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.

翻译：我们从统计视角研究无标注噪声图像的边界检测问题。目标是在没有像素级标注标签或强度分布参数模型的情况下，从原始强度观测中恢复未知物体区域。受基于阈值化误分类损失的稳健吉布斯后验方法启发，我们提出了一种用于边界检测的连续铰链型替代损失。该损失函数适合基于梯度的优化，并能与深度神经网络结合以表示复杂物体边界。我们证明，在温和的分离假设下，所提损失函数具有Fisher一致性，并得到了一个校准不等式，将超额替代风险与估计区域的对称差误差联系起来。在分段光滑边界模型下，我们证明，由此得到的深度神经网络估计器（在对数因子范围内）能达到极小化最优的边界恢复率。分段光滑公式可处理含有角点和折线的边界，从而超越了全局光滑边界模型。数值实验表明，所提方法能在多种噪声水平和形状配置下准确稳定地恢复物体边界，且性能优于现有无监督边界检测方法。