭ r ʋɶ Jacobi ᝑ՛ɨȞʮᷳP. ɀɪȳ՝ʱ᷷ɀɶʫȦɰᷳwp ∈ [1 − δ : 1 + δ] ɨ Ȟʱቭოȳຘ᥏ᇒᲛȣəʝᷳwp = 1 ɨʟ޺ȸɶ‫ݪ‬ ‫ط‬ɰᖧȣᐁഔʾΫȫʱɪᓏȫʭʲʱ᷷ৼɰᷳ஝ࡱ. ҂ɶࢧ័̵̏͢˧ȱʭ૾ʱ᷷‫˧̵̏͢ض‬ɸ 1 × 4 ɶ ޿ȴɄɨȞʮᷳԀзɨɸ P × 4P ɶ޿ȴɄɨȞʱ᷷ ɀʲʾᅋȣɧ়ᷳ (5) ɶᇼᇒᬝతɸ๓ɶʫȦɰɮʱ᷷. (Δr)T W (Δr). ᠞ॄɪฉ྅Ңॄ Dw ɶᬝѢʾʫʮ᠋Ᏸɰඬ៟ɍʱ᷷ ɮȮᷳॳ‫ܬ‬ʾ 1 ɪɍʱ๪ួ‫׈‬ʾᝑʸɮȸɧʟᷳ wp−1 ɶॳ‫ ܬ‬Mw ɸ 1 ɰᦌȣ᷷،ࢉɰɸᷳMw = Trace(Π(F VL F T + F T VR F )ΠT ) ɨȞʱ᷷əɚɈᷳ. VL , VR ɸɕʲɗʲ lp , rp ɶԇՒడᝑ՛ɨȞʮᷳᅖ ӆঌฉᏖɶ΢๓‫ޱ‬றɰʫʮ VL = VR = I ɪɍʱɀ ɪȳɨȴʱ [1]᷷ɀɶɪȴᷳF ɶीΨ 4 ាᏩɶ 2 ϋ٫ ʾ αᷳ‫د‬ΩាᏩɶ 2 ϋʾ β ɪɍʱɪ Mw = 1+α−β ɨȞʱ᷷෎ጦɮ‫طݪ‬ᷳˌ̳ˏͯ˞͋͛ɨɸ α = 0ᷳ ˞͋͛ɶӲ᥊ȳϬॄɍʱ‫طݪ‬ɸ β = 0 ɨȞʱ᷷޺ȸ ɶ‫طݪ‬ɸɀʲʭɰᦌȣႢ໢ɨȞʮ Mw ɸ 1 ɰᦌȣ᷷. 3. .. (20). ،ࢉɰɸᷳɀɶ়ɸ 2 ๓ɶᩁʾဳឿɈəᦌЭɨȞ ʮᷳᇼᇒᬝతɕɶʟɶɪɸ΢ᕶɈɮȣȳᷳɀɀɨɶ ។ഐɰࢧɈɧɸ‫כ‬ՒɨȞʱ᷷ɮȮᷳɀɶʫȦɮᦌ Эɸ Gauss-Newton ໩ɨʟᅋȣʭʲʱʟɶɨȞʱ᷷ ় (2) ɶೋࢵ‫׈‬ɰʫʱ‫ݙ‬ኁᝑ՛஝ࡱɸ়ᷳ (20) ʾೋࢵ‫׈‬ɍʱ Δf ʾஙᏫɍʱɀɪɪʙɮɍɀɪȳ Ռ೽ʱ᷷ɀɶ Δf ɸላɶ‫ݙ‬ኁᝑ՛ȱʭɶɏʲɨȞ ʱȱʭᷳ0 ɰɮʱɀɪȳ೗ʗɈȣ᷷ɀɶɪȴᷳላɶ ‫ݙ‬ኁᝑ՛ȳ஝ࡱɄʲəɀɪɰɮʱȳᷳࡵ᭩ɰɸ΢ ᖑɶ d ɰࢧɈɧ Δf ȳ 0 ɰɮʱɀɪɸɮȣ᷷. ᥢઈᤘ๔̤ᔅ૏̵ಮཌྷ. Ԋзᇒɰᷳ͛ͯ˧ 2 եᏚʾဳឿɈɧ Δf ʾັʝʱ. ᠞ॄɶɮȣӿ֍̻˧̘͟ʾ xT := (· · · (Π lp )T. (Πrp )T · · ·)T ɪɈᷳ‫ض‬ဠɰ֐ʸʱ਋ࢵɮ᠞ॄʾᮀʝ ə̻˧̘͟ʾ dT := [ · · · (ΠΔlp )T (ΠΔr p )T · · · ]. ɪᷳ๓ɶʫȦɰɮʱ: (M T d + JΔf )T W (M T d +. JΔf) ɶೋࢵ‫׈‬ɨȞʱȱʭᷳJ T W (M T d+JΔf ) = 0 ɰढ़ሕɄʲᷳ។ɸᷳ๓ɶ᦯ʮɨȞʱ᷷. ɪɍʱ᷷ӿ֍ x ɰ̥ˑ˼ȳ֐ʸʮ x + d ɪɮɠə. Δf. ౾ᷳ๼ॄ̻˧̘͟ r ɸ r + Δr ɰ‫׈ޱ‬Ɉ (ࡵ᭩ɰɸᷳ ɀɶ r ɸ᠞ॄɶɮȣ lp , rp