## Geometry of the squared distance function to curves and surfaces (2003)

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Venue: | VISUALIZATION AND MATHEMATICS III |

Citations: | 41 - 15 self |

### BibTeX

@INPROCEEDINGS{Pottmann03geometryof,

author = {Helmut Pottmann and Michael Hofer},

title = {Geometry of the squared distance function to curves and surfaces},

booktitle = {VISUALIZATION AND MATHEMATICS III},

year = {2003},

pages = {223--244},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

We investigate the geometry of that function in the plane or 3-space, which associates to each point the square of the shortest distance to a given curve or surface. Particular emphasis is put on second order Taylor approximants and other local quadratic approximants. Their key role in a variety of geometric optimization algorithms is illustrated at hand of registration in Computer Vision and surface approximation.

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Citation Context ... to the approximation of curves and surfaces are active contour models, which are mainly used in Computer Vision and Image Processing. The origin of this technique is the seminal paper by Kass et al. =-=[8]-=-, where a variational formulation of parametric curves, coined snakes, is presented for detecting contours in images. There are various other applications and a variety of extensions of the snake mode... |

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Citation Context ...Hofer parameterization problem. The registration problem we look at is the optimal matching of a point cloud to a surface. The essential difference in our approach to standard techniques, such as ICP =-=[1]-=-, is that we do not have to compute pairs of corresponding points. 2 Graph Surface of the Squared Distance Function to a Planar Curve In Euclidean 3-space R 3 , we consider a planar C 2 curve c(t) wit... |

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Citation Context ...uently appears in problems of geometric computing. Examples include curve and surface approximation in CAD and Geometric Modeling, registration in Computer Vision and positioning problems in Robotics =-=[9]-=-. A variety of contributions deals with the computation of the distance function; in many cases this computation aims towards the singular set of the function, i.e., towards points where the function ... |

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Citation Context ...ms2 Helmut Pottmann and Michael Hofer [4], and this created a large body of literature that deals with its computation and applications in object recognition, registration and mathematical morphology =-=[14, 24, 25]-=-. Also in a discrete setting (pixel plane, voxel space), distance transforms and the skeleton received a lot of attention (see e.g. [18, 24, 25, 28]). The distance function is also the (viscosity) sol... |

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Citation Context ...ametric representation of a curve, one may use an implicit form as zero set (level set) of a bivariate function. The formulation of active contour models via level sets goes back to Osher and Sethian =-=[12]-=-. The level set method [15] has been successfully applied to the solution of a variety of problems, e.g. for segmentation and analysis of medical images [11]. There are also extensions to surfaces. An... |

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Citation Context ... curve, one may use an implicit form as zero set (level set) of a bivariate function. The formulation of active contour models via level sets goes back to Osher and Sethian [12]. The level set method =-=[15]-=- has been successfully applied to the solution of a variety of problems, e.g. for segmentation and analysis of medical images [11]. There are also extensions to surfaces. An application to the surface... |

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Citation Context ... level sets goes back to Osher and Sethian [12]. The level set method [15] has been successfully applied to the solution of a variety of problems, e.g. for segmentation and analysis of medical images =-=[11]-=-. There are also extensions to surfaces. An application to the surface fitting problem has been given by Zhao et al. [16]. In the registration problems outlined above, the moving object (point cloud) ... |

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Citation Context ...riational formulation of parametric curves, coined snakes, is presented for detecting contours in images. There are various other applications and a variety of extensions of the snake model (see e.g. =-=[2]-=-). Instead of a parametric representation of a curve, one may use an implicit form as zero set (level set) of a bivariate function. The formulation of active contour models via level sets goes back to... |

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Citation Context ...errors and to classify the deviations. A wellknown standard algorithm to solve such a registration problem is the iterative closest point (ICP) algorithm which has been introduced by Chen and Medioni =-=[5]-=- and Besl and McKay [1]. In the first step of each iteration, for each point of the data point cloud the closest point in the model shape is computed. As result of this first step one obtains a point ... |

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Citation Context ... since the value of the objective function is decreasing both in steps 1 and 2. An excellent summary with new results on the acceleration of the ICP algorithm has been given by Rusinkiewicz and Levoy =-=[14]-=-, who also suggest that iterative corresponding point is a better expansion for the abbreviation ICP than the original iterative closest point. Actually, we want to apply that motion to the data point... |

