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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 [email protected] 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 ( ) [email protected] 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,

61 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 59 [55] S. Saitoh, D. Alpay, J.A. Ball and T. Ohsawa (eds), Reproducing Kernels and their Applications, (1999), Kluwer Academic Publishers. [56] S. Saitoh and M. Yamamoto, Integral transforms involving smooth functions, Reproducing Kernels and their Applications, (1999), Kluwer Academic Publishers, [57] S. Saitoh, Linear integro-differential equations and the theory of reproducing kernels, Volterra Equations and Applications, C. Corduneanu and I.W. Sandberg (eds), Gordon and Breach Science Publishers (2000), Amsterdam. [58] S. Saitoh, Weighted L p -norm inequalities in convolutions, Survey on Classical Inequalities (T. M. Rassias, ed.), Kluwer Academic Publishers, 2000, pp [59] S. Saitoh, Analytic extension formulas, integral transforms and reproducing kernels, Analytic Extension Formulas and their Applications, (2001), Kluwer Academic Publishers, [60] S. Saitoh, Applications of the reproducing kernel theory to inverse problems, Comm. Korean Math. Soc., 16 (2001), [61] S. Saitoh, Principle of telethoscope, Functional-Analytic and Complex Methods, their Interaction and Applications to Partial Differential Equations, Proceedings of the International Graz Workshop, Graz, Austria, February World Scientific (2001), [62] S. Saitoh, N. Hayashi and M. Yamamoto (eds.), Analytic Extension Formulas and their Applications, (2001), Kluwer Academic Publishers. [63] S. Saitoh, Vu Kim Tuan and M. Yamamoto, Conditional Stability of a Real Inverse Formula for the Laplace Transform, Z. Anal. Anw., 20(2001), [64] S. Saitoh and M. Mori, Representations of analytic functions in terms of local values by means of the Riemann mapping function, Complex Variables, 45 (2001), [65] S. Saitoh, Vu Kim Tuan and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. of Inequalities in Pure and Applied Mathematics, 3 (2002), Article 80. [66] S. Saitoh, T. Matsuura and M. Asaduzzaman, Operator Equations and Best Approximation Problems in Reproducing Kernel Hilbert Spaces, Journal of Analysis and Applications, 1(2003), [67] S. Saitoh, Constructions by Reproducing Kernels of Approximate Solutions for Linear Differential Equations with L 2 Integrable Coefficients, International J. of Math. Sci., 2(2003), [68] S. Saitoh, Approximate Real Inversion Formulas of the Gaussian Convolution, Appli- 21

62 60 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering cable Analysis, 83(2004), [69] S. Saitoh, Best approximation, Tikhonov regularization and reproducing kernels, Kodai. Math. J., 28(2005), [70] S. Saitoh, Tikhonov regularization and the theory of reproducing kernels, Finite or Infinite Dimensional Complex Analysis and Applications (Proceedings of the 12th IC- FIDCAA), Kyushu University Press (2005), [71] S. Saitoh, Applications of reproducing kernels to best approximations, Tikhonov regularizations and inverse problems, Advances in Analysis, Proceedings of the 4th International ISAAC Congress (2005), (World Scientific), [72] S. Saitoh and M. Yamada, Inversion formulas for a linear system determined by input and response relations, by using suitable function spaces, Hokkaido University Technical Report Series in Mathematics, 118(2007), [73] M. Sakai, Analytic functions with finite Dirichlet integrals on Riemann surfaces, Acta. Math., 142 (1979), [74] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer Series in Computational Mathematics, 20, [75] N. Suita and A. Yamada, On the Lu-Qi Keng conjecture, Proc. Amer. Math. Soc., 59 (1976), [76] L. Schwartz, Sous-espaces hilbertiens d espaces vectoriels topologiques et noyaux associés (noyaux reproduisants), J. Analyse Math., 13 (1964), [77] W. Ulmer and W. Kaissl, The inverse problem of a Gaussian convolution and its application to the finite size of measurement chambers/detectors in photon and proton dosimetry, Phys. Med. Biol., 48(2003), [78] V. Vapnik, Statistical Learning Theory, John Wiley & Sons (1998). [79] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, 59, SIAM, Philadelphia, (1990). [80] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, [81] A. Yamada, Fay s trisecant formula and Hardy H 2 reproducing kernels, Reproducing Kernels and their Applications (1999), Kluwer Academic Publishers, [82] A. Yamada, Equality conditions for general norm inequalities in reproducing kernel Hilbert spaces, Advances in Analysis, World Scientific, 2005, [83] D. -X Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49(2003),

63 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 61 [84] /inforcenter/mathsource/4738/ [85] ww2040/abate.html [86] H. Fujiwara, T. Matsuura, S. Saitoh and Y. Sawano, The real inversion of the Laplace transform by numerical singular value decomposition, Journal of Analysis and Applications, Vol.6, 55-68, [87] K. Fukumizu, F.R. Bach and M.I. Jordan, Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces, Journal of Machine Learning Research 5(2004), [88] M. Cuturi, K. Fukumizu and J.P. Vert, Semigroup kernels on measures, Journal of Machine Learning Research, 6(2005), [89] H. Takahasi and M. Mori, Double exponential formulas for numerical integration, Publ. RIMS. Kyoto Univ., 10(1974), ( ) [email protected] 23

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 ( ) [email protected] 18

83 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering DCT Potentiality of Lossy Image Compression Using Haar Transform Keita Ashizawa Maizuru National College of Technology Abstract. A key issue in DCT-based lossy image compression is how to avoid edge artifacts due to the Gibbs phenomenon, such as mosquito noise. Our first approach is to predict each block using a gradient estimation, followed by applying an orthogonal transformation to the prediction error. This method is based on the Haar transform that requires only O(N) operations for an input signal of length N while the DCT requires O(N log 2 N) operations. In this study, we propose a new frequency transform scheme partially based on the Haar transform (HT) to avoid the edge artifacts due to the Gibbs phenomenon. The HT is particularly effective for the reduction of edge artifacts owing to its localized rectangular basis functions. 1. JPEG DCT [1, 2] JPEG [1] 1

84 82 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 1 Fig. 1. Fig. 2 Fig. 2. [3 7] [7] 4K DCT [7 9] 2

85 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 83 DCT M N M log 2 N [10] N JPEG DCT Fig. 3 DCT 8 8 DCT H = P(A) log 2 P(A) A Ω Ω P(A) Ω A Fig DCT DCT DCT Fig. 4 Fig DCT 3

86 84 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (a) (b) DCT (c) Fig. 3. Fig. 5 Fig. 5 8 Fig [ f 0, f 1 ] ( ) F0 = 1 ( )( ) 1 1 f0 (3.1) F f1 ( ) f0 = 1 ( )( ) 1 1 F0 (3.2) f F 1 4

87 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 85 (a) (b) DCT (c) Fig. 4. Fig. 5. F 0 F 1 2 Fig. 6 2 Fig f 0 f 1 f0 L, f 1 L, f 0 R, f 1 R 3) 4) 5

