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Scale-Free Image Keypoints Using Differentiable Persistent Homology
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Proposes MorseDet, a fully convolutional keypoint detector trained with a differentiable persistent-homology loss (discrete Morse theory) that models keypoints as local maxima of a height map, yielding scale-independent repeatability.
Problem
Learning-based keypoint detectors model keypoints as local maxima within fixed-size patches, making keypoint density hard-coded and detection scale-dependent; a principled, scale-independent characterization of local maxima is lacking.
Approach
Model keypoints as the 1-dimensional persistent homology generators (loops born at saddles, closed at maxima) of a CNN height map, and train with a detector loss combining squared persistence and a boundary-similarity penalty across image pairs, differentiated via stratified permutation structure of discrete Morse theory.
Key results
- On HPatches illumination/viewpoint repeatability, MorseDet is best or second-best in nearly all columns (e.g., 44.3 at 250 keypoints under illumination, 40.6 at 250 under viewpoint).
- On scale repeatability averaged over thresholds, MorseDet averages 62.2 vs SIFT 63.6; at 75% resize it reaches 82.2 vs SIFT 75.9.
- Ablation: alpha=0 (no boundary similarity term) collapses to near-zero repeatability (e.g., 1.0 at 250 keypoints), while alpha=10 gives best overall trade-off.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use MorseDet instead of SuperPoint | HPatches illumination, 500 keypoints · HPatches illumination, 500 keypoints | repeatability: 47.7 vs 47.3 | No clear effect |
| use MorseDet instead of DISK | HPatches viewpoint, 4000 keypoints · HPatches viewpoint, 4000 keypoints | repeatability: 47.6 vs 47.2 | Hurt |
| use MorseDet instead of SIFT | HPatches scale repeatability at 75% resized image · HPatches scale repeatability, 75% | repeatability: 75.9 -> 82.2 | Helped |
| use MorseDet instead of SIFT | HPatches scale repeatability at 25% resized image · HPatches scale repeatability, 25% | repeatability: 50.2 vs 41.3 | Hurt |
| add the boundary similarity term (use α=10 instead of α=0) (ablation) | MorseDet · HPatches viewpoint, 250 keypoints | repeatability: 1.0 -> 40.6 | Helped |
| increase α from 10 to 20 (ablation) | MorseDet · HPatches viewpoint, 250 keypoints | repeatability: 40.6 -> 44.0 | Helped |
| increase α from 10 to 20 (ablation) | MorseDet · HPatches viewpoint, 2000 keypoints | repeatability: 46.1 -> 45.1 | Hurt |
| reduce α from 10 to 5 (ablation) | MorseDet · HPatches viewpoint, 250 keypoints | repeatability: 40.6 -> 34.8 | Hurt |
Surprising findings
- Removing the boundary similarity term causes the loss to collapse to a trivial grid-of-ones solution with near-zero repeatability.
Limitations
- Loss implementation runs on CPU and relies on limited discrete Morse theory software, making training slow.
- Only the detection step is addressed; a topological descriptor loss is left for future work.
- Does not beat SIFT on average scale repeatability and is not best across all HPatches settings.
Experimental setup
Datasets: WASF MegaDepth (validation split) HPatches
Benchmarks: HPatches
Metrics: repeatability (mutual nearest neighbor variant) scale repeatability
Baselines: SIFT D2-Net R2D2 SuperPoint DISK ALIKED
Builds on: R2D2 L2Net SuperPoint AdamW
Closest papers by meaning
- Adaptive Topological Feature via Persistent Homology: Filtration Learning for Point Clouds (NeurIPS 2023 · similarity 0.88)
- Towards Scalable Topological Regularizers (ICLR 2025 · similarity 0.85)
- Point-Level Topological Representation Learning on Point Clouds (ICML 2025 · similarity 0.84)
- Topograph: An Efficient Graph-Based Framework for Strictly Topology Preserving Image Segmentation (ICLR 2025 · similarity 0.83)
- CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves (NeurIPS 2026 · similarity 0.83)
- The Flood Complex: Large-Scale Persistent Homology on Millions of Points (NeurIPS 2025 · similarity 0.83)
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