µÚ3ÕµäÐ͵ÄÏ¡Êè½á¹¹¼°Ñ¹Ëõ¸ÐÖªËã·¨


3.1ÒýÑÔ
µÚ2ÕÂϵͳ½éÉÜÁ˽ṹ»¯Ñ¹Ëõ¸ÐÖª»ù±¾ÀíÂÛ£¬°üÀ¨½á¹¹»¯Ï¡Êè±íʾ¡¢½á¹¹»¯¹Û²â¾ØÕóÉè¼ÆºÍ½á¹¹»¯Öع¹¡£Í¨¹ý¶ÔµÚ2ÕµÄѧϰ£¬ÎÒÃÇÁ˽⵽½á¹¹»¯Ñ¹Ëõ¸Ð֪ͨ¹ýÒýÈ븴ÔӵĽṹ»¯Ï¡ÊèÄ£ÐÍ£¬´Ó¶øÄܹ»´ó·ù¶ÈÌá¸ßѹËõ¸ÐÖªÀíÂÛÔÚʵ¼ÊÖеÄÓ¦ÓÃÄÜÁ¦¡£ÔÚÐźŴ¦ÀíÁìÓò£¬¸ù¾ÝÐźÅÄ£ÐÍÏ¡Êè½á¹¹µÄ²»Í¬£¬½á¹¹»¯Ñ¹Ëõ¸ÐÖª¿ÉÒÔ·ÖΪ¿éÏ¡ÊèѹËõ¸ÐÖª¡¢ÁªºÏÏ¡ÊèѹËõ¸ÐÖª¡¢¸ß˹ÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖª¡¢ÈºÏ¡ÊèѹËõ¸ÐÖªµÈ¡£±¾ÕÂÖصã½éÉÜ¿éÏ¡ÊèѹËõ¸ÐÖª¡¢ÁªºÏÏ¡ÊèѹËõ¸ÐÖª¡¢¸ß˹ÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖªÈýÀàµäÐͽṹ»¯Ñ¹Ëõ¸ÐÖª·½·¨¡£¾ßÌåÀ´Ëµ£¬Õë¶ÔÕ⼸ÖÖÌØÊâµÄ½á¹¹»¯Ï¡ÊèÐźÅÄ£ÐÍ£¬×ÅÖؽéÉÜÐźÅÖؽ¨·½·¨¡£
ÓÉÓڽṹ»¯Ï¡ÊèÐźÅÊÇÏ¡ÊèÐźŵÄÌØÊâÐÎʽ£¬Òò´ËÆäÖع¹ÎÊÌâÍêÈ«¿ÉÒÔÓþ­µäµÄѹËõ¸ÐÖªÖع¹Ë㷨ȥ´¦Àí¡£È»¶ø£¬Èç¹ûºöÂÔÁËÏ¡ÊèÐźŵÄÄÚ²¿½á¹¹¶øһζµØʹÓþ­µäµÄѹËõ¸ÐÖªÖع¹ËãÀ´Öع¹£¬½«»á¼«´óµØÓ°ÏìËã·¨ÔËÐÐЧÂʺÍÖع¹¾«¶È¡£ÕýÊÇÔÚÕâÒ»±³¾°Ï£¬¿éÏ¡Êè¡¢ÁªºÏÏ¡ÊèµÈһϵÁнṹ»¯Ñ¹Ëõ¸ÐÖª·½·¨±»Ìá³ö¡£¿éÏ¡ÊèѹËõ¸ÐÖªÖع¹Ëã·¨Õë¶Ô¿éÏ¡ÊèÐźŷÇÁãÔªËسɿé³öÏÖµÄÌص㣬չÏÖ³öÏà±È´«Í³Ñ¹Ëõ¸ÐÖªÖع¹Ëã·¨µÄÓÅÊÆ¡£¿éÏ¡ÊèѹËõ¸ÐÖª·½·¨µÄ³£¼ûÐÎʽÓпé»ìºÏ·¶ÊýÓÅ»¯Ëã·¨¡¢¿éÕý½»Æ¥Åä×·×ÙËã·¨¡¢¿éÆ¥Åä×·×ÙËã·¨¡¢¿éÏ¡Êè×Ó¿Õ¼äѧϰËã·¨ºÍ¿éÏ¡Ê豴Ҷ˹Ëã·¨¡£ÁªºÏÏ¡ÊèѹËõ¸ÐÖª·½·¨³£³£ÀûÓÃÐŵÀµÄÏ¡ÊèÌØÐÔºÍʱ¼äÏà¹ØÐÔ½øÐÐÐźÅÖع¹£¬¿É½øÒ»²½¸ÄÉÆÖع¹ÐÔÄÜ£¬¶Ôʱ±äÐŵÀ¾ßÓиüºÃµÄÊÊÓ¦ÐÔ¡£ÁªºÏÏ¡ÊèѹËõ¸ÐÖª·½·¨°üÀ¨ÈýÖÖÁªºÏÏ¡ÊèÄ£ÐÍϵÄÖع¹Ëã·¨ÒÔ¼°¸ß˹ÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖª¡£
3.2¿éÏ¡ÊèѹËõ¸ÐÖª
±¾½ÚÖصã½éÉÜ¿éÏ¡ÊèѹËõ¸ÐÖªÖع¹·½·¨£¬°üÀ¨¿é»ìºÏL2/L1·¶ÊýÓÅ»¯Ëã·¨¡¢Ï¡ÊèÕý½»Æ¥Åä×·×ÙËã·¨¡¢¿éÏ¡ÊèÆ¥Åä×·×Ù¡¢¿éÏ¡Êè×Ó¿Õ¼äѧϰ¼°»ùÓڷDzÎÊý±´Ò¶Ë¹µÄ¿éÏ¡ÊèÖع¹¡£
3.2.1¿éÏ¡ÊèÐźÅÄ£ÐÍ
Ëæ×ÅѹËõ¸ÐÖªÀíÂÛÑо¿µÄÉîÈ룬Ðí¶àÑо¿Õß¿ªÊ¼Æ«ÏòÒ»ÖÖÌØÊ⵫Ҳ³£¼ûµÄÐźšª¡ª¿éÏ¡ÊèÐźš£Ôںܶàʵ¼ÊÓ¦Óó¡¾°ÖУ¬ÐźŵķÇÁãÔªËس£³Ê´Ø³öÏÖ£¬·ûºÏÕâÖֽṹÌØÐÔµÄÐźųÆΪ¿éÏ¡ÊèÐźţ¬¼ûͼ2.10¡£¿éÏ¡ÊèÐźÅ×îÔç³öÏÖÔÚͳ¼ÆѧÖУ¬µ±Ï¡ÊèϵÊý³ÊÏÖ¿é½á¹¹Ê±£¬Ô¬Ã÷½ÌÊÚÔÚLassoËã·¨µÄ»ù´¡ÉÏÒýÈë¿é½á¹¹Ìص㣬²¢½«ÆäÍƹãΪGroup Lasso¡£Ëæºó£¬¿éÏ¡ÊèÄ£ÐÍÓÖ±»ÓÃÓÚͬ²½Ï¡Êè±Æ½üÒÔ¼°±´Ò¶Ë¹¶à²ãÄ£Ð͵ĶàÈÎÎñѹËõ¸ÐÖªºÍѧϰÎÊÌâ£Û1£Ý¡£ÔÚʵ¼ÊÓ¦ÓÃÖУ¬¿éÏ¡ÊèÄ£ÐÍ´æÔÚÐÎʽ¹ã·º£¬ÈçDNA΢ÕóÁУÛ2£Ý¡¢¶à´øÐźţÛ3£Ý¡¢¶àÖزâÁ¿ÎÊÌâ£Û4£Ý¼°ÐźŵÄƵÆ׸ÐÖªµÈÁìÓò¡£
¿éÏ¡ÊèÐźŵÄѹËõ¸Ð֪ģÐÍÈçÏ£º 


y=Dx+n(3.1)



ÆäÖУ¬y¡ÊRLΪ²âÁ¿ÐźÅ; D¡ÊRL¡ÁNΪ¹Û²â¾ØÕó; x¡ÊRNÊdz¤¶ÈΪNµÄ¿éÏ¡ÊèÐźÅÏòÁ¿; nÊÇÔëÉùÏòÁ¿¡£
ÎÒÃÇÒª½â¾öµÄÎÊÌâÊÇÓÉy¡ÊRLºÍ¾ØÕóD¡ÊRL¡ÁN»Ö¸´¿éÏ¡ÊèÐźÅÏòÁ¿x¡£¸ù¾Ý¿éÏ¡ÊèµÄÐÔÖÊ£¬xÓÉÊý¸ö¼¶ÁªµÄ×Ó¿é×é³É£¬¼Ù¶¨Ã¿¸ö¿éµÄ³¤¶ÈÏàµÈ¾ùΪd£¬½«ÐźÅx»®·ÖΪM¸ö×ӿ飬Áîxl,l¡Ê1,2,¡­,M±íʾÐźÅxµÄµÚl¸ö×ӿ飬¼´


x=£Ûx1¡­xdxT£Û1£Ýxd+1¡­x2dxT£Û2£Ý¡­xN-d-1¡­xNxT£ÛM£Ý£Ý(3.2)






ÓÉÉÏʽ¿ÉÖª£¬ÐźÅx×ܳ¤¶ÈN=Md¡£µ±d=1ʱ£¬¿éÏ¡ÊèÐźÅÍË»¯Îª´«Í³µÄÏ¡ÊèÐźš£ÀàËÆÓÚ´«Í³CSÖжÔÐźÅÏ¡Êè¶ÈµÄ¶¨Ò壬¿éÏ¡Êè¶ÈÖ¸µÄÊÇ¿éÏ¡ÊèÐźÅÖзÇÁã¿éµÄ¸öÊý£¬Í¨³£ÓÃL2,0·¶Êý±íʾÐźÅxµÄ¿éÏ¡Êè¶È£¬ÃèÊöÈçÏ£º


¡¬x¡¬2,0=¡ÆMl=1I¡¬x£Ûl£Ý¡¬2(3.3)

I(¡¬x£Ûl£Ý¡¬2)=1,¡¬x£Ûl£Ý¡¬2>0

0,ÆäËû(3.4)



ÆäÖÐI(¡¤)±íʾ¼ÆÊýº¯Êý£¬¶¨ÒåÈçʽ(3.4)¡£
¼ÙÉèÐźÅx¡ÊRNΪKª²¿éÏ¡ÊèÐźţ¬Ôò¡¬x¡¬2,0=K¡£ÖµµÃ×¢ÒâµÄÊÇ£¬Ò»¸öKª²¿éÏ¡ÊèÐźŵÄʵ¼ÊÏ¡Êè¶ÈΪKd£¬ÆäÖÐdΪһ¸ö¿éµÄ³¤¶È¡£ÀàËÆʽ(3.2)ÖÐÐźÅÏòÁ¿xµÄ¶¨Ò壬¿ÉÒ԰ѹ۲â¾ØÕóD±íʾΪM¸ö¼¶ÁªµÄÁпéD£Ûl£Ý¡ÊRL¡Ád£¬l=1,2,¡­,M£¬ÇÒ¹Û²â¾ØÕóDÐèÒªÂú×ãBlockª²RIPÌõ¼þ£º 


D=£Ûd1¡­ddD£Û1£Ýdd+1¡­d2dD£Û2£Ý¡­¡­dN-d+1¡­dND£ÛM£Ý£Ý(3.5)



ÀûÓÃʽ(3.5)£¬Ê½(3.1)¿É¸ÄдΪ 


y=Dx+n =£Ûd1¡­ddD£Û1£Ý¡­¡­dN-d+1¡­dND£ÛM£Ý£Ý¡¤£Ûx1¡­xdxT£Û1£Ý¡­xN-d+1¡­xNxT£ÛM£Ý£ÝT+n(3.6)



¿éÏ¡ÊèÐźŵÄѹËõ²âÁ¿¹ý³ÌÈçͼ3.1Ëùʾ£Û5£Ý¡£ 


ͼ3.1¿éÏ¡ÊèÐźŵÄѹËõ²âÁ¿Ê¾Òâͼ


¿éÏ¡ÊèÖؽ¨Ëã·¨Òª´¦ÀíµÄÎÊÌâ¾ÍÊÇÔÚ²âÁ¿ÖµyºÍ×ÖµäDÒÑÖªµÄÇé¿öÏ£¬Çó½âÐźŵÄ×îÏ¡Êè±íʾÏòÁ¿x^£¬¼´


x^= arg minx¡ÊRN¡Á1¡¬y-Dx¡¬22   s.t.  ¡¬x¡¬1¡ÜK(3.7)



3.2.2¿é»ìºÏ·¶ÊýÓÅ»¯Ëã·¨
ÔÚ´«Í³CSÖؽ¨Ëã·¨»ù´¡ÉÏ£¬ÒÔEldarµÈÈËΪ´ú±íµÄÑо¿ÕßÃdzä·ÖÀûÓÃÉÏÊö¿éÏ¡Êè½á¹¹£¬Ìá³öÁË»ùÓÚ¿éÏ¡ÊèÄ£Ð͵ÄÖؽ¨Ëã·¨£¬½øÒ»²½½µµÍѹËõ¸ÐÖªµÄÊý¾Ý²ÉÑùƵÂÊ£¬Ìá¸ßÊý¾ÝÖع¹Ð§ÂÊ¡£ÏÂÃæÎÒÃÇ·Ö±ð½éÉÜ͹ÓÅ»¯ÀàµÄLp·¶Êý×îС»¯¡¢Ì°À·ÀàµÄMPËã·¨¡¢OMPËã·¨¼°×Ó¿Õ¼äѧϰ(SSL)Ëã·¨ÔÚ¿éÏ¡ÊèÄ£ÐÍϵÄÀ©Õ¹Ëã·¨¡£
ÀûÓÿéÏ¡ÊèÐźŵĽṹÌØÐÔ£¬EldarµÈÈ˳ɹ¦ÀûÓÃ͹ÓÅ»¯µÄ˼·Ìá³öÁËÒ»ÖÖ»ùÓÚ»ìºÏ·¶ÊýµÄÖؽ¨Ëã·¨£¬³ÆΪL2/L1¿é»ìºÏ·¶ÊýÓÅ»¯Öؽ¨Ëã·¨¡£¸ÃËã·¨µÄºËÐÄ˼·ÊÇÕÒµ½·ûºÏÔ¼ÊøÌõ¼þµÄʹ¸÷¿éx£Ûl£Ý(l=1,2,¡­,M)ÄÜÁ¿Ö®ºÍ×îСµÄÐźÅÏòÁ¿x¡£ÎªÐðÊö·½±ã£¬ÎÒÃÇÊ×Ïȸø³öË÷Òý¼¯¦¤={1,2,¡­,M}ÉÏLp/Lq¿é»ìºÏ·¶ÊýµÄ¶¨Òå


¡¬x¡¬p,q=¡ÆMl=1¡¬xl¡¬qp1/q(3.8)



ÄÇôL2/L1¿é»ìºÏ·¶Êý¶ÔӦΪ


¡¬x¡¬2,1=¡ÆMl=1¡¬xl¡¬12(3.9)



ÔòÄ¿±ê½â¾öµÄÎÊÌâ(²»¿¼ÂÇÔëÉùÓ°Ïì)¿É±í´ïΪ


min¡¬x¡¬2,1 s.t. y=Dx(3.10)



ÔÚµÚ2ÕµÄѹËõ¸ÐÖª»ù±¾ÀíÂÛ½éÉÜÖÐÌáµ½£¬ÒªÏë´ïµ½Ðźŵľ«È·Öؽ¨£¬¹Û²â¾ØÕó±ØÐëÂú×ãRIPÐÔÖÊ¡£ÔÚ¿éÏ¡ÊèÐźÅÖУ¬²âÁ¿¾ØÕóͬÑù±ØÐëÂú×ãÏàÓ¦µÄ¿éÊÜÏ޵ȾàÐÔ(¿éRIP)¡£¸ù¾Ý¸ÃÐÔÖÊ£¬Èô´æÔÚÒ»¸ö³£Êý¦Äk¦¤£¬Ê¹µÃÿһ¸öKª²¿éÏ¡ÊèÐźÅÏòÁ¿x¡ÊRN¾ùÂú×㣺 


1-¦ÄK|¦¤¡Ü¡¬Dx¡¬22¡¬x¡¬22¡Ü1+¦ÄK|¦¤(3.11)



Ôò³Æ¾ØÕóDÂú×ã¿éRIPÐÔÖÊ¡£
EldarµÈÈËÖ¤Ã÷£¬Èç¹û¾ØÕóDÓÐ×㹻СµÄ¿éÊÜÏ޵ȾೣÁ¿£¬Ôò»ìºÏ·¶Êý·½·¨Äܹ»±£Ö¤ÈκοéÏ¡ÊèÐźŵÃÒÔÖع¹£¬²»ÂÛÆä·ÇÁã¿éµÄλÖá£ËûÃÇ»¹Ö¤Ã÷ÁËһЩËæ»ú¾ØÕóÒÔ¼«¸ß¸ÅÂÊÂú×ã¿éRIPÐÔÖÊ£¬ÇÒ×ãÒԸ߹ýÂú×ã±ê×¼RIPµÄ¸ÅÂÊ¡£Êµ¼ÊÉÏ£¬¿éRIPÐÔÖʱȱê×¼RIPÐÔÖʵÄÒªÇóµÍºÜ¶à£¬¼´Âú×ã±ê×¼RIPÐÔÖʵľØÕóÒ»¶¨Âú×ã¿éRIPÐÔÖÊ£¬µ«·´Ö®²»È»¡£ÓÉÉÏÊöÃèÊö¿ÉÖª£¬Õë¶Ô¿éÏ¡ÊèÐźŵĻìºÏ·¶Êý×îÓÅ»¯Ëã·¨ÐèÒªÖªµÀ¿éÏ¡ÊèÐźſ黮·ÖµÄ¾ßÌåÐÅÏ¢¡£
3.2.3¿éÕý½»Æ¥Åä×·×ÙËã·¨
»ùÓÚÌ°À·ÀàµÄOMPËã·¨£¬EldarµÈÈËÔÚ¿éÏ¡ÊèÀíÂÛ¿ò¼ÜÏÂÓÖÌá³öÁËBOMPËã·¨¡£Ó봫ͳµÄÆ¥ÅäÀàËã·¨²»Í¬£¬BOMPËã·¨ÔÚÍ걸×ÖµäÖÐÑ¡ÔñÓë²Ð²îr×îÏà¹ØÆ¥ÅäÔ­×ÓµÄ×¼Ôò±äΪ 


¦Ëk= arg maxi=1,2£¬¡­,M¡´rk-1,D£Ûi£Ý¡µ(3.12)



ÆäÖС´¡¤¡µ±íʾÇóÄÚ»ý¡£
BOMPËã·¨µÄ»ù±¾²½ÖèÈç±í3.1Ëùʾ¡£


±í3.1BOMPËã·¨²½Öè



ÊäÈ룺 ¹Û²â¾ØÕóD¡ÊRL¡ÁN£¬²âÁ¿ÖµÏòÁ¿y¡ÊRL£¬·Ö¿éÊýM£¬¿éÏ¡Êè¶ÈK£» 
³õʼ»¯£º ²Ð²îr0=y£» Ë÷Òý¼¯¦«0=«Á£¬µü´ú´ÎÊýk=1£¬Ô­×Ó¼¯¦µ=«Á£¬¹Û²â¾ØÕóµÄ·Ö¿é½á¹¹D=(D£Û1£Ý,D£Û2£Ý,¡­,D£ÛM£Ý)£» 
Ö÷µü´ú¹ý³Ì£º 
²½Öè1,ÕÒµ½¹Û²â¾ØÕóDµÄij¿éD£Ûi£Ý£¬¸ÃÔ­×Ó¿éÓë²Ð²îrµÄÄÚ»ý×î´ó£¬¼Ç¼¶ÔÓ¦µÄË÷ÒýÖµ£¬Èçʽ(3.12)Ëùʾ£» 
²½Öè2,¸üÐÂË÷Òý¼¯¦«k=¦«k-1¡È¦Ëk£¬¼Ç¼ÕÒµ½µÄ¹Û²â¾ØÕóÖеÄÔ­×ӿ鼯ºÏ¦µk=£Û¦µk-1,D£Û¦Ëk£Ý£Ý£»  
²½Öè3,ÓÃ×îС¶þ³Ë·¨¸üбƽüÐźÅx^k£º 
x^k=arg min¡¬y-¦µkx^¡¬2
²½Öè4£¬¸üввîrk£¬Áîk=k+1£» 
rk=y-¦µkx^k
²½Öè5£¬µ±µü´úÖÕÖ¹Ìõ¼þ£¨²Ð²îÊÕÁ²Çé¿ö»òÕßµü´ú´ÎÊý£©²»Âú×㣬»Øµ½²½Öè1£¬¼ÌÐøÏÂÒ»´Îµü´ú£» ·ñÔòµü´úÖÕÖ¹¡£ 
Êä³ö£º Ô­ÐźŵıƽüÖµx^¡£

BOMPËã·¨¼Ì³ÐÁËOMPË㷨ͨ¹ýÊ©ÃÜÌØÕý½»»¯²Ù×÷±£Ö¤¸üеIJвîÓëËùÓÐÒÑÑ¡Ô­×Ó¿éÕý½»µÄÓŵ㣬ÓÐЧ±£Ö¤Á˵ü´úÐÔÄܵÄÓÅÔ½ÐÔºÍËã·¨µÄµÍ¸´ÔӶȣ¬´ó·ù¶È¼õÉÙÁËËã·¨ÊÕÁ²ËùÐèµÄµü´ú´ÎÊý¡£
ºÍ»ìºÏ·¶Êý×îÓÅ»¯Ëã·¨Ò»Ñù£¬BOMPËã·¨Ò²ÐèÒªÖªµÀ¿éÏ¡ÊèÐźŵĿ黮·ÖÐÅÏ¢£¬¶øÇÒºÍOMPÒ»Ñù£¬BOMPÐèÒªÖªµÀÐźŵĿéÏ¡Êè¶È¡£
3.2.4¿éÆ¥Åä×·×ÙËã·¨
BMPËã·¨¶ÔÓ¦ÓÚBOMPËã·¨µÄÒ»ÖÖÌØÊâÇé¿ö£¬¼´¾ØÕóD¸÷¿éD£Ûl£Ýl=1,2£¬¡­,MÖеÄÁÐÂú×ãÏ໥Õý½»µÄÌØÐÔ(²»Í¬¿é¼äµÄÁв»ÐèÂú×ãÏ໥Õý½»)ʱ£¬¿ÉÒÔÓÃBMPËã·¨»Ö¸´Ðźţ¬BMPÒ²ÐèҪͬʱ֪µÀ¿éÏ¡ÊèÐźŵĿ黮·ÖÐÅÏ¢ºÍÐźŵĿéÏ¡Êè¶È£¬BMPËã·¨µÄ¾ßÌå²½ÖèÈç±í3.2Ëùʾ¡£


±í3.2BMPËã·¨²½Öè



ÊäÈ룺 ¹Û²â¾ØÕóD£¬²âÁ¿ÖµÏòÁ¿y£¬·Ö¿éÊýM£¬¿éÏ¡Êè¶ÈK£» 
³õʼ»¯£º ²Ð²îr0=y£¬Ë÷Òý¼¯¦«0=«Á£¬µü´ú´ÎÊýk=1£¬¹Û²â¾ØÕóµÄ·Ö¿é½á¹¹D=(D£Û1£Ý,D£Û2£Ý,¡­,D£ÛM£Ý)£» 
Ðø±í



Ö÷µü´ú¹ý³Ì£º 
²½Öè1,ÕÒµ½¹Û²â¾ØÕóDµÄij¿éD£Ûi£Ý£¬¸ÃÔ­×Ó¿éÓë²Ð²îrÄÚ»ý×î´ó£¬¼Ç¼Ë÷ÒýÖµ¦Ë£¬Èçʽ(3.12)Ëùʾ£» 
²½Öè2,¸üÐÂË÷Òý¼¯¦«k=¦«k-1¡È¦Ëk£» 
²½Öè3,¸üввÈçʽ(3.1)Ëùʾ£¬Áîk=k+1£» 
²½Öè4,ÅжÏÊÇ·ñÂú×ãk>K£¬ÈôÂú×㣬ÔòÍ£Ö¹µü´ú£» Èô²»Âú×㣬Ôò·µ»Ø²½Öè1¡£ 
Êä³ö£º Ô­ÐźŵĹÀ¼ÆÖµx^¡£

BMPËã·¨³õʼ»¯²Ð²îr0=y£¬ÇÒÔÚµÚk´Îµüʱ£¬Í¬Ñù»ùÓÚʽ(3.12)Ñ¡ÔñÓë²Ð²îrk×îÏà¹ØÆ¥ÅäµÄÔ­×Ó¡£µ«Ôڲвî¸üÐÂÕâÒ»²½£¬BMPËã·¨²»ÔÙÐèÒªÔÚÒÑÑ¡Ôñ¼¯ºÏÉÏÀûÓÃ×îС¶þ³Ë·¨¸üвв¶øÊÇÄÜÖ±½Ó¸üввîrkΪ


rk=rk-1-D£Û¦«k£ÝD¢k£Û¦«k£Ýrk-1(3.13)



