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Distortion-Free Intelligent Sampling of Sparse Surfaces Via Locally Refined T-Spline Metamodelling

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Abstract

Automatic design of the number of sample points and sample locations when measuring surfaces with different geometries is of critical importance to enable autonomous manufacturing. Uniform sampling has been widely used for simple geometry measurement, e.g. planes and spheres. However, there is a lack of appropriate sampling techniques that can be applied to complex freeform surfaces, especially those with sparse topographical features, e.g. cutting edges and other high-curvature features. In this paper, a distortion-free intelligent sampling and reconstruction method with improved efficiency for sparse surfaces is proposed. In this method, a locally-refined T-spline approximation is firstly applied which maps a surface to a simplified T-spline space; then a shift-invariant space sampling method and corresponding reconstruction are applied for the surface measurement. This sampling strategy provides a cost-effective sampling design and guarantees the surface reconstruction without information loss in a T-spline space. Theoretical demonstrations and case studies show that this sampling strategy can provide up to an order of magnitude improvement in accuracy or efficiency over state-of-the-art methods, for the measurement of sparse surfaces, from macro- to nano-scales.

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Abbreviations

ISO:

International standardization organization

CAD:

Computer-aided design

2D, 3D:

Two and three dimensions

NURBS:

Non-uniform rational B-spline

T-mesh:

Genralized B-spline mesh allowing T-junctions

T-patch:

Unit square patches constructing a T-mesh

T-spline:

Spline defined on a T-mesh

RMSE:

Root-mean-square error

SIS:

Shift-invariant space

\( {\mathbb{R}}^{d} \) :

d-Dimensional real number domain

\( C^{k} \) :

Continuous regarding k-th derivative

E :

Half of spline order

v, \( v_{k} \) :

Knot sequence and kth knot of a SIS

\( {\varPhi }, \varphi \left( \cdot \right) \) :

Generating matrix and function of a SIS

\( V_{\varphi } \) :

SIS constructed by generating function \( \varphi ( \cdot ) \)

c, \( c_{k} \) :

Coefficient vector and kth element of a SIS signal

\( l^{p} \) :

p-Norm Lebesgue space

s.j. :

Subject to

\( \left\| \cdot \right\|^{2} \) :

L2-norm or Euclidean norm

X, \( x_{j} \), J :

Sample set, jth sample point and sample index set

\( f(x_{j} ) \) :

Sample value of function \( f( \cdot ) \) at position \( x_{j} \)

δ :

Maximum sample gap

D :

Density of sample set X

\( \cup , \subseteq \), \( \in \) :

Union, subset and element of a union or unions

#:

Element count of a set

inf:

Infimum

(s, t):

Parametric coordinate on a T-mesh

\( \varvec{q}, \varvec{q}_{j} , Q \) :

Spline surface function, jth extracted spline sample point and spline sample point set

\( \varvec{p}_{i} \), P :

ith control point and control point set of a T-spline

\( B_{i} ( \cdot ) \) :

Blending function associated to \( \varvec{p}_{i} \)

\( w_{i} \) :

Weight associated to \( \varvec{p}_{i} \)

\( \varvec{s}_{i} \), \( \varvec{t}_{i} \) :

s- and t-knot vectors associated to \( B_{i} ( \cdot ) \)

\( \otimes \) :

Tensor-product of two vectors

D k :

kth iteratively fitted T-mesh

P k :

kth iteratively fitted T-spline control point set

\( I^{k} \) :

Index set of control point set Pk

\( \hat{q}_{j}^{k} , \hat{Q}^{k} \) :

jth reconstructed sample point and sample set in kth iterative T-spline fitting

\( {\sigma } \) :

Reconstruction error threshold in T-spline fitting

\( \bar{J}^{k} \), R k :

Error-violated sample point index subset and corresponding T-patch subset in kth iteration

\( R_{A}^{k} \) :

Subset of Rk with sufficient sample points

\( J_{ - } \) :

Subset of sample points used in local fittings

\( I_{A}^{k} ,I_{B}^{k} \) :

Changed and kept unchanged control point index subsets in kth iterative fitting

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant nos. 51705178, 51835005).

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Correspondence to Jiang Xu.

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Wang, J., Leach, R., Chen, R. et al. Distortion-Free Intelligent Sampling of Sparse Surfaces Via Locally Refined T-Spline Metamodelling. Int. J. of Precis. Eng. and Manuf.-Green Tech. 8, 1471–1486 (2021). https://doi.org/10.1007/s40684-020-00248-w

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