Fast Sphericity and Roundness approximation in 2D and 3D using Local Thickness

1. Problem Statement

Sphericity and roundness are fundamental morphological descriptors used to assess object uniformity in 2D and 3D images, particularly in microscopy and geology.

• Sphericity measures how close an object is to a perfect sphere (dependent on elongation).
• Roundness measures the smoothness or roughness of an object's edges/corners (dependent on corner curvature).

The Challenge:
While standard shape statistics (e.g., area, perimeter) are fast to compute, calculating exact sphericity and roundness is computationally expensive.

• Sphericity requires accurate surface area (3D) or perimeter (2D) calculations, often necessitating mesh generation (e.g., Marching Cubes).
• Roundness is even more complex, requiring the detection of all corners/ridges and the calculation of their curvature radii. In 3D, this involves handling both corners and ridges.
• Scalability: Existing implementations struggle with large datasets (thousands of objects) common in modern microscopy, making them unsuitable for high-throughput analysis.

2. Methodology

The authors propose a novel approach that leverages Local Thickness (LT) to approximate both metrics. Local thickness is defined as the radius of the largest circle (2D) or sphere (3D) that fits inside an object at any given point.

A. Sphericity Approximation

Instead of calculating the exact surface area via meshing, the authors model objects as geometric primitives based on the mean local thickness:

• 3D: Objects are modeled as spheroids (ellipsoids with two equal semi-axes).
  • The two equal semi-axes ($a_s$) are approximated by the mean local thickness.
  • The third semi-axis ($c_s$) is derived from the object's volume ($V$) and $a_s$.
  • The surface area is calculated analytically using the spheroid formula.
• 2D: Objects are modeled as ellipses.
  • The major axis is approximated by the mean local thickness.
  • The minor axis is derived from the object area.
  • The perimeter is calculated using Ramanujan's approximation.

B. Roundness Approximation

The authors avoid complex corner detection and curvature fitting:

• Concept: Local thickness values along an object's contour/surface naturally approximate the curvature of that boundary.
• 2D: The method extracts the pixel-based contour and collects local thickness values at these boundary points. These values serve as proxies for corner curvature radii.
• 3D: The same logic is applied to the object surface using 3D neighbor counting.
• Normalization: The maximum inscribed circle/sphere radius (equivalent to the maximum local thickness) serves as the denominator ($R$) in the roundness formula ($\bar{r}/R$).
• Note: The resulting values do not strictly adhere to the 0–1 range of the original Wadell definition but exhibit high correlation with it.

3. Key Contributions

1. Algorithmic Efficiency: The proposed methods (abbreviated SLT for Sphericity and RLT for Roundness) reduce computational complexity. Since they rely on a single Local Thickness calculation, they can compute both metrics simultaneously for thousands of objects.
2. Elimination of Meshing: The sphericity method bypasses the need for Marching Cubes or mesh generation, which are the primary bottlenecks in 3D analysis.
3. Simplified Roundness: The roundness method replaces difficult corner detection and ridge integration with a direct sampling of local thickness values on the boundary.
4. Open Source Implementation: The authors provide a Python package (fast_rs) available via pip, making these fast approximations accessible to the community.

4. Experimental Results

The methods were benchmarked against standard baselines (Marching Cubes for sphericity, Zheng & Hryciw for roundness) and visual charts (Krumbein charts).

• Correlation with Ground Truth:
  • Sphericity: SLT achieved a correlation of 0.94 with Krumbein chart values, outperforming the Marching Cubes baseline (0.88).
  • Roundness: RLT achieved a correlation of 0.91 with chart values. In group-mean analysis, RLT achieved 0.986, outperforming the exact Zheng method (0.964) and one approximation method, though slightly trailing another approximation (0.992).
• Real-World Data:
  • Tested on 2D cell microscopy images and 3D mozzarella cheese microstructure (15,503 fat objects).
  • Regression analysis showed strong linear correlation between the proposed methods and baselines, with minor residuals.
• Execution Speed:
  • Single Object: The proposed methods are slower than simple Marching Cubes sphericity but significantly faster than exact roundness.
  • Batch Processing (Key Finding): When processing large groups of objects (e.g., 2,000 objects), the proposed methods scale linearly and are vastly superior.
    • Example: Calculating baseline sphericity for 2,000 objects took 38 minutes, whereas the proposed method took only 14 seconds.
    • The speed advantage is attributed to the ability to process the entire volume's local thickness map in parallel.

5. Significance and Conclusion

This paper addresses a critical bottleneck in quantitative image analysis: the trade-off between morphological accuracy and computational speed.

• Scalability: The methods enable the analysis of massive datasets (thousands of 3D objects) that were previously infeasible to process with exact algorithms.
• Practicality: The authors argue that for most image analysis tasks, an "exact" roundness measurement is unnecessary and often unfeasible due to resolution limits and parameter sensitivity. The LT-based approximation provides a robust, parameter-light alternative that captures the essential geometric properties.
• Versatility: The approach works seamlessly for both 2D and 3D data, unifying the calculation of sphericity and roundness under a single computational framework (Local Thickness).

In summary, the paper demonstrates that approximating sphericity and roundness via local thickness offers a fast, scalable, and highly accurate alternative to traditional exact methods, making high-throughput morphological analysis practical for modern microscopy and imaging pipelines.