The DOI matches its Crossref record; 81 of the 81 references it cites were found in a registry or an index.
Solving Inverse Problems With Piecewise Linear Estimators: From Gaussian Mixture Models to Structured Sparsity
Abstract
A general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper. The approach is based on Gaussian mixture models, which are estimated via a maximum a posteriori expectation-maximization algorithm. A dual mathematical interpretation of the proposed framework with a structured sparse estimation is described, which shows that the resulting piecewise linear estimate stabilizes the estimation when compared with traditional sparse inverse problem techniques. We demonstrate that, in a number of image inverse problems, including interpolation, zooming, and deblurring of narrow kernels, the same simple and computationally efficient algorithm yields results in the same ballpark as that of the state of the art.
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
Boyd, S., & Vandenberghe, L. (2004) · doi:10.1017/cbo9780511804441 · Cited by 56,601
Regression shrinkage and selection via the lasso
Tibshirani, R. (1996) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1996.tb02080.x · Cited by 51,482
Maximum Likelihood from Incomplete Data Via the EM Algorithm
Dempster, A. P., Laird, N. M., & Rubin, D. B. (1977) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1977.tb01600.x · Cited by 50,444
Donoho, D. L. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.871582 · Cited by 23,994
Donoho, D. L. (2004) · Cited by 16,703
A Wavelet Tour of Signal Processing
(1999) · Elsevier eBooks · doi:10.1016/b978-0-12-466606-1.x5000-4 · Cited by 16,227
$rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation
Aharon, M., Elad, M., & Bruckstein, A. (2006) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2006.881199 · Cited by 9,451
Efron, B., Hastie, T., Johnstone, I., & Tibshirani, R. (2004) · The Annals of Statistics · doi:10.1214/009053604000000067 · Cited by 9,357
Image denoising by sparse 3-D transform-domain collaborative filtering
Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2007) · IEEE Transactions on Image Processing · doi:10.1109/tip.2007.901238 · Cited by 9,283
Matching pursuits with time-frequency dictionaries
Mallat, S. G., & Zhang, Z. (1993) · IEEE Transactions on Signal Processing · doi:10.1109/78.258082 · Cited by 9,112
Self-citationAdaptive background mixture models for real-time tracking
Stauffer, C., & Grimson, W. E. L. (2003) · Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149) · doi:10.1109/cvpr.1999.784637 · Cited by 6,923
Atomic Decomposition by Basis Pursuit
Chen, S. S., Donoho, D. L., & Saunders, M. A. (1998) · SIAM Journal on Scientific Computing · doi:10.1137/s1064827596304010 · Cited by 6,845
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
Candes, E. J., & Tao, T. (2006) · IEEE Transactions on Information Theory · doi:10.1109/tit.2006.885507 · Cited by 6,839
Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries
Elad, M., & Aharon, M. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2006.881969 · Cited by 5,321
Image Super-Resolution Via Sparse Representation
Yang, J., Wright, J., Huang, T. S., & Ma, Y. (2010) · IEEE Transactions on Image Processing · doi:10.1109/tip.2010.2050625 · Cited by 5,217
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint.
