Student or educator? Sign up with your school email and get 6 months of Pro free.
CoChat
Sign Up
Cited by 1,633

The DOI matches its Crossref record; 3 of the 3 references it cites were found in a registry or an index.

Splines: a perfect fit for signal and image processing

Open access
Unser, M.·1999·IEEE Signal Processing Magazine
DOI 10.1109/79.799930
Open this paper in CoChat — free account

Abstract

The article provides arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages. To reassure the reader who may be afraid to enter new territory, it is emphasized that one is not losing anything because the traditional theory is retained as a particular case (i.e., a spline of infinite degree). The basic computational tools are also familiar to a signal processing audience (filters and recursive algorithms), even though their use in the present context is less conventional. The article also brings out the connection with the multiresolution theory of the wavelet transform. This article attempts to fulfil three goals. The first is to provide a tutorial on splines that is geared to a signal processing audience. The second is to gather all their important properties and provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature. The third goal is to give a review of the primary applications of splines in signal and image processing.

Overview

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Integrity signals

Registry

Crossref · 10.1109/79.799930

Cites

3 of 3 references verified0 citation errors · 0 not checkable

Statistics

Verification

Claims the paper makes

The claims and statistics of this paper haven’t been checked yet.

Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.

A few free checks a day · papers already verified open instantly · results are public

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

The self-check needs the paper’s full text.