88 86 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (a) f 0 f 1 U a := ( f 0 L + f 1 L) ( f 0 R + f 1 R), 8 (b) f 0 f 1 U b := f L 1 f R 0 3. U a 2 ( ) F0 = 1 ( )( ) ( 1 1 f0 0 V f1 Ua) ( ) f0 = 1 {( )( ) ( )} 1 1 F0 Ua f V U a 2 V := F 1 U a U b (a) Fig. 6. (b) DCT Fig DCT Fig. 4 Fig DCT Fig. 8 DCT 2 Fig. 8 6

89 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 87 (a) (b) DCT (c) Fig. 7. (b) DCT (c) Fig DCT Fig. 9 (Cameraman) DCT Fig. 9(a) Fig. 7

90 88 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 9(b) 10 4 Fig. 9. Fig. 10(a) DCT 8 8 DCT Fig. 10(b) DCT-HT Fig. 10(c) HT-DCT 8 8 DCT 1 [11] 1 Fig. 10(d) HT 8 8 Fig Fig. 11(a) Airplane 1- Fig. 10(b) DCT-HT Fig. 10(c) HT-DCT Fig. 11(b) 1 HT Fig. 11(b) HT 1- Fig ( , 8bits/pixel grayscale) Table. 1 CaseA Fig. 10(d) HT CaseB 2 Table. 1 CaseA CaseB DCT Fig HT 8

91 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 89 (a) DCT (b) DCT-HT (c) HT-DCT Fig. 10. (d) HT 8 8 (a) DCT-HT, HT-DCT, HT (b) DCT Fig DCT HT 8 8 L 9

92 90 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 12. l(= 1, 2,...,L) Fig. 13 Fig. 14 DCT DCT-HT HT-DCT HT 1- ϕ k 4 l f (l) 1- k (l) (= 0, 1, 2, 3) F (l)q F (l)q k (l) k (l) ϕ 1 k (l) A = {a ij } 1 DCT p p = 1 DCT p 0 < p < 1 10

93 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 91 Table Source of Case A Case B Variation DCT HT DCT DCT-HT HT-DCT HT (a)airplane 74.2% 25.8% 40.9% 20.3% 22.3% 16.5% (b)barbara 94.6% 5.4% 77.2% 18.8% 1.7% 2.3% (c)boat 73.2% 26.8% 40.8% 36.9% 9.2% 13.1% (d)brickhouse 78.4% 21.6% 45.6% 16.5% 26.2% 11.7% (e)bridge 53.1% 46.9% 25.4% 22.1% 24.2% 28.3% (f)cameraman 58.3% 41.7% 29.4% 20.2% 24.9% 25.5% (g)goldhill 66.4% 33.6% 31.1% 24.6% 24.4% 19.9% (h)lax 31.2% 68.8% 12.8% 22.6% 15.9% 48.7% (i)lena 83.7% 16.3% 61.4% 15.6% 13.3% 9.7% (j)mandrill 79.9% 20.1% 49.7% 16.2% 24.6% 9.5% (k)parrots 83.5% 16.5% 64.6% 16.4% 10.3% 8.7% (l)pepper 82.5% 17.5% 57.3% 14.7% 18.4% 9.6% (m)sailboat 86.1% 13.9% 65.9% 11.4% 13.8% 8.9% (n)venice 55.4% 44.6% 28.3% 28.1% 15.9% 27.7% (o)woman 85.8% 14.2% 63.2% 18.5% 10.2% 8.1% DCT DCT DCT DCT 11

94 92 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 14. Fig. 13. DCT 2 DCGT DCT JPEG [4, 5] 12

95 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 93 [1] W.B. Pennebaker and J.L. Mitchell, JPEG Still Image Data Compression Standard, Van Nostrand Reinhold, New York, [2] G. K. Wallace, The JPEG still picture compression standard, IEEE Trans. Consumer Electronics, 38, 1992, xviii xxxiv. [3], vol.17 pp , 2007 [4], no.14 pp.71-79, 2009 [5], no.9 pp , 2010 [6] DCT, (A) Vol.J96-A No.7 pp , 2013 [7],, vol.65 pp , 2011 [8],, Vol.64 No.11 pp , 2010 [9],, pp , 2010 [10] W.H.Chen, C.H.Smith, Adaptive Coding of Monochrome and Color Images, IEEE Trans. Commun., COM-25,11, 1977 [11] David Salomon, Data Compression Fourth Edition, Springer Verlag, London, ( ) [email protected] 13

96 94 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering

97 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 95. Improvement of analysis of auditory evoked brain signal applied to objective hearing test Nobuko Ikawa Akira Morimoto Ryuichi Ashino Ryutsu Keizai University Osaka Kyoiku University Abstract. By applying the Karman filter to time series of EEG signal analysis, we extracted an auditory evoked brain response wave pattern shape fast. Furthermore, by using wavelet analysis we analyzed the latency and frequency of the evoked brain response which are diagnosis indecies necessary for audiometry, in a short time. On the other hand, we study from those advanced reports which are the modeling of auditory filter of the cochlea and the frequency variance analysis in functional maps of rat auditory cortex. Since the cochlear nerve and brainstem are relay parts between cochlear and auditory cortices, we verify the analysis of the auditory evoked brain responses by considering the advanced results of both cochlea auditory filter and auditory cortex frequency map. Since the mathematical technique is useful of the analysis of auditory evoked brain responses, furthermore, we hope to apply the mathematical technique to objective audiometry test to make use for extraction in the high speed Hz 30 db 4000 Hz 40 db ( [1], [2]) 125 Hz, 250 Hz, 500 Hz, 1000 Hz, 2000 Hz, 4000 Hz, 8000 Hz 1

98 96 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (pure tone) audiogram (Fig. 1) db 20 db, [ ] Fig. 1. Example of audiogram [2]. objective audiometry test [1] auditory evoked response [3] [4], [5] [6] MRI [7] Auditory Brainstem Response: ABR Auditory Steady-State Response : ASSR ABR [8] ASSR 2

99 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 97 ABR ASSR [9] [10] [1], [11] Rosenthal 2 3

100 98 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (Fig. 2) Fig. 2. Pathway of auditory evoked brain responses [3]. Fig. 3 ( Electrocochleography : Ecoch G) 3 ABR 10 ( ) (Middle Latency Response : MLR ) 100 (Slow Vertex Response : SVR 500 ABR MLR [8] ASSR (sinusoidal amplitude-modulated tone: SAM tone) 4

101 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 99 Fig. 3. Auditory pathway and auditory evoked responses [8]. 2.2 ABR, ASSR msec P50 N100 P200 P1, N1, P2 N P Negative Positive P N ABR I VII 7 On- ABR I V Fig. 4 (ISI) 600 msec 90 db P0 ABR ( slow ABR) P1=Pa P2=Pb 10 msec ABR 100 msec MLR 500 msec SVR ABR MLR [8] ABR Intensity V V peak latency V 5

