## Approximation with Active B-spline Curves and Surfaces (2002)

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Citations: | 42 - 6 self |

### BibTeX

@INPROCEEDINGS{Pottmann02approximationwith,

author = {Helmut Pottmann and Stefan Leopoldseder and Michael Hofer},

title = {Approximation with Active B-spline Curves and Surfaces},

booktitle = {},

year = {2002},

pages = {8--25}

}

### Years of Citing Articles

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### Abstract

An active contour model for parametric curve and surface approximation is presented. The active curve or surface adapts to the model shape to be approximated in an optimization algorithm. The quasi-Newton optimization procedure in each iteration step minimizes a quadratic function which is built up with help of local quadratic approximants of the squared distance function of the model shape and an internal energy which has a smoothing and regularization effect. The approach completely avoids the parametrization problem. We also show how to use a similar strategy for the solution of variational problems for curves on surfaces. Examples are the geodesic path connecting two points on a surface and interpolating or approximating spline curves on surfaces. Finally we indicate how the latter topic leads to the variational design of smooth motions which interpolate or approximate given positions. 1.

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Citation Context ...metric curves, coined snakes, is used for detecting contours in images. An elegant formulation of curve and surface reconstruction and segmentation problems is the concept of geodesic active contours =-=[8, 9, 49]-=-. There, the curve to be reconstructed, e.g. from a medical image, is found as geodesic in a Riemannian space whose metric is derived from the input (image). Analogously, surface reconstruction is ref... |

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240 |
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Citation Context ...metric curves, coined snakes, is used for detecting contours in images. An elegant formulation of curve and surface reconstruction and segmentation problems is the concept of geodesic active contours =-=[8, 9, 49]-=-. There, the curve to be reconstructed, e.g. from a medical image, is found as geodesic in a Riemannian space whose metric is derived from the input (image). Analogously, surface reconstruction is ref... |

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Citation Context ...the region where their respective local quadratic approximants are positive, we have to solve a constrained minimization problem. Such algorithms are known in optimization as trust regions algorithms =-=[28]-=-. s¢ can use any quadratic smoothing functional, which may change in each iteration step. Thus we can also build Greiner’s method for the minimization of nonlinear fairness functionals [19] into our s... |

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Citation Context ...contour of the signed distance function as the reconstructed implicit surface [1, 4]. The formulation of active contour models via level sets goes back to Osher and Sethian [42]. The level set method =-=[41, 53]-=- has been successfully applied to the solution of a variety of problems, e.g. for segmentation and analysis of medical images [36]. There are also several extensions to surfaces. An application to the... |

129 | Implicit and non-parametric shape reconstruction from unorganized points using a variational level set method
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Citation Context ...for segmentation and analysis of medical images [36]. There are also several extensions to surfaces. An application to the surface fitting problem to scattered data sets has been given by Zhao et al. =-=[61, 62]-=-. For further work on the level set method which is related to surface reconstruction, see [17, 18, 60]. In the present investigation, we develop further a concept for parametric curve and surface fit... |

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78 | Mathematical theory of medial axis transform
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Citation Context ...of classical differential geometry, line and sphere geometry (for a modern presentation, see e.g. [47]). For more recent work on distance transforms and the closely related medial axis transform, see =-=[10, 11, 29, 40, 51, 52, 56]-=-. The distance function is also c�¢¡��� the (viscosity) solution of the so-called eikonal equation. Its numerical computation is not trivial because the eikonal equation is a hyperbolic equation and a... |

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Citation Context .... An application to the surface fitting problem to scattered data sets has been given by Zhao et al. [61, 62]. For further work on the level set method which is related to surface reconstruction, see =-=[17, 18, 60]-=-. In the present investigation, we develop further a concept for parametric curve and surface fitting which has recently been developed by the authors [46]. We assume as input a rather dense set of po... |

55 | Artificial life for computer graphics
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Citation Context ...s. This can then be used for motion planning in connection with 5-axis NC machining.sThe present active contour model and its deformability ¡ form the lower levels of a so-called artifical life model =-=[55]-=-. The evaluation points can be seen as ‘sensors’, and the control structure is the layer above that which allows us to coordinate the action of the sensors. Including phyiscal modeling and on top of t... |

48 | Fast Computation of Weighted Distance Functions and Geodesics on Implicit HyperSurfaces
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Citation Context ...hen depends on the application, e.g. on the texture in an image. Recent algorithms for the fast computation of the distance function from a point p in a manifold, a tringulation or even a point cloud =-=[30, 31, 32, 37]-=-, are used to compute the shortest path between p and any other point q in the manifold; one just follows the gradient flow of the distance function. Clearly, this requires the computation of a bivari... |

42 | Geometry of the Squared Distance Function to Curves and Surfaces
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Citation Context ...B-spline representation. The organization of our paper is as follows. In section 2, we review our recent work on local quadratic approximants of the squared distance function�� of curves and surfaces =-=[45]-=-, since it forms the basis for the new approximation technique. In section 3, the new concept is outlined for curve approximation and applied to degree reduction and offset approximation. In section 4... |

35 | A concept for parametric surface fitting which avoids the parametrization problem
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Citation Context ...lated to surface reconstruction, see [17, 18, 60]. In the present investigation, we develop further a concept for parametric curve and surface fitting which has recently been developed by the authors =-=[46]-=-. We assume as input a rather dense set of points or even a given curve or surface representation. We refer to it as model shape. The first situation arises for example when we are processing data of ... |

