| 1. |
- Berglund, Tomas, et al.
(author)
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Epi-convergence of minimum curvature variation B-splines
- 2003
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Reports (other academic/artistic)abstract
- We study the curvature variation functional, i.e., the integral over the square of arc-length derivative of curvature, along a planar curve. With no other constraints than prescribed position, slope angle, and curvature at the endpoints of the curve, the minimizer of this functional is known as a cubic spiral. It remains a challenge to effectively compute minimizers or approximations to minimizers of this functional subject to additional constraints such as, for example, for the curve to avoid obstacles such as other curves. In this paper, we consider the set of smooth curves that can be written as graphs of three times continuously differentiable functions on an interval, and, in particular, we consider approximations using quartic uniform B- spline functions. We show that if quartic uniform B-spline minimizers of the curvature variation functional converge to a curve, as the number of B-spline basis functions tends to infinity, then this curve is in fact a minimizer of the curvature variation functional. In order to illustrate this result, we present an example of sequences of B-spline minimizers that converge to a cubic spiral.
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| 2. |
- Eriksson, J, et al.
(author)
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Regularization methods for uniformly rank-deficient nonlinear least-squares problems
- 2005
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In: Journal of Optimization Theory and Applications. - : Springer Science and Business Media LLC. - 0022-3239 .- 1573-2878. ; 127:1, s. 1-26
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Journal article (peer-reviewed)abstract
- In solving the nonlinear least-squares problem of minimizing ||f(x)||22, difficulties arise with standard approaches, such as the Levenberg-Marquardt approach, when the Jacobian of f is rank-deficient or very ill-conditioned at the solution. To handle this difficulty, we study a special class of least-squares problems that are uniformly rank-deficient, i.e., the Jacobian of f has the same deficient rank in the neighborhood of a solution. For such problems, the solution is not locally unique. We present two solution tecniques: (i) finding a minimum-norm solution to the basic problem, (ii) using a Tikhonov regularization. Optimality conditions and algorithms are given for both of these strategies. Asymptotical convergence properties of the algorithms are derived and confirmed by numerical experiments. Extensions of the presented ideas make it possible to solve more general nonlinear least-squares problems in which the Jacobian of f at the solution is rank-deficient or ill-conditioned.
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| 3. |
- Gulliksson, Mårten, et al.
(author)
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Implicit surface fitting using directional constraints
- 2001
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In: BIT Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 41:1, s. 308-321
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Journal article (peer-reviewed)abstract
- A commonly used technique for fitting curves and surfaces to measured data is that known as orthogonal distance regression, where the sum of squares of orthogonal distances from the data points to the surface is minimized. An alternative has recently been proposed for curves and surfaces which are parametrically defined, which minimizes the sum of squares in given directions which depend on the measuring process. In addition to taking account of that process, it is claimed that this technique has the advantage of complying with traditional fixed-regressor assumptions, enabling standard inference theory to apply. Here we consider extending this idea to curves and surfaces where the only assumption made is that there is an implicit formulation. Numerical results are given to illustrate the algorithmic performance.
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| 4. |
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| 5. |
- Bergström, Per, et al.
(author)
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Efficient computation of the Gauss-Newton direction when fitting NURBS using ODR
- 2012
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 52:3, s. 571-588
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Journal article (peer-reviewed)abstract
- We consider a subproblem in parameter estimation using the Gauss-Newton algorithm with regularization for NURBS curve fitting. The NURBS curve is fitted to a set of data points in least-squares sense, where the sum of squared orthogonal distances is minimized. Control-points and weights are estimated. The knot-vector and the degree of the NURBS curve are kept constant. In the Gauss-Newton algorithm, a search direction is obtained from a linear overdetermined system with a Jacobian and a residual vector. Because of the properties of our problem, the Jacobian has a particular sparse structure which is suitable for performing a splitting of variables. We are handling the computational problems and report the obtained accuracy using different methods, and the elapsed real computational time. The splitting of variables is a two times faster method than using plain normal equations.
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| 6. |
- Berglund, Tomas, et al.
(author)
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An obstacle-avoiding minimum variation B-spline problem
- 2003
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In: Proceedings. - Los Alamitos, Calif : IEEE Communications Society. - 0769519857 ; , s. 156-161
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Conference paper (peer-reviewed)abstract
- We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the minimum variation B-spline problem, which is a linearly constrained optimization problem over curves, defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.
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| 7. |
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| 8. |
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| 9. |
- Berglund, Tomas, et al.
(author)
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Planning smooth and obstacle-avoiding b-spline paths for autonomous mining vehicles
- 2010
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In: IEEE Transactions on Automation Science and Engineering. - 1545-5955 .- 1558-3783. ; 7:1, s. 167-172
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Journal article (peer-reviewed)abstract
- We study the problem of automatic generation of smooth and obstacle-avoiding planar paths for efficient guidance of autonomous mining vehicles. Fast traversal of a path is of special interest. We consider four-wheel four-gear articulated vehicles and assume that we have an a priori knowledge of the mine wall environment in the form of polygonal chains. Computing quartic uniform B-spline curves, minimizing curvature variation, staying at least at a proposed safety margin distance from the mine walls, we plan high speed paths.
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| 10. |
- Berglund, Tomas, et al.
(author)
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The problem of computing an obstacle-avoiding minimum variation B-spline
- 2003
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Reports (other academic/artistic)abstract
- We study the problem of computing a planar curve restricted to lie between two given polygonal chains such that the integral of the square of arc- length derivative of curvature along the curve is minimized. We introduce the Minimum Variation B-spline problem which is a linearly constrained optimization problem over curves defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.
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