Ellipse-preserving Hermite interpolation and subdivision

We introduce a family of piecewise-exponential functions that have the Hermite interpolation property. Our design is motivated by the search for an effective scheme for the joint interpolation of points and associated tangents on a curve with the ability to perfectly reproduce ellipses. We prove that the proposed Hermite functions form a Riesz basis and that they reproduce prescribed exponential polynomials. We present a method based on Green's functions to unravel their multi-resolution and approximation-theoretic properties. Finally, we derive the corresponding vector and scalar subdivision schemes, which lend themselves to a fast implementation. The proposed vector scheme is interpolatory and level-dependent, but its asymptotic behaviour is the same as the classical cubic Hermite spline algorithm. The same convergence properties---i.e., fourth order of approximation---are hence ensured

Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a quadratic regularization associated to a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two.We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method

A dynamic-shape-prior guided snake model with application in visually tracking dense cell populations

This paper proposes a dynamic-shape-prior guided snake (DSP G-snake) model that is designed to improve the overall stability of the point-based snake model. The dynamic shape prior is first proposed for snakes, that efficiently unifies different types of high-level priors into a new force term. To be specific, a global-topology regularity is first introduced that settles the inherent self-intersection problem with snakes. The problem that a snake’s snaxels tend to unevenly distribute along the contour is also handled, leading to good parameterization. Unlike existing methods that employ learning templates or commonly enforce hard priors, the dynamic-template scheme strongly respects the deformation flexibility of the model, while retaining a decent global topology for the snake. It is verified by experiments that the proposed algorithm can effectively prevent snakes from selfcrossing, or automatically untie an already self-intersected contour. In addition, the proposed model is combined with existing forces and applied to the very challenging task of tracking dense biological cell populations. The DSP G-snake model has enabled an improvement of up to 30% in tracking accuracy with respect to regular model-based approaches. Through experiments on real cellular datasets, with highly dense populations and relatively large displacements, it is confirmed that the proposed approach has enabled superior performance, in comparison to modern active-contour competitors as well as the state-of-the-art cell tracking frameworks