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3 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 1

4 2 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (06) JR 10 JR morimoto@cc.osaka-kyoiku.ac.jp tel:

5 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering :00 18:00 13:00 13:10 13:10 14:10 (A-RMW) DWT P-DWT P-DWT 14:25 15:25 Crystal Wavelet Crystal Wavelet Crystal Wavelet 15:40 16:40 16:55 17:55

6 4 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering :00 16:15 10:00 11: :15 12: :15 14:00 14:00 15: ( ) 3 GPS 15:15 16:15

7 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 5 Ý º ¹ÊÅϵ ÏÌ È¹ ÏÌ È¹ ÏÌ È Ö Ø Ö Ø Ï Ú Ð Ø ÌÖ Ò ÓÖÑ Ò Ø ÈÖÓ Ð Ñ ÓÒ Ò Ì ÁÑ ÑÙÖ Ì Ø ÙÓ Å Ý À ÖÓ ÌÓ ÌÓÝÓ ÍÒ Ú Ö ØÝ Ó Ì ÒÓÐÓ Ý Ý Ö Ù Ø Ë ÓÓÐ Ó ÌÓÝÓ ÍÒ Ú Ö ØÝ Ó Ì ÒÓÐÓ Ý ØÖ Øº ÁÑÔ Ø ÓÙÒ ÐÐ Ö ØØÐ ÒÓ ÓÙÖ Ø Ø Ð ØÖ ÈÓÛ Ö ËØ Ö Ò Ý ¹ Ø Ñ È˵ ÓÒ ÖÓ º ÀÓÛ Ú Ö Ù Ò ÐÝ ÆÙÐØ Ù Ó Ø ÓÑÔÐ Ü ØÝ Ó Ñ Ò Ð ØÖ Ò Ö Ô Ø º Á ÒØ Ø ÓÒ Ø Ò ÕÙ Ù Ò Û Ú Ð Ø ÓÒ ØÖÙØ Ö Ð ¹ Ò Ð ÑÓØ Ö Û Ú Ð Ø ÊÅϵ «Ø Ú Ò ÓÖ Ö ØØÐ ÒÓ Ò ÐÝ ÓÒ Ó ÔÔÐ Ø ÓÒ Ó Ø Ñ ¹ Ö ÕÙ ÒÝ Ò ÐÝ Ù Ò Ï Ú Ð Ø ÌÖ Ò ÓÖÑ Û Ø Ò Û ÓÒ ÔØ Ó Ò Ø ÒØ Ò ÓÙ ÓÖÖ Ð Ø ÓÒ ØÓÖ ÏÁ µº ÀÓÛ Ú Ö Ø ÔÖÓ Ò Ø Ñ Ó ÏÁ ÓÒ ÏÌ ÒÖ ÓÖ Ò ØÓ Ø Ø Ð Ò Ø Ó ÊÅÏ ÒÖ º Ï Ú ÐÓÔ Ô Ö Ø Ø Ö Ø Û Ú Ð Ø ØÖ Ò ÓÖÑ È¹ Ï̵ ÓÖ Ô ÔÖÓ Ò Û Ø Ð ÐÙÐ Ø ÓÒ Ó Øº Ò Ø È¹ ÏÌ ÔÔÐ ØÓ ÔÖ Ø Ð Ù Ò Ö Ð¹Ø Ñ Ò ÐÝ Ý Ø Ñ ØÓ ÒØ Ý Ø ÖÖ Ò Ö ØØÐ ÒÓ ÓÙÖ Ò ÓÖ Ö ØÓ ÓÒ ÖÑ Ø Ú Ð Øݺ ½º ÓÐÙÑÒµ ÓÐÙÑÒ¹ØÝÔ Ð ØÖ ÈÓÛ Ö ËØ Ö Ò ¹ È˵ ¹ ÈË ËØ Ö Ò Û Ðµ ¹ ÈË ½

8 6 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ÁÒØ ÖÑ Ø Ó Òص ÅË Öµ ½ Ì ¾ ÓÒ¹ Ø ÒÙ Ò Ï Ú Ð Ø ÌÖ Ò ÓÖÑ ÏÌ Ï Ú Ð Ø ÁÒ Ø ÒØ Ò ÓÙ ÓÖÖ Ð Ø ÓÒ ÏÁ æ ÅÓØ Ö Ï Ú Ð Ø Åϵ ÅÏ ÊÅÏ ÏÁ ÊÅÏ ÊÅÏ ÅÏ ÏÌ ÏÌ ÅÏ ÏÌ ÏÌ È Ö Ø Ö Ø Û Ú Ð Ø ØÖ Ò ÓÖÑ È¹ Ï̵ Ø Û Ú Ð Ø Ò Ø ÒØ Ò ÓÙ ÓÖÖ Ð Ø ÓÒ ¹ÏÁ ¾

9 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 7 ¾º ÊÅÏ Øµ ÏÌ ¾º½µ Û µ ½ ¾ ½ ½ ص Ø ¼µ ½» ص ص ص ÅÏ Ñ Ð ØÝ ÓÒ Ø ÓÒµ ص ½ ¾º¾µ ½ ص Ø ¼ ÅÏ ÏÌ ¾º½µ ÏÌ ½» ¾ ÅÏ ÊÅÏ ÏÌ ½ Û ½ µ ÏÁ µ Ê µ µ Ø ¾º µ Ê µ Û ½ µ ÊÅÏ ½µ ¾µ ÊÅÏ Ê Øµ Ê ½ ¾º µ Ê ½ ½ Ê Øµ ¾ Ø ½ ¾ ½ µ Ê Øµ Ê µ µ ¼ Ê µ ¼ Ê µ ¾ Ê µ º

10 8 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering c 0,k DWT c 1,k d 1,k DWT R u k c 2,k I u k DWT d 2,k c 3,k d 3,k Parasitism filter º ½º ÓÑÔÓ Ø ÓÒ ØÖ Ó Ø Ô Ö Ø Ö Ø Û Ú Ð Ø ØÖ Ò ÓÖÑ È¹ Ï̵ Ò Ø Ô Ö Ø ÐØ Ö R x k I x k µ µ Ö µ µ Õ ¾º µ Ö µ Ê Ö µµ¾ Ê µµ¾ µ µ µ ص Ö Øµ ص ÊÅÏ Ö Øµ ص ÊÅÏ ÊÅÏ ËÝÑÑ ØÖ ÓÑÔÐ Ü Ö Ð¹ Ò Ð ÑÓØ Ö Û Ú Ð Ø Ë ¹ÊÅϵ ¾ Ë ¹ÊÅÏ ÊÅÏ ¹ÊÅϵ ÊÅÏ ÊÅÏ ÏÁ º º½ ÏÌ ÅÏ µ ÊÅÏ ÏÌ ÊÅÏ ÏÌ ÊÅÏ ÏÌ È¹ Ï̵ º ½ ȹ ÏÌ µ ÅÏ ÏÌ È Ö Ø ÌÖ Ò Ð Ø ÓÒ ÁÒÒ Ö Ò ÓÑÔÐ Ü