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Geometric Partial Differential Equations and Image Analysis
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Citation Context ...er belong to the medial axis and are of particular interest. The computation of viscosity solutions with the level set method of Osher and Sethian [19] proved to be a very powerful approach (see e.g. =-=[23, 26, 27]-=-). Distance functions also play a role in other curve and surface evolutions [23] and in the visualization of large sets of unorganized data sets [29]. The vector distance function of an object, which... |

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Citation Context ...on, which also appears in Kimmel et al. [15]. The importance of the skeleton, medial axis and medial axis transform as a shape descriptor has been realized by Blums2 Helmut Pottmann and Michael Hofer =-=[4]-=-, and this created a large body of literature that deals with its computation and applications in object recognition, registration and mathematical morphology [14, 24, 25]. Also in a discrete setting ... |

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Citation Context ...of a variety of problems, e.g. for segmentation and analysis of medical images [11]. There are also extensions to surfaces. An application to the surface fitting problem has been given by Zhao et al. =-=[16]-=-. In the registration problems outlined above, the moving object (point cloud) undergoes a rigid body motion. The motion is linearized in each iteration step and guided by the ‘flow’ imposed by the sq... |

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Computational Line Geometry
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Citation Context ...nce function whose sign is given by the orientation of the curve normal e2, the profile is simply the line (0, u, u) and the generated moulding surface is a developable surface of constant slope (see =-=[13]-=-). For a nonnegative distance, the profile is (0, u, |u|) and the moulding surface is generated from the developable surface described above by reflecting the part below π at π. In case that f is nonn... |

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Citation Context ...s discussed above to the development of geometric optimization algorithms which are based on local quadratic approximants of the function to be minimized, such as quasi-Newton and SQP-type algorithms =-=[3]-=-. We will now outline this idea at hand of two important examples: registration and surface approximation. 6.1 Registration of a Point Cloud to a CAD Model Suppose that we are given a large number of ... |

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Citation Context ...ersections of the surface (folding), smoothing terms must be incorporated into the function to be minimized. A similar shrink wrapping algorithm for polygonal surfaces was described by Kobbelt et. al =-=[9]-=-. In our example the resulting B-spline patch after the first and fifth iteration step is given in Fig. 14, left and right, respectively. Note that the boundary curves of the B-spline patch were not i... |

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Citation Context ...ms2 Helmut Pottmann and Michael Hofer [4], and this created a large body of literature that deals with its computation and applications in object recognition, registration and mathematical morphology =-=[14, 24, 25]-=-. Also in a discrete setting (pixel plane, voxel space), distance transforms and the skeleton received a lot of attention (see e.g. [18, 24, 25, 28]). The distance function is also the (viscosity) sol... |

63 |
Shape Interrogation for Computer Aided Design and Manufacturing
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Citation Context ...10] for the evolution of objects of arbitrary dimension and codimension. The level sets of the distance functions are the offsets which are of particular importance in Computer Aided Design (see e.g. =-=[20, 21]-=- and the literature therein) and also appear in mathematical morphology [24]. Results on the distance function and its application to shape interrogation for CAD/CAM may also be found in the very rece... |

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Citation Context ...]). Here, the parameterization problem is a fundamental one which largely effects the result (see e.g. [10] and the references therein). Therefore, parameter correction procedures have been suggested =-=[7]-=-.s18 Helmut Pottmann and Michael Hofer A different approach to the approximation of curves and surfaces are active contour models, which are mainly used in Computer Vision and Image Processing. The or... |

25 | Skeletonization via distance maps and level sets
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Citation Context ...aphic preimage in the classical language. More recent work on the MAT comes from H. I. Choi [8], especially the decomposition result for the efficient computation, which also appears in Kimmel et al. =-=[15]-=-. The importance of the skeleton, medial axis and medial axis transform as a shape descriptor has been realized by Blums2 Helmut Pottmann and Michael Hofer [4], and this created a large body of litera... |

20 |
Mantegazza C. Curvature and distance function from a manifold
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Citation Context ... quadratic approximants at points different from the medial axis. A contribution dealing with local properties of the squared distance function of a manifold has been given by Ambrosio and Mantegazza =-=[1]-=-. Among other interesting results, the authors describe the relation between third order directional derivatives of the squared distance function (taken at points of the manifold) and the curvature in... |