Citations

Cites

Show all 3 references

Research works printed but not linked: 111

  1. illgner, 1996, Scalable video coding at very low bit rates employing resolution pyramids, Proc EUSIPCO-96, 2, 1343
  2. 10.1137/0143019 · doi:10.1137/0143019
  3. 10.1016/0734-189X(90)90058-4 · doi:10.1016/0734-189x(90)90058-4
  4. 10.1109/83.481672 · doi:10.1109/83.481672
  5. 10.1080/03610928408828695 · DOI
  6. 10.1109/18.382009 · DOI · Paper page
  7. 10.1109/83.536895 · DOI
  8. 10.1109/TASSP.1978.1163154 · DOI · Paper page
  9. 10.1109/18.119740 · DOI
  10. 10.1007/BF01456326 · DOI · Paper page
  11. 10.1007/BF00123164 · DOI · Paper page
  12. de boor, 1985, Bivariate cardinal interpolation by splines on a three-direction mesh, Ill J Math, 29, 533, 10.1215/ijm/1256045495 · doi:10.1215/ijm/1256045495
  13. diercks, 1995, Curve and Surface Fitting with Splines
  14. 10.2307/2153941 · doi:10.2307/2153941
  15. 10.1109/83.350818 · DOI
  16. 10.1002/cpa.3160450502 · DOI · Paper page
  17. 10.1016/0021-9045(72)90080-9 · DOI · Paper page
  18. 10.1109/78.340787 · DOI
  19. 10.1007/BF01890564 · doi:10.1007/bf01890564
  20. 10.1109/83.366474 · doi:10.1109/83.366474
  21. de boor, 1978, A practical Guide to Splines, 10.1007/978-1-4612-6333-3 · DOI · Paper page
  22. mallat, 1989, Multiresolution approximations and wavelet orthogonal bases of L<superscript>2</superscript>(R), Trans Amer Math Soc, 315, 69
  23. 10.1109/34.192463 · DOI · Paper page
  24. 10.1007/BF01183840 · doi:10.1007/bf01183840
  25. paeth, 1986, A fast algorithm for general raster rotation, Proceedings of Graphics Interface 86 and Vision Interface, 77
  26. 10.1097/00004728-199709000-00029 · DOI
  27. 10.1109/ICIP.1996.559473 · doi:10.1109/icip.1996.559473
  28. 10.1109/78.709545 · doi:10.1109/78.709545
  29. 10.1109/83.650115 · doi:10.1109/83.650115
  30. 10.1109/83.413171 · doi:10.1109/83.413171
  31. menet, 1990, B-snakes: implementation and application to stereo, Proc Image Understanding Workshop, 720
  32. jia, 1993, Approximation by multi-integer translates of functions having global support, J Approximation Theory, 72, 10.1006/jath.1993.1002 · doi:10.1006/jath.1993.1002
  33. 10.1090/S0002-9939-96-03255-8 · doi:10.1090/s0002-9939-96-03255-8
  34. ahlberg, 1967, The Theory of Splines and Their Applications
  35. 10.1080/01630569408816545 · doi:10.1080/01630569408816545
  36. 10.1080/01630569308816532 · DOI
  37. 10.1137/S0036141092234086 · doi:10.1137/s0036141092234086
  38. 10.1109/34.142909 · DOI · Paper page
  39. 10.1016/0165-1684(92)90030-Z · DOI
  40. 10.1109/83.136597 · DOI · Paper page
  41. lee, 1998, High quality image re-sizing using oblique projection operators, IEEE Trans Image Processing, 7, 679, 10.1109/83.668025 · doi:10.1109/83.668025
  42. lancaster, 1986, Curve and Surface Fitting An Introduction
  43. mallat, 1998, A Wavelet Tour of Signal Processing
  44. lemari�, 1988, Ondelettes � localisation exponentielles, J Math pures et appl, 67, 227
  45. 10.1080/00207729608929301 · doi:10.1080/00207729608929301
  46. 10.1109/78.382410 · doi:10.1109/78.382410
  47. 10.1109/5.687833 · doi:10.1109/5.687833
  48. 10.1007/BF00133570 · DOI · Paper page
  49. 10.1007/BF02761973 · doi:10.1007/bf02761973
  50. 10.1137/1.9781611970555 · DOI · Paper page
  51. 10.1073/pnas.52.4.947 · DOI · Paper page
  52. 1946, Contribution to the problem of approximation of equidistant data by analytic functions, Quart Appl Math, 4, 45, 10.1090/qam/15914 · doi:10.1090/qam/15914
  53. 10.1109/JRPROC.1949.232969 · DOI · Paper page
  54. 10.1109/78.258085 · DOI · Paper page
  55. schumaket, 1981, Spline Functions - Basic Theory
  56. 10.1109/79.733498 · doi:10.1109/79.733498
  57. 10.1109/PROC.1976.10110 · DOI · Paper page
  58. 10.1137/1031128 · DOI · Paper page
  59. 10.1109/TMI.1983.4307610 · DOI · Paper page
  60. 10.1016/0885-064X(88)90024-6 · DOI
  61. 10.1038/317314a0 · DOI · Paper page
  62. powell, 1992, Advances in Numerical Analysis II Wavelets Subdivision Algorithms and Radial Functions, 105
  63. prenter, 1975, Splines and Variational Methods
  64. 10.1007/BF02162161 · DOI · Paper page
  65. 10.1137/0523086 · doi:10.1137/0523086
  66. rioni, 1992, Fast algorithms for discrete and continuous wavelet transforms, IEEE Trans Information Theory, it 38, 569
  67. rosenfeld, 1984, Multiresolution Image Promsing, 10.1007/978-3-642-51590-3 · DOI
  68. 10.1081/STA-120002644 · doi:10.1081/sta-120002644
  69. said, 1996, A new fast and efficient image codec based on set partitioning in hierarchical trees, IEEE Trans Circuits Syst Video Technol, 6, 243, 10.1109/76.499834 · DOI · Paper page
  70. 10.1007/s00041-001-4025-4 · DOI
  71. vetterli, 1995, Wavelets and Subband Coding
  72. 10.1109/18.108253 · DOI
  73. 10.1109/78.157221 · DOI · Paper page
  74. 10.1109/18.119742 · DOI · Paper page
  75. 10.1109/76.109149 · DOI · Paper page
  76. 10.1109/34.75514 · DOI · Paper page
  77. 10.1109/82.718806 · DOI
  78. 10.1109/78.330352 · DOI · Paper page
  79. 10.1109/83.465102 · doi:10.1109/83.465102
  80. unser, 1992, Wavelets-A Tutorial in Theory and Applications, 91
  81. 10.1109/97.481163 · doi:10.1109/97.481163
  82. 10.1117/12.292801 · DOI
  83. 10.1109/78.599940 · doi:10.1109/78.599940
  84. unser, 0, Fractional splines and wavelets, SIAM Review, 10.1137/S0036144598349435 · DOI · Paper page
  85. 10.1016/S0165-1684(96)00189-2 · doi:10.1016/s0165-1684(96)00189-2
  86. 10.1109/78.564177 · doi:10.1109/78.564177
  87. 10.1117/12.255253 · doi:10.1117/12.255253
  88. 10.1016/0165-1684(93)90144-Y · DOI · Paper page
  89. 10.1109/34.206956 · DOI
  90. 10.1109/78.193221 · DOI · Paper page
  91. 10.1109/78.193220 · DOI · Paper page
  92. bartels, 1987, Splines for Use in Computer Graphics
  93. 10.1007/BF01205550 · DOI · Paper page
  94. 10.1109/ICIP.1998.999014 · doi:10.1109/icip.1998.999014
  95. 10.1006/acha.1998.0249 · doi:10.1006/acha.1998.0249
  96. 10.1109/78.790660 · doi:10.1109/78.790660
  97. 10.1109/78.790659 · DOI · Paper page
  98. ?rigger, 0, B-spline snakes: a flexible tool for parametric contour detection, IEEE Trans Image Processing
  99. str�mberg, 1983, A modified Franklin system and higher-order spline system of R, Proc Conference in Harmonic Analysis in honor of Antoni Zygmund, ii, 475
  100. ?rigger, 0, Centered pyramids, IEEE Trans Image Processings, 6, 1254
  101. strang, 1996, Wavelets and Filter Banks
  102. th�venaz, 0, Image interpolation and resampling, Handbook of Medical Image Processing
  103. 10.1109/83.661182 · DOI · Paper page
  104. 10.1007/s002110050068 · doi:10.1007/s002110050068
  105. 10.1002/mrm.1910210213 · DOI · Paper page
  106. strang, 1971, Constructive Aspect of Functional Analysis, 796
  107. 10.1109/78.489025 · doi:10.1109/78.489025
  108. 10.1109/83.650848 · DOI · Paper page
  109. th�venaz, 1997, Separable least-squares decomposition of affine transformations, Proc IEEE Int Conf on Image Processing
  110. 10.1016/0031-3203(88)90062-3 · DOI
  111. 10.1016/0165-1684(92)90007-J · DOI

Cited by

See all 1,633

Metadata

Authors
Unser, M.
Venue
IEEE Signal Processing Magazine
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Published
1999
Volume
16
Issue
6
Pages
22-38
ISSN
1053-5888 (ISSN-L)
DOI
10.1109/79.799930
OpenAlex
W2003410902
Cited by
1,633
References
3
FWCI
49.04
Type
journal article
Language
en
Open access
green
Licence
ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html (vor)

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review

How to cite

Cite CoChat for the checks on this page, and the paper for its own findings.

This page
CoChat. (2026). Citation check: Splines: a perfect fit for signal and image processing. https://app.cochat.ai/papers/10.1109/79.799930
The paper
Unser, M. (1999). Splines: a perfect fit for signal and image processing. IEEE Signal Processing Magazine. https://doi.org/10.1109/79.799930

Data sources

Bibliographic data: Semantic Scholar, OpenAlex, Crossref.

Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).

Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.