11 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 9 c j, k (RMW) DWT c j 1, k x k d, + u k DWT xout j 1 k (a) Decomposition (b) Reconstruction º ¾º Ò Ñ Ø Ó Ó Ø Ô Ö Ø ÐØ Ö Ö Ø Ï Ú Ð Ø ÌÖ Ò ÓÖÑ ÈÌÁ¹ Ï̵ ½¼ ÏÌ ÈÌÁ¹ ÏÌ ÏÌ ÊÅÏ ÏÌ ÅÏ ÅÏ ÏÌ µ µ Ü Ê ÜÁ ÙÊ Ù Á ÊÅÏ º½ ¾ º½ ȹ ÏÌ ÏÌ È¹ ÏÌ ÏÌ ÏÌ µ º¾ ½µ ¾µ ¾ ÊÅÏ ÏÌ ½ Ù µ º¾ ½ ¼ ½ ¼ Ü Æ ¼ ÃÖÓÒ Ö ÐØ µ Ü ÓÙØ µ Ö Ñ Ò Ü ÓÙØ ÊÅÏ Ù ÊÅÏ Ù µ ÊÅÏ Øµ Ö Øµ ص ÊÅÏ Ù Ê ÙÁ º½µ ¼ ÀÞ ½¼¼ ÀÞ ¾¼¼ ÀÞ

12 10 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Amplitude Amplitude º º Data Number u (a) Parasitic filter } { I k Data Number (d) Imaginary component of RMW Ü ÑÔÐ Ó Ø Ò Ô Ö Ø ÐØ Ö R I X ( f ) ix ( f ) ˆ ( f ) out out Power [db] A( f ) Aliasing elements Frequency [Hz] º º Ü ÑÔÐ Ó Ø Ö ÕÙ ÒÝ Ö Ø Ö Ø Ó Ø ÊÅÏ µ Ò Ê ÓÙØ µ Á ÓÙØ µ º º½µ ص Ò ½¼¼ ص ¼ Ò ¾¼¼ ص ¼ Ò ¼¼ ص ¼¼ ÀÞ ÊÅÏ ½¾ ¾ Ë ¹ÊÅÏ ÅÏ ËÝÑÐ Ø ÊÅÏ ¾ º µ ÊÅÏ Ù Á ÊÅÏ º µ ÊÅÏ Øµ µ Ü Ê ÓÙØ ÜÁ ÓÙØ Ê ÓÙØ µ Á ÓÙØ µ º º ÅÏ µ º µ ÓÙØ Ê µ Á ÓÙØ µ ÊÅÏ µ Ê µ Á µ Ù Ê ÙÁ º ¼ ÀÞ ½½¼¼ ÀÞ

13 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 11 Amplitude R(b), R(k) Time [s] (a) Analysis signal R(k) R(b) º º Time [s] (b) R (b) and R(k) Ü ÑÔÐ Ó Ê µ Ó Ø Ò ÖÓÑ Ø È¹ ÏÌ Ò Ê µ Ó Ø Ò ÖÓÑ Ø ÏÌ ¼ ¾¼¼ ÀÞ ¾¼ º¾ Ø Û Ú Ð Ø Ò Ø ÒØ Ò ÓÙ ÓÖÖ Ð Ø ÓÒ ¹ÏÁ µ Õ º¾µ Ê µ Ü Ê µ¾ Ü Á µ¾ Ø Ø ¾ Ê µ Ø ÏÌ Ê µ Ê µ Ü Ê Ü Á Ê µ ÊÅÏ ÊÅÏ ÊÅÏ Ä º µ º µ Ä ½¼ ÐÓ ½¼ ¼ È È µ ¾ ص ¾ º µ º½ ½µ ½ ½

14 12 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 0,k discrete data 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,253 0,254 0,255 0,256 0,257 0,258 0,259 0,260 j= -1-1,1-1,2-1,3-1,4-1,127-1,128-1,129-1,130 j= -2 RMW length = 256 Parastic level = -2 º º -2,1-2,2-2,64-2,65 -{u k } R x 1 -{u k } length = 64 R x 2 ÓÒ ÔØ Ó ¹ÏÁ ÔÖÓ Ò º µ Ù Á ÙÊ º µ ÊÅÏ Ê µ ÊÅÏ Ä ¾ ÊÅÏ º µ ÏÌ Ê µ º º µ µ Ê µ Ê µ º µ Ê µ Ê µ Ê µ Ê µ Ê µ Ê µ È Ê µ Ê µ ¾ Æ ÅË µ ¾ ½ ȹ ÏÌ ¹ÏÁ Ù Ê ÙÁ ÊÅÏ ÏÌ ÏÁ º º½ ÏÌ ¼ Ò Òµ ¼ Ò ¼ ¹ÏÁ Ê µ º ÊÅÏ Ä ¾ ¾ ¼ ¾ ¾ ¾ ¼ ¾ Ä º¾µ ½ ¹ÏÁ

15 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 13 Ì Ð ½º ÆÙÑ Ö Ó ÑÙÐØ ÔÐ Ø ÓÒ ÊÅÏ Ð Ò Ø Ð Ú Ð ½¾ ¾ ½¾ ½¼¾ ¹½ ½ ¾ ¾ ¼ ¾ ½¼ ¹¾ ½ ¾ ¾ ¼ ¹ ¼ ½¼ ½ ¾¼¾ ¼ ¹ ½ ¾ ½ ¼ ½ ¾½ ¾ ¾ ÏÌ ½ ¾ ¼ ½ ½¼¾ ¾¼ ¾ º¾µ ¾ ¼ ¼ ¾ ¼ ½ ¾ Ê µ ¼ ¹ÏÁ ¹ÏÁ º¾ ¹ÏÁ Ê µ ½ º½µ É ½¼ ½ Ô ¾ ½ ¾ Ô ½ Ä Ô Ô ¼ µ Ä ÊÅÏ º½µ ½ ÏÌ ¾ ¹ÏÁ ÏÌ ÏÁ º¾µ É ¾Ä Ì Ð ½ ȹ ÏÌ ¹ÏÁ Ê µ ÏÌ ÏÁ Ê µ Ì Ð ½ ÊÅÏ ÏÁ ¹ÏÁ ȹ ÏÌ ¹ÏÁ Ê µ ÏÌ

16 14 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Ì Ð ¾º ÌÖ Ò Ø ÓÒ Ø Ñ ÓÖ Ê µ Ò Ê µ ÊÅÏ Ð Ò Ø Ð Ú Ð ½¾ ¾ ½¾ ½¼¾ ¾¼ ¼ ¹½ ¾º¼ ½¼ º¼ ½¼ º¾ ½¼ º¾ ½¼ ½ º ½¼ º ½¼ ¹¾ ½º ½¼ ¾º ½¼ º ½¼ º ½¼ º ½¼ ½ º ½¼ ¹ ½º ½¼ ¾º½ ½¼ º¾ ½¼ º½ ½¼ º¼ ½¼ ½¼º ½¼ ¹ ¾º ½¼ ¾º ½¼ º ½¼ º ½¼ º ½¼ º ½¼ ÏÌ ¾º ½¼ º ½¼ º ½¼ ½ º ½¼ º ½¼ º ½¼ º º½ Ï Ò ÓÛ ÇË ÊÅÏ Ä ¹ÏÁ ÏÌ ¾ ½ ÈÍ ÓÖ ¾º ÀÞ Ê Å º¼ Ì Ð ¾ ÊÅÏ Ä ¾ ¹¾ ȹ ÏÌ ¹ÏÁ ¾º ½¼¹ ÏÌ ÏÁ º ½¼¹ ¹ÏÁ ÏÌ ¾ ÊÅÏ Ä È¹ ÏÌ ¹ÏÁ ½ º½µ º º½ Ä ÎÁ Ï ¹ÏÁ º ¹ÏÁ º½ º ÊÅÏ Ä ¾ ¹¾ ¾ ¹ÏÁ ½¼

