Boundary Distance Regression (BDR)

• Boundary Distance Regression is a family of techniques that replaces binary boundary classification with continuous regression of distance-to-boundary fields for enhanced learning and inference.
• It integrates Euclidean and signed distance computations with deep learning, local polynomial regression, and geometric analysis to improve segmentation and causal estimation.
• Applications span medical imaging, OCTA retinal analysis, point cloud geometry, and temporal action localization, achieving measurable gains such as higher Dice coefficients and sharper boundary detections.

Boundary Distance Regression (BDR) refers to a broad family of techniques that leverage the notion of Euclidean or signed distance to domain boundaries in order to structure, estimate, or regularize learning or inference tasks involving spatial, temporal, or covariate boundaries. Common to all BDR approaches is the replacement of binary "hard" boundary classification with the regression of a continuous distance-to-boundary field, which can then be used for downstream inference, segmentation, or causal estimation along or across boundaries. BDR methodologies have found wide application in image analysis, causal inference (regression discontinuity), point cloud geometry, and temporal action localization.

1. Core Mathematical Principles of Boundary Distance Regression

Fundamentally, BDR operates by defining, for each data sample or grid point $x$, a boundary distance $d(x)$: $d(x) = \min_{b \in \partial \Omega} \|x - b\|_2,$ where $\partial \Omega$ denotes the boundary of the region of interest. This definition appears in spatial contexts such as cell or organ segmentation, and can be generalized to signed distances in segmentation,

$$d(x) = \begin{cases} -\min_{b\in \partial\Omega} \|x - b\|_2, & x \text{ interior} \ 0, & x\in \partial\Omega \ \min_{b\in \partial\Omega} \|x - b\|_2, & x \text{ exterior} \end{cases}$$

as in signed distance map constructions.

These distances serve as regression targets or features, converting a classification task into a regression problem and enabling the exploitation of rich geometric regularity. In causal inference, the boundary is a manifold $\mathcal{B}$ in $\mathbb{R}^d$ (e.g., a treatment assignment boundary) and the signed distance is used to transfer the multivariate problem into a univariate coordinate normal to $\mathcal{B}$ (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025).

2. Algorithms and Architectures

BDR implementations vary by domain, but several core algorithmic motifs recur:

A. Spatial Image Analysis and Segmentation

• Boundary Distance Map Computation: Use manual or automatic boundary tracing, oversampling boundaries (e.g., fitting periodic splines and evaluating at 1000+ points), and Euclidean distance transforms (EDT) for accurate pixel-wise computation of $d(x)$ (Choudhury et al., 2010).
• Regression Networks: Deep neural architectures predict per-pixel distance maps, commonly employing a feature encoder (e.g., VGG16 or ResNeSt50), followed by upsampling decoders (using transposed convolutions) to restore full image resolution (Yin et al., 2019, Yin et al., 2018, Lin et al., 2021).
• Normalization: An exponential potential mapping, $d_i = \exp(-D_i)$, is frequently applied to bound the range and concentrate sensitivity near the boundary (Yin et al., 2019, Yin et al., 2018).

B. Temporal Action Localization

• Signed Distance Regression: Each temporal sample $t$ is assigned the signed distance to the nearest ground-truth boundary, enabling the model to regress piecewise-linear fields with sharp zero-crossings at action boundaries (Shihab et al., 6 Nov 2025).
• Architecture: BDR head is typically a 3-layer MLP appended to standard backbones, with minimal impact on existing architectures.

C. Causal Inference Along Boundaries

• Distance-to-Boundary Variable: For each observation, compute signed distance $D_i(x)$ to the nearest point on the assignment boundary (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025).
• Local Polynomial Regression: Treatment effect at $x$ is estimated by fitting local polynomials in $D_i(x)$ on both sides (treatment and control) using kernel weights. Proper choice of kernel, bandwidth, and polynomial order is essential for convergence and bias control.

D. Geometric Point Cloud Analysis

• Local Moment Estimators: For each point, local averages of position vectors in a ball of fixed radius yield empirical normal vectors and boundary distances (Calder et al., 2021).
• Error Analysis: Nonasymptotic bounds $O_p(r)$ for normal estimation and $O_p(r^2)$ for distance estimation, with $r \sim (\log n / n)^{1/(d+2)}$.

3. Statistical Properties and Theoretical Guarantees

Bias, Variance, and Rate Analysis

• Nonparametric Regression: Boundary regression under one-sided error models admits uniform rate $O(h^\beta) + O_P((n h^d)^{-1/\alpha})$ for boundary H\"older smoothness $\beta$, ambient dimension $d$, and error tail index $\alpha$ (Selk et al., 2020).
• Causal Boundary Estimation: For distance-based local polynomial estimators, leading bias is typically $O(h^{p+1})$ for smooth boundaries but exhibits an irreducible $O(h)$ bias at kinks or non-differentiable points, regardless of polynomial order $p$ (Cattaneo et al., 8 May 2025).
• Variance: Variance declines as $(n h^d)^{-1}$, where $h$ is the bandwidth and $n$ the sample size; the asymptotic normality of pointwise estimators is established under mild conditions.

