Splines: A Perfect Fit for Signal and Image Processing

IEEE Signal Processing Magazine

Volumn 16, Issue 6, 1999, Pages 22-38

  Unser, Micheal ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

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EID: 0002999362     PISSN: 10535888     EISSN: None     Source Type: Journal    
DOI: 10.1109/79.799930     Document Type: Article

References (114)

1. P. Abry and A. Aldroubi, "Designing multiresolution analysis-type wavelets and their fast algorithms," J. Fourier Analysis and Applications, vol. 2, no. 2, pp. 135-159, 1995.
2. G.G. Agarwal, "Splines in statistics," Bull. Allahabad Math.Soc., vol. 4, pp. 1-55, 1989.
3. J.H. Ahlberg, E.N. Nilson, and J.L. Walsh, The Theory of Splines and Their Applications, New York: Academic Press, 1967.
4. A. Aldroubi, "Oblique projections in atomic spaces," Proc. Amer. Math. Soc., vol. 124, no. 7, pp. 2051-2060, 1996.
5. A. Aldroubi and M. Unser, "Families of multiresolution and wavelet spaces with optimal properties," Numerical Functional Analysis and Optimization, vol. 14, no. 5-6, pp. 417-446, 1993.
6. A. Aldroubi and M. Unser, "Sampling procedures in function spaces and asymptotic equivalence with Shannon's sampling theory," Numer. Funct. Anal. and Optimiz., vol. 15, no. 1 & 2, pp. 1-21, 1994.
7. A. Aldroubi, M. Unser, and M. Eden, "Cardinal spline filters: stability and convergence to the ideal sine interpolator," Signal Processing, vol. 28, no. 2, pp. 127-138, 1992.
8. 2," SIAM J. Math. Anal., vol. 25, no. 5, pp. 1412-1432, 1994.
9. M. Antonini, M. Barlaud, P. Mathieu, and I. Daubechies, "Image coding using wavelet transform," IEEE Trans. Image Processing, vol. 1, no. 2, pp. 205-220, 1992.
10. R.H. Bartels, J.C. Beatty, and B.A. Barsky, Splines for Use in Computer Graphics, Los Altos, CA: Morgan Kaufmann, 1987.
11. G. Battle, "A block spin construction of ondelettes, Part I: Lemarié functions," Commun. Math. Phys., vol. 110, pp. 601-615, 1987.
12. T. Blu and M. Unser, "Minimum support interpolators with optimum approximation properties," in Proc. Int. Conf. Image Processing, Santa Barbara, CA, 1998, vol. III, pp. 242-245.
13. T. Blu and M. Unser, "Approximation error for quasi-interpolators and (multi-) wavelet expansions," Applied and Computational Harmonic Analysis, vol. 6, no. 2, pp. 219-251, 1999.
14. T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part II-wavelets," IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2796-2806.
15. T. Blu and M. Unser, "Quantitative Fourier analysis of approximation techniques: Part I-interpolators and projectors," IEEE Trans. Signal Processing, vol. 47, no. 10, pp. 2783-2795.
16. P. Brigger, J. Hoeg, and M. Unser, "B-spline snakes: a flexible tool for parametric contour detection," IEEE Trans. Image Processing, in press.
17. P. Brigger, F. Müller, K. Illgner, and M. Unser, "Centered pyramids," IEEE Trans. Image Processing, vol. 6, no. 9, pp. 1254-1264.
18. J.F. Canny, "A computational approach to edge detection," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-8, no. 6, pp. 679-697, 1986.
19. A. Chambolle, R.A. DeVore, N.-Y. Lee, and B.J. Lucier, "Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage," IEEE Trans. Image Processing, vol. 7, no. 33, pp. 319-335, 1998.
20. C.K. Chui and J.Z. Wang, "On compactly supported spline wavelets and a duality principle," Trans. Amer. Math. Soc., vol. 330, no. 2, pp. 903-915, 1992.
21. A. Cohen, I. Daubechies, and J.C. Feauveau, "Bi-orthogonal bases of compactly supported wavelets," Comm. Pure Appl. Math., vol. 45, pp. 485-560, 1992.