Óɱí3.2¿ÉÒÔ¿´³ö£¬ÓÉÓÚûÓнøÐÐÔ­×Ó¿éµÄÕý½»»¯´¦Àí£¬BMPË㷨ÿ´Îµü´úÑ¡³öµÄÔ­×Ó¿éÓëÇ°ÃæÒÑÑ¡³öµÄÔ­×Ó¿é·ÇÕý½»£¬¿ÉÄܻᵼÖÂÖظ´Ñ¡ÔñÏàͬԭ×Ó¿éµÄ½á¹û£¬Ê¹µÃËã·¨ÐèÒªºÜ¶à´Îµü´ú²ÅÄÜ´ïµ½ÊÕÁ²£¬Ëã·¨¸´ÔÓÇÒÐźÅÖؽ¨ÐÔÄܵÍÏ¡£ËùÒÔ£¬µ±¾ØÕóD²»Âú×ãÕý½»Ìõ¼þʱ£¬Ö®Ç°½éÉܵÄBOMPËã·¨ÊǸü¼ÓÊʺϵÄÑ¡Ôñ¡£
´ÓÒÔÉÏ¿éÏ¡ÊèµÄ·ÖÎöºÍ¼¸ÖÖÏÖÓлùÓÚ¿éÏ¡Êè½á¹¹ÐźŵÄÖؽ¨Ëã·¨¼ò½é¿É×ܽáµÃµ½£¬»ùÓÚ¿éÏ¡Êè½á¹¹µÄÐźÅÖؽ¨£¬Äܹ»ÒÔ¸üµÍµÄÒªÇóÂú×㾫ȷÖؽ¨ËùÐèÂú×ãµÄÌõ¼þ¡£µ«ËüÃǾùÐèÒªÖªµÀ¿éÏ¡ÊèµÄ¿é»®·ÖÐÅÏ¢¡£¶ÔÓÚÌ°À·ÀàËã·¨£¬ºÍÆÕͨÖؽ¨Ëã·¨ÀàËÆ£¬»¹ÐèÒª¶îÍâÒÑÖªÐźŵĿéÏ¡Êè¶È¡£²»¹ý£¬ÏÖÓм¸ÖÖÖؽ¨Ëã·¨²¢Ã»ÓÐÍÚ¾ò¿éÏ¡Êè½á¹¹¸üÉîÈëµÄÌØÐÔ¡£ÓÉ´ËÒý·¢ÁËÖÚ¶àѧÕßÔÚ´Ë»ù´¡ÉÏÌá³öеÄËã·¨¡£
3.2.5¿éÏ¡Êè×Ó¿Õ¼äѧϰËã·¨
ÏÂÃæÒÔÊÓƵÊý¾ÝΪÀý£¬½éÉÜBSSL¿ò¼ÜÀ´»ñµÃ¿éÏ¡Êè±íʾÓÃÓÚѹËõÊÓƵ²ÉÑù¡£Ê×ÏÈ£¬Í¨¹ýÏû³ý×Ó¿Õ¼äµÄ½»¼¯£¬ÀûÓýṹ»¯Ï¡ÊèÀ´ÓÅ»¯¿éÏ¡Êè±íʾ¡£ÔÚ×Ó¿Õ¼äѧϰÖУ¬²ÉÓÃÇó½â»ùÓÚ¿éÏà¹ØÐÔÔ¼Êø×îС»¯ÎÊÌâµÄ·½·¨À´¹²Í¬×îС»¯×ӿռ䲢ÁªºÏ¸÷×Ó¿Õ¼ä»ù¾ØÕóÖ®¼äµÄ½üËÆÎó²îºÍÏà¹ØÐÔ¡£ÑµÁ·µÃµ½µÄ×Ó¿Õ¼äÖ®¼äÏ໥Õý½»Ê¹µÃÏ¡Êè±íʾ¿É±»½øÒ»²½Ñ¹Ëõ£¬¶ÔÓ¦ÓÚÏàÓ¦×Ó¿Õ¼äµÄ»ù¡£Ö¤Ã÷ͨ¹ý¸Ã¿ò¼Ü¿É½¥½øµÃµ½×îÓŽüËƽ⣬ÆäµÈ¼ÛÓÚÄâÕóÔ¼ÊøϵÄͶӰ×î´ó»¯ÎÊÌ⣻ Ö¤Ã÷BSSL·½·¨ÔÚ¿éRIPÌõ¼þϵÄÎȶ¨Öع¹¿ÉµÃµ½±£Ö¤¡£¸Ã·½·¨Äܹ»ÓÐЧ¿Ë·þÊý¾ÝÇý¶¯×Ó¿Õ¼äÁªºÏÄ£ÐÍ(Union of Dataª²Driven Subspace,UoDS)ÓÉÓÚ×Ó¿Õ¼ä¹ØÁªÎÞ·¨»ñµÃ¿éÏ¡ÊèµÄÎÊÌ⣬ͨ¹ýÓÅ»¯ÊÓƵÐźžֲ¿¼¸ºÎ½á¹¹Ö¸Ê¾µÄ½á¹¹»¯Ï¡ÊèÐ͸ĽøUoDSÄ£ÐÍ¡£
½øÒ»²½µØ£¬BSSL¿ò¼Ü¿ÉÒÔ½øÒ»²½À©Õ¹µ½»ùÓÚKronecker»ý¿ò¼ÜµÄ·º»¯¿éÏ¡Êè×Ó¿Õ¼äѧϰ(Generalized Block Sparse Subspace Learning,GBSSL)£¬²¢½«ÆäÓ¦Óõ½Ñ¹ËõÕÅÁ¿²ÉÑùÖС£Í¨¹ýÔÚ¸÷¸öÕÅÁ¿Ä£Ê½ÏÂѵÁ·µÃµ½µÄ¶àÏßÐÔ»ùÉÏÕûºÏ¿éÏ¡Êè±íʾÓÅ»¯»ùÓÚÕÅÁ¿µÄ±íʾ¡£ÊµÑé½á¹û±íÃ÷£¬¸Ã¿ò¼ÜÓÃÓÚѹËõÊÓƵ²ÉÑù¿É»ñµÃ¸ü¸ßµÄÎȶ¨ÐÔºÍЧÂÊ¡£
´«Í³Ñ¹Ëõ¸ÐÖªÓÉÓÚ¶àάÐźÅÏòÁ¿»¯µÄ±íʾÎÞ·¨¾«È·²¶»ñÄÚÔڵĽṹ¡£¾¡¹Ü»ùÓÚÕÅÁ¿µÄ¸÷ÖÖѹËõ¸ÐÖª·½·¨¿ÉÒÔ±£³Ö¶àάÐźÅÄÚÔڵĸ÷Öֽṹ£¬È»¶øÆä»ùÓÚµ¥×Ó¿Õ¼äÄ£Ð͵ļÙÉèµ¼ÖÂÆä²ÉÑùЧÂʺÍÐÔÄÜÊܵ½ÏÞÖÆÔ¼Êø¡£Îª´Ë£¬ÎÄÏ×£Û6£ÝÌá³öÁËUoDSÄ£ÐÍ£¬ÀûÓÿéÏ¡ÊèÐÔÈ¥»ñµÃ×ÔÊÊÓ¦ºÍ·Ç¾Ö²¿µÄ»ù²¢ÓÃÔÚѹËõÊÓƵ²ÉÑùÖС£UoDSÄ£Ðͼٶ¨ÁªºÏÖеÄ×Ó¿Õ¼äÖ®¼äÏ໥¶ÀÁ¢£¬¿éÏ¡Êè±íʾµÄ½ô´ÕÐÔ»áÊܵ½ÁªºÏÖи÷×Ó¿Õ¼äÏ໥¶ÀÁ¢¼ÙÉèÏÂ×ÖµäѵÁ·µÄÓ°Ïì£Û7£Ý£¬È»¶øÏÖʵÖÐÐí¶à´¦ÔÚ²»Í¬×Ó¿Õ¼äµÄ²ÉÑùÐźż¯ºÏÍùÍù´æÔÚ½»¼¯¡£ÕâÒ»ÊÂʵÒâζ×Å×ӿռ佫ÓÉ»ùÖй²Í¬µÄ¼¸¸öÔ­×ÓÕųɣ¬Òò´Ë¿éÏ¡Êè½á¹¹»á±»ÑÚ¸ÇʹµÃÖع¹ÐÔÄܽµµÍ¡£ÓÐЩ³¡¾°Ï¡ÊèÏòÁ¿µÄ·ÇÁãϵÊý±é²¼ÔÚËùÓлùÉÏ¡£Òò´Ë£¬±¾½Ú½éÉÜ¿éÏ¡Êè×Ó¿Õ¼äѧϰÒÔÏû³ý×Ó¿Õ¼äµÄ½»¼¯£¬Í¬Ê±±£³Ö×Ó¿Õ¼äÁªºÏµÄ»ùÖ®¼äµÄ²»Ïà¹ØÐÔ¡£½ô´ÕµÄ¿éÏ¡Êè±íʾ¿ÉÒÔͨ¹ýÇó½â×îС»¯¿éÏà¹ØÎÊÌâÀ´»ñµÃ¡£´ËÍ⣬Õë¶Ô·º»¯µÄѹËõÕÅÁ¿²ÉÑù£¬¿É²ÉÓ÷º»¯µÄÊý¾ÝÇý¶¯ÕÅÁ¿×Ó¿Õ¼äÁªºÏÄ£ÐÍ£¬ÔÚÕÅÁ¿±íʾÖнøÈëÁ˽ṹ»¯Ï¡Êè²¢¸ø³öÁË¿éÏ¡ÊèÌõ¼þϵÄÈ¡µÃ×îÓŽâµÄÌõ¼þ¡£ÁíÍ⣬ÓйØÕÅÁ¿µÄ»ù±¾¸ÅÄÔÚ3.8½Úµ¥¶À½éÉÜ£¬±¾½Ú²»×öÏêϸ½éÉÜ¡£
ÉèѵÁ·¼¯ÎªX£¬¸ø¶¨t¸ö¾ÛÀà´ØG=£ÛG1,G2,¡­,Gt£Ý£¬ÀûÓÃÎÄÏ×£Û6£ÝÖеÄUoDSÄ£ÐÍ£¬¿ÉÒÔÉú³ÉÒ»¸ö°üº¬r¸ö»ùÏòÁ¿µÄ»ù¾ØÕó¦·*=£Û¦·*1,¦·*2,¡­,¦·*t£Ý¡ÊRn¡Ár£¬ÆäÖÐr=¡Æti=1di>n£¬¾ÛÀà´ØGi±»ÈÏΪ°üº¬ÏàËƵÄÐźţ¬Æä¶ÔÓ¦µÄ×Ó¿Õ¼äÓɦ·*iÕųɡ£Òò´Ë£¬±¾½Ú½éÉÜBSSL·½·¨£¬ÒÔ½øÒ»²½ÓÅ»¯½á¹¹»¯Ï¡ÊèÐÔϵÄѵÁ·ºÍ±íʾ¡£ÎªÁËÒÖÖƴصĽ»¼¯²¢±£Ö¤Ï¡Êè±íʾÖеĿéÏ¡ÊèÐÔ£¬½«Ã¿¸ö´ØµÄ¿éÏà¹ØÐÔ×îС»¯£¬Æä×îÓÅ»ù¾ØÕóD=£ÛD1,D2,¡­,Dt£ÝµÄÇó½â¿ÉÃèÊöΪÏÂÃæµÄÓÅ»¯ÎÊÌ⣺ 


D= arg min¦·*i,Ci¡Æti=1¡¬Gi-¦·*iCi¡¬22+¦Ë¡Æti=1¡Æpij=1¡¬cji¡¬1+¦Î¡Æi¡Ùj¦¸(¦·*i,¦·*j)
(3.14)



ÆäÖÐCi=£Ûc1i,c2i,¡­,cpii£Ý¡ÊRdi¡ÁpiΪµÚi¸ö´ØµÄÏ¡Êè±íʾ£¬ÆäÁÐÏòÁ¿cji¶ÔÓ¦ÓÚ²ÉÑùÐźÅxji,i¡Ê£Û1,t£Ý,j¡Ê£Û1,pi£Ý£» ¿éÏà¹ØϵÊý¦¸(¦·*i,¦·*j)=¡¬(¦·*i)T¦·*j¡¬2F£» ÓÃÓÚƽºâÖع¹Îó²îµÄ²ÎÊý¦Ë¼°¦Î·Ö±ð±íʾϡÊè¶ÈÒÔ¼°¿éÏà¹ØÐԳͷ£ÏµÊý¡£
ÓÉÓÚ¿éÏà¹ØÐԳͷ£ÏîÓëtd2i³ÉÕý±È£¬Òò´Ë¦Î×ÔÊÊÓ¦µ÷Õû£¬³õʼֵΪ¦Î0¡£ÔÚʽ(3.14)ÖУ¬µÚÒ»Ïî»ùÓÚ¶ÔÓ¦µÄ»ùDi£¬¶ÔµÚi¸ö´Ø½øÐÐ×îС»¯Öع¹Îó²î£¬¶øµÚ¶þÏîºÍµÚÈýÏî·Ö±ðͨ¹ýÔ¼ÊøCiµÄÏ¡ÊèÐÔÒÔ¼°×Ó¿Õ¼äÖ®¼äµÄ¿éÏà¹ØÐÔÀ´ÓÅ»¯Ï¡Êè±íʾ¡£Ó¦¸Ã×¢ÒâµÄÊÇ£¬CiµÄ·ÇÁãϵÊý½«±»ºÏ²¢Îª¿é£¬ÒòΪ×Ó¿Õ¼äÖ®¼äµÄ½»¼¯±»²»Ïà¹ØµÄ»ùËùÏû³ý¡£
ʽ(3.14)ÊÇ͹ÎÊÌ⣬¿ÉÒÔͨ¹ý½»ÌæÖ´ÐÐÏ¡Êè±àÂëºÍ»ù¸üÐÂÀ´½â¾ö¡£ÔÚÿ´Îµü´úÖУ¬¹Ì¶¨»ù¦·*i²»±ä£¬Ï¡Êè±íʾCi=£Ûc1i,c2i,¡­,cpii£Ýͨ¹ý¶ÔµÚi¸öÀà´Ø(¸ÃÀà´Ø°üº¬ÁËpi¸öÊý¾Ýµã)Gi=£Ûx1i,x2i,¡­,xpii£ÝµÄÏ¡Êè±àÂëµÃµ½£¬²¢ÇÒ¿ÉÃèÊöΪÈçÏÂÓÅ»¯ÎÊÌ⣺ 


cji= arg min{cji}j=1,2,¡­,pi¡Æpij=1¡¬xji-¦·*icji¡¬22+¦Ë¡¬cji¡¬1(3.15)



Ëæºó£¬¸ù¾ÝµÃµ½µÄcji£¬¶ÔµÚi¸öÀà´Ø½øÐлùµÄ¸üÐÂѧϰ£¬Ñ§Ï°¹ý³ÌËù¶ÔÓ¦µÄÓÅ»¯ÎÊÌâÈçÏ£º 


Di= arg min¦·*i¡Æpij=1¡¬xji-¦·*icji¡¬22+¦Ë¡¬cji¡¬1+¦Î¡Æj¡Ùi¦¸(¦·*i,¦·*j)(3.16)



Òò´Ë£¬Ã¿¸ö¾ÛÀà´Ø¶¼±»¶ÀÁ¢µØÓÅ»¯ÒÔ²úÉú×Ó¿Õ¼äÁªºÏ»ù¾ØÕóD=£ÛD1,D2,¡­,Dt£Ý¡£
BSSL¿ÉÒÔÏÔÖø½µµÍ¿éµÄÏà¹ØÐÔ£¬´Ó¶ø´Ù½ø¿éÏ¡ÊèÐÔ£¬Èçͼ3.2Ëùʾ(×Ó¿Õ¼äÊýÁ¿Îª50²¢ÇÒÿ¸ö×Ó¿Õ¼äÓÉ10¸ö»ùÏòÁ¿×é³É)£¬ÆäÖÐBC±íʾ¿éÏ¡ÊèϵÊý(Block coherence)£¬¸Ãͼ±íÃ÷ͨ¹ýBSSLËã·¨¸÷¸ö×Ó¿Õ¼äÓëÆäÓà×Ó¿Õ¼äµÄ×î´óºÍ×îС¿éÏà¹ØÐÔ±»ÒÖÖÆ¡£Í¼3.3ÁгöÁËÓÉUoDSºÍBSSLµÃµ½µÄÏàÓ¦»ù£¬ÆäÖÐͨ¹ýBSSLÓÅ»¯µÃµ½µÄ»ùµÄÐÐÊÇ¿ÉÇø·ÖµÄ¡£ÕâÒ»ÊÂʵÒâζ×ÅBSSLÖеĿéÏà¹ØÐÔ×îС»¯Ïû³ýÁË×Ó¿Õ¼äµÄ½»¼¯¡£´ËÍ⣬ͼ3.4ÌṩÁËÁ½¸öÀý×ÓÀ´ÏÔʾËùÌá³öµÄ·½·¨µÄÓÐЧÐÔ¡£ÓÉBSSLµÃµ½µÄÖع¹±íʾÏòÁ¿µÄÏ¡ÊèÐÔÊǽô´ÕµÄ£¬½ø¶øµÃµ½¾«È·Öع¹£¬¶øUoDSµÄ·ÇÁãϵÊý±é²¼ÔÚ¶à¸ö×Ó¿Õ¼äÖС£


ͼ3.2ÔÚAkiyoÐòÁÐÉÏ£¬ÀûÓÃUoDSºÍBSSLµÃµ½µÄ¿éÏà¹ØÐԱȽÏ




ͼ3.3UoDSÄ£Ð͵õ½µÄ»ùÓë¾­¹ýBSSLµÃµ½µÄÓÅ»¯ºóµÄ»ùµÄ±È½Ï




ͼ3.4BSSL·½·¨ÓëUoDSÄ£Ð͵ıȽÏ


¶¨Òå3.1(ÄâÕó)M=(G,T)ÊÇÒ»¸ö¶þÔª×飬ÆäÖÐGΪÓÐÏÞ¼¯£¬TΪÓÉGµÄ×Ó¼¯¹¹³ÉµÄ¼¯×壬Èô¶ÔÈÎÒâµÄ×Ó¼¯X,X¡äª¼G£¬Âú×ãÈçÏÂÌõ¼þ£º 
(1) ÒÅ´«ÐÔ£º Èç¹ûX¡ÊT£¬Xª¼X¡ä£¬ÄÇôX¡ä¡ÊT£» 
(2) Ôö¹ãÐÔ£º Èç¹ûX,X¡ä¡ÊT£¬»ùÊýX¡ä<X£¬Ôò´æÔÚÔªËØX¡ÊX£ÜX¡ä£¬Ê¹µÃX¡ä¡È{X}¡ÊT£¬Ôò³ÆÕâ¸ö¶þÔª×éM=(G,T)Ϊһ¸öÄâÕó¡£
¶¨Àí3.1µ±M=(G,T)ÊÇÒ»¸öÄâÕ󣬲¢ÇÒ¶ÔÓÚi,j¡ÊI£¬¦¸(¦·*i,¦·*j)=0ʱ£¬BSSLµÄ¾Ö²¿×îÓŽâ¿ÉÒÔͨ¹ý¿éÕý½»Æ¥Åä×·×Ù(Bª²OMP)Ëã·¨Çó½âµÃµ½¡£
Ö¤Ã÷£º Èç¹ûÁ*i={¦·1,¦·2,¡­,¦·K}Ϊһ¸ö×Ó¼¯£¬¸ÐÐËȤÐźÅΪx¡ÊG£¬ÄÇô¸ù¾ÝÏ£¶û²®ÌØͶӰ¶¨ÀíºÍ¹´¹É¶¨Àí¿ÉµÃ£Û8£Ý£¬


¡¬px(¦Å)¡¬2=¡ÆKi=1¡´x,¦×i¡µ2(3.17)



ÒòΪBª²OMPË㷨ѡÔñÔÚGÖеÄËùÓÐÏòÁ¿ÉϵÄK¸ö×î´óµÄͶӰ£¬Òò´Ë×îÓŽâµÄÇó½âµÈ¼ÛÓÚBª²OMPÇó½â¡£
¶¨Àí3.1Ϊ±£Ö¤BSSLÄ£Ð͵Ä×îÓÅÐÔÌṩÁËÀíÏëÌõ¼þ¡£µ«ÊÇ£¬ÕâÔÚʵ¼ùÖÐÊDz»¿ÉÐеġ£Òò´Ë£¬¿éÏà¹ØÐÔ×îС»¯Í¨¹ýʽ(3.14)µü´úÖ´ÐУ¬ÒÔÓÅ»¯Ã¿¸ö×Ó¿Õ¼äµÄ»ù£¬µÃµ½½ô´ÕµÄ¿éÏ¡Êè±íʾ¡£¸ù¾Ý¶¨Àí3.1£¬¿ÉÒԵõ½Ò»¸ö½üËÆ×îÓŽ⣬¼°½üËÆÎó²îµÄÉϽ硣
¶¨Àí3.2BSSLÎÊÌâ¿ÉÒԵõ½Ò»¸ö(1-(1-sin2¦¼/K)K)ª²½üËÆ×îÓŽ⣺ 


f(TK)¡Ý1-1-sin2¦¼KKf(OPT)(3.18)



½«¶Ô¸ø¶¨µÄX¡ÊSi£¬ÔÚÕýÈ·µÄ¿éÉ϶Իù½øÐÐÓÅ»¯£¬Èç¹û¹¹³ÉD=£ÛD1,D2,¡­,Dt£ÝµÄ»ùÖеÄÔ­×ÓÂú×ã


maxd¡¬D¢kid¡¬1<1(3.19)


ÆäÖÐD¢kiΪDi£¬i=1,2,¡­,tµÄMooreª²PenroseαÄ棬d±íʾDjÖеÄÔ­×ÓÇÒj¡Ùi£¬¦¼Îª×Ó¿Õ¼äÓëËüÃÇÔÚU*ÖеIJ¹Ö®¼ä×îСµÄÖ÷½Ç£¬f£º2X¡úRΪ¶¨ÒåÔÚXÃݼ¯Éϵĺ¯Êý¡£
¸ù¾Ý¶¨Àí3.2£¬¿ÉÒԵõ½ÒÔ(1-(1-sin2¦¼/K)K)ºÍ×îÓŽâf(OPT)À´½ç¶¨µÄ½âf(TK)ÆäÖÐ12<1-(1-sin2¦¼/K£©K<1)
¡£ÕâÒ»ÊÂʵÒâζ×Å£¬¼´Ê¹×ӿռ䲻Ï໥Õý½»£¬Ò²¿ÉÒԵõ½½Ó½ü×îÓŵĽ⡣
ÔÚÓÅ»¯µÄ»ù¾ØÕóDÉÏ»ñµÃ½ô´Õ¿éÏ¡Êèºó£¬¿É²ÉÓÃ͹ÓÅ»¯Ëã·¨µÃµ½ÓÐЧµÄ¾«È·Öع¹¡£ÏÂÃæµÄ¶¨ÀíÖ¤Ã÷Á˳£Êý¦Ä2k<2-1£¬´«¸Ð¾ØÕóA*=¦µDÂú×ã¿éRIPÌõ¼þʱ£¬´æÔÚΨһµÄ¿éÏ¡ÊèÏòÁ¿c¡£
¶¨Àí3.3µ±³£Êý¦Ä2k<2-1ʱ£¬A*Âú×ã¿éRIPÌõ¼þ£¬ÄÇô×îС»¯ÎÊÌâ


mincª³¡¬cª³¡¬2,I s.t.y=A*cª³(3.20)



ÖеÄÏòÁ¿cª³¿ÉÒÔ°´ÕÕ͹¶þ½×׶¹æ»®»ñµÃ£Û9£Ý£¬ÆäÖÐI±íʾ×é¿éµÄË÷Òý¼¯¡£
¶¨Àí3.3Òâζ×ÅÖ»Òª¦Ä2k×㹻С£¬ÎÊÌâ(3.20)µÄΨһ½â´æÔÚ¡£Óë±ê×¼µÄCSÏà±È£¬SOCPÇó½â¿ÉÃ÷È·µØºÍ¼òµ¥µØ±£Ö¤cµÄ¿éÏ¡ÊèÐÔ¡£ÔÚʵ¼ùÖУ¬ÎÒÃÇͨ¹ý×é»ù×·×Ù(Group BP)Ëã·¨Çó½âʽ(3.20)£¬»ñµÃ¿éÏ¡ÊèÏòÁ¿cª³£Û10£Ý¡£BSSLËã·¨Èç±í3.3Ëùʾ¡£


±í3.3»ùÓÚBSSL·½·¨µÄѹËõÊÓƵ²ÉÑùËã·¨



ÊäÈ룺 ѵÁ·¼¯X£¬²ÉÑù²âÁ¿Öµyª³
³õʼ»¯£º ²ÎÊý¦Ë£¬¦Î0
Öع¹£º 

²½Öè1,¶ÔѵÁ·¼¯X¾ÛÀà³Ét¸ö´ØG1,G2,¡­,Gt£» 

²½Öè2,»ñµÃ¸÷¸ö´ØG1,G2,¡­,Gt¶ÔÓ¦µÄ»ù¾ØÕó¦·*=£Û¦·*1,¦·*2,¡­,¦·*t£Ý£» 

²½Öè3,For´ØÊýj=1µ½t

 Do

 ¸ù¾Ýʽ£¨3.2£©µÃµ½Cj£» 

 ¸ù¾Ýʽ£¨3.3£©µÃµ½Dj£» 

 Ö±ÖÁÊÕÁ²

 End For

²½Öè4,¼¶Áª¸÷¸ö×Ó×ÖµäDj¹¹³É×ÖµäD=£ÛD1,D2£¬¡­,Dt£Ý£» 

²½Öè5,¸ù¾Ýʽ(3.20)ºÍ²ÉÑù²âÁ¿Öµy£¬Öع¹¿éÏ¡ÊèÏòÁ¿cª³£¬½ø¶øÖع¹ÐźÅx^=Dc*¡£ 
Êä³ö£º Öع¹µÄÊÓƵͼÏñ¿éx^¡£

ÏÂÃ潫BSSLËã·¨ÍØÕ¹µ½ÕÅÁ¿Êý¾Ý´¦ÀíÖС£ÀàËƵأ¬Í¨¹ýÔÚÕÅÁ¿ÑµÁ·¼¯XÉϽøÐÐ×Ó¿Õ¼ä¾ÛÀ࣬µÃµ½t¸ö´Ø£¬ÄÇôÊý¾ÝÇý¶¯µÄÕÅÁ¿×Ó¿Õ¼äÁªºÏUª³=¡ÈiSiµÄ»ù¿ÉÒÔ±íʾΪ¦·(n)=£Û¦·(n)1,¦·(n)2,¡­,¦·(n)t£Ý¡ÊRIn¡Árn£¬ÆäÖжÔÓÚµÚnģʽ£¬r=¡Æti=1Pi>In£¬²¢ÇÒÿ¸ö´ØGiÖаüº¬ÁËÏàËƵÄÕÅÁ¿Êý¾Ý£¬ÕâЩÕÅÁ¿Êý¾Ý´¦ÔÚͬһ¸öÕÅÁ¿×Ó¿Õ¼äSiÖУ¬¸Ã×Ó¿Õ¼äÓÉ»ù¦·(1)i«á¦·(2)i«á¡­«á¦·(N)iÕųɡ£ÔÚ´Ë»ù´¡ÉÏ£¬ÀûÓ÷º»¯µÄ¿éÏ¡Êè×Ó¿Õ¼äѧϰ½øÒ»²½ÑµÁ·£¬ÓÅ»¯½á¹¹»¯Ï¡ÊèµÄÕÅÁ¿±íʾ¡£ÔÚµÚnģʽÖУ¬¶Ôÿ¸ö´Ø½øÐпéÏà¹ØÐÔ×îС»¯Ô¼ÊøÀ´ÒÖÖƴؼäÏཻ¶ÔѵÁ·µÄ²»ÀûÓ°Ï죬½ø¶ø±£Ö¤ÔÚÕÅÁ¿±íʾϵĿéÏ¡ÊèÐÔ£¬ÆäÃèÊöÈçÏ£º 


