Second Order Kinematic Surface Fitting in Anatomical Structures

TL;DR

The paper tackles the limitation of first order kinematic surface fitting in capturing curved rotational symmetries of anatomical structures. It introduces a second order velocity field $\mathbf{v}(\mathbf{p})=(\mathbf{t}\times\mathbf{p})\times\mathbf{p}+\mathbf{r}\times\mathbf{p}+\gamma\mathbf{p}+\mathbf{c}$, with a 10D parameter vector $\mathbf{m}=[\mathbf{t},\mathbf{r},\mathbf{c},\gamma]$, and demonstrates a robust fitting framework based on a Student-$t$-driven EM scheme to estimate $\mathbf{m}$ from oriented points. The method enables extraction of convergence points, core lines, and center lines, and supports morphological classification by deriving intrinsic parameters such as $\|\mathrm{proj}_{\mathbf{r}}\mathbf{t}\|$ that relate to shape tilt (e.g., cochlear coiling). Validation on synthetic examples and real anatomical data (aorta, cochlea, semicircular canal, heart chambers) shows that the second order model captures curved symmetries that first order fields miss and that robustness to outliers improves with the proposed approach. Overall, the work advances medical image analysis by providing a mathematically grounded, robust, and globally descriptive framework for symmetry detection and morphometric classification of complex anatomical structures.

Abstract

Symmetry detection and morphological classification of anatomical structures play pivotal roles in medical image analysis. The application of kinematic surface fitting, a method for characterizing shapes through parametric stationary velocity fields, has shown promising results in computer vision and computer-aided design. However, existing research has predominantly focused on first order rotational velocity fields, which may not adequately capture the intricate curved and twisted nature of anatomical structures. To address this limitation, we propose an innovative approach utilizing a second order velocity field for kinematic surface fitting. This advancement accommodates higher rotational shape complexity and improves the accuracy of symmetry detection in anatomical structures. We introduce a robust fitting technique and validate its performance through testing on synthetic shapes and real anatomical structures. Our method not only enables the detection of curved rotational symmetries (core lines) but also facilitates morphological classification by deriving intrinsic shape parameters related to curvature and torsion. We illustrate the usefulness of our technique by categorizing the shape of human cochleae in terms of the intrinsic velocity field parameters. The results showcase the potential of our method as a valuable tool for medical image analysis, contributing to the assessment of complex anatomical shapes.

Figures (8)

• Figure 1: Kinematic surface fitting of a logarithmic spiral. Displayed are the results for the first order rotational (yellow curves) and second order rotational (red curves) velocity fields. The dashed curves indicate the detected core lines, and convergence points $\mathbf{p}_0$ are shown as circles. Continuous curves represent example center lines $\mathcal{C}$ traced from a central seed point $\mathbf{p}_\mathrm{s}$ (black diamonds). a) Ideal case: both velocity fields converge to the same solution. b) Non-robust fitting with added cylindrical structure as outlier (arrow). c) Robust fitting of shape with outlier (arrow). d) Visualization of confidence weights $z_i$ estimated for robust fitting of the second order velocity field.
• Figure 2: Kinematic surface fitting of a bent helix. Displayed are the results for the first order rotational (yellow curves) and second order rotational (red curves) velocity fields. The dashed curves indicate the detected core lines. Continuous curves represent example center lines $\mathcal{C}$ traced from a central seed point $\mathbf{p}_\mathrm{s}$ (black diamonds). a) Ideal case: Second order fitting can capture the bending, in contrast to the first order approach. b) Non-robust fitting with added cylindrical structure as outlier (arrow): the second order fit does not converge properly. c) Robust fitting with outliers (arrow): the second order fitting method converges to the correct solution. d) Visualization of confidence weights $z_i$ estimated for robust fitting of the second order velocity field.
• Figure 3: Kinematic surface fitting of a human aortic arch onlineA. Displayed are the results for the first order rotational (yellow curves) and second order rotational (red curves) velocity fields. The dashed curves indicate the detected core lines, and convergence points $\mathbf{p}_0$ are shown as circles. Continuous curves represent example center lines $\mathcal{C}$ traced from a central seed point $\mathbf{p}_\mathrm{s}$ (black diamonds).
• Figure 4: Kinematic surface fitting of a human left ventricle bai2015bi. Displayed are the results for the first order rotational (yellow curves) and second order rotational (red curves) velocity fields. The dashed curves indicate the detected core lines, and convergence points $\mathbf{p}_0$ are shown as circles. Continuous curves represent example center lines $\mathcal{C}$ traced from a central seed point $\mathbf{p}_\mathrm{s}$ (black diamonds).
• Figure 5: Kinematic surface fitting of a human rib onlineR. Displayed are the results for the first order rotational (yellow curves) and second order rotational (red curves) velocity fields. The dashed curves indicate the detected core lines, and convergence points $\mathbf{p}_0$ are shown as circles. Continuous curves represent example center lines $\mathcal{C}$ traced from a central seed point $\mathbf{p}_\mathrm{s}$ (black diamonds)