Hermite Snakes With Control of Tangents

Abstract

Abstract-We introduce a new model of parametric contours defined in a continuous fashion. Our curve model relies on Hermite spline interpolation and can easily generate curves with sharp discontinuities; it also grants direct access to the tangent at each location. With these two features, the Hermite snake distinguishes itself from classical spline-snake models and allows one to address certain bioimaging problems in a more efficient way. More precisely, the Hermite snake construction allows introducing sharp corners in the snake curve and designing directional energy functionals relying on local orientation information in the input image. Using the formalism of spline theory, the model is shown to meet practical requirements such as invariance to affine transformations and good approximation properties. Finally, the dependence on initial conditions and the robustness to the noise is studied on synthetic data in order to validate our Hermite snake model, and its usefulness is illustrated on real biological images acquired using brightfield, phase-contrast, differentialinterference-contrast, and scanning-electron microscopy.

Support and approximation properties of Hermite splines

“…Similary, (14) with ω = ω 0 and (ω 0 + 2π) implies that A(ω 0 ) = C(ω 0 ) = 0. Injecting this in (11) with ω = ω 0 , we deduce that β 2 (ω 0 ) = α 3 = 0, which contradicts our initial assumption. We now study det(ω) around the origin.…”

General surface energy for spinal cord and aorta segmentation

“…In essence, the proposed energy aligns the snake to the surface detected by the steerable filter and is reminiscent of the 2D approach used in [15]. Energy minimization is efficiently achieved by computing its gradient with respect to {c[k, l]} k,l2Z , the control points defining the snake.…”

Dictionary Learning for Two-Dimensional Kendall Shapes

“…This extended framework applies in particular to spline curves generated by piecewise polynomials. They constitute a common representation for parametric curves [30,31,32,33,34,35]. Our work is hence related to [35], which relies on the isometry between spline curves and configurations of landmarks to apply dictionary learning after pre-alignment.…”

“…Upper row: landmarks with = 20. Lower row: closed Hermite spline curves with = 16[33]. Left column: original pre-shapes (centered and normalized) in blue and in orange.…”

“…The distance ([ ], [ ]) then corresponds to the angle measured from the center of the pre-shape sphere along this geodesic arc. Here, we illustrate this concept with open Hermite-spline curves[33,53].…”