D(n)= arg min¦·(n)i,¦¨(n),i¡Æti=1¡¬Gi-¦·(n)i¦¨(n),i¡¬22+¦Ë¡Æti=1¡Æpij=1¡¬¦Èji¡¬1+¦Î¡Æj¡Ùi¦¸(¦·(n)i,¦·(n)j)
(3.21)



ÆäÖЦ¨(n),i=£Û¦È1i,¦È2i,¡­,¦Èpii£Ý¡ÊRPi¡Ápi±íʾµÚi¸ö´ØÀ࣬µÚnģʽÁÐÏòÁ¿¦Èji¶ÔÓ¦ÓÚÕÅÁ¿Xji£¬i¡Ê£Û1,t£Ý£¬j¡Ê£Û1,pi£Ý¡£
ÀàËÆÓÚBSSL£¬ÓÃÓÚÔÚ¸÷¸öģʽÏÂƽºâ»Ö¸´Îó²îµÄ²ÎÁ¿¦ËÒÔ¼°¦Î·Ö±ð±íʾϡÊèÐԺͿéÏà¹ØÐԵijͷ£ÏµÊý£¬¿éÏà¹ØÐÔÅбðÏîÓëtP2i³É±ÈÀý£¬¦ÎÓɳõʼֵ¦Î0½øÐж¯Ì¬µÄµ÷½Ú¡£ÔÚʽ(3.21)ÖУ¬µÚÒ»Ïî¶ÔµÚi¸ö´ØÒÔ¼°ÏàÓ¦µÚnģʽϵĻùD(n)i½øÐÐÖع¹Îó²î×îС»¯¡£Í¬Ê±£¬µÚ¶þÏîºÍµÚÈýÏî·Ö±ðÓÃÓÚÓÅ»¯Ô¼Êø¿éÏ¡Êè±íʾ¦¨(n),iºÍÕÅÁ¿×Ó¿Õ¼äÖ®¼äµÄ¿éÏà¹ØÐÔ¡£ÐèÒªÖ¸³öµÄÊÇ£¬µ±ÕÅÁ¿×Ó¿Õ¼äÖ®¼äµÄ½»¼¯±»È¥³ýʱ£¬ÔÚ·ÇÏà¹Ø»ùϵıíʾ¦¨(n),iÖеķÇÁãϵÊý»á±»ºÏ²¢Îª¿é¡£Í¬ÑùµØ£¬ÀàËÆÓÚBSSL£¬Ê½(3.21)µÄ½â¿ÉÒÔÔÚ¸÷¸öģʽÏ£¬Í¨¹ýÀûÓý»ÌæµÄÏ¡Êè±àÂëºÍ»ù¸üйý³ÌÀ´½â¾ö¡£ÔÚÿ¸öµü´ú¹ý³ÌÖУ¬ÔÚ¸ø¶¨µÄ»ù¦·(n)iÏ£¬¦¨(n),i=£Û¦È1i,¦È2i,¡­,¦Èpii£Ý¿ÉÒÔͨ¹ýÔÚµÚi¸ö´ØÀàGi=£ÛX1i,X2i,¡­,Xpii£ÝÖнøÐÐÏ¡Êè±àÂëµÃµ½£¬ÆäÖÐGi°üº¬ÁËpi¸öÕÅÁ¿Êý¾Ý£¬ÆäÏ¡Êè±àÂëÇó½âÎÊÌâÈçÏ£º 


¦Èji= arg min{¦Èji}j=1,2,¡­,pi¡Æpij=1¡¬Xji-¦·(n)i¦Èji¡¬22+¦Ë¡¬¦Èji¡¬1(3.22)



½ô½Ó×Å£¬ÀûÓÃʽ(3.22)µÃµ½µÄ¦Èji£¬¿ÉÒÔ¶ÔµÚi¸ö´ØÀà½øÐлù¾ØÕóµÄ¸üУ¬¸üз½Ê½¿ÉÃèÊöΪÏÂÃæµÄÓÅ»¯ÎÊÌ⣺ 


D(n)i= arg min¦·(n)i¡Æpii=1¡¬Xji-¦·(n)i¦¨(n),i¡¬22+¦Ë¡¬¦Èji¡¬1+¦Î¡Æj¡Ùi¦¸(¦·(n)i,¦·(n)j)

(3.23)



ÖÁ´Ë£¬¾ÍʵÏÖÁ˶Ôÿ¸ö¾ÛÀà´Ø¶ÀÁ¢µØ½øÐÐÓÅ»¯µÃµ½µÄ»ùD(n)=£ÛD(n)1,D(n)2,¡­,D(n)t£Ý£¬²¢½«Ö®ÓÃÓÚ¶ÔÕÅÁ¿×Ó¿Õ¼äÁªºÏÄ£Ð͵ĻùµÄÓÅ»¯¡£
3.2.6¿éÏ¡Êè·Ç²ÎÊý±´Ò¶Ë¹¹À¼Æ
½üÄêÀ´£¬¶ÔÓÚ¾ßÓпé×´½á¹¹µÄÏ¡ÊèÐźŵĻָ´£¬Ç°ÃæÒѾ­½éÉÜÁËBOMP¡¢BMP¡¢BSSLµÈ¾­µäËã·¨¡£µ«ÊÇ£¬ÕâЩËã·¨¶¼ÐèÒª·Ö¿éµÄÏà¹ØÐÅÏ¢²ÅÄÜ׼ȷµØ»Ö¸´¿éÏ¡ÊèÐźš£ÁíÍâµÄ»Ö¸´Ëã·¨£¬ÀýÈçStructª²OMP£Û11£Ý£¬ËäÈ»²»ÐèÒª·Ö¿éÐÅÏ¢µ«ÊÇÐèÒªÒÑÖªÆäËûµÄһЩÏÈÑéÐÅÏ¢£¬ÈçÏ¡ÊèÐźÅÖзÇÁãÔªËصĸöÊý¡£ÔÚѹËõ¸ÐÖªµÄ»·¾³Ï£¬Èç¹û·Ö¿éµÄÏà¹ØÐÅϢδ֪£¬ÄÇôҪÏë´Ó²âÁ¿ÖµÖо«È·µØ»Ö¸´Ô­Ê¼Ðźŷdz£À§ÄÑ¡£ÔÚʵ¼ÊÓ¦ÓÃÖУ¬ÐźÅÄ£Ð͵ĺܶà²ÎÊýÄÑÒÔÔ¤Öª£¬ËùÒÔ¶ÔÓÚ¿é½á¹¹Ï¡Êè»Ö¸´ÎÊÌ⣬ÎÞ²ÎÊýµÄËã·¨·Ç³£ÓÐÓᣵ«Êǵ½Ä¿Ç°ÎªÖ¹£¬¿ÉÒÔ¾«È·µØ»Ö¸´¿éÏ¡ÊèÐźŵÄÕâÀàÎÞ²ÎÊýµÄËã·¨»¹ÊÇÁÈÁÈÎÞ¼¸¡£×î½ü£¬CluSSª²MCMC£Û12£ÝÒÔ¼°BMª²MAPª²OMP£Û13£ÝËã·¨±»Ìá³ö£¬ÕâЩËã·¨¿ÉÒÔÓ÷dz£ÉÙµÄÏÈÑéÐÅÏ¢À´¾«È·µØ»Ö¸´¿éÏ¡ÊèÐźš£
ÔÚ¾­µäµÄËæ»ú±´Ò¶Ë¹·½·¨ÖУ¬³£³£»áͨ¹ýͼģÐÍ(GM£Û14£Ý)ÒÔ¼°Òþ±äÁ¿À´ÃèÊö²âÁ¿ÖµÓëÄ£ÐͲÎÊýÖ®¼äµÄÁªºÏ¸ÅÂÊ·Ö²¼(»òÏà¹ØÐÔ)¡£ÕâÖÖ²»½ö²ÉÓÃÁËͼģÐͶøÇÒÒýÈëÁËÒþ±äÁ¿µÄ±´Ò¶Ë¹·½·¨³ÆΪÒþ±äÁ¿·ÖÎö(LVA£Û15£Ý)¡£²¢ÇÒ£¬ÎÞ²ÎÊýµÄ±´Ò¶Ë¹¹À¼ÆÆ÷Ò²¿ÉÓÉÒþ±äÁ¿·ÖÎöµÃµ½¡£ÁíÍ⣬ϡÊèϵÊýµÄ½á¹¹Ô¼Êø¿ÉÒÔ²ÉÓÃͼģÐ͵ķ½Ê½ÒýÈëÒþ±äÁ¿·ÖÎöµÄÀíÂÛ¿ò¼ÜÀï£Û16ª²18£Ý¡£ÀýÈ磬»ùÓÚͼģÐ͵ľۼ¯½á¹¹Ï¡Êè»Ö¸´Ëã·¨(LaMPËã·¨£Û18£Ý)£¬ÕâÖÖËã·¨ÊÇÔÚµü´úµÄ¹ý³ÌÖÐÓÉÒþ±äÁ¿½øÐÐÔ¼ÊøÆ¥Å䣬²¢ÇÒÓÉͼÇиî(Graph Cut)À´ÓÅ»¯ËüµÄÔ¼Êø×¼Ôò£¬ËùÒÔLaMPËã·¨²¢²»ËãÊÇÕæÕýÒâÒåÉϵı´Ò¶Ë¹·½·¨¡£¸üÖØÒªµÄÊÇ£¬¶ÔÓÚ²¡Ì¬ÏßÐÔÄæÎÊÌâÀ´Ëµ£¬¿ÉÓɱ´Ò¶Ë¹·½·¨µÃµ½ÎÒÃÇÏëÒªµÄ·Ç²ÎÊýËã·¨£Û18ª²19£Ý¡£
±¾½ÚΪÁËÐðÊöµÄÁ¬¹áÐÔºÍϵͳÐÔ£¬Ê×Ïȼòµ¥½éÉܱ´Ò¶Ë¹Ñ¹Ëõ¸ÐÖªÀíÂÛ£» È»ºóÓñ´Ò¶Ë¹¿ò¼Ü¸ø³ö¿éÏ¡ÊèÐźŵÄÏÈÑéÄ£ÐÍ£» ×îºó¶ÔÓÚ»ùÓÚMCMC²ÉÑùµÄ¿éÏ¡ÊèÐźű´Ò¶Ë¹»Ö¸´Ëã·¨½øÐÐÏêϸ½éÉÜ¡£
1. ±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖª
ÒÑÖªÐźÅxÊÇNά¿ÉѹËõµÄ£¬²¢ÇÒËüµÄÏ¡Êè±í´ï»ùΪ¦·¡ÊRM¡ÁN,Ò²¾ÍÊÇx=¦·s¡£¼ÙÉèss´ú±íÒ»¸ö¾ßÓÐM¸ö·ÇÁãÖµµÄNάÏòÁ¿£¬Æä·ÇÁãÖµÓësÖеÄM¸ö·ùÖµ×î´óµÄÔªËØÏàͬ£» ¼ÙÉèse´ú±íÒ»¸ö¾ßÓÐN-M¸ö·ÇÁãÖµµÄNάÏòÁ¿£¬ÆäÁãÖµÓësÖеÄN-M¸ö·ùÖµ×îСµÄÔªËØÏàͬ¡£ËùÒÔ£¬¿ÉµÃµ½s=ss+se£¬²¢ÇÒ£º 


y=¦¨s=¦¨ss+¦¨se=¦¨ss+ne


ÆäÖЦ¨=A¦·Îª·ûºÏRIPÌõ¼þµÄ¸ÐÖª¾ØÕó£¬ne¿ÉÒÔ¿´×÷Ò»¸öÁã¾ùÖµ¸ß˹ÔëÉùÏòÁ¿¡£
ÔÚʵ¼ÊÖУ¬²âÁ¿ÖµÒ²ÓпÉÄܱ»ÔëÉùÎÛȾ£¬ËùÒÔÎÒÃÇÒ²ÓÃÒ»¸öÁã¾ùÖµ¸ß˹·Ö²¼nmÀ´±íʾ²âÁ¿ÔëÉù¡£ÔòÉÏʽ¿ÉÒÔ¸ÄдΪ


y=¦¨ss+ne+nm=¦¨ss+n(3.24)



ÆäÖеÄÔëÉùn±»µ±×÷ÊÇ·½²î¦Ò2δ֪Áã¾ùÖµ¸ß˹ÔëÉù¡£
ÖµµÃ×¢ÒâµÄÊÇ£¬ÒòΪѹËõ¸ÐÖª¶ÔÐźŵIJÉÑùºÍѹËõ¶¼ÔÚÒ»¸ö²½Ö裬ËùÒÔÕâÀïµÄMªîN£¬Òò´Ë»Ö¸´Ï¡ÊèÐźÅssµÄÎÊÌâʵ¼ÊÉÏÊÇÇó½âÒ»¸öÇ·¶¨ÏßÐÔ·½³ÌµÄÎÊÌ⡣ΪÁË·½±ã£¬´Ó´ËÎÒÃÇÓÃs´úÌæss£¬µ«ÎÒÃǸÐÐËȤµÄ»¹ÊÇϵÊýµÄÏ¡Êè½â¡£ÁíÍ⣬¼ÇÁã¾ùÖµ¸ß˹ÔëÉùnµÄ·½²î¦Ò2µÈÓÚ¦Á-10£¬Ôò¹ØÓÚ²âÁ¿µÄ¸ß˹ËÆȻģÐÍ¿ÉÒÔ±»ÃèÊöΪÈçÏÂÐÎʽ£º 


p(ys)¡Øexp-¦Á02¡¬y-¦¨s¡¬2(3.25)



È»ºó¿ÉÒÔÓÉ×î´óËÆÈ»¹À¼ÆÆ÷»ñµÃÒ»¸ö×îС¶þ³Ë»Ø¹éËã·¨£º 


s^= arg mins¡¬y-¦¨s¡¬2 (3.26)



ÏÔÈ»£¬ÉÏʽµÄ×îС¶þ³Ë»Ø¹éËã·¨»¹µÃÐèÒªÌí¼ÓÒ»¶¨µÄÔ¼ÊøÌõ¼þ£¬²ÅÓпÉÄÜÇó³öʽ(3.24)µÄ½â¡£
Òò´Ë£¬ÔÚʽ(3.26)ÖÐÒýÈëÏ¡ÊèÏÈÑé£Û20ª²22£Ý£¬ÀýÈ磺 


p(s)¡Øexp-¡ÆNi=1sip(3.27)



ÆäÖвÎÊýpÈ¡Öµ·¶Î§Îª£Û0,1£Ý¡£¼ÙÉè¸ø¶¨µÄ²âÁ¿yÂú×ãʽ(3.25)ËùÃèÊöµÄ¸ß˹ËÆÈ»¸ÅÂÊÄ£ÐÍ£¬²¢ÇÒÒýÈëʽ(3.27)µÄÏ¡ÊèÏÈÑ飬ÄÇô¶Ôʽ(3.25)µÄ×î´óºóÑé¸ÅÂʽøÐÐÇó½â£¬¿ÉµÃ


s^= arg min ¦Ës¡¬y-¦¨s¡¬2+¡ÆNi=1sip(3.28)



ÆäÖвÎÊý¦Ë=¦Á0Ϊһ¸öƽºâ²ÎÊý£¬ËüÔÚÏ¡ÊèÐԺ͹۲âÆ¥Åä¶ÈÖ®¼äÆðµ÷ºÍ×÷Óá£
µ±²ÎÊýpµÄÈ¡ÖµÇ÷ÓÚÁãʱ£¬ÃèÊöÏ¡ÊèÏÈÑéµÄʽ(3.27)½«»áÇ÷½üÓÚL0·¶Êý£¬Ò²¾ÍÊǶÔÏòÁ¿sÖеķÇÁãÔªËصÄÊýÄ¿½øÐж¨Ò壺 


¡¬s¡¬0=¡ÆNi=1Isi¡Ù0


ÆäÖеķûºÅI¡¤´ú±íµÄÊÇָʾº¯Êý¡£
ʵ¼ÊÉÏ£¬ÈôÔÚÕâÖÖp¡ú0Ìõ¼þ϶Ôʽ(3.27)½øÐÐÇó½â½«ÊǷdz£À§Äѵģ¬ÒòΪÕâÊôÓÚÒ»¸öNPª²ÄÑÎÊÌâ¡£
Èôµ±²ÎÊýp=1ʱ£¬Ê½(3.28)ËùÃèÊöµÄÓÅ»¯ÎÊÌâ¾Í»áת»¯ÎªÒ»¸ö͹ÓÅ»¯ÎÊÌâ¡£ Ä¿Ç°£¬¶ÔÓÚÇó½âÕâÖÖ͹ÓÅ»¯ÎÊÌâÀ´Ëµ£¬ÒѾ­ÓкܶàÏàÓ¦µÄ¾­µäµÄËã·¨£¬ÀýÈç»ù×·×ÙËã·¨£Û20£Ý¡£ÁíÍ⣬ÔÚÎÄÏ×£Û23£ÝÖÐÒѾ­Ö¤Ã÷£¬ÈôԭʼÐźÅ×㹻ϡÊ裬Ôò»ù×·×ÙËã·¨ÊÇ¿ÉÒÔ»ñµÃԭʼÐźŵÄÕæʵ½âµÄ¡£¶øµ±²ÎÊýpÈ¡Öµ·¶Î§Îª(0,1)ʱ£¬Ôòʽ(3.28)ËùÃèÊöµÄÓÅ»¯ÎÊÌâ¾Í»áת»¯ÎªÒ»¸ö·Ç͹ÓÅ»¯ÎÊÌ⣬¶øÔÚijЩÌض¨µÄÌõ¼þÏ£¬´ËÀàÎÊÌâµÄ½âÓëÕæʵϡÊèÐźű¾Éí¸ü¼Ó½Ó½ü£Û24£Ý¡£
ËäÈ»ÔÚѹËõ¸ÐÖªÀíÂÛ¿ò¼ÜÏ£¬¿ÉÒÔͨ¹ý×î´óºóÑé¹À¼ÆÀ´Çó½âԭʼϡÊèÐźŵĽüËÆÖµ¡£µ«ÊÇ£¬ÓÃÕâÖÖ·½Ê½Çó½âµÄͬʱ»¹ÐèÒª¹À¼Æʽ(3.28)ÖеÄƽºâ²ÎÊý¦Ë»òÕßÌáÇ°¶ÔËü½øÐÐÔ¤Éè¡£ÁíÍ⣬ҪÊÇÔÚʽ(3.27)ÖÐÒýÈ볬¸ß˹ÏÈÑ飬һ°ã»áµ¼ÖÂƽºâ²ÎÊý¦ËµÄºóÑé·Ö²¼²»¿É·ÖÎö£¬Ôò»áÖÂʹԭÀ´ÎÊÌâµÄÇó½â¸ü¼Ó¸´ÔÓ»¯¡£Òò´Ë£¬ÍƼöʹÓ÷DzÎÊýµÄ·½·¨À´½â¾öÏ¡ÊèÏßÐÔÄæÎÊÌâ¡£
ΪÁËÔÚѹËõ¸ÐÖªÀíÂÛ¿ò¼ÜÏÂ


ͼ3.5±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖª·Ö²ãÄ£ÐÍ

ʹÓñ´Ò¶Ë¹·½·¨À´ÃèÊöѹËõ¸ÐÖªµÄ²âÁ¿¹ý³Ì,¿ÉÒÔ²ÉÓ÷ֲãµÄ±´Ò¶Ë¹Ä£ÐÍ£Û25ª²27£Ý£¬ËüÒ²±»³ÆΪ±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖª¡£
¸üÖØÒªµÄÊÇ£¬·Ö²ã±´Ò¶Ë¹Ä£Ð͵ÄʹÓÿÉÒÔʹϡÊèÏßÐÔÄæÇó½âµÄ¹ý³Ì·Ç²ÎÊý»¯£¬Ê¹µÃÕâÖÖÎÊÌâµÄ½â¾ö¸ü¼òµ¥»¯£¬Èçͼ3.5Ëùʾ¡£Èç¹ûÔÚʵ¼ÊµÄ±´Ò¶Ë¹Ñ¹Ëõ¸Ð֪ģÐÍÖÐÒýÈ볬¸ß˹ÏÈÑ飬½«»áµ¼Ö¸ü¸´ÔÓµÄÎÊÌâÇó½â¡£ÎªÁËʹÎÊÌâ¼òµ¥»¯£¬¼ÙÉèÏ¡ÊèÐźŷþ´Ó¸ß˹ÏÈÑ飬²¢¼ÇÕâ¸ö¸ß˹ÏÈÑéµÄ¾«¶È²ÎÊýΪ¦Á=£Û¦Á1,¦Á2,¡­,¦Ái,¡­,¦ÁN£Ý£¬×îºó¼ÙÉè¦ÁÂú×㳬ÏÈÑéÙ¤Âí·Ö²¼£º 


p¦Áia,b¡Ø¦Áa-1ie-b¦Ái(3.29)



ÆäÖÐa>0£¬b>0£¬ÇÒËüÃÇÒ»°ã»á±»ÉèÖÃΪ·Ç³£Ð¡µÄÊýÖµ¡£
Ôò¿ÉµÃµ½ÒÔϵÄÙ¤Âíª²¸ß˹ģÐÍ£º 


ps¦Á¡Ø¡ÇNi=1exp-¦Ái2s2i(3.30)




¶øʽ(3.29)ÖеIJÎÊýa,b>0£¬²¢ÇÒËüÃÇÒ»°ã»á±»ÉèÖÃΪ·Ç³£Ð¡µÄÊýÖµ¡£
ÏÂÃæ¶Ôʽ(3.30)¶¨ÒåµÄÙ¤Âíª²¸ß˹ģÐͽøÐзÖÎö¡£ÓÃʽ(3.29)¶Ôʽ(3.30)Öеij¬²ÎÊý¦Á½øÐбßÔµ»¯£¬Ôò¿ÉµÃµ½Ï¡ÊèÐźÅsµÄÒþº¬ÏÈÑ飺 


psa,b=¡Òps¦Áp¦Áa,bd¦Á

¡Ø¡ÇNi=1b+s2i2-(a+0.5)(3.31)



ʵ¼ÊÉÏ,psa,bÓëStudentª²T·Ö²¼³ÉÕý±È¡£Èô´ËÏÈÑé²ÎÊýaºÍbÇ÷½üÓÚÁ㣬ÄÇôʽ(3.31)¿ÉÒÔ¼ò»¯Îª 


ps¡Ø¡ÇNi=11si(3.32)



Ò²¾ÍÊÇ˵£¬Ê½(3.32)ʵ¼ÊÉÏÃèÊöÁËÒ»¸öÏ¡ÊèÏÈÑé¸ÅÂÊ£Û27ª²28£Ý¡£
ÓÉÓÚÔÚѹËõ¸ÐÖªÖеIJâÁ¿Öµ»á±»ÔëÉùÎÛȾ£¬Òò´ËÐèÒªÔڴ˲âÁ¿¹ý³ÌÖÐÉ趨һ¸ö²âÁ¿ÔëÉù²ÎÊý¦Á0£¬Ëüʵ¼ÊÉÏΪÔëÉù·½²î¦Ò0µÄµ¹ÊýͬʱҲ·þ´ÓÙ¤Âí·Ö²¼£º 

p¦Á0c,d¡Ø¦Ác-10e-d¦Á0

ÓÖÒòΪ¸ß˹ËÆÈ»·Ö²¼ÓëÉÏʽ¶¨ÒåµÄÙ¤Âí·Ö²¼¹ØÓÚ²âÁ¿ÔëÉù²ÎÊý¦Á0¹²éÔò¹ØÓÚ¦Á0µÄºóÑé¸ÅÂÊ·Ö²¼¿ÉÒԺܷ½±ãµØ»ñµÃ¡£È»ºó,¿ÉÒÔʹÓñä·Ö±´Ò¶Ë¹(VB)£Û28£ÝÍÆÀí»òÕßÏà¹ØÏòÁ¿»ú£Û27£Ý¹À¼Æ³ö¦Á0µÄ½üËÆÖµ¡£Ôڴ˹ý³ÌÖУ¬Æ½ºâ²ÎÊý¦Ë²»ÐèÒªÌáÇ°ÉèÖöøÆä¿ÉÒÔ×Ô¶¯µØ¸üС£Òò´Ë£¬Ó봫ͳµÄÀûÓÃ×î´óºóÑé¹À¼Æ(MAP)À´½øÐÐÏ¡ÊèÇó½âµÄËã·¨Ïà±È,ÕâÖֱȽÏÖÇÄܵÄËã·¨ÔÚʵ¼ÊµÄÓ¦ÓÃÖоßÓиü´óµÄÓÅÊÆ¡£
ÁíÍ⣬ҲÄÜÒÔÏÂÃæÕâÖÖ·½Ê½À´Çó½â¸Õ²ÅÌáµ½µÄ±´Ò¶Ë¹Ñ¹Ëõ¸Ð֪ģÐ͵Ĵú¼Ûº¯Êý£º 


py¦Á0,¦Á=¡Òpys,¦Á0ps¦Áds

¡ØN(0,¦²0)(3.33)



Ò²¾ÍÊÇ˵£¬´Ë´ú¼Ûº¯ÊýÊÇÓɱßÔµËÆÈ»·Ö²¼µÄ»ý·ÖÇóµÃ¡£ÆäÖУ¬Á0=¦Á-10I+ABAT²¢ÇÒB¦¤diag¦Á£¬Ôò´ú¼Ûº¯Êý¿ÉÒÔдΪ 


L=In¦²0+yT¦²-10y(3.34)



