The L2 polynomial spline wavelet transforms: A signal and image processing perspective

IETE Technical Review (Institution of Electronics and Telecommunication Engineers, India)

Volumn 15, Issue 3, 1998, Pages 153-164

  Panda, Rutuparna ^a   Chatterji, B N ^a  

^a INDIAN INSTITUTE OF TECHNOLOGY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

INTERPOLATION; POLYNOMIALS; WAVELET TRANSFORMS;

POLYNOMIAL SPLINE FUNCTIONS;

IMAGE PROCESSING;

EID: 0032057870     PISSN: 02564602     EISSN: 09745971     Source Type: Journal    
DOI: 10.1080/02564602.1998.11416744     Document Type: Article

References (93)

1. S D, Conte & C, de Boor, Elementary numerical analysis, Mc Graw Hill, Kogakusha, 1972.
2. E V, Krishnamurty & S K, Sen, Computer based numerical algorithms, Affiliated East-West Press, New Delhi, 1992.
3. J H, Ahlberg, E N, Nilson & J L, Walsh, The theory of splines and their applications. Academic Press, New York, 1967.
4. T N E, Greville, Theory and applications of spline functions, Academic Press, New York, 1969.
5. A, Sand & S, Weintraub, A book of splines, Wiley, New York, 1971.
6. M H, Schultz, Spline Analysis, Prentice-Hall, Englewood Cliffs NJ, 1973.
7. L L, Schumaker, Spline functions:Basic theory, Wiley, New York, 1981.
8. A, Meir & A, Sharma, Spline Functions and Approximation Theory, Birkhauser Verlag Basel, 1973.
9. P M, Prenter, Splines and Variational Methods, Wiley:New York, 1975.
10. AG, Law & B N, Sahey, Theory of Approximation with Applications, Academic Press:New York, 1976.
11. C, de Boor, A practical guide to splines, Springer-Verleg, New York, 1978.
12. T M R, Ellis & D H, Mclain, A new method of cubic curve fitting using local data, ACM Trans Math Software, vol 3, pp 175–181, 1977.
13. A A, Ball, A simple specification of the parametric cubic segment, Computer Aided Design, vol 10, pp 181–185, 1978.
14. H, Akima, A new method of interpolation and smooth curve fitting based on local procedure, Journal ACM, vol 17, pp 589–602, 1970.
15. 2 approximation of functions of one and two variables, IEEE Trans Computer, vol 24, pp 98–102, January 1975.
16. G, Birkhoff & C D, Boor, Error bounds for spline interpolation, J Math Mach, vol 13, pp 827–836, 1964.
17. C D, Boor, On uniform approximation by splines, Approx Theory, vol 1, pp 219–235, 1968.
18. H, Akima, A method of bivariate interpolation and smooth surface fitting based on local procedure, Comm ASS Comput Mach, vol 17, pp 18–20, 1974.
19. M J, Munteanu & L L, Schumaker, Some multidimensional spline approximation methods, J Approx Theory, vol 10, pp 23–40, 1974.
20. C D, Boor, On calculating with B-splines. J Approx Theory, vol 6, pp 50–62, 1972.
21. B, Asker, The spline curve, a smooth interpolating function used in numerical design of shiplines, Nord Tidscar Inform Behandling, vol 2, pp 76–82, 1962.
22. R E, Barnhill & R F, Rosenfeld, Computer Aided Geometric Design, Academic Press:New York, 1974.
23. G, Wahbi, A polynomial algorithm for density estimation, Annual Math Statistics, vol 42, pp 1870–1886, 1971.
24. G, Wahba, Interpolating spline methods for density estimation—I:Equispaced data, Annual Math Statistics, vol 13, pp 30–48, 1975.
25. L L, Horowitz, The effects of spline interpolation on power spectral density, IEEE Trans Acoust Speech Signal Processing, vol 22, pp 22–27, February 1974.
26. T, Pavlidis, Curve fitting as a pattern recognition problem, Proc 6th Int Conf Pattern Recognition, pp 853–859, 1982.
27. B N, Chatterji, Use of splines for data reduction in pattern recognition, Proc 6th Int Conf Pattern Recognition, pp 1181–1187, 1982.
28. G, Medioni & Y, Yasumoto, Corner detection and curve representation using cubic B-splines, Computer Vision Graphics and Image Processing, vol 39, pp 267–278, 1987.
29. D D, Majumder & K S, Ray, Recognition and position determination of partially occluded object for a computer vision system. Journal of the IETE, vol 37, pp 419–442, September 1991.
30. R, Panda, N, Mahallik & G S, Rath, An efficient algorithm for recognition of space curves, Proc Int Conf Automation, Robotis and Computer Vision, Singapore, Dec 1994.
31. H S, Hou & H C, Andrews, Least square image restoration using spline basis functions, IEEE Trans Computer, vol C-26, pp 856–873, September 1977.
