Contour Reconstruction and Matching Using Recursive Smoothing Splines

Karasalo, Maja; Hu, Xiaoming; Martin, Clyde F.

doi:10.1007/978-3-540-73570-0_16

Cited by 12 publications

(11 citation statements)

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“…This paper follows the convention used in [9][10][11]. To clarify, the relation between the smoothing parameters in (E.1) and (E.14) is ε 2 = 1/λ .Ŝ(ε) should satisfy 20) where r N,ε (t) is the optimal solution to Problem E.2.1, given the data set (t i , z i ) and the smoothing parameter ε.…”

Section: E42 Gcv For Periodic Control Theoretic Smoothing Splinesmentioning

confidence: 99%

“…The particular type of periodic control theoretic smoothing spline explored in this paper has been previously presented in [9][10][11]. These publications cover error convergence properties for a recursive formulation of the smoothing spline problem.…”

Section: E1 Introductionmentioning

confidence: 99%

“…Control theoretic smoothing splines have been studied in [1][2][3][4][5][6][7] and it has been shown in [5] that such splines, where the curve is found through minimizing a cost function, act as filters and are better suited for noisy measurements. A thorough treatment of control theoretic smoothing splines is provided in the book [8].The particular type of periodic control theoretic smoothing spline explored in this paper has been previously presented in [9][10][11]. These publications cover error convergence properties for a recursive formulation of the smoothing spline problem.…”

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confidence: 99%

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An Estimated General Cross Validation Function for Periodic Control Theoretic Smoothing Splines

Karasalo

Martin

2010

Perspectives in Mathematical System Theory, Control, and Signal Processing

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In this paper, a method is developed for estimating the optimal smoothing parameter ε for periodic control theoretic smoothing splines. The procedure is based on general cross validation (GCV ) and requires no a priori information about the underlying curve or level of noise in the measurements. The optimal ε is the minimizer of a GCV cost function, which is derived based on a discretization of the L 2 smoothing problem for periodic control theoretic smoothing splines. Keywords: general cross validation, optimal smoothing, influence matrix, periodic control theoretic smoothing splines E.1 IntroductionIn this paper, we consider the problem of estimating representations of objects or contours using a type of continuous closed curves, the periodic control theoretic smoothing splines. The splines are retrieved from noisy measurements of an unknown, underlying contour. Intended applications include mapping, identification and path planning for autonomous agents. The focus of this paper is the issue of the level of smoothing, determined by the magnitude of the so called smoothing parameter ε.It is well known that an interpolating spline generated from noisy measurements yields a poor estimate of the underlying curve, as the spline will pass through every measurement point. An interpolating spline may be regarded as a smoothing spline with ε = ∞, while a periodic smoothing spline with ε → 0 approaches a circle. The theory of regular smoothing splines, and the particular issue of choice of smoothing parameter, is treated in [13][14][15]. Control theoretic smoothing splines have been studied in [1][2][3][4][5][6][7] and it has been shown in [5] that such splines, where the curve is found through minimizing a cost function, act as filters and are better suited for noisy measurements. A thorough treatment of control theoretic smoothing splines is provided in the book [8].The particular type of periodic control theoretic smoothing spline explored in this paper has been previously presented in [9][10][11]. These publications cover error convergence properties for a recursive formulation of the smoothing spline problem. Experimental results indicate that the convergence is fairly robust with respect to the choice of smoothing parameter ε, but a formal 109

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Section: E42 Gcv For Periodic Control Theoretic Smoothing Splinesmentioning

confidence: 99%

Section: E1 Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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An Estimated General Cross Validation Function for Periodic Control Theoretic Smoothing Splines

Karasalo

Martin

2010

Perspectives in Mathematical System Theory, Control, and Signal Processing

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“…The main contributions of this paper are the derivation of an implementable discrete version of the smoothing spline problem both for the closed form and the recursive approach, a contour reconstruction algorithm including both forms, and a series of experiments that validate convergence of the recursive approach and the applicability of the methods in practice. For an extensive theoretical background on the type of smoothing splines studied in this paper we refer to [15], [17], [19], and [20] and the references therein.…”

Section: D2 Related Work and Contributionsmentioning

confidence: 99%

“…By making a least squares fit between two consecutive data sets from the same object, we find and compensate for the odometry drift at each time step. For details of the procedure we refer to [17].…”

Section: D52 Converting Measurements To Data Pointsmentioning

confidence: 99%

Contour reconstruction using recursive smoothing splines - Algorithms and experimental validation

Karasalo

Piccolo

Kragić

et al. 2009

Robotics and Autonomous Systems

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In this paper, a recursive smoothing spline approach for contour reconstruction is studied and evaluated. Periodic smoothing splines are used by a robot to approximate the contour of encountered obstacles in the environment. The splines are generated through minimizing a cost function subject to constraints imposed by a linear control system and accuracy is improved iteratively using a recursive spline algorithm. The filtering effect of the smoothing splines allows for usage of noisy sensor data and the method is robust with respect to odometry drift. The algorithm is extensively evaluated in simulations for various contours and in experiments using a SICK laser scanner mounted on a PowerBot from ActivMedia Robotics.Keywords: mapping, optimal control, recursive smoothing splines, implementation D.1 IntroductionToday robots are expected to operate in at least partially unknown, open-ended environments, detecting and moving around obstacles, building maps of the environment and localizing their position. The area of autonomous mapping has during the past few years matured and there has been a significant amount of work reported using both visual and laser sensors, [8], [26]. However, most methods are still concerned with detecting and maintaining a dense set of discrete features of the environment that are suitable for navigation and localization without retrieving any specific knowledge about the detailed shape of the encountered objects.In this paper, we consider the problem of estimating representations of objects using a type of continuous closed curves, the recursive periodic smoothing splines. The splines are retrieved from noisy measurements of the true contour. Intended applications include mapping, identification and path planning. It is well known that interpolating splines from noisy measurement data will give a poor result, as the resulting curve will go through every measurement point. Data smoothing has been a classical problem in system and control history [1], [13], [18]. The theory of regular smoothing splines is thoroughly treated in [29] and [30]. It has been further shown in [35] that control theoretic smoothing splines, where the curve is found through minimizing a cost function, act as filters and are better suited for noisy measurements. It is also noted in [27] that smoothing 89

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Recursive construction of optimal smoothing splines with constraints

Fujioka

Kano

2010

Proceedings of the 2010 American Control Conference

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Contour Reconstruction and Matching Using Recursive Smoothing Splines

Cited by 12 publications

References 12 publications

An Estimated General Cross Validation Function for Periodic Control Theoretic Smoothing Splines

An Estimated General Cross Validation Function for Periodic Control Theoretic Smoothing Splines

Contour reconstruction using recursive smoothing splines - Algorithms and experimental validation

Recursive construction of optimal smoothing splines with constraints

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