Convex Hull-Based Semantic Masks

• Convex hull-based semantic masks are representations that regularize semantically meaningful regions using convexity constraints to enhance geometric consistency.
• They employ optimization techniques like ADMM and quadratic programming to robustly extract features from noisy data, benefiting segmentation and trajectory planning.
• Applications include improved object segmentation, precise SLAM, and safe autonomous vehicle planning by ensuring trajectories lie within dynamically defined convex corridors.

Convex hull-based semantic masks are a class of representations and constraints applied in image processing, computer vision, robotics, and 3D scene understanding, where semantically meaningful regions or objects are regularized by their convex hulls for better geometric consistency, tractable optimization, or shape prior enforcement. These masks leverage convexity both as a geometric guarantee (e.g., trajectory containment, object silhouette regularization) and as a means of extracting reliable features from noisy or imprecise semantic data. Applications include robust object segmentation, accurate mapping and localization in SLAM, autonomous vehicle motion planning within safe spatio-temporal corridors, and guided manipulation of point cloud data for data augmentation or scene editing.

1. Theoretical Foundations of Convex Hull-Based Semantic Masks

Convex hull-based semantic masks originate from classical convexity theory: the convex hull $Conv(S)$ of a set $S$ in $\mathbb{R}^n$ is the smallest convex superset containing $S$. In semantic masking, convexity serves to regularize object boundaries or enforce shape priors. A rigorous binary characterization of convexity states that the indicator function $u(x) = 1_{D}(x)$ for region $D$ is convex if and only if, for all $r > 0$,

$(b_r * u)(x)[1 - u(x)] \leq \frac{1}{2}[1 - u(x)]$

where $b_r$ is a non-negative, compactly supported kernel and $*$ denotes convolution. This allows convexity to be imposed as a quadratic inequality constraint on binary (semantic) masks, enabling both exact and approximate convex hull computation even in the presence of noise (Luo et al., 2022).

For parametric trajectories or regions (e.g., Bézier curves in spatio-temporal corridors), the convex hull property ensures that the curve remains inside the convex hull determined by its control points. A sufficient condition for this property is obtained by sampling upper and lower boundary functions (concave $f_{ub}$ and convex $f_{lb}$) at equal time intervals and enforcing control points to lie in $[f_{lb}(t_i), f_{ub}(t_i)]$. This guarantees that the entire curve is contained within the convex corridor, with a provable approximation gap scaling as $O(1/n^2)$, where $n$ is the curve degree (Zhang et al., 2021).

2. Algorithmic Construction and Optimization Approaches

There are several algorithmic pipelines for building convex hull-based semantic masks, including:

• Binary Convex Hull Optimization:
  • Indicator functions are initialized from foreground-background scribbles or noisy point samples.
  • Convexity constraints are linearized at each iteration.
  • The constrained problem is solved using a proximal alternating direction method of multipliers (ADMM), with projection onto the simplex and explicit slack variables for constraint satisfaction.
  • Per-instance runtime is on the order of seconds for typical 512×512 grids (Luo et al., 2022).
• Convex Hull Extraction in Image Space:
  • Object detections yield pixel contours (via instance segmentation frameworks such as YOLOv8 + SAM).
  • The convex hull is computed using algorithms like Quick-Hull, optionally pruned (Douglas–Peucker simplification) to avoid overfitting.
  • The convex polygon is rasterized into a binary hull mask (Yu et al., 3 Mar 2025).
• Spatio-Temporal Semantic Mask Generation (Trajectory Planning):
  • Predict dynamic obstacles’ reachable sets in a given planning domain.
  • Compute upper/lower envelopes by convex/concave hulls.
  • Enforce Bézier control points within the sample envelopes (using linear QP constraints) to ensure planned trajectories remain safely inside the mask (Zhang et al., 2021).
• Point Cloud Range Image Masking:
  • Semantically label a LiDAR point cloud, project to a 2D range image, and extract a pixel-object mask.
  • Compute the convex hull of object pixels and rasterize as a semantic mask guiding further operations (e.g., conditional diffusion models) (Uppur et al., 21 Nov 2025).

3. Applications Across Perception and Robotics

Convex hull-based semantic masks have been demonstrated in a variety of high-impact domains:

• Image Segmentation:
  • Convexity priors incorporated as quadratic constraints in variational segmentation frameworks improve geometric regularity, noise robustness, and accuracy, especially for objects with natural convexity (Luo et al., 2022).
  • Multi-class extensions are accomplished by enforcing simplex constraints on the collection of indicator functions.
• Visual SLAM (Simultaneous Localization and Mapping):
  • Convex hulls of semantic segmentation boundaries are used to regularize the reconstruction and pose refinement of 3D quadrics, yielding tighter and more consistent multi-view fits compared to bounding-box and ellipse-based priors (Yu et al., 3 Mar 2025).
  • The resulting algebraic plane-tangency constraints improve both mapping accuracy and localization, as measured by ATE and SIoU.
• Autonomous Vehicle Motion Planning:
  • Spatio-temporal convex hull masks define the safe reachable region for parametric trajectories. By leveraging a sufficient condition for the convex hull property, generated motions remain entirely within dynamic safety corridors, reducing harsh braking events and increasing comfort (Zhang et al., 2021).
• Semantic Scene Editing and Point Cloud Synthesis:
  • In diffusion-based generation for LiDAR data, convex hull-based semantic masks provide geometric priors for region-guided inpainting and object removal/insertion, ensuring that the resulting synthetic samples retain realistic geometric alignment with real-world objects (Uppur et al., 21 Nov 2025).