ɰࢧɍʱ๼ॄɨȞʱȱ. 3.2. ʭ 0 ɨȞʱ)ᷳ‫ݙ‬ኁᝑ՛ɶ஝ࡱҐɸ f + Δf ɪɮʮᷳ. =. −(J T W J)− J T W M T d .. (21). ᤘ‫ޟ‬̵̡᫈ಳ۫ 2. ់ས᠞ॄ d ȳᷳȞʱᐸৣᨃՒዴᬗɰࣗɍʱ‫طݪ‬. ˙̰෎ɶ஝ࡱҐɸ e + Δe ɰɮʱɪɍʱ᷷. ɰɸ Δr = M T d + JΔf = 0 ɪɮʱ Δf ȳࡑ‫ܤ‬Ɉᷳ. ஝ࡱଚ໩ɶᏁঋʾᡱᠳɍʱ‫طݪ‬ɰɸᷳɀʲʭɶ Δ ɨ᝝Ʉʲʱ᠞ॄ (ɶ΢๓ɶ‫׈ޱ‬Ւ) ʾԊзᇒɰ d ɶᬝతɪɈɧ᝝Ɉᷳɕɶᐌៜᇒਵᣏ (ॳ‫ܬ‬ͶԇՒడ ᝑ՛) ʾᅋȣʱ [5]᷷АΩɨɸᷳ΢๓ɶ‫׈ޱ‬ᩁɶෲǻ. ᇼᇒᬝతɸ 0 ɰɮʱȳᷳΔf ɸ 0 ɨɸɮȣəʝᷳ‫ݙ‬. ɮࢴՌ໩ɪᬝᦶɍʱᓏ࢕ʾᦗʍʱɪɪʟɰᷳ[5] ɶ. ɨɸᷳɀɶዴᬗɶ‫ݙ‬অ̻˧̘͟ʾ෬૾ɍʱɪ‫ػ‬౾. ჼᠳɶූ់ɪងᣵʾᝑȦ᷷. ɰᷳ។ɶ΢๓ɶ‫֧ޱ‬ʾៜ፡ɍʱ՞ɶూ໩ʾࢴȸ᷷. ኁᝑ՛ɸ๪Ɉȸ஝ࡱɄʲɮȣ᷷ᦢɰᷳd ȳɀɶዴ ᬗɰࣗɍʱ૾Ւʾ‫ه‬ʗɮȣ‫طݪ‬ɰɸᷳᇼᇒᬝతɶ ๼ॄɸ 0 ɨɸɮȣȳᷳ஝ࡱ᠞ॄɸᅇɋɮȣ᷷ɀɀ. əɚɈᷳ᠞ॄ d ɸᷳᅖӆΨɨɶ᠞ॄɶ͕͹˧̏͝. ʗ ɏ়ᷳ (9) ɨ ࡱ ᒳ Ɉ ə ̻ ˧ ̘ ͟ ʾ ᅋ ȣ ɧ ᷳ. ̚᣿ᮍɰᇾ஛ᬝѢɍʱॽи࡚ᇒɮચٟʾ୨ɢ̻˧. ላ ɶ។ ᦌ ҳ ɰ Ȯ Ƚ ʱ ។ɶ 7 ɢɶ ᕳ ᅐ ঋ ɰ ࢧ ਚ. ̘͟ɨȞʱȱʭᷳˠ˔˺Ւॏɰ਀ȦɪВࡱɈᷳॳ. ɍ ʱ๪ ួᇾ Ϭ‫ ݙ‬অʾ ෬૾ ɍ ʱ᷷ɀ ʲɸ H7 :=. 4 −158−.

(6) [h1 , h2 , h3 , h5 , h6 , h7 , σ2h4 − σ1 h8 ] ɶ՛̻˧̘͟ ɨ෬૾ɄʲᷳΔf ɸɀʲʭɶ 7 ɢɶ̻˧̘͟ɶᐸ ৣᐁ‫ط‬ɨȞʱ᷷΢ూᷳ[σ1 h4 + σ2 h8 , h0 ] ɪɸᇾϬ ɈɮȽʲɺɮʭɮȣ᷷1 ɢᇼɶ̻˧̘͟ɸ f ɕɶ ʟɶɨȞʮᷳf T Δf = 0 ɸ f + Δf ɶ̥͉͟ʾ (1. Ґȳ 0 ɨɸɮȸɮʱ᷷͛ͯ˧եᏚȳȞʱ‫طݪ‬ɰɸ. d0 ɶ৬ᯮɸ h0 А޷ɶ૾Ւɰ᥇࠯Ʉʲᷳೋࢵܑ೎ Ґɸ 0 ɰѰəʲʱ᷷. 3.3. ᤘ‫ޟ‬̵̡᫈ಳ۫ 3. Δf ʾࢴȸೋʟᎋѡɮూ໩ɸᷳАΩɰᦗʍʱ୅. ๓ɶᩁɰᬝɈɧ)1 ɰѰɢ೹ЕɨȞʱ᷷hT0 Δf = 0. ᢏᇒɮూ໩ɨȞʴȦ᷷ɈȱɈᷳձᦗɶ 2 ɢɶూ໩. ɸᷳ͛ͯ˧ 2 եᏚɰࢧਚɍʱ᷷΢ూᷳ͛ͯ˧եᏚ. ɪᅲɮʮ Δf ȳᅇɋʱ‫؂‬ჼɸиʟኊ‫ڔ‬Ɉɮȣ᷷. ʾဳឿɈə‫طݪ‬ɸᷳH8 := [H7 |h0 ] ɶ 8 ɢɶ՛̻ ˧̘͟ɶ৐ʱዴᬗɰ Δf ȳࣗɍʱ᷷. Δf ʾ Hd ɶ՛ዴᬗɰୖ೷Ɉɧ় (20) ʾೋࢵ‫׈‬ ɍʱɰɸᷳg = HTd Δf ɪȮȴᷳΔf = Hd g ʾ় (20) ɰЎӿɈᷳɀʲʾ g ɨ਋ՒɈəʟɶȳೋ᧜។ ɰࢧɈɧ 0 ɪɮʱȱʭᷳ๓়ȳਁʭʲʱ᷷ Δf = − Hd (HTd HHd )−1 HTd J T W M T d . (22). Hd∗. ় (12) ʾ਋ՒɍʱɀɪɰʫʮᷳH, f , h ɶ΢๓ɶ ‫׈ޱ‬Ւ (Δ ៥‫ر‬ɨኊɍ) ȳྀəɍూው়ȳਁʭʲʱ:. ((ΔH) + ν((Δh)hT + h(Δh)T ) − (Δλ)I)f +(H + νhhT − λI)(Δf ) = 0 . (24) ላɶ។ f ɶٛᦄɨɸᷳλ = 0ᷳνhT f = 0 ፅʾ ᅋȣɧᎋ‫ש‬ɰɍʱɀɪȳɨȴʱ᷷ೀɰᷳΔhT f =. (Δe )T F e+eT F (Δe) = ε((Δe )T e +eT (Δe)) = 0 ɨȞʱ᷷ɀʲɸᷳe ɪ e ȳ‫ש‬вᬌɰୖ೷Ʉʲɧ. ɀɶ Hd∗ ɰᬝɈɧᷳH8∗ = H − ᷳH7∗ = H † ɨȞʱ᷷. Ȯʮᷳ΢๓ɶ‫׈ޱ‬Ւȳʟɪɶ̻˧̘͟ɪᇾϬɍʱ. ɀɶፅ়ɸ๓ɶʫȦɰɈɧਁʭʲʱ᷷ʗɏᷳεᦄȳ. əʝɨȞʱ᷷ʗə Δλ = f T ΔHf = 0 ɨȞʱ᷷ɮ. ԇ᦯ɶᮔዴᬗ [f , h] ʾ୨ɢɀɪɸࢂ౨ɰ՝ʱ᷷ɀʲ. ȮᷳΔH = ΔJ T W J + J T W ΔJ ɨȞʱ᷷ɀʲʭʾ. †. А޷ɶዴᬗɰɢȣɧᷳH , H. −. , Hd∗ T. ɶ‫د‬ȱʭ HHd −. ᅋȣɧᷳ. T. ʾகȽʱɪᷳɕʲɗʲ (I − f f − αH hh )Hd , (I − f f T )Hd , Hd (X)−1 (X) ɪɮʮᷳȣɏʲʟ Hd ɪɮʱ (d ɸ 7 ȱ 8 ȱ᧜Փɮూʾ᧤ʆ᷷ʗəᷳα = T. −. HTd HHd. ɨȞʱ)᷷ 1/h H h, X = ˆ k ɪɈɧ়ᷳ (22) ๓ɰᷳHd ɶ‫ض‬՛̻˧̘͟ʾ h ∗ T T ˆ k = −H J W M d ˆ k ɪɮʱʫȦɮ d ˆk ɰ‫ݙ‬ɤȴ h. ˆ k ʾ k = 1 ȱʭ 7 ʗɨζʍəʟɶɸ๓ ʾ෬૾ɍʱ᷷d. Δf = − H † J T W M T d. (25). ɀʲɸ় (22) ɪ‫ػ‬ɋᐁഔɨȞʱ়᷷ (21) ɶᐁഔɪ ຘʍʱɪ H − ȳ H † ɰᒛȴறʸɠɧȣʱ᷷ɀʲɸ ͛ͯ˧ 2 եᏚɰʫʱ។ɶ᧐ȣɨȞʱ᷷. 