The DOI matches its Crossref record; 17 of the 17 references it cites were found in a registry or an index.
A Closed-Form Solution to Natural Image Matting
Abstract
Interactive digital matting, the process of extracting a foreground object from an image based on limited user input, is an important task in image and video editing. From a computer vision perspective, this task is extremely challenging because it is massively ill-posed -- at each pixel we must estimate the foreground and the background colors, as well as the foreground opacity ("alpha matte") from a single color measurement. Current approaches either restrict the estimation to a small part of the image, estimating foreground and background colors based on nearby pixels where they are known, or perform iterative nonlinear estimation by alternating foreground and background color estimation with alpha estimation. In this paper we present a closed-form solution to natural image matting. We derive a cost function from local smoothness assumptions on foreground and background colors, and show that in the resulting expression it is possible to analytically eliminate the foreground and background colors to obtain a quadratic cost function in alpha. This allows us to find the globally optimal alpha matte by solving a sparse linear system of equations. Furthermore, the closed-form formula allows us to predict the properties of the solution by analyzing the eigenvectors of a sparse matrix, closely related to matrices used in spectral image segmentation algorithms. We show that high quality mattes for natural images may be obtained from a small amount of user input.
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
GrabCut: Interactive foreground extraction using iterated graph cuts
Rother, C., Kolmogorov, V., & Blake, A. (2004) · ACM Transactions on Graphics · doi:10.1145/1015706.1015720 · Cited by 5,703
Interactive graph cuts for optimal boundary & region segmentation of objects in N-D images
Boykov, Y. Y., & Jolly, M.-P. (2002) · Proceedings Eighth IEEE International Conference on Computer Vision. ICCV 2001 · doi:10.1109/iccv.2001.937505 · Cited by 3,674
Advances in Neural Information Processing Systems 18
Murray, I., Mackay, D., Zoubin Ghahramani, A., & Skilling, J. (2006) · Cited by 1,927
Colorization using optimization
Levin, A., Lischinski, D., & Weiss, Y. (2004) · ACM Transactions on Graphics · doi:10.1145/1015706.1015780 · Cited by 1,411
Self-citationLi, Y., Sun, J., Tang, C.-K., & Shum, H.-Y. (2004) · ACM Transactions on Graphics · doi:10.1145/1015706.1015719 · Cited by 1,097
Normalized cuts and image segmentation
Shi, J., & Malik, J. (2002) · Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition · doi:10.1109/cvpr.1997.609407 · Cited by 849
A Bayesian approach to digital matting
Chuang, Y.-Y., Curless, B., Salesin, D. H., & Szeliski, R. (2005) · Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001 · doi:10.1109/cvpr.2001.990970 · Cited by 796
Sun, J., Jia, J., Tang, C.-K., & Shum, H.-Y. (2004) · ACM Transactions on Graphics · doi:10.1145/1015706.1015721 · Cited by 583
Colorization using optimization
Levin, A., Lischinski, D., & Weiss, Y. (2004) · ACM SIGGRAPH 2004 Papers · doi:10.1145/1186562.1015780 · Cited by 478
Self-citationAlpha estimation in natural images
Ruzon, M. A., & Tomasi, C. (2002) · Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662) · doi:10.1109/cvpr.2000.855793 · Cited by 348
Cited by
DehazeNet: An End-to-End System for Single Image Haze Removal
Cai, B., Xu, X., Jia, K., Qing, C., & Tao, D. (2016) · IEEE Transactions on Image Processing · doi:10.1109/tip.2016.2598681 · Cited by 3,367
Random Walks for Image Segmentation
Grady, L. (2006) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2006.233 · Cited by 2,624
Underwater Image Restoration Based on Image Blurriness and Light Absorption
Peng, Y.-T., & Cosman, P. C. (2017) · IEEE Transactions on Image Processing · doi:10.1109/tip.2017.2663846 · Cited by 1,250
Automatic Red-Channel underwater image restoration
Galdran, A., Pardo, D., Picón, A., & Alvarez-Gila, A. (2015) · Journal of Visual Communication and Image Representation · doi:10.1016/j.jvcir.2014.11.006 · Cited by 995
Pattern-Affinitive Propagation Across Depth, Surface Normal and Semantic Segmentation
Zhang, Z., Cui, Z., Xu, C., Yan, Y., Sebe, N., & Yang, J. (2019) · 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2019.00423 · Cited by 331
Metadata
- Authors
- Levin, A., Lischinski, D., & Weiss, Y.
- Venue
- IEEE Transactions on Pattern Analysis and Machine Intelligence
- Published
- 2008-02
- Volume
- 30
- Issue
- 2
- Pages
- 228-242
- ISSN
- 0162-8828 (ISSN-L)
- OpenAlex
- W2035773017
- Cited by
- 1,616
- References
- 17
- FWCI
- 64.15
- 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
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: A Closed-Form Solution to Natural Image Matting. https://app.cochat.ai/papers/10.1109/tpami.2007.1177
- The paper
- Levin, A., Lischinski, D., & Weiss, Y. (2008). A Closed-Form Solution to Natural Image Matting. IEEE Transactions on Pattern Analysis and Machine Intelligence. https://doi.org/10.1109/tpami.2007.1177
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.