22. F.S. Cohen, Z. Huang, and Z. Yang, "Invariant matching and identification of curves using B-splines curve representation," IEEE Trans. Image Processing, vol. 4, no. 1, pp. 1-10, 1995.
23. I. Daubechies, Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics, 1992.
24. C. de Boor, "On calculating with B-splines," J. Approximation Theory, vol. 6, pp. 50-62, 1972.
25. C. de Boor, A Practical Guide to Splines, New York: Springer-Verlag, 1978.
26. C. de Boor, "The polynomials in the linear span of integer translates of a compactly supported function," Constructive Approximation, vol. 3, pp. 199-208, 1987.
27. C. de Boor, K. Höllig, and S. Riemenschneider, "Bivariate cardinal interpolation by splines on a three-direction mesh," IL J. Mathematics, vol. 29, no. 4, pp. 533-566, 1985.
28. R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comput. Vis., vol. 1, no. 2, pp. 167-187, 1987.
29. P. Diercks, Curve and Surface Fitting With Splines, Oxford: Oxford University Press, 1995.
30. D.L. Donoho, "De-noising by soft-thresholding," IEEE Trans. Information Theory, vol. 41, no. 3, pp. 613-627, 1995.
31. R.L. Eubank, "Approximate regression models and splines," Commun. Stat. Theory Math., vol. 13, pp. 433-484, 1984.
32. M. Flickner, J. Halner, E.J. Rodriguez, and J.L.C. Sanz, "Periodic quasi-orthogonal spline bases and applications to least-squares curve fitting of digital images," IEEE Trans. Image Processing, vol. 5, no. 1, pp. 71-88, 1996.
33. A. Goshtasby, F. Cheng, and B.A. Barsky, "B-spline curves and surfaces viewed as digital filters," Computer Vision, Graphics, and Image Processing, vol. 52, pp. 264-275, 1990.
34. A. Haar, "Zur Theorie der orthogonalen Funktionensysteme," Math. Ann., vol. 69, pp. 331-371, 1910.
35. D.M. Healy, Jr. and J.B. Weaver, "Two applications of wavelet transforms in magnetic resonance," IEEE Trans. Information Theory, vol. 38, no. 2, pp. 840-860, 1992.
36. H.S. Hou and H.C. Andrews, "Cubic splines for image interpolation and digital filtering," IEEE Trans. Acoust., Speech, and Signal Processing, vol. ASSP-26, no. 6, pp. 508-517, 1978.
37. Z. Huang and F.S. Cohen, "Affine-invariant B-spline moments for curve matching," IEEE Trans. Image Processing, vol. 5, no. 10, pp. 1473-1480, 1996.
38. R. Hummel, "Sampling for spline reconstruction," SIAM J. Appl. Math., vol. 43, no. 2, pp. 278-288, 1983.
39. K. Illgner and F. Müller, "Scalable video coding at very low bit rates employing resolution pyramids," in Proc. EUSIPCO-90, Trieste, Italy, 1996, vol. 2, pp. 1343-1346.
40. R.-Q. Jia and J. Lei, "Approximation by multi-integer translates of functions having global support," J. Approximation Theory, vol. 72, no. 2-23, 1993.
41. M. Kamada, K. Toraichi, and R.E. Kalman, "A smooth signal generator based on quadratic B-spline functions," IEEE Trans. Signal Processing, vol. 43, no. 5, pp. 1252-1255, 1995.
42. M. Kamada, K. Toraichi, and R.E. Kalman, "Quadratic spline interpolator," Int. J. Systems Science, vol. 27, no. 10, pp. 977-983, 1996.
43. M. Kass, A. Witkin, and D. Terzopoulos, "Snakes: active contour models," Int. J. Comput. Vision, vol. 1, pp. 321-331, 1987.
44. A.K. Katsaggelos, L.P. Kondi, F.W. Meier, J. Ostermann, and G.M. Schuster, "MPEG-4 and Rate-Distortion-Based Shape-Coding Techniques," Proc. IEEE, vol. 86, no. 6, pp. 1126-1154, 1998.
45. P. Lancaster and K. Salkauskas, Curve and Surface Fitting: An Introdution, London: Academic Press, 1986.
46. C. Lee, M. Eden, and M. Unser, "High quality image re-sizing using oblique projection operators," IEEE Trans. Image Processing, vol. 7, no. 5, pp. 679-692, 1998.
