Advanced Distribution Theory for Significance in Scale Space

TL;DR

This work tackles the challenge of inferring significance in multiscale smoothing by extending SiZer to two dimensions with a new distribution theory based on extreme value theory. It builds a field-wise framework that links kernel-smoothed surface derivatives to Gaussian random fields and derives asymptotic thresholds for both slope and curvature tests, enabling valid inference across image scales. Through simulations and Gamma camera analyses, the method demonstrates robust Type I error control while preserving power to detect true features, outperforming classical SSS in avoiding spurious scale-space signals. The approach provides practical, scalable, and reliable surface-feature detection for imaging and other high-dimensional smoothing contexts, with accompanying data and code availability.

Abstract

Smoothing methods find signals in noisy data. A challenge for Statistical inference is the choice of smoothing parameter. SiZer addressed this challenge in one-dimension by detecting significant slopes across multiple scales, but was not a completely valid testing procedure. This was addressed by the development of an advanced distribution theory that ensures fully valid inference in the 1-D setting by applying extreme value theory. A two-dimensional extension of SiZer, known as Significance in Scale Space (SSS), was developed for image data, enabling the detection of both slopes and curvatures across multiple spatial scales. However, fully valid inference for 2-D SSS has remained unavailable, largely due to the more complex dependence structure of random fields. In this paper, we use a completely different probability methodology which gives an advanced distribution theory for SSS, establishing a valid hypothesis testing procedure for both slope and curvature detection. When applied to pure noise images (no true underlying signal), the proposed method controls the Type I error, whereas the original SSS identifies spurious features across scales. When signal is present, the proposed method maintains a high level of statistical power, successfully identifying important true slopes and curvatures in real data such as gamma camera images.

Figures (4)

• Figure 1: Comparison of curvature identification results for a pure noise example from the proposed advanced SSS method (top row) and the original SSS method (bottom row) across four bandwidths $h = 2, 4, 8, 16$. The raw image is shown in the leftmost panel. Each colored dot represents a statistically significant curvature classification: purple for ridge and orange for valley structures. Note the original SSS indicates spurious structure for three of the four scales.
• Figure 2: Comparison of slope and curvature identification results from the proposed Advanced SSS method (rows 1 and 3) and the original SSS method (rows 2 and 4) across four bandwidths $h = 2, 4, 8, 16$. The Peaks and Valleys dataset with random noise added is shown in the leftmost panel. In the top rows, the green streamlines follow statistically significant gradients. In the bottom rows, each colored dot represents a statistically significant curvature classification: blue for peaks, yellow for holes, purple and orange for ridge and valley structures, and red for saddle points. These demonstrate comparable statistical power.
• Figure 3: Comparison of slope and curvature identification results from the proposed Advanced SSS method (rows 1 and 3) and the original SSS method (rows 2 and 4) across four bandwidths $h = 2, 4, 8, 16$. The Gamma camera image is shown in the leftmost panel. In the top rows, the green streamlines follow statistically significant gradients. In the bottom rows, each colored dot represents a statistically significant curvature classification: blue for peaks, yellow for holes, purple and orange for ridge and valley structures, and red for saddle points. These demonstrate comparable statistical power.
• Figure 4: Curve of function $f_1$ and $f_2$.

Theorems & Definitions (9)

• Theorem 1: french2013asymptotic
• Theorem 2
• Corollary 2.1
• proof
• Theorem 3
• Corollary 3.1
• proof
• proof
• proof