32. H S, Hou & H C, Andrews, Cubic splines for image interpolation and digital filtering, IEEE Trans Acoust Speech Signal Processing, vol 22, pp 508–517, December 1978.
33. D G, McCaughy & H C, Andrews, Image approximation by variable knot bicubic splines, IEEE Trans Patt Anal Machine Intell, vol 32, pp 229–310, 1981.
34. M J, Peyrovian, Image restoration by spline functions. University of Southern California, Los Angeles CA, August 1976.
35. N, Ishimura, Spline approximation of line images by modified dynamic programming, Trans IECE, Japan, vol 68, pp 169–176, February 1985.
36. T C, Chen & R J P, deFigueiredo, Two-dimensional interpolation by generalized spline filters based on partial differential equations, IEEE Trans Acoust Speech Signal Processing, vol 33, pp 631–642, June 1985.
37. P V, Sankar & L A, Ferrari, Simple algorithms and architectures for B-spline interpolation, IEEE Trans Patt Anal Machine Intell, vol 10, pp 271–276, 1988.
38. M L, Liou, Spline fit made easy, IEEE Trans Computer, vol C-25. pp 522–527, May 1976.
39. K., Toraichi, S, Yang, M, Kamada & R, Mori, Two dimensional spline interpolation for image reconstruction, Pattern Recognition, vol 22, pp 275–284, December 1988.
40. T, Poggio, H, Voorthees & Yuille, Regularizing edge detection, Artificial Intelligence Lab Memo 776, Massachusetts Institute of Technology, Cambridge:MA, 1984.
41. V T T, Poggio & C, Koch, Computational vision and regularization theory, Nature, vol 317, pp 314–319, 1985.
42. M, Unser, A, Aldroubi & M, Eden, Fast B-spline transforms for continuous image representation and interpolation, IEEE Trans Patt Anal Machine Intel, vol 13, pp 277–285, March 1991.
43. M, Unser, A, Aldroubi, & M, Eden, Polynomial spline signal approximation:Filter design and asymptotic equivalence to Shannon's sampling theory, IEEE Trans Information Theory, vol 38, p 1992, January 1992.
44. M, Unser, A, Aldroubi & M, Eden, B-spline signal processing:Part I—Theory, IEEE Trans Signal Processing, vol 41, pp 821–833, Feb 1993.
45. M, Unser, A, Aldroubi & M, Eden, B-spline signal processing:Part II—Efficient design and applications, IEEE Trans Signal Processing, vol 41, pp 834–848, Feb 1993.
46. 2 polynomial spline pyramid, IEEE Trans Patt Anal Machine Intel, vol 15, pp 364–378, April 1993.
47. H, Olkonen, Discrete binomial splines, CVGIP:Graphical Model and Image Processing, vol 57, pp 101–106, March 1995.
48. G, Chen & Y H H, Yang, Edge detection by regularized cubic B-spline fitting, IEEE Trans System Man Cybern, vol 25, pp 636–643, April 1995.
49. R, Panda & G S, Rath, Computer oriented approach to impulse response identification, Journal of the Institution of Engineers (India), vol 73, pp 70–73, August 1992.
50. G S, Rath, On splines, spline transform and its application to handprinted Oriya character recognition, PhD Thesis, IIT Kharagpur (India), 1984.
51. R, Panda & G S, Rath, Computation and effectiveness of discrete spline transform, Journal of the Institution of the Engineers (India), vol 74, pp 74–79, March 1994.
52. R, Panda, G S, Rath & B N, Chatterji, Generalized B-spline signal processing, Signal Processing, vol 55, pp 1–14, November 1996.
53. G G, Walter, A sampling theorem for wavelet subspaces, IEEE Trans Information Theory, vol 38, pp 881–84, 1992.
54. S G, Mallat, A theory for multiresolution signal decomposition:The wavelet representation, IEEE Trans Patt Anal Machine Intell, vol 11, pp 674–693, July 1989.
55. S G, Mallat, Multifrequency channel decomposition of images and wavelet models, IEEE Trans Acoust Speech Signal Processing, vol 37, pp 2091–2110, Dec 1989.
56. I, Duaubechies, Orthonormal bases of compactly supported wavelets, Comm Pure Appl Math, vol 41, pp 909–996, 1988.
57. I, Duaubechies, The wavelet transform, time-frequency localization and singal analysis, IEEE Trans Information Theory, vol 36, pp 961–1005, September 1990.
58. W, Grimson, Computational experiments with a feature based stereo algorithm. IEEE Trans Patt Anal Machine Intell, vol 7, pp 17–34, 1985.
59. E, Hall, J, Rough & R, Wong, Hierarchical search for image matching, Proc Conf Decision Control, pp 791–796 1976.
