# Representing 3D Faces with Learnable B-Spline Volumes

arXiv: https://arxiv.org/abs/2604.12894 · HTML: https://mlsanity.com/paper/2604.12894/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 56 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 30 of 38)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Efficient Geometry-aware 3D Generative Adversarial Networks](https://mlsanity.com/paper/2112.07945/) (2021)
- [Face2Face: Real-time Face Capture and Reenactment of RGB Videos](https://mlsanity.com/paper/2007.14808/) (2020)
- [Accurate 3D Face Reconstruction with Weakly-Supervised Learning: From Single Image to Image Set](https://mlsanity.com/paper/1903.08527/) (2019)
- [Learning an Animatable Detailed 3D Face Model from In-The-Wild Images](https://mlsanity.com/paper/2012.04012/) (2020)
- [XCiT: Cross-Covariance Image Transformers](https://mlsanity.com/paper/2106.09681/) (2021)
- [3D Morphable Face Models -- Past, Present and Future](https://mlsanity.com/paper/1909.01815/) (2019)
- [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018)
- [GaussianAvatars: Photorealistic Head Avatars with Rigged 3D Gaussians](https://mlsanity.com/paper/2312.02069/) (2023)
- [Fake It Till You Make It: Face analysis in the wild using synthetic data alone](https://mlsanity.com/paper/2109.15102/) (2021)
- [Instant Volumetric Head Avatars](https://mlsanity.com/paper/2211.12499/) (2022)
- [FlashAvatar: High-fidelity Head Avatar with Efficient Gaussian Embedding](https://mlsanity.com/paper/2312.02214/) (2023)
- [Efficient Learning on Point Clouds with Basis Point Sets](https://mlsanity.com/paper/1908.09186/) (2019)
- [Emotional Speech-Driven Animation with Content-Emotion Disentanglement](https://mlsanity.com/paper/2306.08990/) (2023)
- [Learning Formation of Physically-Based Face Attributes](https://mlsanity.com/paper/2004.03458/) (2020)
- [Learning Neural Parametric Head Models](https://mlsanity.com/paper/2212.02761/) (2022)
- [ImFace: A Nonlinear 3D Morphable Face Model with Implicit Neural Representations](https://mlsanity.com/paper/2203.14510/) (2022)
- [NPGA: Neural Parametric Gaussian Avatars](https://mlsanity.com/paper/2405.19331/) (2024)
- [Media2Face: Co-speech Facial Animation Generation With Multi-Modality Guidance](https://mlsanity.com/paper/2401.15687/) (2024)
- [Instant Multi-View Head Capture through Learnable Registration](https://mlsanity.com/paper/2306.07437/) (2023)
- [3D Face Modeling From Diverse Raw Scan Data](https://mlsanity.com/paper/1902.04943/) (2019)
- [3D Face Tracking from 2D Video through Iterative Dense UV to Image Flow](https://mlsanity.com/paper/2404.09819/) (2024)
- [3D Shape Variational Autoencoder Latent Disentanglement via Mini-Batch Feature Swapping for Bodies and Faces](https://mlsanity.com/paper/2111.12448/) (2021)
- [Shape My Face: Registering 3D Face Scans by Surface-to-Surface Translation](https://mlsanity.com/paper/2012.09235/) (2020)

## Built on by (most cited first, 1 of 1)

- [PDB: Point-Based Deformation Blending for Facial Animation Retargeting](https://mlsanity.com/paper/2610.08672/) (2026)

## Abstract (arXiv)

We present CUBE (Control-based Unified B-spline Encoding), a new geometric representation for human faces that combines B-spline volumes with learned features, and demonstrate its use as a decoder for 3D scan registration and monocular 3D face reconstruction. Unlike existing B-spline representations with 3D control points, CUBE is parametrized by a lattice (e.g., 8 x 8 x 8) of high-dimensional control features, increasing the model's expressivity. These features define a continuous, two-stage mapping from a 3D parametric domain to 3D Euclidean space via an intermediate feature space. First, high-dimensional control features are locally blended using the B-spline bases, yielding a high-dimensional feature vector whose first three values define a 3D base mesh. A small MLP then processes this feature vector to predict a residual displacement from the base shape, yielding the final refined 3D coordinates. To reconstruct 3D surfaces in dense semantic correspondence, CUBE is queried at 3D coordinates sampled from a fixed template mesh. Crucially, CUBE retains the local support property of traditional B-spline representations, enabling local surface editing by updating individual control features. We demonstrate the strengths of this representation by training transformer-based encoders to predict CUBE's control features from unstructured point clouds and monocular images, achieving state-of-the-art scan registration results compared to recent baselines.