3.4. Δf ̵ષࡰً̤‫ږ‬൥ᢕ‫ڟ‬. Δf ɸᷳӿ֍̻˧̘͟ɰ਋ࢵ̥ˑ˼ d ȳ֐ʸɠ. ɶʫȦɰɮʱ: D7 := [d1 , d2 , d3 , d5 , d6 , d7 , σ2 d4 − ˆk = 0 σ1 d8 ]᷷əɚɈᷳdk ɸ Δr = M T dk + J T h. ɀʲɸ d ɶᐸৣᬝతɨȞʱȱʭᷳॳ‫ܬ‬ɸ 0 ɨȞʮᷳ. ʾྀəɍ๓ɶ̻˧̘͟ɨȞʱ᷷. ΢๓ɶᩁɰᬝɈɧɸ஝ࡱҢॄɸɮȣ᷷. dk. := −M (M T M )−1 Jhk ⎡ .. . ⎢ T T ⎢ (u ΠF r p ⎢ i lp )(v j r p ) = ⎢ ⎢ ΠF T lp |ΠF T lp |2 + |ΠF rp |2 ⎣ .. .. ə౾ɰুȴᣮɀɄʲʱ f ɶ‫֧ޱ‬ɶ΢๓ᦌЭɨȞʱ᷷. ԇՒడᝑ՛ E[(Δf )(Δf )T ] ɸᷳ๓ɶʫȦɰɮʱ᷷. (23) ⎤ ⎥ ⎥ ⎥ ⎥ . ⎥ ⎦. ͛ͯ˧եᏚʾဳឿɈə‫طݪ‬ɸ D8 := [D7 |d0 ] ʾᓏ ȫʱ᷷ɀʲʭȳᷳձᦗɶȈȞʱᐸৣᨃՒዴᬗȊɶ ˆ k ɶᐸৣᐁ ‫ݙ‬অɨȞʱ᷷΢ᖑɶ d ɸᷳɀʲʭɶ d ‫ط‬ɪ๼ʮɶ૾ՒɰՒ។Ʉʲʱ᷷əɚɈᷳ‫ݙ‬অᏖ Dd ɸᇾϬɨʟ๪ួɨʟɮȣəʝᷳᇾϬՒ។Ʉʲʱʸ. V (W ). :=. 2 H † J T W M T M W JH † (26). ɀʲɸ W ɶᬝతɨȞʱ᷷ɈȱɈᷳ஝ࡱ᠞ॄȳΪፅ ় Δf T V (W )−1 Δf ≤ θ ɨࡱᒳɄʲʱලԐзɰ‫ه‬ ʗʲʱቭოɸᷳW ɰɸјʭɏ θ ɶʙɨ (χ2 Ւॏɰ ʫɠɧ) ໃʗʱ᷷ ɀɀɨᷳWopt = (M T M )−1 ɶȈೋ᧜ਵȊɰɢȣ ɧᓏȫʱ᷷Зચɶ W ʾᅋȣə஝ࡱᩁɰ‫ه‬ʗʲʱ᠞ ॄ Δf ɪᷳWopt ʾᅋȣə஝ࡱᩁɰ‫ه‬ʗʲʱ᠞ॄ Δf¯ ɶॄɶԇՒడᝑ՛ʾៜ፡ɍʱɪ (ᬌȣៜ፡ɶৼ ɰ) ๓ɶʫȦɰɮʱ᷷. Ƚɨɸɮȣ᷷ ͛ͯ˧եᏚʾဳឿɍʱɪ d0 ɮʱ૾Ւȳ‫ݙ‬ኁᝑ՛ ɶ஝ࡱᩁʾ h0 ɚȽ‫֧ޱ‬Ʉɑᷳ‫ݙ‬ኁᝑ՛ɶೋࢵܑ೎ −159− 5. X := V (W ) − V (Wopt ). (27). ɀɀɨᷳ V (Wopt ) = 2 H † (28).

(7) ɀɶ X ɸԇՒడᝑ՛ɨȞʮᯀᢣҐɨȞʱȱʭᷳ. ɀɶ౾ɶ᠞ॄ Δf ɸ়ᷳ (24) ιɶ W ʾ W +ΔW. Зચɶ̻˧̘͟ a ɰࢧɈɧ a Xa ≥ 0 ɨȞ. ɨᒛȴறȫʱɀɪɰʫʮ๓ɶʫȦɰਁʭʲʱ (ላɶ. ʮᷳaT V (W )−1 a. ។ f ɸᨿʙɶ᧤଼ɰјʭɮȣɀɪɰ໳ચ)᷷. T. ≤ aT V (Wopt )−1 a ɨ Ȟ ʱ1 ᷷ T ɀʲɸᷳලԐз a V (W )−1 a ≤ θ ȳᷳලԐз. aT V (Wopt )−1 a ≤ θ ʾࡪԀɰ‫ه‬ʚɀɪʾኊɈɧ ȣʱ᷷ʫɠɧᷳWopt ʾᅋȣə౾ɶ஝ࡱ᠞ॄɶՒॏ ȳᷳɭɶෲɮ W ʾᅋȣəɪȴɶՒॏʫʮʟላɰࢵ ɄȣəʝᷳWopt ȳȈೋ᧜Ȋɪ៙ȫʱ᷷ [5] ɨɸᷳɀʲʾ V (W ) V (Wopt ) ɶʫȦɰ ೃȴᷳ‫ᦄد‬ʾ KCR Ӭኡɪ٤ˀɨȣʱ᷷ɮȮᷳ[6] ɨᦗʍʭʲɧȣʱԇՒడᝑ՛ɶΩᅝʾᷳೠዠɶ៥ ‫ر‬ɨೃȸɪ 2 (I − hhT )H(I − hhT ) ɸ. 2. (H7 HT7 HH7 HT7 )−. − 2. Ȟʱȣ †. Δf. . H + J T ΔW J. Δf ɶॳ‫ܬ‬ɸ 0 ɨȞʱ᷷Ւడʾ 2 ɨտɠəʟɶɸᷳ. =. ɀɀɨɸᷳ͛ͯ˧եᏚɶ೎ဳᷳॽи࡚ᇒ/Ўతᇒ. =. ՛ʾັʝɧຘ᥏ɍʱ᷷ʗəᷳ˙̰෎ɶԇՒడᝑ՛. † H − H † J T ΔW JH † J T (W + ΔW )M T ·M (W + ΔW )J H † − H † J T ΔW JH † † H − H † J T ΔW JH † (H + 2J T (ΔW )J) 2 ) · H † − H † J T ΔW JH † + O(Dw 2 H † + O(Dw ). (33). ᦪιᷳАΩɶᬝѢ়ʾᅋȣə᷷. ʟ‫ػ‬ෲɰຘ᥏ɍʱ᷷. 4.1. †. 2 = H † − H † J T ΔW J H † + O(Dw ) (32). 2 H † ɶৣɶూȳ។ഐɰɸᨇ‫ط‬ȳᖧȣ᷷. ᠞ॄɶ᧐ȣɰʫʱ 4 ଚ໩ɰࢧɍʱ Δf ɶԇՒడᝑ. (31). ɀɀɨ় (14) ɸ๓ɶʫȦɰଡ଼৐Ʉʲɧȣʱ2 ᷷. ɪɮʱȳᷳᐁࣉ H ɪ΢. ً‫ږ‬൥ᢕ‫̵ڟ‬࿜᪓. − H † − H † J T ΔW JH † 2 ·J T (W + ΔW )M T d + O(Dw ). ᕶɍʱ᷷ɈȱɈᷳᦢᝑ՛ྯ፡ɶвᒛɶ᧐ȣɶəʝᷳ. 4. =. ુռঞ጖ᥢઈ̵ࢮ‫ݻ‬. ͛ͯ˧ 2 եᏚʾဳឿɈɧᷳೋ᧜ɮᨿʙ Wopt ʾᅋ. (W + ΔW )W −1 (W + ΔW ) = W + 2(ΔW ) + (ΔW )W −1 (ΔW ). ȣə‫طݪ‬ɶԇՒడᝑ՛ɸ. 2 H −. (34). (29). ় (33) ɨᷳΔW ɶ΢๓ɶ᯲ɸԀɧଝɝ༢Ɉ‫ط‬ȣᷳ. ɨȞʮᷳ͛ͯ˧ 2 եᏚʾᅋȣə‫طݪ‬ɶ় (28) ɪɶ. ʙɨȞʱ (H † ɸ় (28) ʾ 2 ɨտɠəʟɶɨȞʱ)᷷. ॄ᧐ɸᷳ๓়ɶ᦯ʮɨȞʱ᷷. 2. −. T. H hh H hT H − h. 2 ೋ᧜ɮ W ʾᅋȣə‫طݪ‬ɪ O(Dw ) ɶ᯲ȳᅲɮʱɶ. ͛ͯ˧ 2 եᏚʾဳឿɈəЎతᇒ᠞ॄೋࢵ‫( ׈‬ᐸৣ. −. .. (30). 8 ဠ໩) ɶԇՒడᝑ՛ʾັʝʱɰɸᷳH † ʾ H − ɰ ᒛȴறȫɧ‫ػ‬ෲɶᡱᠳʾᝑȫɺᖧȣ᷷ɕɶᐁഔɸᷳ. 2 2 H − + O(Dw ) .. ձጜɶ Wopt ɶೋ᧜ਵɶᡱᠳɰʫʮ়ᷳ (28) ɶՒ ॏ (Ȟʱቭოɨᅇɋʱ᠞ॄʾ‫ه‬ʚೋࢵɶලԐз) ɸ. (35). ় (29) ɶՒॏɰԏ‫׀‬ɄʲɧȮʮᷳ͛ͯ˧ 2 եᏚʾ ဳឿɍʱɀɪɰʫɠɧᷳԀɧɶూ‫ـ‬ɰࢧɈɧᏁঋ ȳгΩɍʱɀɪȳ՝ʱ᷷. 