Feature-Centered First Order Structure Tensor Scale-Space in 2D and 3D

TL;DR

This work tackles the parameter sensitivity of the first-order structure tensor scale-space for images with features spanning multiple sizes. It reframes scale-space from a feature-centered viewpoint, linking the derivative-filter width to feature size and replacing the conventional smoothing with a ring filter that emphasizes feature centers. It also introduces a scale-map correction that blends isotropic and anisotropic scale estimates using feature shape metrics, enabling more accurate and consistent scale assignment in both 2D and 3D. The approach is validated on artificial 2D data and real 3D CT scans, showing improved anisotropy, orientation accuracy, and robust scale estimation, and is released as open-source Python code. Overall, it provides an out-of-the-box, robust tool for dense structural analysis across a wide range of feature sizes.

Abstract

The structure tensor method is often used for 2D and 3D analysis of imaged structures, but its results are in many cases very dependent on the user's choice of method parameters. We simplify this parameter choice in first order structure tensor scale-space by directly connecting the width of the derivative filter to the size of image features. By introducing a ring-filter step, we substitute the Gaussian integration/smoothing with a method that more accurately shifts the derivative filter response from feature edges to their center. We further demonstrate how extracted structural measures can be used to correct known inaccuracies in the scale map, resulting in a reliable representation of the feature sizes both in 2D and 3D. Compared to the traditional first order structure tensor, or previous structure tensor scale-space approaches, our solution is much more accurate and can serve as an out-of-the-box method for extracting a wide range of structural parameters with minimal user input.

Figures (15)

• Figure 1: Features and their approximate width on a simplified fiber image, defined for edge detection and structural analysis. In edge detection, the feature is usually defined as a gradient of the fiber's edge. In structural analysis, the feature is the fiber itself.
• Figure 2: Gaussian derivative filters aligned with a feature edge. With default normalization, filters with $\sigma\rightarrow 0$ return a higher convolution response (red and blue). Setting $\gamma>1$ yields a higher value for $\sigma > 0$ (orange and cyan).
• Figure 3: Effect of running the structure tensor algorithm on a sample image with a circular (isotropic), a linear (anisotropic), and an elliptical feature, all with the same width (20px), using a single appropriate scale. The raw derivative filter response is placed on the edges of the features. Both strong smoothing and a ring filter shift the response to the object center, but the ring filter provides stronger peaks on object centers and a lower smoothing effect. The isotropic feature gets a lower response in all cases.
• Figure 4: Difference between the slope angles of the outer and inner sides of the ring filter, depending on the $k$ parameter, with visualization of 1D ring filters for two values of $k$. The slopes of the filter are more uniform for $k$ approaching $1$.
• Figure 5: Numerical quantification of the relation between the width of an anisotropic feature (rectangular binary line) and detected feature width (at the center of it), caused by the alignment with the ring filter. The relation is independent of the feature width. The small deviations can most likely be attributed to numerical errors caused by representations of even and odd-sized features.

Theorems & Definitions (1)

• Proposition 1