4. Comparative Evaluation and Metrics

Quantitative studies demonstrate the impact of convex hull-based semantic masks on task performance:

Area	Baseline	Convex-Hull-Based Mask	Improvement
Monocular ATE (cm) [SLAM]	2.58 (ORB-SLAM2)	1.78 (Yu et al., 3 Mar 2025)	↓ 31%
SIoU (Shape IoU) [SLAM]	0.662 (OA-SLAM)	0.719	+0.057
Segmentation Shape-Error (%)	7.6 (level-set)	5.7 (Luo et al., 2022)	↓ ~25%
Point Cloud JSD	1.880e-2 (no hull mask)	1.089e-2 (Uppur et al., 21 Nov 2025)	↓ ~42%
Vehicle Deceleration [AV]	–5.23 m/s² (trapezoidal)	–4.46 m/s² (Zhang et al., 2021)	Smoother braking

Ablative analyses confirm that the removal of convex hull masking degrades metrics across visual, geometric, and perceptual loss functions. In point cloud editing, region-focused losses applied to convex hull mask regions accelerate convergence and improve object-level generative fidelity.

5. Integration into End-to-End Systems

Convex hull-based semantic masks are compatible with existing pipelines and frameworks:

• Post-Processing for Neural Segmentation:
  • Given a soft probability map, thresholding and convex hull enforcement refine the mask, improving regularity without requiring direct integration with the neural model (Luo et al., 2022).
• SLAM Frontend/Backend:
  • Plane tangent constraints induced by hulls act as priors in both pose initialization (frontend) and bundle adjustment (backend) without disrupting standard factor graph or optimization architectures (Yu et al., 3 Mar 2025).
• Safety Masking for Autonomous Planning:
  • Linear bounds for QPs can be directly derived from convex hull envelopes, making hull-based masking a drop-in for common trajectory optimization frameworks (e.g., using OSQP) (Zhang et al., 2021).
• Latent Diffusion Models for Scene Editing:
  • Convex hull masks, when rasterized to appropriate spatial scales, serve as channelwise condition vectors in U-Net or VAE encoder pipelines, enforcing explicit shape priors during generative training (Uppur et al., 21 Nov 2025).

6. Advantages, Limitations, and Prospective Extensions

Advantages of convex hull-based semantic masking include:

• Geometric Regularity: Enforces natural convexity, mitigating label noise or non-convex artifacts.
• Optimization Tractability: Convexity and hull constraints are efficiently implementable via ADMM or QP, with convergence guarantees.
• Robustness to Outliers: Especially in noisy or weakly-supervised contexts, convex hull masking avoids overfitting to spurious boundaries or isolated points.
• Multi-Class and Multi-Component Handling: Methods support both multiclass segmentation and disconnected components under flexible constraint schemes (Luo et al., 2022).

Limitations are largely dictated by constraint tightness:

• Multiple closely located objects may be forced under a single convex hull unless handled by instance-level or indicator-matrix extensions.
• In tasks requiring non-convex segmentation (e.g., articulated shapes), pure convex hull constraints may undesirably merge separate regions or overly simplify boundaries.

Prospective extensions include learning convexity radius parameters in a data-driven fashion, integrating hull-based losses with deep network outputs, and employing learned or adaptive kernels for “soft convexity” priors (Luo et al., 2022).

7. Representative Implementations and Empirical Results

Implementations span both open-source and research codebases:

• Segmentation and Convex Hull Computation: Proximal-ADMM based solvers available for shape completion and multi-class convex segmentation.
• Visual SLAM: End-to-end integration with YOLOv8, SAM, and factor graph optimization via G2O/Ceres. The simplification and convex hull computation is performed by Quick-Hull and Douglas–Peucker, with results reported on standard TUM and KITTI datasets (Yu et al., 3 Mar 2025).
• LiDAR Scene Editing: Range-EDIT pipeline achieves region-wise MAE gains (0.283→0.313), improved JSD and MMD metrics, with code and ablation analysis provided (Uppur et al., 21 Nov 2025).

These systems demonstrate that convex hull-based semantic masks deliver improvements in both geometric fidelity and downstream task accuracy, supporting their growing use in computer vision, robotics, and autonomous systems.