857:. Since the designed filter has a lower regularity, worse flatness and wider passband, the resulting dual low pass filter has a higher regularity, better flatness and narrower passband. Besides, if the passband of the starting biorthogonal coiflet is narrower than the target synthesis filter G0, then its passband is widened only enough to match G0 in order to minimize the impact on smoothness (i.e. the analysis filter H0 is not invariably the design filter). Similarly, if the original coiflet is wider than the target G0, than the original filter's passband is adjusted to match the analysis filter H0. Therefore, the analysis and synthesis filters have similar bandwidth.
471:
In order to construct a biorthogonal nearly coiflet base, the
Pixstream Incorporated begins with the (max flat) biorthogonal coiflet base. Decomposing and reconstructing low-pass filters expressed by Bernstein polynomials ensures that the coefficients of filters are symmetric, which benefits the
99:
are not sufficient for excellent image compression. Common filter banks prefer filters with high regularity, flat passbands and stopbands, and a narrow transition zone, while
Pixstream Incorporated proposed filters with wider passband by sacrificing their regularity and passband flatness.
51:
Nowadays, a large amount of information is stored, processed, and delivered, so the method of data compressing—especially for images—becomes more significant. Since wavelet transforms can deal with signals in both space and frequency domains, they compensate for the deficiency of
868:
and undershoot) and shift-variance of image compression might be alleviated by balancing the passband of the analysis and synthesis filters. In other word, the smoothest or highest regularity filters are not always the best choices for synthesis low pass filters.
472:
image processing: If the phase of real-valued function is symmetry, than the function has generalized linear phase, and since the human eyes are sensitive to symmetrical error, wavelet base with linear phase is better for image processing application.
615:
790:
877:
The idea of this method is to obtain more free parameters by despairing some vanishing elements. However, this technique cannot unify biorthogonal wavelet filter banks with different taps into a
286:
173:
466:
409:
352:
318:
831:
1089:
Yang, X (January 2011). "General framework of the construction of biorthogonal wavelets based on
Bernstein bases: theory analysis and application in image compression".
202:
135:
660:
432:
375:
248:
225:
485:
1073:
668:
75:, and can be achieved by maximizing the total number of vanishing moments and distributing them between the analysis and synthesis
1162:
807:
coefficients determines the vanishing moments of wavelet functions. By sacrificing a zero of the
Bernstein-basis filter at
1001:
Tian, J (1997). "Coifman
Wavelet Systems: Approximation, Smoothness, and Computational Algorithms". In M. Bristeau (ed.).
36:
1157:
854:
1050:
937:
878:
1116:
Liu, Zaide (2007). "Parametrization construction of biorthogonal wavelet filter banks for image coding".
253:
140:
974:
929:
804:
476:
92:
29:
942:
437:
380:
323:
291:
17:
810:
1133:
882:
865:
1069:
955:
850:
178:
111:
84:
64:
53:
40:
1125:
1098:
1061:
1032:
947:
861:
987:
627:
983:
76:
71:
are such a kind of filter which emphasizes the vanishing moments of both the wavelet and
1023:
L. Winger, Lowell (2001). "Biorthogonal nearly coiflet wavelets for image compression".
933:
414:
357:
230:
207:
1036:
1151:
905:
Ke, Li. "The
Correlation between the Wavelet Base Properties and Image Compression".
610:{\displaystyle B_{k}^{n}(x)=(_{k}^{n})x^{k}(1-x)^{n-k}{\text{ for }}k=1,2,\ldots ,n,}
88:
80:
79:. The property of vanishing moments enables the wavelet series of the signal to be a
1137:
920:
Villasenor, John (August 1995). "Wavelet filter evaluation for image compression".
907:
2007 International
Conference on Computational Intelligence and Security Workshops
95:
with maximized vanishing moments have also been proposed. However, regularity and
1065:
842:
1129:
1102:
621:
96:
63:
filter design prefers filters with high regularity and smoothness to perform
959:
846:
833:(which sacrifices its regularity and flatness), the filter is no longer
838:
834:
72:
68:
60:
32:
25:
951:
662:. Besides, the Bernstein form of a general polynomial is expressed by
35:
wavelet bases, but sacrifices its regularity to increase the filter's
841:. Then, the magnitude of the highest-order non-zero Bernstein basis
785:{\displaystyle H1(x)=\sum _{k=0}^{n}d(k)(_{k}^{n})x^{k}(1-x)^{n-k},}
803:) are the Bernstein coefficients. Note that the number of zeros in
83:
presentation, which is the reason why wavelets can be applied for
1060:. Geometry and Computing. Vol. 1. 2008. pp. 249–260.
