The DOI matches its Crossref record; 36 of the 36 references it cites were found in a registry or an index.
Directionlets: anisotropic multidirectional representation with separable filtering
Open accessAbstract
In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis unctions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N(-1.55)), which, while slower than the optimal rate O(N(-2)), is much better than O(N(-1)) achieved with wavelets, but at similar complexity.
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.
Integrity signals
Registry
Cites
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.
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
A Wavelet Tour of Signal Processing
(1999) · Elsevier eBooks · doi:10.1016/b978-0-12-466606-1.x5000-4 · Cited by 16,227
A new, fast, and efficient image codec based on set partitioning in hierarchical trees
Said, A., & Pearlman, W. A. (1996) · IEEE Transactions on Circuits and Systems for Video Technology · doi:10.1109/76.499834 · Cited by 5,353
Embedded image coding using zerotrees of wavelet coefficients
Shapiro, J. M. (1993) · IEEE Transactions on Signal Processing · doi:10.1109/78.258085 · Cited by 4,815
Computer graphics: principles and practice
(2014) · Choice Reviews Online · doi:10.5860/choice.51-2713 · Cited by 4,651
The contourlet transform: an efficient directional multiresolution image representation
Do, M. N., & Vetterli, M. (2005) · IEEE Transactions on Image Processing · doi:10.1109/tip.2005.859376 · Cited by 3,952
Self-citationSphere Packings, Lattices and Groups.
Coxeter, H. S. M., Conway, J. H., & Sloane, N. J. A. (1989) · The American Mathematical Monthly · doi:10.2307/2323992 · Cited by 3,377
Vetterli, M., & Kovačević, J. (1995) · Infoscience (Ecole Polytechnique Fédérale de Lausanne) · Cited by 2,898
Self-citationSphere Packings, Lattices and Groups
Conway, J. H., & Sloane, N. J. A. (1999) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-1-4757-6568-7 · Cited by 2,223
Algorithm for computer control of a digital plotter
Bresenham, J. E. (1965) · IBM Systems Journal · doi:10.1147/sj.41.0025 · Cited by 2,172
The Nonsubsampled Contourlet Transform: Theory, Design, and Applications
Da Cunha, A. L., Zhou, J., & Do, M. N. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2006.877507 · Cited by 2,035
New edge-directed interpolation
Li, X., & Orchard, M. T. (2001) · IEEE Transactions on Image Processing · doi:10.1109/83.951537 · Cited by 2,023
The JPEG 2000 still image compression standard
Skodras, A., Christopoulos, C., & Ebrahimi, T. (2001) · IEEE Signal Processing Magazine · doi:10.1109/79.952804 · Cited by 1,787
Complex Wavelets for Shift Invariant Analysis and Filtering of Signals
Kingsbury, N. (2001) · Applied and Computational Harmonic Analysis · doi:10.1006/acha.2000.0343 · Cited by 1,675
New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
Candès, E. J., & Donoho, D. L. (2004) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.10116 · Cited by 1,545
Mathematical Methods and Algorithms for Signal Processing
Moon, T. K., & Stirling, W. C. (1999) · Cited by 1,491
Shiftable multiscale transforms
Simoncelli, E. P., Freeman, W. T., Adelson, E. H., & Heeger, D. J. (1992) · IEEE Transactions on Information Theory · doi:10.1109/18.119725 · Cited by 1,333
Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges
Candès, E. J., & Donoho, D. L. (2000) · Cited by 1,330
Woods, J., & O'Neil, S. (1986) · IEEE Transactions on Acoustics, Speech, and Signal Processing · doi:10.1109/tassp.1986.1164962 · Cited by 1,016
A filter bank for the directional decomposition of images: theory and design
Bamberger, R. H., & Smith, M. J. T. (1992) · IEEE Transactions on Signal Processing · doi:10.1109/78.127960 · Cited by 885
Sphere Packings, Lattices and Groups
Conway, J. H., & Sloane, N. J. A. (1993) · Grundlehren der mathematischen Wissenschaften · doi:10.1007/978-1-4757-2249-9 · Cited by 880