102 100 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 4. Pathway of ABR and MLR [8]. I-L curve (Fig. 5) conductive deafness V I-L ( sensory deafness V I-L ( ) Fig. 5. Example of I-L curves for fifth ABR waveform used to diagnose hard of hearing patients. ASSR ASSR 40-Hz ASSR 80-Hz ASSR 40-Hz ASSR 6

103 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 101 slow ABR (P0) + MLR P0+Pa+Pb 80-Hz ASSR slow ABR P0 40-Hz ASSR 80-Hz ASSR [8] ABR 80-Hz ASSR far field potential 1/5 1/100 (Fig. 6) ABR 2000 Fig. 6. Image of averaging for the observed signal. ABR K = {k(i), 1 i m} k(i) v k(i) (t) ABR S ABR (t) k(i) x k(i) (t), 0 t 10(ms) 7

104 102 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (2.1) x k(i) (t) = S ABR (t) + v k(i) (t) k(i) v k(i) (t) 0 k(i) = 2000 x 2000 (t) = S ABR (t) [9], [10] ABR ABR ASSR 3.1 (autoregressive moving average model, ARMA) n a n b (3.1) y(t) = a i y(t i) + b i u(t i) + v(t) i=1 i=1 y(t) t u(t) v(t) v 0 σ 2 ν v(t) (3.1) a i (i = 1, 2,...,na), b i (i = 1, 2,...,nb) θ = [a T b T ] T t θ ˆθ(t) (3.2) ˆθ(t + 1) = ˆθ(t) + k(t + 1)[y(t + 1) z T (t + 1)ˆθ(t)], (3.3) z(t + 1) = [ y(t),..., y(t na + 1), u(t),...,u(t nb + 1)] T 8

105 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 103 (3.4) ˆv(t + 1) = y(t + 1) z T (t + 1)ˆθ(t), (3.5) ˆσ 2 v(t + 1) = 1 t + 1 t ˆv 2 (i + 1) i=0 (3.6) k(t + 1) = R(t)z(t + 1) ˆσ 2 v(t + 1) + z T (t + 1)R(t)z(t + 1) (3.7) R(t + 1) = [I k(t + 1)z T (t + 1)]R(t)], R(0) = γi T AIC (3.8) AIC(n) = N In σ 2 v + 2n + N(1 + In 2π) N π σ 2 v 2 n AIC n (3.1) n = n a = n b (3.9) H(z) = b 1z b n z n 1 + a 1 z a n z n z = e jω (0 ω 2π) (3.10) H ( e jω) = A(ω) = 1 + a 1 cos ω + a 2 cos 2ω a n cos nω, B(ω) = a 1 sin ω + a 2 sin 2ω a n sin nω, C(ω) = b 1 cos ω + b 2 cos 2ω +...b n cos nω, D(ω) = b 1 sin ω + b 2 sin 2ω b n sin nω (3.11) C(ω) jd(ω) A(ω) jb(ω) = H ( e jω) e jφ(ω) H ( e jω) = C 2 (ω) + D 2 (ω) A 2 (ω) + B 2 (ω) ( ) B(ω)C(ω) A(ω)D(ω) (3.12) Φ(ω) = tan 1 A(ω)C(ω) + B(ω)D(ω) 9

106 104 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ABR 2000 ABR 2000 ABR ±σ 5 6msec V Fig. 7 4 (3.2) (3.9) Fig. 7. Experimental data of the ABR waveform of types 1(a), 2(b), 3-1(c), 3-2(d) for 2000 sweep times. AIC ABR { 1 (t = 0), (3.13) u(t) = 0 (t = 1, 2, 3,...). ˆθ(0) = 0, R(0) = γi, γ= Fig. 8 a b n a = n b = ABR 4 AIC n = 8 Fig. 9 a i b i 10

107 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 105 Fig. 8. Relationship between AIC value and degree when experimental model is applied to four types of ABRs and sweep time is Hz ASSR 80-Hz ASSR *1 80-Hz ASSR Picton [12] MASTER (Multiple Auditory Stady-State Response MASTER ) 80 Hz *1 (1) 20 (2) (3) (4) No

108 106 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 9. Convergence value of coefficients a i (upper) and b i (lower). MASTER 80-Hz ASSR FFT 4 Carrier frequency: CF Hz ( ) Hz ( ) SAM (Sinusoidally Amplitude-Modulated tone) SAM, Modulation frequency: MF F-test 4 (Fig. 10) MASTER 80-Hz ASSR (3.2) (3.9) AM (3.13) ABR AIC (Fig. 11) ABR n a = n b = 8 12

109 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 107 Fig. 10. Relationship CF, MF and response power spectrum [8]. 1 n a = n b 60 n a = 11 n a = 11 and 1 n b 11 n b n b = 8 Fig. 11. Optimum degree of the transfer function of 80-Hz ASSR using MASTER data. 40-Hz ASSR 40-Hz ASSR 80-Hz ASSR 80-Hz ASSR 40-Hz ASSR 40-Hz ASSR ( [16]) 40-Hz ASSR 13

110 108 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering MASTER AM (3.13) ABR 40-Hz ASSR (3.2) (3.9) AIC (Fig. 12) n a = n b = 25 Fig. 13 a 1 = b i, i = 1, 2,...,25 0 Fig. 12. Optimum degree of the transfer function of 40-Hz ASSR using our observed data. Fig. 13. Simulation results of coefficients of the 40-Hz ASSR transfer function. 14

111 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ABR ASSR ABR Fig ABR Fig. 14. Comparison of correlation coefficients between measured and calculated ABRs when model of degree 8 is used. 80-Hz ASSR Fig. 15 MASTER db nhl Decibel Normal Hearing Level - indicates a person s hearing relative to accepted standards for normal hearing 15

112 110 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 16 Fig. 15. Comparison correlation coefficient between measured and calculated 80-Hz AS- SRs when model of degree 8 is used. Fig. 16. Simulation results of 80-Hz ASSR waveform. 40-Hz ASSR 40-Hz ASSR 50% Fig db nhl 60 db nhl 30dB nhl 3.3 ABR ABR Fig. 5 V I-L 16

113 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 111 Fig. 17. Simulation results of the measured and calculated 40-Hz ASSRs. V ABR Fig. 7 1 (Fig. 18) (Fig. 19) Fig. 18. Scheme of a discrete wavelet transform. s is the input signal, Hi D is the high pass filter, Lo D is the low-pass filter, A is the approximation, D is the detail, ca and cd are the DWT coefficients. MATLAB 1 Discrete, Stationary Wavelet DWT SWT (One-dimensional Discrete Stationary Wavelet Analysis) 1 Stationary 17

114 112 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 19. Scheme of a MRA. Wavelet SWT DWT Wavelet Wavelet Bi-orthogonal Wavelet ABR ABR ABR (de-noising) ABR DWT I-L ABR Wilson [21] ABR 1500 Hz Hz Hz Hz 1 ABR III V V 2 I III V VI VII 3 I II III IV V VI VII [23] ABR Hz Hz Hz A Hz B ( Hz) C ( Hz) A I II III IV V B I III C V ABR 1 18