ÉÏʽÓÉÁ½Ïî×é³É£¬ºÜÃ÷ÏÔµÚÒ»ÏîµÄ×÷ÓÃÊǽøÐÐÏ¡ÊèÔ¼Êø£¬¶øµÚ¶þÏîÔòÊÇÆðµ½¹Û²âÔ¼ÊøµÄ×÷Óá£
ËäÈ»ÔÚ¸Õ²ÅÌÖÂ۵ı´Ò¶Ë¹Ñ¹Ëõ¸ÐÖªÖУ¬½ö½öÖ»¶ÔÐźŵÄÏ¡ÊèÐÔ½øÐÐÁËÔ¼Êø,¶ø²¢Ã»ÓжÔÐźŵÄÆäËû¸½¼Ó½á¹¹ÐÅÏ¢¼ÓÒÔ¿¼ÂÇ£¬µ«ÊÇÖµµÃ×¢ÒâµÄÊÇ£¬ÓÉÓڷֲ㱴Ҷ˹ģÐ͵ÄÒýÈ룬ѹËõ¸ÐÖª»Ö¸´Ëã·¨µÄ·Ç²ÎÊý»¯ÒѳÉΪ¿ÉÄÜ¡£Òò´Ë£¬ÎÒÃǾͿÉÒÔÔÚÕâ¸öÄ£Ð͵Ļù´¡ÉÏͨ¹ý¼ÓÈëÐźžßÓеÄÆäËû¸½¼Ó½á¹¹ÐÅÏ¢À´¸ü½øÒ»²½Ìá¸ß·Ç²ÎÊý»Ö¸´Ëã·¨ÐÔÄÜ¡£
2. ¿éÏ¡ÊèÐźŵÄÏÈÑé¸ÅÂÊÄ£ÐÍ
1£© ¿éÏ¡Êè
Ï¡Êè±í´ïÔÚ½üЩÄêÀ´µÃµ½Á˳ä·ÖµÄ¹Ø×¢¡£ÔÚѹËõ¸ÐÖª»·¾³Ï£¬ÐźŵÄÏ¡ÊèÏÈÑéÊÇÐźŴÓÉÙÁ¿²âÁ¿ÖµÖаÑԭʼÐźŽøÐÐΨһ»Ö¸´µÄ±£Ö¤Ìõ¼þ¡£Ôںܶàʵ¼ÊµÄÐźÅÖУ¬Ï¡ÊèÐÔÖ»ÊÇÆäÖÐÒ»¸öÌØÐÔ£¬ÆäÍùÍù»¹ÓµÓÐÆäËû½á¹¹ÌØÐÔ¡£ÆäÖÐÒ»ÖÖ¼òµ¥²¢ÇÒºÜÖØÒªµÄÐźű»³ÆΪ¿éÏ¡ÊèÐźÅ,ÆäÌصãÊÇËüµÄ·ÇÁãÔªËØÊÇÒÔ¿éÐÎʽ³öÏֵģ¬¶ø²»ÊÇÒÔËæ»úÀ©É¢µÄÐÎʽ³öÏֵġ£ÕâÀàÐźÅÒ²³£¼ûÓÚ¶àƵ´øÐźÅÒÔ¼°¶àÖزâÁ¿ÏòÁ¿(MMV)ÎÊÌâ¡£
ÕâÖֹ㷺±»Ñо¿µÄ¿éÏ¡ÊèÐźſÉÒÔ±íʾÈçÏ£º 


x= £Ûx1,x2,¡­,xd1xT1,¡­,xdg-1+1,¡­,xdgxTg£ÝT


ÆäÖÐd1,d2,¡­,dg²»Ò»¶¨Ïàͬ¡£
ÔÚÈ«²¿µÄÕâЩ¿éÖУ¬Ö»ÓкÜÉٵĿéΪ·ÇÁã¿é£¬µ«ÊÇËüÃǵľßÌåλÖÃδ֪¡£Í¼3.6¸ø³öÁ˼¸¸öµäÐ͵ĿéÏ¡ÊèÐźÅÀý×Ó£¬ÆäÖÐͼ3.6(a)±íʾµÄÊÇÔÚ¸ÐÖªÎÞÏߵ糡¾°Öо­³£¼ûµ½µÄ¶àƵ´øÐźŵķù¶ÈÆ×£¬¿ÉÒÔ¿´³öÔÚÆäƵ´øÖУ¬Ö»ÓÐÎå¸öÕ­´øÐźŵķù¶ÈÆײ»ÎªÁ㣬¶øÆäËûƵ´øµÄ·ù¶ÈÆ׶¼ÎªÁ㣬ÕâÓë¿éÏ¡ÊèÐźŵĶ¨ÒåºÜÏà·û£» ͼ3.6(b)±íʾµÄÊÇһЩ»Ò¶ÈͼÏñÖг£³öÏÖµÄ×Öĸͼ£¬ºÜÃ÷ÏÔÔÚ»Ò¶ÈͼÏñµÄÖмäÖ»Óдóд×ÖĸH´ú±íµÄ»Ò¶ÈֵΪ·ÇÁã,¶øÆäËûÏñËصãµÄ»Ò¶ÈÖµ¶¼ÎªÁ㣬ÕâÒ²Óë¿éÏ¡ÊèÐźŵĶ¨ÒåÏà·ûºÏ¡£




ͼ3.6¿éÏ¡ÊèÐźÅ


2£© ÏÈÑé¸ÅÂÊÄ£ÐÍ
¼ÇÎÒÃǸÐÐËȤµÄÄ£ÄâÐźÅΪx(t)£¬²¢ÇÒ0¡Üt¡ÜT¡£¼ÙÉèËüÊÇÓÉÓÐÏÞ¶à¸ö»ùº¯Êý(ÀýÈ磬¸µÀïÒ¶»ù)¦×i(t)µÄȨÖغÍ×é³É£¬Ôò¿É±íʾÈçÏ£º 


x(t)=¡ÆNi=1si¦×i(t)(3.35)



¼ÙÉè´ËÐźÅx(t)¾ßÓÐÏ¡ÊèÌØÐÔ£¬Ôò´ó²¿·ÖµÄ»ùº¯ÊýµÄϵÊýsiΪÁã¡£ÔÚÀëɢʱ¼äѹËõ¸ÐÖªÀíÂÛ¿ò¼ÜÏ£¬Ê½(3.35)¿ÉÒÔдΪx=¦×s£¬ÆäÖдóСΪN¡Á1µÄÏòÁ¿x±»µ±×÷ÊÇÄ£ÄâÐźÅx(t)µÄÄοü˹ÌØËÙÂÊÐźţ¬¶ø¦×=¦×1,¦×2,¡­,¦×NTΪһ¸öN¡ÁN µÄÏ¡Êè±í´ï¾ØÕó£¬s=s1,s2,¡­,sNT±íʾµÄÊÇÒ»¸ö´óСΪN¡Á1µÄÏ¡Êè±í´ïϵÊýÏòÁ¿¡£Ò²¾ÍÊÇ˵£¬ÔÚËùÓеÄÏ¡ÊèϵÊýÏòÁ¿ÖУ¬ÆäÖÐÖ»ÓÐk¸öÔªËØΪ·ÇÁãÖµ£¬²¢ÇÒ kªîN¡£ÔÚÎÄÏ×£Û29£ÝºÍ£Û30£ÝÖÐÒÑÖ¤Ã÷£¬Èç¹ûÏ¡Êè±í´ï¾ØÕó¦×Óë³ß´çΪM¡ÁNµÄѹËõ¸ÐÖª²ÉÑù¾ØÕóA³ä·Ö²»Ïà¹Ø£¬Ôò¿ÉÒÔ´ÓM=kOlogN/k¸ö·ÇÏßÐÔ²âÁ¿Öо«È·µØ»Ö¸´Ï¡ÊèÐźÅx(t)¡£±»¹ã·ºÊ¹ÓõĵäÐ͵IJÉÑù¾ØÕóAµÄÿ¸öÔªËضÀÁ¢µØÈ¡×ÔËæ»ú¸ß˹·Ö²¼¡£¸ù¾ÝѹËõ¸ÐÖªÀíÂÛ,¿ÉÒÔд³öÈçϹØÓÚ²âÁ¿ÏòÁ¿µÄ±í´ïʽ£º 


y=Ax=A¦·s=¦¨s(3.36)



ÆäÖУ¬¦·¿ÉÒÔΪһ¸öN¡ÁNµÄÀëÉ¢¸µÀïÒ¶±ä»»(DFT)¾ØÕó£¬Æä¿ÉÒ԰ѸÐÐËȤÐźŴÓʱÓòÓ³É䵽ƵÓòÖÐÀ´£¬¶ø¾ØÕó¦¨=A¦·ÎªÒ»¸ö´óСΪM¡ÁN¸ÐÖª¾ØÕó¡£ÎªÁË´Ó²âÁ¿ÖµyÖеõ½ÐźŵĻָ´Á¿s^£¬Ôòʽ(3.36)µÄÄæ¹ý³Ì¿É±»ÊÓΪÇó½âÒÔÏÂL1·¶Êý×îÓÅ»¯ÎÊÌ⣺ 


s^=arg min¡¬s¡¬1s.t.y=¦¨s(3.37)



Òò´Ë£¬ºÜ¶àÏֳɵÄËã·¨¶¼¿ÉÒÔÓÃÀ´Çó½â´Ëʽ£¬ÀýÈ磬ÊôÓÚÏßÐԹ滮·½·¨µÄ»ù×·×Ù(BP)Ëã·¨£Û20£Ý£¬ÓÖÀýÈçÊôÓÚµü´úÌ°À·Ëã·¨µÄÕý½»Æ¥Åä×·×Ù(OMP)Ëã·¨£Û31£Ý¡¢ ÕýÔò»¯µÄÕý½»Æ¥Åä×·×Ù(ROMP£Û32£Ý)Ëã·¨ÒÔ¼°Ñ¹Ëõ²ÉÑùÆ¥Åä×·×Ù(CoSaMP£Û33£Ý)Ëã·¨¡£µ«ÊÇ£¬ÕâЩ´«Í³µÄ¾­µäËã·¨¶¼Ö»ÊÇ»ùÓÚÐźŵÄÏ¡ÊèÐÔ±»Ìá³öµÄ¡£ÏÂÃ潫עÒâÁ¦¼¯ÖÐÔÚ¿éÏ¡ÊèÐźŵĻָ´ÉÏ£¬ÎÒÃÇÌá³öÒ»ÖÖ»ùÓÚË«²ã¶þ²æÊ÷Ä£µÄ¿éÏ¡ÊèÐźŻָ´Ëã·¨£¬´ËËã·¨ÊÇ»ùÓÚ±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖªÀíÂÛ¿ò¼ÜÌá³öµÄ¡£Ê×ÏÈÀ´¿´Ò»Ï¶ԿéÏ¡ÊèÐźŵÄÏ¡ÊèÏÈÑéµÄÃèÊö¡£
¶ÔÓÚ¸ÐÐËȤ¿éÏ¡ÊèÐźŵÄÏ¡ÊèÐÔ£¬¿ÉÒÔÀûÓÃÒ»¸öspikeª²andª²slabÏÈÑéÄ£ÐÍ£¨Ò²±»³ÆΪխ¸ß˹·Ö²¼Óë¿í¸ß˹·Ö²¼µÄ»ìºÏÄ£ÐÍ£©¶ÔÆäÏ¡ÊèÐÔ½øÐн¨Ä£¡£¼ÙÉè¿éÏ¡ÊèÐźÅÖеÄÿһ¸öÔªËØsi¶¼·þ´Ó´Ëspikeª²andª²slabÏÈÑéÄ£ÐÍ£¬Ôò¿ÉÒÔ½«´ËÄ£ÐͱíʾÈçÏ£º 


si¡«(1-¦Ði)¦Ä0+¦ÐiN(0,¦Á-1i)(3.38)



ºÜÃ÷ÏÔ£¬ÎÒÃǹ¹½¨µÄ´ËÏÈÑéÄ£ÐÍÓÉÁ½²¿·Ö×é³É¡£ÔÚµÚÒ»²¿·ÖÀ·ûºÅ¦Ä0±íʾµÄÊÇ´¦ÓÚÁãλÖõĵã·Ö²¼£¬ËüµÄ×÷ÓÃÊǶÔsÖеÄÁãϵÊý½øÐÐÃèÊö¡£ÔÚµÚ¶þ²¿·ÖÀ·ûºÅN(0,¦Á-1i)±íʾµÄÊÇÒ»¸öÁã¾ùÖµ¸ß˹·Ö²¼(¦ÁiΪ¸ß˹·Ö²¼µÄ¾«¶È²ÎÊý)£¬ËüµÄ×÷ÓÃÊǶÔsÖеķÇÁãϵÊý½øÐÐÃèÊö¡£¶øÔÚʽ(3.38)µÄÁ½¸ö²¿·ÖÖж¼´æÔڵĻìºÏȨÖزÎÊý¦ÐiÊÇÒ»¸ö±êÁ¿£¬ÆäÈ¡Öµ·¶Î§Îª£Û0,1£Ý¡£×Ðϸ¹Û²ìʽ(3.38)µÄÁ½¸ö²¿·Ö¿ÉÒÔ·¢ÏÖ£¬Êµ¼ÊÉÏ»ìºÏȨÖزÎÊý¦Ði±íʾµÄÊÇÐźÅÖеĵÚi¸öÔªËØsi¿ÉÄÜΪ·ÇÁã״̬µÄ¸ÅÂÊ¡£Ò²¾ÍÊÇ˵£¬µ±»ìºÏȨÖزÎÊý¦ÐiÈ¡½Ï´óµÄֵʱ(ÀýÈ磬½Ó½üÓÚ1)£¬¾Í»áÒԽϴóµÄ¸ÅÂʵõ½Ò»¸öȡֵΪ·ÇÁãµÄϵÊýsi£» Ïà·´£¬µ±»ìºÏȨÖزÎÊý¦ÐiÈ¡½ÏСµÄֵʱ(ÀýÈ磬½Ó½üÓÚ0)£¬¾Í»áÒԽϴóµÄ¸ÅÂʵõ½Ò»¸öȡֵΪÁãµÄϵÊýsi¡£ÁíÍ⣬ÎÒÃÇÀûÓ÷ֲãµÄÏÈÑéÀ´µÃµ½ÏÔʾºóÑé¸ÅÂÊÃܶȣ¬¼´É趨¾«²ÎÊý¦ÁiÂú×ãÏÂÃæµÄÌõ¼þ£º 


¦Ái¡«Gammaa,b


ʵ¼ÊÉÏ£¬ÕâÖÖÏ¡ÊèÏÈÑé¾ÍÊÇ·Ö²ãµÄ¸ß˹ª²Ù¤ÂíÏÈÑéÄ£ÐÍ¡£
3£© Ë«²ã¶þ²æÊ÷½á¹¹
ÈçÇ°ËùÊö£¬ÀûÓø½¼ÓµÄ½á¹¹ÐÅÏ¢¿ÉÒÔÌá¸ßÏ¡ÊèÐźŻָ´Ëã·¨µÄÐÔÄÜ¡£Ò²¾ÍÊÇ˵£¬ÎÒÃÇ¿ÉÒÔÀûÓÃÏ¡ÊèϵÊýsiÖ®¼äµÄ¹ØϵÀ´Ìá¸ß»Ö¸´¾«¶È²¢½ÚÔ¼»Ö¸´Ê±¼ä¡£¶ÔÓÚ¿éÏ¡ÊèÐźÅÀ´Ëµ£¬Æä·ÇÁãϵÊýÍùÍù»á¾Û¼¯ÔÚÒ»ÆðÐγÉÒ»¸ö»òÈô¸É¸ö·ÇÁã¿é£» ¶øÆäÁãϵÊýÒ²ÍùÍù¾Û¼¯ÔÚÒ»ÆðÐγÉÒ»¸ö»òÈô¸É¸öÁã¿é¡£ÀàËƵأ¬Ä³¸ö·ÇÁãϵÊýµÄ(Ò»½×)ÁÚÓòϵÊý³£³£Ò²Îª·ÇÁ㣻 ¶øij¸öÁãϵÊýµÄ(Ò»½×)ÁÚÓòϵÊý³£³£Ò²ÎªÁã¡£»ùÓÚÒÔÉÏÊÂʵ£¬ÎÒÃÇÌá³öÁËÒ»ÖÖË«²ã¶þ²æÊ÷½á¹¹À´ÃèÊöϵÊýÓëÆäÁìÓòÖ®¼äµÄ¡°ÒÅ´«¡±ÌØÐÔ¡£ÔÚÎÒÃÇÌá³öµÄË«²ã¶þ²æÊ÷½á¹¹Ä£ÐÍÖУ¬¿éÏ¡ÊèÐźŵÄÿÈý¸öÁ¬Ðø²»ÖصþµÄϵÊý±»µ±×÷Ò»¸öË«²ã¶þ²æÊ÷½á¹¹(DBT)ssubi£¬²¢ÇÒÿһ¸össubiÓÉÒ»¸ö¸ù½ÚµãRiºÍÁ½¸öÒ¶×Ó½ÚµãLi1ºÍLi2×é³É£¬Èçͼ3.7ÖеÄÐéÏß¿òËùʾ¡£Òò´Ë£¬¿éÏ¡ÊèÐźŵÄÈ«²¿ÏµÊý½«»áÐγÉÒ»Á¬´®µÄÈçͼ3.7µÄË«²ã¶þ²æÊ÷£¬²¢ÇÒÆä¿ÉÒÔ±»ÃèÊöΪs=L11,R1,L12ssub1,¡­,Li1,Ri,Li2ssubi,¡­£¬Lk1,Rk,Lk2ssubkT¡£


ͼ3.7Ë«²ã¶þ²æÊ÷Ä£ÐÍ 


ÓÚÊÇ£¬¾Í¿ÉÒÔÔÚ±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖªµÄ¿ò¼ÜϼÓÈëËùÌá³öDBTÄ£ÐÍÀ´Ìá¸ß»Ö¸´Ëã·¨µÄÐÔÄÜ¡£

¼ÙÉ覸=1,2,¡­,NΪ¿éÏ¡ÊèÐźÅϵÊýsµÄÿһ¸öÔªËصÄλÖü¯ºÏ¡£²¢ÇÒ¶¨Òå ¦¸R=3¡Áj-1,j¡Ê1,kºÍ¦¸L=3¡Áj-2,3¡Áj,j¡Ê1,k·Ö±ðΪDBTµÄ¸ù½ÚµãÒÔ¼°Ò¶×ӽڵ㡣Òò´Ë£¬ºÜÃ÷ÏÔ¦¸R¡È¦¸L=¦¸²¢ÇÒ¦¸R¡É¦¸L=«Á¡£Ò²¾ÍÊÇ˵£¬ÔÚÎÒÃÇÌá³öµÄÕâÖÖDBT½á¹¹ÖÐsµÄ³¤¶ÈNÐèÒªÂú×ãÌõ¼þmodN,3=0¡£µ«ÊÇÔÚʵ¼ÊÓ¦ÓÃÖУ¬µ±modN,3=1ʱ£¬ÎÒÃÇ¿ÉÒÔ¶¨ÒåÓësÏà¶ÔÓ¦µÄ×îºóÒ»¸öDBTÖ»¾ßÓиù½Úµã£» ¶øµ±modN,3=2ʱ£¬ÎÒÃÇ¿ÉÒÔ¶¨ÒåÓësÏà¶ÔÓ¦µÄ×îºóÒ»¸öDBT²»½ö¾ßÓиù½Úµã£¬¶øÇÒ¾ßÓÐÒ»¸öÒ¶×ӽڵ㡣ʵ¼ÊÉÏ£¬ÕâÖÖ²¹³äÐԵĶ¨Òå¶ÔÎÒÃÇÌá³öµÄDBT½á¹¹µÄÕûÌåÊÊÓÃÐÔûÓÐÈκÎÓ°Ïì¡£ÔÚͼ3.7ÖУ¬Ri´ú±í×ܵÄk¸öDBTµÄµÚi¸öDBTµÄ¸ù½Úµã£» ¶øLi1ºÍLi2·Ö±ð±íʾµÚi¸öDBTµÄ¸ù½ÚµãµÄ×óÒ¶×Ó½ÚµãºÍÓÒÒ¶×ӽڵ㡣¸ù¾ÝÕâÖֽṹģÐÍ£¬sµÄÿÈý¸öÔªËØ(²»Öصþ)±»ÊÓ×÷siµÄÒ»¸ö×Ó¼¯ssubi£¬²¢ÇÒÕâ¸ö×Ó¼¯µÄÖмäλÖñ»¶¨ÒåΪDBTµÄ¸ù½Úµã¶øÆäËûÁ½¸öλÖñ»¶¨ÒåΪDBTµÄÒ¶×ӽڵ㡣ÕýÈçͼ3.5Ëùʾ£¬sµÄËùÓÐÔªËØ¿ÉÒÔ±»·Ö³ÉÈçÏÂÁ½²ã£º 


L1¡«sR={Ri|i¡Ê£Û1,k£Ý}

L2¡«sL={Li1,Li2|i¡Ê£Û1,k£Ý}(3.39)



ÔÚʽ(3.39)ÖУ¬sR´¦ÔÚµÚÒ»²ã£¬Ëü°üº¬ÁËËùÓеĸù½Úµã£» sL´¦ÔÚµÚ¶þ²ã£¬Ëü°üº¬ÁËËùÓеÄÒ¶×ӽڵ㡣Òò´Ë£¬ÎÒÃÇ°ÑÒ»¸ö¸ù½Úµãs3i-1ÓëÆäÁ½¸öÒ¶×ӽڵ㶨ÒåΪϵÊýsµÄÒ»¸ö×Ó¼¯£º 


ssubi={Li1,Ri,Li21¡Üi¡Ük}(3.40)



ÏÖÔÚ¿É°Ñÿһ¸ö×Ó¼¯Öеĸù½Úµã·ÖΪÁ½ÖÖÀàÐÍ£º 


type(1)={Ri¡Ù0,i¡Ê£Û1,k£Ý}

type(2)={Ri=0,i¡Ê£Û1,k£Ý}


ÄÇô¸ù¾Ý²»Í¬µÄ¾Û¼¯Ä£ÐÍ£¬Ôò»ìºÏȨÖزÎÊý¦Ði¿ÉÒÔ¸ù¾ÝÒÔϵÄÄ£ÐÍÑ¡Ôñ¹ý³Ì½øÐÐÑ¡Ôñ£º 


¦Ð0r,si´¦ÔÚµÚÒ»²ã

¦Ð1i,si´¦ÔÚµÚ¶þ²ã£¬¸ù½ÚµãΪtype£¨1£©ÐÍ

¦Ð2i,si´¦ÔÚµÚ¶þ²ã£¬¸ù½ÚµãΪtype£¨2£©ÐÍ(3.41)



µÚÒ»²ãµÄϵÊý£¬ÓÉÒ»¸öÏÈÑé²ÎÊý¦Ð0rÉ趨£¬ËüÊÇÒ»¸öÇ÷ÏòÓÚ²úÉú(½Ó½üÓÚ1)·ÇÁãÖµµÄÏÈÑé¡£Èç¹ûÏÈÑé²ÎÊýÔª¦Ð1i(ÒԽϴó¸ÅÂÊÈ¡½Ï´óÖµ)±»Ñ¡Ôñ£¬ÔòÒâζ×ÅÈô¸ù½ÚµãÇ÷½üÓÚ(½Ï´óÖµ)·ÇÁ㣬ÄÇô´Ë¸ù½ÚµãµÄÁ½¸öÒ¶×Ó½ÚµãÒ²»áÒԸ߸ÅÂÊÇ÷½üÓÚ(½Ï´óÖµ)·ÇÁã¡£Èç¹ûÏÈÑé²ÎÊý¦Ð2i(ÒԽϴó¸ÅÂÊÈ¡½ÏСֵ)±»Ñ¡Ôñ£¬ÔòÒâζ×ÅÈô¸ù½ÚµãÇ÷½üÓÚÁ㣬ÄÇô´Ë¸ù½ÚµãµÄÁ½¸öÒ¶×Ó½ÚµãÒ²»áÒԸ߸ÅÂÊÇ÷½üÓÚÁã¡£Ò²¾ÍÊÇ˵£¬ÔÚÕâ¸öģʽµÄÑ¡Ôñ¹ý³ÌÖÐDBTµÄÒ¶×Ó½ÚµãÓëËüµÄ¸ù½ÚµãÓÐÒ»ÖÖ±¾ÖʵÄÒÅ´«¹Øϵ¡£ÄÇôÔÚ¿éÏ¡ÊèÐźŵĻָ´¹ý³ÌÖУ¬´ËÄ£ÐÍ¿ÉÒÔʹÿһ¸öDBT½øÐÐÏàÓ¦µÄ¿é¾Û¼¯£¬Ò²¿ÉÒÔÀí½âΪÔÚÐźŻָ´Ê±¸ø¿éÑ°ÕҵĹý³ÌÌí¼ÓÁËÒ»¸öÈõÔ¼Êø¡£
ÓÉÓÚBeta·Ö²¼ÊDz®Å¬ÀûËÆÈ»µÄ¹²éîÏÈÑ飬Ôòʽ(3.41)ÖеĻìºÏȨÖئÐi¿ÉÒÔ´ÓÈçÏÂBeta·Ö²¼µÃµ½£º 


¦Ð0r¡«Betae0,f0
¦Ð1i¡«Betae1,f1
¦Ð2i¡«Betae2,f2


ÆäÖÐe0,f0¡¢e1,f1ÒÔ¼°e2,f2ΪBeta·Ö²¼µÄÐÎ×´²ÎÊý¡£Èô¶ÔÕâЩ²ÎÊý½øÐкÏÊʵÄÉèÖã¬ÔòÒÔÉÏģʽѡÔñ¹ý³Ì²»½ö¿ÉÒÔÌá¸ß¿éµÄ¾Û¼¯ÐÔ£¬¶øÇÒ¿ÉÒÔÆðµ½ÒÖÖƹÂÁ¢µãµÄ×÷Óá£ÀýÈ磬µ±e<fʱ£¬Betae,f·Ö²¼ÇãÏòÓÚ²úÉúÒ»¸ö(½ÏС)Ç÷ÓÚÁãµÄ²ÉÑùÖµ£» µ±e>fʱ£¬ÆäÇãÏòÓÚ²úÉúÒ»¸ö(½Ï´ó)·ÇÁãµÄ²ÉÑùÖµ¡£Òò´Ë£¬ÎªÁ˼ÓÇ¿¿éµÄ¾Û¼¯ÐÔ²¢ÒÖÖƹÂÁ¢µãµÄ³öÏÖ£¬¶ÔÓÚ²ÎÊýe1,f1ÒªÉèÖÃe1>f1ÒÔʹBetae1,f1·Ö²¼²úÉúÒ»¸ö½Ï´óµÄ¸ÅÂʦÐ1iÀ´´Ù½ø¿é¾Û¼¯£» ¶ÔÓÚ²ÎÊýe2,f2ÒªÉèÖÃe2<f2ÒÔʹBetae2,f2·Ö²¼²úÉúÒ»¸ö½ÏСµÄ¸ÅÂʦÐ2iÀ´ÒÖÖƹÂÁ¢µãµÄ´æÔÚ¡£
¸ù¾ÝÒÔÉϵĶ¨Òå¼ÙÉ裬¾Í¿ÉÒÔ¸ø³ö·Ö²ãµÄÏÈÑéÄ£ÐÍ£¬Èçͼ3.8Ëùʾ£¬ÆäÖЦÐi¿É¸ù¾ÝsÖеÄÿһ¸öÔªËØËùÊô²»Í¬ÀàÐÍÒÔ¼°Ëù´¦²»Í¬µÄ²ã(¾Û¼¯Ä£ÐÍÑ¡Ôñ¹ý³Ì)½øÐÐÑ¡Ôñ¡£Í¬Ê±³¬²ÎÊý¦Ái¿ÉÓÉGamma(a,b)·Ö²¼²úÉú¡£×îºó£¬¾Í¿ÉÒÔÒÔÒÑÖªµÄ»ìºÏȨÖزÎÊý¦ÐiÒÔ¼°¾«¶È³¬²ÎÊý¦Ái£¬ÓÉspikeª²andª²slabÏÈÑé²úÉúsµÄÿһ¸öÔªËØsi¡£


