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| Version | Summary | Created by | Modification | Content Size | Created at | Operation |
|---|---|---|---|---|---|---|
| 1 | Sirius Huang | -- | 4920 | 2022-11-24 01:42:13 |
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The convolutional sparse coding paradigm is an extension of the global Sparse Coding model, in which a redundant dictionary is modeled as a concatenation of circulant matrices. While the global sparsity constraint describes signal [math]\displaystyle{ \mathbf{x}\in \mathbb{R}^{N} }[/math] as a linear combination of a few atoms in the redundant dictionary [math]\displaystyle{ \mathbf{D}\in\mathbb{R}^{N\times M}, M\gg N }[/math], usually expressed as [math]\displaystyle{ \mathbf{x}=\mathbf{D}\mathbf{\Gamma} }[/math] for a sparse vector [math]\displaystyle{ \mathbf{\Gamma}\in \mathbb{R}^{M} }[/math], the alternative dictionary structure adopted by the Convolutional Sparse Coding model allows the sparsity prior to be applied locally instead of globally: independent patches of [math]\displaystyle{ \mathbf{x} }[/math] are generated by "local" dictionaries operating over stripes of [math]\displaystyle{ \mathbf{\Gamma} }[/math]. The local sparsity constraint allows stronger uniqueness and stability conditions than the global sparsity prior, and has shown to be a versatile tool for inverse problems in fields such as Image Understanding and Computer Vision. Also, a recently proposed multi-layer extension of the model has shown conceptual benefits for more complex signal decompositions, as well as a tight connection the Convolutional Neural Networks model, allowing a deeper understanding of how the latter operates.
References
- Jianchao Yang; Wright, John; Huang, Thomas S; Yi Ma (November 2010). "Image Super-Resolution Via Sparse Representation". IEEE Transactions on Image Processing 19 (11): 2861–2873. doi:10.1109/TIP.2010.2050625. PMID 20483687. Bibcode: 2010ITIP...19.2861Y. https://dx.doi.org/10.1109%2FTIP.2010.2050625
- Wetzstein, Gordon; Heidrich, Wolfgang; Heide, Felix (2015). Fast and Flexible Convolutional Sparse Coding. pp. 5135–5143. https://www.cv-foundation.org/openaccess/content_cvpr_2015/html/Heide_Fast_and_Flexible_2015_CVPR_paper.html.
- Wohlberg, Brendt (2017). "SPORCO: A Python package for standard and convolutional sparse representations". Proceedings of the 16th Python in Science Conference: 1–8. doi:10.25080/shinma-7f4c6e7-001. http://conference.scipy.org/proceedings/scipy2017/brendt_wohlberg.html.
- Mairal, Julien; Bach, Francis; Ponce, Jean; Sapiro, Guillermo (2009). "Online Dictionary Learning for Sparse Coding". Proceedings of the 26th Annual International Conference on Machine Learning (ACM): 689–696. doi:10.1145/1553374.1553463. ISBN 9781605585161. https://dl.acm.org/citation.cfm?id=1553463.
- Papyan, Vardan; Sulam, Jeremias; Elad, Michael (1 November 2017). "Working Locally Thinking Globally: Theoretical Guarantees for Convolutional Sparse Coding". IEEE Transactions on Signal Processing 65 (21): 5687–5701. doi:10.1109/TSP.2017.2733447. Bibcode: 2017ITSP...65.5687P. https://dx.doi.org/10.1109%2FTSP.2017.2733447
- Wohlberg, Brendt (6–8 March 2016). "Convolutional sparse representation of color images". 2016 IEEE Southwest Symposium on Image Analysis and Interpretation (SSIAI): 57–60. doi:10.1109/SSIAI.2016.7459174. ISBN 978-1-4673-9919-7. https://dx.doi.org/10.1109%2FSSIAI.2016.7459174
- Papyan, Vardan; Romano, Yaniv; Elad, Michael (2017). "Convolutional Neural Networks Analyzed via Convolutional Sparse Coding". J. Mach. Learn. Res. 18 (1): 2887–2938. ISSN 1532-4435. Bibcode: 2016arXiv160708194P. http://dl.acm.org/citation.cfm?id=3122009.3176827.
- Beck, Amir; Teboulle, Marc (January 2009). "A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems". SIAM Journal on Imaging Sciences 2 (1): 183–202. doi:10.1137/080716542. https://dx.doi.org/10.1137%2F080716542
- Boyd, Stephen (2010). "Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers". Foundations and Trends in Machine Learning 3 (1): 1–122. doi:10.1561/2200000016. ISSN 1935-8237. https://dx.doi.org/10.1561%2F2200000016