17 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 15 LabVIEW discrete data RMW data length: 256 time Calculation part [C language] C C 2,1 2,64 x out put Calculation Down sampling { u k } filter length : 64 º º ÐÓÛ Ó ¹ÏÁ ØÖ Ò Ø ÓÒ Ì Ð º ÌÖ Ò Ø ÓÒ Ø Ñ ÓÖ Ê µ ØÊÅÏÄ ¾ Ä Ú Ð ÌÖ Ò º Ø Ñ ÐÙÐ Ø ÓÒ Ø Ñ Ù«Ö Ò Ø Ñ ½ ¼º¼¼ ¼ ½¼ ½¾ ½¼ ½¾ ¾ ¼º¼¼ ¾ ½¼ º ½¼ ¼º¼¼¾ ½ ½¼ ¾ º ½¼ ¾ ¼º¼¼½¾ ½ ¼ ½¼ ¼ ½ º ½¼ ½ ÏÁ ¼º¼½ ½¼ ¾ º ½¼ ¾ Ä ÎÁ Ï ÏÁ ÊÅÏ ÊÅÏ Ì Ð ¾ Ì Ð ¾ ÊÅÏ Ì Ð Ì Ð Ê µ Ê µ ÊÅÏ ½½

18 16 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (a) (b) (c) º º Ì Ñ Ò ÖØ Ò Ø Ö º¾ Ä ÎÁ Ï ÈÍ º Ø Ø ½¾ ÀÞ ÊÅÏ Ä ¾ Ø Ø Ä ¼º¼¾ ½ ¼º¼¼ Ø Ø ½» ÏÌ ¼º¼½ Ø ¹ÏÁ ½¾

19 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 17 º º Ö Ñ Ó Ñ ÙÖ Ñ ÒØ Ý Ø Ñ º º½ º¾ º ÆÁ ÓÑÔ Ø É È È Ä ÎÁ Ï ÓÑÔ Ø É ÆÁ ÍË È Ä ÎÁ Ï ÊÅÏ È ÊÅÏ ½¾ ¼ ½¾ ¾ ¾ ½ ÀÞ ½¼ ÀÞ ¾ º ÀÞ ½

20 18 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Ì Ð º Ê ÙÐØ Ó Ê Ð¹Ø Ñ Ô Ö ÓÖÑ Ò Ë ÑÔÐ Ò ÊÅÏ Ð Ò Ø Ø Ð Ò Ø Ø Ì Ñ ÀÞ ½ ¾ ¼ ¼ ¼º½ ¾¼ ¾¼ ¼º¼ ½¼¾ ½¼¾ ¼º¼ ¾ º ½¾ ½¾ ¼º¼¾ ¾ ¾ ¼º¼½ ½¾ ½¾ ¼º¼¼ ½ ½ Ä ÎÁ Ï Ì Ð ¾ º ÀÞ ÊÅÏ Ø Ø Ø Ø Ø ½ ¾ Ì Ð ¾ ¾ Ø ¼º¼½ ½¾ Ø ¾ ÆÁ ÓÑÔ Ø Õ È Á Ä ÎÁ Ï Î Öº º ½¼¼ ÀÞ ¼ ÀÞ È Ï Ò ÓÛ È ÓÖ ¾º ÀÞ ½¼ ÀÞ»½ ¹ÏÁ µ ÝØ ÓÑÔ Ø É È º½¼ È ÓÑÔ Ø É ÍË ÓÑÔ Ø É ¾ ÊÅÏ ¹ÏÁ ½

21 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 19 º ½¼º Ú Ö Ð¹Ø Ñ ÒÓ Ý Ø Ñ º ½½º ËØ Ö Ò Ý Ø Ñ Û Ø Ø ÈË Ò ÜÔ Ö Ñ ÒØ Ð ØÙÔ Û Ö Ò ÓÖ Ö ¹ Ñ Ò ÓÒ Ð Ð Ö Ø ÓÒ Ò ÓÖ º ½¼ ½ Ñ» ÊÅÏ ½ ¹ÏÁ ÏÌ ÏÁ ¹ ÈË ÅË º½½ ÅË Ü Ý Þ º½¾ µ ½

22 20 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (a) Example of sound signal on bad road º ½¾º (b) CWT of sound signal using the Gabor function Ü ÑÔÐ Ó Ø ÏÌ Ó ÓÙÒ Ò Ð Ù Ò Ø ÓÖ ÙÒØ ÓÒ º ½ º Ü ÑÔÐ Ó Ø Ê µ Ó Ú Ö Ø ÓÒ Ò Ð º½¾ µ ÓÖ ÏÌ ÊÅÏ º½¾ µ ÅÏ Ú Ö ¹ÊÅÏ º½ ¹ÏÁ ÏÌ º½ ½ ¹ÏÁ ÏÌ ÏÁ ÅË ÅË ¹ÏÁ ½

23 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 21 º ½ º (a) R(k) of MS gear measured from sensor A (b) R(k) of intermediate joint measured from sensor B (c) R(k) of column measured from sensor C ÓÑÔ Ö ÓÒ ØÛ Ò ¹ÏÁ Ð Ú Ð ¹½ Ò Ç«¹Ð Ò ÏÌ Ö ÙÐØ º ȹ ÏÌ ¹ÏÁ Ä ÎÁ Ï ½µ ¹ÏÁ ÊÅÏ Ä ¾ ¹¾ ¹ÏÁ ÏÁ ¾ ÊÅÏ Ä ¹ÏÁ ¾µ Ä ÎÁ Ï ¾ ½ µ ½

24 22 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering À¾¾ Ä ÎÁ Ï ½ ½¹ ½½ ¾¼¼ µ ÔÔº ¹ º ¾ ¾ ÎÓк ÆÓº½¾ ½ µ Ôº½ ¹½ ½¹ ½ µ ÔÔº½ ¹½ ÎÓк ¼ ÆÓº ¾¼¼ µ ÔÔº¾ ¼æ¾ º µ ¹ ¼ ¾¼¼ º µ ÔÔº½ ¹½ º ½¼¹ ¾¼½½ µ ÔÔº ¹ ½ º ÎÓк ÆÓº ¾¼¼ º µ ÔÔº¾ ¾ ¹¾ º Å ÐÐ Ø Ëº º Û Ú Ð Ø ØÓÙÖ Ó Ò Ð ÔÖÓ Ò Ñ ÈÖ ½ µ Ôº¾ º ½¼ ÂÓÙÖÒ Ð Ó Ë Ò Ð ÈÖÓ Ò ÎÓк½¾ ÆÓº ¾¼¼ µ ½ æ½ º µ ½¹ ¼ ½¹½ ¹Ñ Ð Þ Ò ºÑ ºØÙغ º Ô µ ½

25 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 23 ½¹ ¼ ½¹½ ¹Ñ Ð Ñ ºÑ ºØÙغ º Ô µ ½¹ ¼ ½¹½ ¹Ñ Ð Ñ Ý ºÑ ºØÙغ º Ô µ ½¹ ¼ ½¹½ ¹Ñ Ð ÔÜؼ¼½ Ò ØݺÓÑ ½

26 24 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering

27 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 25. Crystal Wavelet Application to microcrystalline structure of the signal processing Tomoko Endo Naoki Mukawa Tokyo Denki University Abstract. The authors have proposed methods that extract crystal grain boundaries from three-dimensional (3D) crystal structure data using 3D Fourier analysis of the orthogonal coordinate system and 3D wavelet analysis of oblique coordinate system. For method using 3D Fourier analysis, we contrive an objective method based on frequencydomain processing to resolve the problem presuming that crystals have highly periodic structures and grain boundaries are disruptions of the repetitive structure, On the other hand, we introduce 3D wavelet defined on crystal lattice based on crystal structure theory. In this paper we first compare with Fourier and wavelet analysis, and then consider two coordinate systems, e.g. oblique and orthogonal. Next, we will discuss advantages and disadvantages of the systems. Finally, we summarize future works for analysis of microcrystalline structures. 1. [1] 1