4.2. 4.3. 4 ౞ိ̵࿜᪓. АΨɶᐁഔʾ᝝ 1 ɰʗɪʝʱ᷷᝝ιᷳीΨɸ͛. Ւ൨጖ᥢઈ̵ࢮ‫ݻ‬. ๓ɰᷳW = (M T M )−1 ȳ W + ΔW ɰ‫׈ޱ‬Ɉ ə‫طݪ‬ʾᓏȫʱ᷷Ԋзᇒɰɸᷳɕɶ‫֧ޱ‬ɰʫɠɧ. W + ΔW ȳ‫ש‬вᝑ՛ɶࡱత҃ɪɮʱ‫طݪ‬ɨȞʱ᷷ ɀɶɪȴᷳΔW ɶ‫ض‬ាᏩɸࡵ֜ᇒɰɸᷳձɶጜɨ ᡱᠳɈəᇿࢧҢॄ Dw ɶ˝͹̉ɨȞʱ᷷ 1 T (A. − B) ≥ 0 ɸᷳ T A / T B ȳ 1 АΨɨȞʱɀ ɪɪፅњɨȞʮᷳʫɠɧ (A − λB) = 0 ɶ΢ᖑܑ೎Ґɸ 1 А ΨɨȞʱ᷷ɀɶ΢ᖑܑ೎Ґ‫ڦ‬ᰊɸ (B −1 − λA−1 )(A ) = 0 ɪ ೃȴறȫʭʲʱ᷷λ ɸԇ᦯ɨȞʱȱʭᷳ T B −1 / T A−1 ɸ 1 АΨɨȞʮᷳ T (B −1 − A−1 ) ≥ 0 ɪɮʱ᷷. −160− 6. ͯ˧ 2 եᏚʾᓏૂɈəॽи࡚ᇒ᠞ॄೋࢵ‫׈‬ɨȞʮᷳ. KCR Ωᅝ A ʾ᧏૾Ɉɧȣʱ᷷͛ͯ˧ 2 եᏚʾဳ ឿɍʱɪᷳԇՒడᝑ՛ɸ B ɚȽȈ‫)׈֑( ֐ޅ‬Ȋɍ ʱ (‫د‬Ψ๋)᷷΢ూᷳॽи࡚ᇒ᠞ॄɶЎʸʮɰЎత 2 ᇒ᠞ॄʾᅋȣʱɪԇՒడᝑ՛ɸ O(Dw ) ɚȽ֑‫׈‬ɍ. ʱ (ीΩ๋)᷷‫د‬Ωɸᷳᐸৣ 8 ဠˌ͟˰͝˼͉ (͛ͯ 2 ɀɶᝑ՛ɶࢴՌɰȮȣɧᷳೠబɨɸ. H7 ɰ៙ؓɈɧȣɮȣ ȳᷳᅋȣəూȳᎋ‫ש‬ɨȞʱ: (A + B)−1 ≈ A−1 − A−1 BA−1 T −1 HT ≈ ʾᅋȣɧᷳH7 (HT H† − 7 J (W + ΔW )JH7 ) 7. . . H † J T ΔW J H † ᷷.

(8) ᝝ 1: Δf ɶԇՒడᝑ՛ / 2 ᇼᇒᬝత. ͛ͯ˧ 2 եᏚȞʮ. ͛ͯ˧ 2 եᏚɮɈ. ॽи࡚ᇒ. A 2 A + O(Dw ). A+B 2 A + B + O(Dw ). Ўతᇒ. ⎛ ⎝ B = A =. ⎞ H − hhT H − hT H − h ⎠ H− − B. ᝝ 2: Δe ɶԇՒడᝑ՛ / 2 ᇼᇒᬝత. ͛ͯ˧ 2 եᏚȞʮ. ͛ͯ˧ 2 եᏚɮɈ. ॽи࡚ᇒ. A 2 A + O(Dw ). A+B 2 A + B + O(Dw ). Ўతᇒ. ˧ 2 եᏚʾဳឿɈəЎత᠞ॄೋࢵ‫ )׈‬ɨȞʮᷳɀʲ. ⎛ ⎝ B = A =. ⎞ H − hhT H − T K ⎠ hT H − h − T KH K − B. K. Dw 10. ʭɶεూɶ֑‫׈‬ȳᣮɀʱ᷷ 2 ΢๓ɶ᠞ॄ B ɪ O(Dw ) ɶ޿ȴɄʾჼᠳᇒɰຘ. 1. ᥏ɍʱɀɪɸᮎɈȣ᷷ЎʸʮɰᷳDw ɰᬝɍʱʟȦ. 