Support and approximation properties of Hermite splines

Fageot, Julien; Aziznejad, Shayan; Unser, Michaël; Uhlmann, Virginie

doi:10.1016/j.cam.2019.112503

Cited by 13 publications

(8 citation statements)

References 37 publications

(65 reference statements)

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“…As shown in Figure 1, the spline curve is determined by the Hermite interpolation function based on the coordinates of each node P1, P2 etc. The interpolation curve not only strictly passes through each node, but also meet the requirement that the derivatives at each node are continuous [24,25]. The upper part of a unit model is the arch generated by 12 adaptive points as in Figure 2(a), within which the first 6 adaptive points shape the front section of the unit model, the latter 6 adaptive points shape the rear section.…”

Section: Parameterized Bim Tunnel Modelmentioning

confidence: 99%

Tunnel Point Cloud and BIM Model Integration for Cross-section Monitoring

Zhang¹,

Hao²,

Dong³

et al. 2020

Preprint

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This paper introduces a method for tunnel point cloud and BIM model integration and cross-section monitoring, providing information to analyse tunnel cross-sections and surrounding rock deformation, and support for tunnel maintenance and reconstruction. Three types of data are processed for the integration: laser scanning point cloud, BIM tunnel model and terrain model from oblique photogrammetry. An adaptive BIM modelling scheme is proposed for tunnels with alien structures. Precise spatial registration of the data sets is conducted by applying singular value decomposition (SVD) algorithm to calculate transformation parameters from the point cloud model to the BIM model. Since the tunnel central line has high-order derivability, a cross-section calculation method based on tangent vector is proposed to obtain the cross-sectional profile of tunnels at any mileage. The proposed method has been verified by applying it to a tunnel reconstruction project. The experiment results show that the tunnel point cloud and the BIM model were highly coincident after the integration. The developed program can effectively get the cross-section of the tunnel at any mileage, and correctly express the spatial relationship between the BIM tunnel, the point cloud of tunnel and the external mountainous terrain.

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Section: Parameterized Bim Tunnel Modelmentioning

confidence: 99%

Tunnel Point Cloud and BIM Model Integration for Cross-section Monitoring

Zhang¹,

Hao²,

Dong³

et al. 2020

Preprint

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“…In this work, B-splines and monomial splines are evaluated together. Both spline types have different advantages and disadvantages, but comparable approximation properties [17,20]. On the one hand, univariate B-splines of degree 3 are used in the work to realize the required continuity.…”

Section: Relevant Types Of Splinementioning

confidence: 99%

Performance Evaluation of Offline Motion Preparation Approaches on the Example of a Non-Linear Kinematics

2020

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In motion control, the generation of motion profiles for non-linear kinematics is usually computationally complex. In order to minimize the workload on the machine’s control system, the approach pursued is outsourcing complex calculation tasks to the offline area. In this offline motion preparation, predefined criteria have to be taken into account to guarantee process stability on the real machine. During the motion preparation, a high performance is desired, characterized by less data generated and at the same time little computing effort. The evaluation will use the example of a motion specification, which is characterized by a large amount of data compared to conventional motion specifications. Thus, the demands on performance become even higher. This paper examines the performance of different motion preparation approaches known from literature. On the one hand, selected spline-based algorithms are discussed and compared. A recursive algorithm based on monomial splines is recommended for use in the example. On the other hand, a very simple approach based on the linearization of the non-linear workspace of the mechanism is presented and applied on the algorithms. With this, the performance increased significantly again.

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“…where f = f (1) is the derivative of f . The excellent approximation capabilities and minimal-support property of the Hermite splines [30] give a strong incentive to investigate more general multi-spline spaces. The bicubic Hermite splines are the backbone of many computer-graphics applications and closely linked to Bézier curves [31,32,33,34].…”

Section: Multi-splinesmentioning

confidence: 99%

“…The space S(φ) is said to have an approximation power of order M if any sufficiently smooth and decaying function can be approached by an element of S h (φ) with an error decaying as O(h M ). The so called "Strang-Fix conditions" give sufficient conditions to have a space with an approximation power of order M [30,38,39]. In particular, for compactly supported and integrable generating functions, it is sufficient to have the space S(φ) reproduce polynomials of degree up to (M − 1).…”

Section: Reproducing Polynomialsmentioning

confidence: 99%

Shortest-support Multi-Spline Bases for Generalized Sampling

Goujon¹,

Aziznejad²,

Naderi³

et al. 2020

Preprint

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Generalized sampling consists in the recovery of a function f , from the samples of the responses of a collection of linear shift-invariant systems to the input f . The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M . While this property allows for an approximation power of order (M + 1), it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M +1). Following this result, we introduce the notion of shortest basis of degree M , which is motivated by our desire to minimize the computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power.

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Support and approximation properties of Hermite splines

Cited by 13 publications

References 37 publications

Tunnel Point Cloud and BIM Model Integration for Cross-section Monitoring

Tunnel Point Cloud and BIM Model Integration for Cross-section Monitoring

Performance Evaluation of Offline Motion Preparation Approaches on the Example of a Non-Linear Kinematics

Shortest-support Multi-Spline Bases for Generalized Sampling

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