115 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Hz Hz (slow component) Hz (fast component) [8] DWT SWT ABR DWT SWT Wavelet 10 ms SWT ms 500 SWT Hz SWT Hz 8 detail scales (D1-D8) final approximation (A8) Table 1 Wilson Zhang [22] Wavelet CWT Daubechies Bi-orthogonal Wavelets ABR [19] Bi-orthogonal Wavelets DWT Bior5.5 SWT Bior2.6 (Fig. 20) Table 1 wilson, Zhang DWT Fig. 20. Bi-orthogonal 2.6 and 5.5 wavelet functions. 3.4 ABR ABR 1 Wavelet 1 Wavelet 8 s=a8+d8+d7+d6+d5+d4+d3+d2+d1 D Hz ABR 19

116 114 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Table 1. Relationship between decomposition level and its frequency range. Decomposition level Wilson et al. s frequency Zhang et al. s frequency Our frequency D Hz Hz Hz D Hz Hz Hz D Hz Hz Hz D Hz Hz Hz D Hz Hz Hz D Hz A5:0-315 Hz Hz D7 A6:0-267 Hz Hz D Hz A Hz Fig ABR D5 10 ABR I V ABR I V DWT D5 ABR D6 D7 - V I VII Hz ASSR ASSR linear phase ABR Bior. 2.6 MATLAB2010a Table 2 Fig Hz ASSR 70 db nhl 2 sweeps (20 epochs) 2 sweeps 20

117 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 115 Fig. 21. Left side figures show original ABR signals recorded by conventional averaging procedure. And right side figures show the ABR signals reconstructed using D5. The cases of types 2, 3-1 and 3-2 normal ABR and the display waves are shown from top to bottom, with decreasing number of averagings from 2000 to 1. 21

118 116 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Table 2. Decomposition levels and their major frequencies of the case of 40-Hz ASSR. Level Major frequency (Hz) D D D D D D D7 4-8 A7 0-4 FFT FFT 40-Hz D1, D2,..., D7, A7 FFT D4 40-Hz ASSR D4 40-Hz ASSR D Hz ASSR ABR 40-Hz ASSR epoch Fig. 23 FFT D4 40-Hz ASSR 40 Hz Meyer [29] ψ supp ψ [A/α, Aα] α >1 A [A/α, Aα] [A/α, A] [A, Aα] [Aα n 1, Aα n+1 ] A = 38.9, α = 1.4 n = 0 Fig. 24 Fig Hz (n = 5) 110 Hz (n = 3) n = Hz g m 22

119 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 117 Fig. 22. Example of reconstracted SWT waveform and its FFT power spectrum. The case of 70 db nhl of 40-Hz ASSR averaged waveform. 23

120 118 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 23. Example of reconstracted SWT waveform and its FFT power spectrum. The case of 70 db, not averaged waveform, 5 epoches overlapped. 24

121 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 119 Fig. 24. Window function and band pass filter Hz f n (3.14) (3.14) ( f g) k = length(g m ) m=1 f k m g m. 40-Hz ASSR epoch Fridman [30] CSM component synchrony measure CSM (Synchrony Measure method) M sweep k s M,k = (s M,k [t]) t=0,..., Hz 512 s M,k S M,k [m] = 511 t=0 ( ) 2πimt s M,k [t] exp 512 S M,k [m] 2m Hz s M,k s M,k, k = 1,...,10 2m Hz angle (S M,k [m]) CSM M sweep 2m Hz CSM 1 (3.15) CS M M (m) = k=1 sin ( angle (S M,k [m]) ) k=1 cos ( angle (S M,k [m]) ) CS M M (m)

122 120 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 2m Hz n = 10 CS M M (m) > 1 n 1 n + 3 = 1 9 n m Hz 40-Hz ASSR m = 20 CS M M (m) > m 50 CS M M (m) (3.15) epoch Fig. 25 (3.14) Fig. 25 Fig. 25. Example of power spectrum of original evoked brain wave (upper graph) and applied the filter (lower graph). 4. ABR AIC ABR ABR 26

123 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering ABR I-L - V 10 Wavelet 40-Hz ASSR ABR Meyer CSM 40-Hz ASSR ABR MLR [8] [7] MEG 40-Hz ASSR (40 Hz ) 100 ms N1m AEF(Auditory Evoked Field ) N1m m MEG N1m ASSR ASSR 1 5. ABR ASSR ABR 40-Hz ASSR 27

124 122 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (C) , (C) [1], 2,, [2],, [3],,, [4],, IEICE, Technical survey, Vol.77, No.9, , [5],, ASJ tutorial,66 10, , [6],,, C 133 (3), , [7] S. Kuriki, Y. Kobayashi, T. Kobayashi, K. Tanaka, Y. Uchikawa, Steady-state MEG responses elicited by a sequence of amplitude-modulated short tones of different carrier frequencies, Hearing Research, 296, 25 35, [8],,, [9],, Journal of Signal Processing, Vol.8, No.4, pp , [10] N. Ikawa, Automated averaging of auditory evoked response waveforms using wavelet analysis, International Journal of Wavelets, Multiresolution and Information Processing (IJWMIP), Vol. 11, No. 4, (21 pages), [11],,ASJ,66 9, , [12] M. S. John, et al., MASTER: a Window program for recording multiple auditory steadystate responses, Comput. Methods Programs Biomed., 61, , [13] Audera, ( ), [14], 40-Hz, ASJ2008 (A), , [15], (AMFR), Audiology Japan, 51(5), , [16], PXI-4461, CFME, [17], 40-Hz 28

125 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 123, ASJ2009 (A), , [18] N. Ikawa, et al., A new automated audiometry device of measurement and analysis of 40-Hz auditory steady-state response, Proc. of the NCSP10, , [19] N. Ikawa, T. Yahagi and H. Jiang, Waveform analysis based on latency-frequency characteristics of auditory brainstem response using wavelet transform, Proc. of the NCSP05, , [20] N. Ikawa, T.Yahagi and H. Jiang, Waveform analysis based on latency-frequency characteristics of auditory brainstem response using wavelet transform, Journal of Signal Processing, 9 (6), , [21] W.J. Wilson, The relationship between the auditory brain-stem response and its re- constructed waveforms following discrete wavelet transformation, Clinical Neuro- physiology, , [22] R. Zhang, G. McAllister, B. Scottney, S. McClean and G. Houston, Feature extraction and classification of the auditory brainstem response using wavelet analysis, Knowledge Exploration in Life Science Informatics International Symposium KELSI, , [23],,,, ME, BME 9, 6 23, [24], Kusuma,,,, ABR ASSR Wavelet, Audiology Japan, 49(5), , [25],,,, ASSR Wavelet, Audiology Japan, 50(5), , [26],,,,,,,,, 19(3), 53 74, [27],,, 40Hz, JSIAM, 47 48, [28] N. Ikawa, A. Morimoto and R. Ashino, Waveform analysis of 40-Hz auditory steadystate response using wavelet analysis, IEEE conference of Wavelet Analysis and Pattern Recognition (ICWAPR), , [29] N. Ikawa, A. Morimoto and R. Ashino, An application of wavelet analysis to procedure of averaging waveform of 40-Hz auditory steady-state response, IEEE conference of Wavelet Analysis and Pattern Recognition (ICWAPR), 79 84, [30] J. Fridman, R. Zappulla, M. Bergeison, E. Greenblatt, L. Malis, F. Morrell, and T. Hoeppner, Application of Phase Spectral Analysis for Brain Stem Auditory Evoked Potential Detection in Normal Subjects and Patients with Posterior Fossa Tumors, Au- 29

