機械学習によるナンバープレート数字画像認識精度の向上

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(1)Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report. ᶵᲔᏛ⩦࡟ࡼࡿࢼࣥࣂ࣮ࣉ࣮ࣞࢺᩘᏐ⏬ീㄆ㆑⢭ᗘࡢྥୖ ㎷ᗈ⏕†1†2 ⚟Ỉὒᖹ†2 㐨㛵㝯ᅜ†2 ᒣෆᐶ⣖†2 ᒣᓮྐ⣫†2 ർᕝṌ†3 ᴫせ㸸㜵≢࣓࢝ࣛ⏬ീ࡟ᫎࡗࡓࢼࣥࣂ࣮ࣉ࣮ࣞࢺࡢᩘᏐ᝟ሗࡣ㸪≢⨥ᤚᰝ࡟࠾࠸࡚᭷⏝࡞᝟ሗ࡜࡞ࡿࡓࡵ㸪ࢼࣥࣂ࣮ ࣉ࣮ࣞࢺᩘᏐ᝟ሗࢆㄞࡳྲྀࡿࡓࡵࡢ⏬ീㄆ㆑ᢏ⾡ࡀᥦ᱌ࡉࢀ࡚ࡁࡓ㸬ࡋ࠿ࡋࠊ≢⨥ᤚᰝࡢᑐ㇟⏬ീ࡟࠾࠸࡚㸪⏬㉁ࡢ ຎ໬ࡀ㢧ⴭ࡞ሙྜࡀከࡃ㸪㧗࠸ㄆ㆑⢭ᗘࢆᚓࡿࡇ࡜ࡀᅔ㞴࡛࠶ࡿ㸬ຎ໬⏬ീ࡛ࡣ㸪≉࡟ᩘᏐ㡿ᇦࡢษࡾฟࡋ⢭ᗘࡀప ୗࡍࡿࡇ࡜ࡸග㔞ࡢ೫ࡾࡀၥ㢟࡜࡞ࡿ㸬ࡲࡓ㸪⏬ീฎ⌮ࣃ࣓࣮ࣛࢱࡢ᭱㐺໬ࡀㄢ㢟࡜࡞ࡗ࡚࠾ࡾ㸪ฎ⌮ᐇ᪋⪅ࡢ୺ほ ࡟ࡼࡽ࡞࠸ᡭἲࡀᚲせ࡛࠶ࡿ㸬ࡑࡇ࡛㸪ᮏ✏࡛ຎ໬⏬ീ࡟࠾ࡅࡿㄆ㆑⢭ᗘྥୖ࡜㸪ᐈほⓗᡭἲ࡟ࡼࡿࣃ࣓࣮ࣛࢱ᭱㐺 ໬ࢆ┠ⓗ࡜ࡋ࡚㸪ᶵᲔᏛ⩦࡟ࡼࡿࢼࣥࣂ࣮ࣉ࣮ࣞࢺᩘᏐ⏬ീㄆ㆑⢭ᗘࡢྥୖᡭἲࢆᥦ᱌ࡍࡿ㸬ᥦ᱌ᡭἲࡣ➨୍࡟㸪ΰ ྜṇつศᕸࣔࢹࣝ࡟ᇶ࡙ࡃ㍤ᗘ್ࢡࣛࢫࢱࣜࣥࢢࢆᑐ㇟ࣈࣟࢵࢡෆ࡛⾜࠸㸪ᑐ㇟ࣈࣟࢵࢡࢆࣛࢫࢱࢫ࢟ࣕࣥࡋ࡚ࢼࣥ ࣂ࣮ࣉ࣮ࣞࢺ⏬ീ඲యࢆࢡࣛࢫࢱࣜࣥࢢࡍࡿᡭἲࢆ⏝࠸㸪ᩘᏐ㡿ᇦࡢษࡾฟࡋࢆ㧗⢭ᗘ࡟⾜࠺㸬ࡑࡋ࡚➨஧࡟㸪ࢡࣛ ࢫࢱࣜࣥࢢ࡟ࡼࡾᚓࡽࢀࡓ㡿ᇦศ๭⏬ീ࡟ᑐࡋ࡚⏬ീㄆ㆑ࢆ⾜࠺ࡇ࡜࡟ࡼࡾග㔞ࡢ೫ࡾࡀㄆ㆑⢭ᗘ࡟ཬࡰࡍᙳ㡪ࢆ పῶࡍࡿ㸬ࡲࡓ㸪ࢡࣛࢫࢱࣜࣥࢢฎ⌮ࣃ࣓࣮ࣛࢱࢆ㸪ᶵᲔᏛ⩦࢔ࣝࢦࣜࢬ࣒࡟ࡼࡾタᐃࡍࡿࡇ࡜࡛㸪ㄆ㆑⤖ᯝࡢᐈほ ᛶࢆ㧗ࡵࡿ㸬ᮏ✏࡛ࡣᐇ෗⏬ീࢆ⏝࠸࡚ᥦ᱌ἲࡢ᭷ຠᛶ࡜ጇᙜᛶࢆ☜ㄆࡍࡿࡓࡵࡢᐇ㦂ࢆ⾜ࡗࡓ㸬ᐇ㦂ࡢ⤖ᯝ㸪ᥦ᱌ ᡭἲ࡟ࡼࡾㄆ㆑ࢆ⾜ࡗࡓሙྜ㸪ᚑ᮶ᡭἲࡢࢡࣛࢫࢱࣜࣥࢢ࡛ᩘᏐ㡿ᇦࢆษࡾฟࡋ࡚ㄆ㆑ࢆ⾜ࡗࡓሙྜࡼࡾࡶ㸪ㄆ㆑ࡢ ṇ⟅⋡ࡀᖹᆒ⣙ 50.4 ࣃ࣮ࢭࣥࢺྥୖࡋ㸪ᥦ᱌ᡭἲ࡟᭷ຠᛶࢆㄆࡵࡓ㸬ࡲࡓ㸪ᥦ᱌ᡭἲ࡛タᐃࡋࡓࣃ࣓࣮ࣛࢱ࡟࠾࠸ ࡚㸪ㄆ㆑ࡢṇ⟅⋡ࡀ᭱኱࡜࡞ࡾ㸪ᥦ᱌ᡭἲ࡟ጇᙜᛶࢆㄆࡵࡓ㸬 ࣮࣮࢟࣡ࢻ㸸⏬ീㄆ㆑㸪㡿ᇦศ๭㸪ࢡࣛࢫࢱࣜࣥࢢ㸪ΰྜṇつศᕸ㸪㍤ᗘ⿵ṇ㸪ຎ໬⏬ീ㸪ᶵᲔᏛ⩦. A Machine Learning Approach to Improve Image Recognition Accuracy for License Plate Numbers HIROO TSUJI†1†2 YOHEI FUKUMIZU†2 TAKAKUNI DOUSEKI†2 HIRONORI YAMAUCHI†2 FUMIHIRO YAMASAKI†2 and AYUMI YOSHIKAWA†3 Abstract: Various image recognition methods are proposed for reading the numbers of the license plate in security cameras because its numbers are useful for criminal investigations. However, it is difficult to obtain high recognition accuracy in the target criminal image which quality is deteriorated remarkably. In deteriorated images, cutouts accuracy degradation of the numbers area and light intensity bias becomes a problem. The optimization of image processing parameters also becomes a problem, so that the approach that does not depend on the subjectivity of the process practitioner is required. In order to solve these problems, we propose a machine learning approach to improve image recognition accuracy for license plate numbers. Firstly, the proposed method cuts out numbers area accurately using image intensity clustering based on Gaussian mixture model in the target block with a raster scan technique. And secondly, the proposed method reduces the effect of the light intensity bias on the recognition accuracy using the segmented image. In addition, the proposed method adopt a machine learning approach for setting the clustering parameters to increase the objectivity of the recognition result. In order to confirm the effectiveness and the validity of the proposed method, we have experimented with real images. From the experimental results, the recognition rate has increased by about 50.4% on average using the proposed method, and the effectiveness of the proposed method was confirmed. Furthermore, the recognition rate was maximized in the parameters of the proposed method, and the proposed method was validated. Keywords: image recognition, image segmentation, clustering, Gaussian mixture model, luminosity compensation, deteriorated image, machine learning. 1. ᗎㄽ. ࡢሙྜ㸪㜵≢࣓࢝ࣛ⏬ീ࡟࠾࠸࡚㸪ࡰࡅ㸪ග㔞ࡢ೫ࡾ㸪⏬ ⣲ᩘ୙㊊㸪ࡪࢀ㸪ࣞࣥࢬṍࡳ➼ࡢຎ໬せᅉࡀ」ྜⓗ࡟㔜␚. ⾤㢌ࡸᗑ㢌࡟࠾ࡅࡿ㜵≢࣓࢝ࣛタ⨨ྎᩘࡣᖺࠎቑຍࡋ࡚. ࡍࡿࡓࡵ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺࡢᩘᏐ᝟ሗࢆどㄆࡍࡿࡇ࡜ࡣ. ࠾ࡾ㸪⌧ᅾ㸪㜵≢࣓࢝ࣛ⏬ീࡣ≢⨥ᤚᰝ࡟Ḟ࠿ࡏ࡞࠸Ꮡᅾ. ᅔ㞴࡛࠶ࡿ㸬. ࡜࡞ࡗ࡚࠸ࡿ㸬㜵≢࣓࢝ࣛ⏬ീ࠿ࡽᚓࡽࢀࡿ᝟ሗࡢ࠺ࡕ㸪. ⌧ᅾࡲ࡛࡟㸪どㄆᅔ㞴࡞ࢼࣥࣂ࣮ࣉ࣮ࣞࢺࡢᩘᏐ᝟ሗࢆ㸪. ㌴୧ࢼࣥࣂ࣮ࣉ࣮ࣞࢺࡢᩘᏐ᝟ሗࡣ㸪㌴୧ࡢ≉ᐃ࡟ࡘ࡞ࡀ. ྛ✀⏬ീㄆ㆑ᢏ⾡࡛ㄞࡳྲྀࡿᡭἲࡀᥦ᱌ࡉࢀ࡚ࡁࡓ[1][2]㸬. ࡿࡓࡵ㸪≢⨥ᤚᰝ࡟࠾࠸࡚≉࡟㔜せ࡛࠶ࡿ㸬ࡋ࠿ࡋ㸪ከࡃ. ࡋ࠿ࡋ㸪᪤ᡭἲ[1][2]ࢆ⏝࠸ࡓሙྜ࡛ࡶ㸪⏬㉁ຎ໬ࡢ✀㢮ࡸ. †1 ⁠㈡┴㆙ᐹᮏ㒊 ⛉Ꮫᤚᰝ◊✲ᡤ Forensic Science Laboratory, Shiga Prefectural Police Headquarters †2 ❧࿨㤋኱Ꮫ኱Ꮫ㝔 ⌮ᕤᏛ◊✲⛉ Graduate School of Science and Engineering, Ritsumeikan University †3 ⏥༡኱Ꮫ ඹ㏻ᩍ⫱ࢭࣥࢱ࣮ Center for Education in General Studies, Konan University. ‫ף‬2016 Information Processing Society of Japan. ⛬ᗘ࡟ࡼࡗ࡚ࡣ༑ศ࡞ㄆ㆑⢭ᗘࢆᚓࡽࢀࡎ㸪ㄆ㆑⢭ᗘࡢྥ ୖࡀㄢ㢟࡜࡞ࡗ࡚࠸ࡿ㸬 ᪤ᡭἲ[1][2]ࡣ࠸ࡎࢀࡶᡭື࡛ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീࡢ ᅄゅᙧ࡟࠾ࡅࡿ㡬Ⅼ఩⨨ࢆ㑅ᢥࡋ㸪㑅ᢥࡋࡓ㡬Ⅼ఩⨨࡟ᇶ ࡙ࡁᩘᏐ㡿ᇦࢆษࡾฟࡍ㸬ࡋ࠿ࡋ㸪ࡰࡅ➼ࡢ⏬㉁ຎ໬ࡀ㢧. 1.