60. P J, Burt & E H, Adelson, The Laplacian pyramid as a compact image code, IEEE Trans Communication, vol 31, pp 532–540, 1983.
61. J, Crowley, A representation for visual informaton, Robotics. Carnegie-Mellon University Technical Report, 1987.
62. O, Rioul, Wavelets and signal processing, IEEE Signal Processing Magazine, vol 8, pp 14–38, 1991.
63. O, Rioul, A discrete-time multiresolution theory, IEEE Trims Signal Processing, vol 41, pp 2591–2606, August 1993.
64. G, Pirani & V, Zingarelli, Analytical formula for design of qudrature mirror filters, IEEE Trans Acoust Speech Signal Processing, vol 32, 1984.
65. A, Harr, Zur theory der orthogonalen funktionen-system, Math Annalen, vol 69, pp 331–371, 1910.
66. n as unconditional bases for Hardy spaces, Proc Conf on Harmonic Analysis in honour of Antoni Zygmund, pp 475–493, 1983.
67. G, Battle, A Block spin construction on ondolettes—Part I:Lemaric functions, Commun Math Physics, vol 110, pp 601–615, 1987.
68. P G, Lemarie & Y, Meyer, Ondelettes et bases Hilbertiennes, Radista Matematica Ibero Americanas, vol 2, 1986.
69. M, Unser, A Aldroubi & M Eden, On the asymptotic convergence of B-spline wavelets to Gabor functions, IEEE-IT, vol 38, pp 864–872, 1992.
70. C K, Chui, An Introduction to wavelets, Boston, MA:Academic, 1992.
71. M, Unser, A, Aldroub & M, Eden, A family of polynomial spline wavelet transforms, Signal Processing, vol 30, pp 141–162, Jan 1993.
72. I, Daubechies, Ten Lectures on Wavelets, Philadelphia, PA:SIAM, 1992.
73. Y, Meyer, Wavelets:Algorithms and Applications, Philadelphia, PA:SIAM, 1993.
74. M, Vetterli & J, Kovacevic, Wavelets and Subband Coding, Englewood Cliffs, NJ:Prentice-Hall, 1995.
75. G, Strang & T, Nguyen, Wavelets and Filter Banks, Wellesley, MA:Wellesley-Cambridge, 1996.
76. C, Herley & M, Vetterli, Wavelets and recursive filter banks. IEEE Trans Signal Processing, vol 41, pp 2536–2556, August 1993.
77. M, Unser & M, Eden, FIR approximation of inverse ers and perfect reconstruction filter banks, Signal Processing, 1994.
78. P L, Ainsleigh & C K, Chui, A B-wavelet-based noise reduction algorithm, IEEE Trans Signal Processing, vol 44, May 1996.
79. P, Abry & A, Aldroubi, Designing multiresolution analysis-type wavelets and their fast algorithms, J Fourier Anal Appl, 1995.
80. A, Cohen, I, Daubecjies & J C, Feauveau, Bi-orthogonal bases of compactly supported wavelets, Comm Pure Appl Math, vol 45, pp 485–560, 1992.
81. M, Vetterli & C, Herley, Wavelets and filter banks:Theory and design, IEEE Trans Signal Processing, vol 40, pp 2207–2232, September 1992.
82. M G, Strintzis, Optimal Biorthogonal wavelet bases for signal decomposition, IEEE Trans Signal Processing, vol 44, pp 1406–1417, June 1996.
83. M, Unser, Approximation power of Biorthogonal wavelet expansions, IEEE Trans Singal Processing, vol 44, pp 519–527, March 1996.
84. M, Unser, P, Thevenaz & A, Aldroubi, Shift-orthogonal wavelet bases using splines, IEEE Signal Processing Letters, vol 3, pp 85–88, March 1996.
85. A M M, Muth, Sequential method for spline approximation with variable knots, Int J of System Science, vol 9, pp 1055–1067, 1978.
86. D G, Schweikert, An interpolating curve using a spline in tension, J Math Physics, vol 45, pp 312–317, 1966.
87. A K, Cline, Scalar and planar valued curve fitting using splines under tension, Comm Assoc Comput Mach, vol 17, pp 218–220, 1974.
88. H, Spath, Spline Algorithms for Curves and Surfaces, Util Math Publ Inc, Winnipeg, Manitoba, 1974.
89. R W, Soans, VP-splines—an extenstion of twice differentiate interpolation, Proc Conference on Army Numerical Analysis and Computer, ARO Report 76, US Army, 1976.
90. G S, Rath & B N, Chatterji, A novel B-spline interpolation technique for bandlimited signal, Proc International Symposium on Microwave and Communication, Kharagpur (India), 1981.
91. P, Dierchx, Algorithms for smoothing data with periodic and parametric splines, Computer Graphic Image Processing, vol 20, pp 171–184, 1982.
92. K, Haradae & E, Nakamae, An isotropic four point interpolation based on cubic spline, Computer Graphics Image Processing, vol 20, pp 283–287, 1982.
93. L D, Wu, Consistent piecewise linear approximation, Proc 6th Int Conf Pattern Recognition, 1982.