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ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ØØØ   DDDaaa{{{000 x** Department of Computer Science **University December 3, 2010 xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Jj 1 A Big Idea 2 ��C† 3 Ø Da 4 ¯„Ž{ xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ µe 1 A Big Idea 2 ��C† 3 Ø Da 4 ¯„Ž{ xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ A Big Idea u¾êÆ[>óZ¶d"4ŒÆ�ÚOÆ[David Donoho�Ç‰Ñ ­‡��z UC Berkerley�Yu Bin¶UIUC�Ma Yi Compressive Sensing, CS Sparse Representation ...... xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ µe 1 A Big Idea 2 ��C† 3 Ø Da 4 ¯„Ž{ xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ��C† ‰½x ∈ RN×1˜‘&Ò, e&E´P{�, KŒ±ÏL�� C†?1Ø . y = Ψx , ‡C†x = ΨHy , ÷vΨHΨ = I , Ψ ∈ CN×N , Iü  . éuy ∈ CN×1, Uþ�x8¥, �Ÿþ�Ø ƒ '5. Ï , I�3k‡�Œ�©þ, rÙ§N − k‡˜ 0"ÏL‡C†, U C��{�­ï�©&Ò, Ϗ @N − k‡C†Xê��z, ‡�ه. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ��C† äNöŠµ ?è: �EΨ, ‰��C†y = Ψx , �3y¥k‡Xê9 ÙéA� ˜; )è: rk©þ˜£�éA� ˜, Ù§ ˜0§¦‡ C†xˆ = ΨH yˆ ; ·‚F"‖x − xˆ‖2 = ‖y − yˆ‖2 ≤ δ, ½ ö‖y − yˆ‖2/‖y‖2 ≤ δ. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ��C† ù«?è�":µ ŠâShannonæ�½n, æ�m…ŒU¬é�, Ï ¬ �—�©&Ò�݌U¬é, C†L§žÑ�ž mŒUé; k­‡�©þ� ˜‘X&Ò�ØÓ ØÓ; e3DÑL§¥, k‡©þ¥k¿”�, ¬é­�E¤ î­��J. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ µe 1 A Big Idea 2 ��C† 3 Ø Da 4 ¯„Ž{ xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Ø Da éux ∈ RN×1, é�§�M‚5ÿþs = Φx , Φ ∈ RM×N . Φ�z˜1Œ±w‰´˜‡Daì, §†&҃¦, Œ±¼ �&Ò�Ü©&E. k Φ, w˜eXÛ­��©&Ò. min ‖yˆ‖0 s.t. ΦΨH yˆ = s (1) ½ö´ min ‖s − ΦΨH yˆ‖2 + λ‖yˆ‖0 (2) xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Ø Da Ÿo´0-‰êQ? ù¿›X:�3Š­‡�K‡©þ, ¿ …´g·Aé��. ΦAT÷vŸo5Ÿ? ^4©Û{. b½Φ†Ψ4àƒq(Coherence), =, <ÑΨ�cM1ŠΦ, K^ù‡Ž{¬��: yˆ = ( sM×1 0 ) (3) ùw,´†Ø�. Ÿo? xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Ø Da duK‡©þ� ˜¯kØ�, ¤±þ¡��E{Ø 1. �,, Œ±òΦ�E¤­‡�K1, �´“d´C kN . …, ·‚ØU(½(J˜½´é�. u´, ·‚ÀJΦ†Ψ4à؃q(extremely incoherence)⇒‘Å(randomness): Œ±´pdxD(Ý ; �ãp©Ù�±1Ý (noiselet); Ӟ, F" 1− δ ≤ ‖ΦΨ H yˆ‖2 ‖y‖2 ≤ 1 + δ (4) xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Ø Da nþ, CS�?èüÑ: �EΦ, )¤ÿþs = Φx , �3s; �EÓ��Φ, �E?˜«��C†ΨH , Šâs­�x . xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ Ø Da Ø Da�`³: Œ±DÑ�á�&Ò, $–â»æ�½n�4; |Z6r. s¥�?ۘ‘Ñ´­‡�, ½ö`´Ø­ ‡�, ¿” ,A‘, E,Œ±�{­�. A^2: ÀªÑªæ8��; U©ã”; �ä; �¯ �, {I�󧓮²�OÑ ¢S��¬. ":: s�ݘ„‡¦´:M ≈ 4K½öM ≥ K log(NK ); ­�Ž{´NP�. =´ò0-‰ê=†1-‰ê, du٠،‡5, Ž{�E,ÝE,ép. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ µe 1 A Big Idea 2 ��C† 3 Ø Da 4 ¯„Ž{ xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ ·‚25)º˜eŸo´1-‰ê, Ø´2-‰ê. -T = ΦΨH , K�ª�å Tyˆ = s (5) yˆ�™êkN‡, §kM << N‡, �káõ‡). l AÛþ`, Tyˆ − s = 0´˜‡‡²¡, 32D¯K¥, Œ@ ´˜^†‚(dž, yˆ¥kü‡�ƒ–¦). xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ é‰ê��å: 0-‰ê 1-‰ê 2-‰ê Figure: norms xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ y3¯K2=z: min ‖s − ΦΨH yˆ‖2 + λ‖yˆ‖1 (6) ˜‡š‚5�à`z¯K. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ éu`z¯K, ˜„^FÝ�{5¦). éu‖yˆ‖1, 30:´ØŒ��, u´, Œ±æ^fFÝ{, ²wCq{ �. ó§+¥, ~^�´��š�Jl(orthogonal matching pursuit). duK‡©þ­‡, KÙXê�ýéŠAT'Ù §N − K‡©þŒ�õ. ؔb�K = 1, ˜�š"� ƒyˆq3yˆ¥éA� ˜q. KTyˆÒ´¡EÝ T� 1q�Tq†yˆq�¦È Tq yˆq = sq (7) xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ ÷v ‖s − sq‖2/‖s‖2 ≤ δ (8) Ò´`, Tq†s�ƒq݁p, = | < Tq, s > | >> | < Tr , s > |, r 6= q (9) Ï , I‡OŽ¡EÝ T�¤k�†s�SÈ, é�ýé ŠŒ�@‡=Œ. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ Šâ��¦ yˆq = (T H q Tq) −1THq s (10) k:”Schimidt��z{, {þ rn = s − < Tq, s > < Tq,Tq > Tq (11) ©ªÓTq��. ù´”��š�Jl”¶¡�d5. xxx** ØØØ   DDDaaa{{{000 ØØØ   DDDaaa{{{000 xxx** JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ JJJjjj A Big Idea ������CCC††† ØØØ   DDDaaa ¯¯¯„„„ŽŽŽ{{{ ¯„Ž{ XJK > 1, K1�‡Ò´{þrnÓT�¤k��SȁŒ �@˜�. Xd E, †�é�Œ�K‡©þ. Figure: norms xxx** ØØØ   DDDaaa{{{000 A Big Idea 正交变换 压缩传感 快速算法