126 124 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering diology, 23(1), , [31] K. V. Mardia, Statistics of Directional Data, Academic Press, New York, [32] K. V. Mardia, Statistics of Directional Data, Journal of the Royal Statistical Society, Series B (Methodological), 37(3), , [33] R. Galambos et al, A 40-Hz auditory potential recorded from the human scalp, Proc. Nati. Acad. Sci. USA, 78(4), , ( ) [email protected] ( ) [email protected] ( ) [email protected] 30

127 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 125 ( ) FNS GPS AIC BIC MDL Analysis of Land Deformation of Great East Japan Earthquake by Hierachical Motion Model Fitting Kenichi Kanatani Chikara Matsunaga Okayama University FOR-A IBE Co., Ltd. Abstract. Given 3-D sensing data of points slightly moving in space, we pose and discuss the problem of discerning whether or not translation, rotation, and scale change take place and to what extent. Here, we propose a new method for fitting various motion models to 3-D noisy data. Based on the observation that subgroups of the 3-D affine transformations are defined by imposing various internal constraints on the variables, our method fits 3-D affine transformations with internal constraints using the scheme of EFNS, which, unlike conventional methods, dispenses with introducing particular parameterizations for particular motion models. We apply our method to the GPS geodetic data of the land deformation in northeast Japan, where a massive earthquake took place on 11 March 2011, and show how model selection using the geometric AIC, the geometric BIC and the geometric MDL works

128 126 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 3 AIC [5] BIC [7] MDL [6] x, y, z [9,14] [12] FNS [11] 2 F [3] F det F = 0 [3] FNS FNS 3 FNS 3 FNS 2

129 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 127 [13] 2 /3 3 GPS [8] 2. 3 r = (x, y, z), r = (x, y, z ) A t (2.1) r = Ar + t A t u 1 u 2 u 3 (2.2) A = u 4 u 5 u 6, t = u 7 u 8 u 9 u 10 L 0 u 11 L 0 u 12 L 0 L 0 r, r r, r 0 1 (2.1) (2.3) u 0 x = u 1 x + u 2 y + u 3 z + u 10 L 0, u 0 y = u 4 x + u 5 y + u 6 z + u 11 L 0, u 0 z = u 7 x + u 8 y + u 9 z + u 12 L 0 u 0 = 1 u 0 (2.1) 1 13 u, ξ (1), ξ (2), ξ (3) (2.4) u = (u 1, u 2, u 3, u 4, u 5, u 6, u 7, u 8, u 9, u 10, u 11, u 12, u 0 ), (2.5) ξ (1) = (x/l 0, y/l 0, z/l 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, x /L 0 ), ξ (2) = (0, 0, 0, x/l 0, y/l 0, z/l 0, 0, 0, 0, 0, 1, 0, y /L 0 ), ξ (3) = (0, 0, 0, 0, 0, 0, x/l 0, y/l 0, z/l 0, 0, 0, 1, z /L 0 ) (2.3) (2.6) (ξ (1), u) = 0, (ξ (2), u) = 0, (ξ (3), u) = 0 a, b (a, b) 3

130 128 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 3. 3 φ 1 (u),..., φ r (u) φ 1 (u) = 0,..., φ r (u) = 0 1 A (2.3) A 2 0 (3.1) φ 1 (u) = u 1 u 4 + u 2 u 5 + u 3 u 6, φ 2 (u) = u 4 u 7 + u 5 u 8 + u 6 u 9, φ 3 (u) = u 7 u 1 + u 8 u 2 + u 9 u 3, φ 4 (u) = u u2 2 + u2 3 u2 4 u2 5 u2 6, φ 5 (u) = u u2 5 + u2 6 u2 7 u2 8 u2 9, φ 6 (u) = u u2 2 + u2 3 u2 0 φ 6 (u) = 0 A (2.3) 2 A = I, t = 0, s = φ 7 (u) = u 2, φ 8 (u) = u 3, φ 9 (u) = u 4, φ 10 (u) = u 6, φ 11 (u) = u 7, φ 12 (u) = u 8, φ 13 (u) = φ 13 (u) = u 1 u 5, φ 14 (u) = u 5 u 9, φ 15 (u) = u 1 u 0, (3.2) φ 16 (u) = u 10, φ 17 (u) = u 11, φ 18 (u) = u 12 φ 15 (u) = 0 A I φ 16 (u) = 0, φ 17 (u) = 0, φ 18 (u) = 0 u U 13 R 13 u 0 = 1 (2.6) u u = 1 u u 0 = 1 U (3.3) U = {u u = 1,φ 1 (u) = 0,..., φ r (u) = 0} R 13 u = 1 R 13 φ k (u) = 1 R 13 U r r U 4

131 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 129 u = 1 u φ k (u) = 0 u φ k { } L (3.4) N u = {u, u φ 1,..., u φ r } L U u T u (U) T u (U) N u M u (3.5) M u = { u φ 1,..., u φ r } L N u (3.6) N u = {u} L M u FNS φ k (u) = 0 ( u φ k, u) = 0 φ k (u) D k t φ k (tu) = t D k φ k (u) t ( u φ k (tu), u) = D k t Dk 1 φ k (u) t = 1 ( u φ k, u) = D k φ k (u) (3.3) U (3.7) U = {u u = 1, u M u } 4. 3 N r α, r α (α = 1,..., N) σ 2 V 0 [r α ], σ 2 V 0 [r α] σ V 0 [r α ], V 0 [r α] σ V 0 [r α ] σ V 0 [r α ] r α, r α (2.5) ξ(k) ξ (k) α Δξ (k) α (2.5) Δx α, Δy α, Δx α, Δy α (4.1) Δξ (1) α = T 1 Δx α Δy α Δz α Δx α Δy α Δz α, Δξ (2) α = T 2 5 Δx α Δy α Δz α Δx α Δy α Δz α, Δξ (3) α = T 3 Δx α Δy α Δz α Δx α Δy α Δz α

