# Spherical Regression: Learning Viewpoints, Surface Normals and 3D Rotations on n-Spheres

arXiv: https://arxiv.org/abs/1904.05404 · HTML: https://mlsanity.com/paper/1904.05404/

## Key facts

- Cited by 37 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2019.

## Built on by (most cited first, 30 of 37)

- [Metric3Dv2: A Versatile Monocular Geometric Foundation Model for Zero-shot Metric Depth and Surface Normal Estimation](https://mlsanity.com/paper/2404.15506/) (2024)
- [Estimating and Exploiting the Aleatoric Uncertainty in Surface Normal Estimation](https://mlsanity.com/paper/2109.09881/) (2021)
- [Deep Regression on Manifolds: A 3D Rotation Case Study](https://mlsanity.com/paper/2103.16317/) (2021)
- [Implicit-PDF: Non-Parametric Representation of Probability Distributions on the Rotation Manifold](https://mlsanity.com/paper/2106.05965/) (2021)
- [Wide-Baseline Relative Camera Pose Estimation with Directional Learning](https://mlsanity.com/paper/2106.03336/) (2021)
- [Projective Manifold Gradient Layer for Deep Rotation Regression](https://mlsanity.com/paper/2110.11657/) (2021)
- [IronDepth: Iterative Refinement of Single-View Depth using Surface Normal and its Uncertainty](https://mlsanity.com/paper/2210.03676/) (2022)
- [PoseContrast: Class-Agnostic Object Viewpoint Estimation in the Wild with Pose-Aware Contrastive Learning](https://mlsanity.com/paper/2105.05643/) (2021)
- [Image to Sphere: Learning Equivariant Features for Efficient Pose Prediction](https://mlsanity.com/paper/2302.13926/) (2023)
- [FisherMatch: Semi-Supervised Rotation Regression via Entropy-based Filtering](https://mlsanity.com/paper/2203.15765/) (2022)
- [Delving into Discrete Normalizing Flows on SO(3) Manifold for Probabilistic Rotation Modeling](https://mlsanity.com/paper/2304.03937/) (2023)
- [PolyMaX: General Dense Prediction with Mask Transformer](https://mlsanity.com/paper/2311.05770/) (2023)
- [Neural Fourier Transform: A General Approach to Equivariant Representation Learning](https://mlsanity.com/paper/2305.18484/) (2023)
- [DejaVu: Conditional Regenerative Learning to Enhance Dense Prediction](https://mlsanity.com/paper/2303.01573/) (2023)
- [A Laplace-inspired Distribution on SO(3) for Probabilistic Rotation Estimation](https://mlsanity.com/paper/2303.01743/) (2023)
- [Lifting 2D Object Locations to 3D by Discounting LiDAR Outliers across Objects and Views](https://mlsanity.com/paper/2109.07945/) (2021)
- [Equivariant Single View Pose Prediction Via Induced and Restricted Representations](https://mlsanity.com/paper/2307.03704/) (2023)
- [PanoNormal: Monocular Indoor 360deg Surface Normal Estimation](https://mlsanity.com/paper/2405.18745/) (2024)
- [Semi-supervised Viewpoint Estimation with Geometry-aware Conditional Generation](https://mlsanity.com/paper/2104.01103/) (2021)
- [Towards Robust Probabilistic Modeling on SO(3) via Rotation Laplace Distribution](https://mlsanity.com/paper/2305.10465/) (2023)
- [Multi-view Surface Reconstruction Using Normal and Reflectance Cues](https://mlsanity.com/paper/2506.04115/) (2025)
- [Image to Icosahedral Projection for SO(3) Object Reasoning from Single-View Images](https://mlsanity.com/paper/2207.08925/) (2022)
- [3D Equivariant Pose Regression via Direct Wigner-D Harmonics Prediction](https://mlsanity.com/paper/2411.00543/) (2024)
- [Learning ABCs: Approximate Bijective Correspondence for isolating factors of variation with weak supervision](https://mlsanity.com/paper/2103.03240/) (2021)
- [Polymorphic-GAN: Generating Aligned Samples across Multiple Domains with Learned Morph Maps](https://mlsanity.com/paper/2206.02903/) (2022)
- [ViewNet: Unsupervised Viewpoint Estimation from Conditional Generation](https://mlsanity.com/paper/2212.00435/) (2022)
- [Multi-task Geometric Estimation of Depth and Surface Normal from Monocular 360deg Images](https://mlsanity.com/paper/2411.01749/) (2024)
- [FOUND: Foot Optimization with Uncertain Normals for Surface Deformation Using Synthetic Data](https://mlsanity.com/paper/2310.18279/) (2023)
- [Semi-Supervised Unconstrained Head Pose Estimation in the Wild](https://mlsanity.com/paper/2404.02544/) (2024)
- [Alignist: CAD-Informed Orientation Distribution Estimation by Fusing Shape and Correspondences](https://mlsanity.com/paper/2409.06683/) (2024)

## Abstract (arXiv)

Many computer vision challenges require continuous outputs, but tend to be solved by discrete classification. The reason is classification's natural containment within a probability n-simplex, as defined by the popular softmax activation function. Regular regression lacks such a closed geometry, leading to unstable training and convergence to suboptimal local minima. Starting from this insight we revisit regression in convolutional neural networks. We observe many continuous output problems in computer vision are naturally contained in closed geometrical manifolds, like the Euler angles in viewpoint estimation or the normals in surface normal estimation. A natural framework for posing such continuous output problems are n-spheres, which are naturally closed geometric manifolds defined in the ℝ(n+1) space. By introducing a spherical exponential mapping on n-spheres at the regression output, we obtain well-behaved gradients, leading to stable training. We show how our spherical regression can be utilized for several computer vision challenges, specifically viewpoint estimation, surface normal estimation and 3D rotation estimation. For all these problems our experiments demonstrate the benefit of spherical regression. All paper resources are available at https://github.com/leoshine/SphericalRegression.