1058:
Pythagorean-Hodograph Curves: Algebra and
Geometry Inseparable
108:
The biorthogonal wavelet base contains two wavelet functions,
28:
bases proposed by Lowell L. Winger. The wavelet is based on
56:
and emerged as a potential technique for image processing.
976:
Coiflet-type wavelets: Theory, design, and applications
853:
and reconstruction, analysis filters are determined by
813:
671:
630:
488:
440:
417:
383:
360:
326:
294:
256:
233:
210:
181:
143:
114:
825:
784:
654:
609:
460:
426:
403:
369:
346:
312:
280:
242:
219:
196:
167:
129:
982:(PhD thesis). The University of Texas at Austin.
8:
1003:Computational Science for the 21st Century
941:
812:
767:
745:
732:
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629:
572:
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520:
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209:
180:
145:
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113:
288:relates to the lowpass synthesis filter
894:
204:relates to the lowpass analysis filter
1025:Signal Processing: Image Communication
922:IEEE Transactions on Image Processing
845:is increased, which leads to a wider
7:
1018:
1016:
1014:
1012:
1005:. New York: Wiley. pp. 831–840.
900:
898:
320:and the high pass synthesis filter
281:{\displaystyle {\tilde {\psi }}(t)}
168:{\displaystyle {\tilde {\psi }}(t)}
1118:Signal, Image and Video Processing
227:and the high pass analysis filter
14:
354:. For biorthogonal wavelet base,
22:biorthogonal nearly coiflet bases
849:. On the other hand, to perform
764:
751:
738:
724:
720:
714:
684:
678:
649:
637:
557:
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118:
1:
1037:10.1016/S0923-5965(00)00047-3
620:which can be considered as a
461:{\displaystyle {\tilde {H0}}}
404:{\displaystyle {\tilde {G0}}}
347:{\displaystyle {\tilde {G0}}}
313:{\displaystyle {\tilde {H}}0}
39:, which might lead to better
1066:10.1007/978-3-540-73398-0_11
826:{\displaystyle \omega =\pi }
1179:
411:are orthogonal; Likewise,
1130:10.1007/s11760-007-0001-z
1103:10.1049/iet-cvi.2009.0083
881:expression based on one
197:{\displaystyle \psi (t)}
130:{\displaystyle \psi (t)}
624:f(x) over the interval
137:and its couple wavelet
827:
786:
710:
656:
611:
479:are defined as below:
462:
428:
405:
371:
348:
314:
282:
244:
221:
198:
169:
131:
1163:Biorthogonal wavelets
1051:"The Bernstein Basis"
828:
787:
690:
657:
655:{\displaystyle x\in }
612:
477:Bernstein polynomials
468:are orthogonal, too.
463:
429:
406:
372:
349:
315:
283:
245:
222:
199:
170:
132:
93:biorthogonal wavelets
811:
669:
628:
486:
438:
415:
381:
358:
324:
292:
254:
231:
208:
179:
141:
112:
1091:IET Computer Vision
934:1995ITIP....4.1053V
737:
530:
503:
18:applied mathematics
973:Wei, Dong (1998).
823:
782:
723:
652:
607:
516:
489:
458:
427:{\displaystyle G0}
424:
401:
370:{\displaystyle H0}
367:
344:
310:
278:
243:{\displaystyle G0}
240:
220:{\displaystyle H0}
217:
194:
165:
127:
54:Fourier transforms
1158:Image compression
1075:978-3-540-73397-3
952:10.1109/83.403412
883:degree of freedom
855:synthesis filters
851:image compression
575:
455:
398:
341:
304:
266:
153:
85:image compression
65:image compression
41:image compression
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1007:
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981:
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964:
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832:
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829:
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749:
736:
731:
709:
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616:
614:
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608:
576:
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571:
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542:
529:
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502:
497:
475:Recall that the
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77:low pass filters
73:scaling function
1178:
1177:
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1147:
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1115:
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1048:
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1022:
1021:
1010:
1000:
999:
995:
979:
972:
971:
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943:10.1.1.467.5894
919:
918:
914:
904:
903:
896:
891:
875:
809:
808:
763:
741:
667:
666:
626:
625:
574: for
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387:
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177:
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110:
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49:
12:
11:
5:
1176:
1174:
1166:
1165:
1160:
1150:
1149:
1144:
1143:
1108:
1081:
1074:
1042:
1031:(9): 859–869.