0.1. ΢ɢɶਵᣏʾኊ‫ڔ‬ɍʱࡵᱸᐁഔʾኊɍ (܏ 2)᷷ɀʲ 0.01. ɸ̥͉ᷳ͟ʾ 1 ɰ๪ួ‫׈‬Ɉə˙̰෎ɶ z ঌฉɪ Dw ɶᬝѢʾኊɈɧȣʱ᷷ɀɶ܏ȱʭᷳz ঌฉɪ Dw ɰ. 0.001. ɸ৑ȣᇿᬝȳȞʮᷳ˙̰෎ɶ z ঌฉȳ Dw ɶᖧȣ 0.001. ୪ฉɪɮʱɪ៙ȫʱ᷷ɀʲɸᷳ˙̰෎ȳᅖӆι਒ ɰᦌȣȱᷳဳᭆ᧖ɰᦌȣȱɰʫɠɧ஝ࡱᏁঋȳᅲ. 0.01. 0.1. 1z. ܏ 2: wp−1 ɶՒడɪ˙̰෎ɶ z ঌฉɶᬝѢ. ɮʱɀɪʾ᝝Ɉɧȣʱ᷷ɕɈɧᷳ޺ȸɶ‫طݪ‬ᷳ˙̰ ෎ɸᅖӆι਒ʫʮʟဳᭆ᧖ɰᦌȸᷳz ɸࢵɄȸ Dw 2 ʟࢵɄȣ᷷ɀɶෲɮ‫طݪ‬ɰɸᷳO(Dw ) ɶ᯲ɸ B ɶ. ɀɀɨᷳV ɸ Δf ɶԇՒడᝑ՛ɨȞʱ᷷ձᦗɶ 4 ɢ. ᯲ʫʮʟࢵɄȸɮʮᷳ͛ͯ˧ 2 ɶեᏚɸೋ᧜ɮᨿ. ɶ‫طݪ‬ԀɧɰɢȣɧᷳɀɶৣɨȞʱ᷷ʫɠɧᷳΔf ɶ. ʙЇȽ (ॽи࡚ᇒ᠞ॄɶщᅋ) ʫʮʟᨿាɪɮʱ᷷. ‫طݪ‬ɪ‫ػ‬ෲɰ᝝ 2 ɶʫȦɰʗɪʝʱɀɪȳɨȴʱ᷷. 4.4. 5. e ̵ً‫ږ‬൥ᢕ‫ڟ‬. e ɸ F e = 0 ɶూው়ʾྀəɍɶɨᷳɀɶ΢๓ ɶ‫׈ޱ‬Ւɸ๓ɶʫȦɰ᝝Ʉʲʱ (ձጜɶ Δf ɶጽ 3 ɶ។ഐ໩ɪ‫ػ‬ɋ‫؂‬ჼɨȞʱܑ᷷೎̻˧̘͟ɶ௉֧ ࡱჼʾᓏȫɧʟᖧȣ)᷷Მ๓ɶ‫׈ޱ‬Ւɰɢȣɧɸᓏ ȫɮȣ᷷. হᶼ АΩɨɸ, ܏ 3 ʾ‫ه‬ʚ 37 ഓɶࡵᅖӆɰ‫ݙ‬ɤȸࡵ. ᱸᐁഔʾኊɍ᷷܏ιɶी 2 ഓᷳ‫ د‬2 ഓʾࢧɪɈə‫ݪ‬ ‫ط‬ɶ˙̰͂͹͛ᐸʟኊɈɧȞʱ᷷႒਍ဠɸ 100 ဠ ɨȞʮᷳ7 ဠȱʭ 52 ဠɶԇ᦯ဠʾ‫ه‬ʚᅖӆࢧȳ 397 ᦯ʮȞʱ᷷37 ഓι 30 ഓɸႏзʾ‫܉‬ʱԐ᥁᧎ȱʭႏ. Δe = −F − B T Δf. (36). зʾ௠৬ɈəʟɶɨȞʮᷳ˙̰͂͹͟ȳ ±(1, 0, 0). Аৼᷳɀɶ F −B T ʾ K ɪೃȸɀɪɰɍʱ᷷ɀʲ. ЇᦌɰȞʱᅖӆȳ޺ȣ᷷ࡵᅖӆɰࢧɈɧላɶ‫ݙ‬ኁᝑ. ɸᷳ๓ɶʫȦɰ႒ᅲҐՒ។ɨȴʱ᷷. K. := F − B T. v 1 hT3 v 2 hT6 = + . (37) σ1 σ2. Δe ɸᐁࣉ d ɶᐸৣᬝతɨȞʱȱʭॳ‫ܬ‬ɸᷳ E[Δe] = 0 .. KV K. ᠞ॄʾೋࢵ‫׈‬ɍʱ‫ݙ‬ኁᝑ՛ʾ஝ࡱɈᷳɕɶ‫ݙ‬ኁᝑ՛ ɰࢧɈɧ๪ɈȸɮʱʫȦɰࢧਚဠঌฉʾѶ๪Ɉə᷷ ɀɶ̗͹̇ɰ‫ݙ‬ɤȸࡵᱸɸᷳላɶ‫ݙ‬ኁᝑ՛ɶ՝ɠ ɧȣʱࡵᅖӆɰʫʱࡵᱸɪʙɮɍɀɪȳɨȴʱ᷷ ΫȫʭʲəࢧਚဠᒱɰࢧɈɧೋ᧜ɮ‫ݙ‬ኁᝑ՛ɸᷳ. (38). ˙̰෎ɶᬝతɪɈɧ΢ચɰࡱʗʱ [2] ɶɨᷳ˙̰෎. ԇՒడᝑ՛ɸᷳ๓ɶʫȦɰɮʱ᷷ T. ՛ɸ՝ʭɮȣɶɨᷳʗɏࡵᅖӆɰࢧɈɧ Sampson. ɶԇՒడᝑ՛ɶʙʾᓏૂɍʲɺᖧȣ᷷. (39) −161− 7. ጽ 1 ɶюʾ᝝ 3 ɰኊɍ᷷Δe ɶԇՒడᝑ՛ɶȦ.