28 26 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [2 4] [5] [6] Crystal wavelets [7] [8] Crystal wavelets Crystal wavelets 2

29 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 27 Fig. 1. Fig. 2. A θ A ϕ A Crystal wavelets J. Kovačević [9] Fig.1 A B B A z 1. 3

30 28 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering A B Fig.3(a) Fig.3(b) 1 A B 4. A 1 θ A ϕ A Fig.2 A 5. A Fig.4(a) B 6. Fig.4(b) 1 7. B θ B ϕ B 2 1. A A A A A 2. A t(i, j, z) 3. 3 f(x, y, z) A t(i, j, z) A R(x, y, z) (2.1) R(x, y, z) = a i ai bj c k f(x + i, y + j, z + k) t(i, j, k) bj c k f(x + i, y + j, z + ai bj c k)2 k t(i, j, k) A 1 A B 2 4

31 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 29 Fig. 3. A B a b Fig. 4. B a b a 0.287nm μm TEM 10nm [10] A B C A α 0,0,0 B β 0,45,0 C γ 45, 45,45 3 α β A B 3 δ 3 5

32 30 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 5. α β 3 a a b 2 α β Fig.5 a α β β α m Fig.5(b) δ 128 δ 6

33 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Crystal wavelets 1 [11,12] 3 Crystal wavelets 3.1 Crystal wavelets (3.1) Λ={t : t = 3 n i t i n i Z, i = 1, 2, 3} i=1 [6] 2 j {c j [t]} t Λ Λ 8 Λ m (3.2) Λ m = {t : t = 2(n 1 t 1 + n 2 t 2 + n 3 t 3 ) + t m n i Z, i = 1, 2, 3}, m = 0, 1,, 7 (3.2) {c j [t]} t Λ Λ m 8 ĉ m, j (ω) = c j [2t + t m ]e iω t, ω R 3 t Λ ω t ω t 1 ĉ j 1 (ω) ĉ 0, j (ω) ˆd 1, j 1 (ω) ĉ 1, j (ω) ˆd 2, j 1 (ω) ĉ 2, j (ω) ˆd (3.3) 3, j 1 (ω) = P(ω) ˆd 4, j 1 (ω) ĉ 3, j (ω) ĉ 4, j (ω) ˆd 5, j 1 (ω) ĉ 5, j (ω) ˆd 6, j 1 (ω) ĉ 6, j (ω) ˆd 7, j 1 (ω) ĉ 7, j (ω) P(ω) 8 8 7

34 32 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (3.4) P(ω) 2 = û 1 (ω) û 2 (ω) û 3 (ω) û 4 (ω) û 5 (ω) û 6 (ω) û 7 (ω) ˆp 1 (ω) ˆp 2 (ω) ˆp 3 (ω) ˆp 4 (ω) ˆp 5 (ω) ˆp 6 (ω) ˆp 7 (ω) P(ω) 7 predictor ˆp m (ω) updater û m (ω), m = 1, 2,...,7 3.3 ĉ j 1 (ω) ˆd m, j 1 (ω), m = 1, 2,...7 P(ω) = P(ω) 1 (3.5) P(ω) P(ω) = I LP HP Λ B Z Λ B 3 Λ B = {t : t = n i t i n i B, i = 1, 2, 3} i=1 LP {h[t]} t ΛB ĥ(ω) = h[t]e iω t, ω R 3 t Λ B ĥ(ω) ω ω + 2πλ λ 8

35 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 33 P(ω) 3.1 ĥ(ω) 1 ĝ 1 (ω) e iω t 1 ĝ 2 (ω) e iω t 2 ĝ (3.6) 3 (ω) = P(ω) ĝ 4 (ω) e iω t 3 e iω t 4 ĝ 5 (ω) e iω t 5 ĝ 6 (ω) e iω t 6 ĝ 7 (ω) e iω t 7 φ L 2 (R 3 ) MRA φ(r) = t Λ B 2 2h[t]φ(2r t), r R 3 r = xt 1 + yt 2 + zt 3 R 3 (3.7) ˆφ(ω) = 1 2 ĥ( ω 2 2 ) ˆφ( ω ), ω R3 2 ĥ(ω/2) ˆφ(ω) = 1 2 ĥ( ω 2 2 j ), ˆφ(0) = 1 j=1 ψ m (r), m = 1, 2,...,7, r R 3 ˆψ m (ω) = 1 2 ω 2ĝm( 2 ) ˆφ( ω ), ω R crystal wavelets crystal wavelets Haar crystal wavelets Haar crystal wavelets Haar crystal wavelets 3.4 ˆp m (ω) = 1, û m (ω) = 1, m = 1, 2,...,7 8 9

36 34 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 3.6 (3.8) ĥ(ω) ĝ 1 (ω) ĝ 2 (ω) ĝ 3 (ω) ĝ 4 (ω) ĝ 5 (ω) ĝ 6 (ω) ĝ 7 (ω) = e iω t 1 e iω t 2 e iω t 3 e iω t 4 e iω t 5 e iω t 6 e iω t 7 Haar crystal wavelets e iω t m, m = 1, 2,...,7, t m crystal wavelets Crystal wavelets Crystal wavelets 2.21 a 1 Fig Haar x y z ζ η ξ a ζ η ξ ζ = 0 η ξ 2 b η = 0 c ξ = Crystal wavelets Crystal wavelets Crystal wavelets 10

37 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 35 Fig. 6. Haar a ξ = 0 b η = 0 c ζ = 0 Crystal wavelets 4. Crystal wavelets Crystal wavelets Crystal wavelets 11

38 36 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering δ [13] [14] 2011 Al-Cu-Fe 10 12

39 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 37 Crystal wavelets 5. Crystal wavelets [1], [2] H. Jinnai, K. Yasuda and T. Nishi, Three-Dimensional Observations of Grain Boundary Morphologies in a Cylinder-Forming Block Copolymer, Macromolecular Symposia, , pp , [3] K. Inoke, K. Kaneko, M. Weyland, P.A. Midgley, K. Hogashida and Z. Horita, Severe local strain and the plastic deformation of Guinier-Preston zones in the Al-Ag system revealed by three-dimensional electron tomography, Acta Materialia, vol.54, pp , [4] K. Kaneko, R. Nagayama, K. Inoke, E. Noguchi and Z. Horita, Application of threedimensional electron tomography using bright-field imaging Two types of Si-phases in Al-Si alloy, Science and Technology of Advanced Materials, vol.7, pp , [5] [6] G. Grosso and G. P. Parravicini, Solid State Physics, Elsevier Academic Press, [7]. A, J92-A(8), pp ,

40 38 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [8] S. Mallat: A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, [9] J. Kovačević and W. Sweldens, Wavelet Families of Increasing Order in Arbitrary Dimensions, IEEE Trans. On Image Processing, vol.9, no.3, pp , Mar [10],, [11] W. Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, J. Appl. Comput. Harmonic Analysis, vol.3, no.2, pp , [12] G. Uytterhoeven, D. Roose, and A. Bultheel, Wavelet transforms using the Lifting Scheme, ITA-Wavelets Report WP 1.1, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, November [13] Gunther D. Schaaf, Ralf Mikulla, Johannes Roth, H.-R. Trebin, Numerical simulation of dislocation motion in an icosahedral quasicrystal, Materials Science and Engineering , pp , [14], ( ) tomoko.endo.yamagishi@gmail ( ) mukawa@sie.dendai.ac.jp 14