1008:
993:
965:
928:(8): 1053–60.
912:
893:
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890:
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874:
871:
822:
819:
816:
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579:
569:
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563:
559:
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552:
549:
546:
541:
537:
533:
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523:
519:
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512:
509:
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501:
496:
492:
454:
450:
447:
423:
420:
397:
393:
390:
366:
363:
340:
336:
333:
309:
303:
300:
277:
274:
271:
265:
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239:
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216:
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193:
190:
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184:
164:
161:
158:
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149:
126:
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105:
102:
91:filter banks,
48:
45:
13:
10:
9:
6:
4:
3:
2:
1175:
1164:
1161:
1159:
1156:
1155:
1153:
1139:
1135:
1131:
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1123:
1119:
1112:
1109:
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1096:
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1082:
1077:
1071:
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1063:
1059:
1052:
1046:
1043:
1038:
1034:
1030:
1026:
1019:
1017:
1015:
1013:
1009:
1004:
997:
994:
989:
985:
978:
977:
969:
966:
961:
957:
953:
949:
944:
939:
935:
931:
927:
923:
916:
913:
908:
901:
899:
895:
888:
886:
884:
880:
872:
870:
867:
863:
858:
856:
852:
848:
844:
840:
836:
820:
817:
814:
806:
802:
798:
779:
774:
771:
768:
760:
757:
754:
746:
742:
733:
728:
717:
711:
706:
701:
698:
695:
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687:
681:
675:
672:
665:
664:
663:
646:
643:
640:
634:
631:
623:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
567:
564:
561:
553:
550:
547:
539:
535:
526:
521:
513:
507:
499:
494:
490:
482:
481:
480:
478:
473:
469:
448:
445:
421:
418:
391:
388:
364:
361:
334:
331:
307:
298:
272:
260:
250:. Similarly,
237:
234:
214:
211:
188:
182:
159:
147:
121:
115:
103:
101:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
57:
55:
46:
44:
43:performance.
42:
38:
34:
31:
27:
23:
19:
1121:
1117:
1111:
1097:(1): 50–67.
1094:
1090:
1084:
1057:
1045:
1028:
1024:
1002:
996:
975:
968:
925:
921:
915:
906:
876:
859:
800:
796:
794:
619:
474:
470:
107:
59:Traditional
58:
50:
30:biorthogonal
21:
15:
879:closed-form
843:coefficient
837:but nearly
1152:Categories
889:References
622:polynomial
97:smoothness
89:orthogonal
87:. Besides
47:Motivation
1124:: 63–76.
938:CiteSeerX
866:overshoot
821:π
815:ω
805:Bernstein
772:−
758:−
692:∑
635:∈
596:…
565:−
551:−
453:~
396:~
339:~
302:~
264:~
261:ψ
183:ψ
151:~
148:ψ
116:ψ
37:bandwidth
1138:46301605
960:18291999
873:Drawback
864:effect (
847:passband
175:, while
69:Coiflets
988:2698147
930:Bibcode
862:ringing
839:coiflet
835:coiflet
61:wavelet
33:coiflet
26:wavelet
1136:
1072:
986:
958:
940:
795:where
104:Theory
81:sparse
1134:S2CID
1054:(PDF)
980:(PDF)
1070:ISBN
956:PMID
860:The
434:and
377:and
24:are
1126:doi
1099:doi
1062:doi
1033:doi
948:doi
16:In
1154::
1132:.
1120:.
1093:.
1068:.
1056:.
1029:16
1027:.
1011:^
984:MR
954:.
946:.
936:.
924:.
897:^
885:.
67:.
20:,
1140:.
1128::
1122:1
1105:.
1101::
1095:5
1078:.
1064::
1039:.
1035::
990:.
962:.
950::
932::
926:4
909:.
818:=
801:i
799:(
797:d
780:,
775:k
769:n
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761:x
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752:(
747:k
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739:)
734:n
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725:(
721:)
718:k
715:(
712:d
707:n
702:0
699:=
696:k
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685:)
682:x
679:(
676:1
673:H
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644:,
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602:n
599:,
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270:(
238:0
235:G
215:0
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192:)
189:t
186:(
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160:t
157:(
125:)
122:t
119:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.