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.
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