Daubechies, I., Defrise, M., & De Mol, C. (2004) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.20042 · Cited by 4,958
On the Statistical Analysis of Dirty Pictures
Besag, J. (1986) · Journal of the Royal Statistical Society Series B: Statistical Methodology · doi:10.1111/j.2517-6161.1986.tb01412.x · Cited by 4,191
A Review of Image Denoising Algorithms, with a New One
Buades, A., Coll, B., & Morel, J. M. (2005) · Multiscale Modeling & Simulation · doi:10.1137/040616024 · Cited by 4,083
Greed is Good: Algorithmic Results for Sparse Approximation
Tropp, J. A. (2004) · IEEE Transactions on Information Theory · doi:10.1109/tit.2004.834793 · Cited by 3,635
Cubic convolution interpolation for digital image processing
Keys, R. (1981) · IEEE Transactions on Acoustics, Speech, and Signal Processing · doi:10.1109/tassp.1981.1163711 · Cited by 3,255
How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis
Fraley, C. (1998) · The Computer Journal · doi:10.1093/comjnl/41.8.578 · Cited by 2,549
Image denoising using scale mixtures of gaussians in the wavelet domain
Portilla, J., Strela, V., Wainwright, M. J., & Simoncelli, E. P. (2003) · IEEE Transactions on Image Processing · doi:10.1109/tip.2003.818640 · Cited by 2,255
Mixture models : inference and applications to clustering
McLachlan, G. J., & Basford, K. E. (1988) · M. Dekker eBooks · Cited by 2,094
A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration
Bioucas-Dias, J. M., & Figueiredo, M. A. T. (2007) · IEEE Transactions on Image Processing · doi:10.1109/tip.2007.909319 · Cited by 2,064
New edge-directed interpolation
Li, X., & Orchard, M. T. (2001) · IEEE Transactions on Image Processing · doi:10.1109/83.951537 · Cited by 2,023
Sparse Representation for Color Image Restoration
Mairal, J., Elad, M., & Sapiro, G. (2008) · IEEE Transactions on Image Processing · doi:10.1109/tip.2007.911828 · Cited by 1,716
Self-citationNon-local sparse models for image restoration
Mairal, J., Bach, F., Ponce, J., Sapiro, G., & Zisserman, A. (2009) · 2009 IEEE 12th International Conference on Computer Vision · doi:10.1109/iccv.2009.5459452 · Cited by 1,666
Self-citationSplines: a perfect fit for signal and image processing
Unser, M. (1999) · IEEE Signal Processing Magazine · doi:10.1109/79.799930 · Cited by 1,633
Schäfer, J., & Strimmer, K. (2005) · Statistical Applications in Genetics and Molecular Biology · doi:10.2202/1544-6115.1175 · Cited by 1,561
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
Nonlocal Operators with Applications to Image Processing
Gilboa, G., & Osher, S. (2009) · Multiscale Modeling & Simulation · doi:10.1137/070698592 · Cited by 1,423
Kernel Regression for Image Processing and Reconstruction
Takeda, H., Farsiu, S., & Milanfar, P. (2007) · IEEE Transactions on Image Processing · doi:10.1109/tip.2006.888330 · Cited by 1,340
Block-Sparse Signals: Uncertainty Relations and Efficient Recovery
Eldar, Y. C., Kuppinger, P., & Bolcskei, H. (2010) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2010.2044837 · Cited by 1,313
Model-Based Compressive Sensing
Baraniuk, R. G., Cevher, V., Duarte, M. F., & Hegde, C. (2010) · IEEE Transactions on Information Theory · doi:10.1109/tit.2010.2040894 · Cited by 1,308
Iteratively reweighted least squares minimization for sparse recovery
Daubechies, I., DeVore, R., Fornasier, M., & Güntürk, C. S. (2010) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.20303 · Cited by 1,295
Mixture Models: Inference and Applications to Clustering.
Lindsay, B., McLachlan, G. L., Basford, K. E., & Dekker, M. (1989) · Journal of the American Statistical Association · doi:10.2307/2289892 · Cited by 1,062
Robust Recovery of Signals From a Structured Union of Subspaces
Eldar, Y. C., & Mishali, M. (2009) · IEEE Transactions on Information Theory · doi:10.1109/tit.2009.2030471 · Cited by 1,044
Generalized principal component analysis (GPCA)
Vidal, R., Ma, Y., & Sastry, S. (2005) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2005.244 · Cited by 1,038
An edge-guided image interpolation algorithm via directional filtering and data fusion
Zhang, L., & Wu, X. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2006.877407 · Cited by 1,020
Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA)
Elad, M., Starck, J.-L., Querre, P., & Donoho, D. L. (2005) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2005.03.005 · Cited by 904