41 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 39. The theory of reproducing kernels and its applications Tsutomu Matsuura Gunma University Abstract. The theory of reproducing kernels is grand and beautiufl in itself. But recently we and some other researchers have tried to apply this theory to some concrete problems (especially to inverse problems ). Here we describe the outline of this theory. And I submit a method to apply this theory to real inversion of the Laplace transform. Furthermore I would like to discuss the applicability of the theory of reproducing kernels to real inversion of the Laplace transform. 1. S. Bergman G. Szegö Szegö Bergman 1950 Aronszajn [6] Schwartz [76] Schwartz 1983 [43] Bergman Szegö ([68]) Tikhonov 1

42 40 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 2 3, 4 Tikhonov 5,6 2. (cf. [23]) F (E) E H (, ) H. h : E H H E., f H F (E) L : (2.1) f (p) = (L f)(p) = ( f, h(p)) H. (2.1) f (p) H f f (p) 2

43 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 41 E E (2.2) K(p, q) = (h(q), h(p)) H R(L) L H E (2.3) f R(L) = inf{ f H ; f = L f} 1 (2.3) R(L) [R(L), (, ) R(L) ] (2.2) K(p, q) (i) q E, K(p, q) p R(L), (ii) f R(L) q E, f (q) = ( f ( ), K(, q)) R(L). (i) (ii) K(p, q) R(L). L H R(L) {h(p); p E} H. K(p, q) (i) (ii) R(L) K(p, q) K(p, q) ii f RKHS (reproducing kernel Hilbert space). 1 (2.2) K(p, q) K(p, q) H K (2.1) (2.2) R(L) = H K 1 2 (2.1) H { f (p)} (2.2) K(p, q) H K. f HK f H f H K H f E f (p) = ( f, h(p)) H f HK = f H. 2 H 2 (2.2) K(p, q) 3

44 42 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (2.1) H H K T E : H = L 2 (T, dm), H K L 2 (E, dμ). ( dm, dμ T, E dm, dμ L 2. (2.4) f (p) = F(t)h(t, p)dm(t) T h(t, p) T E h(, p) L 2 (T, dm) F F L 2 (T, dm). K(p, q) = h(t, q)h(t, p)dm(t) on E E. T H K L 2 (E, dμ). 3 E {E N } N=1 : (a) E 1 E 2, (b) N=1 E N = E, (c) K(p, p)dμ(p) <. E N, f H K N f (p)h(t, p)dμ(p) L E 2 (T, dm) N { } (2.5) f (p)h(t, p)dμ(p) E N N=1 (2.4) 2 F L 2 (T, dm). L 2 (T, dm) 2, L E K(p, q) H K H L (2.6) g L (q) = LK(, q) L 4

45 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 43 4 (2.7) f (p) = ( f, g L (p)) H, for f H f HK = f H. (2.7) L {g L (p); p E} H. (2.4) H K H f. (2.1) ill-posed problem 3 2.6,,. ([79]). 5 E { f (p)} H, q E, f f (q) H 5

46 44 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering { f n } f H K E K(p, p) E E E k(p, q) E E E X(p) p,q X(p)X(q)k(p, q) 0. K(p, q) E, 6 E K(p, q), H K 7 E H K(p, q) H {v j (p)} j E E K(p, q) = j v j (p)v j (q). H K H K -. E K(p, q), K(p, q) H K E K(p, q) E E 1, E 1 E 1 K (1) (p, q) = K(p, q) E1 E 1 K (1) (p, q) K (1) (p, q) H K (1) H K (1) H K 8 f (1) H K (1) H K f f (1) = f E1 H K (1) f (1) HK (1) = min { f HK ; f } E1 = f (1), f H K. E K (1) (p, q) K (2) (p, q), E E K(p, q) = K (1) (p, q) + K (2) (p, q) E H K, H K (1) H K (2) 6

47 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 45 9 f H K f (1) H K (1) f (2) H K (2) f (p) = f (1) (p) + f (2) (p) on E H K : f 2 H K = min { f (1) 2 H K (1) + f (2) 2 H K (2) ; f (p) = f (1) (p) + f (2) (p) on E, f (1) H K (1), f (2) H K (2)}. E K (1) (p, q) K (2) (p, q), K(p, q) = K (1) (p, q)k (2) (p, q) Schur E H K H K (1) H K (2), 8 10 { f (1) j } j { f (2) j } j H K (1) H K (2), H K E (2.8) f (p) = α i, j f (1) i (p) f (2) j (p) on E, α i, j 2 <. i, j i, j H K (8) {α i, j } f 2 H K = min i, j α i, j 2. ([52]) (1) ([48]) (2) ([60]). 3.. (3) ([56]).. (4) ([45]). (2) Paley-Wiener. (5) ([49],[52]).. (6) ([52])... (7) ([57]).. ([25]). 7

48 46 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (8) ([57]).. (9) ([12],[13]).. (10) ([50]).,.. [64]. (11) ([53])... ([58,63,65]). (12) ([51], 4,2 )... (13) ([46]).. Pick ([4]). (14) ([47],[59],[62]).. ([21]). (15) ([66]).. (16) ([61]) L H K H. H d (4.1) inf f H K Lf d H. Moore-Penrose, 8

49 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ([13]) H d, (4.2) inf Lf d H = L f d H f H K H K f k(p, q) = (L LK(, q), L LK(, p)) HK H k (4.3) L d H k. (4.2) f H K f d (4.4) f d (p) = (L d, L LK(, p)) Hk on E. L L (L d)(p) = (L d, K(, p)) HK = (d, LK(, p)) H d, L, K(p, q) H. 11 (4.1).. f d Lf = d Moore-Penrose L d 11 Moore-Penrose d. Tikhonov [17] L [20] {E λ } L L. L L 1 (L L) 1 = λ de λ. Moore-Penrose (4.4) f d (p) = 1 λ de λl d. R(L) d D(L ), Lf = d Moore-Penrose α >0 1 f d,α (p) = λ + α de λl d 9

50 48 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering. d d δ H δ d δ f d,α f δ d,α (p) = 1 λ + α de λl d δ 12 D(L ) d (4.5) lim α 0 (L L + αi) 1 L d = lim α 0 f d,α = f d. Lf d,α Lf δ d,α H δ, f d,α f δ d,α H K δ α. 13 d d δ H δ D(L ) d δ f δ d,α (4.6) inf f H K {α f 2 H K + d δ Lf 2 H }. α = α(δ) lim α(δ) = 0, lim δ 0 δ 0 δ 2 α(δ) = 0 (4.7) lim δ 0 f δ d,α = f d = L (d).. Tikhonov Tikhonov f d,α f δ d,α. L.. K L (, p; α) = (L L + αi) 1 K(, p) α>0 : (4.8) ( f, g) HK (L;α) = α( f, g) HK + (Lf, Lg) H. H K H K (L; α). 10

51 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ([68]) Tikhonov (4.9) inf f H K {α f 2 H K + d Lf 2 H } f d,α (p) K L (p, q; α) (4.10) f d,α (p) = (d, LK L (, p; α)) H. K L (p, q; α) H K (L; α) (4.11) K(p, q; α) + 1 α (L K q, LK p ) H = 1 K(p, q) α K(p, q; α). K q = K(, q; α) H K for q E, K p = K(, p) for p E. (4.10), d. 15 ([35],[25]). (4.10) f d,α (p) 1 α K(p, p) d H. α d H.. α. α α α α ([17],[20]). ([7,29-37,66-71]). 5.. L 2 (R n ) F } 1 ξ x 2 (5.1) u F (x, t) = (L t F)(x) = F(ξ)exp { dξ (4πt) n/2 R n 4t 11