132 130 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering T k (4.2) T 1 = 1 L 0 ( IOOO 0 OOOO i T 2 = 1 L 0 ( O I O O 0 OOOO j T 3 = 1 L 0 ( OO I O 0 OOOO k ), ), ) O i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) ξ (k) α ξ (l) α (4.3) σ 2 V (kl) 0 [ξ α ] = E[Δξ (k) α Δξ (l) α ] = σ 2 T k ( V0 [r α ] O O V 0 [r α] ) T l 5. ξ (k) α (5.1) J = ξ (k) α N ( α=1 ξ (1) α ξ (1) α ξ (2) α ξ (2) α ξ (3) α ξ (3) α, ξ (k) α λ (k) α ξ (1) α, ξ (2) α, ξ (3) α (5.2) V (11) 0 [ξ α ] V (12) 0 [ξ α ] V (13) V (21) 0 [ξ α ] V (22) 0 [ξ α ] V (23) V (31) 0 [ξ α ] V (32) 0 [ξ α ] V (33) 0 [ξ α ] 0 [ξ α ] 0 [ξ α ] N α=1 0 V (11) 0 [ξ α ] V (12) 0 [ξ α ] V (13) V (21) 0 [ξ α ] V (22) 0 [ξ α ] V (23) V (31) 0 [ξ α ] V (32) 0 [ξ α ] V (33) 0 [ξ α ] 0 [ξ α ] 0 [ξ α ] 1 ξ (1) α ξ (1) α ξ (2) α ξ (2) α ξ (3) α ξ (3) α (2.6) 3 k=1 λ(k) α ( ξ (k) α, u) 1 ξ (1) α, ξ (2) α, ξ (3) α (5.3) ξ (k) α = ξ (k) α + ( ξ (k) α, u) = 0 3 l=1 ξ (1) α ξ (1) α ξ (2) α ξ (2) α ξ (3) α ξ (3) α λ (l) α V (kl) 0 [ξ α ]u λ (1) α u λ (2) α u λ (3) α u = 0 ) (5.4) 3 l=1 λ (l) α (u, V (kl) 0 [ξ α ]u) = (ξ (k) α, u) λ (k) α 1 6

133 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 131 (5.5) λ (k) α = (u, V (kl) 0 [ξ α ]u) (kl) 3 l=1 W α (kl) (ξ (l) α, u) (5.6) V α = ( (u, V (kl) 0 [ξ α ]u) ) W (kl) α V 1 α (5.7) W (kl) α (kl) = ( (u, V (kl) 0 [ξ α ]u) ) 1 (5.5) (5.3) (5.1) J (5.8) J = N 3 α=1 k,l=1 W α (kl) (ξ (k) α, u)(ξ (l) α, u) u U 6. J J u U (Fig. 1(a)). A 3 3 AA = I, det A = 1 [4] u A [9, 14] U J (Fig. 1(b)) ( U (Fig. 1(c)) U U U Fig. 1. (a) (b) (c) U (a) (b) (c) 7

134 132 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering FNS [11] F det F = 0 [3] FNS (5.8) J u (6.1) u J = N 3 α=1 k,l=1 u W α (kl) (ξ (k) α, u)(ξ (l) α, u) + N 3 α=1 k,l=1 W α (kl) (ξ (l) α, u)ξ (k) α u W α (kl) W α (5.6) V α V α W α = I u i (6.2) V α u i W α + V α W α u i = O (6.3) W α u i = V 1 V α V α α W α = W α W α u i u i (kl) (5.6) (6.4) u W (kl) α = 3 m,n=1 W α (km) u (u, V (mn) 0 [ξ α ]u)w α (nl) = 2 (6.1) (6.5) u J = ( N 3 α=1 k,l=1 (6.6) v (k) α = M, L ) ( N W α (kl) ξ α (k) ξ (l) α u = 3 l=1 3 α=1 m,n=1 W α (kl) (ξ (l) α, u) 3 m,n=1 W (km) α v (m) α v (n) α V (mn) 0 [ξ α ] ) u W α (nl) V (mn) 0 [ξ α ]u (6.7) M = N 3 α=1 k,l=1 W α (kl) ξ (k) α ξ (l) α, L = N 3 α=1 k,l=1 v (k) α v (l) α V (kl) 0 [ξ α ] (6.5) (6.8) u J = (M L)u 8

135 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering J u U ( R 13 ) J u J u U U J (3.4) N u u J N u (5.8) 0 u J u J u (3.6) u J N u (7.1) u J M u M u M u P M (7.2) P M u J = 0 (3.7) u U u M u P M (7.3) P M u = u (7.2), (7.3) u (7.2) (6.8) (7.4) P M (M L)u = 0 (7.3) u (7.5) P M (M L)P M u = 0 P M X (7.6) X = P M (M L)P M (7.2), (6.8) u (7.7) Xu = 0, P M u = u u 1. u 2. (6.7) M, L 3. u φ 1 (u),..., u φ r (u) {u 1,..., u r } P M r (7.8) P M = I u k u k k=1 9

136 134 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 4. (7.6) X 5. Xv = λv r + 1 v 0,..., v r 6. u ˆN u = {v 0,..., v r } L û (7.9) û = r (u, v k )v k k=0 7. u (7.10) u = N[P M û] N[ ] (N[a] = a a ) 8. u u u u N[u + u ] (2) (5) r + 1 r + 1 Chojnacki [2] FNS Kanatani [11] FNS [10, 11]. (8) u u (2) (u + u)/2 N[u + u] Chojnacki [2] FNS Kanatani [11] FNS 8. (6) ˆN u X (7.8) P M P M u k = 0, k = 1,..., r (7.6) X u 1,..., u r X 0 (5) r + 1 v 0,..., v r r 0 ˆN u X v 0 λ ( 0) M u = {u 1,..., u r } L v M u (7.9) û ˆN u = {v } L M (7.10) u ˆN u M u u ±v u = u = ±v v λ Xu = λ u u (8.1) (u, Xu) = λ ( 0) 10

137 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 135 u (= ±v ) X 0 M u (8.2) P M u = u. (8.3) (u, Xu) = (u, P M (M L)P M u) = (u, (M L)u) = (u, Mu) (u, Lu) (u, Mu) = (u, Lu) (8.4) (8.5) (u, Mu) = (u, Lu) = = = = N 3 α=1 k,l=1 N 3 α=1 k,l=1 N α=1 k,l=1 m=1 N W α (kl) (ξ (k) α, u)(ξ (l) α, u), v (k) α v (l) α (u, V (kl) 0 [ξ α ]u) 3 ( 3 3 ( 3 α=1 m,n=1 k,l=1 N 3 α=1 m,n=1 W (mn) α W (km) α W (mk) α (ξ (m) α V (kl) α, u) )( 3 W (ln) α n=1 (ξ (m) α, u)(ξ (n) α, u) W α (ln) (ξ (n) α, u) ) V α (kl) ) (ξ (m) α, u)(ξ (n) α, u) W α = V 1 α W αv α W α = W α (u, Xu) = 0 (8.1) ˆN u X 0 ˆN u X u = u = ±v ˆN u Xu = 0 (8.2) u (7.7) 9. GPS GPS Table X, Y, and Z ( ) *1 Table 1 ID 0036, 0172, 0175, 0549, 0550, 0914, 0916, 0918 Fig. 2 σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 Table 2 σ 31 σ 32 σ *1 ( 11