ͼ3.8DBTÄ£Ð͵Äͼ½á¹¹


ʵ¼ÊÉÏ£¬ÔÚ²âÁ¿¹ý³ÌÖвâÁ¿Öµ²»Ãâ»á±»ÔëÉùËùÎÛȾ¡£Òò´Ë£¬ÎªÁËÈ·±£²âÁ¿¹ý³ÌµÄ¾«È·ÐÔ£¬ÎÒÃǶԲâÁ¿ÖµÌí¼ÓÒ»¸öÔëÉù²¢ÓɲÎÊý¦Å±íʾ£¬Ôò²âÁ¿Öµ¿ÉÒÔдΪÈçÏÂÐÎʽ£º 


y=¦¨s+¦Å


ÆäÖЦÅΪһ¸öÁã¾ùÖµ²»Ïà¹Ø¸ß˹ÔëÉùÏòÁ¿£¬²¢ÇÒÆäÔªËصķ½²îΪ¦Ò2¡£
ÔÚÕâÖÖÇé¿öÏ£¬²âÁ¿ÖµµÄÌõ¼þ·Ö²¼p(ys)Ó¦¸Ã±»ÖØиÄдΪp(ys,¦Án)£¬²¢ÇÒ¦Án=¦Ò-2ΪÔëÉùµÄ¾«¶È²ÎÊý¡£Í¬Àí£¬Èó¬²ÎÊý¦ÁnÒ²·þ´ÓÈçÏÂÙ¤Âí·Ö²¼£º 


p(¦Ánc,d)=¡ÇMi=1Gamma(¦Áinc,d)


Ôò¿ÉÒԵõ½ËÆÈ»º¯Êý(Ìõ¼þ·Ö²¼)£º 


y¡«N(¦¨s,¦Á-1n¦©)


ÔÚ±¾Ð¡½ÚÖУ¬ÎÒÃÇÔÚ±´Ò¶Ë¹Ñ¹Ëõ¸ÐÖªµÄ¿ò¼ÜÏ£¬ÀûÓ÷ֲãµÄÏÈÑéÄ£ÐÍͨ¹ýÒ»¸öÄ£ÐÍÑ¡Ôñ¹ý³Ì°Ñ¿é×´ÏÈÑé½á¹¹½øÐÐÒýÈ룬ÆäÓëÎÄÏ×£Û26£ÝËùÌá³öµÄ·½Ê½²»Í¬£¬ÕâÖÖÄ£Ð͵ľۼ¯½á¹¹ÏÈÑéÊÇÓÉÂí¶û¿Æ·òËæ»ú³¡½øÐÐÒýÈëµÄ¡£
3. »ùÓÚMCMC²ÉÑùµÄ±´Ò¶Ë¹¹À¼Æ
ÓÉÓÚʹÓñ´Ò¶Ë¹¹À¼ÆÆ÷¶ÔÎÒÃÇËùÌá³öÄ£Ð͵ÄËæ»ú±äÁ¿½øÐÐÏÔʽ±í´ïʽµÄÇó½âºÜÀ§ÄÑ¡£Òò´Ë±¾½Ú½«²ÉÓÃMCMC(Markov Chain Monte Carlo£Û34£Ý)·½·¨¶Ôδ֪ģÐͲÎÊý½øÐÐÍÆÀí¡£ÓÚÊÇ£¬Î´Öª²ÎÊý¿ÉÃèÊöΪ¦Ã=s,¦Án,¦Á,¦Ð£¬¶øÄ£ÐͲÎÊýÔò¼ÇΪv=a,b,c,d,e,f£¬ÆäÖÐe={e0,e1,e2}ÒÔ¼°f={f0,f1,f2}¡£Ôòδ֪²ÎÊý ¦Ã=s,¦Án,¦Á,¦ÐµÄºóÑé¸ÅÂÊÂú×ãÏÂʽ£º 


p(¦Ãy,v)¡Øp(y¦Ã)p(¦Ãv)


È»ºó£¬¿ÉÒÔÒÀ¾Ý·Ö²ãµÄÏÈÑéÄ£ÐÍ(¼ûͼ3.8)À´µÃ³öδ֪ģÐͲÎÊý¦Ã=s,¦Án,¦Á,¦ÐÌõ¼þºóÑé¸ÅÂÊ£º 


p(¦Ãy,v)=p(s¦Án,¦Á,¦Ð,y)p(¦Ás,a,b)p(¦Ðs,e,f)p(¦Áns,y,c,d) (3.42)



1) δ֪²ÎÊýµÄºóÑé¸ÅÂÊ
¼ÙÉèÏ¡ÊèÏòÁ¿sÖÐÿ¸öÔªËصÄÏÈÑé¸ÅÂʶ¼ÊǶÀÁ¢µÄ£¬ÄÇô¿ÉÒÔ°ÑsµÄÈ«ÏÈÑé¸ÅÂÊ·Ö²¼ÃèÊöÈçÏ£º 


p(s¦Ð,¦Á)=¡ÇNi£Û(1-¦Ði)¦Ä0+¦ÐiN(si0,¦Á-1i)£Ý(3.43)



½áºÏÌõ¼þ·Ö²¼p(ys,¦Án)£¬Ôò¿ÉÒԵõ½sµÄºóÑé¸ÅÂÊ£º  


p(s¦Ð,¦Á,¦Án,y)¡Øp(s¦Ð,¦Á)p(ys,¦Án)
=¡ÇNi£Û(1-¦Ði)¦Ä0+¦ÐiN(si0,¦Á-1i)£ÝN(yAs,¦Á-1n¦©)
(3.44)



¼ÙÉè¾ØÕó¦¨ÌÞ³ýµÚiÁкó³ÉΪ×Ó¾ØÕó¦¨-i£¬²¢ÇÒ¼Çs-iΪÌÞ³ýµÚi¸öÔªËصÄÏòÁ¿£¬Ôò¿ÉÒÔÉè¼ÆÒ»¸ö¼ª²¼Ë¹(Gibbs)²ÉÑùÆ÷À´¸ù¾ÝÏÂʽµÄºóÑé¸ÅÂʶÔÏ¡ÊèÏòÁ¿µÄÿһ¸öÔªËØsi½øÐвÉÑù£º 


p(si¦Ì~i,¦Á~i)=(1-¦Ð~i)¦Ä0+¦Ð~iN(si¦Ì~i,¦Á~-1i)(3.45)



ÆäÖеÄÏà¹Ø²ÎÊý¶¨ÒåÈçÏ£º 


¦Á~i=¦Ái+¦Án¦¼Ti¦¼i

¦Ì~i=¦Á~-1i¦Án¦¼Ti(y-¦¨-is-i)

¦Ð~i1-¦Ð~i=¦Ði1-¦Ði¡¤N(00,¦Á-1i)N(0¦Ì~i,¦Á~-1i)


ÓÉÓÚÏ¡ÊèÄ£Ð͵ķ´·½²î¦ÁµÄÏÈÑé·Ö²¼ÎªÙ¤Âí·Ö²¼£¬Ôò¸ù¾ÝÆä¹²éîÐÔ¿ÉÖªËüµÄºóÑé¸ÅÂÊÒÀȻΪһ¸öÙ¤Âí·Ö²¼£º 


p(¦Áissubi)=Gammaa+¡¬si¦¸¡¬0/2,b+1/2¡¬si¦¸¡¬22


ÆäÖÐsi¦¸±íʾµÄÊǵÚi¸ö¸ù½ÚµãÓëÆäÁ½¸öÒ¶×Ó½ÚµãµÄ¼¯ºÏ£¬²¢ÇÒi¡Ê£Û1,k£Ý¡£
ÈçÇ°ËùÊö£¬ÎÒÃÇ¿ÉÒÔͨ¹ýÄ£ÐÍÑ¡ÔñµÄ¹ý³ÌÀ´¸ø»ìºÏȨÖزÎÊýµÄÿ¸öÔªËئÐiÑ¡ÔñÏàÓ¦µÄ²ÎÊý£¬Ò²¾ÍÊǶÔÓÚÈýÖÖÏ¡Êèģʽq¡Ê0,1,2¿ÉÒÔÑ¡Ôñ²»Í¬µÄ¦Ði=¦Ðqi¡£ÁíÍ⣬ÓÉÓÚÕâÈýÖÖÏ¡ÊèģʽÆ䳬²ÎÊý¶¼Âú×ãBeta·Ö²¼£¬ÔòËüÃÇÊDz®Å¬Àû·Ö²¼µÄ¹²éîÏÈÑé¡£Òò´Ë£¬ÕâÈýÖÖÏ¡Êèģʽ¦ÐqiµÄºóÑé·Ö²¼¿ÉÓÉÏÂʽµÃ³ö£º 


p(¦Ð0rssubi)=Beta(e0+¡¬siR¡¬0,f0+siR-¡¬siR¡¬0)

p(¦Ð1issubi)=Beta(e1+¡¬siLV¡¬0,f1+siLV-¡¬siLV¡¬0)

p(¦Ð2issubi)=Beta(e2+¡¬siLV¡¬0,f2+siLV-¡¬siLV¡¬0)


ÆäÖÐsiLV±íʾµÄÊǵÚi¸ö¸ù½ÚµãµÄÁ½¸öÒ¶×Ó½ÚµãµÄ¼¯ºÏ£¬²¢ÇÒi¡Ê£Û1,k£Ý£¬¶øsiR±íʾµÄÊǵÚi¸ö¸ù½ÚµãµÄ¼¯ºÏ¡£
ÓÉÓÚÔëÉù·½²î¦ÁnµÄÏÈÑé¸ÅÂÊÓë¸ß˹ËÆÈ»·Ö²¼Îª¹²éîµÄ£¬Ôò¿ÉµÃ³ö¦ÁnµÄºóÑé¸ÅÂÊ£º 


p(¦Áns,y)=Gamma(c+M/2,d+1/2¡¬y-As¡¬22)


2) »ùÓÚ¼ª²¼Ë¹²ÉÑùµÄÏ¡ÊèÏòÁ¿¹À¼Æ
¸ù¾Ýδ֪ģÐͲÎÊýµÄºóÑé¸ÅÂÊ£¬¿ÉÒÔÀûÓüª²¼Ë¹²ÉÑùÀ´²úÉúÏà¹ØÑù±¾¡£Æä¾ßÌå²ÉÑù¹ý³ÌÈçÏ£º 
£¨1£© ´Óp(sis-i,¦Á,¦Án,¦Ð,y)ÖнøÐÐsi²ÉÑù£» 
£¨2£© ´Óp(¦Áis,a,b)ÖнøÐЦÁi²ÉÑù£» 
£¨3£© ´Óp(¦Ðis,e,f)ÖнøÐЦÐi²ÉÑù,¶ÔÓÚi=1,2,¡­,N½øÐв½Öè(1)µ½²½Öè(3)µÄÑ­»·£» 
£¨4£© ´Óp(¦Áns,y,c,d)ÖнøÐЦÁn²ÉÑù¡£
ÔÚÀûÓÃÉÏÊö¼ª²¼Ë¹²ÉÑù¹ý³Ì¶Ô²ÎÊýs½øÐб´Ò¶Ë¹ÍÆÀíµÄͬʱ£¬ÆäÓàδ֪²ÎÊý¦Ð,¦Á,¦ÁnÒ²¿ÉÔÚ´ËÍÆÀíµÄ¹ý³ÌÖеõ½¹À¼Æ¡£Ò²¾ÍÊÇ˵£¬¿ÉÒÔÓɼª²¼Ë¹²ÉÑù»ñµÃÒ»¸ö¹ØÓÚ´ý¹À²ÎÊý¦Ã={s,¦Án,¦Á,¦Ð}µÄ²ÉÑù¼¯£¬¶øËüµÄÑù±¾·Ö²¼ÓëÎÒÃÇÏëÒª»ñµÃµÄÓÉʽ(3.42)±íʾµÄÁªºÏºóÑé¸ÅÂÊÏà½ü¡£ÔÚ´Ë£¬¼ÙÉ輪²¼Ë¹²ÉÑùµÄÈ«ÌåÑù±¾Îª 


¦Ã= ¦Ã(j)j=1,2£¬¡­,tNi,¡­,tMC


ÆäÖУ¬j±íʾ¼ª²¼Ë¹²ÉÑùµÄµÚj²½£» ¶øtNiΪ¼ª²¼Ë¹²ÉÑùBurnª²inµÄµü´ú´ÎÊý£» tMCΪÆä×ܵĵü´ú´ÎÊý¡£
ÔÚÕâÀïÈô¶Ô²ÉÑù¼¯½øÐÐÇóƽ¾ù£¬Ôò¿ÉÄÜ»áʹ¹À¼Æ³öµÄsÖµ²»Ï¡Ê裬ËùÒÔÎÒÃDz¢²»Ê¹ÓÃ×îС¾ù·½Îó²î¹À¼Æ£¬¶øÊDzÉÓÃ×î´óºóÑé(MAP)¹À¼ÆÀ´½øÐÐsµÄ¹À¼Æ¡£ÒÀ¾Ýʽ(3.42)ËùʾµÄÈ«ºóÑé¸ÅÂÊ£¬²¢¶ÔÆäÓàµÄ³¬²ÎÊý(¦Ð,¦Á,¦Án)½øÐлý·Ö£¬Ôò¿ÉµÃÒÔϵıßÔµ·Ö²¼£º 


p(sy)¡Ø¬íp(s¦Á,¦Ð,¦Án,y)d¦Ád¦Ðd¦Á0(3.46)



¸ù¾Ýʽ(3.46)£¬Ï¡ÊèÏòÁ¿sµÄ×î´óºóÑé¹À¼Æ¿ÉÒÔдΪÒÔÏÂÐÎʽ£º 


s^¡Öarg maxs¡Ê¦Ã p(sy)


ÆäÖЦÃ=P{¦Ã(j)}j=1,2,¡­,tNi,¡­tMC¡£
ÕâÑù¾Í¿ÉÒԵõ½Ï¡ÊèÏòÁ¿µÄ¹À¼Æs^£¬¼´ËùÓеÄδ֪ģÐͲÎÊý¾ù¿É»ñµÃ¡£
3.3ÁªºÏÏ¡ÊèѹËõ¸ÐÖª
ÔÚʵ¼ÊÓ¦ÓÃÖУ¬²»Í¬µÄϵͳ½á¹¹µ¼ÖÂÐźż¯ºÏÖÐÏ¡ÊèÐźÅÓв»Í¬ÐÎʽµÄ¹ØÁª·½Ê½£¬ËùÒÔBaronµÈÈ˸ù¾ÝÐźŲ»Í¬µÄÏ¡Êè½á¹¹£¬½«ÁªºÏÏ¡ÊèÄ£ÐÍ(Joint Sparsity Mode,JSM)·ÖΪÈýÖÖ²»Í¬ÀàÐÍ£º JSMª²1¡¢JSMª²2ºÍJSMª²3£¬²¢Ìá³öÁËÏàÓ¦µÄÖؽ¨Ëã·¨¡£¶ÔÓÚÿһÖÖÁªºÏÏ¡ÊèÀàÐ͵Ķ¨ÒåÔÚ3.3½ÚÒѸø³ö¡£
ÉèÒ»¸öÏ¡ÊèÐźż¯ºÏ{xj}(j¡Ê{1,2,¡­,J}£©£¬ÆäÖÐÿ¸öÐźÅxj¡ÊRN£¬¾ùÓɹ«¹²²¿·ÖzcºÍ¸÷×Ô¶ÀÁ¢²¿·Özj×é³É£¬¼´ÓÐxj=¦·j(zc+zj)¡£¸÷½ÚµãÐźÅxj¶ÔÓ¦µÄ²âÁ¿¾ØÕó¼ÇΪ¦µj£¬Ò»°ãÒÔ¶ÀÁ¢Í¬·Ö²¼µÄËæ»ú¸ß˹¾ØÕóΪÖ÷¡£ÎªÁËÐðÊö¼ò±ã£¬±¾½ÚÉèÐźÅÏ¡Êè»ùΪ¦·=I£¬ÆäÖÐI±íʾµ¥Î»¾ØÕó¡£Ã¿¸ö·¢ËͶ˶ÀÁ¢·¢ËÍÐźÅxjµÄѹËõ²ÉÑùÖµ£¬½ÓÊÕ¶ËÔò¸ù¾ÝÐźżäÏà¹ØÐÔÀ´½øÐÐÐźÅÁªºÏÖؽ¨¡£¶ÔÓÚÿ¸ö½Úµã£¬ÓÐyj=¦µjxj¡£ÏÂÃæÕë¶ÔÈýÖÖ²»Í¬µÄÏ¡ÊèÀàÐÍ·Ö±ð½øÐнéÉÜ¡£
3.3.1JSMª²1Ä£Ðͼ°Öع¹Ëã·¨
1. JSMª²1Ä£Ðͼò½é


JSMª²1Ä£ÐÍÖÐÐźŵĹ«¹²²¿·ÖzcºÍ¸÷×Ô¶ÀÁ¢²¿·Özj·Ö±ð¿ÉÒÔÏ¡Êè±íʾ£¬ÇÒ¹«¹²²¿·ÖµÄ·ÇÁãÏ¡ÊèϵÊýλÖü°ÏµÊý´óСһÖ£¬¸÷×Ô¶ÀÁ¢²¿·ÖµÄÏ¡ÊèϵÊýλÖûòϵÊý´óС²»Í¬£¬ÌØÓв¿·ÖµÄÏ¡Êè¶È²»ÒªÇóÒ»Ö¡£
JSMª²1µÄÊýѧģÐÍΪ


xj=zc+zj(3.47)

zc=¦·sc(3.48)

zj=¦·sj(3.49)

¡¬sc¡¬0=Kc,¡¬sj¡¬0=Kj(3.50)



ÆäÖÐ,xj±íʾµÚj¸öͨµÀµÄÐźţ» zcºÍzj·Ö±ð±íʾÐźŵĹ«¹²²¿·ÖºÍ¸÷×ԵĶÀÁ¢²¿·Ö£» scºÍsj·Ö±ð±íʾÐźŹ«¹²²¿·ÖºÍÌØÓв¿·ÖµÄÏ¡Êè±íʾÏòÁ¿£» KcºÍKj·Ö±ðΪÐźŶÀÁ¢²¿·ÖºÍÌØÓв¿·ÖµÄÏ¡Êè¶È¡£
µ±²»ÄܽøÐÐÏ¡Êè¶È½µµÍʱ£¬Ðźż¯ºÏµÄÁªºÏÏ¡Êè¶ÈD=Kc+¡Æj¡Ê¦«Kj¡£
JSMª²1ÊʺϽڵãÍøÂçÖеÄÐźÅÊÜͬһ¸ö´ó»·¾³Ó°Ï죬ÇÒ¾Ö²¿»¹·Ö±ð´æÔÚ²»Í¬µÄ¡¢Óë±¾µØÏà¹ØµÄÓ°Ïì¡£ÀýÈ磬Õû¸öÉ­ÁÖζȵı仯£¬Î¶ȴ«¸ÐÆ÷ÏÔʾɭÁÖζȼÈÊÜʱ¼ä(ÐźÅÄÚ²¿)Ó°Ï죬ҲÊÜ¿Õ¼ä(ÐźÅÍⲿ)Ó°Ïì¡£´ó·¶Î§µÄÆøÎÂÓ°ÏìÒòËØ£¨ÈçÌ«Ñô¹âÕպͷçµÈ£©»á¶ÔÐźŹ«¹²²¿·ÖzcÔì³ÉÓ°Ï죬ÊÊÓÃÓÚËùÓд«¸ÐÆ÷¡£¾Ö²¿ÒòËØ£¨ÈçÒõÓ°¡¢Ë®Ô´ºÍ¶¯ÎïµÈ£©Ö»»áÓ°ÏìÉÙÁ¿´«¸ÐÆ÷½ÚµãµÄ¶ÀÁ¢²¿·ÖzjµÄ¶ÁÊý¡£ÀàËƽṹµÄÐźŻ¹¿É³öÏÖÔÚ²âÁ¿¹âÇ¿¶È¡¢ÆøѹµÄ´«¸ÐÆ÷ÍøÂçÖС£ÕâЩӦÓó¡¾°¾ù¶ÔӦһЩÎïÀí¹ý³ÌµÄÖ¸±ê²âÁ¿£¬ËüÃÇÔÚʱ¼äºÍ¿Õ¼äÉϾù±ä»¯»ºÂý£¬ËùÒÔÓи߶ÈÏà¹ØÐÔ¡£
2. JSMª²1ѹËõ¸ÐÖªÖؽ¨Ëã·¨
¾­Ñо¿±íÃ÷£¬¿ÉÒÔͨ¹ýÇó½âÒ»¸ö»ùÓÚȨÖصÄL1·¶Êý×îС»¯ÎÊÌ⣺ 


Z^=arg min ¦Ãc¡¬zc¡¬1+¡ÆJi=1¦Ãi¡¬zi¡¬1  s.t. Y=¦µZ(3.51)



ÆäÖЦÃcºÍ¦Ãi¾ù´óÓÚ0¡£
Õâ¸öJSMª²1ѹËõ¸ÐÖªÖؽ¨Ëã·¨³ÆΪ»ùÓÚ¦ÃȨÖصÄL1·¶ÊýËã·¨¡£
ÒªÏëËã·¨´ïµ½½ÏºÃÐÔÄÜ£¬ÐèÒª¸ù¾ÝzcºÍzjµÄÏ¡Êè¶È´óСµÈÐÅÏ¢½øÐЦÃȨÖØÓÅ»¯¡£Èô¸÷zjµÄÏ¡Êè¶ÈÏàͬ£¬Ôò¿ÉÒ԰ѦÃi¶¼ÉèΪ1£¬¶Ô¦Ãc½øÐе¥¶ÀÓÅ»¯Ê¹Ëã·¨ÐÔÄÜ´ïµ½×î¼Ñ¡£BaronµÈÈ˶ԸÃËã·¨½øÐÐÁË·ÂÕæÊÔÑ飬֤Ã÷ÆäÒòÀûÓÃÐźżäµÄÏà¹ØÐÔ¶ø²úÉúµÄÏà¶ÔѹËõ¸ÐÖª¾ßÓжÀÁ¢Öؽ¨µÄ¼«´óÐÔÄÜÓÅÊÆ¡£µ«Ëü±¾ÖÊÊÇ»ùÓÚ͹ÓÅ»¯ÀàµÄLp ·¶Êý×îС»¯£¬Í¹ÓÅ»¯Ëã·¨µÄÌصãÊÇÖؽ¨ÐÔÄÜÓÅÔ½£¬µ«¼ÆË㸴ÔӶȸߡ£ÓÉ´Ë£¬ÖÚ¶àѧÕßÃǶԴËÕ¹¿ªÁËÑо¿£¬ÆäÖÐÑо¿±È½Ï¶àµÄÊÇ»ùÓÚ±ßÐÅÏ¢µÄÑÜÉúËã·¨£¬½ÓÏÂÀ´Ö÷Òª½éÉÜ»ùÓÚ±ßÐÅÏ¢µÄÕý½»Æ¥Åä×·×Ù(Side Information Orthogonal Matching Pursuit,SIOMP)Ëã·¨£Û35£Ý¡£SIOMP Ë㷨ͨ¹ýÔÚÖÚ¶àµÄ¹Û²âÖµÖÐÖع¹ÆäÖеÄÒ»¸öÐźŵõ½±ßÐÅÏ¢£¬ÓÃÆäËûÐźÅÓëËùµÃµ½µÄ±ßÐÅÏ¢×ö²î£¬ÏȻָ´²îÖµÔÙ»Ö¸´Ô­Ðźţ¬´Ó¶ø¼õÉÙÆäËûÐźÅËùÐèµÄ¹Û²â´ÎÊý¡£¼ÙÉèÓÐÁ½¸öÐźÅx1ºÍx2·þ´ÓJSMª²1Ä£ÐÍ£¬SIOMPÖؽ¨Ëã·¨µÄ»ù±¾²½ÖèÈç±í3.4Ëùʾ¡£


±í3.4SIOMPËã·¨²½Öè



²½Öè1,ÊäÈëx1µÄ¹Û²âÏòÁ¿y1£¬¹Û²â¾ØÕó¦µ1£¬ÓÉOMPËã·¨µÃµ½x*1×÷Ϊ±ßÐÅÏ¢£» 
²½Öè2,ÀûÓñßÐÅÏ¢»Ö¸´x2£» 
ÊäÈ룺 x2µÄ¹Û²â¾ØÕó¦µ2£¬x2µÄ¹Û²âÏòÁ¿y2£¬±ßÐÅÏ¢x*1£» 

£¨1£© ³õʼ»¯Ö§³ÅË÷Òý¼¯¦¸=«Á£¬²Ð²îrt=y2-¦µ2x*1£¬µü´ú´ÎÊýt=0£¬×î´óµü´ú´ÎÊýT£» 

£¨2£©  ͨ¹ýʽw={wjwj=¡´r,¦Õj¡µ,j=1,2,¡­,N}Ñ¡Ôñ²Ð²îÖµÓ릵2×î´óÏà¹ØϵÊýµÄÔ­×Ó£¬Ô­×ÓË÷Òý¼ÇΪ¦Ë£» 
Ðø±í



£¨3£©  ¸üÐÂÖ§³ÅË÷Òý¼¯¦¸¡û¦¸¡È¦Ë£¬¸üÐÂÖ§³Å¼¯¦µ2¦¸¡û¦µ2¦¸¡È¦Õ¦Ë,¦µ2¦¸±íʾµÄÊǦµ2 ÖÐË÷ÒýΪ¦¸µÄÁÐÏòÁ¿×é³ÉµÄ¾ØÕ󡣱ƽü²Ð²îÐźţº x*=arg min¡¬y-¦µx¡¬2²Ð²î¸üУº rnew=y-¦µx*¡£Áît=t+1£¬·µ»Ø²½Öè(2)£¬µü´úT´Î£» 

£¨4£© Í£Ö¹µü´ú£¬¼ÆËãԭʼ²Ð²îÐźÅx*t=pinv(¦µ2¦¸)rt,pinv(¡¤)±íʾ¾ØÕóµÄαÄ棻 

Êä³ö£º  x2=x1+x*t¡£


JSMª²1ºËÐÄ˼ÏëÊÇÏÈ»ñÈ¡±ßÐÅÏ¢£¬ÀûÓñßÐÅÏ¢Öع¹ÆäËûÐźš£
3.3.2JSMª²2Ä£Ðͼ°Öع¹Ëã·¨
1. JSMª²2Ä£Ðͼò½é


