Scale Dependent Differential Geometry for the Measurement of Center Line and Diameter in 3D Curvilinear Structures

We introduce a 3D tracing method based on differential geometry in Gaussian blurred images. The line point detection part of the tracing method starts with calculation of the line direction from the eigenvectors of the Hessian matrix. The sub-voxel center line position is estimated from a second order Taylor approximation of the 2D intensity profile perpendicular to the line. In curved line structures the method turns out to be biased. We model the bias in center line position using the first order Taylor expansion of the gradient in scale and position. Based on this model we found that the bias in a torus with a generalized line profile was proportional to σ2. This result was applied in a procedure to remove the bias and to measure the radius of curvature in a curved line structure. The line diameter is obtained using the theoretical scale dependencies of the 0-th and 2nd order Gaussian derivatives at the line center. Experiments on synthetic images reveal that the localization of the centerline is mainly affected by line curvature and is well predicted by our theoretical analysis. The diameter measurement is accurate for diameters as low as 4 voxels. Results in images from a confocal microscope show that the tracing method is able to trace in images highly corrupted with noise and clutter. The diameter measurement procedure turns out to be accurate and largely independent of the scale of observation.