Rate-optimal neural boundary detection from unlabeled noisy images

2026-05-30

AI Summary

This study addresses the problem of recovering unknown object boundaries from noisy, unlabeled images in an unsupervised and nonparametric setting. To this end, the authors propose a boundary detection method that integrates a continuous hinge-type surrogate loss with deep neural networks, embedded within a robust Gibbs posterior framework based on a thresholded misclassification loss. Theoretical analysis demonstrates that the resulting estimator achieves minimax optimal convergence rates—up to logarithmic factors—for piecewise smooth boundaries that may include corners and kinks, while also enjoying Fisher consistency and calibration properties. Extensive experiments confirm the method’s stability and superior performance across varying noise levels and complex boundary shapes, significantly outperforming existing unsupervised boundary detection approaches.

Abstract

We study boundary detection for unlabeled noisy images from a statistical perspective. The aim is to recover an unknown object region from raw intensity observations without pixel-wise annotating labels or a parametric model for the intensity distributions. Motivated by robust Gibbs posterior approaches based on thresholded misclassification losses, we propose a continuous hinge-type surrogate loss for boundary detection. The proposed loss is amenable to gradient-based optimization and can be combined with deep neural networks to represent complex object boundaries. We prove that the proposed loss function is Fisher consistent under a mild separation assumption and obtain a calibration inequality linking excess surrogate risk to the symmetric difference error of the estimated region. Under a piecewise smooth boundary model, we prove that the resulting deep neural network estimator achieves the minimax-optimal boundary recovery rate, up to logarithmic factors. The piecewise smooth formulation accommodates boundaries with corners and kinks, thereby extending beyond globally smooth boundary models. Numerical experiments demonstrate that the proposed method accurately and stably recovers object boundaries across a range of noise levels and shape configurations, and compares favorably with existing unsupervised boundary detection methods.

Problem

• boundary detection
• unlabeled images
• noisy images
• object region recovery
• minimax-optimal rate

Innovation

• boundary detection
• unsupervised learning
• deep neural networks
• Fisher consistency
• minimax optimality