# Hierarchical Discrete Distribution Decomposition for Match Density Estimation

arXiv: https://arxiv.org/abs/1812.06264 · HTML: https://mlsanity.com/paper/1812.06264/

## Key facts

- Cited by 93 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Dec 2018.

## Built on by (most cited first, 30 of 93)

- [RAFT-Stereo: Multilevel Recurrent Field Transforms for Stereo Matching](https://mlsanity.com/paper/2109.07547/) (2021)
- [GMFlow: Learning Optical Flow via Global Matching](https://mlsanity.com/paper/2111.13680/) (2021)
- [TAPIR: Tracking Any Point with per-frame Initialization and temporal Refinement](https://mlsanity.com/paper/2306.08637/) (2023)
- [FlowFormer: A Transformer Architecture for Optical Flow](https://mlsanity.com/paper/2203.16194/) (2022)
- [Unifying Flow, Stereo and Depth Estimation](https://mlsanity.com/paper/2211.05783/) (2022)
- [Learning to Estimate Hidden Motions with Global Motion Aggregation](https://mlsanity.com/paper/2104.02409/) (2021)
- [CroCo v2: Improved Cross-view Completion Pre-training for Stereo Matching and Optical Flow](https://mlsanity.com/paper/2211.10408/) (2022)
- [CFNet: Cascade and Fused Cost Volume for Robust Stereo Matching](https://mlsanity.com/paper/2104.04314/) (2021)
- [SMD-Nets: Stereo Mixture Density Networks](https://mlsanity.com/paper/2104.03866/) (2021)
- [FlowFormer++: Masked Cost Volume Autoencoding for Pretraining Optical Flow Estimation](https://mlsanity.com/paper/2303.01237/) (2023)
- [Learning Accurate Dense Correspondences and When to Trust Them](https://mlsanity.com/paper/2101.01710/) (2021)
- [AutoFlow: Learning a Better Training Set for Optical Flow](https://mlsanity.com/paper/2104.14544/) (2021)
- [VideoFlow: Exploiting Temporal Cues for Multi-frame Optical Flow Estimation](https://mlsanity.com/paper/2303.08340/) (2023)
- [PDC-Net+: Enhanced Probabilistic Dense Correspondence Network](https://mlsanity.com/paper/2109.13912/) (2021)
- [LIGA-Stereo: Learning LiDAR Geometry Aware Representations for Stereo-based 3D Detector](https://mlsanity.com/paper/2108.08258/) (2021)
- [Global Matching with Overlapping Attention for Optical Flow Estimation](https://mlsanity.com/paper/2203.11335/) (2022)
- [IGEV++: Iterative Multi-range Geometry Encoding Volumes for Stereo Matching](https://mlsanity.com/paper/2409.00638/) (2024)
- [SKFlow: Learning Optical Flow with Super Kernels](https://mlsanity.com/paper/2205.14623/) (2022)
- [High-Resolution Optical Flow from 1D Attention and Correlation](https://mlsanity.com/paper/2104.13918/) (2021)
- [Learning Optical Flow from a Few Matches](https://mlsanity.com/paper/2104.02166/) (2021)
- [RoMa v2: Harder Better Faster Denser Feature Matching](https://mlsanity.com/paper/2511.15706/) (2025)
- [Correlate-and-Excite: Real-Time Stereo Matching via Guided Cost Volume Excitation](https://mlsanity.com/paper/2108.05773/) (2021)
- [Stereo Anywhere: Robust Zero-Shot Deep Stereo Matching Even Where Either Stereo or Mono Fail](https://mlsanity.com/paper/2412.04472/) (2024)
- [Revisiting Domain Generalized Stereo Matching Networks from a Feature Consistency Perspective](https://mlsanity.com/paper/2203.10887/) (2022)
- [DIP: Deep Inverse Patchmatch for High-Resolution Optical Flow](https://mlsanity.com/paper/2204.00330/) (2022)
- [ITSA: An Information-Theoretic Approach to Automatic Shortcut Avoidance and Domain Generalization in Stereo Matching Networks](https://mlsanity.com/paper/2201.02263/) (2022)
- [Learning Depth Estimation for Transparent and Mirror Surfaces](https://mlsanity.com/paper/2307.15052/) (2023)
- [Learning Optical Flow with Adaptive Graph Reasoning](https://mlsanity.com/paper/2202.03857/) (2022)
- [Disentangling Architecture and Training for Optical Flow](https://mlsanity.com/paper/2203.10712/) (2022)
- [TransFlow: Transformer as Flow Learner](https://mlsanity.com/paper/2304.11523/) (2023)

## Abstract (arXiv)

Explicit representations of the global match distributions of pixel-wise correspondences between pairs of images are desirable for uncertainty estimation and downstream applications. However, the computation of the match density for each pixel may be prohibitively expensive due to the large number of candidates. In this paper, we propose Hierarchical Discrete Distribution Decomposition (HD^3), a framework suitable for learning probabilistic pixel correspondences in both optical flow and stereo matching. We decompose the full match density into multiple scales hierarchically, and estimate the local matching distributions at each scale conditioned on the matching and warping at coarser scales. The local distributions can then be composed together to form the global match density. Despite its simplicity, our probabilistic method achieves state-of-the-art results for both optical flow and stereo matching on established benchmarks. We also find the estimated uncertainty is a good indication of the reliability of the predicted correspondences.