(2) Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report ⴭ࡞ሙྜ࡟ࡣ㸪ほ ⏬ീࡢどㄆᛶపୗ࡟క࠸㸪㑅ᢥࡋࡓ㡬. ‫ݖ‬௜௞ =. Ⅼ఩⨨࡟ㄗᕪࡀ⏕ࡌࡿ㸬ࡑࡋ࡚㸪㡬Ⅼ఩⨨ࡢㄗᕪ࡟ࡼࡾ㸪 ษࡾฟࡋࡓᩘᏐ㡿ᇦࡀᮏ᮶ࡢᩘᏐ㡿ᇦ࠿ࡽࡎࢀ࡚ࡋࡲ࠸㸪 ⤖ᯝ࡜ࡋ࡚ㄆ㆑⢭ᗘࡀపୗࡍࡿ㸬ࡋࡓࡀࡗ࡚㸪➨୍࡟㸪⏬. ଵ ߨ ܰ(࢞|ࣆ௞ , ઱௞ ), ௓೔ ௞. (3). σ௞ ‫ݖ‬௜௞ = 1.. (4). ࢆィ⟬ࡍࡿ㸬. ㉁ຎ໬ࡀ㢧ⴭ࡞ሙྜ࡛ࡶᩘᏐ㡿ᇦࢆṇ☜࡟ษࡾฟࡍᢏ⾡ࡀ. 㸰 ᭱኱໬ࢫࢸࢵࣉ㸦M ࢫࢸࢵࣉ㸧. ㄆ㆑⢭ᗘࡢྥୖࡢࡓࡵ࡟ᚲせ࡛࠶ࡿ㸬ࡲࡓ㸪᪤ᡭἲ[1][2]ࡣ. ḟᘧ࡟ࡼࡾࣃ࣓࣮ࣛࢱࢆ᭦᪂ࡍࡿ㸬. ࠸ࡎࢀࡶග㔞࡟೫ࡾࡢ࡞࠸⏬ീࢆᩍᖌ⏬ീ࡜ࡋ࡚࠸ࡿࡓࡵ㸪 ࢼࣥࣂ࣮ⅉ➼ࡢᙳ㡪࡟ࡼࡾ㸪ࢸࢫࢺ⏬ീࡢග㔞࡟೫ࡾࡀ⏕ ࡌࡿ࡜㸪ㄆ㆑⢭ᗘࡀపୗࡍࡿ㸬ࡼࡗ࡚㸪➨஧࡟㸪ග㔞ࡢ೫ ࡾࡀㄆ㆑⢭ᗘ࡟ཬࡰࡍᙳ㡪ࢆపῶࡍࡿᢏ⾡ࡀㄆ㆑⢭ᗘࡢྥ. ઱௞ =. ୖࡢࡓࡵ࡟ᚲせ࡛࠶ࡿ㸬ࡲࡓ㸪⏬ീㄆ㆑ࡢࣃ࣓࣮ࣛࢱࢆᐈ. ଵ ேೖ. ଵ ேೖ. σ௜ ‫ݖ‬௜௞ ࢞௜ .. (5). σ௜ ‫ݖ‬௜௞ (‫ݔ‬௜ െࣆ௞ ) (‫ݔ‬௜ െࣆ௞ )்.. (6). ࣆ௞ =. ߨ௞ =. ほⓗ࡟᭱㐺໬ࡍࡿᡭἲࡀ㸪ㄆ㆑⤖ᯝࡢᐈほᛶࢆ㧗ࡵࡿࡓࡵ. ேೖ , ே. ܰ௞ = σ௜ ‫ݖ‬௜௞.. ࡟ᚲせ࡛࠶ࡿ㸬. (7) (8). ௨ୖࡢᚲせᛶࢆ⪃៖ࡋ㸪ᶵᲔᏛ⩦࡟ࡼࡿࢼࣥࣂ࣮ࣉ࣮ࣞ ࢺᩘᏐ⏬ീㄆ㆑ࡢྥୖᡭἲࢆᥦ᱌ࡍࡿ㸬ᮏ✏࡟࠾࠸࡚㸪ᥦ. ᑠ⃝ࡽࡣ㸪⾨ᫍ⏬ീࡢ㞼ᇦࢆศ㢮ࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡋ࡚. ᱌ࡢ㡿ᇦศ๭⏬ീࢆ⏝࠸ࡿࡇ࡜࡟ࡼࡾ㸪ᩘᏐ㡿ᇦษࡾฟࡋ. ୖグ㡿ᇦศ๭ࢆ⾜ࡗ࡚࠸ࡿ[5]㸬ᑠ⃝ࡽࡢᡭἲ[5]࡛ࡣ㸪⏬ീ. ⢭ᗘࡀྥୖࡋ㸪ග㔞ࡢ೫ࡾࡀㄆ㆑⢭ᗘ࡟୚࠼ࡿᙳ㡪ࢆపῶ. ࢆᒁᡤ㡿ᇦ࡟ศ๭ࡋ࡚ࢡࣛࢫࢱࣜࣥࢢࢆ⾜ࡗ࡚࠾ࡾ㸪ᒁᡤ. ࡛ࡁࡿࡇ࡜ࢆ㸪⏬ീㄆ㆑ࡢṇ⟅⋡ࢆᣦᶆ࡜ࡋ࡚♧ࡍ㸬ࡲࡓ㸪. 㡿ᇦ࡛ࡣ≉ᚩ㔞ศᕸ࡟ΰྜṇつศᕸࢆ௬ᐃࡍࡿࡇ࡜ࡀ࡛ࡁ. ᥦ᱌ᡭἲ࡛ࡣ㸪㡿ᇦศ๭ࡢࣃ࣓࣮ࣛࢱࢆᶵᲔᏛ⩦࡟ࡼࡾ᭱. ࡿ࡜ࡋ࡚࠸ࡿ㸬ࡑࡋ࡚㸪ᒁᡤ㡿ᇦ୰࡟ࡣ㸪㞼ᇦ㸪㝣ᇦ㸪ᾏ. 㐺໬ࡍࡿࡇ࡜࡟ࡼࡾ㸪ㄆ㆑⤖ᯝࡢᐈほᛶࢆ㧗ࡵࡿ㸬. ᇦࡢ࠺ࡕ㸪ࡓ࠿ࡔ࠿ 2 ࡘࡢ㡿ᇦࡋ࠿ྵࡲࢀ࡞࠸࡜⪃࠼㸪ࢡ. ᮏ✏ࡢᵓᡂࢆ௨ୗ࡟♧ࡍ㸬ࡲࡎ㸪➨ 2 ❶࡛⏬ീࡢ㡿ᇦศ ๭࡟㛵ࡍࡿ㛵㐃◊✲࡟ࡘ࠸࡚ㄝ᫂ࡍࡿ㸬ḟ࡟㸪➨ 3 ❶࡛ᶵ ᲔᏛ⩦࡟ࡼࡿࢼࣥࣂ࣮ࣉ࣮ࣞࢺᩘᏐ⏬ീㄆ㆑ࡢྥୖᡭἲࢆ ᥦ᱌ࡍࡿ㸬ࡑࡋ࡚㸪➨ 4 ❶࡛ᥦ᱌ᡭἲࡢ᭷ຠᛶ㸪ጇᙜᛶࢆ ☜ㄆࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡋ࡚⾜ࡗࡓᐇ㦂࡟ࡘ࠸࡚㏙࡭ࡿ㸬᭱ ᚋ࡟㸪➨ 5 ❶࡛ᮏ✏ࡢ⤖ㄽࢆ㏙࡭ࡿ㸬. ࣛࢫࢱᩘࢆ 2 ࡟タᐃࡋ࡚࠸ࡿ㸬 ࡲࡓ㸪ᒁᡤ㡿ᇦࡢ኱ࡁࡉࢆኚ໬ࡉࡏ࡞ࡀࡽ㸪ᑓ㛛ᐙࡀᡭ ື࡛ศ㢮ࡋࡓ⤖ᯝ࡟ᑐࡋ࡚㸪 ݉ܽ‫= ݁ݐܽݎ_݄݃݊݅ܿݐ‬. ௠௔௧௖௛௜௡௚_௣௜௫௘௟௦ × 100 ௜௠௔௚௘_௣௜௫௘௟௦. (9). ࡛ᐃ⩏ࡋࡓ݉ܽ‫⟬ࢆ݁ݐܽݎ_݄݃݊݅ܿݐ‬ฟࡍࡿࡇ࡜࡟ࡼࡾ㸪᭱㐺࡞ ᒁ ᡤ 㡿 ᇦ ࡢ ኱ ࡁ ࡉ ࢆ ᑟ ࠸ ࡚ ࠸ ࡿ 㸬 (9) ᘧ ࡟ ࠾ ࠸ ࡚ 㸪. 2. 