47. P.-G. Lemarié, "Ondelettes à localisation exponentielles," J. Math. pures et appl., vol. 67, no. 3, pp. 227-236, 1988.
48. S. Mallat, A Wavelet Tour of Signal Processing, San Diego: Academic Press, 1998.
49. S. Mallat and S. Zhong, "Characterization of signals from multiscale edges," IEEE Trans. Patt. Anal. Machine Intell., vol. 14, no. 7, pp. 710-732, 1992.
50. 2(R)," Trans. Amer. Math. Soc., vol. 315, no. 1, pp. 69-87, 1989.
51. S.G. Mallat, "A theory of multiresolution signal decomposition: the wavelet representation," IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-11, no. 7, pp. 674-693, 1989.
52. S. Menet, P. Saint-Marc, and G. Medioni, "B-snakes: implementation and application to stereo," in Proc. Image Understanding Workshop, DARPA, 1990, 720-726.
53. P. Moulin, "A multiscale relaxation algorithm for SNR maximization in nonorthogonal subband coding," IEEE Trans. Image Processing, vol. 4, no. 9, pp. 1269-1281, 1995.
54. P. Moulin, R. Krishnamurthy, and J.W. Woods, "Multiscale modeling and estimation of motion fields for video coding," IEEE Transactions on Image Processing, vol. 6, no. 12, pp. 1606-1620, 1997.
55. F. Müller, P. Brigger, K. Illgner, and M. Unser, "Multiresolution approximation using shifted splines," IEEE Trans. Signal Processing, vol. 46, no. 9, pp. 2555-2558, 1998.
56. F. Müller and K. Illgner, "Embedded Laplacian pyramid image coding using conditional arithmetic coding," in Proc. Int. Conf. Image Processing, Lausanne, Switzerland, 1996, vol. 1, pp. 221-224.
57. J.L. Ostuni, A.K.S. Santha, V.S. Mattay, D.R. Weinberger, R.L. Levin, and J.A. Frank, "Analysis of interpolation effects in the reslicing of functional MR-images," J. Computer Assisted Tomography, vol. 21, no. 5, pp. 803-810, 1997.
58. A.W. Paeth, "A fast algorithm for general raster rotation," in Proc. Graphics Interface '86 - Vision Interface '86, 1986, 77-81.
59. D. Pang, L.A. Ferrari, and P.V. Sankar, "B-spline FIR filters," Circuits, Systems, and Signal Processing, Vol. 13, no. 1, pp. 31-64, 1994.
60. J.A. Parker, R.V. Kenyon, and D.E. Troxel, "Comparison of interpolating methods for image resampling," IEEE Trans. Med. Imaging, vol. MI-2, no. 1, pp. 31-39, 1983.
61. T. Poggio, V. Torre, and C. Koch, "Computational vision and regularization theory," Nature, vol. 317, pp. 314-319, 1985.
62. T. Poggio, H. Voorhees, and A. Yuille, "A regularized solution to edge detection," Journal of Complexity, vol. 4, no. 2, pp. 106-123, 1988.
63. M.J.D. Powell, "The theory of radial basis function approximation in 1990," in Advances in Numerical Analysis II: Wavelets, Subdivision Algorithms and Radial Functions, W.A. Light, Ed., Oxford, U.K.: Oxford Univ. Press, pp. 105-210, 1992.
64. P.M. Prenter, Splines and Variational Methods, New York: Wiley, 1975.
65. C.H. Reinsh, "Smoothing by spline functions," Numer. Math., vol. 10, pp. 177-183, 1967.
66. O. Rioul, "Simple regularity criteria for subdivision schemes," SIAM J. Math. Anal., vol. 23, pp. 1544-1576, 1992.
67. O. Rioul and P. Duhamel, "Fast algorithms for discrete and continuous wavelet transforms," IEEE Trans Information Theory, vol. IT-38, no. 2, pp. 569-586, 1992.
68. A. Rosenfeld, Multiresolution Image Processing, New York: Springer-Verlag, 1984.
69. A. Said and W.A. Pearlman, "A new fast and efficient image codec based on set partitioning in hierarchical trees," IEEE Trans. Circuits Syst. Video Technol., vol. 6, pp. 243-250, 1996.
