To cite this version

This paper addresses smoothing spline estimation of complex functions subject to shape and/or dynamics constraints. Such estimation problems receive growing interest in engineering and statistics, particularly newly emerging areas such as systems biology. In this paper, we formulate the estimation problem as an optimal control problem subject to convex control constraints. By exploring techniques from convex and variational analysis, the existence and uniqueness of optimal solutions is established and explicit optimality conditions are obtained. It is shown that the optimality conditions are given in term of a two-point boundary value problem for a complementarity system. To compute an optimal solution, we formulate the optimality conditions as a B-differentiable equation. A nonsmooth Newton's method is exploited to solve this equation; global convergence of this method is established.

Journal of Computational and Applied Mathematics, 2009

A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation. Optimality conditions are derived for both the fixed and free-knot problems. These conditions are expressed in terms of jumps in certain (discrete) derivatives. A computational algorithm is given that applies to constraints whose boundaries are either piecewise linear or spherical. The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.

Journal of Approximation Theory, 1976

Journal of Approximation Theory, 2004

In this paper, the theory of abstract splines is applied to the variational refinement of (periodic) curves that meet data to within convex sets in R d. The analysis is relevant to each level of refinement (the limit curves are not considered here). The curves are characterized by an application of a separation theorem for multiple convex sets, and represented as the solution of an equation involving the dual of certain maps on an inner product space. Namely, T * Tf +˜ * w (f) = 0. Existence and uniqueness are established under certain conditions. The problem here is a generalization of that studied in (Kersey, Near-interpolatory subdivided curves, author's home page, 2003) to include arbitrary quadratic minimizing functionals, placed in the setting of abstract spline theory. The theory is specialized to the discretized thin beam and interval tension problems.

Journal of Algorithms & Computational Technology, 2019

Total variation smoothing methods have been proven to be very efficient at discriminating between structures (edges and textures) and noise in images. Recently, it was shown that such methods do not create new discontinuities and preserve the modulus of continuity of functions. In this paper, we propose a Galerkin-Ritz method to solve the Rudin-Osher-Fatemi image denoising model where smooth bivariate spline functions on triangulations are used as approximating spaces. Using the extension property of functions of bounded variation on Lipschitz domains, we construct a minimizing sequence of continuous bivariate spline functions of arbitrary degree, d, for the TV-L 2 energy functional and prove the convergence of the finite element solutions to the solution of the Rudin, Osher, and Fatemi model. Moreover, an iterative algorithm for computing spline minimizers is developed and the convergence of the algorithm is proved.

Mathematical and Computer Modelling, 1994

An extension of the one-parameter class of functionals considered in the Smoothing Spline Estimate framework is proposed to better deal with scaling problems or anisotropies of data to be approximated. In particular, a matrix, L, of regularizing parameters is introduced in place of the typical parameter X. By means of an appropriate coordinates transformation, the resulting variational problem can be rephrased in terms of the "traditional" Smoothing Spline Estimate method, and the Generalized Cross Validation technique can consequently be applied to find optimal values for L. Following this approach, numerical experiments have been performed on synthetic 2D data, and the results compared with those obtained with Smoothing Spline Estimate and Hyper Basis Function methods. *The research work of B.C. is done within MAIA, the leading AI project being presently developed at 1-T. The authors feel deeply indebted to L. Tubaro, for many crucial suggestions and the constant support provided. C. Furlanello and F. Girosi carefully read the manuscript and made several useful comments.

Journal of Mathematical Analysis and Applications, 2008

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C 0 approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.

Computer Aided Geometric Design, 2005

Given an m-dimensional surface Φ in R n , we characterize parametric curves in Φ, which interpolate or approximate a sequence of given points p i ∈ Φ and minimize a given energy functional. As energy functionals we study familiar functionals from spline theory, which are linear combinations of L 2 norms of certain derivatives. The characterization of the solution curves is similar to the well-known unrestricted case. The counterparts to cubic splines on a given surface, defined as interpolating curves minimizing the L 2 norm of the second derivative, are C 2 ; their segments possess fourth derivative vectors, which are orthogonal to Φ; at an end point, the second derivative is orthogonal to Φ. Analogously, we characterize counterparts to splines in tension, quintic C 4 splines and smoothing splines. On very special surfaces, some spline segments can be determined explicitly. In general, the computation has to be based on numerical optimization.

Journal of Computational and Applied Mathematics, 2008

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of D m-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.

Computer Graphics Forum, 2011

We present a linear system for modeling 3D surfaces from curves. Our system offers better performance, stability, and precision in control than previous non-linear systems. By exploring the direct relationship between a standard higher-order Laplacian editing framework and Hermite spline curves, we introduce a new form of Cauchy constraint that makes our system easy to both implement and control. We introduce novel workflows that simplify the construction of 3D models from sketches. We show how to convert existing 3D meshes into our curve-based representation for subsequent editing and modeling, allowing our technique to be applied to a wide range of existing 3D content.