㛵㐃◊✲. ݅݉ܽ݃݁_‫ ࡣ ݏ݈݁ݔ݅݌‬㸪 ᒁ ᡤ 㡿 ᇦ ࡢ ඲ ⏬ ⣲ ᩘ ࡛ ࠶ ࡾ 㸪. ⏬ീࡢ㡿ᇦศ๭ᡭἲࡢ 1 ࡘ࡟㸪⏬ീ≉ᚩ㔞ࡢศᕸ࡟ΰྜ. ݉ܽ‫ࡣݏ݈݁ݔ݅݌_݄݃݊݅ܿݐ‬㸪ᒁᡤ㡿ᇦ୰࡛ᡭື࡟ࡼࡿศ㢮⤖ᯝ࡜ྠ. ṇつศᕸࢆ௬ᐃࡋ࡚≉ᚩ㔞ࢆࢡࣛࢫࢱࣜࣥࢢࡋ㸪ࢡࣛࢫࢱ. ࡌ࢝ࢸࢦ࣮ࣜ࡜࡞ࡗࡓ⏬⣲ᩘࢆ♧ࡍ㸬. ࣜࣥࢢ⤖ᯝ࡟ᇶ࡙ࡁ㡿ᇦศ๭ࡍࡿᡭἲࡀ࠶ࡿ[3]㸬 ࢡࣛࢫࢱࣜࣥࢢࡣ㸪≉ᚩ㔞ࢆ᭱ࡶᙜ࡚ࡣࡲࡾࡢⰋ࠸ࢡࣛ. 3. ᥦ᱌ᡭἲ. ࢫࢱ࡟ศ㢮ࡍࡿၥ㢟࡜ゝ࠺ࡇ࡜ࡀ࡛ࡁࡿ㸬ศ㢮ၥ㢟ࡣ㸪☜. ᥦ᱌ᡭἲࡣ㸪ḟࡢ 2 ࢫࢸࢵࣉ࡟኱ูࡍࡿࡇ࡜ࡀ࡛ࡁࡿ㸬. ⋡ࣔࢹࣝࢆᑟධࡍࡿ࡜㸪ΰྜṇつศᕸࡢࣃ࣓࣮ࣛࢱࢆ᥎ᐃ ࡍࡿၥ㢟࡟ᖐ╔ࡍࡿ㸬‫ܯ‬ḟඖࡢ≉ᚩ㔞ܰಶ࠿ࡽ࡞ࡿࢹ࣮ࢱ ‫=ܦ‬. ൛࢞(ଵ), ࢞(ଶ), ‫ ڮ‬, ࢞(ே) ൟ. (1). 㸯 ᩘᏐ㡿ᇦࡢษࡾฟࡋࢆ㧗⢭ᗘ໬ࡍࡿࢫࢸࢵࣉ 㸰 ග㔞ࡢ೫ࡾࡀㄆ㆑࡟୚࠼ࡿᙳ㡪ࢆపῶࡍࡿࢫࢸࢵࣉ. ࡀ୚࠼ࡽࢀࡓ࡜ࡋ࡚,݇␒┠ࡢࢡࣛࢫࢱࡀ㸪ᖹᆒࣆ௞ 㸪ඹศᩓ ⾜ิ઱௞ ࡢṇつศᕸܰ(࢞|ࣆ௞ , ઱௞ )࡛⾲ࡉࢀࡿ࡜ࡍࡿ࡜㸪ΰྜṇ. ᮏ✏࡛ࡣ㸪ࢫࢸࢵࣉ 1 ࡢࡳࢆ㐺⏝ࡋࡓሙྜࢆᥦ᱌ᡭἲ 1㸪. つศᕸᐦᗘ㛵ᩘࡣ㸪ߨ௞ ࢆΰྜಀᩘ࡜ࡋ࡚㸪. ࢫࢸࢵࣉ 1 ࡜ࢫࢸࢵࣉ 2 ࡢ୧᪉ࢆ㐺⏝ࡋࡓሙྜࢆᥦ᱌ᡭἲ. ࢖(࢞|{ߨ௞ , ࣆ௞ , ઱௞ }) = σ௞ ߨ௞ ܰ(࢞|ࣆ௞ , ઱௞ ). (2). ࡜⾲ࡉࢀࡿ㸬 ᮍ▱ࡢΰྜࣃ࣓࣮ࣛࢱ߆ = {ߨ௞ , ࣆ௞ , ઱௞ }ࢆ᥎ᐃࡍࡿࡓࡵ㸪 EM ࢔ࣝࢦࣜࢬ࣒[4]࡛ࡣḟࡢ 2 ࢫࢸࢵࣉࢆ཯᚟ࡍࡿ㸬. 2 ࡜ࡍࡿ㸬ᥦ᱌ᡭἲࡢᴫせࢆᅗ 1 ࡟♧ࡍ㸬 ࡞࠾㸪ᅗ 1 ࡢᩘᏐ㡿ᇦษࡾฟࡋࡣ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺࡢ ᅄゅᙧ࡟࠾ࡅࡿ㡬Ⅼ఩⨨࡟ᇶ࡙ࡁᢳฟࡋࡓᩘᏐ㡿ᇦೃ⿵࡜ 㡿ᇦศ๭⏬ീ࡟ᇶ࡙ࡁᢳฟࡋࡓᩘᏐ㡿ᇦೃ⿵ࡢඹ㏻㒊ࢆᢳ ฟࡍࡿࡇ࡜࡟ࡼࡾ⾜࠺㸬. 㸯 ᮇᚅ್ィ⟬ࢫࢸࢵࣉ㸦E ࢫࢸࢵࣉ㸧 ≉ᚩ㔞࢞௜ ࡀ݇␒┠ࡢṇつศᕸࢡࣛࢫࢱ࠿ࡽ⏕ᡂࡉࢀࡿ ᗘྜ࠸ࡢ᥎ᐃ್࡛࠶ࡿ㈇ᢸ⋡. ᅗ 1 ࡢࢫࢸࢵࣉ 1㸪ࢫࢸࢵࣉ 2 ࡣ㸪࠸ࡎࢀࡶ㡿ᇦศ๭⏬ ീ࡟ᇶ࡙࠸࡚⾜࠺㸬㡿ᇦศ๭ࡣ㸪ࢡࣛࢫࢱࣜࣥࢢ࡟ᇶ࡙࠸ ࡚⾜࠸㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീࡢ㍤ᗘ್ࢆࢡࣛࢫࢱࣜࣥࢢ ࡢ≉ᚩ㔞࡜ࡋ㸪≉ᚩ㔞ศᕸ࡟ΰྜṇつศᕸࢆ௬ᐃࡍࡿ㸬ᥦ. ‫ף‬2016 Information Processing Society of Japan. 2.