70. I.J. Schoenberg, "Contribution to the problem of approximation of equidistant data by analytic functions," Quart. Appl. Math., vol. 4, pp. 45-99, 112-141, 1946.
71. I.J. Schoenberg, "Spline functions and the problem of graduation," Proc. Nat. Acad. Sci., vol. 52, pp. 947-950, 1964.
72. I.J. Schoenberg, Cardinal Spline Interpolation, Philadelphia, PA: Society of Industrial and Applied Mathematics, 1973.
73. I.J. Schoenberg, "Notes on spline functions III: on the convergence of the interpolating cardinal splines as their degree tends to infinity," Israel J. Math., vol. 16, pp. 87-92, 1973.
74. L.L. Schumaker, Spline Functions: Basic Theory, New York: Wiley, 1981.
75. G.M. Schuster, G. Melnikov, and A.K. Katsaggelos, "Operationally optimal vertex-based shape coding," IEEE Signal Processing Magazine, vol. 15, no. 6, pp. 91-108, 1998.
76. C.E. Shannon, "Communication in the presence of noise," Proc. I.R.E., vol. 37, pp. 10-21, 1949.
77. J. Shapiro, "Embedded image coding using zerotrees of wavelet coefficients," IEEE Trans. Acoustics, Speech and Signal Processing, vol. 41, no. 12, pp. 3445-3462, 1993.
78. D. Slepian, "On bandwidth," Proc. IEEE, vol. 64, no. 3, pp. 292-300, 1976.
79. G. Strang, "Wavelets and dilation equations: a brief introduction," SIAM Review, vol. 31, pp. 614-627, 1989.
80. G. Strang and G. Fix, "A Fourier analysis of the finite element variational method," in Constructive Aspect of Functional Analysis, Rome: Edizioni Cremonese, pp. 796-830, 1971.
81. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley, MA: Wellesley-Cambridge, 1996.
82. n as unconditional bases for Hardy spaces," in Proc. Conference in Harmonic Analysis in honor of Antoni Zygmund, 1983, vol. II, pp. 475-493.
83. W. Sweldens and R. Piessens, "Asymptotic error expansions for wavelet approximations of smooth functions II," Numer. Math., vol. 68, no. 3, pp. 377-401, 1994.
84. P. Thévenaz, T. Blu, and M. Unser, "Image interpolation and resampling," in Handbook of Medical Image Processing, in press.
85. P. Thévenaz, U.E. Ruttimann, and M. Unser, "A pyramid approach to subpixel registration based on intensity," IEEE Trans. Image Processing, vol. 7, no. 1, pp. 27-41, 1998.
86. P. Thévenaz and M. Unser, "Separable least-squares decomposition of affine transformations," in Proc. IEEE Int. Conf. on Image Processing, Santa Barbara, October 26-29, 1997, CD-ROM paper no. 200.
87. K. Toraichi, S. Yang, M. Kamada, and R. Mori, "Two-dimensional spline interpolation for image reconstruction," Pattern Recognition, vol. 21, no. No. 3, pp. 275-284, 1988.
88. M. Unser, "An improved least squares Laplacian pyramid for image compression," Signal Processing, vol. 27, no. 2, pp. 187-203, 1992.
89. M. Unser, "Approximation power of biorthogonal wavelet expansions," IEEE Trans. Signal Processing, vol. 44, no. 3, pp. 519-527, 1996.
90. M. Unser, "Ten good reasons for using spline wavelets," Proc. SPIE vol. 3169, Wavelet Applications in Signal and Image Processing V, San Diego, CA, August 6-9, 1997, pp. 422-431.
91. M. Unser and A. Aldroubi, "Polynomial splines and wavelets-A signal processing perspective," in Wavelets-A Tutorial in Theory and Applications, C.K. Chui, Ed., San Diego: Academic Press, pp. 91-122, 1992.
92. M. Unser and A. Aldroubi, "A general sampling theory for non-ideal acquisition devices," IEEE Trans. Signal Processing, vol. 42, no. 11, pp. 2915-2925, 1994.
93. M. Unser, A. Aldroubi, and M. Eden, "Fast B-spline transforms for continuous image representation and interpolation," IEEE Trans. Pattern Anal. Machine Intell., vol. 13, no. 3, pp. 277-285, 1991.