Gaussian parsimonious clustering models
Celeux, G., & Govaert, G. (1995) · Pattern Recognition · doi:10.1016/0031-3203(94)00125-6 · Cited by 872
Roth, S., & Black, M. J. (2009) · International Journal of Computer Vision · doi:10.1007/s11263-008-0197-6 · Cited by 847
Image Segmentation in Video Sequences: A Probabilistic Approach
Friedman, N., & Russell, S. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1302.1539 · Cited by 833
A classification EM algorithm for clustering and two stochastic versions
Celeux, G., & Govaert, G. (1992) · Computational Statistics & Data Analysis · doi:10.1016/0167-9473(92)90042-e · Cited by 772
Supervised learning from incomplete data via an EM approach
Zoubin Ghahramani, A., & Jordan, M. I. (1993) · Cited by 552
Nonlocal Discrete Regularization on Weighted Graphs: A Framework for Image and Manifold Processing
Elmoataz, A., Lezoray, O., & Bougleux, S. (2008) · IEEE Transactions on Image Processing · doi:10.1109/tip.2008.924284 · Cited by 466
Image restoration by sparse 3D transform-domain collaborative filtering
Dabov, K., Foi, A., Katkovnik, V., & Egiazarian, K. (2008) · SPIE Proceedings · doi:10.1117/12.766355 · Cited by 461
Learning Multiscale Sparse Representations for Image and Video Restoration
Mairal, J., Sapiro, G., & Elad, M. (2008) · Multiscale Modeling & Simulation · doi:10.1137/070697653 · Cited by 460
Self-citationForWaRD: Fourier-Wavelet Regularized Deconvolution for Ill-Conditioned Systems
Neelamani, R., Choi, H., & Baraniuk, R. (2004) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2003.821103 · Cited by 442
Structured Variable Selection with Sparsity-Inducing Norms
Jenatton, R., Audibert, J.-Y., & Bach, F. (2009) · arXiv (Cornell University) · doi:10.48550/arxiv.0904.3523 · Cited by 431
Nonparametric Bayesian Dictionary Learning for Analysis of Noisy and Incomplete Images
Zhou, M., Chen, H., Paisley, J., Ren, L., Li, L., Xing, Z., Dunson, D., Sapiro, G., & Carin, L. (2012) · IEEE Transactions on Image Processing · doi:10.1109/tip.2011.2160072 · Cited by 378
Self-citationCombining Mixture Components for Clustering
Baudry, J.-P., Raftery, A. E., Celeux, G., Lo, K., & Gottardo, R. (2010) · Journal of Computational and Graphical Statistics · doi:10.1198/jcgs.2010.08111 · Cited by 348
Inpainting and Zooming Using Sparse Representations
Fadili, M. J., Starck, J.-L., & Murtagh, F. (2008) · The Computer Journal · doi:10.1093/comjnl/bxm055 · Cited by 333
Super-Resolution With Sparse Mixing Estimators
Mallat, S., & Yu, G. (2010) · IEEE Transactions on Image Processing · doi:10.1109/tip.2010.2049927 · Cited by 296
Self-citationClustering-Based Denoising With Locally Learned Dictionaries
Chatterjee, P., & Milanfar, P. (2009) · IEEE Transactions on Image Processing · doi:10.1109/tip.2009.2018575 · Cited by 280
Another interpretation of the EM algorithm for mixture distributions
Hathaway, R. J. (1986) · Statistics & Probability Letters · doi:10.1016/0167-7152(86)90016-7 · Cited by 257
Non-local Regularization of Inverse Problems
Peyré, G., Bougleux, S., & Cohen, L. (2008) · Lecture Notes in Computer Science · doi:10.1007/978-3-540-88690-7_5 · Cited by 249
Towards a Coherent Statistical Framework for Dense Deformable Template Estimation
Allassonnière, S., Amit, Y., & Trouvé, A. (2007) · Journal of the Royal Statistical Society Series B (Statistical Methodology) · doi:10.1111/j.1467-9868.2007.00574.x · Cited by 222
A New Approach to Collaborative Filtering: Operator Estimation with Spectral Regularization
Abernethy, J., Bach, F., Evgeniou, T., & Vert, J.-P. (2008) · arXiv · doi:10.48550/arxiv.0802.1430 · Cited by 210
MOMS: maximal-order interpolation of minimal support
Blu, T., Thcvenaz, P., & Unser, M. A. (2001) · IEEE Transactions on Image Processing · doi:10.1109/83.931101 · Cited by 206
Guleryuz, O. G. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2005.863057 · Cited by 193
Learning with structured sparsity
Huang, J., Zhang, T., & Metaxas, D. (2009) · International Conference on Machine Learning · doi:10.1145/1553374.1553429 · Cited by 190
Total Variation-Based Image Deconvolution: a Majorization-Minimization Approach
Bioucas-Dias, J. M., Figueiredo, M. A. T., & Oliveira, J. P. (2006) · 2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings · doi:10.1109/icassp.2006.1660479 · Cited by 183
