Fast polynomial spline approximation for large scattered data sets via L1 minimization

28/08/2013
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Fast polynomial spline approximation for large scattered data sets via L1 minimization

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(3) is satisfied (3) is not satisfied Fig.10. The two cases of proposition 1. Fast Polynomial Spline Approximation for Large Scattered Data Sets via L1 Minimization Laurent Gajny, Éric Nyiri, Olivier Gibaru Laurent.GAJNY@ensam.eu, Eric.NYIRI@ensam.eu, Olivier.GIBARU@ensam.eu References [Her13] F.Hernoux, R.Béarée, L.Gajny, E.Nyiri, J.Bancalin, O.Gibaru, Leap Motion pour la capture de mouvement 3D par spline L1. Application à la robotique. GTMG 2013, Marseille, 27-28 mars 2013 [Gib1899] J. Willard Gibbs, lettre à l’éditeur, Nature 59 (April 27, 1899) 606. [Lav00] J.E. Lavery, Univariate Cubic Lp splines and shape-preserving multiscale interpolation by univariate cubic L1 splines. CAGD 17 (2000) 319 – 336. [Lav00bis] J.E. Lavery, Shape-preserving, multiscale tting of univariate data by cubic L1 smoothing splines, Comput. Aided Geom. Design, 17 (2000), 715-727. [NGA11] E. Nyiri, O. Gibaru, P. Auquiert, Fast L1 kCk polynomial spline interpolation algorithm with shape preserving properties. CAGD 28(1), 2011, 65 – 74. Behavior on noisy data set Context of the study The proposed method A brief state of the art We propose to develop a fast method of approximation by polynomial spline with no Gibbs phenomenon near to abrupt changes in the shape of the data [Gib1899]. Comparison between L1 and L2 regression line Fig2. Stability of L1 regression line against outliers . Cubic spline interpolation The L1 norm is robust against outliers. Sliding window for L1 interpolation The Lp smoothing splines Aims : Using theoretical results, develop a spline approximation method : • With shape-preserving properties. L1 norm. • With prescribed error. The control. • Fast for real-time use. Sliding window process. The problem and existence theory Fig3. A cubic spline is defined by its knots and associated first derivative values. Let (xi,yi), i=1,2,…,n, be n points in the plane, the Lp regression line problem is to find a line y=a*x+b* solution of : Let qi, ui i=1,2,…,n, be respectively n points in Rd and the associated parameters, The cubic spline interpolation problem is to find a curve  such that: • (ui) = qi for all i=1,2,…,n. •  is a polynomial function of degree at most 3 on each interval [ui,ui+1], The solution set is infinite. The C2 cubic spline interpolation problem leads to a unique solution which satisfies the following minimization property : Fig4. Gibbs phenomenon with cubic spline interpolation. It is a least square method or a L2 method. The L1 cubic spline interpolation Fig5. No Gibbs phenomenon with the L1 cubic interpolation. The L1 cubic spline interpolation introduced in [Lav00] consist in mininizing the following functional : over the set of C1 cubic splines which interpolates the points qi. + No Gibbs phenomenon. - Non-linear problem with non-unique solution. A sliding window process proposed in [Nyi2011] admits a linear complexity with the number of data points. Fig6. the sliding window process. We solve a sequence of local L1 problems and we keep only the derivative value at the middle point of each window. This method enables to obtain an algebraic solution and to manage the non-unicity of solutions. Fig7. Global method (left) and local method (right). The Lp smoothing splines are obtained by minimization of : Fig8. Overshoot with L2 smoothing spline. In this method, the parameter is not easy to choose. We have no control to the initial data points. Algebraic resolution on three points Iteration 1 Iteration 3 Iteration 6 Iteration 1 Iteration 3 Iteration 6 The three-point algorithm The -controlled L1 regression line for non-unicity cases Behavior of the method on a Heaviside data set When (3) is satisfied, the solution set may be infinite. To compute a relevant solution, we extend the window with supplementary more neighbours and solve : We consider the case n = 5 and we give more importance to the three middle point. Then we choose w2 = w3 = w4 > w1 = w5 = 1. Fig.11. The -controlled L1 regression line step in the three-point algorithm. Fig.12. Approximation of a Heaviside data set (left) and zoom on the top part of the jump (right). After applying the algorithm on the initial noisy data set, the spline solution is not smooth. However we can iterate the method. At step k+1, data points are the approximation points computed at step k. Using this process, we emphasize a smoothing phenomenon while keeping jumps. We are currently studying the proposed approach for the problem of approximation of functions. Fig.13. Approximation over noisy data sets. We apply these results to the treatment of sensor data from a low-cost computer vision system, the Leap Motion (See [Her13]). This system is able to capture very accurately fingers motion. When the user’s moves are too fast, we may observe holes in the data. Fig1. Motion capture by the leap motion and industrial robot steering. Fig.9. We look for spline solutions that do not deviate to initial data more than a given tolerance.