DeVore, R. A. (1998) · Acta Numerica · doi:10.1017/s0962492900002816 · Cited by 877
Sparse geometric image representations with bandelets
Le Pennec, E., & Mallat, S. (2005) · IEEE Transactions on Image Processing · doi:10.1109/tip.2005.843753 · Cited by 860
Wedgelets: nearly minimax estimation of edges
Donoho, D. L. (1999) · The Annals of Statistics · doi:10.1214/aos/1018031261 · Cited by 638
Adaptive homogeneity-directed demosaicing algorithm
Hirakawa, K., & Parks, T. W. (2005) · IEEE Transactions on Image Processing · doi:10.1109/tip.2004.838691 · Cited by 439
Data compression and harmonic analysis
Donoho, D. L., Vetterli, M., DeVore, R. A., & Daubechies, I. (1998) · IEEE Transactions on Information Theory · doi:10.1109/18.720544 · Cited by 432
Self-citationThe cortex transform: Rapid computation of simulated neural images
Watson, A. B. (1987) · Computer Vision, Graphics, and Image Processing · doi:10.1016/s0734-189x(87)80184-6 · Cited by 394
A new class of two-channel biorthogonal filter banks and wavelet bases
Phoong, S.-M., Kim, C. W., Vaidyanathan, P. P., & Ansari, R. (1995) · IEEE Transactions on Signal Processing · doi:10.1109/78.370620 · Cited by 390
Space-frequency quantization for wavelet image coding
Xiong, Z., Ramchandran, K., & Orchard, M. T. (1997) · IEEE Transactions on Image Processing · doi:10.1109/83.568925 · Cited by 366
Brushlets: A Tool for Directional Image Analysis and Image Compression
Meyer, F. G., & Coifman, R. R. (1997) · Applied and Computational Harmonic Analysis · doi:10.1006/acha.1997.0208 · Cited by 252
Rate-distortion optimized tree-structured compression algorithms for piecewise polynomial images
Shukla, R., Dragotti, P. L., Do, M. N., & Vetterli, M. (2005) · IEEE Transactions on Image Processing · doi:10.1109/tip.2004.840710 · Cited by 152
Self-citationImage compression with geometrical wavelets
Pennec, E. L., & Mallat, S. (2002) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2000.901045 · Cited by 114
Wavelet-domain approximation and compression of piecewise smooth images
Wakin, M. B., Romberg, J., Choi, H., & Baraniuk, R. G. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2005.864175 · Cited by 85
Multiscale wedgelet image analysis: fast decompositions and modeling
Romberg, J., Wakin, M. B., & Baraniuk, R. G. (2003) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2002.1039038 · Cited by 58
Rate-distortion optimized image compression using wedgelets
Wakin, M. B., Romberg, J., Choi, H., & Baraniuk, R. G. (2003) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2002.1038949 · Cited by 41
Approximation and compression of piecewise smooth images using a wavelet/wedgelet geometric model
Romberg, J., Wakin, M. B., & Baraniuk, R. G. (2004) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2003.1246895 · Cited by 23
Approximation power of directionlets
Velisavljević, V., Beferull‐Lozano, B., Vetterli, M., & Dragotti, P. L. (2005) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2005.1529857 · Cited by 11
Self-citation
Cited by
Dictionaries for Sparse Representation Modeling
Rubinstein, R., Bruckstein, A. M., & Elad, M. (2010) · Proceedings of the IEEE · doi:10.1109/jproc.2010.2040551 · Cited by 1,306
Tetrolet transform: A new adaptive Haar wavelet algorithm for sparse image representation
Krommweh, J. (2010) · Journal of Visual Communication and Image Representation · doi:10.1016/j.jvcir.2010.02.011 · Cited by 130
Metadata
- Authors
- Velisavljevic, V., Beferull-Lozano, B., Vetterli, M., & Dragotti, P. L.
- Venue
- IEEE Transactions on Image Processing
- Published
- 2006-07
- Volume
- 15
- Issue
- 7
- Pages
- 1916-1933
- ISSN
- 1057-7149 (ISSN-L)
- OpenAlex
- W2169244200
- Cited by
- 326
- References
- 36
- FWCI
- 28.23
- Type
- journal article
- Language
- en
- Open access
- green
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
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.