(3) Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report. ┠ⓗ࡜ࡍࡿࡀ㸪B 㒊ࡢᖜࡀ A 㒊ࡼࡾ▷࠸ࡓࡵ㸪B 㒊࡜㞄᥋. ࢫࢸࢵࣉ 1 㡿ᇦศ๭⏬ീ. ほ ⏬ീ ㍤ᗘ್ࢡࣛࢫࢱࣜࣥࢢ. ᩘᏐ㡿ᇦ ᩘᏐ㡿ᇦษࡾฟࡋ. ษࡾฟࡋ. ๭ࡍࡿࡓࡵ࡟ࡣ㸪ࣛࢫࢱࢫ࢟ࣕࣥࡢ඲ᑐ㇟ࣈࣟࢵࢡ࡟࠾࠸ ࡚㸪B 㒊࡜㞄᥋ A 㒊࡜ࢆ᭱㐺࡟ศ๭ࡍࡿᚲせࡀ࠶ࡾ㸪B 㒊 ࡜㞄᥋ A 㒊ࢆ࡜ࡶ࡟ෆໟࡍࡿ㡿ᇦ㸦ᅗ 2(a), (b), (c)ࡢ㉥Ⰽᅄ ゅᙧ࡛ᅖࢇࡔ㡿ᇦࢆෆໟࡍࡿ㡿ᇦ㸧ࢆᑐ㇟ࣈࣟࢵࢡ࡜ࡍࡿ. ࢫࢸࢵࣉ 2. ࡢࡀ㐺ᙜ࡜⪃࠼ࡽࢀࡿ㸬. ほ ⏬ീ࠿ࡽ. 㡿ᇦศ๭⏬ീ࠿ࡽ. ษࡾฟࡉࢀࡓᩘᏐ⏬ീ. ษࡾฟࡉࢀࡓᩘᏐ⏬ീ ᩘᏐㄆ㆑. ᩘᏐㄆ㆑. A 㒊࡜ࡢศ๭ࡀㄢ㢟࡜࡞ࡿ㸬B 㒊࡜㞄᥋ A 㒊࡜ࢆ᭱㐺࡟ศ. ௚᪉㸪୍⯡ⓗ࡟㸪ᑐ㇟ࣈࣟࢵࢡࢧ࢖ࢬࡀᑠࡉ࠸࡯࡝㸪ග 㔞ࡢ೫ࡾࡀ㡿ᇦศ๭࡟୚࠼ࡿᙳ㡪ࢆపῶ࡛ࡁࡿ㸬 ௨ୖࢆ⪃៖ࡍࡿ࡜㸪ᅗ 2(a), (b), (c)ࡢ㉥Ⰽᅄゅᙧ࡛ᅖࢇࡔ 㡿ᇦ࡛ᐃ⩏ࡍࡿ㡿ᇦศ๭ᑐ㇟㒊ࢆෆໟࡋ㸪࠿ࡘ㸪㠃✚ࡀ᭱. ㄆ㆑⤖ᯝ. ㄆ㆑⤖ᯝ. ᑠ࡜࡞ࡿṇ᪉ᙧࢧ࢖ࢬࡀ᭱㐺࡞ࣈࣟࢵࢡࢧ࢖ࢬ࡟㏆ఝࡍࡿ. 㸦ᥦ᱌ᡭἲ 1㸧. 㸦ᥦ᱌ᡭἲ 2㸧. ࡜⪃࠼ࡽࢀࡿ㸬ࡓࡔࡋ㸪᭱㐺࡞ࣈࣟࢵࢡࢧ࢖ࢬࢆᥦ᱌ᡭἲ. ᅗ 1 ᥦ᱌ᡭἲࡢᴫせ. ࡢࣃ࣓࣮ࣛࢱ᭱㐺໬ᡭἲ࡟㐺ྜࡉࡏࡿࡓࡵ㸪㡿ᇦศ๭ᑐ㇟. ᱌ᡭἲ࡛ࡣ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ࡟࠾ࡅࡿග㔞ࡢ೫ࡾࡀ㡿ᇦ. 㒊ࢆᅄゅᙧ㸪᭱㐺࡞ࣈࣟࢵࢡࢧ࢖ࢬࢆṇ᪉ᙧ࡟㝈ᐃࡋࡓ㸬 ௨ୖࡼࡾ㸪ᅗ 2(a), (b), (c)ࡢ඲㡿ᇦศ๭ᑐ㇟㒊ࢆෆໟࡋ㸪. ศ๭࡟୚࠼ࡿᙳ㡪ࢆపῶࡍࡿࡓࡵ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീ ࡟ᑐࡋ࡚ᒁᡤⓗ࡟ࢡࣛࢫࢱࣜࣥࢢᑐ㇟ࣈࣟࢵࢡࢆタᐃࡍࡿ㸬. ࠿ࡘ㠃✚ࡀ᭱ᑠ࡜࡞ࡿṇ᪉ᙧ࡟࠾ࡅࡿ㎶ࡢ㛗ࡉࢆࣈࣟࢵࢡ. ᑐ㇟ࣈࣟࢵࢡෆ࡛ࢡࣛࢫࢱࣜࣥࢢࢆ⾜࠸㸪ᑐ㇟ࣈࣟࢵࢡࢆ. ࢧ࢖ࢬึᮇ್‫(ܮ‬0)࡜ࡍࡿ㸬. ࣛࢫࢱࢫ࢟ࣕࣥࡋࡓᚋ㸪ྛࢡࣛࢫࢱࣜࣥࢢ⤖ᯝࡢᖹᆒ್ࢆ. ᥦ᱌ᡭἲࡢࢡࣛࢫࢱࣜࣥࢢ࡛ࡣ㸪᭱ึ࡟㸪ࢼࣥࣂ࣮ࣉࣞ. ⟬ฟࡋ࡚ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീ඲యࡢࢡࣛࢫࢱࣜࣥࢢ⤖ᯝ. ࣮ࢺ⏬ീࡢᅄゅᙧ࡟࠾ࡅࡿ㡬Ⅼ఩⨨࡟ᇶ࡙ࡁ㸪ୖグࡢ࡜࠾. ࡜ࡍࡿ㸬ࣛࢫࢱࢫ࢟ࣕࣥࢆᐇ⾜ࡍࡿ㝿࡟ࡣ㸪ࢼࣥࣂ࣮ࣉࣞ. ࡾᐃ⩏ࡋࡓ㡿ᇦศ๭ᑐ㇟㒊ࢆ᥎ᐃࡋࣈࣟࢵࢡࢧ࢖ࢬึᮇ್. ࣮ࢺ⏬ീ➃㒊࡛ቃ⏺㒊ࢆᢡࡾ㏉ࡋ࡚ᣑᙇࡍࡿ㸬. ࢆỴᐃࡋࡓᚋ㸪ḟࡢ 2 ࢫࢸࢵࣉࢆᐇ⾜ࡍࡿ㸬 ࢫࢸࢵࣉ A ࡣ㸪. ᑐ㇟ࣈࣟࢵࢡࢧ࢖ࢬࡢ᭱㐺್ࡣ㸪㡿ᇦศ๭ᑐ㇟㒊ࡢ኱ࡁ. ࣈࣟࢵࢡࢧ࢖ࢬึᮇ್ࢆධຊࡍࡿࡇ࡜࡟ࡼࡾ㛤ጞࡍࡿ㸬. ࡉ࡟౫Ꮡࡍࡿ㸬㡿ᇦศ๭ᑐ㇟㒊ࢆㄝ᫂ࡍࡿࡓࡵࡢࢼࣥࣂ࣮ ࣉ࣮ࣞࢺ⏬ീᶍᘧᅗࢆᅗ 2 ࡟♧ࡍ㸬. 㸿 ࣃ࣓࣮ࣛࢱ᭱㐺໬ࢫࢸࢵࣉ ࢡࣛࢫࢱᩘ࡜ࣈࣟࢵࢡࢧ࢖ࢬࢆ᭱㐺໬ࡍࡿ㸬㹀 ᐇ⾜ࢫࢸࢵࣉ ࢫࢸࢵࣉ A ࡛᭱㐺໬ࡋࡓࣃ࣓࣮ࣛࢱࢆ⏝࠸࡚ࢡࣛࢫࢱ ࣜࣥࢢࢆᐇ⾜ࡍࡿ㸬. (a). ࢫࢸࢵࣉ A ࡟࠾࠸࡚㸪࣋࢖ࢬ᝟ሗ㔞ᇶ‽㸦BIC㸧ࢆᑟධ ࡍࡿ㸬ࢡࣛࢫࢱᩘ‫࡟ܭ‬㛵ࡍࡿ BIC ࡣ㸪≉ᚩ㔞ࡢศᕸ࡟ΰྜ ṇつศᕸࢆ௬ᐃࡋࡓሙྜ㸪(2)ᘧ࡟࠾࠸࡚, ߨ௞ ࡟‫ܭ‬ಶ㸪ࣆ௞ ࡟ ‫ ܯܭ‬ಶ㸪 ઱௞ ࡟‫ ܯ(ܯܭ‬+ 1)Τ2 ಶࡢࣃࣛ ࣓࣮ࢱࢆ౑࠺ ࡢ࡛㸪 ‫ܮ‬൫߆෠ ห‫ܦ‬൯ࢆᮍ▱ࣃ࣓࣮ࣛࢱࡢᑐᩘᑬᗘ࡜ࡋ࡚㸪 BICΰྜṇつศᕸ = െ2‫ܮ‬൫߆෠ ห‫ܦ‬൯ +. (b). ‫ܭ‬ (‫ ܯ‬+ 1)(‫ ܯ‬+ 2)lnܰ (10) 2. ࡜࡞ࡿ[6]㸬 ࢫࢸࢵࣉ A ࢆ㸪BICΰྜṇつศᕸ ࡟ࡼࡾ⟬ฟࡉࢀࡿホ౯㛵ᩘ ࡟㛵ࡍࡿ໙㓄ἲ࡛⾜࠺㸬ᮏ໙㓄ἲ࡛ࡣ㸪ࣈࣟࢵࢡࢧ࢖ࢬึ ᮇ್‫(ܮ‬0)࡟࠾࠸࡚BICΰྜṇつศᕸ ࢆ㐺⏝ࡋ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞ ࢺ⏬ീ඲యࡢ᭱㐺ࢡࣛࢫࢱᩘ‫ܭ‬᭱㐺 ࢆồࡵࡓᚋ㸪݅ࢆᩚᩘ࡜ (c) ᅗ 2 ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീᶍᘧᅗ. ࡋ࡚݅ = 0࠿ࡽ᪼㡰࡟ࣈࣟࢵࢡࢧ࢖ࢬ‫ࢆ)݅(ܮ‬᭦᪂ࡋ㸪᭱㐺࡞ ࣈࣟࢵࢡࢧ࢖ࢬࢆ᥈⣴ࡍࡿ㸬 ࢫࢸࢵࣉ A ࡟࠾࠸࡚ୗグ⾲グἲࢆ⏝࠸ࡿ㸬. 㯮Ⰽᅄゅᙧ㸸ᩘᏐ㸪グྕ㒊㉥Ⰽᅄゅᙧ㸸㡿ᇦศ๭ᑐ㇟㒊 ᮏ◊✲࡛ࡣ㸪㞄᥋ࡍࡿᩘᏐ㸪グྕ㒊㸦ᅗ 2(a), (b), (c)ࡢ A 㒊㸧࡜ቃ⏺㒊㸦ᅗ 2(a), (b), (c)ࡢ B 㒊㸧࡜ࢆศ๭ࡍࡿࡇ࡜ࢆ. ‫ף‬2016 Information Processing Society of Japan. z. ݅ᅇ┠ࡢ⧞ࡾ㏉ࡋ࡟࠾ࡅࡿ㸪݆␒┠ࡢᑐ㇟ࣈࣟࢵࢡ࡟ᑐ ࡍࡿ᭱㐺ࢡࣛࢫࢱᩘࢆ‫ܭ‬௝ (‫))݅(ܮ‬㸪඲ࢫ࢟ࣕࣥᅇᩘࢆܵ㸪 ᭱㐺ࢡࣛࢫࢱᩘࡢᖹᆒ್ࢆ‫ࡿࡍ࡜)݅(ܣ‬㸬ࡍ࡞ࢃࡕ㸪. 3.