52 50 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering. u(x, 0) = F(x) u t (x, t) = u xx (x, t) u(x, t). [77]. n = 1 [44] 2 3, u F (x, t) ([62]).. F t = 1 e D2 [(L 1 F)(x)] = F(x) pointwisely on R ([26], p. 182). 5 Sobolev [68,33], [35], Paley-Wiener. Paley-Wiener Stenger[74] sampling theory. sinc method. L 2 (R n, ( π/h, +π/h) n ), (h > 0) g f (z) = 1 (2π) n χ h (t)g(t)e iz t dt R n. z = (z 1, z 2,..., z n ), t = (t 1, t 2,..., t n ), dt = dt 1 dt 2 dt n, z t = z 1 t 1 + +z n t n ( π/h, +π/h) χ χ h (t) =Π n ν=1 χ(t ν). K h (z, u) = 1 (2π) n χ h (t)e iz t e iu t dt =Π n ν R n 1 π(z ν u ν ) sin π h (z ν u ν ) Paley-Wiener W h ν C ν z ν ( ) π zν f (z 1,..., z ν, z ν+1,..., z n ) C ν exp, f (x) 2 dx < h R n. j = ( j 1, j 2,..., j n ) Z n 1 (2π) n g(t) 2 dt = h n f ( jh) 2 = f (x) 2 dx R n j R n. f (x) = ( f ( ), K h (, x)) HKh = h n j f ( jh)k h ( jh, x) = R n f (ξ)k h (ξ, x)dξ 12

53 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 51 f (x) { f ( jh)} j sampling theorem). {hj} j [23,45]. 16 ([35]) L 2 (R n ) g α>0, { } inf α F 2 HKh + g u F (, t) 2 F H L 2 (R n ) Kh (5.2) = α F t,α,h,g 2 H Kh + g u F t,α,h,g (, t) 2 L 2 (R n ) F t,α,h,g (5.3) F t,α,h,g (x) = R n g(ξ)q t,α,h (ξ x)dξ. Q t,α,h (ξ x) = 1 (2π) n R n χ h (p)e ip (ξ x) dp αe p 2 t. + e p 2 t H Kh F u F (x, t) g u F (ξ, t) α 0 Ft,α,h,g F. Sobolev, α = 0 (5.3) ([68],[33]), Paley-Wiener W h α = 0 (5.3). 16, α = 0. Tikhonov (L t Ft,0,h,g )(x) = (g( ), K h(, x)) L2 (R n ) (L t Ft,0,h,g )(x) g Paley-Wiener W h, Ft,0,h,g g L t Ft,0,h,g g L 2 (R n ) Ft,0,h,g L 2(R) g W h F L t F = g Moore-Penrose Paley-Wiener W h Tikhonov Moore-Penrose Tikhonov Ft,α,h,g α = 0. Sobolev H S Paley-Wiener W h α 0 h 0 30 h 0. [33,35]. L 2 (R) g. 13

54 52 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 6. F (LF)(p) = f (p) = 0 e pt F(t)dt, p > 0.. ( 1) n ( n ) n+1 ( lim f (n) n ) ( = F(t), lim Π n n n! t t n k=1 1 + t ) d [ n ( n )] k dt t f = F(t) t ([40,80]). [11,36] [84,85] [27,28]. [36]. R + { F (t) 2 1 } 1/2 t et dt 0 F(0) = 0 F H K. (6.1) K(t, t ) =. min(t,t ) 0 ξe ξ dξ (6.2) 0 (LF)(p)p 2 dp 1 2 F 2 H K ; H K L 2 (R +, dp) = L 2 (R + ) (LF)(p)p. ([86]). 17 ([36]). g L 2 (R + ) α>0, { α inf F H K 0 F (t) 2 1 t et dt + (LF)(p)p g 2 L 2 (R + ) 14 }

55 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 53 (6.3) = α Fα,g(t) t et dt + (LFα,g)(p)p g 2 L 2 (R + ) F α,g (6.4) F α,g(t) = 0 g(ξ) (LK α (, t)) (ξ)ξdξ. K α (, t) K α,t = K α (, t ), K t = K(, t) (6.5) K α (t, t ) = 1 α K(t, t ) 1 α ((LK α,t )(p)p, (LK t)(p)p) L2 (R + ). F α,g(t) (6.4). (6.5) t t t (6.6) (LK α (, t))(ξ) = 1 α (LK(, t))(ξ) 1 α ((LK α,t)(p)p, (L(LK )(p)p))(ξ)) L2 (R + ). { te K(t, t t e t + 1 for t t ) = t e t e t + 1 for t t. (LK(, t ))(p) = e t p e t [ t p(p + 1) + 0 e qt (LK(, t ))(p)dt = 1 ] 1 + p(p + 1) 2 p(p + 1). 2 1 pq(p + q + 1) 2. (LK α (, t))(ξ)ξ = H α (ξ, t) αh α (ξ, t) e tξ H α (p, t) (p + ξ + 1) dp = e t 2 ξ + 1 ( t + 1 ) + ξ (ξ + 1) 2. ([37]) sinc method ([38]) ([3,28]) [19]) DE formula ([89]) Kryzhniy. (6.3) H K 15

56 54 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering. α Paley-Wiener h 1/300 α H ([36]) ([34]). ([3,28]) 7. [8,18,22] D.A. Hejhal[22] J.D. Fay[18] [6,45]. [55] S. Smale[15] ([83]) [78], Support Vector Machines, MIT [14] ([87-88]). [5] 16

57 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (C)(No , No , No , No , No , No ) 36,, No. 20 ( ). [1], II -, (1976). [2],, (2002). [3],,,,, 55(2003), [4] J. Agler and J. E. McCarthy, Pick Interpolation and Hilbert Function Spaces, Graduate Studies in Mathematics, Vol. 44 (2002). [5] D. Alpay, The Schur Algorithm, Reproducing Kernel Spaces and System Theory, American Mathematical Society (1998). [6] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), [7] M. Asaduzzaman, T. Matsuura, and S. Saitoh, Constructions of approximate solutions for linear differential equations by reproducing kernels and inverse problems, Advances in Analysis, Proceedings of the 4th International ISAAC Congress (2005), (World Scientific). [8] S. Bergman, The kernel function and conformal mapping, Amer. Math. Soc. Providence, R. I. (1950, 1970). [9] A. Berlinet and C. Thomas-Agnan, Reproducing kernel Hilbert spaces in probability and statistics, Kluwer Akademic Publishers, Boston/Dordrecht/London (2004). 17

58 56 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [10] A. Berlinet, Reproducing kernels in probability and statistics, The ISAAC Catani Congress Proceedings (to appear). [11] D.-W. Byun and S. Saitoh, A real inversion formula for the Laplace transform Z. Anal. Anw., 12(1993), [12] D.-W. Byun and S. Saitoh, Approximation by the solutions of the heat equation, J. Approximation Theory, 78 (1994), [13] D.-W. Byun and S. Saitoh, Best approximation in reproducing kernel Hilbert spaces, Proc. of the 2th International Colloquium on Numerical Analysis, VSP-Holland, (1994), [14] N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and other Kernel-based Learning Methods, Cambridge Univ. Press, Cambridge (2001). [15] F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc., 39 (2001), [16] G. Doetsch, Handbuch der Laplace Transformation, Vol. 1., Birkhäuser Verlag, Basel, [17] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications, 376(2000), Kluwer Academic Publishers. [18] J.D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Math., Vol. 352, Springer-Verlag (1973). [19] H. Fujiwara, T. Matsuura and S. Saitoh, Numerical real inversion formulas of the Laplace transform by using a Fredholm integral equation of the second kind, RIMS. Kyoto Univ., 1566(2007), [20] C. W. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg & Sohn Verlagsgesellschaft mbh, Braunschweig/Wiesbaden (1993). [21] N. Hayashi, Analytic function spaces and their applications to nonlinear evolution equations, Analytic Extension Formulas and their Applications, (2001), Kluwer Academic Publishers, [22] D.A. Hejhal, Theta functions, kernel functions and Abel integrals, Memoirs of Amer. Math. Soc., Vol. 129, Providence, R. I. (1972). [23] J. R. Higgins, A sampling principle associated with Saitoh s fundamental theory of linear transformations, Analytic Extension Formulas and their Applications, (2001), Kluwer Academic Publishers, [24] H. Imai, T. Takeuchi and M. Kushida, On numerical simulation of partial differential 18