33 | Deformable organisms for automatic medical image analysis
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Citation Context ...ing and on top of that behavioral, perceptual and cognitive modeling yields an artificial life model. Those models play an increasingly important role in Computer Graphics [55] and Medical s¢ Imaging =-=[20]-=-. We expect that this is also a promising research direction for the creation of ‘intelligent’ modeling and reverse engineering systems, which include shape understanding abilities. Acknowledgements T... |

32 |
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31 | and topology constraints on parametric active contours. Computer Vision and Image Understanding
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Citation Context ...is not possible to approximate those inner offsets of our model curve which consist of two closed curves. Detection of a topology change and the corresponding adaption of the active curve is possible =-=[13]-=-. 8 Figure 8. Closed planar curve and five of its offset curves approximated by active B-spline curves. 3.3. Approximation of a helix segment by a B-spline curve For the illustration of space curve ap... |

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Citation Context ...hen depends on the application, e.g. on the texture in an image. Recent algorithms for the fast computation of the distance function from a point p in a manifold, a tringulation or even a point cloud =-=[30, 31, 32, 37]-=-, are used to compute the shortest path between p and any other point q in the manifold; one just follows the gradient flow of the distance function. Clearly, this requires the computation of a bivari... |

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Citation Context ...metric curves, coined snakes, is used for detecting contours in images. An elegant formulation of curve and surface reconstruction and segmentation problems is the concept of geodesic active contours =-=[8, 9, 49]-=-. There, the curve to be reconstructed, e.g. from a medical image, is found as geodesic in a Riemannian space whose metric is derived from the input (image). Analogously, surface reconstruction is ref... |

21 |
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Citation Context ...s algebraic surfaces. For algebraic surface fitting, especially the fitting of algebraic tensor-product spline surfaces to scattered data, we refer the reader to the recent paper of Jüttler and Felis =-=[26]-=-. Some approaches to surface reconstruction via implicit surfaces define a signed distance function to the data set and denote the zero contour of the signed distance function as the reconstructed imp... |

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Citation Context ...hen depends on the application, e.g. on the texture in an image. Recent algorithms for the fast computation of the distance function from a point p in a manifold, a tringulation or even a point cloud =-=[30, 31, 32, 37]-=-, are used to compute the shortest path between p and any other point q in the manifold; one just follows the gradient flow of the distance function. Clearly, this requires the computation of a bivari... |

16 |
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Citation Context ...of classical differential geometry, line and sphere geometry (for a modern presentation, see e.g. [47]). For more recent work on distance transforms and the closely related medial axis transform, see =-=[10, 11, 29, 40, 51, 52, 56]-=-. The distance function is also c�¢¡��� the (viscosity) solution of the so-called eikonal equation. Its numerical computation is not trivial because the eikonal equation is a hyperbolic equation and a... |

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Citation Context ...for segmentation and analysis of medical images [36]. There are also several extensions to surfaces. An application to the surface fitting problem to scattered data sets has been given by Zhao et al. =-=[61, 62]-=-. For further work on the level set method which is related to surface reconstruction, see [17, 18, 60]. In the present investigation, we develop further a concept for parametric curve and surface fit... |

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Citation Context ...algorithms [28]. s¢ can use any quadratic smoothing functional, which may change in each iteration step. Thus we can also build Greiner’s method for the minimization of nonlinear fairness functionals =-=[19]-=- into our surface approximation technique. 4.1. Approximation with B-spline surfaces In the following we give an example for the approximation of a given surface by a B-spline surface. In this example... |

11 |
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Citation Context ...tion of the squared distance function with such a spatial data structure would be important. The concept is applicable to approximation with sub¡ division surfaces. One can use an initial shape as in =-=[34]-=-, but other choices and an appropriate handling of details require a lot of future research. An interesting extension concerns the incorporation of ¡ shape constraints such as convexity. For example, ... |

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8 |
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Citation Context ...eeds to be investigated inPSfrag more detail replacements in future research. We need to get more information about the typical distribution of local minima. For a first result in this direction, see =-=[12]-=-. s �� s � s � Figure 3. Active curve flow of the B-spline curve c � towards target curve ¨ . Even if the initial curve intersects the medial axis of ¨ , we obtain good results in most cases. For an e... |

8 |
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Citation Context ... ¡ shape constraints such as convexity. For example, we can use the sufficient linear convexity conditions which have been derived by B. Jüttler for surface fitting with convex tensor product splines =-=[24, 25]-=-. In our framework, we would then have to solve a quadratic programming problem in each iteration step. Special interpolation and approximation problems ¡ which appear in geometric modeling, for examp... |

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Citation Context .... An application to the surface fitting problem to scattered data sets has been given by Zhao et al. [61, 62]. For further work on the level set method which is related to surface reconstruction, see =-=[17, 18, 60]-=-. In the present investigation, we develop further a concept for parametric curve and surface fitting which has recently been developed by the authors [46]. We assume as input a rather dense set of po... |

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Citation Context ...roblem has received a lot of attention in various scientific communities (Computer Graphics, CAGD, Robotics, Computational Geometry). We do not review the literature, but just point to a survey paper =-=[48]-=- and � to a recent paper [21], where a new concept is presented which is based on the following idea. It is simple to solve the problem with an affine motion, i.e., to at first admit affine distortion... |

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