19 |
New algorithm for medial axis transform of plane domain. Graphical Models and Image Processing 59
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Citation Context .... [21]). This approach is also naturally related to the medial axis transform (MAT) which appears as cyclographic preimage in the classical language. More recent work on the MAT comes from H. I. Choi =-=[8]-=-, especially the decomposition result for the efficient computation, which also appears in Kimmel et al. [15]. The importance of the skeleton, medial axis and medial axis transform as a shape descript... |

16 |
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Citation Context ...rmulation with a regularization term that expresses the fairness of the final result (see e.g. [6]). Here, the parameterization problem is a fundamental one which largely effects the result (see e.g. =-=[10]-=- and the references therein). Therefore, parameter correction procedures have been suggested [7].s18 Helmut Pottmann and Michael Hofer A different approach to the approximation of curves and surfaces ... |

12 |
Fast surface reconstruction and deformation using the level set method Proceedings of IEEE workshop on variational and level set methods in computer vision
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Citation Context ...of a variety of problems, e.g. for segmentation and analysis of medical images [17]. There are also extensions to surfaces. An application to the surface fitting problem has been given by Zhao et al. =-=[30, 31]-=-. In the registration problems outlined above, the moving object (point cloud) undergoes a rigid body motion. The motion is linearized in each iteration step and guided by the ‘flow’ imposed by the sq... |

11 |
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Citation Context ...t recognition, registration and mathematical morphology [14, 24, 25]. Also in a discrete setting (pixel plane, voxel space), distance transforms and the skeleton received a lot of attention (see e.g. =-=[18, 24, 25, 28]-=-). The distance function is also the (viscosity) solution of the so-called eikonal equation. Its numerical computation is not trivial because it is a hyperbolic equation and an initially smooth front ... |

9 | A hamiltonian approach to the eikonal equation
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(Show Context)
Citation Context ...er belong to the medial axis and are of particular interest. The computation of viscosity solutions with the level set method of Osher and Sethian [19] proved to be a very powerful approach (see e.g. =-=[23, 26, 27]-=-). Distance functions also play a role in other curve and surface evolutions [23] and in the visualization of large sets of unorganized data sets [29]. The vector distance function of an object, which... |

8 | Dynamic shapes of arbitrary dimension: the vector distance functions
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Citation Context ...tion of large sets of unorganized data sets [29]. The vector distance function of an object, which connects each point in space to its closest point on the object, has been used by Faugeras and Gomes =-=[10]-=- for the evolution of objects of arbitrary dimension and codimension. The level sets of the distance functions are the offsets which are of particular importance in Computer Aided Design (see e.g. [20... |

6 |
A study of optimal-criteria identification based on the small-displacement screw model
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Citation Context ...approximants results in a quadratic approximant of the function F . The linearization of the motion is equivalent to using instantaneous kinematics, an idea that already appeared in a similar form in =-=[4]-=-. We estimate the displacements of points via their velocity vectors. The velocity vector field of a rigid body motion is known to have the form v(x) = ¯c + c × x. (13) The distance of the point xi + ... |

6 |
Techniques for fair and shape preserving surface fitting with tensor-product B-splines
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Citation Context ...d of points by a NURBS surface is a widely investigated problem. The main approach uses a least square formulation with a regularization term that expresses the fairness of the final result (see e.g. =-=[11]-=-). Here, the parameterization problem is a fundamental one which largely effects the result (see e.g. [16] and the references therein). Therefore, parameter correction procedures have been suggested [... |

2 |
Geometrie-Rekonstruktion aus Meßpunktwolken mit glatten B-Spline-Flächen
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Citation Context ...d of points by a NURBS surface is a widely investigated problem. The main approach uses a least square formulation with a regularization term that expresses the fairness of the final result (see e.g. =-=[6]-=-). Here, the parameterization problem is a fundamental one which largely effects the result (see e.g. [10] and the references therein). Therefore, parameter correction procedures have been suggested [... |

1 |
2002): Analysis and visualization of large sets of unorganized points using the distance function
- Zhao
(Show Context)
Citation Context ... to be a very powerful approach (see e.g. [23, 26, 27]). Distance functions also play a role in other curve and surface evolutions [23] and in the visualization of large sets of unorganized data sets =-=[29]-=-. The vector distance function of an object, which connects each point in space to its closest point on the object, has been used by Faugeras and Gomes [10] for the evolution of objects of arbitrary d... |