The inherent challenge of detecting symmetries stems from arbitrary orientations of symmetry patterns; a reflection symmetry mirrors itself against an axis with a specific orientation while a rotation symmetry matches its rotated copy with a specific orientation. Discovering such symmetry patterns from an image thus benefits from an equivariant feature representation, which varies consistently with reflection and rotation of the image. In this work, we introduce a group-equivariant convolutional network for symmetry detection, dubbed EquiSym, which leverages equivariant feature maps with respect to a dihedral group of reflection and rotation. The proposed network is built end-to-end with dihedrally-equivariant layers and trained to output a spatial map for reflection axes or rotation centers. We also present a new dataset, DENse and DIverse symmetry (DENDI), which mitigates limitations of existing benchmarks for reflection and rotation symmetry detection. Experiments show that our method achieves the state of the arts in symmetry detection on LDRS and DENDI datasets.

Overall architecture

Figure 1. Illustration of the proposed symmetry detection network, EquiSym. After an input image I passed a group-equivariant encoder Enc, group-equivariant decoders Decref and Decrot predict intermediate predictions Sref and Srot for rotation and reflection, respectively. Auxillary tasks for the rotation and reflection symmetry are the order(N) of the rotation fold and the orientation of the reflection axis. The foreground logits are pooled to Pref and Prot and stacked with the scores Sref and Srot, respectively. The final score Yref and Yrot for the rotation center and the reflection axis are predicted using a group-equivariant 1 × 1 convolution.

Quantitative results

1. Ablation study.

Table 1. Ablation on the symmetry detection network in reflection, rotation, and joint model on DENDI dataset.

2. Comparison with the state-of-the-art methods.

Table 2. Comparison with the state-of-the-art methods on DENDI.

Qualitative results

1. Reflection symmetry.

Figure 2. Qualitative results of the reflection symmetry detection on DENDI-ref test.

2. Rotation symmetry.

Figure 3. Qualitative results of the rotation symmetry detection on DENDI-rot test


This work was supported by Samsung Advanced Institute of Technology (SAIT) and also by the NRF grant (NRF-2021R1A2C3012728) and the IITP grant (No.2021-0-02068: AI Innovation Hub, No.2019-0-01906: Artificial Intelligence Graduate School Program at POSTECH) funded by the Korea government (MSIT). We like to thank Yunseon Choi for her contribution to DENDI.


Reflection and Rotation Symmetry Detection via Equivariant Learning
Ahyun Seo, Byungjin Kim, Suha Kwak, Minsu Cho
CVPR, 2022
[arXiv] [Bibtex]

Code & dataset

Code is available online. [github (EquiSym)]

Access our new dataset (DENDI) here. [project page (DENDI)] [github (DENDI)]