Local Adaptivity to Variable Smoothness for Exemplar-Based Image Regularization and Representation
Kervrann, C., & Boulanger, J. (2007) · International Journal of Computer Vision · doi:10.1007/s11263-007-0096-2 · Cited by 160
Guleryuz, O. G. (2006) · IEEE Transactions on Image Processing · doi:10.1109/tip.2005.863055 · Cited by 140
Gaussian mixture models of texture and colour for image database retrieval
Permuter, H. H., Francos, J. M., & Jermyn, I. H. (2003) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2003.1199538 · Cited by 121
Image Restoration Using Space-Variant Gaussian Scale Mixtures in Overcomplete Pyramids
Guerrero-Colon, J. A., Mancera, L., & Portilla, J. (2007) · IEEE Transactions on Image Processing · doi:10.1109/tip.2007.911473 · Cited by 118
Least-squares image resizing using finite differences
Munoz, D. C., Blu, T., & Unser, M. A. (2001) · IEEE Transactions on Image Processing · doi:10.1109/83.941860 · Cited by 117
Image restoration through l0 analysis-based sparse optimization in tight frames
Portilla, J. (2009) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2009.5413975 · Cited by 98
Orthogonal bandelet bases for geometric images approximation
Gabriel Peyré, A., & Mallat, S. (2008) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.20242 · Cited by 92
Self-citationStatistical Compressed Sensing of Gaussian Mixture Models
Yu, G., & Sapiro, G. (2011) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2011.2168521 · Cited by 92
Self-citationImage modeling and enhancement via structured sparse model selection
Yu, G., Sapiro, G., & Mallat, S. (2010) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2010.5653853 · Cited by 78
Self-citationModeling Multiscale Subbands of Photographic Images with Fields of Gaussian Scale Mixtures
Lyu, S., & Simoncelli, E. P. (2009) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2008.107 · Cited by 74
Shape-adaptive DCT for denoising and image reconstruction
Foi, A., Dabov, K., Katkovnik, V. Y., et al. (2006) · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · doi:10.1117/12.642839 · Cited by 70
Image interpolation using multiscale geometric representations
Mueller, N. A., Lu, Y. M., & Do, M. N. (2007) · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · doi:10.1117/12.714510 · Cited by 69
Lou, Y., Bertozzi, A. L., & Soatto, S. (2010) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-010-0220-8 · Cited by 65
EM Demystified: An Expectation-Maximization Tutorial
Chen, Y., & Gupta, M. R. (2010) · Cited by 56
Image denoising using mixtures of Gaussian scale mixtures
Guerrero-Colon, J. A., Simoncelli, E. P., & Portilla, J. (2008) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2008.4711817 · Cited by 40
A Bayesian MAP-EM Algorithm for PET Image Reconstruction Using Wavelet Transform
Zhou, J., Coatrieux, J.-L., Bousse, A., Shu, H., & Luo, L. (2007) · IEEE Transactions on Nuclear Science · doi:10.1109/tns.2007.901200 · Cited by 28
Non-local image smoothing by applying anisotropic diffusion PDE's in the space of patches
Tschumperlé, D., & Brun, L. (2009) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.2009.5413453 · Cited by 25
Sparse Super-Resolution with Space Matching Pursuits
Yu, G., & Mallat, S. (2009) · HAL · Cited by 13
Self-citation
Cited by
The Little Engine That Could: Regularization by Denoising (RED)
Romano, Y., Elad, M., & Milanfar, P. (2017) · SIAM Journal on Imaging Sciences · doi:10.1137/16m1102884 · Cited by 934
Turning a denoiser into a super-resolver using plug and play priors
Brifman, A., Romano, Y., & Elad, M. (2016) · 2016 IEEE International Conference on Image Processing (ICIP) · doi:10.1109/icip.2016.7532589 · Cited by 91
Stochastic Image Denoising by Sampling from the Posterior Distribution
Kawar, B., Vaksman, G., & Elad, M. (2021) · 2021 IEEE/CVF International Conference on Computer Vision Workshops (ICCVW) · doi:10.1109/iccvw54120.2021.00213 · Cited by 42
Metadata
- Authors
- Yu, G., Sapiro, G., & Mallat, S.
- Venue
- IEEE Transactions on Image Processing
- Published
- 2012-05
- Volume
- 21
- Issue
- 5
- Pages
- 2481-2499
- ISSN
- 1057-7149 (ISSN-L)
- 1941-0042
- OpenAlex
- W2109320267
- Cited by
- 561
- References
- 81
- FWCI
- 35.89
- Type
- journal article
- Language
- en
- Open access
- closed
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.
