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Implicit Representations for Constrained Image Segmentation
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Proposes implicit neural representations (input convex networks, diffeomorphism compositions, polar star-shape parametrizations) that provably enforce geometric constraints such as convexity, star-shape, symmetry, periodicity and path-connectedness in image segmentation, used sequentially or jointly in training.
Problem
Enforcing geometric properties (convexity, path-connectedness, star-shape, symmetry) on segmentations is very hard for pixel-wise discrete representations, since the constraints grow quadratically in the number of pixels and binary optimization is often NP-hard, especially under occlusion or scarce
Approach
Represent the segmentation as a coordinate-based implicit function (input convex network, possibly composed with a diffeomorphism or in polar coordinates) so that the constraint holds by architecture, and use it as a fitting target or soft penalty in variational methods and network training.
Key results
- On the convexity dataset, joint training with a convex implicit prior raises CNN RGB+semantic+spatial IoU from 0.818 (predicted seg.) to 0.899 with convex segmentation (joint).
- On FBMS-59 the path-connected prior gains about 1% IoU sequentially and about 2% IoU jointly over the UNet baseline.
- Convex projection of SAM improves clean-image IoU from 0.7275 to 0.7407 and improves IoU under corruptions in Table 2.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Fit a convex implicit representation to the semantic-feature CNN prediction | CNN on the Convexity dataset · Convexity dataset | IoU: 0.726 -> 0.843 | Helped |
| Fit a convex implicit representation to the semantic-feature FCN prediction | FCN on the Convexity dataset · Convexity dataset | IoU: 0.714 -> 0.851 | Helped |
| Jointly train the CNN with a convex implicit representation using semantic features | CNN on the Convexity dataset · Convexity dataset | IoU: 0.818 -> 0.899 | Helped |
| Jointly train the FCN with a convex implicit representation using semantic features | FCN on the Convexity dataset · Convexity dataset | IoU: 0.635 -> 0.894 | Helped |
| Use joint rather than sequential convex fitting with semantic features | FCN on the Convexity dataset · Convexity dataset | IoU: 0.851 -> 0.894 | Helped |
| Project SAM output onto a convex implicit representation | SAM on the clean Convexity dataset · Convexity dataset, clean | IoU: 0.7275 -> 0.7407 | Helped |
| Project SAM output onto a convex implicit representation | SAM under Brightness corruption · Convexity dataset, Brightness corruption | IoU: 0.6456 -> 0.6516 | Helped |
| Fit a path-connected prior sequentially | UNet on FBMS-59 cars3 · FBMS-59, cars3 | IoU: 0.779 -> 0.880 | Helped |
| Apply a path-connected prior to the cats05 segmentation | UNet on FBMS-59 cats05 · FBMS-59, cats05 | IoU: 0.607 -> 0.571 | Hurt |
| Jointly train with the path-connected prior | UNet on FBMS-59 cats05 · FBMS-59, cats05 | IoU: 0.406 -> 0.534 | Helped |
Surprising findings
- Sequential projection onto a convex set does not always help; for some input configurations (e.g. FCN with spatial+semantic features) the convex projection lowers IoU relative to the predicted segmentation.
- Ground truth annotations that are not connected or convex can make the path-connected prior score lower IoU than the unconstrained baseline (e.g. cats05: 57.8% vs 62.3%).
Limitations
- Path-connected priors can be satisfied via infinitely thin connections that vanish when evaluated on a discrete grid, requiring weight decay for stability.
- Constraints are only beneficial when the assumed geometric property holds in the ground truth; the path-connected prior hurts when ground truth is disconnected.
- Experiments are limited to single-object, data-scarce settings and the authors note they did not evaluate on large datasets or SOTA data-driven models.
Experimental setup
Datasets: Convexity scribble dataset (Gorelick et al., 2014) FBMS-59 (Brox & Malik, 2010; Ochs et al., 2013) Hendrycks & Dietterich corruptions
Benchmarks: Convexity scribble dataset FBMS-59
Metrics: IoU pixel accuracy
Baselines: Unconstrained CNN/FCN segmentation Kardoost & Keuper UNet Chen et al. (2021) convex orientation-based lifting SAM
Builds on: Input convex neural networks Dröge & Moeller (2021) scribble-based segmentation Kardoost & Keuper (2021) motion segmentation Real NVP SAM
Closest papers by meaning
- Implicit Geometric Regularization for Learning Shapes (ICML 2020 · similarity 0.85)
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- Implicit Zoo: A Large-Scale Dataset of Neural Implicit Functions for 2D Images and 3D Scenes (NeurIPS 2024 · similarity 0.83)
- Hierarchy-Agnostic Unsupervised Segmentation: Parsing Semantic Image Structure (NeurIPS 2024 · similarity 0.83)
- Few-Shot Unsupervised Implicit Neural Shape Representation Learning with Spatial Adversaries (ICML 2024 · similarity 0.83)
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