59 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 57 equations in infinite precision, Advances in Mathematical Sciences and Applications, 9(1999), [25] H. Itou and S. Saitoh, Analytical and numerical solutions of linear singular integral equations, Functional Equations, Integral Equations and Differential Equations with Applications (The Tricentennial Birthday Anniversary of Leonhard EULER), Taylor & Francis Books, US [26] I.I. Hirschman and D.V. Widder, The Convolution Transform, Princeton University Press, Princeton, New Jersey, (1955). [27] V.V. Kryzhniy, Regularized inversion of integral transformations of Mellin convolution type, Inverse Problems, 19(2003), [28] V. V. Kryzhniy, Numerical inversion of the Laplace transform: analysis via regularized analytic continuation, Inverse Problems, 22 (2006), [29] T. Matsuura and S. Saitoh, Analytical and Numerical Solutions of the Inhomogeneous Wave Equation, Australian J. of Math. Anal. and Appl., 1(2004), Volume 1, Issure 1, Article 7. [30] T. Matsuura, S. Saitoh and D.D. Trong, Numerical solutions of the Poisson equation, Applicable Analysis, 83(2004), [31] T. Matsuura and S. Saitoh, Analytical and numerical solutions of linear ordinary differential equations with constant coefficients, Journal Analysis and Applications, 3(2005), [32] T. Matsuura and S. Saitoh, Numerical inversion formulas in the wave equation, Journal of Computational Mathematics and Optimization, 1(2005), [33] T. Matsuura, S. Saitoh and D.D. Trong, Approximate and analytical inversion formulas in heat conduction on multidimensional spaces, J. of Inverse and Ill-posed Problems, 13 (2005), [34] T. Matsuura and S. Saitoh, Dirichlet s Principle Using Computers, Applicable Analysis, 84(2005), [35] T. Matsuura and S. Saitoh, Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces, Applicable Analysis, 85(2006), [36] T. Matsuura and S. Saitoh, Analytical and numerical real inversion formulas of the Laplace transform, More Progresses in Analysis, World Scientific, , [37] T. Matsuura, A. Al-Shuaibi, H. Fuijiwara and S. Saitoh, Numerical real inversion formulas of the Laplace transform by using a Fredholm integral equation of the second kind, Journal of Analysis and Applications, Vol.5, ,

60 58 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [38] T. Matsuura, A. Al-Shuaibi, H. Fuijiwara, S. Saitoh and M. Sugihara, Numerical real inversion formulas of the Laplace transform by a sinc method, Far East Journal of Mathematical Sciences, Vol.27(1), 1-14, [39] Z. Nehari, Conformal mapping, McGraw-Hill Book Company Inc., New York (1952). [40] E. L. Post, Generalized diffentiation, Trans. Amer. Math. Soc., 32(1930), [41] Th. M. Rassias and S. Saitoh, The Pythagorean theorem and linear mappings, PanAmerican Math. J., 12 (2002), [42] S. Saitoh, The Bergman norm and the Szegö norm, Trans. Amer. Math. Soc., 249 (1979), [43] S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc., 89 (1983), [44] S. Saitoh, The Weierstrass transform and an isometry in the heat equation, Applicable Analysis, 16(1983), 1-6. [45] S. Saitoh, Theory of Reproducing Kernels and its Applications, Pitman Research Notes in Mathematics Series, 189 (1988), Longman Scientific & Technical, UK. [46] S. Saitoh, Interpolation problems of Pick-Nevanlinna type, Pitman Research Notes in Mathematics Series, 212 (1989), [47] S. Saitoh, Representations of the norms in Bergman-Selberg spaces on strips and half planes, Complex Variables, 19 (1992), [48] S. Saitoh, One approach to some general integral transforms and its applications, Integral Transforms and Special Functions, 3 (1995), [49] S. Saitoh, Natural norm inequalities in nonlinear transforms, General Inequalities 7(1997), Birkhäuser Verlag, Basel, Boston. [50] S. Saitoh, Representations of inverse functions, Proc. Amer. Math. Soc., 125 (1997), [51] S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications, Pitman Research Notes in Mathematics Series, 369 (1997), Addison Wesley Longman, UK. [52] S. Saitoh, Nonlinear transforms and analyticity of functions, Nonlinear Mathematical Analysis and Applications, (1998), Hadronic Press, Palm Harbor. [53] S. Saitoh, Various operators in Hilbert space induced by transforms, International J. of Applied Math., 1 (1999), [54] S. Saitoh, Applications of the general theory of reproducing kernels, Reproducing Kernels and their Applications, (1999), Kluwer Academic Publishers,

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64 62 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering

65 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 63 * * Wavelet applications for pattern recognition, in particular, for human gait of moving picture analysis Kohei Arai, Graduate School of Science and Engineering, Saga University Abstract. Wavelet applications for pattern recognition, in particular for human gait of moving picture analysis are discussed. There are some problems on pattern recognition, in particular for human gait recognition. One of these is effective feature extraction from acquired human gait of moving picture. Wavelets are applicable for time-frequency analysis. Therefore, it is expected that wavelet analysis is effective for feature extraction from the acquired moving picture. Through experiments, it is found that specific orders and levels of wavelets are effective for human gait recognitions. (Human Gait) [1] [2] / 1

66 64 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [3] [4] [5] [6] ( ) [3] [7],[8] CASIA [9]. 2.1 (1) [10]-[24] [25] [26] CASIA HFB Database 1 25fps

67 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 65 (2) ( ) 1 Figure 1. Background subtraction for silhouettes images RGB 2 1 Image_current( ) Image_background( ) 2 Image_dimension (Image_row image_column) 3

68 66 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 3 (work_thresh) 4 R G B (1) 0 5 1E R G B a) b) (2) (3) c) (4) 7, 4) 6) 8 (5) 9 O_thresh 10) O_thresh (6) 12 C(R) = A(R) B; C(G) = A(G) B; C(B) = A(B) B (7) 13 C 1 2 I (0) I (0) 3 1 t = 1 I I 4 2 4

69 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 67 Figure 2. Silhouettes results 2.2 ( ) ( ) ( ) 3 6 [29] G1 G2 G3 p 3 2 G1 G2 G3 2 p [6]

70 68 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Figure 3. Proporsional average size of human body 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 4 d 4 c 6

71 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 69 (a) (b) (c) (d) Figure 4. (a) Skeleton image per frame, (b) Skeleton image per video sequence (c) Skeleton per frame,(d) Skeleton per frame sequence. 2.3 (1) 7

72 70 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 5(a) 5(b) (a) Figure 5. (a) Skeleton image per frame, (b) Skeleton image per video sequence (b) (2) (. ) (DWT) DWT [7],[8],[12],[13],[15] [10] 2D DWT (decomposition) 2D (IDWT) (reconstruction) DWT LL( ) HL( ) LH( ) HH( ) 4 8