138 136 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 39 00' Kesennuma ' Towa ' Shizugawa 0175 Minamikata 0916 Kahoku ' Onagawa 0036 Yamoto 0549 Oshika ' 38 00' km /3/11 M ' ' ' ' ' Fig : 1. : φ 1 (u),..., φ 5 (u) 2. : φ 1 (u),..., φ 6 (u) 3. : φ 1 (u),..., φ 5 (u), φ 16 (u), φ 17 (u), φ 18 (u) 4. : φ 7 (u),..., φ 14 (u) 5. : φ 1 (u),..., φ 6 (u), φ 16 (u), φ 17 (u), φ 18 (u) 6. : φ 7 (u),..., φ 15 (u) 7. : φ 7 (u),..., φ 14 (u), φ 16 (u), φ 17 (u), φ 18 (u) 8. : φ 7 (u),..., φ 18 (u) Fig. 3 R t s ( [9, 14] t affinity similarity rigid motion rotation+scale translation+scale rotation translation scale identity Fig

139 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 137 Table GPS ( April 2010 ID x y z January 2011 ID x y z January 2012 ID x y z s R [4] R = I t = 0 s = 1 L 0 = 1000 AIC, BIC, MDL [5 7] BIC MDL (9.1) G-AIC = ˆ J + 2(3N + p)ˆσ 2, G-BIC = G-MDL = ˆ J (3N + p)ˆσ 2 log σ2 L 2 0 ˆ J J N 8 p 0, 1,.., 8 p = 12, 7, 6, 4, 4, 3, 3, 1, 0 ˆσ 2 σ 2 1,...,

140 138 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Table 2. 1 GPS ( 10 8 ) April 2010 ID σ σ σ σ σ σ January 2011 ID σ σ σ σ σ σ January 2012 ID σ σ σ σ σ σ [4] (9.2) ˆσ 2 = ˆ J 0 3N 12 ˆ J 0 0 AIC AIC [1] AIC Kullback-Leibler N σ 0 AIC [5] BIC Schwarz BIC [16] Schwarz BIC N σ 0 BIC [7] MDL Rissanen MDL [15] Rissanen MDL N σ 0 MDL [6] ( , , ) L 0 =

141 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 139 (9.3) x y z = x y z GPS WGS84 (World Geodetic System 1984) GMT (Generic Mapping Tools) *2 Fig J G-AIC, G-BIC (= G-MDL) Table 3 AIC BIC MDL BIC MDL AIC Kesennuma 0172 Towa 0914 Minamikata 0916 Shizugawa 0175 Kahoku 0918 Yamoto 0549 Onagawa 0036 Oshika 0550 Fig *2 ( 15

142 140 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Table J, G-AIC, G-BIC (= G-MDL). model J G-AIC G-BIC/MDL x y = z (9.4) Fig. 4 Fig Fig. 2 Table 4 AIC BIC MDL x y z Kesennuma 0172 Towa 0914 Minamikata 0916 Shizugawa 0175 Kahoku 0918 Yamoto 0549 Onagawa 0036 Oshika 0550 Fig

143 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 141 Table J, G-AIC, G-BIC (= G-MDL). model J G-AIC G-BIC/MDL FNS GPS GPS Istanbul Orhan Akyilmaz ) [1] H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control ( ),

144 142 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering [2] W. Chojnacki, M. J. Brooks, A. van den Hengel and D. Gawley, On the fitting of surfaces to data with covariances, IEEE Trans. Patt. Anal. Mach. Intell., ( ), [3] R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision, 2nd ed., Cambridge University Press, Cambridge, U.K., [4] K. Kanatani, Statistical Optimization for Geometric Computation: Theory and Practice, Elsevier Science, Amsterdam, the Netherlands, 1996; Reprinted, Dover, New York, NY, U.S.A., [5] K. Kanatani, Geometric information criterion for model selection, Int. J. Comput. Vis., 26-3 ( /03), [6] K. Kanatani, Uncertainty modeling and model selection for geometric inference, IEEE Trans. Patt. Anal. Mac. Intell., ( ), [7] K. Kanatani, Geometric BIC, IEICE Trans. Inf. & Syst., Vol. E93-D-1 (2010-1), [8] K. Kanatani and C. Matsunaga, Computing internally constrained motion of 3-D sensor data for motion interpretation, Pattern Recognition, 46-6 (2013-6), [9] K. Kanatani and H. Niitsuma, Optimal computation of 3-D similarity: Gauss-Newton vs. Gauss-Helmert. Comp. Stat. Data Anal., ( ), [10] K. Kanatani and Y. Sugaya, Performance evaluation of iterative geometric fitting algorithms. Comp. Stat. Data Anal ( ), [11] K. Kanatani and Y. Sugaya, Compact fundamental matrix computation, IPSJ Trans. Comput. Vis. Appl. 2 (2010-3), [12],,, 3,, 2011-CVIM (2011-3), 1 8. [13], 2 /3 18, , IS4-04, pp [14] H. Niitsuma and K. Kanatani, Optimal computation of 3-D rotation under inhomogeneous anisotropic noise, Proc. 12th IAPR Conf. Machine Vis. Appl. June 13 15, 2011, Nara, Japan, pp [15] J. Rissanen, Stochastic Complexity in Statistical Inquiry, World Scientific, Singapore, [16] G. Schwarz, Estimating the dimension of a model, Annals Statis., 6-2 (1987-7),

145 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 143 ( ) [email protected] (( ) ) [email protected] 19

146 144 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering

147 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 145. Digital watermarking methods for still images based on the wavelet transform Teruya Minamoto Saga University Abstract. Digital watermarking refers to specific information hiding techniques whose purpose is to embed secret information inside multimedia content, like images, video, or audio data. The watermark is typically added to a specific field in the original content to protect its copyright. We have developed several digital watermarking methods for copyright protection and authentication based on the wavelet transforms. We must develop a robust watermarking method against several attacks for copyright protection, whereas we must develop a semi-fragile watermarking for authentication. We present our digital watermarking methods for both copyright protection and authentication, and discuss the problems what kind of the wavelet transforms we should use, or how to utilize frequency components produced by the wavelet transforms. 1. [1] Robust 1

148 146 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fragile JPEG Semi-Fragile [3], [4] (Dual-Tree Complex Discrete Wavelet Transform, DT-CDTW) 2. DT-CDWT (Interval Arithmetic, IA) 2.1 (DT-CDWT) [7] { f l } φ R (t k) φ I (t k) f n = f (n), n Z f (t) = {c R 0,k φr (t k) + c I 0,k φi (t k)}, (2.1) φ(t) φ(t) k c R 0,k = 1 f l φ 2 R (l k), c0,k I = 1 f l φ 2 I (l k), l DT-CDWT c R j 1,n = a R 2n k cr j,k, dr j 1,n = b R 2n k cr j,k, (2.2) k c I j 1,n = a I 2n k ci j,k, k k di j 1,n = b I 2n k ci j,k, k l j = 0, 1, 2,..., {a R n}, {b R n} {a I n}, {b I n} DT- 2