Confidence Bands and Inference

• Uniform Bands: Gaussian multiplier bootstrap construction provides joint confidence bands along complex boundaries (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025).
• Pointwise vs Uniform Inference: Bias control is more demanding in uniform settings; practical recommendations include undersmoothing and robust t-statistic construction especially near boundary irregularities.

4. Application Domains

Domain	BDR Target	Key Implementation Elements
Biological imaging	Protein/organ boundaries	EDT, spline smoothing, kernel regression (Choudhury et al., 2010, Yin et al., 2019, Yin et al., 2018)
OCTA retinal analysis	FAZ segmentation	Signed distance maps, joint learning (boundary, segmentation, distance) (Lin et al., 2021)
Causal inference (RD)	Discontinuous treatment edge	Local polynomial on signed distances, confidence banding, R package rd2d (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025)
Point cloud geometry	Physical boundary estimation	Local moments, Taylor approximations, density normalization (Calder et al., 2021)
Action localization	Temporal event boundary	Signed temporal distance, MLP distance head, L1+smoothness loss (Shihab et al., 6 Nov 2025)

5. Empirical Performance and Implementation Considerations

A. Empirical Gains

• Medical image segmentation with BDR networks achieves Dice coefficients as high as $0.94 \pm 0.03$ for kidney ultrasound, outperforming pixel-masking-only models by 6–16 pp and reducing mean boundary error by more than 1 pixel (Yin et al., 2019, Yin et al., 2018).
• In OCTA images, adding an SDM regression branch improves Dice by $\sim0.7\%$ and reduces average surface distance by $\sim0.13$ px over strong baselines (Lin et al., 2021).
• Temporal action localization with BDR yields 1.8–3.1% [email protected] improvements and 43% sharper boundary peak gradients relative to classification heads, confirmed on multiple benchmarks (Shihab et al., 6 Nov 2025).

B. Computational and Practical Aspects

• Image BDR: Efficient on modern GPUs; forward-backward pipelines amenable to end-to-end training; typical inference time $\sim$0.18s/image (Yin et al., 2019).
• Causal BDR: Core code implemented in the R package rd2d, with automated bandwidth selection and support for both pointwise and uniform inference (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025).
• Point cloud BDR: Scales with point cloud density and dimension; local neighborhood search and kernel computations critical (Calder et al., 2021).

C. Limitations and Domain-Specific Caveats

• Boundary curvature and reach: For accurate geometric regression, smoothness ($C^2$, $C^3$) and positive reach of $\partial \Omega$ are required for consistency and to minimize bias (Calder et al., 2021).
• BDR in regression discontinuity: Methods incur irreducible $O(h)$ bias at non-differentiable boundaries; in such settings, bivariate local polynomial estimators are theoretically preferable (Cattaneo et al., 8 May 2025).
• High dimensions: Boundary estimation rates degrade substantially as $d$ increases, necessitating larger samples and careful tuning (Selk et al., 2020).

6. Advancements, Variants, and Theoretical Optimality

• Adaptive Bandwidth and Smoothing: Variable bandwidths decreasing as sampling density increases (e.g., bandwidth $\propto$ [density of $d$]${}^{-1/4}$ in smoothing splines) mitigate variance inflation near boundaries (Choudhury et al., 2010).
• Piecewise Linear Models: Adaptive knot search enables BDR to capture heterogeneity in intra-, peri-, and extra-zonal boundary gradients (e.g., protein marker drop-off inside vs. outside nuclei) (Choudhury et al., 2010).
• Discriminant Analysis with Regularization: Ridge-penalized within-group scatter matrices enable robust class separation even when the dimensionality of boundary profiles is large compared to sample size (Choudhury et al., 2010).
• Information-Theoretic Optimality: For temporal BDR, Cramér–Rao lower bounds demonstrate a variance reduction proportional to $(\kappa/\Delta t)^2$ versus classic classification, where $\kappa$ is the plateau width of softmax outputs and $\Delta t$ the resolution of time bins (Shihab et al., 6 Nov 2025).

7. Software and Reproducibility

Several BDR approaches provide mature, open-source codebases:

• rd2d (R): Bandwidth selectors, local polynomial fits, and bootstrap-based confidence bands for distance-based regression discontinuity designs (Cattaneo et al., 30 Oct 2025, Cattaneo et al., 8 May 2025).
• Medical Imaging BDR: PyTorch and TensorFlow implementations with pretrained weights for VGG-16/DeepLab or ResNeSt50 backbones, with demonstrated reproducibility in end-to-end segmentation pipelines (Yin et al., 2019, Lin et al., 2021).
• Action Localization BDR: Drop-in BDR loss retrofitting for major temporal localization models (BMN, ActionFormer, TriDet), minimal code additions, and maintained performance and scaling documented in ablation experiments (Shihab et al., 6 Nov 2025).
• Point Cloud BDR: Efficient ball query implementations and least-squares normal estimation with range search scaling to hundreds of thousands of points (Calder et al., 2021).

In summary, Boundary Distance Regression is a principled set of techniques leveraging local or signed distances to domain boundaries as a statistical, geometric, or deep-learning signal. It offers robust, theoretically justified improvements in a range of tasks, provided care is taken with regularity, parameter tuning, and recognition of setting-specific limitations.