JSMª²2Ä£ÐÍÖй«¹²²¿·ÖzcΪ0£¬¼´²»´æÔÚ¹«¹²²¿·Ö¡£¸÷ÐźŶÀÁ¢²¿·Özj¹²ÏíÖ§³Å¼¯£¬µ«¶ÔÓ¦·ù¶È¸÷²»Ïàͬ¡£Ã¿¸öÐźÅÏ¡Êè±íʾÏòÁ¿µÄÏ¡Êè¶ÈÏàͬ¡£JSMª²2ÊýѧģÐÍÈçÏ£º 


xj=¦·sj(3.52)



JSMª²2Ä£Ð͹㷺ӦÓÃÓÚÉùÕóÁкÍÉäƵ(RF)´«¸ÐÆ÷ÕóÁС£ÆäÖи÷´«¸ÐÆ÷»ñÈ¡Ò»¸ö¸µÀïҶϡÊèµÄÐźţ¬µ«Òò¶à¾¶´«²¥£¬Êܵ½²»Í¬³Ì¶ÈµÄÏàλƫÒƺÍË¥¼õµÄÓ°Ïì¡£ÔÚMIMOͨÐÅϵͳÖУ¬Ò²³£ÄÜÓ¦ÓÃJSMª²2Ä£ÐÍ¡£
2. JSMª²2ѹËõ¸ÐÖªÖؽ¨Ëã·¨
JSMª²2Ä£ÐÍ¿ÉÒÔͨ¹ýSOMPËã·¨»Ö¸´Ðźš£SOMP¿ÉÒÔ³ä·ÖÀûÓÃJSMª²2Ä£ÐÍÖÐÐźŵÄÏ¡ÊèÐÔ¡£ÔÚ·Ö²¼Ê½³¡¾°ÖУ¬¶Ôµ¥¸öÐźÅÖؽ¨OMPËã·¨µÄÌ°À·Ñ¡Ôñ˼ÏëºÍ²Ð²î×îÏà¹ØÔ­×Ó˼Ïë½øÐÐÁËÍØÕ¹¡£ÔÚSOMPËã·¨ÖУ¬ÈÚºÏÖÐÐĽ«Ã¿¸öλÖÃÉÏ£¬¸÷½Úµã¶ÔÓ¦²âÁ¿¾ØÕóµÄÏàÓ¦ÁÐÓë²Ð²îµÄÏà¹ØϵÊýÇóºÍºó£¬×÷Ϊ¸ÃλÖöÔÓ¦µÄÏà¹ØϵÊý£¬½øÐÐÔ­×ÓÑ¡Ôñ¡£DCSSOMPËã·¨²½ÖèÈç±í3.5Ëùʾ¡£


±í3.5DCSSOMPËã·¨²½Öè



£¨1£© ³õʼ»¯µü´ú´ÎÊýt=1£¬Ö§³ÅË÷Òý¼¯¦¸=«Á£¬Õý½»ÏµÊýÏòÁ¿¦Âj=0,¦Âj¡ÊRM£¬rj,t ±íʾµÚt ´Îµü´úºóyjµÄ²Ð²î£¬³õʼ»¯²Ð²îrj,0=yj£» 
£¨2£© Ñ¡ÔñÄÚ»ýÄÜÁ¿Ö®ºÍ×î´óµÃµ½Ô­×ÓË÷Òý£¬Ìí¼Óµ½Ë÷Òý¼¯¦¸£» 
nt= arg maxn¡Ê1,2,¡­,N¡ÆJj=1¡´rj,t-1,¦Õj,n¡µ¡¬¦Õj,n¡¬2£¬¦¸=¦¸0nt
£¨3£© ½«ËùÑ¡Ô­×Ó½øÐÐÊ©ÃÜÌØÕý½»»¯£» 
rj,t=¦Õj,nt-¡Æt-1l=0¡´¦Õj,nt,¦Ãj,l¡µ¡¬¦Ãj,l¡¬00¦Ãj,l
£¨4£© ¸üÐÂËùÑ¡ÐźŶÔÓ¦µÄÕý½»ÏµÊýÓë²Ð²î£¬Í¨¹ý¼ÆËã²Ð²îÓëÕý½»ºóµÄÔ­×ÓÀ´¸üÐÂÕý½»ÏµÊý£» 
¦Âj(t)=¡´rj,t-1,¦Ãj,t¡µ¡¬¦Ãj,t¡¬00,rj,t=rj,t-1-¦Âj(t)¦Ãj,t
£¨5£© ÅжÏÊÕÁ²Ìõ¼þ£¬µ±ËùÓÐÐźŲвrj,t¡¬0>¦Å¡¬yj¡¬0(¦ÅΪ²Ð²îãÐÖµ)ʱ£¬t=t+1£¬·µ»Ø²½Ö裨2£©£¬·ñÔòÖ´Ðв½Ö裨6£©¡£
£¨6£© ½âÕý½»»¯£¬ÔËÓÃQR·Ö½â˼Ï룬¦£j=£Û¦Ãj,1,¦Ãj,2,¡­,¦Ãj,M£ÝÓ릵jÂú×ã¹Øϵ¦µj,¦¸=¦£jRj£¬ÆäÖЦµj,¦¸=£Û¦Õj,n1,¦Õj,n2,¡­,¦Õj,nM£Ý¬®yj=¦£j¦Âj=¦µj,¦¸xj,¦¸=¦£jRjxj,¦¸£¬µÃµ½»Ö¸´ÐźÅ
x^j,¦¸=R-1j¦Âj


3.3.3JSMª²3Ä£Ðͼ°Öع¹Ëã·¨
1. JSMª²3Ä£Ðͼò½é


JSMª²3Ä£ÐÍÖÐÐźÅÓɹ«¹²²¿·ÖºÍÌØÓв¿·Ö×é³É¡£ºÍJSMª²1µÄÇø±ðÔÚÓÚ£¬ÐźŵĹ«¹²²¿·Ö²»Ï¡Ê裬¼´ÔÚÈκλù϶¼²»ÄÜÏ¡Êè±íʾ¡£JSMª²3¿ÉÒÔ¿´×÷JSMª²1µÄÒ»ÖÖÀ©Õ¹£¬½µµÍÁËJSMª²1ÐźŶԹ«¹²²¿·ÖҲҪϡÊè±íʾµÄÑϸñÒªÇó¡£JSMª²3ÐźŵÄÊýѧģÐÍÈçÏ£º 


xj=zc+zj(3.53)

zj=¦·sj(3.54)



JSMª²3µÄÌØÓÐÏ¡ÊèϵÊýÏòÁ¿ºÍJSMª²1ÌØÓв¿·ÖµÄÏ¡ÊèϵÊýÏòÁ¿ÀàËÆ£¬¼´Ï¡ÊèϵÊýµÄλÖûò´óС²»Í¬¡£ 
2. JSMª²3ѹËõ¸ÐÖªÖؽ¨Ëã·¨
ÓëJSMª²1¡¢JSMª²2²»Í¬µÄÊÇ£¬JSMª²3ÐźÅÊÇ·ÇÏ¡ÊèµÄ£¬²»ÄÜÓô«Í³µÄCSËã·¨½øÐÐÖع¹¡£ÏÂÃæ½éÉÜÒ»ÖÖÄܹ»¶ÔJSMª²3ʵÏÖÁªºÏÖع¹µÄËã·¨£¬»ùÓÚ±ßÐÅÏ¢µÄTexas DOIËã·¨£¬ÐèҪעÒâµÄÊÇ£¬Õë¶ÔJSMª²1Ìá³öµÄÖع¹Ëã·¨SiOMP¿ÉÒÔÓõ½JSMª²3µÄÖع¹ÖУ¬µ«´ËʱËùÐèµÄ±ßÐÅÏ¢²»ÄÜѹËõ»Ö¸´£¬ÇÒTexas DOIÒªÇóËùÓÐÐźŵĹ۲â¾ØÕóÒ»Ö¡£Texas DOIËã·¨Èç±í3.6Ëùʾ¡£


±í3.6Texas DOIËã·¨²½Öè



ÊäÈ룺 ¹Û²â¾ØÕ󦵣¬Ï¡Êè¾ØÕó¦·£¬ÐźŸöÊýJ£» 
£¨1£© ¹«¹²²¿·Ö¹Û²âÏòÁ¿µÄ½üËÆ¡¢±ßÐÅÏ¢ÌØÓв¿·Ö¹Û²âÏòÁ¿µÄ½üËÆ£¬¿ÉÒÔͨ¹ýÇóºÍ×ö²îµÃµ½
yc=1J¡ÆJj=1yj

yI,1=y1-yc
£¨2£© ÀûÓÃyI,1¡¢¦µ¡¢¦·µÈÐÅϢʹÓÃÏàÓ¦µÄËã·¨(OMP)Öع¹±ßÐÅÏ¢µÄÏ¡ÊèϵÊýÏòÁ¿sI,1£» 
£¨3£© ÒÀ´Î»Ö¸´ÆäËûÐźţº 
ydiff,j=yj-y1

yI,j=ydiff,j+yI,1
ÀûÓÃyI,j¡¢¦µ¡¢¦·µÈÐÅÏ¢»Ö¸´ÆäËûÐźŵĽüËÆÏ¡ÊèϵÊýÏòÁ¿sI,j¡£Ê¹ÓÃsJ=s1-sI,1+sI»Ö¸´ÆäËûÐźŵÄÍêÕûϵÊý¡£

3.4¸ß˹ÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖª
±¾½Ú֮ǰµÄ²¿·Ö½éÉܵÄÐźŶ¼ÊÇÒÔһάÏòÁ¿»ò¶þά¾ØÕóµÄÐÎʽ´æÔڵģ¬µ±Éæ¼°²ÊɫͼƬ¡¢ÊÓƵÐòÁС¢Ò½Ñ§³ÉÏñÒÔ¼°¸ß¹âÆ×ͼÏñµÈÈýάÉõÖÁ¶àάÐźÅʱ£¬Èç¹û°´ÕÕ´«Í³µÄѹËõ¸ÐÖªÀíÂÛ¶ÔһάµÄÏòÁ¿½øÐвÙ×÷£¬ÄÇôÆäÒ»°ãµÄ×ö·¨ÊÇÏȶÔԭʼÊý¾Ý½øÐÐÏòÁ¿»¯²Ù×÷£¬È»¶øÕâÖÖ·½·¨²»½öÆÆ»µÁËÊý¾Ý×ÔÉíµÄÏà¹ØÐÔ£¬ÀË·ÑÁ˱¾¾Í¼«ÆäÏ¡ÉÙµÄÐÅÏ¢£¬¶øÇÒ¼«´óµØÔö¼ÓÁËËã·¨µÄʱ¼ä¸´ÔӶȺͿռ临ÔӶȡ£ÀýÈ磬¶ÔÓÚÒ»¸ö100¡Á100¡Á100ÏñËصĸ߹âÆ×ͼÏñ£¬Èô²ÉÓô«Í³µÄѹËõ¸ÐÖª·½·¨£¬Ëã·¨µÄ¸´ÔÓ¶ÈÖÁÉÙΪO(106)£¬¶ø¸ß¹âÆ×ͼÏñ±¾Éí¾ßÓеĿռäºÍÆ׼伫ǿµÄÏà¹ØÐÔÒ²²»¸´´æÔÚ¡£Òò´Ë£¬Ì½Ë÷еÄѹËõ¸ÐÖªµÄÄ£Ð;ßÓÐÖØÒªÒâÒ壬¶øÕÅÁ¿µÄ·½·¨Îª´ËÌṩÁËÒ»ÖÖ¿ÉÐеķ½·¨¡£ÕÅÁ¿Ñ¹Ëõ¸ÐÖªÓйػù±¾ÀíÂÛÔÚ2.6½ÚÒѽéÉܹý¡£±¾½ÚÒÔÊÓƵÊý¾ÝΪÀý½éÉÜÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖª¡£

3.4.1ÕÅÁ¿±íʾ¼°Æä·Ö½â
Ò»¸öÕÅÁ¿¿ÉÒÔ±»ÊÓΪһ¸ö¶àË÷ÒýÊýÖµÊý×飬һ¸öd½×ÕÅÁ¿±íʾΪA¡ÊRI1¡ÁI2¡Á¡­¡ÁId£¬¶ÔÓ¦ÔªËرíʾΪai1i2¡­in-1inin+1¡­id£¬ÆäÖÐ1¡Üin¡ÜIn£¬1<n<d¡£ÄÇô¶ÔÓÚÒ»¸öd½×µÄÕÅÁ¿A£¬ËüµÄnª²Ä£Õ¹¿ª¾ØÕó±íʾΪAn=unfoldnA¡ÊRIn¡ÁI1I2¡­Id£¬ÆäÁÐÓÉÑØ×ÅAµÄnª²Ä£µÄËùÓÐInάÏòÁ¿¹¹³É£» Ïà·´£¬ÑØnª²Ä£µÄÕ¹¿ª¾ØÕó¿ÉÒÔͨ¹ýA=foldnAn±ä»»»ØÕÅÁ¿¡£
ÕÅÁ¿A¡ÊRI1¡ÁI2¡Á¡­¡ÁIdºÍ¾ØÕóB¡ÊRJn¡ÁInµÄnª²Ä£³Ë·¨Ï൱ÓÚÆäËùÓÐnª²Ä£µÄÏòÁ¿ÏËάÓëBÏà³Ë£¬½á¹ûÒ²ÊÇÒ»¸öd½×ÕÅÁ¿C¡ÊRI1¡ÁI2¡Á¡­¡ÁId£¬ÆäÔªËØ°´ÕÕÒÔϼÆË㣺  


ci1i2¡­in-1jn,in+1¡­id=¡Æinai1¡­in-1inin+1¡­idbj,in


Ä¿Ç°ÓÐÁ½ÖÖÌØÊâµÄÕÅÁ¿·Ö½âÐÎʽ£º Tucker·Ö½â£Û36£ÝºÍCANDECOMP/PARAFAC(CP)·Ö½â£Û37£Ý¡£
ÔÚTucker·Ö½âÖУ¬d½×ÕÅÁ¿A¡ÊRI1¡ÁI2¡Á¡­¡ÁIdд×÷£º 


A=S¡Á1U1¡Á2U2¡Á¡­¡ÁdUd


ÆäÖÐ,S¡ÊRr1¡Ár2¡Á¡­¡Árd³ÆΪºËÐÄÕÅÁ¿£» Un¡ÊRIn¡Árn1¡Üi¡ÜdÓÉÑØ×ÅAµÄnª²Ä£µÄrn¸öÕý½»»ù×é³É¡£
Tucker·Ö½â¿¼Âǵ½ÁËÑØ×Åÿ¸öģʽչ¿ªµÄÏòÁ¿×Ó¿Õ¼äµÄµÍÖÈÐÔÖÊ¡£
CP·Ö½âÊÔͼ½«d½×ÕÅÁ¿A¡ÊRI1¡ÁI2¡Á¡­¡ÁId·Ö½â³ÉһϵÁÐKronecker»ù(ÖÈΪ1µÄÕÅÁ¿µÄ)µÄÏßÐÔ×éºÏ£¬Ð´×÷£º 


A=¡Æri=1ciVi=¡Æri=1ciÿðþ
½›i1¬®ÿðþ
½›i2¬®¡­¬®ÿðþ
½›iN


ÆäÖÐci±íʾʩ¼ÓÔÚKroneckerÄÚ»ýÉϵÄϵÊý¡£
ͨ¹ý½«Ã¿¸öKroneckerϵÊýÅÅÁе½ºËÐÄÕÅÁ¿ÏàÓ¦µÄλÖã¬CP·Ö½â¿ÉÒÔд³É Tucker·Ö½âµÄÐÎʽ£¬Èçͼ3.9Ëùʾ¡£ÓëTucker·Ö½âÏà±È£¬CP·Ö½â¶ÔÓ¦µÄºËÐÄÕÅÁ¿·Ç³£Ï¡Ê裬µ«ºöÂÔÁËÑØÆäģʽµÄÏòÁ¿×Ó¿Õ¼äµÄµÍÖÈÐÔÖÊ¡£


ͼ3.9ÕÅÁ¿Tucker·Ö½âÒÔ¼°¶ÔÓ¦µÄKronecker»ù±íʾ



3.4.2ÄÚÔÚÕÅÁ¿Ï¡Êè¶ÈÁ¿
ÕÅÁ¿µÄÏ¡ÊèÐÔ¿ÉÒÔͨ¹ý·Ö½âºóµÄºËÐÄÕÅÁ¿ºÍÑØÿ¸öģʽչ¿ªµÄÏòÁ¿×Ó¿Õ¼äµÄÏ¡ÊèÐÔÀ´¶ÈÁ¿£¬¶øÿ¸öģʽϵÄÏòÁ¿×Ó¿Õ¼äÊÇÒ»¸ö¾ØÕóÐÎʽ£¬¿ÉÒÔͨ¹ýÖÈÀ´ºâÁ¿¾ØÕóµÄÏ¡ÊèÐÔ¡£Èç3.6.1½ÚËùÊö£¬Tucker·Ö½â¿¼Âǵ½ÑØÆäÿ¸öģʽչ¿ªµÄÏòÁ¿×Ó¿Õ¼äµÍÖÈÊôÐÔ(Ï¡ÊèÐÔ)£¬µ«ÊÇÈ´ºöÂÔºËÐÄÕÅÁ¿µÄÏ¡ÊèÐÔ¡£ÓëTucker·Ö½âÇé¿öÏà·´£¬CP·Ö½â¾ßÓÐÏ¡ÊèµÄºËÐÄÕÅÁ¿£¬¶øÑØÿ¸öÕÅÁ¿Ä£Ê½µÄ×Ó¿Õ¼äÈ´²»ÊǵÍÖȵġ£Í¨¹ý´ÓÁ½ÖÖ·Ö½âÐÎʽÖÐÕûºÏ¸üºÏÀíµÄÏ¡ÊèÀí½â£¬ÃϵÂÓîµÈÈËÌá³öÁËÒ»ÖÖеÄÕÅÁ¿Ï¡Êè²â¶È¡ª¡ªÄÚÔÚÕÅÁ¿Ï¡Êè(Intrinsic Tensor Sparsity,ITS)²â¶È£¬¸Ã²â¶Èͬʱ¿¼ÂÇÁ˵äÐ͵ÄTucker·Ö½âºÍCPµÍÖÈ·Ö½â¶ÔÏ¡ÊèÐÔµÄÀí½â¡£ÕÅÁ¿XµÄÏ¡Êè¶ÈS(X)¶¨ÒåÈçÏ£Û38£Ý£º 


S(X)=t¡¬S¡¬0+(1-t)¡ÇNi=1rankX(i)(3.55)



ÆäÖÐ,SÊÇ´ÓTucker·Ö½âÖлñµÃµÄXµÄºËÐÄÕÅÁ¿£¬t(0<t<1)ÊÇÕÛÖÐÁ½ÏîµÄȨÖØϵÊý¡£
ITS²â¶È×ۺϿ¼ÂÇÁËTucker·Ö½âºÍCP·Ö½âµÄÄÚÔÚÏ¡ÊèÐÔ£¬²»½ö¿¼ÂÇÁ˺ËÐÄÕÅÁ¿µÄÏ¡Ê裬¶øÇÒ¿¼ÂÇÁËÿ¸öÕÅÁ¿Ä£Ê½ÏÂÕÅÁ¿×Ó¿Õ¼äµÄµÍÖÈÌØÐÔ¡£
3.4.3»ùÓÚCACTIµÄ½á¹¹»¯²âÁ¿
±¾½ÚµÄ¹Û²âÊÓƵͨ¹ýÀûÓÃʱ¼äѹËõ¿×¾¶±àÂë³ÉÏñ(Coded Aperture Compressive Temporal Imager,CACTI)²âÁ¿¶ø»ñµÃ£¬ÕâÖֽṹ»¯²âÁ¿ÃèÊöÈçÏ¡£
ÉèX¡ÊRNx¡ÁNy¡ÁD±íʾԭʼÀëÉ¢Ðźţ¬Y¡ÊRNx¡ÁNy±íʾ¶þάµÄCACTI²âÁ¿£¬Í¨¹ý¶þÖµÑÚÄ£¾ØÕó(Ai,j,n)¹Û²âXµÄÁ¬ÐøDÖ¡²¢µþ¼Ó£¬»ñµÃÒ»Ö¡²âÁ¿Y£º 


Yi,j=£ÛA1,j,1,A1,j,2,¡­,A1,j,D£Ý£Ûxi,j,1,xi,j,2,¡­,xi,j,D£ÝT


ÆäÖÐYi,j±íʾ²âÁ¿YÖеĵÚiÐÐjÁÐÔªËØ£¬xi,j,nÓë¶ÔÓ¦µÄ¿Õ¼ä(i,j)ºÍʱ¼änµÄÏñËØÏà¹ØÁª£¬Ai,j,n¡Ê0,1ÊÇÒ»¸öËæ»ú¶þÖµÑÚÄ£¡£
ÉÏʽÒâζ×ÅYÖеÄÿ¸öÏñËØÊÇÊÓƵÐòÁÐx=£ÛxT1,xT2,¡­,xTK£ÝTµÄDÖ¡ÖÐÏàӦλÖÃÏñËصļÓȨ£¬ÆäȨÖØÊÇ0»ò1¡£
½«Í¼Ïñ·Ö¸î³ÉͼÏñ¿éºó£¬Í¼ÏñÖдæÔڵĽṹ¸üÈÝÒ×±»½¨Ä££¬Òò´Ë£¬ÔÚÏÂÃæµÄÄæÎÊÌâÖУ¬½«Y·ÖΪP¡ÁP´óСµÄ¿é£¬Ã¿Ò»¿é¶ÔÓ¦Ò»¸öÆÚÍû»Ö¸´µÄÈýάʱ¿ÕÐźÅÏòÁ¿xk£¬¶ÔÓÚµÚk¸ö¿éyk¡ÊRPÓÐ 


yk=¦µkxk=¡ÆDi=1¦µikxik(3.56)



ÆäÖУ¬xk=x1Tk,x2Tk,¡­,xDTkT¡ÊRPD¡Á1,xik¡ÊRP£» ¦µk=¦µ1k,¦µ2k,¡­,¦µDk¡Ê{0,1}P¡ÁPD£» xik¶ÔÓ¦ÊÓƵÐòÁÐxµÄµÚiÖ¡ÖеĵÚk¸ö¶þάͼÏñ¿é£» ¦µikÊÇÒ»¸ö¶Ô½Ç¾ØÕó£¬Æä¶Ô½ÇÔªËضÔÓ¦¶þÖµÑÚÄ£ÕÅÁ¿AµÄµÚiÖ¡µÄµÚk¿é£¬ÆäÖÐÈýάͼÏñ¿éxkºÍ¶þά¿éyk¾ù±»ÏòÁ¿»¯£¬CACTI µÄ¹Û²âʾÒâͼÈçͼ3.10Ëùʾ¡£



ͼ3.10CACTI ¹Û²âʾÒâͼ


3.4.4»ùÓÚ¸ß˹ÁªºÏÏ¡ÊèÄ£Ð͵ÄÖع¹·½·¨
×ÔȻͼÏñµÄ×ÔÏàËÆÐÔ¿ÉÒÔÓÃÓÚ¸ÄÉÆͼÏñµÄÖع¹½á¹û£¬Í¬ÑùÔÚÊÓƵÖÐÒ²´æÔÚ×ÔÏàËÆÐÔ£¬²¢ÇÒÆä×ÔÏàËÆÐÔ¿ÉÒÔ·ÖΪÈýÀࣺ 
(1) ¶ÔÓÚÊÓƵµÄÿһ֡¶øÑÔ£¬Ã¿Ò»¸öͼÏñ¿é×ÜÄܹ»ÔÚͬһ֡ÖÐÕÒµ½ËüµÄÏàËƿ飬¼´Ã¿Ò»Ö¡Í¼ÏñÄÚµÄ×ÔÏàËÆÐÔ£¬Èçͼ3.11Öа×ɫʵÏß·½¿òͼÏñ¿é×é¡£


ͼ 3.11ÊÓƵÐòÁÐÖеÄÈýÖÖÀàÐÍÏàËÆͼÏñ¿éʾÒâͼ


(2) ¶ÔÓÚÊÓƵÖеľ²Ì¬³¡¾°£¬Èç¹ûÔÚij¸öʱ¿Ì³öÏÖ³¡¾°£¬ÏàͬµÄ³¡¾°½«³öÏÖÔÚÒÔºóʱ¿ÌÖ¡ÖеÄÏàͬλÖã¬Èçͼ3.11Öеİ×É«Ïà¿òͼÏñ¿é×é¡£
(3) ¶ÔÓÚÊÓƵÖеĶ¯Ì¬³¡¾°£¬Èç¹ûÒ»¸öÄ¿±êÔÚijһ֡³öÏÖ(¼´¦¤tʱ¿Ì³öÏÖ)£¬ÄÇô¿ÉÒÔÔÚ½ÓÏÂÀ´µÄ¼¸Ö¡ÖÐÕÒµ½Õâ¸öÄ¿±ê£¬Ä¿±êµÄλÖÿÉÄÜÓÐËù¸Ä±ä£¬Ò²¾ÍÊÇ˵£¬¶¯Ì¬Ä¿±ê»á³öÏÖÔÚ²»Í¬Ö¡µÄÏàÁÚ¿Õ¼äλÖã¬Èçͼ3.11Öеİ×É«ÐéÏß¿òͼÏñ¿é×é¡£
µÚÒ»ÀàÊôÓÚÖ¡ÄÚ×ÔÏàËÆÐÔ£¬µÚ¶þÀàºÍµÚÈýÀàÊôÓÚÊÓƵµÄÖ¡¼äÏàËÆÐÔ£¬ÏÂÃ潫½áºÏͼ3.12Ëùʾ¿ò¼Üͼ£¬½éÉÜÈçºÎÀûÓÃÊÓƵ֡ÄÚºÍÖ¡¼äµÄÏàËÆÐÔÌá¸ßÊÓƵÖع¹ÖÊÁ¿¡£



ͼ3.12»ùÓÚITSµÄÊÓƵÖع¹Ëã·¨¿ò¼Üͼ


1. »ùÓÚÄÚÔÚÕÅÁ¿Ï¡Êè¶ÈÁ¿µÄÖع¹·½·¨
±¾½Úͨ¹ý¶ÔÊÓƵÖÐÏàËƵĶþάͼÏñ¿é½¨Á¢ÕÅÁ¿Ï¡ÊèÄ£ÐÍ£¬²¢²ÉÓÃITS×÷ΪÕÅÁ¿Ï¡ÊèÐÔ¶ÈÁ¿£¬Ìá³öÁËÒ»ÖÖ»ùÓÚITSµÄÊÓƵÖع¹Ä£ÐÍ¡£
¶ÔÓÚÿ¸ö¶þάͼÏñ¿éxim£¬ÔÚÒ»¸öZ¡ÁZ¡ÁD´óСµÄʱ¿ÕÁÚÓò´°¿ÚÖ´ÐÐk½üÁÚËÑË÷£¬ÕÒµ½ÓëÆäÏàËƵÄQ¸ö¶þάͼÏñ¿é£¬ÕâЩͼÏñ¿éµÄË÷Òý¼¯±íʾÈçÏ£º 


