Article
| Title: | A framework to combine vector-valued metrics into a scalar-metric: Application to data comparison (English) |
| Author: | Piella, Gemma |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 68 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 143-152 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Distance metrics are at the core of many processing and machine learning algorithms. In many contexts, it is useful to compute the distance between data using multiple criteria. This naturally leads to consider vector-valued metrics, in which the distance is no longer a real positive number but a vector. In this paper, we propose a principled way to combine several metrics into either a scalar-valued or vector-valued metric. We illustrate our framework by reformulating the popular structural similarity (SSIM) index and a simple case of the Wasserstein distance used for optimal transport. (English) |
| Keyword: | generalized metric |
| Keyword: | vector-valued metric |
| Keyword: | scalarization |
| Keyword: | image comparison |
| Keyword: | structural similarity index |
| MSC: | 46A40 |
| MSC: | 94A08 |
| idZBL: | Zbl 07675563 |
| idMR: | MR4574650 |
| DOI: | 10.21136/AM.2021.0090-21 |
| . | |
| Date available: | 2023-03-31T09:33:36Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151609 |
| . | |
| Reference: | [1] Amini-Harandi, A., Fakhar, M.: Fixed point theory in cone metric spaces obtained via the scalarization method.Comput. Math. Appl. 59 (2010), 3529-3534. Zbl 1197.54055, MR 2646324, 10.1016/j.camwa.2010.03.046 |
| Reference: | [2] Athar, S., Wang, Z.: A comprehensive performance evaluation of image quality assessment algorithms.IEEE Access 7 (2019), 140030-140070. 10.1109/ACCESS.2019.2943319 |
| Reference: | [3] Brunet, D., Vrscay, E. R., Wang, Z.: On the mathematical properties of the structural similarity index.IEEE Trans. Image Process. 21 (2012), 1488-1499. Zbl 1373.94051, MR 2959465, 10.1109/TIP.2011.2173206 |
| Reference: | [4] Bures, D.: An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite $w^*$-algebras.Trans. Am. Math. Soc. 135 (1969), 199-212. Zbl 0176.11402, MR 0236719, 10.1090/S0002-9947-1969-0236719-2 |
| Reference: | [5] Chen, G.-H., Yang, C.-L., Xie, S.-L.: Gradient-based structural similarity for image quality assessment.IEEE International Conference on Image Processing IEEE, Piscataway (2006), 2929-2932. 10.1109/ICIP.2006.313132 |
| Reference: | [6] Dosselmann, R., Yang, X. D.: A comprehensive assessment of the structural similarity index.Signal Image Video Process. 5 (2011), 81-91. 10.1007/s11760-009-0144-1 |
| Reference: | [7] Horé, A., Ziou, D.: Image quality metrics: PSNR vs. SSIM.20th International Conference on Pattern Recognition IEEE, Piscataway (2010), 2366-2369. 10.1109/ICPR.2010.579 |
| Reference: | [8] Huang, L.-G., Zhang, X.: Cone metric spaces and fixed point theorems of contractive mappings.J. Math. Anal. Appl. 332 (2007), 1468-1476. Zbl 1118.54022, MR 2324351, 10.1016/j.jmaa.2005.03.087 |
| Reference: | [9] Kang, L., Ye, P., Li, Y., Doermann, D.: Convolutional neural networks for no-reference image quality assessment.IEEE Conference on Computer Vision and Pattern Recognition IEEE, Piscataway (2014), 1733-1740. 10.1109/CVPR.2014.224 |
| Reference: | [10] Khani, M., Pourmahdian, M.: On the metrizability of cone metric spaces.Topology Appl. 158 (2011), 190-193. Zbl 1206.54026, MR 2739889, 10.1016/j.topol.2010.10.016 |
| Reference: | [11] Li, C., Bovik, A. C.: Content-partitioned structural similarity index for image quality assessment.Signal Process., Image Commun. 25 (2010), 517-526. 10.1016/j.image.2010.03.004 |
| Reference: | [12] Li, Y., Wang, G., Nie, L., Wang, Q., Tan, W.: Distance metric optimization driven convolutional neural network for age invariant face recognition.Pattern Recognition 75 (2018), 51-62. 10.1016/j.patcog.2017.10.015 |
| Reference: | [13] Ma, Z., Zhou, S., Wu, X., Zhang, H., Yan, W., Sun, S., Zhou, J.: Nasopharyngeal carcinoma segmentation based on enhanced convolutional neural networks using multi-modal metric learning.Phys. Med. Biol. 64 (2019), Article ID 025005. 10.1088/1361-6560/aaf5da |
| Reference: | [14] Villani, C.: Optimal Transport: Old and New.Grundlehren der Mathematischen Wissenschaften 338. Springer, Berlin (2009). Zbl 1156.53003, MR 2459454, 10.1007/978-3-540-71050-9 |
| Reference: | [15] Wainwright, M. J., Schwartz, O., Simoncelli, E. P.: Natural image statistics and divisive normalization.Probabilistic Models of the Brain: Perception and Neural Function MIT Press, Cambridge (2002), 203-222. |
| Reference: | [16] Wang, C., Peng, G., Baets, B. De: Deep feature fusion through adaptive discriminative metric learning for scene recognition.Inf. Fusion 63 (2020), 1-12. 10.1016/j.inffus.2020.05.005 |
| Reference: | [17] Wang, Z., Bovik, A. C.: A universal image quality index.IEEE Signal Process. Lett. 9 (2002), 81-84. 10.1109/97.995823 |
| Reference: | [18] Wang, Z., Bovik, A. C., Lu, L.: Why is image quality assessment so difficult?.IEEE International Conference on Acoustics, Speech, and Signal Processing IEEE, Piscataway (2002), IV-3313--IV-3316. MR 0642901, 10.1109/ICASSP.2002.5745362 |
| Reference: | [19] Wang, Z., Bovik, A. C., Sheikh, H. R., Simoncelli, E. P.: Image quality assessment: From error visibility to structural similarity.IEEE Trans. Image Process. 13 (2004), 600-612. 10.1109/TIP.2003.819861 |
| Reference: | [20] Wang, Z., Simoncelli, E. P., Bovik, A. C.: Multiscale structural similarity for image quality assessment.37th Asilomar Conference on Signals, Systems & Computers IEEE, Piscataway (2003), 1398-1402. 10.1109/ACSSC.2003.1292216 |
| Reference: | [21] Zhao, H., Gallo, O., Frosio, I., Kautz, J.: Loss functions for image restoration with neural networks.IEEE Trans. Comput. Imaging 3 (2017), 47-57. 10.1109/TCI.2016.2644865 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
