Multiple testing, uncertainty and realistic pictures

We study statistical detection of grayscale objects in noisy images. The object of interest is of unknown shape and has an unknown intensity, that can be varying over the object and can be negative. No boundary shape constraints are imposed on the object, only a weak bulk condition for the object’s interior is required. We propose an algorithm that can be used to detect grayscale objects of unknown shapes in the presence of nonparametric noise of unknown level. Our algorithm is based on a nonparametric multiple testing procedure. We establish the limit of applicability of our method via an explicit, closed-form, non-asymptotic and nonparametric consistency bound. This bound is valid for a wide class of nonparametric noise distributions. We achieve this by proving an uncertainty principle for percolation on finite lattices.

💡 Research Summary

The paper addresses the fundamental problem of detecting an object in a noisy image when virtually no prior information about the object’s shape, size, or intensity is available. Unlike most existing approaches that assume smooth boundaries, binary (black‑and‑white) objects, or Gaussian noise, the authors develop a fully non‑parametric framework capable of handling grayscale objects with spatially varying (and possibly negative) intensities, embedded in noise of unknown distribution and unknown variance.

The methodological core is a percolation‑based statistical test. The image is discretized onto a planar graph (typically a triangular lattice). For a given threshold τ, each pixel’s observed value Y(s) is compared to τ; pixels exceeding τ are declared “occupied” (value 1) and the rest “vacant” (value 0). This yields a random site‑percolation configuration with occupation probability p that depends on the unknown noise distribution and the chosen τ. Connected occupied sites form clusters; the size T of the largest cluster serves as the test statistic. The null hypothesis H0 (no object) is rejected whenever T exceeds a data‑driven critical value φ(N), where N denotes the lattice size.

Because the true signal intensity is unknown, a single τ is insufficient. The authors therefore introduce a collection of thresholds {τ1,…,τK}. For each τk a maximum‑cluster test is performed, producing a p‑value. These p‑values are then combined using a multiple‑testing correction (e.g., Bonferroni, Holm, or FDR). This “multiple‑threshold” scheme guarantees control of the overall Type I error while preserving power across a wide range of possible signal strengths.

A major theoretical contribution is the derivation of an “uncertainty principle” for finite lattices. By relating the percolation critical probability pc to the noise distribution’s cumulative function F, the authors obtain an explicit condition: if the noise’s quantiles satisfy F(m±c)=1−pc for some interval