# Modeling Depth Ambiguity: A Mixture-Density Representation for Flying-Point-Free Depth Estimation

arXiv: https://arxiv.org/abs/2606.02552 · HTML: https://mlsanity.com/paper/2606.02552/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 36 references that are in the map.
- First posted on arXiv 1 Jun 2026.

## Builds on (most cited first, 30 of 31)

- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Vision Transformers for Dense Prediction](https://mlsanity.com/paper/2103.13413/) (2021)
- [DUSt3R: Geometric 3D Vision Made Easy](https://mlsanity.com/paper/2312.14132/) (2023)
- [VGGT: Visual Geometry Grounded Transformer](https://mlsanity.com/paper/2503.11651/) (2025)
- [Towards Robust Monocular Depth Estimation: Mixing Datasets for Zero-shot Cross-dataset Transfer](https://mlsanity.com/paper/1907.01341/) (2019)
- [Depth Anything: Unleashing the Power of Large-Scale Unlabeled Data](https://mlsanity.com/paper/2401.10891/) (2024)
- [Depth Map Prediction from a Single Image using a Multi-Scale Deep Network](https://mlsanity.com/paper/1406.2283/) (2014)
- [Grounding Image Matching in 3D with MASt3R](https://mlsanity.com/paper/2406.09756/) (2024)
- [Depth Anything 3: Recovering the Visual Space from Any Views](https://mlsanity.com/paper/2511.10647/) (2025)
- [Hypersim: A Photorealistic Synthetic Dataset for Holistic Indoor Scene Understanding](https://mlsanity.com/paper/2011.02523/) (2020)
- [Continuous 3D Perception Model with Persistent State](https://mlsanity.com/paper/2501.12387/) (2025)
- [DL3DV-10K: A Large-Scale Scene Dataset for Deep Learning-based 3D Vision](https://mlsanity.com/paper/2312.16256/) (2023)
- [TartanAir: A Dataset to Push the Limits of Visual SLAM](https://mlsanity.com/paper/2003.14338/) (2020)
- [Virtual KITTI 2](https://mlsanity.com/paper/2001.10773/) (2020)
- [π3: Permutation-Equivariant Visual Geometry Learning](https://mlsanity.com/paper/2507.13347/) (2025)
- [MapAnything: Universal Feed-Forward Metric 3D Reconstruction](https://mlsanity.com/paper/2509.13414/) (2025)
- [Neural RGB-D Surface Reconstruction](https://mlsanity.com/paper/2104.04532/) (2021)
- [DeepMVS: Learning Multi-view Stereopsis](https://mlsanity.com/paper/1804.00650/) (2018)
- [PointOdyssey: A Large-Scale Synthetic Dataset for Long-Term Point Tracking](https://mlsanity.com/paper/2307.15055/) (2023)
- [ReFusion: 3D Reconstruction in Dynamic Environments for RGB-D Cameras Exploiting Residuals](https://mlsanity.com/paper/1905.02082/) (2019)
- [Aria Digital Twin: A New Benchmark Dataset for Egocentric 3D Machine Perception](https://mlsanity.com/paper/2306.06362/) (2023)
- [DynamicStereo: Consistent Dynamic Depth from Stereo Videos](https://mlsanity.com/paper/2305.02296/) (2023)
- [ClearGrasp: 3D Shape Estimation of Transparent Objects for Manipulation](https://mlsanity.com/paper/1910.02550/) (2019)
- [SMD-Nets: Stereo Mixture Density Networks](https://mlsanity.com/paper/2104.03866/) (2021)
- [STream3R: Scalable Sequential 3D Reconstruction with Causal Transformer](https://mlsanity.com/paper/2508.10893/) (2025)
- [OmniWorld: A Multi-Domain and Multi-Modal Dataset for 4D World Modeling](https://mlsanity.com/paper/2509.12201/) (2025)
- [Pixel-Perfect Depth with Semantics-Prompted Diffusion Transformers](https://mlsanity.com/paper/2510.07316/) (2025)
- [Open Challenges in Deep Stereo: the Booster Dataset](https://mlsanity.com/paper/2206.04671/) (2022)
- [Seeing and Seeing Through the Glass: Real and Synthetic Data for Multi-Layer Depth Estimation](https://mlsanity.com/paper/2503.11633/) (2025)
- [MoE3D: A Mixture-of-Experts Module for 3D Reconstruction](https://mlsanity.com/paper/2601.05208/) (2026)

## Built on by (most cited first, 2 of 2)

- [PXDepth: Pixel-Space Modeling for Structure Preserving Monocular Depth Estimation](https://mlsanity.com/paper/2608.16984/) (2026)
- [Depth Any Seen: Which Surfaces and How Far?](https://mlsanity.com/paper/2609.32027/) (2026)

## Abstract (arXiv)

Despite advances in depth estimation, flying points remain a persistent failure mode: near object boundaries, depth estimators often predict spurious 3D points in the empty space between foreground and background surfaces. We trace this artifact to a standard modeling choice: assigning each pixel a single depth hypothesis. At boundaries, a pixel can straddle a foreground and a background surface, so its true depth is ambiguous between the two. A model that predicts a single depth cannot keep both possibilities, so training instead pulls the prediction toward an intermediate depth that lies on neither surface. We address this with MDA, a mixture-density representation that lets the model predict multiple depth hypotheses and their associated probabilities for each pixel. Near boundaries, different hypotheses can align with different surfaces, and the decoded depth is selected from one of these hypotheses rather than placed in the empty space between them. Across different backbones, MDA substantially improves boundary reconstruction and largely removes flying-point artifacts even under severe input blur, while adding negligible runtime overhead. The same mixture-density framework naturally extends to transparent objects, where it predicts multiple depth layers at transparent pixels, and to sky regions, where a dedicated component separates the unbounded sky from finite-depth regions, producing flying-point-free skylines. Project Page: https://biansy000.github.io/mda-site/.