Pixel-wise Confidence Region for a 2D Change-Boundary

Boundary detection methods typically provide an estimated boundary without quantifying the uncertainty associated with its spatial location. This study develops a pixel-wise uncertainty quantification framework for boundaries in noisy images. Local discrepancy scan scores are computed around the estimated boundary and directly used as calibration data to construct a score-based confidence region for the unknown true boundary. Under increasing-resolution asymptotics and suitable conditions on boundary localization and score regularity, we show that the empirical score distribution along the estimated boundary consistently approximates the corresponding distribution along the true boundary, yielding asymptotic average pixel-wise coverage close to the nominal level. Simulation studies under piecewise-constant Gaussian image models demonstrate the effects of image resolution and signal strength on coverage and localization accuracy. The proposed method extends boundary segmentation from point estimation to statistically interpretable spatial uncertainty assessment.