(4) Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report. ‫= )݅(ܣ‬. σೄೕసభ ௄ೕ (௅(௜)) ௌ. (11). ௦⏝ࡋࡓࡶࡢࢆ᪤ᡭἲ 2 ࡜ࡍࡿ㸬 ᐇ㦂 2 ࡛ࡣ㸪ᥦ᱌ᡭἲࡢࢡࣛࢫࢱࣜࣥࢢ࡟࠾࠸࡚㸪ࢡࣛ. . ࡜࡞ࡿ㸬. ࢫࢱᩘ࡜ࣈࣟࢵࢡࢧ࢖ࢬࢆኚ໬ࡉࡏࡓ࡜ࡁࡢṇ⟅⋡ࢆ ᐃ. z. ݅ᅇ┠ࡢ⧞ࡾ㏉ࡋ࡟࠾ࡅࡿ㸪᭱㐺ࡉࡢᗘྜ࠸ࢆ⾲ࡍホ. ࡋࡓ㸬 4.1 ᐇ㦂᪉ἲ. ౯㛵ᩘࢆ㸪 ‫ )݅(ܣ = )݅(ܬ‬െ ‫ܭ‬᭱㐺. (12). ࣂ࣮ࣉ࣮ࣞࢺࢆ᧜ᙳࡋ࡚↓ᅽ⦰࡛ಖᏑࡋࡓ⏬ീ࡟㸪⏬ീฎ. ࡜ࡍࡿ㸬 ࢫࢸࢵࣉ A ࡢ࢔ࣝࢦࣜࢬ࣒ࢆḟ࡟♧ࡍ㸬. ⌮ࢯࣇࢺ࢙࢘࢔ࢆ౑⏝ࡋ࡚⦰ᑠ࡜ࡰࡅࡢ⏬㉁ຎ໬ࢆຍ࠼ࡓ ⏬ീࢆᑐ㇟⏬ീ࡜ࡋࡓ㸬. 㸯 ݅ = 0࡟タᐃࡍࡿ㸬㸰 BICΰྜṇつศᕸ࡟ࡼࡾ‫ܭ‬௝ (‫(ܮ‬0))ࢆィ⟬ࡍࡿ㸬㸱 (11)ᘧ࡟ࡼࡾ㸪‫(ܣ‬0)ࢆィ⟬ࡍࡿ㸬㸲 ‫ܭ‬᭱㐺 = [‫(ܣ‬0) + 0.5]. ࢼࣥࣂ࣮ⅉࡣ㸪ᕥྑ࡟ 1 ࡘࡎࡘ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ࡟ᑐ ࡋ࡚ᑐ⛠࡞఩⨨࡟タ⨨ࡋࡓ㸬ᕥഃࢼࣥࣂ࣮ⅉ࡟ࡘ࠸࡚㸪タ (13). ࢆồࡵࡿ㸬㸳. ᐇ㦂 1㸪ᐇ㦂 2 ࡛ࡣ㸪ࢼࣥࣂ࣮ⅉࢆⅬⅉࡋࡓ≧ែࡢࢼࣥ. ⨨⟠ᡤࢆᅗ 3 ࡢ༳ d ࡜ࡋ㸪➃Ⅼ࠿ࡽࡢ㊥㞳ࢆᅗ 3 ࡢ㊥㞳 ab㸪Ỉᖹ㊥㞳ࢆᅗ 3 ࡢ㊥㞳 bc㸪ᆶ┤㊥㞳ࢆᅗ 3 ࡢ㊥㞳 cd ࡜ ᐃ⩏ࡍࡿ㸬. (12)ᘧ࡟ࡼࡾ㸪ホ౯㛵ᩘ‫ࢆ್ࡢ)݅(ܬ‬ィ⟬ࡍࡿ㸬. 㸴㸦㸯㸧‫(ܬ‬0) < 0ࡢሙྜ㸪ࡶࡋࡃࡣ݅ ് 0 ࡢ᮲௳ୗ࡛ |‫ ݅(ܬ| < |)݅(ܬ‬െ 1)|࠿ࡘ‫ < )݆(ܬ‬0ࡢሙྜ㸪 ݅ࢆ 1 ⧞ࡾୖࡆ㸪. 4. ᐇ㦂. ‫ ݅(ܮ = )݅(ܮ‬െ 1) + 2. (14). ࡟ࡼࡾ‫ࢆ)݅(ܮ‬᭦᪂ࡋ㸪㸳࡟ᡠࡿ㸬㸦㸰㸧‫(ܬ‬0) > 0ࡢሙྜ㸪ࡶࡋࡃࡣ݅ ് 0 ࡢ᮲௳ୗ࡛ |‫ ݅(ܬ| < |)݅(ܬ‬െ 1)|࠿ࡘ‫ > )݆(ܬ‬0ࡢሙྜ㸪 ݅ࢆ 1 ⧞ࡾୖࡆ㸪 ‫ ݅(ܮ = )݅(ܮ‬െ 1) െ 2. (15). ᅗ 3 ᕥഃࢼࣥࣂ࣮ⅉࡢ఩⨨. ࡟ࡼࡾ‫ࢆ)݅(ܮ‬᭦᪂ࡋ㸪㸳࡟ᡠࡿ㸬. d. 㸦㸱㸧‫(ܬ‬0) = 0ࡢሙྜ㸪ࡶࡋࡃࡣ݅ ് 0 ࡢ᮲௳ୗ࡛. ㊥㞳ab㸸➃Ⅼ࠿ࡽࡢ㊥㞳㸦x᪉ྥ㸧. |‫ |)݅(ܬ‬൒ |‫ ݅(ܬ‬െ 1)|ࡢሙྜ㸪. ㊥㞳bc㸸Ỉᖹ㊥㞳㸦y᪉ྥ㸧. ‫ ݅(ܮ‬െ 1)ࢆ᭱㐺ࣈࣟࢵࢡࢧ࢖ࢬ㸪‫ ݅(ܣ‬െ 1)ࢆ᭱㐺 ࢡࣛࢫࢱᩘ࡜ࡋ࡚⤊஢ࡍࡿ㸬. 4. ᐇ㦂 ᮏᐇ㦂ࡣ㸪ᥦ᱌ᡭἲࡢ᭷ຠᛶ㸪ጇᙜᛶࢆ☜ㄆࡍࡿࡇ࡜ࢆ ┠ⓗ࡜ࡋ࡚⾜ࡗࡓ㸬ᥦ᱌ᡭἲࡢ᭷ຠᛶࢆ☜ㄆࡍࡿᐇ㦂ࢆᐇ 㦂 1㸪ጇᙜᛶࢆ☜ㄆࡍࡿᐇ㦂ࢆᐇ㦂 2 ࡜ࡍࡿ㸬 ᐇ㦂 1 ࡛ࡣ㸪᪤ᡭἲ࡜ࡢẚ㍑ᐇ㦂ࢆ⾜ࡗࡓ㸬᪤ᡭἲࡢ㡿 ᇦศ๭࡜ࡋ࡚㸪ᑠ⃝ࡽࡢᡭἲ[5]ࢆ⏝࠸ࡓ㸬ᑠ⃝ࡽࡢᡭἲ[5] ࡟ᇶ࡙ࡁ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ⏬ീࡢᑠ㡿ᇦ୰࡟ࡣ㸪ᩘᏐࠊ ᩥᏐ㡿ᇦ࡜⫼ᬒ㡿ᇦࡢࡓ࠿ࡔ࠿ 2 ࡘࡢ㡿ᇦࡋ࠿ྵࡲࢀ࡞࠸ ࡜⪃࠼㸪᪤ᡭἲࡢࢡࣛࢫࢱᩘࢆ 2 ࡜ࡋࡓ㸬ࣈࣟࢵࢡࢧ࢖ࢬ ࡟ࡘ࠸࡚㸪ᑠ⃝ࡽࡢᡭἲ[5]࡛ࡣ㸪ᑓ㛛ᐙࡀᡭື࡛ศ㢮ࡋࡓ ⤖ᯝ࡜ࡢẚ㍑࡟ࡼࡾ᭱㐺໬ࡋ࡚࠸ࡿࡀ㸪ࢼࣥࣂ࣮ࣉ࣮ࣞࢺ ⏬ീࡢሙྜ㸪㡿ᇦศ๭ࡢᑓ㛛ᐙࡀᏑᅾࡋ࡞࠸㸬ࡋࡓࡀࡗ࡚㸪 ᪤ᡭἲࡢࣈࣟࢵࢡࢧ࢖ࢬ࡟ࡘ࠸࡚ࡣ㸪ᥦ᱌ἲࡢึᮇ್࡛௦ ⏝ࡋࡓࡶࡢ࡜㸪ࢡࣛࢫࢱᩘ 2 ࡢ᮲௳ୗ࡛ BIC ࡟ࡼࡾ᭱㐺໬ ࡋࡓ್࡛௦⏝ࡋࡓࡶࡢࢆ⏝࠸ࡓ㸬ᮏ✏࡛ࡣ㸪ࣈࣟࢵࢡࢧ࢖ ࢬࢆᥦ᱌ἲࡢࣈࣟࢵࢡࢧ࢖ࢬึᮇ್࡛௦⏝ࡋࡓࡶࡢࢆ᪤ᡭ. 