Understanding image representations by measuring their equivariance and equivalence
Abstract
Despite the importance of image representations such as histograms of oriented gradients and deep Convolutional Neural Networks (CNN), our theoretical understanding of them remains limited. Aiming at filling this gap, we investigate three key mathematical properties of representations: equivariance, invariance, and equivalence. Equivariance studies how transformations of the input image are encoded by the representation, invariance being a special case where a transformation has no effect. Equivalence studies whether two representations, for example two different parametrisations of a CNN, capture the same visual information or not. A number of methods to establish these properties empirically are proposed, including introducing transformation and stitching layers in CNNs. These methods are then applied to popular representations to reveal insightful aspects of their structure, including clarifying at which layers in a CNN certain geometric invariances are achieved. While the focus of the paper is theoretical, direct applications to structured-output regression are demonstrated too.
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A vision model trained on ImageNet can be aligned with a model trained on Places-365 while maintaining good performance.
“In their study, Lenc & Vedaldi (2015) made two notable findings: (1) A vision model trained on ImageNet (Russakovsky et al., 2015) can be aligned with a model trained on Places-365 (Zhou et al., 2017) while maintaining good performance;”
From The Platonic Representation Hypothesis (2024) · cites this paper as Lenc & Vedaldi (2015)
The early layers of convolutional networks are more interchangeable than later layers.
“(2) The early layers of these convolutional networks are more interchangeable than later layers.”
From The Platonic Representation Hypothesis (2024) · cites this paper as Lenc & Vedaldi (2015)
Cites
ImageNet classification with deep convolutional neural networks
Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2017) · Communications of the ACM · doi:10.1145/3065386 · Cited by 112,155 · see citing works
Distinctive Image Features from Scale-Invariant Keypoints
Lowe, D. G. (2004) · International Journal of Computer Vision · doi:10.1023/b:visi.0000029664.99615.94 · Cited by 55,589 · see citing works
Histograms of Oriented Gradients for Human Detection
Dalal, N., & Triggs, B. (2005) · 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) · doi:10.1109/cvpr.2005.177 · Cited by 33,584 · see citing works
Object recognition from local scale-invariant features
Lowe, D. G. (1999) · Proceedings of the Seventh IEEE International Conference on Computer Vision · doi:10.1109/iccv.1999.790410 · Cited by 16,472 · see citing works
Video Google: a text retrieval approach to object matching in videos
Sivic, & Zisserman (2003) · Proceedings Ninth IEEE International Conference on Computer Vision · doi:10.1109/iccv.2003.1238663 · Cited by 6,367 · see citing works
Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps
Simonyan, K., Vedaldi, A., & Zisserman, A. (2013) · arXiv · doi:10.48550/arxiv.1312.6034 · Cited by 5,370 · see citing works
Self-citationDeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition
Donahue, J., Jia, Y., Vinyals, O., Hoffman, J., Zhang, N., Tzeng, E., & Darrell, T. (2013) · arXiv · doi:10.48550/arxiv.1310.1531 · Cited by 4,019 · see citing works
Locality-constrained Linear Coding for image classification
Wang, J., Yang, J., Yu, K., Lv, F., Huang, T., & Gong, Y. (2010) · 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition · doi:10.1109/cvpr.2010.5540018 · Cited by 3,100 · see citing works
Representing and Recognizing the Visual Appearance of Materials Using Three-Dimensional Textons
Leung, T., & Malik, J. (2001) · International Journal of Computer Vision · doi:10.1023/a:1011126920638 · Cited by 1,498 · see citing works
HOGgles: Visualizing Object Detection Features
Vondrick, C., Khosla, A., Malisiewicz, T., & Torralba, A. (2013) · 2013 IEEE International Conference on Computer Vision · doi:10.1109/iccv.2013.8 · Cited by 282 · see citing works
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Deformable Convolutional Networks
Dai, J., Qi, H., Xiong, Y., Li, Y., Zhang, G., Hu, H., & Wei, Y. (2017) · 2017 IEEE International Conference on Computer Vision (ICCV) · doi:10.1109/iccv.2017.89 · Cited by 7,513 · see citing works
Cross Modal Distillation for Supervision Transfer
Gupta, S., Hoffman, J., & Malik, J. (2016) · 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2016.309 · Cited by 530 · see citing works
Why do deep convolutional networks generalize so poorly to small image transformations?
Azulay, A., & Weiss, Y. (2018) · arXiv · doi:10.48550/arxiv.1805.12177 · Cited by 428 · see citing works
Cheng, G., Han, J., Zhou, P., & Xu, D. (2019) · IEEE Transactions on Image Processing · doi:10.1109/tip.2018.2867198 · Cited by 408 · see citing works
Cheng, G., Zhou, P., & Han, J. (2016) · 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) · doi:10.1109/cvpr.2016.315 · Cited by 175 · see citing works
COOKIE: Contrastive Cross-Modal Knowledge Sharing Pre-training for Vision-Language Representation
Wen, K., Xia, J., Huang, Y., Li, L., Xu, J., & Shao, J. (2021) · 2021 IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv48922.2021.00221 · Cited by 37 · see citing works
The Platonic Representation Hypothesis
Huh, M., Cheung, B., Wang, T., & Isola, P. (2024) · arXiv · doi:10.48550/arxiv.2405.07987 · Cited by 35 · see citing works
Back into Plato's Cave: Examining Cross-modal Representational Convergence at Scale
Koepke, A. S., Zverev, D., Ginosar, S., & Efros, A. A. (2026) · arXiv · doi:10.48550/arxiv.2604.18572
- Authors
- Lenc, K., & Vedaldi, A.
- Affiliations
- University of Oxford (GB)
- Venue
- 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
- Publisher
- IEEE
- Published
- 2015-06
- Pages
- 991-999
- ISSN
- 1063-6919 (ISSN-L)
- OpenAlex
- W1912570122
- Cited by
- 366
- References
- 10
- Citations by year
- 2026: 7 · 2025: 22 · 2024: 26 · 2023: 32 · 2022: 36 · 2021: 50 · 2020: 45 · 2019: 50 · 2018: 40 · 2017: 31 · 2016: 18 · 2015: 9
- FWCI
- 13.40
- Type
- conference paper
- Language
- en
- Open access
- closed
- Locations
- IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings (publishedVersion)
- Topics
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Neural Network Applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Image and Video Retrieval Techniques
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Image Retrieval and Classification Techniques
- Keywords
- image representation, equivariance, invariance, representation equivalence, convolutional neural networks, geometric invariance, histogram of oriented gradients
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