149 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 147 CDWT c R j,n = {g R n 2k cr j 1,k + hr n 2k dr j 1,k }, (2.3) k c I j,n = {g I n 2k ci j 1,k + hi n 2k di j 1,k }. k {g R n }, {hr n } {gi n }, {hi n } (2.3) 0 {c R 0,n }, {ci 0,n } (2.1) f n = f (n) { f n } 2.2 (IA) A = [a 1, a 2 ] = {t a 1 t a 2, a 1, a 2 R} A inf(a) = a 1 sup(a) = a 2 w(a) = a 2 a 1 2 A = [a 1, a 2 ] B = [b 1, b 2 ] [5] (2.4) A + B = [a 1 + b 1, a 2 + b 2 ], A B = [a 1 b 2, a 2 b 1 ], A B = [min{a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 }, max{a 1 b 1, a 1 b 2, a 2 b 1, a 2 b 2 }], A/B = [a 1, a 2 ] [1/b 2, 1/b 1 ], 0 B. (2.4), [2] 3. DT-CDWT (2.2) DT-CDWT I(c R j 1,n ) = I(Δ k )a R 2n k cr j,k, I(dR j 1,n ) = I(Δ k )b R 2n k cr j,k, (3.1) k I(c I j 1,n ) = I(Δ k )a I 2n k ci j,k, k k I(dI j 1,n ) = I(Δ k )b I 2n k ci j,k, I(Δ k ) = [1 Δ k, 1 +Δ k ] Δ k c R j 1,n I(cR j 1,n ), dr j 1,n I(dR j 1,n ), ci j 1,n I(cI j 1,n ), d I j 1,n I(dI j 1,n ) (2.2), (3.1) (2.3) Fig.1 C D, E, F R I 3 k

150 148 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig DT-CDWT 4. [3] C 0 DT-CDWT 16 I(C RR 1 ) I(DRR 1 ) I(ERR 1 ) I(FRR 1 ) I(C RI 1 ) I(DRI 1 ) I(ERI 1 ) I(FRI 1 ) I(CIR 1 ) I(DIR 1 ) I(EIR 1 ) I(FIR 1 ) I(CII 1 ) I(DII 1 ) I(E II 1 ) I(FII 1 ) 2. Step1 16 I(S i )(i = 1, 2,..., T, 1 < T < 16) 3. (4.1) S i = sup(i(s i)) 4. C 0 DT-CDWT 16 C 1 RR DRR 1 ERR 1 FRR 1 CRI 1 DRI 1 ERI 1 FRI 1 CIR 1 DIR 1 EIR 1 4

151 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 149 F IR 1 CII 1 DII 1 EII 1 FII 1 5. (2K + 1) (2L + 1) (4.2) S i (m, n) = sgn(s i (m, n)) 1 1 2K + 1 2L + 1 K L k= K l= L K L sgn(x) (4.3) 1 (x > 0) sgn(x) = 0 (x = 0). 1 (x < 0) S i (m + k, n + l), 6. S i W (4.4) S i (m, n) = S i (m, n)(1 + βw(m, n)) 0 <β<1 7. Step4 DT-CDWT Step5 DT-CDWT C C 0 DT-CDWT 16 C 1 RR RR D 1 ẼRR RR 1 F 1 C 1 RI RI D 1 ẼRI 1 F 1 RI IR IR C 1 D 1 ẼIR IR II II 1 F 1 C 1 D 1 ẼII II 1 F 1 S i 2. S i (2K + 1) (2L + 1) S i 3. S i S i 5

152 150 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering (4.5) (4.6) (4.7) W i = sgn( S i S i ), T W e = W i, i=1 W = sgn( W e ). 5. [4] Wavelet Transform Modulus Maxima (WTMM) 1 1 Fig.2 Fig.3 Components except S i The host image DT-CDWT Compute WTMM Generate watermark Embed DT-CDWT and IA S i Inverse DT-CDWT The watermarked image Fig. 2. The test image DT-CDWT Extracted Watermark W~ Compute WTMM Generate image W Authentic? (d<?) Yes The tampered areas Locate tamperd areas No Fig. 3. 6

153 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig.4 SIDBA bit Boat Pepper Lenna Woman Fig MATLAB 1 INTLAB [6] Boat Papper Lenna Woman Fig. 4. Fig PSNR 30 Δ k = 0.02 β = 0.9 K = L = 10 I(D RR 1 ) I(DRI 1 ) I(DIR 1 ) I(DII 1 ) Fig.6 Fig.6 PSNR K 2 7

154 152 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering PSNR = PSNR = PSNR = PSNR = Fig Fig.7 Fig.7 K K Fig Fig.8 Fig.8 K 8

155 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 153 Fig JPEG Fig.9 JPEG Fig.9 Fig.9 JPEG 15% K K 18.88% 19.53% 17.41% 21.96% 15.43% 14.52% 14.36% 17.25% Fig. 9. Lenna Woman Boat Pepper JPEG JPEG2000 Fig.10 JPEG2000 Fig.10 Fig.10 JPEG % K K 9

156 154 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 10.81% 9.82% 7.56% 10.79% 6.71% 7.71% 5.41% 7.61% Fig. 10. Lenna Woman Boat Pepper ) JPEG Fig.11 Fig.11 K 5 Fig Fig.12 Fig.12 K 10

157 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 155 Fig bit RGB G I(D RR 1 ), I(DRI 1 Fig.13 WTMM ), I(DIR 1 ), I(DII 1 ) Fig. 13. WTMM Fig.14 Fig.15 11

158 156 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering Fig. 14.,,, Fig Fig.16, 17,. 12

159 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering 157 Fig. 16. Fig

160 158 Proceedings of the MEXT & OKU 2013 Workshop on Wavelet Theory and its Applications to Engineering PSNR Wavelet Wavelet JSPS [1] Cox, I. J., Miller, M. L., Bloom, J. A., Fridrich, J., Kalker, T.: Digital Watermarking and Steganography. Morgan Kaufmann Publishers (2008) [2] Minamoto, T., Aoki, K.: A blind digital image watermarking method using interval wavelet decomposition. International Journal of Signal Processing, Image Processing and Pattern Recognition 3(2), (2010) [3] Minamoto, T., Ohura, R.: A Blind Digital Image Watermarking Method Based on the Dual-Tree Complex Discrete Wavelet Transform and Interval Arithmetic, Lecture Notes in Computer Science 7259, , Springer-Verlag (2012) [4] Minamoto, T., Ohura, R. : A Digital Image Watermarking for Authentication Based on The Dual-Tree Complex Discrete Wavelet Transform and Interval Arithmetic, International Journal of Wavelets, Multiresolution and Information Processing, Vol.11, Issue 4, , 19pages, DOI: /S (2013) [5] Moore, R. E., Kearfott, R. B., Cloud, M. J.: Introduction to interval Analysis. SIAM (2009) [6] Rump, S. M.: INTLAB - interval Laboratory. rump/intlab/ [7] Toda, H., Zhang, Z.: Perfect translation invariance with a wide range of shapes of Hilbert transform pairs of wavelet bases. International Journal of Wavelets, Multiresolution and Information Processing 8(4), (2010) 14

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