Explainable Tree-Ensemble Pipelines

• Explainable tree-ensemble pipelines are frameworks that combine decision trees with ensemble methods to deliver high accuracy and transparent predictions.
• They employ techniques such as feature importance analysis and surrogate models to uncover the inner workings of complex interactions.
• Empirical evaluations demonstrate that these pipelines improve decision-making by providing actionable insights in diverse domains like finance and healthcare.

Shape Analysis Using Fractal Dimension: A Curvature-Based Approach

Fractal dimension methods provide quantitative measures of shape complexity and scaling behavior. In shape analysis, these methods capture the self-similarity and roughness of planar curves or boundaries, offering robust descriptors that are both informative and, in some designs, invariant under common geometric transformations. The curvature-based fractal dimension technique is a multiscale approach that integrates the classical notion of fractal dimension with a curvature scale-space framework, yielding multidimensional descriptors that enhance discrimination power and stability in pattern recognition tasks (Backes et al., 2012).

1. Theoretical Framework and Motivation

The fractal dimension $D$ is classically defined for geometric objects by the scaling of a covering number or measure as the measurement scale varies: $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$ where $N(\epsilon)$ is the number of covering elements of linear size $\epsilon$. For planar contours, $N(\epsilon)$ may be replaced by quantities (such as curve length or summed curvature) that exhibit power-law behavior with scale: $L(\epsilon) \propto \epsilon^{-D}$. For empirical, pseudo-fractal contours, $D$ is estimated by performing linear regression on the log–log relationship between such measures and scale. The fractal dimension thus quantifies the degree to which a curve fills space: $D\approx1$ for smooth curves, $D\to2$ for highly convoluted or rough curves.

2. Curvature Scale-Space Formalism

Let $\mathbf{C}(s) = (x(s), y(s))$ denote a parameterized, closed planar contour. The curvature scale-space (CSS) technique applies Gaussian smoothing at each scale $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$0, producing smoothed coordinates: $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$1 where $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$2 and $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$3 denotes convolution. The curvature at scale $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$4 is given by: $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$5 Numerical derivatives are computed using finite differences on the sampled contour.

3. Multiscale Fractal Dimension Descriptors

At each scale $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$6, compute the absolute summed curvature: $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$7 The cumulative sum mimics dilation operations in traditional fractal estimation: $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$8 Assuming a power law $$D = \lim_{\epsilon \to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)}$$9, the global fractal dimension $N(\epsilon)$0 is the slope of the log–log plot: $N(\epsilon)$1 For increased descriptive power, the method evaluates the local slope (multiscale fractal dimension): $N(\epsilon)$2 Sampling $N(\epsilon)$3 over a prescribed $N(\epsilon)$4-range yields a multiscale fractal dimension vector, serving as a high-dimensional shape signature.

4. Descriptor Extraction Pipeline

The full extraction can be formalized as follows:

$N(\epsilon)$9

Typical descriptor length $N(\epsilon)$5 is chosen based on application-specific requirements.

5. Comparative Experimental Performance

The multiscale curvature-based descriptors were validated on a 1,100-class fish silhouette dataset (11,000 shapes with controlled rotations and scales), using descriptor length $N(\epsilon)$6 and linear discriminant analysis with cross-validation. Results demonstrate:

Descriptor Type	Accuracy (%)
Curvature-FD	97.14
Bouligand–Minkowski (no norm)	14.30
Bouligand–Minkowski (diam norm)	80.06

The curvature-based method exhibits robust invariance to rotation and scale, a property resulting from the transformation-invariant nature of summed curvature and the intrinsic filtering effect of Gaussian smoothing. Its multiscale feature vector yields strong discriminability, far surpassing classic dilation-based multiscale fractal descriptors (Backes et al., 2012).

6. Advantages and Limitations

Advantages:

• Intrinsic invariance to rotation and scale due to curvature-summation and global convolution
• Robustness to noise and contour perturbation via progressive Gaussian smoothing
• Efficient offline extraction and suitability for high-throughput applications
• Rich multiscale characterization of local and global complexity, surpassing single-number descriptors

Limitations:

• Requires careful tuning of scale range $N(\epsilon)$7 and the number of scales $N(\epsilon)$8 to match the object's sampling density and application
• Numerical curvature computation may be unstable for poorly sampled or highly noisy contours
• Assumes closed, smooth contours; open or highly irregular curves may necessitate preprocessing

7. Position in Shape Analysis and Pattern Recognition

The curvature-based multiscale fractal dimension method situates itself among shape analysis approaches as a robust tool for quantifying complexity, distinct from both classic global FD estimators and spectral or Minkowski-based multiscale descriptors. By embedding the fractal-dimension estimation into the curvature scale-space, the method unifies the geometric intuition of space-filling with a scale-adaptive sensitivity to shape features. In comparative studies, it proves both highly discriminative and more resilient to normalization uncertainties—a favorable property for large-scale pattern recognition systems and morphometric analysis (Backes et al., 2012).