94. M. Unser, A. Aldroubi, and M. Eden, "On the asymptotic convergence of B-spline wavelets to Gabor functions," IEEE Trans. Information Theory, vol. 38, no. 2, pp. 864-872, 1992.
95. M. Unser, A. Aldroubi, and M. Eden, "Polynomial spline signal approximations: filter design and asymptotic equivalence with Shannon's sampling theorem," IEEE Trans. Information Theory, vol. 38, no. 1, pp. 95-103, 1992.
96. M. Unser, A. Aldroubi, and M. Eden, "B-spline signal processing: Part II-efficient design and applications," IEEE Trans. Signal Processing, vol. 41, no. 2, pp. 834-848, 1993.
97. M. Unser, A. Aldroubi, and M. Eden, "B-spline signal processing: Part I-theory," IEEE Trans. Signal Processing, vol. 41, no. 2, pp. 821-833, 1993.
98. M. Unser, A. Aldroubi, and M. Eden, "A family of polynomial spline wavelet transforms," Signal Processing, vol. 30, no. 2, pp. 141-162, 1993.
99. 2 polynomial spline pyramid," IEEE Trans. Pattern Anal Mach. Intell., vol. 15, no. 4, pp. 364-379, 1993.
100. M. Unser, A. Aldroubi, and M. Eden, "Enlargement or reduction of digital images with minimum loss of information," IEEE Trans. Image Processing, vol. 4, no. 3, pp. 247-258, 1995.
101. M. Unser, A. Aldroubi, and S.J. Schiff, "Fast implementation of the continuous wavelet transform with integer scales," IEEE Trans. Signal Processing, vol. 42, no. 12, pp. 3519-3523, 1994.
102. M. Unser and T. Blu, "Fractional splines and wavelets," SIAM Review, in press.
103. M. Unser and I. Daubechies, "On the approximation power of convolution-based least squares versus interpolation," IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1697-1711, 1997.
104. M. Unser, P. Thévenaz, and A. Aldroubi, "Shift-orthogonal wavelet bases using splines," IEEE Signal Processing Letters, vol. 3, no. 3, pp. 85-88, 1996.
105. M. Unser, P. Thévenaz, and L. Yaroslavsky, "Convolution-based interpolation for fast, high-quality rotation of images," IEEE Trans. Image Processing, vol. 4, no. 10, pp. 1371-1381, 1995.
106. M. Unser and J. Zerubia, "A generalized sampling without bandlimiting constraints," IEEE Trans. Circuits and Systems II: Analog and Digital Signal Processing, vol. 45, no. 8, pp. 959-969, 1998.
107. K.M. Uz, M. Vetterli, and D.J. LeGall, "Interpolative multiresolution coding of advanced television with compatible subchannel," IEEE Trans. Circuits Syst. Video Tech., vol. 1, no. 1, pp. 86-99, 1991.
108. M. Vetterli and C. Herley, "Wavelets and filter banks: theory and design," IEEE Trans. Signal Processing, vol. 40, no. 9, pp. 2207-2232, 1992.
109. M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Englewood Cliffs, NJ: Prentice Hall, 1995.
110. M.J. Vrhel, C. Lee, and M. Unser, "Comparison of algorithms for the fast computation of the continuous wavelet transform," Proc. SPIE, vol. 2825, Wavelet Applications in Signal and Image Processing IV, Denver, CO, August 6-9, 1996, pp. 422-431.
111. M.J. Vrhel, C. Lee, and M. Unser, "Fast continuous wavelet transform: a least squares formulation," Signal Processing, vol. 57, no. 2, pp. 103-119, 1997.
112. M.J. Vrhel, C. Lee, and M. Unser, "Rapid computation of the continuous wavelet transform by oblique projection," IEEE Trans. Signal Processing, vol. 45, no. 4, pp. 891-900, 1997.
113. G. Wahba, Spline Models for Observational Data" Philadelphia: Society for Industrial and Applied Mathematics, 1990.
114. J.B. Weaver, X. Yansun, D.M. Healy, and J.R. Driscoll, "Filtering noise from images with wavelet transforms," Magnetic Resonance in Medicine, vol. 21, no. 2, pp. 288-95, 1991.