73 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 71 Figure 6. 1-Level Decomposition 2D DWT DWT f(t) (8) (t) (t) [7] (8) (1) (11) (9) (10) 9

74 72 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (11) (12) ( ) (SVM). 2.1 (USF) (SOTON) (CASIA) CASIA CASIA [9] 90 B CASIA B CASIA 10

75 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 73 Figure 7. Frame sample of CASIA Human Gait Dataset 2.2 (1) , ( ) eaeh 2 eaev 3 eaed 4 ehev 5 ehed 6 eved Haar o + ehev 11

76 74 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Figure 8. Result from single skeleton frame of 2 persons ehed Figure 9. Result from skeleton frame sequence of 2 persons

77 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 75 Figure 10. Result from skeleton frame sequence of 4 persons Figure 11. Result from single motion frame of 2 persons ehev 13

78 76 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Figure 12. Result from single motion frame of 4 persons ehev (2) Figure 13. Result from motion frame sequence of 4 persons 1 CASIA 14

79 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 77 TABLE I. COMPARISON OF GENDER CLASSIFICATION PERFORMANCES AMONG THE PROPOSED METHOD AND THE CONVENTIONAL METHODS Method Dataset CCR Lee and Grimson [6] 25 males & 25 females 85.0% Huang and Wang [28] 25 males & 25 females 85.0% 2D DWT Energy proposed [17] 31 males & 31 females 92.9% Li et al. [30] 31 males & 31 females 93.28% GEM Proposed 31 males & 31 females 97.63% CCR Correct Classification Ratio Lee [6] [28] CASIA B D DWT [17] Li [30] Li Gait Energy Motion: GEM (3)GEM GEM 14 Gait ( ) Figure 14 Accumulated Gait Energy Motion: GEM 15

80 78 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering f ( 13) 15 ( ) Figure 15 F-value image of the Gait Energy Motion: GEM (13) x GEM F F 3. ( ) ( ) F 16

81 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 79 [1] X. Qinghan, Technology review Biometrics Technology, Application, Challenge, and Computational Intelligence Solutions, IEEE Computational Intelligence Magazine, vol. 2, pp. 5-25, [2] Jin Wang, Mary She, Saeid Nahavandi, Abbas Kouzani, A Review of Vision-based Gait Recognition Methods for Human Identification, IEEE Computer Society, 2010 International Conference on Digital Image Computing: Techniques and Applications, pp , 2010 [3] N. V. Boulgouris, D. Hatzinakos, and K. N. Plataniotis, Gait recognition: a challenging signal processing technology for biometric identification, IEEE Signal Processing Magazine, vol. 22, pp , [4] M. S. Nixon and J. N. Carter, "Automatic Recognition by Gait", Proceedings of the IEEE, vol. 94, pp , [5] Y. Jang-Hee, H. Doosung, M. Ki-Young, and M. S. Nixon, Automated Human Recognition by Gait using Neural Network, in First Workshops on Image Processing Theory, Tools and Applications, 2008, pp [6] Wilfrid Taylor Dempster, George R. L. Gaughran, Properties of Body Segments Based on Size and Weight, American Journal of Anatomy, Volume 120, Issue 1, pages 33 54, January 1967 [7] Gilbert Strang and Truong Nguen, Wavelets and Filter Banks. Wellesley-Cambridge Press, MA, 1997, pp , [8] I. Daubechies, Ten lectures on wavelets, Philadelphis, PA: SIAM, [9] CASIA Gait Database, English/index.asp [10] Edward WONG Kie Yih, G. Sainarayanan, Ali Chekima, "Palmprint Based Biometric System: A Comparative Study on Discrete Cosine Transform Energy, Wavelet Transform Energy and Sobel Code Methods", Biomedical Soft Computing and Human Sciences, Vol.14, No.1, pp.11-19, 2009 [11] Dong Xu, Shuicheng Yan, Dacheng Tao, Stephen Lin, and Hong-Jiang Zhang, Marginal Fisher Analysis and Its Variants for Human Gait Recognition and Content- Based Image Retrieval, IEEE Transactions On Image Processing, Vol. 16, No. 11, November 2007 [12] Hui-Yu Huang, Shih-Hsu Chang, A lossless data hiding based on discrete Haar wavelet transform, 10th IEEE International Conference on Computer and Information Technology, 2010 [13] Kiyoharu Okagaki, Kenichi Takahashi, Hiroaki Ueda, Robustness Evaluation of Digital Watermarking Based on Discrete Wavelet Transform, Sixth International Conference on Intelligent Information Hiding and Multimedia Signal Processing, 2010 [14] Bogdan Pogorelc, Matjaž Gams, Medically Driven Data Mining Application: Recognition of Health Problems from Gait Patterns of Elderly, IEEE International Conference on Data Mining Workshops, 2010 [15] B.L. Gunjal, R.R.Manthalkar, Discrete Wavelet Transform based Strongly Robust Watermarking Scheme for Information Hiding in Digital Images, Third International Conference on Emerging Trends in Engineering and Technology, 2010 [16] Turghunjan Abdukirim, Koichi Niijima, Shigeru Takano, Design Of Biorthogonal Wavelet Filters Using Dyadic Lifting Scheme, Bulletin of Informatics and Cybernetics Research Association of Statistical Sciences, Vol.37, 2005 [17] Seungsuk Ha, Youngjoon Han, Hernsoo Hahn, Adaptive Gait Pattern Generation of Biped Robot based on Human s Gait Pattern Analysis, World Academy of Science, Engineering and Technology [18] Maodi Hu, Yunhong Wang, Zhaoxiang Zhang and Yiding Wang, Combining Spatial and Temporal Information for Gait Based 17

82 80 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Gender Classification, International Conference on Pattern Recognition 2010 [19] Xuelong Li, Stephen J. Maybank, Shuicheng Yan, Dacheng Tao, and Dong Xu, Gait Components and Their Application to Gender Recognition, IEEE Transactions On Systems, Man, And Cybernetics Part C: Applications And Reviews, Vol. 38, No. 2, March 2008 [20] Shiqi Yu,, Tieniu Tan, Kaiqi Huang, Kui Jia, Xinyu Wu, A Study on Gait-Based Gender Classification, IEEE Transactions On Image Processing, Vol. 18, No. 8, August 2009 [21] M.Hanmandlu, R.Bhupesh Gupta, Farrukh Sayeed, A.Q.Ansari, An Experimental Study of different Features for Face Recognition, International Conference on Communication Systems and Network Technologies, 2011 [22] S. Handri, S. Nomura, K. Nakamura, Determination of Age and Gender Based on Features of Human Motion Using AdaBoost Algorithms, 2011 [23] Massimo Piccardi, Background Subtraction Techniques: Review, ~massimo/backgroundsubtractionreview-piccardi.pdf [24] Bakshi, B., "Multiscale PCA with application to MSPC monitoring," AIChE J., 44, pp , 1998 [25] G. Huang, Y. Wang, Gender Classification Based on Fusion of Multi-view Gait Sequences, Proceedings of the Asian Conference on Computer Vision, [26] W. Kusakunniran et al., Multi-view Gait Recognition Based on Motion Regression Using Multilayer Perceptron, Proceedings of the IEEE International Conference on Pattern Recognition, pp , ~ ~1990 ( JAXA) ( ) A 2008 International Journal of Advanced Computer Science and Applications Best Paper Award 30 ( ) 31 ( ) arai@is.saga-u.ac.jp 18