㸸ࢼࣥࣂ࣮ⅉタ⨨⟠ᡤ. ㊥㞳cd㸸ᆶ┤㊥㞳㸦z᪉ྥ㸧 4.1.1 ᐇ㦂 1 ᑐ㇟⏬ീ࡟࠾ࡅࡿࢼࣥࣂ࣮ⅉࡢ఩⨨ࢆ⾲ 1㸪⏬ീฎ⌮᮲ ௳ࢆ⾲ 2㸪ᑐ㇟⏬ീ✀ู㸪ᯛᩘࢆ⾲ 3 ࡟♧ࡍ㸬ᐇ㦂 1 ࡛ࡣ㸪 ᪤ᡭἲ࡜ᥦ᱌ᡭἲࡢᛶ⬟ࢆᩘᏐㄆ㆑ࡢṇ⟅⋡࡟ࡼࡾホ౯ࡋ ࡓ㸬ᩘᏐㄆ㆑࡛ࡣ㸪ᶓ‫⏬[ܯ‬⣲]⦪ܰൣ⏬⣲൧ࡢࢸࢫࢺ⏬ീ ‫݅(ܫ‬, ݆)࡜‫⏬[ܯ‬⣲]⦪ܰൣ⏬⣲൧ࡢᩍᖌ⏬ീܶ(݅, ݆)࡟ᑐࡋ࡚㸪ḟ ᘧ࡛ᐃ⩏ࡉࢀࡿṇつ໬┦஫┦㛵ܴே஼஼ ࢆ⟬ฟࡋ[7]㸪ܴே஼஼ ࡀ᭱ ࡶ኱ࡁࡃ࡞ࡿᩘᏐࢆㄆ㆑⤖ᯝ࡜ࡋࡓ㸬 ܴே஼஼ =. ಾషభ σಿషభ ೕసబ σ೔సబ ூ(௜,௝)்(௜,௝) ಿషభ ಾషభ ಾషభ మ మ ටσಿషభ ೕసబ σ೔సబ ூ(௜,௝) × σೕసబ σ೔సబ ்(௜,௝). (16). 4.1.2 ᐇ㦂 2 ᐇ㦂 2 ࡢࢧࣥࣉࣝ࡟ࡘ࠸࡚㸪ᩘᏐ⏬ീࠕ5ࠖ㸪 ࠕ6ࠖࡣ㸪ࣈ ࣟࢵࢡࢧ࢖ࢬࡢึᮇ್タᐃᇶ‽࡜࡞ࡿ㡿ᇦศ๭ᑐ㇟㒊࡟୍ ᵝ࡟ศᕸࡍࡿࡇ࡜࠿ࡽᥦ᱌ࡢࣔࢹࣝ࡟ᑐࡍࡿ㐺ྜࡀⰋዲ࡜ ⪃࠼ࡽࢀ㸪࠿ࡘ㸪඲ᩘᏐ࡜ࡢẚ㍑࡟࠾࠸࡚ᖹᆒⓗ࡞㍤ᗘ್ ࡛࠶ࡾࢧࣥࣉࣝ࡜ࡋ࡚㐺ᙜ࡜⪃࠼ࡽࢀࡿࡓࡵ㸪ᩘᏐ⏬ീࠕ5ࠖ㸪 ࠕ6ࠖࢆࢧࣥࣉࣝ࡜ࡋࡓ㸬. ἲ 1㸪ࢡࣛࢫࢱᩘ 2 ࡢ᮲௳ୗ࡛ BIC ࡟ࡼࡾ᭱㐺໬ࡋࡓ್࡛. ‫ף‬2016 Information Processing Society of Japan. 4.

(5) Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report ᐇ㦂 2 ࡣ 4 ᯛࡢᩘᏐ⏬ീࠕ5ࠖ࡜ 4 ᯛࡢᩘᏐ⏬ീࠕ6ࠖࢆ. ⾲5. ⏝࠸࡚⾜ࡗࡓ㸬ᑐ㇟⏬ീ࡟࠾ࡅࡿࢼࣥࣂ࣮ⅉࡢ఩⨨ࢆ⾲ 4㸪 ⏬ീฎ⌮᮲௳ࢆ⾲ 5 ࡟♧ࡍ㸬⾲ 1 ᐇ㦂 1 ࡢᑐ㇟⏬ീ࡟࠾ࡅࡿࢼࣥࣂ࣮ⅉࡢ఩⨨ ➃Ⅼ࠿ࡽࡢ. ࢼࣥࣂ࣮ⅉ. ࡰࡅ⏬ീ. ࢧ࢖ࢬ. ⏕ᡂ⏝. [⏬⣲]. ࣇ࢕ࣝࢱ. ᶆ‽೫ᕪ [⏬⣲]. 㸦ᅗ 3 ㊥㞳 ab㸧. ᕥഃ. ✀㢮. 8.5. ㊥㞳[cm] Ỉᖹ㊥㞳[cm]. 㡿ᇦศ๭. ࣭2.5. ฎ⌮. 㸦ᅗ 3 ㊥㞳 bc㸧 ࣭4.5 ᆶ┤㊥㞳[cm] 㸦ᅗ 3 ㊥㞳 cd㸧 ྑഃ. ࢼࣥࣂ࣮ⅉ. ᐇ㦂 2 ࡢ⏬ീฎ⌮᮲௳. ࢡࣛࢫࢱᩘ ࣈࣟࢵࢡ ࢧ࢖ࢬ[⏬⣲]. ࢞࢘ࢩ࢔ࣥࣇ࢕ࣝࢱ ᶓ 10⦪ 10 1.2 2, 3, 4 6, 10, 14, 18, 22. 4.2 ᐇ㦂⤖ᯝ. 1. 4.2.1 ᐇ㦂 1. ᕥ ഃ ࢼ ࣥ ࣂ ࣮ ⅉ ࡜ ᕥྑ. ᪤ᡭἲ࡜ᥦ᱌ᡭἲ࡟࠾ࡅࡿᩘᏐㄆ㆑ࡢṇ⟅⋡ࢆᅗ 4㸪඲. ᑐ⛠࡜࡞ࡿ఩⨨. ᑐ㇟⏬ീ࡟࠾ࡅࡿᖹᆒṇ⟅⋡࡜ᖹᆒṇ⟅⋡ࡢᕪศࢆ⾲ 6 ࡟ ♧ࡍ㸬 ᑐ㇟⏬ീ࡜㡿ᇦศ๭⏬ീࡢ୍౛ࢆᅗ 5 ࡟♧ࡍ㸬. ⾲ 2 ᐇ㦂 1 ࡢ⏬ീฎ⌮᮲௳ ⦰ᑠ᪉ἲ. ࣂ࢖࣮࢟ࣗࣅࢵࢡἲ ᶓ 49⦪ 23. ⦰ᑠᚋࡢᑐ㇟⏬ീࢧ࢖ࢬ[⏬⣲] ✀㢮. ࢞࢘ࢩ࢔ࣥࣇ࢕ࣝࢱ. ࢧ࢖ࢬ. ࡰࡅ⏬ീ. ᶓ 10⦪ 10. [⏬⣲]. ⏕ᡂ⏝ ࣇ࢕ࣝࢱ. ࣭1.1. ᶆ‽೫ᕪ. ࣭1.2. [⏬⣲]. ࣭1.3. ⾲ 3 ᐇ㦂 1 ࡢᑐ㇟⏬ീ✀ู㸪ᯛᩘ ࢼࣥࣂ࣮ⅉ. ࢞࢘ࢩ࢔ࣥ. ᩘᏐ⏬ീ. ఩⨨[cm]. ࣇ࢕ࣝࢱࡢ. ࠕ0ࠖ࠿ࡽ. ᅗ3. ᶆ‽೫ᕪ. ࠕ9ࠖࡢ. ㊥㞳 bc. [⏬⣲]. ྛᯛᩘ. 1.1. 4. 1.2. 4. ᑐ㇟⏬ീ 3. 1.3. 4. ᑐ㇟⏬ീ 4. 1.1. 4. 1.2. 4. 1.3. 4. ᑐ㇟⏬ീ 1 ᑐ㇟⏬ീ 2. 