={mqi¡¬xim-xmqi¡¬22¡ÜT,q=1,2,¡­,Q}


ÆäÖÐT±íʾãÐÖµ¡£
ÓÃRmix=£ÛRm1ix,Rm2ix,¡­,Rmqix,¡­,RmQix£Ý¡ÊRp¡Áp¡ÁQ±íʾÓÉËùÓжþάÏàËÆͼÏñ¿é¶Ñµþ³ÉµÄÈý½×ÕÅÁ¿£¬Rmqi±íʾ´ÓÊÓƵÐòÁÐxÖÐÌáÈ¡µÄµÚmqi¸öͼÏñ¿éxmQi£¬QÊÇ´ÓÊÓƵÐòÁÐxÌáÈ¡ÑùÀý¿éµÄ×ÜÊý¡£
¶Ôÿһ×éÏàËƵÄͼÏñ¿éRmix²ÉÓÃITS½øÐÐÕÅÁ¿Ï¡ÊèÔ¼Êø£¬ÓÉÓÚʽ(3.55)ÖеÄL0·¶ÊýºÍµÍÖÈÏîÊÇ×éºÏÓÅ»¯ÎÊÌ⣬ºÜÄÑÇó½â£¬Òò´Ë½«ITSËɳÚΪ¶ÔÊýºÍµÄÐÎʽ£º 


(x^,L^mi)= minx,L¡¬y-¦µx¡¬22+¦Ç¡Æm¡ÆDi=1¦Â2¡¬Rmix-Lmi¡¬2F+¦Ë¡Ç3j=1P*ls(Lmi(j))+Pls(S)(3.57)



ÆäÖÐ,¦µÎª¸ÐÖª¾ØÕó£¬x=£ÛxT1,xT2,¡­,xTK£ÝT£¬y=£ÛyT1,yT2,¡­,yTK£ÝT£¬¦Ë=1-tt£¬¦Â=¦Ãt£¬Pls(A)=¡Æi1,i2,i3log(a1,i2,i3+¦Å)£¬P*ls(A)=¡Æjlog(¦Òj(A)+¦Å)£¬¦Òj(A)±íʾAµÄµÚj¸öÆæÒìÖµ¡£
ʽ(3.57)µÄµÚÒ»²¿·ÖÊDzвîÔ¼ÊøÏµÚ¶þ²¿·ÖÊÇͨ¹ýËÑË÷Èýά¿Õ¼äÖеÄÏàËÆÐÔ¶þάͼÏñ¿é½¨Á¢ITS³Í·£¡£²ÉÓÃÏàËƶþάͼÏñ¿é½¨Ä£±È½ö½«ÊÓƵ·Ö½â³ÉÈýάͼÏñ¿é¸üÓÐЧ£¬ÕâÊÇÒòΪ²ÉÓöþάͼÏñ¿é¸üÈÝÒײ¶×½ÊÓƵÖÐÏàËƵĽṹÐÅÏ¢¡£
ʽ(3.57)ÖеÄÄ¿±êº¯Êý¿ÉÒÔͨ¹ý½»Ìæµü´úÓÅ»¯ÊÓƵÐòÁÐXºÍÏ¡ÊèÕÅÁ¿Lmi½øÐÐÇó½â£¬ÏÂÃæÏêϸ½éÉÜÇó½â·½·¨¡£
1£© ¹Ì¶¨ÊÓƵÐòÁлָ´Ï¡ÊèÕÅÁ¿
¹Ì¶¨ÊÓƵÐòÁÐXÓÅ»¯Ï¡ÊèÕÅÁ¿Lmi£¬Ï൱ÓÚÔÚ³õʼ¹À¼ÆµÄÊÓƵÐòÁÐÉÏ£¬Ìáȡÿһ¸öÑùÀý¿éximµÄËùÓÐÏàËƿ飬ȻºóÇó½âÒÔÏÂÓÅ»¯ÎÊÌâµÃµ½Lmi¡ÊRP¡ÁP¡ÁQ£º 


Lmi= arg minLmi¦Â2¡¬Rmix-Lmi¡¬2F+¦Ë¡Ç3j=1P*ls(Lmi(j))+Pls(S)(3.58)



²ÉÓý»Ìæ·½Ïò³Ë×ÓËã·¨(Alternating Direction Method of Multipliers£¬ADMM)Çó½âÉÏÊöÎÊÌâ£Û38,39£Ý¡£Ê½(3.58)µÄÔö¹ãÀ­¸ñÀÊÈÕº¯ÊýΪ 


L¦Ì(S,M1,M2,M3,U1,U2,U3)=Pls(S)+¦Ë¡Ç3j=1P*ls(Mj(j))+

¦Â2¡¬Rmix-S¡Á1U1¡Á2U2¡Áu3¡¬2F+

¡Æ3j=1¡´S¡Á1U1¡Á2U2¡Á3U3-Mj,Pj¡µ+

¡Æ3j=1¦Ì2¡¬S¡Á1U1¡Á2U2¡Á3U3-Mj¡¬(3.59)



ÆäÖÐMj(j)=unfoldj(Mj)£¬Mjj=1,2,3ÊÇÈý¸ö¸¨ÖúÕÅÁ¿£¬Pjj=1,2,3ΪÀ­¸ñÀÊÈÕ³Ë×ÓÕÅÁ¿£¬¦ÌΪһ¸öÕý±êÁ¿£¬¾ØÕóUjÂú×ãUTjUj=I£¬ªÐj=1,2,3¡£
ʽ(3.59)µÄÓÅ»¯°üÀ¨ÒÔϵü´ú¹ý³Ì¡£
£¨1£© ¹Ì¶¨ÆäËû²ÎÊý£¬Í¨¹ýÇó½âÒÔÏÂ×ÓÎÊÌâ¸üÐÂÕÅÁ¿S£º 


minSbPls(S)+12¡¬S-O¡Á1UT1¡Á2UT2¡Á3UT3¡¬2FF


ÆäÖÐ,b=1¦Â+3¦Ì£¬O=¦Â(Rmix)+¡Æj(¦Ì-Pj)¦Â+3£¬¸Ã¹«Ê½¾ßÓбÕʽ½â£Û40£Ý£º 


S+=Db,¦Å(O¡Á1UT1¡Á2UT2¡Á3UT3)


ÆäÖÐDb,g¡¤ÎªãÐÖµ²Ù×÷£º 


Db,¦Å(x)=0,c2¡Ü0

c1+c22sign(x),c2>0


ÆäÖÐ,c1=|x|-¦Å£¬c2=c12-4(b-¦Å|x|)¡£
£¨2£©  ¹Ì¶¨ÆäËû²ÎÊý£¬¸üÐÂUjj=1,2,3£º 


U+j=BjCTj


ÆäÖÐA+j=BjDCTjÊÇAjµÄSVD·Ö½â¡£µ±j=1ʱ£¬A1=(O(1)unfold1(S¡Á2U2¡Á3U3))¡£
£¨3£© ¹Ì¶¨ÆäËû²ÎÊý£¬¸üÐÂMjj=1,2,3£º 


minMjajP*ls(Mj(j))+12¡¬L+1¦ÌPj-Mj¡¬2F


ÆäÖÐaj=¦Ë¦Ì¡Çk¡ÙjP*ls(Mk(k))£¬L=S¡Á1U1¡Á2U2¡Á3U3¡£Mj¿ÉÒÔͨ¹ýÒÔϹ«Ê½¸üУº 


M+j= foldjV1¦²akVT2


Æä ÖÐ ¦²ak=diag£ÛDak,¦Å(¦Ò1),Dak,¦Å(¦Ò2),¡­,Dak,¦Å(¦Òn)£Ý£¬V1diag£Û¦Ò1,¦Ò2,¡­,¦Òn£ÝVT2Ϊunfoldj(L+Pj/¦Ì)µÄSVD·Ö½â¡£
£¨4£© ¸üгË×Ó£º 


¦Ì+=¦Ñ¦Ì


2) ¹Ì¶¨Ï¡ÊèÕÅÁ¿Çó½âÊÓƵÐòÁÐ
¹Ì¶¨ÉÏÊö¼ÆËã³öµÄLmi£¬¿ÉÒÔͨ¹ýÏÂÃæµÄ×ÓÓÅ»¯ÎÊÌâÇó½âÊÓƵÐòÁÐx£º 


x= arg minx¡¬y-¦µx¡¬22+¦Ç¡Æm¡ÆDi=1¡¬Rmix-Lmi¡¬2F


Æä¾ßÓбÕʽ½â£º 


x=¦µT¦µ+¦Ç¡Æm¡ÆDi=1RTmiRmi-1¦µTy+¦Ç¡Æm¡ÆDi=1RTmiLmi


ÆäÖÐRTmiRmi=¡ÆRTmqiRmqi¡£
ÔÚÕâÀ¾ØÕóÌ«´óÒÔÖÁÓÚÎÞ·¨Ö±½ÓÇóÆäÄæ¾ØÕó£¬Òò´Ë²ÉÓ÷ֿéCSµÄ˼Ïë¶ÔÆä½øÐмò»¯Çó½â£Û44£Ý¡£Èçͼ3.13Ëùʾ£¬¸ÐÖª¾ØÕó¦µÊÇÒ»¸ö¿é¶Ô½Ç¾ØÕ󣬦µ=diag£Û¦µ1,¦µ2,¡­,¦µk,¡­,¦µK£Ý£¬ÆäÖжԽǿé¾ØÕó¦µKµÄάÊýÈ¡¾öÓÚ¼ÆËã»úµÄ¼ÆËãÄÜÁ¦¡£LmiÊÇÓëxjmÏàËƵÄͼÏñ¿é×éµÄÒ»¸öÏ¡ÊèÕÅÁ¿½üËÆ£¬ÒòΪÏàËÆͼÏñ¿éµÄË÷ÒýÒÑÖª£¬ËùÒÔ¿ÉÒÔͨ¹ýÔÚÿ¸öÏñËØ´¦¾ÛºÏLmiÀ´»ñµÃÕû¸öÊÓƵµÄÏ¡ÊèÕÅÁ¿²¿·ÖL£¬¶ÔӦΪԭʼÊÓƵÐòÁдóС¡£¿ÉÒÔ·Ö½âΪһϵÁжÀÁ¢µÄ×ÓÎÊÌ⣺ 


xk= arg minxk¡¬yk-¦µkxk¡¬22+¦Ç¡¬xk-Lk¡¬2F 


ÆäÖÐLk=Lmi(k)ÊÇ´ÓLÖÐÌáÈ¡µÄ¶ÔÓ¦ÈýάÊÓƵ¿éxmλÖã¬xkºÍLk¾ù±»ÏòÁ¿»¯¡£¸Ã×ÓÎÊÌâºÜÈÝÒ×Çó½â£º 


xk=((¦µk)T¦µk+¦ÇI)-1((¦µk)Tyk+¦ÇLmi(k))


µ±Çó½âÍêËùÓÐÈýάÊÓƵ¿éxk1¡Ük¡ÜK£¬¿ÉÒÔÖØн«ÆäÕûºÏΪÕû¸öÊÓƵÐòÁС£ 
»ùÓÚITSµÄÊÓƵÖع¹¹ý³Ì¿É¼ûËã·¨3.7£¬¿ò¼Üͼ¼ûͼ3.13¡£


±í3.7»ùÓÚITSµÄÊÓƵÖع¹¹ý³Ì



»ùÓÚITSµÄÊÓƵÖع¹
³õʼ»¯£º 

²½Öè1,¹À¼Æ³õʼÊÓƵÐòÁÐx^ 

²½Öè2,ÉèÖòÎÊý¦Ë£¬¦Ç£¬Q

Ñ­»·£º 




Foriter=1,2£¬¡­,max_iter do

²½Öè3£¬¶ÔÓÚÿһ¸öÑùÀýͼÏñ¿éxjm£¬ÔÚ Z ¡ÁZ ¡ÁD ´óСµÄ¾Ö²¿´°¿ÚÖÐËÑË÷kª²½üÁÚͼÏñ¿é£¬µÃµ½Ã¿¸öÑùÀýͼÏñ¿éµÄÏàËÆ¿é×éLmi£» 

²½Öè4£¬Forÿ¸öÏàËÆ¿é×éLmido

//ͨ¹ýADMMËã·¨Çó½â

While not convergence do

¸üÐÂS

¸üÐÂUjj=1,2,3

¸üÐÂMjj=1,2,3 

¸üгË×Ó¦Ì+=¦Ñ¦Ì£» 

End while

End for

²½Öè5£¬Í¨¹ýÔÚÿ¸öÏñËصã¾ÛºÏLmiµÃµ½Ï¡ÊèÕÅÁ¿L£» 

²½Öè6,¹Ì¶¨Ï¡ÊèÕÅÁ¿Lͨ¹ýÇó½â¸üÐÂx£º 

Fork=1,2,¡­,K do

¼ÆËãµÚk¸öÈýάÊÓƵͼÏñ¿é£º 

xk=((¦µk)T¦µk+¦ÇI)-1((¦µk)Tyk+¦ÇLmi(k))

End for

²½Öè7,ͨ¹ýÔÚÿ¸öÏñËصã¾ÛºÏxk1¡Ük¡ÜKµÃµ½¹À¼ÆÊÓƵÐòÁÐx^£» 

Endfor

Êä³ö£º ×îÖÕÖع¹ÊÓƵÐòÁÐx^¡£



ͼ3.13CACTI µÄ·Ö¿éÊÓƵ¸ÐÖª¿ò¼Üͼ


2.  »ùÓÚ¸ß˹ÁªºÏÏ¡ÊèÄ£Ð͵ÄÊÓƵÖع¹
ÓÉËã·¨3.1µÄ²½Öè1ÖпÉÒÔ¿´³ö£¬»ùÓÚITSµÄÖع¹Ä£ÐÍÐèÒªÒ»¸ö³õʼ¹À¼ÆµÄÊÓƵÐòÁС£±¾½Úͨ¹ý²ÉÓÃÏàÁÚÖ¡Ö®¼äµÄÏàËÆÐÔ£¬Ìá³öÁËÓÃÓÚ³õʼÊÓƵ¹À¼ÆµÄ¸ß˹ÁªºÏÏ¡ÊèÄ£ÐÍ(Gaussian Joint Sparsity model£¬GJS)¡£
1) ·Ö¶ÎÏßÐÔ¹À¼Æ
YuµÈÈ˽éÉÜÁ˲ÉÓ÷ֶÎÏßÐÔ¹À¼Æ(Piecewise Linear Estimations£¬PLE)½â¾ö¶þάͼÏñÄæÎÊÌâµÄÒ»°ã¿ò¼Ü£Û41£Ý¡£»ùÓÚ»ìºÏ¸ß˹ģÐÍ(Mixture Gaussian Model,GMM)µÄPLE·½·¨Í¨¹ý×î´ó»¯ºóÑé¸ÅÂÊÀ´Çó½â£¬ÎªÄæÎÊÌâÌṩÁËÒ»°ãÇÒÓÐЧµÄ½â¾ö·½°¸£¬ÔÚÒÑ֪ÿ¸öͼÏñ¿éµÄѹËõ²âÁ¿Ê±£¬¿ÉÒÔͨ¹ýάÄÉÂ˲¨Æ÷Öع¹¶þάͼÏñ¿é£Û42£Ý¡£
ÔÚPLEµÄ³õʼ»¯ÖУ¬K-1¸ö¸ß˹·ÖÁ¿¶ÔÓ¦ÓÚ´Ó0µ½¦Ð¾ùÔȲÉÑùµÄK-1¸ö½Ç¶È£¬ÕâЩK-1¸ö·ÖÁ¿µÄPCA¿Õ¼äÊÇ»ùÓÚÑØÏàͬ·½ÏòµÄºÏ³ÉºÚ°×±ßԵͼÏñѧϰµÃµ½µÄ£¬µÚK¸ö¸ß˹·ÖÁ¿²ÉÓÃDCT×÷Ϊ»ùÓÃÀ´²¶»ñ¸÷ÏòͬÐÔͼÏñͼ°¸¡£
2£© ¸ß˹ÁªºÏÏ¡ÊèÄ£ÐÍ
PLEÊÇÒ»ÖÖÖع¹¶þάͼÏñµÄ·½·¨£¬µ«²¢²»ÄÜÖ±½ÓÍØÕ¹µ½ÊÓƵÖع¹ÁìÓò£¬Ò»·½ÃæÊÇÒòΪËü¶¨ÒåµÄ²»ÊÇÈýάͼÏñ¿é£» ÁíÒ»·½ÃæÊÇÒòΪËü²»Äܲ¶×½ÊÓƵ֡¼äµÄÏàËÆÐÔ¡£½ÓÏÂÀ´£¬ÎªÁ˽â¾öÕâÁ½¸öÎÊÌ⣬Õë¶ÔÈýάÊÓƵ¿é»ùÓÚÖ¡¼äµÄÏàËÆÐÔÌá³öÁËÒ»ÖÖÊÓƵ³õʼ»¯Öع¹·½·¨¡£
Óë¶þάͼÏñÏà±È£¬ÊÓƵÐòÁÐÖаüº¬¸ü¶àµÄÏÈÑ飬ÀýÈç¾ßÓÐÏàËƽṹµÄÏàͬ³¡¾°ÒÔÏ൱´óµÄ¸ÅÂʳöÏÖÔÚÏàÁÚÖ¡µÄÏàͬλÖô¦¡£¶ÔÓÚ²»Í¬ÁÚÖ¡µÄÏàͬλÖã¬Èç¹ûËüÃǾßÓÐÏàËƵĽṹ£¬ÄÇôËüÃDZãÊÇÁªºÏÏ¡ÊèµÄ£¬Ôò¿ÉÒÔͨ¹ýÏàͬµÄ·Ö²¼½¨Ä£¡£Òò´Ë£¬¼ÙÉèÔÚÏàÁÚÖ¡µÄÏàͬλÖÃÉϵÄͼÏñ¿é·þ´ÓÏàͬµÄ¸ß˹·Ö²¼£¬Ôò³Æ֮Ϊ¸ß˹ÁªºÏÏ¡Êè(Gaussian Joint Sparse,GJS)¡£
ÔÚCACTI²âÁ¿Ï£¬¹Û²âµ½µÄ¶þάͼÏñ¿éym1¡Üm¡ÜMÈçÏ£º 


ym=¦µmxm+¦Åm=¡ÆDi=1¦µimxim+¦Åm£¬ªÐm=1,2£¬¡­,M 


ÆäÖÐxm=£Ûx1Tm,x2Tm,¡­,xDTm£ÝTÊÇ´ýÖع¹µÄÈýάÊÓƵ¿é£¬ÔëÉùÏòÁ¿¦Åm¡ÊRp¡£
´Ë¹«Ê½¿ÉÒÔ¿´×÷Ò»¸öÄæÎÊÌâ¡£
¼ÙÉè¶þάͼÏñ¿é¿ÉÒÔͨ¹ýÒ»¸ö»ìºÏ¸ß˹·Ö²¼±íʾ£¬ÉèÓÐK¸ö¸ß˹·Ö²¼{N(¦Ìk,¦²k)}1¡Ük¡ÜK£¬ÕâK¸ö¸ß˹·Ö²¼¿ÉÒÔÓÉPCA³õʼ»¯·½²îÉú³É¡£¶þάͼÏñ¿éximÒÔ¶ÀÁ¢µÈ¸ÅÂÊ·þ´ÓÈÎÒ»¸ß˹·Ö²¼¡£¼ÙÉèͼÏñ¿éxim·þ´ÓµÚki¡Ê£Û1,K£Ý¸ö¸ß˹·Ö²¼£¬¼ÇΪ£º N(xim|¦Ìkim,¦²kim)£¬Æä¸ÅÂÊÃܶȺ¯Êý±íʾÈçÏ£º 


p(xim)=(2¦Ð)-N/2¦²kim-1/2exp-12(xim-¦Ìkim)T(¦²kim)-1(xim-¦Ìkim)


ÈýάÊÓƵ¿éÓÉÏàÁÚÖ¡ÏàͬλÖõĶþάͼÏñ¿é×é³Éxm=£Ûx1Tm,x2Tm,¡­,xDTm£ÝT£¬¼ÙÉèximÖ®¼äÏ໥¶ÀÁ¢£¬ÔòÈýάÊÓƵ¿éµÄ¸ÅÂÊÃܶȺ¯ÊýΪ 


p(xm)=¡ÇDi=1p(xim)=N(xm|¦Ìkm,¦²-km)


ÆäÖоùÖµºÍЭ·½²î¾ØÕó·Ö±ðΪ


¦Ìkm=¦Ìk1m
¦Ìk2m
¦ó
¦ÌkDm£¬¦²-km=¦²k1m00¡­00
0¦²k2m0¡­¦ó¦ó
¦ó00ª÷00
000¡­0¦²kDm


ÕâÀïÊǦÌk,¦²kµÄÒ»¸ö×éºÏ£¬ÓÉÓÚ¼ÙÉèÁËK¸ö¸ß˹·Ö²¼{N(¦Ìk,¦²k)}1¡Ük¡ÜK£¬Òò´Ë(¦Ìkm,¦²-km)µÄ×éºÏµÄ¿ÉÄÜÐÔÓÐKDÖÖ£¬¿É¼ûΪÈýάÊÓƵ¿éxmÑ¡ÔñÒ»¸öºÏÊʵĸß˹·Ö²¼ÊÇÒ»¸ö¸´ÔÓµÄ×éºÏÓÅ»¯ÎÊÌâ¡£
ÖµµÃ×¢ÒâµÄÊÇ£¬Ò»°ãÇé¿öÏÂÔÚÏàÁÚÖ¡µÄÏàͬλÖõĶþάͼÏñ¿é¾ßÓÐÏàͬµÄ½á¹¹£¬¿ÉÒÔͨ¹ýÏàͬµÄ¸ß˹·Ö²¼½¨Ä££¬³ÆÆäΪ¸ß˹ÁªºÏÏ¡Êè¡£ÓÉÓÚÈýάÊÓƵ¿éxmÖÐËùÓеÄxim·þ´Óͬһ¸ö¸ß˹·Ö²¼Õâ¸öÌØÊâÊôÐÔ£¬KDÖÖ×éºÏµÄ¿ÉÄÜÐÔ½µÎªKÖÖ£¬ÈýάÊÓƵ¿éKÖÖ¸ß˹·Ö²¼Îª{N(¦Ì-k,¦²-k)}1¡Ük¡ÜK£º 


¦Ì-k=¦Ìk
¦Ìk
¦ó
¦Ìk£¬¦²-k=¦²k00¡­00

0¦²k0¡­¦ó¦ó

¦ó00ª÷00

000¡­0¦²k


3£© »ùÓÚPLEµÄGJSÄ£ÐÍÇó½â
ÔÚ»ñµÃ±íʾÈýάµÄ¸ß˹ģÐÍÖ®ºó£¬²ÉÓÃPLEÀ´Çó½âGJSÄ£ÐÍ£Û43£Ý¡£ÓÉÓÚͼÏñ¿éµÄ¾ùÖµ×Ü¿ÉÒÔ¹éÒ»»¯£¬Òò´ËΪÁ˼ò»¯·ûºÅ£¬¼ÙÉè¸ß˹·Ö²¼¾ßÓÐÁã¾ùÖµÏòÁ¿£¬¼´¦Ìk=0£¬¶ÔÓÚµÚm¸öÈýάÊÓƵ¿é£¬ÐźŹÀ¼ÆºÍÄ£ÐÍÑ¡Ôñͨ¹ý×î´ó»¯¶ÔÊý¸ÅÂÊÀ´¼ÆË㣺 


(xm,k¡«m)= arg maxxm,k ln p(xm|ym)

= arg maxxm,k(ln p(ym|xm)+lnp(xm))

= arg maxxm,k(lnN(ym|¦µmxm,¦Ò2IP)+lnN(xm|0,¦²-k))

= arg minxm,k(¡¬ym-¦µmxm¡¬2+¦Ò2xTm(¦²-k)-1xm+¦Ò2ln¦²-k)


ÆäÖЦÅm¡«N(0,¦Ò2IP)£¬IPΪpάµ¥Î»¾ØÕó¡£
Ê×Ïȹ̶¨k¼ÆËãxm£¬¼´µÚk¸ö¸ß˹·Ö²¼Ä£ÐÍÏÂxm¡«N(0,¦²-k)¹À¼ÆÊÓƵ¿é£º 


xkm=arg  minxm(¡¬¦µmxm-ym¡¬2+¦Ò2(xm)T¦²--1k(xm))


²ÉÓÃMAPµÃµ½xkmµÄ¹À¼ÆÖµ£º 


xkm=¦²-k¦µTm(¦µm¦²-k¦µTm+¦Ò2Ip)-1ym


¹À¼ÆÍêËùÓиß˹·Ö²¼Ä£ÐÍ{N(¦Ì-k,¦²-k)}1¡Ük¡ÜK ºó£¬×îÓŵÄÄ£ÐÍkmΪËùÓÐÄ£ÐÍÖоßÓÐ×î´ó¸ÅÂʵÄÄ£ÐÍ£º 


k¡«m=arg mink(¡¬¦µmxkm-ym¡¬2+¦Ò2(xkm)T¦²--1k(xkm)+¦Ò2log¦²-k)


×îºó£¬ÈýάÊÓƵÐźÅÓÉ×îÓÅÄ£ÐÍkm¹À¼ÆµÃµ½ 


xm=xkmm


ÕâÖÖ½«ÊÓƵ½¨Ä£Îª¸ß˹ÁªºÏÏ¡ÊèÄ£ÐÍ£¬²¢²ÉÓÃPLE·½·¨Çó½âµÄ·½·¨³ÆΪGJS_PLEËã·¨£¬ÓÃÓÚÊÓƵÖع¹µÄGJS_PLEËã·¨µÄϸ½Ú¼û±í3.8¡£


±í3.8GJS_PLEËã·¨



GJS_PLEËã·¨
²½Öè1,²ÉÓÃPLEÖеijõʼ»¯·½·¨»ñµÃK¸ö¸ß˹·Ö²¼{N(¦Ì-k,¦²-k)}1¡Ük¡ÜK£¬K-1¸ö¸ß˹·ÖÁ¿¶ÔÓ¦´Ó0µ½¦Ð¾ùÔȲÉÑùµÄK-1¸ö½Ç¶È£¬µÚK¸ö¸ß˹·ÖÁ¿²ÉÓÃDCT»ù¡£