(9) ܏ 3: ᅖӆ՛ (37 ഓι 4 ഓᷳጽ 0,1,10,19 ᅖӆ). ᝝ 3: Δe ɶԇՒడᝑ՛ɶܑ೎Ґ (ю 1ᷳᅖӆ 0 ɪ 1ᷳࢧਚဠ 36 ࢧ) ᇼᇒᬝత. ͛ͯ˧ 2 եᏚȞʮ. ͛ͯ˧ 2 եᏚɮɈ. ॽи࡚ᇒ. (15.299 7.094) (15.355 7.114). (15.355 7.213) (15.397 7.232). Ўతᇒ. ܏ 4: ᝝ 3 ɶලԐ᝝ኊ. ᝝ 4: Δe ɶԇՒడᝑ՛ɶܑ೎Ґ (ю 2ᷳᅖӆ 10 ɪ 19ᷳࢧਚဠ 13 ࢧ) ᇼᇒᬝత. ͛ͯ˧ 2 եᏚȞʮ. ͛ͯ˧ 2 եᏚɮɈ. ॽи࡚ᇒ. (0.949 0.294) (0.954 0.295). (6.120 0.380) (6.167 0.381). Ўతᇒ. ܏ 5: ᝝ 4 ɶලԐ᝝ኊ. ɝᷳ2 ɢɶܑ೎Ґʾ᝝Ɉɧȣʱ (3 ɢʝɸਓɏ 0 ɨȞ. ʾࡱਵᇒͶࡱᩁᇒɰ៼њɈə᷷ɕɶᐁഔᷳ޺ȸɶ. ʱ)᷷ɀɶతҐȳࢵɄȣʐɭ᠞ॄȳՌɰȸȣ᷷͛ͯ. ‫طݪ‬ᷳ͛ͯ˧ 2 եᏚɶూȳᨿាɨȞʮᷳᇼᇒᬝత. ˧ 2 եᏚʾဳឿɈəɀɪɰʫʱ֑‫׈‬ɸ (0.06, 0.12). ɶ᧤଼ɶ৬ᯮɸࢵɄȣɀɪʾ౦ʭȱɰɈə᷷ೠዠ. ɨȞʮ (ɀʲɸᷳձ፯ɶ B ૾Ւɰࢧਚɍʱ)ᷳɀɶ. ɨɸࡵᱸɰᰆʭɆʱʾਁɮȱɠə Dw ɶჼᠳ។ഐ. ҐɸɭɝʭɶᇼᇒᬝతɨʟʐʒፅɈȣ᷷΢ూᷳЎ. ʣᷳೠዠɨɸሂᅪɈə 2 ๓ɶᩁɶ។ഐɸϾৼɶᠦ. త᠞ॄɶщᅋɰʫʱ֑‫׈‬ɸ (0.06, 0.02) ውঋɨȞʮ. ᰊɨȞʱ᷷. 2 (ɀʲɸᷳձ፯ɶ O(Dw ). ɰࢧਚɍʱ)ᷳ͛ͯ˧եᏚ. ɶ೎ဳɰјʭɏʐʒፅɈȣ᷷͛ͯ˧ 2 եᏚɸᇼᇒ ᬝతɶ᧤଼ʫʮʟᨿាɨȞʱɀɪɸ՝ʱȳᷳʟɪ ʟɪɶܑ೎Ґ (15, 7) ɰࢧɈɧɀɶ‫֧ޱ‬ɸࢵɄȣ᷷ ܏ 4 ɰᷳԇՒడᝑ՛ɰʫʮоʭʲʱලԐʾኊɍ᷷4 ɢɶලԐʾኊɈɧȣʱȳᷳ๹ˀɭ‫ח‬՞ɨȴɮȣ᷷ ጽ 2 ɶю (᝝ 4) ɨɸᷳ͛ͯ˧ 2 եᏚɸ (5, 0) ው ঋɶ֑‫׈‬ʾুȴᣮɀɍ΢ూᷳᇼᇒᬝతɸ๹ˀɭ֑ ‫׈‬ɶ‫܊؂‬ɪɸɮɠɧȣɮȣ᷷܏ 5 ɰᷳԇՒడᝑ՛ ɰʫʮоʭʲʱලԐʾኊɍ᷷܏ 4 ɪ‫ػ‬΢ɶᑕࣃɨ Ȟʱ᷷4 ɢɶලԐʾኊɈɧȣʱȳᷳ2 ࢧɰɮɠɧȮ ʮᷳԏҬȳ͛ͯ˧եᏚȞʮᷳ޷ҬȳɮɈɶ‫طݪ‬ɨ Ȟʱ᷷ᇼᇒᬝతɶ᧐ȣɸ܏ɨɸ՝՞ɨȴɮȣ᷷ɀ ɶ‫طݪ‬ɸ͛ͯ˧եᏚɸ‫כ‬Ւɰ޿ȴɮ֜ഔȳȞʱɪ ៙ȫʱ᷷. 6. ‫ݓ‬ᘓ൰ሊ [1] Hartley, R. I., In Defence of the 8-point Algorithm, IEEE Trans. PAMI, 19, 6 (1997), 580–593. [2] Hartley, R. I. and Zisserman, A., Multiple View Geometry in Computer Vision, Cambridge University Press, ISBN: 0521540518, second edition (2004). [3] Zhang, Z., Determining the Epipolar Geometry and its Uncertainty: A Review, IJCV, 27, 2 (1998), 161–198. [4] ‫د‬ᅏ, ࣃᬌ, ೞൈ๪ᅖӆࢧιɶဠࢧਚɰ‫ݙ‬ɤȸ˙̰ ͂͹͟ɶ 1 ๓ӬஙᏫ໩, ੸Հቆ‫ ݩ‬CVIM, 153, 64 (2006), 413–420. [5] ᩃᢃ, ೋࢽ஝ࡱɶೋ᧜ਵɪ KCR Ωᅝ, ੸Հቆ‫ݩ‬ CVIM, 147, 8 (2005), 59–66. [6] Χँ, ᩃᢃ, ‫ݙ‬ኁᝑ՛ɶೋ᧜ៜ፡ɪɕɶѲᰆਵ៼њ, ੸Հቆ‫ ݩ‬CVIM, 118, 10 (1999), 67–74.. ˋΙ΋̬ ೠዠɨɸᷳ‫ݙ‬ኁᝑ՛஝ࡱɰȮȽʱ͛ͯ˧ 2 եᏚ. ɶᨿាਵɪᷳॽи࡚ᇒ/Ўతᇒ᠞ॄɶ᧤଼ɶᨿាਵ. 8 −162−.

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