2.5. ᑐ㇟⏬ീ 5. 4.5. ᑐ㇟⏬ീ 6. ⾲ 4 ᐇ㦂 2 ࡢᑐ㇟⏬ീ࡟࠾ࡅࡿࢼࣥࣂ࣮ⅉࡢ఩⨨ ➃Ⅼ࠿ࡽࡢ ㊥㞳[cm]. 8.5. 㸦ᅗ 3 ㊥㞳 ab㸧. ᕥഃ ࢼࣥࣂ࣮ⅉ. Ỉᖹ㊥㞳[cm] 㸦ᅗ 3 ㊥㞳 bc㸧 ᆶ┤㊥㞳[cm] 㸦ᅗ 3 ㊥㞳 cd㸧 ྑഃ. ࢼࣥࣂ࣮ⅉ. ᅗ 4 ᪤ᡭἲ࡜ᥦ᱌ᡭἲ࡟࠾ࡅࡿᩘᏐㄆ㆑ࡢᖹᆒṇ⟅⋡ ⾲ 6 ඲ᑐ㇟⏬ീ࡟࠾ࡅࡿᖹᆒṇ⟅⋡࡜ᖹᆒṇ⟅⋡ࡢᕪศ ᪤ᡭἲࡢᖹᆒṇ⟅⋡[%] ᪤ᡭἲ 1 ࡜᪤ᡭἲ 2 ࡜ࡢ. 23.5. ᖹᆒ್ ᥦ᱌ᡭἲ 2 ࡢᖹᆒṇ⟅⋡[%]. 73.9. ᖹᆒṇ⟅⋡ࡢᕪศ[%]. 50.4. 4.2.2 ᐇ㦂 2 ᥦ᱌ᡭἲࡢࢡࣛࢫࢱࣜࣥࢢ࡟࠾࠸࡚㸪ࣈࣟࢵࢡࢧ࢖ࢬ ‫⏬[ܮ‬⣲]࡜ࢡࣛࢫࢱᩘ‫ࢆܭ‬ኚ໬ࡉࡏࡓ࡜ࡁࡢṇ⟅⋡ࢆᅗ 6㸪⾲ 7 ࡟♧ࡍ㸬 4.3 ⪃ᐹ 4.3.1 ᐇ㦂 1 ᥦ᱌ᡭἲࡢ᭷ຠᛶ࡟ࡘ࠸࡚㸪ᅗ 4 ࡼࡾ඲ᑐ㇟⏬ീ࡟࠾࠸. 2.5. ࡚ᥦ᱌ᡭἲ 1 ࡣ᪤ᡭἲ 1㸪2 ࡼࡾࡶᖹᆒṇ⟅⋡ࡀྥୖࡋ࡚ ࠾ࡾ㸪ᥦ᱌ᡭἲ 2 ࡣᥦ᱌ᡭἲ 1 ࡼࡾࡶࡉࡽ࡟㧗࠸ᖹᆒṇ⟅. 1. ⋡࡜࡞ࡗ࡚࠸ࡿ㸬ࡲࡓ㸪⾲ 6 ࡼࡾ㸪඲ᑐ㇟⏬ീ࡟࠾ࡅࡿᖹ ᆒṇ⟅⋡࡟ࡘ࠸࡚ࡶ㸪ᥦ᱌ᡭἲ 2 ࡢᖹᆒṇ⟅⋡ࡀ᪤ᡭἲࡢ. ᕥ ഃ ࢼ ࣥ ࣂ ࣮ ⅉ ࡜ ᕥྑ. ᖹᆒṇ⟅⋡ࢆୖᅇࡗ࡚࠸ࡿ㸬. ᑐ⛠࡜࡞ࡿ఩⨨. ‫ף‬2016 Information Processing Society of Japan. 5.

(6) Vol.2016-CVIM-202 No.2 2016/5/12. ᝟ሗฎ⌮Ꮫ఍◊✲ሗ࿌ IPSJ SIG Technical Report ㆑⢭ᗘࡢ⤖ᯝ࡜୍⮴ࡋ࡚࠸ࡿ㸬 4.3.2 ᐇ㦂 2. ᥦ᱌ᡭἲࡢጇᙜᛶ࡟ࡘ࠸࡚㸪ᮏᐇ㦂౑⏝⏬ീࡢሙྜ㸪ᥦ ᱌ᡭἲ࡟࠾ࡅࡿࣈࣟࢵࢡࢧ࢖ࢬࡢึᮇ್ࡣ 14 ⏬⣲㸪᭱㐺 ࢡࣛࢫࢱᩘࡣ 3 ࡜࡞ࡿ㸬ᅗ 6㸪⾲ 7 ࡼࡾᖹᆒṇ⟅⋡ࡣࣈࣟ. (a)ᑐ㇟⏬ീ 4. ࢵࢡࢧ࢖ࢬ 14 ⏬⣲㸪ࢡࣛࢫࢱᩘ 3 ࡢ࡜ࡁ࡟᭱኱್࡜࡞ࡗ ࡚࠾ࡾ㸪ᥦ᱌ᡭἲࡢጇᙜᛶࢆ☜ㄆ࡛ࡁࡿ㸬ࡓࡔࡋ㸪ᖹᆒṇ ⟅⋡ࡣ㸪ࣈࣟࢵࢡࢧ࢖ࢬ 10 ⏬⣲㸪ࢡࣛࢫࢱᩘ 3 ࡢ࡜ࡁ࡟ࡶ ᭱኱್࡜࡞ࡗ࡚࠾ࡾ㸪ᥦ᱌ᡭἲࡼࡾࡶ᭱㐺࡞ࣈࣟࢵࢡࢧ࢖ (b)᪤ᡭἲ 1 ࡢ㡿ᇦศ๭⏬ീ. (c)᪤ᡭἲ 2 ࡢ㡿ᇦศ๭⏬ീ. ࢬึᮇ್ࡀᏑᅾࡍࡿྍ⬟ᛶࡀ࠶ࡿ㸬. 5. ⤖ㄽ ᮏ✏࡛㸪ᶵᲔᏛ⩦࡟ࡼࡿࢼࣥࣂ࣮ࣉ࣮ࣞࢺᩘᏐㄆ㆑⢭ᗘ ࡢྥୖᡭἲࢆᥦ᱌ࡋࡓ㸬 ᥦ᱌ᡭἲࡢ᭷ຠᛶࢆ☜ㄆࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡋࡓᐇ㦂ࡢ. (d)ᥦ᱌ᡭἲࡢ㡿ᇦศ๭⏬ീ. ⤖ᯝ㸪ᥦ᱌ᡭἲࢆ㐺⏝ࡋࡓሙྜ㸪᪤ᡭἲࢆ㐺⏝ࡋࡓሙྜࡼ. ᅗ 5 ᑐ㇟⏬ീ࡜㡿ᇦศ๭⏬ീࡢ୍౛. ࡾࡶㄆ㆑⢭ᗘࡀྥୖࡋ㸪ᥦ᱌ᡭἲ࡟᭷ຠᛶࢆㄆࡵࡓ㸬ᐇ㦂 ࡣᐇ⎔ቃࢆ᝿ᐃࡋࡓ᮲௳ୗ࡛⾜ࡗ࡚࠾ࡾ㸪ᮏ✏ࡢ⤖ᯝࡣ㸪 ᐇ⎔ቃ࡟࠾ࡅࡿᥦ᱌ᡭἲࡢ᭷ຠᛶࢆ♧၀ࡍࡿࡶࡢ࡛࠶ࡿ㸬 ᥦ᱌ᡭἲࡢጇᙜᛶࢆ☜ㄆࡍࡿࡇ࡜ࢆ┠ⓗ࡜ࡋࡓᐇ㦂ࡢ⤖ ᯝ㸪ᥦ᱌ᡭἲࡢࣈࣟࢵࢡࢧ࢖ࢬึᮇ್㸪᭱㐺ࢡࣛࢫࢱᩘ࡟ ࠾࠸࡚ṇ⟅⋡ࡀ᭱኱࡜࡞ࡗ࡚࠾ࡾᥦ᱌ᡭἲ࡟ጇᙜᛶࢆㄆࡵ ࡓ㸬ࡋ࠿ࡋ㸪ᐇ㦂⤖ᯝ࡟࠾࠸࡚ᥦ᱌ᡭἲࡢࣈࣟࢵࢡࢧ࢖ࢬ ึᮇ್㸪᭱㐺ࢡࣛࢫࢱᩘ௨እ࡟ࡶṇ⟅⋡ࡀ᭱኱࡜࡞ࡿࣈࣟ ࢵࢡࢧ࢖ࢬ㸪ࢡࣛࢫࢱᩘࡀᏑᅾࡋ࡚࠾ࡾ㸪ᥦ᱌ᡭἲࡼࡾࡶ ᭱㐺࡞ึᮇ್ࡀᏑᅾࡍࡿྍ⬟ᛶࡀ࠶ࡿ㸬௒ᚋࡢㄢ㢟ࡣ㸪ࣈ. ࢡࣛࢫࢱᩘ ᩘ K ࡀ ࡢሙྜ. ࣟࢵࢡࢧ࢖ࢬึᮇ್ࡢ᭱㐺໬࡟ࡘ࠸᳨࡚ウࡍࡿࡇ࡜࡛࠶ࡿ㸬. ࢡࣛࢫࢱᩘ ᩘ K ࡀ ࡢሙྜ ࢡࣛࢫࢱᩘ ᩘ K ࡀ ࡢሙྜ ࣔࢹࣝ࡟ ࡟୙㐺ྜ. ཧ⪃ᩥ⊩. ᅗ 6 ࣈࣟࢵࢡࢧ࢖ࢬ‫⏬[ܮ‬⣲]࡜ࢡࣛࢫࢱᩘ‫ࢆܭ‬ኚ໬ࡉࡏࡓ. [1]. ർᕝṌ㸪ർᕝዉ⥴Ꮚ㸪⸨⏣࿴ᘯ㸪グྕ㆑ู⿦⨨㸪グྕ㆑ู᪉ ἲ㸪࠾ࡼࡧࢥࣥࣆ࣮ࣗࢱࣉࣟࢢ࣒ࣛ㸬≉チ➨ 4243941 ྕ㸬 2004-05-13㸬. [2]. ⚟㇂♸㈗㸪㟷ᮌ೺ኴ㑻㸪⚟Ỉὒᖹ㸪ᒣෆᐶ⣖㸪͆ᶵᲔᏛ⩦࡟ ࡼࡿຎ໬ࢼࣥࣂ࣮ࣉ࣮ࣞࢺᩘᏐㄆ㆑ࡢ᪉ἲ࡜ᛶ⬟ẚ㍑㸪͇㟁 Ꮚ᝟ሗ㏻ಙᏛ఍ᢏ⾡◊✲ሗ࿌㸪Vol.112㸪No.472㸪pp103-106㸪 2013.. [3]. Richard Szeliski㸪ࢥࣥࣆ࣮ࣗࢱࣅࢪࣙࣥ㸪ඹ❧ฟ∧㸪ᮾி㸪 2013.. [4]. A㸬P㸬Dempster㸪N㸬M㸬Laird and D㸬B㸬Rubin㸪 ͆Maximum Likelihood from Incomplete Data via the EM Algorithm㸪͇Journal of the Royal Statistical Society㸪Vol.39㸪No.1㸪pp1-38㸪1977.. [5]. ᑠ⃝᠇⛅㸪㟷ᮌಇᚨ㸬ຍ⸨ᑀ㸪᰿ඖ⩏❶㸪͆ᒁᡤ㡿ᇦ࡛ࡢࢡ ࣛࢫࢱࣜࣥࢢ࡟ࡼࡿ⾨ᫍ⏬ീࡢ㞼ᇦ⮬ືศ㢮㸪͇㟁Ꮚ᝟ሗ㏻ ಙᏛ఍ㄽᩥㄅ㸪Vol.J84-D2㸪No.8㸪pp1608-1617㸪2001.. [6]. ஭ฟ๛㸪ධ㛛 ᶵᲔᏛ⩦࡟ࡼࡿ␗ᖖ᳨▱㸪ࢥࣟࢼ♫㸪ᮾி㸪 2015㸬. [7]. ዟᐩṇᩄ㸦⦅㸧㸪ࢹ࢕ࢪࢱࣝ⏬ീฎ⌮㸪⏬ീ᝟ሗᩍ⫱᣺⯆༠ ఍㸪ᮾி㸪2015㸬. ᅗ 7 ࡜ࡁࡢᖹᆒṇ⟅⋡[%] ⾲ 7 ࣈࣟࢵࢡࢧ࢖ࢬ‫⏬[ܮ‬⣲]࡜ࢡࣛࢫࢱᩘ‫ࢆܭ‬ኚ໬ࡉࡏࡓ ᅗ 7 ࡜ࡁࡢᖹᆒṇ⟅⋡[%]. L. K 2. 6 50. 10. 14. 18. 22. 50. 50. 50. 50. 3. 100. 100. 50. 12.5. 4. 37.5. 0. 0. 0. ௨ୖࡼࡾ㸪ᮏᐇ㦂࡟࠾࠸࡚㸪᪤ᡭἲࢆ㐺⏝ࡋࡓሙྜ࡜ẚ ㍑ࡋ࡚㸪ᥦ᱌ᡭἲ 1 ࢆ㐺⏝ࡋࡓሙྜ㸪ㄆ㆑⢭ᗘࡀᨵၿࡋ㸪 ᥦ᱌ᡭἲ 2 ࢆ㐺⏝ࡋࡓሙྜ㸪ᥦ᱌ᡭἲ 1 ࢆ㐺⏝ࡋࡓሙྜࡼ ࡾࡶࡉࡽ࡟ㄆ㆑⢭ᗘࡀᨵၿࡍࡿ࡜ゝ࠼ࡿ㸬 ࡞࠾㸪ᅗ 5 ࡼࡾ㸪ᥦ᱌ᡭἲࡢሙྜ㸪4 ࡘࡢᩘᏐࡀศ㞳࡛ ࡁ࡚࠸ࡿࡢ࡟ᑐࡋ࡚㸪᪤ᡭἲࡢሙྜ㸪4 ࡘࡢᩘᏐࢆศ㞳࡛ ࡁ࡚࠾ࡽࡎ㸪どぬⓗ࡟ࡶᥦ᱌ᡭἲࡢ㡿ᇦศ๭⏬ീࡣ㸪᪤ᡭ ἲࡢ㡿ᇦศ๭⏬ീࡼࡾࡶⰋዲ࡛࠶ࡾ㸪どぬⓗ࡞⤖ᯝࡣ㸪ㄆ. ‫ף‬2016 Information Processing Society of Japan. 6.

(7) ṇㄗ⾲. 4 ࣮࣌ࢪᕥẁ 16 ⾜┠ 4 ࣮࣌ࢪᕥẁ 21 ⾜┠ 4 ࣮࣌ࢪᕥẁ 25 ⾜┠࠿ࡽ 28 ⾜┠ࡲ࡛. ㄗ. ṇ. ‫ܬ‬ሺ݆ሻ ൏ Ͳ. ‫ܬ‬ሺ݅ሻ ൏ Ͳ. ‫ܬ‬ሺ݆ሻ ൐ Ͳ. ‫ܬ‬ሺ݅ሻ ൐ Ͳ. 㸦㸱㸧‫ܬ‬ሺͲሻ ൌ Ͳࡢሙྜ㸪ࡶࡋࡃࡣ ݅ ് Ͳ ࡢ᮲௳ୗ࡛ ȁ‫ܬ‬ሺ݅ሻȁ ൒ ȁ‫ܬ‬ሺ݅ െ ͳሻȁࡢሙྜ㸪. 㸦㸱㸧‫ܬ‬ሺ݅ሻ ൌ Ͳࡢሙྜ㸪 ‫ܮ‬ሺ݅ሻ ࢆ ᭱ 㐺 ࣈ ࣟ ࢵ ࢡ ࢧ ࢖ ࢬ 㸪 ‫ܣ‬ሺ݅ሻࢆ᭱㐺ࢡࣛࢫࢱᩘ࡜ࡋ࡚. ‫ܮ‬ሺ݅ െ ͳሻࢆ᭱㐺ࣈࣟࢵࢡࢧ࢖ࢬ㸪. ⤊஢ࡍࡿ㸬. ‫ܣ‬ሺ݅ െ ͳሻࢆ᭱㐺ࢡࣛࢫࢱᩘ࡜ࡋ࡚. 㸦㸲㸧݅ ് Ͳ ࡢ᮲௳ୗ࡛. ⤊஢ࡍࡿ㸬. ȁ‫ܬ‬ሺ݅ሻȁ ൒ ȁ‫ܬ‬ሺ݅ െ ͳሻȁࡢሙྜ㸪 ‫ܮ‬ሺ݅ െ ͳሻ ࢆ ᭱ 㐺 ࣈ ࣟ ࢵ ࢡ ࢧ ࢖ ࢬ 㸪 ‫ܣ‬ሺ݅ െ ͳሻࢆ᭱㐺ࢡࣛࢫࢱᩘ࡜ࡋ࡚ ⤊஢ࡍࡿ㸬.

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