²½Öè2,¸ù¾Ý¶þάͼÏñ¿éµÄ¸ß˹ģÐÍ{N(¦Ì-k,¦²-k)}1¡Ük¡ÜK½¨Á¢ÈýάÊÓƵ¿éµÄÄ£ÐÍ{N(¦Ì-k,¦²-k)}1¡Ük¡ÜK£º 
Ðø±í



¦Ì-km=¦Ìk1m
¦Ìk2m
¦ó
¦ÌkDm£¬¦²-km=¦²k1m00¡­00

0¦²k2m0¡­¦ó¦ó

¦ó00ª÷00

000¡­0¦²kDm

¹À¼Æÿһ¸öÈýάÊÓƵ¿éµÄԭʼÐźź͸ß˹ģÐÍ£º 

²½Öè3,ͨ¹ý¹À¼Æÿ¸öÄ£ÐÍkϵÄÐźţº 

xkm=¦²-k¦µTm¦µm¦²-k¦µTm+¦Ò2IP-1ym

²½Öè4,ͨ¹ýÑ¡Ôñ×îÓÅÄ£ÐÍk¡«m£º 

k¡«m=arg mink(¡¬¦µmxkm-ym¡¬2+¦Ò2(xkm)T¦²--1k(xkm)+¦Ò2log¦²-k)

²½Öè5,ͨ¹ý×îÓÅÄ£ÐÍk¡«m¹À¼ÆÈýάÊÓƵ¿é£º xm=xkmm¡£

4) ·ÂÕæʵÑé¼°½á¹û·ÖÎö
±¾½Ú²ÉÓÃËÄÖֶԱȷ½·¨½øÐзÖÎö£¬Ò»ÖÖÊÇÔÚÎÄÏ×£Û42£ÝÖÐÌáµ½µÄGMM£¬ÁíÍâÈýÖÖÊÇÓë±¾ÕÂÌá³öËã·¨Ïà½áºÏµÄ·½·¨£¬¾ßÌåÈçÏ£º 
GMM¡ª¡ªYang µÈÈËÌá³öµÄ»ìºÏ¸ß˹ģÐÍÇó½âËã·¨£Û42£Ý¡£
GJS_PLE¡ª¡ª»ùÓÚ¸ß˹ÁªºÏÏ¡ÊèµÄÊÓƵÐòÁгõʼ»¯Öع¹·½·¨¡£
GMM_ITS¡ª¡ª²ÉÓÃGMMÀ´»ñµÃ×î³õµÄÊÓƵÐòÁУ¬È»ºó²ÉÓûùÓÚITSµÄÄ£ÐÍ»ñµÃ×îÖÕµÄÖع¹ÊÓƵÐòÁС£
GJS_PLE _ITSV¡ª¡ªÕâÊDZ¾Õ½éÉܵÄÊÓƵÐòÁÐÖع¹µÄÍêÕûËã·¨£¬¸ù¾ÝGJS_PLEËã·¨µÃµ½³õʼ»¯ÊÓƵÐòÁУ¬È»ºó²ÉÓûùÓÚITSµÄÄ£ÐÍ»ñµÃ×îÖÕµÄÖع¹ÊÓƵÐòÁС£
(1) ²ÎÊýÉèÖá£
¶ÔÓÚGJS_PLE·½·¨£¬K=20¸ö¸ß˹·Ö²¼{N(¦Ì-k,¦²-k)}1¡Ük¡ÜKÓÉYuµÈÈËÌá³öµÄPCA³õʼ»¯·½·¨Éú³É£Û41£Ý£¬Nx¡ÁNy´óСµÄ²âÁ¿£¬ÒÔ²½³¤ÿðþ
½›ͨ¹ýˮƽºÍ´¹Ö±»¬¶¯À´·Ö¸î³ÉһϵÁÐÖصþµÄ´óСΪP¡ÁPµÄ¿é£¬ÆäÖÐv¡Ê{1,2£¬¡­£¬P}¡£ÔÚʵÑéÖУ¬¶Ô¼ò±ãÐÔºÍ׼ȷÐÔ½øÐÐÕÛÖУ¬É趨v=P/2£¬P=8£¬¸ß˹ÔëÉù·þ´Ó¦Åm¡«N(0,¦Ò2IP)£¬¦Ò=0.01£¬ÕýÔò»¯²ÎÊý¦Ç=0.5¡£
ÕÅÁ¿Ï¡ÊèÄ£Ð͵ÄÖ÷Òª²ÎÊýÉèÖÃÈçÏ£º ¿é´óСP¡ÁP=8¡Á8£¬Ã¿¸öÑùÀý¿éÔÚZ¡ÁZ¡ÁD=25¡Á25¡ÁD´óСµÄ´°¿ÚÖÐÑ¡ÔñÓëÖ®ÏàËƵÄQ=50¸öͼÏñ¿é£¬ÕýÔò»¯²ÎÊý¼òµ¥µØÉèÖÃΪ¦Ë=5£¬¦Ñ=1.2£¬¦Ç=0.5£¬ÆäËû²ÎÊý²ÎÕÕÎÄÏ׵IJ¹³ä²ÄÁÏ£Û38£Ý¡£ÎªÁ˼õÉÙ¼ÆË㸴ÔӶȣ¬ÊµÑéÖÐÑØˮƽºÍ´¹Ö±·½Ïòÿ¸ô5¸öÏñËØÌáÈ¡Ò»¸öÑùÀýͼÏñ¿é¡£
(2) ʵÑé½á¹û¡£
ͨ¹ýÎå¸öÊÓƵ½øÐÐCACTI¹Û²â¼°Öع¹£¬ÉèÖÃʱ¼äѹËõ±ÈD=8£¬Îå¸öÊÓƵ·Ö±ðÊÇ£º TrafficÊÓƵ£¬´óСΪ256¡Á256¡Á96ÏñËØ£» WindmillÊÓƵ£¬´óСΪ256¡Á256¡Á128ÏñËØ£» DroplightÊÓƵ£¬´óСΪ288¡Á352¡Á500ÏñËØ£» WheelÊÓƵ£¬´óСΪ288¡Á352¡Á602ÏñËØ£» FountainÊÓƵ£¬´óСΪ288¡Á352¡Á782ÏñËØ¡£
ͼ3.14ºÍͼ3.15¡¢Í¼3.16ºÍͼ3.17¡¢Í¼3.20ºÍͼ3.21¡¢Í¼3.22ºÍͼ3.23¡¢Í¼3.24ºÍͼ3.25·Ö±ðչʾÁËGMM¡¢GJS_PLE¡¢GMM_ITSºÍGJS_PLE_ITS¶Ô Traffic¡¢Windmill¡¢Droplight¡¢Wheel¡¢FountainµÚ1Ö¡ºÍµÚ8Ö¡µÄÖع¹½á¹ûͼ¡£´ÓʵÑé½á¹û¿ÉÒÔ¿´³ö£¬GJS_PLE_ITSËã·¨¿ÉÒÔÌṩ¸ßÊÓ¾õÖÊÁ¿£¬ÓÐЧµØ½µµÍÁ˱ßÔµµÄÔëÒô¡£»ùÓÚGJS_PLEµÄ³õʼ»¯
Öع¹Ëã·¨¿ÉÒÔ»ñµÃ±ÈGMMËã·¨¸ü¾«È·µÄ±ßÔµ£¬Äܹ»¶ÔÊÓƵÖÐÒƶ¯³¡¾°µÄ±ßÔµ½øÐнϺõĹÀ¼Æ£¬Ëù»ñµÃµÄ³õʼͼÏñ½á¹¹ÐÔ¸üÇ¿¡£ÓëûÓÐÕÅÁ¿Ï¡ÊèÔ¼ÊøÄ£Ð͵ķ½·¨Ïà±È£¬GMM_ITSºÍGJS_PLE_ITSÖؽ¨½á¹û¸ü¼Óƽ»¬ºÍÒ»Ö£¬ÊÓƵÖÐÒƶ¯Ä¿±êµÄ±ßÔµ½á¹¹¸ü¼ÓÇåÎú£¬ÀýÈçÒƶ¯



ͼ3.14TrafficµÚ1Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ




ͼ3.15TrafficµÚ8Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ





ͼ3.16WindmillµÚ1Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ





ͼ3.17WindmillµÚ8Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ




ͼ3.18²»Í¬Öع¹·½·¨¶ÔTrafficÊÓƵµÄÖع¹PSNR




ͼ3.19²»Í¬Öع¹·½·¨¶ÔTrafficÊÓƵµÄÖع¹SSIM 




ͼ3.20DroplightµÚ1Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ 




ͼ3.21DroplightµÚ8Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ



µÄÆû³µ¡¢×ª¶¯µÄ·çÒ¶¡¢ÐýתµÄ³µÂÖºÍÅçÓ¿µÄȪˮ¡£ÁíÒ»·½Ã棬¾­¹ý¶Ô±Èͼ3.16ºÍͼ3.17Öв»Í¬Öع¹·½·¨ÏÂWindmill·çÒ¶µÄλÖ㬿ÉÒÔ¿´³öGMMË㷨ģºýÁËÒƶ¯Ä¿±êµÄÈ·¶¨Î»Ö㬶øGJS_PLEÄܹ»½ÏΪ¾«È·µØչʾ³öÒƶ¯Ä¿±êµÄλÖá£Í¼3.18ºÍͼ3.19ΪTrafficÊÓƵÊý¾Ý»æÖÆÁËGMM¡¢GJS_PLE¡¢GMM_ITSºÍGJS_PLE_ITS·½·¨µÄPSNRºÍSSIMÕÛÏßͼ¡£´Óͼ3.20ºÍͼ3.21µÄ¶Ô±ÈÖÐÒ²¿ÉÒÔ¿´³ö£¬³õʼ»¯Öع¹Ëã·¨GJS_PLEÄܹ»¸ü׼ȷµØÖع¹°Ú¶¯µõµÆµÄ±ßÔµÇøÓò¡£×ÛºÏÀ´Ëµ£¬ÔÚGJSª²PLEÄ£ÐÍÌṩ·½Ïò½á¹¹ÐÔ½ÏÇ¿µÄ³õʼÖع¹Í¼Ïñ»ù´¡ÉÏ£¬²ÉÓûùÓÚITSµÄÖع¹Ä£ÐÍ£¬Äܹ»ÓÐЧÌá¸ßͼÏñÖع¹ÖÊÁ¿£¬¶þÕßÏà½áºÏÄܹ»ÓÐЧÒÖÖÆÒƶ¯±ßÔµµÄÔëÉù£¬ÌṩÇåÎú¿É¿¿µÄÒƶ¯±ßÔµ£¬±£Ö¤¹â»¬ÇøÓòµÄÒ»ÖÂÐÔ¡£


ͼ3.22WheelµÚ1Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ 




ͼ3.23WheelµÚ8Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ





ͼ3.24FountainµÚ1Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ 




ͼ3.25FountainµÚ8Ö¡²»Í¬Öع¹·½·¨µÄ½á¹ûͼ


±í3.9ºÍ±í3.10·Ö±ðչʾÁËTrafficÊÓƵºÍWindmillÊÓƵµÄµÚ1~8Ö¡Öع¹µÄ·åÖµÐÅÔë±È(Peak Signal to Noise Ratio,PSNR)¼°½á¹¹ÏàËÆÐÔ¶ÈÁ¿Ö¸±êSSIM(Structural Similarity)¡£¿ÉÒÔ¿´³ö£¬ËùÓÐÈýÖÖÌá³öµÄ·½·¨¡ª¡ªGJS_PLE£¬GMM_ITSºÍGJS_PLE_ITSµÄPSNR¼°SSIMÖµ¾ù±ÈGMMËã·¨µÄÏàÓ¦Öµ¸ß¡£¶ÔÓÚTrafficÊÓƵÀ´Ëµ£¬»ùÓÚÕÅÁ¿Ï¡ÊèµÄÔ¼Êø¶ÔÊÓƵµÄÖÊÁ¿ÌáÉýЧ¹û¸ü´ó£» ¶ÔÓÚWindmillÊÓƵÀ´Ëµ£¬GJS_PLEÄܸüÓÐЧ²¶×½ÊÓƵÖÐÒƶ¯µÄ½á¹¹¡£


±í3.9TrafficµÚ1~8Ö¡Öع¹½á¹û±í



Ö¡Êý

·½·¨/PSNR(SSIM)
GMM
GJS_PLE
GMM_ITS
GJS_PLE_ITS

1
21.06(0.6908)
21.63(0.6857)
23.79(0.7949)
25.01(0.8235)
2
21.36(0.7057)
21.77(0.7051)
24.04(0.8029)
24.88(0.8241)
3
21.28(0.7088)
21.68(0.7069)
23.88(0.8033)
24.60(0.8230)
4
21.37(0.7157)
21.56(0.7110)
23.84(0.8018)
24.57(0.8233)
5
21.22(0.7092)
21.69(0.7128)
23.67(0.7996)
24.43(0.8242)
6
21.24(0.7101)
21.47(0.7074)
23.80(0.8013)
24.47(0.8223)
7
21.07(0.7062)
21.35(0.6943)
23.58(0.7961)
24.34(0.8202)
8
20.68(0.6856)
21.16(0.6795)
23.27(0.7854)
24.17(0.8118)



±í3.10WindmillµÚ1~8Ö¡Öع¹½á¹û±í



Ö¡Êý

·½·¨/PSNR(SSIM)
GMM
GJS_PLE
GMM_ITS
GJS_PLE_ITS

1
24.41(0.8217)
27.50(0.8847)
24.79(0.8357)
28.51(0.9102)
2
25.28(0.8316)
27.44(0.8879)
25.61(0.8441)
28.48(0.9131)
3
25.50(0.8349)
27.79(0.8945)
25.85(0.8471)
28.85(0.9191)
4
25.48(0.8385)
27.77(0.8966)
25.83(0.8515)
28.86(0.9216)
5
25.70(0.8407)
27.65(0.8958)
26.02(0.8522)
28.74(0.9209)
6
25.52(0.8411)
27.82(0.8932)
25.84(0.8525)
28.87(0.9186)
7
25.19(0.8361)
27.78(0.8887)
25.54(0.8488)
28.86(0.9136)
8
24.23(0.8204)
27.84(0.8848)
24.61(0.8336)
28.88(0.9105)

±¾½ÚÔÚCACTI²âÁ¿Ï£¬Ìá³öÁËÒ»ÖÖ»ùÓÚITSÊÓƵÖؽ¨Ëã·¨¡£Í¨¹ý½«ÊÓƵÖع¹ÎÊÌâת»¯ÎªÕÅÁ¿Ï¡Êè½üËÆÎÊÌ⣬¸ÃËã·¨¾ßÓÐÒÔÏÂÓŵ㣺 
(1) ÕÅÁ¿Ï¡ÊèÄ£ÐÍ£¬³ä·ÖÀûÓÃÖ¡ÄÚºÍÖ¡¼ä(¿Õ¼äºÍʱ¼ä)µÄÕÅÁ¿Ï¡Êè³Í·£À´²¶»ñÊÓƵµÄ×ÔÏàËÆÐÔ£¬²¢ÇÒITS¶ÈÁ¿±»ÓÃ×÷ÕÅÁ¿Ï¡Êè¶ÈÁ¿£¬Ëü½áºÏÁËTuckerºÍCP·Ö½â¶ÔÓÚÕÅÁ¿Ï¡Êè¶ÈÁ¿µÄÓÅÊÆ£¬ÔÚ»Ö¸´ÎÆÀí¡¢±ßÔµµÈ½á¹¹·½Ãæ±íÏÖÁ¼ºÃ¡£
(2) Ìá³öÁËGJSÄ£ÐÍÓÃÀ´Öع¹³õʼ»¯ÊÓƵÐòÁУ¬½«¸ß˹»ìºÏÄ£ÐÍÓëÁªºÏÏ¡ÊèÄ£ÐͽáºÏÆðÀ´½¨Á¢¸ß˹ÁªºÏÏ¡ÊèÄ£ÐÍÀ´Çó½âÈýάÊÓƵ£¬¸ÃÄ£ÐÍÄܹ»ºÜºÃµØ·´Ó³ÊÓƵ֡¼äµÄÏàËÆÐÔ¡£
×îºóʵÑé½á¹ûºÍ·ÖÎö±íÃ÷£¬ËùÌá³öµÄGJS³õʼ»¯·½·¨ºÍ»ùÓÚITSµÄÊÓƵÖع¹·½·¨¾ßÓÐÁ¼ºÃµÄ±íÏÖÐÔÄÜ¡£
3.5±¾ÕÂС½á
±¾Õ½éÉÜÁ˼¸ÖÖ¾­µäµÄ½á¹¹»¯Ï¡ÊèѹËõ¸Ð֪ģÐͼ°ÆäÖع¹Ëã·¨£¬Ö÷ÒªÓпéÏ¡ÊèѹËõ¸ÐÖª·½·¨¡¢ÁªºÏÏ¡ÊèѹËõ¸ÐÖª·½·¨£¬ÆäÖпéÏ¡ÊèѹËõ¸ÐÖª°üÀ¨¿é»ìºÏ·¶ÊýÓÅ»¯Ëã·¨¡¢¿éÕý½»Æ¥Åä×·×ÙËã·¨¡¢¿éÆ¥Åä×·×ÙËã·¨¡¢¿éÏ¡Êè×Ó¿Õ¼äѧϰËã·¨¼°¿éÏ¡Ê豴Ҷ˹Ëã·¨£» ÁªºÏÏ¡ÊèѹËõ¸ÐÖª·½·¨°üÀ¨ÈýÖÖÁªºÏÏ¡ÊèÄ£ÐÍϵÄÖع¹Ëã·¨ÒÔ¼°¸ß˹ÁªºÏÏ¡ÊèÕÅÁ¿Ñ¹Ëõ¸ÐÖª£¬Îª´ÓÊÂÐźŴ¦Àí¹¤×÷µÄ¹ã´ó¶ÁÕßÌṩ²Î¿¼ºÍ½è¼ø¡£
²Î¿¼ÎÄÏ×

£Û1£ÝJi S H,Dunson D,Carin L.Multitask compressive sensing£ÛJ£Ý.IEEE Transactions on Signal Processing,2009,57(1)£º  92ª²106.

£Û2£ÝParvaresh F,Vikalo H,Misra S,et al.Recovering sparse signals using sparse measurement matrices in compressed DNA microarrays£ÛJ£Ý.IEEE Journal of Selected Topics in Signal Processing,2008,2(3)£º  275ª²285.

£Û3£ÝMishali M,Eldar Y C.From theory to practice£º  subª²nyquist sampling of sparse wideband analog signals£ÛJ£Ý.IEEE Journal of Selected Topics in Signal Processing,2010,4(2)£º  375ª²391.

£Û4£ÝDavies M E,Eldar Y C.Rank awareness in joint sparse recovery£ÛJ£Ý.IEEE Transactions on Information Theory,2012,58(2)£º  1135ª²1146.

£Û5£ÝÐùÆôÔË.»ùÓÚ×ÖµäѧϰµÄÁªºÏ¿éÏ¡Êè·Ö½âËã·¨Ñо¿£ÛD£Ý.¹þ¶û±õ£º  ¹þ¶û±õ¹¤Òµ´óѧ,2019.

£Û6£ÝYong L,Dai W,Zou J,et al.Structured sparse representation with union of dataª²driven linear and multilinear subspaces model for compressive video sampling£ÛJ£Ý.IEEE Transactions on Signal Processing,2017,65(99)£º 5062ª²5077.

£Û7£ÝRamirez I,Sprechmann P,Sapiro G.Classification and clustering via dictionary learning with structured incoherence and shared features£ÛC£Ý.Computer Vision and Pattern Recognition,2010 IEEE Conference on.IEEE,2010.

£Û8£ÝZhang Z,Wang Y,Chong E,et al.Subspace selection for projection maximization with matroid constraints£ÛJ£Ý.IEEE Transactions on Signal Processing,2017,65(5)£º  1339ª²1351.

£Û9£ÝEldar Y C,Member S,Member S,et al.Robust recovery of signals from a structured union of subspaces£ÛJ£Ý.IEEE Transactions on Information Theory,2009,55(11)£º 5302ª²5316.

£Û10£ÝEwout V,Friedlander M P.Sparse optimization with leastª²squares constraints£ÛJ£Ý.Siam Journal on Optimization,2011,21(4)£º  1201ª²1229.

£Û11£ÝHuang J Z,Zhang T,Metaxas D.Learning with structured sparsity£ÛP£Ý.Machine Learning,2009£º  417ª²424.

£Û12£ÝYu L,Sun H,Barbot J P,Zheng G.Bayesian compressive sensing for cluster structured sparse signals£ÛJ£Ý.Signal Processing,2011,92(1)£º  259ª²269.

£Û13£ÝPeleg T,Eldar Y,Elad M.Exploiting statistical dependencies in sparse representations for signal recovery£ÛJ£Ý.IEEE Trans.on Signal Processing,2012,60(5)£º  2286ª²2303.

£Û14£ÝBishop C M,Pattern recognition and machine learning(Information Science and Statistics)£ÛM£Ý.Secaucus,NJ,USA£º  Springerª²Verlag New York,Inc.,2006.

£Û15£ÝBlei D M,Ng A Y,Jordan M I.Latent dirichlet allocation£ÛJ£Ý.Journal of Machine Learning Research,2003,3£º  993ª²1022.

£Û16£ÝHe L,Carin L.Exploiting structure in waveletª²based Bayesian compressive sensing£ÛJ£Ý.IEEE Transactions on Signal Processing,2009,57(9)£º  3488ª²3497.

£Û17£ÝCevher V,Duarte M,Hegde C,el at.Sparse signal recovery using Markov random fields£ÛC£Ý.Advances in Neural Information Processing Systems 21,Proceedings of the Twentyª²Second Annual Conference on Neural Information Processing Systems,Vancouver,British Columbia,Canada,Dec.2008.

£Û18£ÝCevher V,Indyk P,Carin L,el at.Sparse signal recovery and acquisition with graphical models£ÛJ£Ý.IEEE Signal Processing Magazine,2010,27(6)£º  92ª²103.

£Û19£ÝMohammadª²Djafari A A A.Bayesian inference for inverse problems£ÛM£Ý.San Diego,California.1998.

£Û20£ÝChen S,Donoho D,Saunders M.Atomic decomposition by basis pursuit£ÛJ£Ý.Siam Journal on Scientific Computing,1998,20(1)£º  33ª²61.

£Û21£ÝDaubechies I,DeVore R,Fornasier M,et al.Iteratively reª²weighted least squares minimization£º  proof of faster than linear rate for sparse recovery£ÛC£Ý.Conference on Information Sciences & Systems,2008£º  26ª²29.

£Û22£ÝGorodnitsky I F,Rao B.D.Sparse signal reconstruction from limited data using FOCUSS£º  a reª²weighted minimum norm algorithm£ÛJ£Ý.IEEE Transactions on Signal Processing,1997,45(3)£º  600ª²616.

£Û23£ÝCand¨¨s E,Tao T.Decoding by linear programming£ÛJ£Ý.IEEE Transactions on Information Theory,2005,51(12)£º  4203ª²4215.

£Û24£ÝChartrand R,Yin W.Iteratively reweighted algorithms for compressive sensing£ÛC£Ý.In Proc.ICASSP,2008£º  3869ª²3872.

£Û25£ÝJi S,Ya X,Carin L.Bayesian compressive sensing£ÛJ£Ý.IEEE Transactions on Signal Processing,2008,56(6)£º  2346ª²2356.

£Û26£ÝBabacan S D,Molina R,Katsaggelos A K.Bayesian compressive sensing using Laplace priors£ÛJ£Ý.IEEE Transactions on Image Processing,2010,19(1)£º  53ª²63.

£Û27£ÝTipping M E.Sparse Bayesian learning and the relevance vector machine£ÛJ£Ý.Journal of Machine Learning Research,2001,1£º  211ª²244.

£Û28£ÝBeal M.Variational algorithms for approximate Bayesian inference£ÛD£Ý.Phd Thesis University of London,2003.

£Û29£ÝBaraniuk R G,E Cand¨¨s,Nowak R,et al.Compressive sampling£ÛJ£Ý.IEEE Signal Processing Magazine,2008,25(2)£º  1433ª²1452.

£Û30£ÝDonoho D.Compressed sensing£ÛJ£Ý.IEEE Transactions on Information Theory,2006,52(4)£º  1289ª²1306.

£Û31£ÝTropp J,Gilbert A.Signal recovery from partial information via orthogonal matching pursuit£ÛJ£Ý.IEEE Transactions on Information Theory,2007,53(12)£º  4655ª²4666.

£Û32£ÝDonoho D L,Tsaig Y,Drori I,et al.Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit£ÛJ£Ý.IEEE Transactions on Information Theory,2012,58(2)£º 1094ª²1121.

£Û33£ÝNeedell D,Tropp J.CoSaMP£º  iterative signal recovery from incomplete and inaccurate samples£ÛJ£Ý.Applied and Computational Harmonic Analysis,2009,26(3)£º  301ª²321.

£Û34£ÝRobert C P,Casella G.Monte Carlo statistical methods£ÛM£Ý.Springer Verlag,2004.

£Û35£ÝZhang W,Ma C,Wang W,et al.Side information based orthogonal matching pursuit in distributed compressed sensing£ÛC£Ý.IEEE International Conference on Network Infrastructure and Digital Content,2010£º  80ª²84.

£Û36£ÝTucker L.Some mathematical notes on threeª²mode factor analysis£ÛJ£Ý.Psychometrika,1966,31(3)£º  279ª²311.

£Û37£ÝKolda T G,Ba Der B W.Tensor decompositions and applications£ÛJ£Ý.Siam Review,2009,51(3)£º 455ª²500.

£Û38£ÝQi X,Qian Z,Meng D,et al.Multispectral images denoising by intrinsic tensor sparsity regularization£ÛC£Ý.IEEE Conference on Computer Vision & Pattern Recognition,2016.

£Û39£ÝBoyd S,Parikh N,Hu E C,et al.Distributed optimization and statistical learning via the alternating direction method of multipliers£ÛJ£Ý.Foundations & Trends in Machine Learning,2010,3(1)£º  1ª²122.

£Û40£ÝGong P,Zhang C,Lu Z,et al.A general iterative shrinkage and thresholding algorithm for nonª²convex regularized optimization problems£ÛJ£Ý.Applied Mathematics,2013.


£Û41£ÝYu G.Solving inverse problems with piecewise linear Estimators£º  from Gaussian mixture models to structured sparsity£ÛJ£Ý.IEEE Transactions on Image Processing,2012,21(5)£º  2481ª²2499.

£Û42£ÝYang J,Yuan X,Liao X,et al.Video compressive sensing using Gaussian mixture models£ÛJ£Ý.IEEE Transactions on Image Processing,2014,23(11)£º  4863ª²4878.

£Û43£ÝYu G,Sapiro G.Statistical Compressed Sensing of Gaussian mixture models£ÛJ£Ý.IEEE Transactions on Signal Processing,2011,59(12): 5842ª²5858.