EP2904587A2

EP2904587A2 - Method for processing data defining an element in a space e of dimensions d, and associated computer program

Info

Publication number: EP2904587A2
Application number: EP13774392.8A
Authority: EP
Inventors: Bruno Levy
Original assignee: Institut National de Recherche en Informatique et en Automatique INRIA
Current assignee: Institut National de Recherche en Informatique et en Automatique INRIA
Priority date: 2012-10-05
Filing date: 2013-10-03
Publication date: 2015-08-12
Also published as: FR2996669A1; US10460006B2; CN104937640B; WO2014053606A2; US20150254209A1; CN104937640A

Description

Data processing method defining an element in a space E of dimensions d, associated computer program

The present invention relates to methods of data processing defining an element in a space E of dimensions d, comprising the following steps:

al defines a set of n points in the space E;

b / calculating the intersection between said element and Voronoi cells each determined from an associated point respectively among said n defined points.

Such techniques are used, inter alia, to generate a mesh of an element such as a surface or a volume.

Surface meshes are used in many technical applications, for example in the numerical simulation of flow for aeronautical design or for oil exploration, in the mechanics of structures for the design of frames, bridges and other structures. art, in the mechanics of deformations for the simulation of automobile impact tests etc.

The mesh consists of cutting the element in question into a set of basic cells. Each base cell and the set of basic cells as a whole must satisfy predefined validity conditions, for example relating to the geometric shape of the basic cells (square, triangle, tetrahedron, hexahedron, cube, etc.) or on angle values at vertices (respect of minimum / maximum angle values).

These predefined conditions depend to a large extent on the application to which the mesh is intended.

There are many methods for generating a mesh of a domain.

Certain methods based on the optimization of an "objective" function depending on the coordinates at the vertices of the cells are known. Such methods are sometimes referred to as "variational methods".

Processes use the barycentric tiling of Voronoi (in English "Centroidal

Voronoi Tessellation ", or" CVT ") and the optimal Delaunay triangulation (Optimal Delaunay Triangulation" or "ODT").

They make it possible to efficiently and robustly generate isotropic meshes. These methods are especially particularly effective when it comes to generating an isotropic mesh from triangular base cells (mesh of a surface) or tetrahedral (mesh of a volume).

Moreover, some applications may require the generation of an anisotropic mesh, that is to say which has elements that differ from each other in size. and / or in orientation, in predefined areas of the element. An anisotropy matrix is defined for each base cell specifying these orientation and size constraints.

The document FR 2962582 describes an anisotropic mesh generation technique.

Techniques have emerged according to which the anisotropy is replaced by additional dimensions of the space: an anisotropic element of dimension d0, for example 3D, is represented by a corresponding isotropic element of dimension d, for example 6D. We then speak of embedding the dimension element dO into a corresponding element of dimension d.

By way of illustration, with reference to FIGS. 1 and 2, to generate a 2D anisotropic mesh governed by the 2D anisotropic metric shown in part A of FIG. 1, a corresponding isotropic 3D surface is generated (see part A of FIG. 2).

In part A of FIG. 1, each deformed circle represents the equidistant points of the center of the deformed circle in terms of anisotropic distance defined by the anisotropic metric considered.

According to these techniques, an isotropic Voronoi diagram is then generated for this 3D surface, represented in part B of FIG. 2. Then an isotropic mesh by Delaunay triangulation, represented in part C of FIG. 2, is deduced from this diagram of FIG. Voronoï.

The isotropic 3D surfaces shown in parts A, B, C of FIG. 2 are projected on the (xOy) plane to generate the corresponding anisotropic 2D surfaces of parts A, B, C of FIG.

Some of these techniques are notably exposed in the following documents, named references_1:

- J.F. Nash, The imbedding problem for riemannian manifolds. Annals of Mathematics, 63: 20-63, 1956;

- F. Labelle and J.-R. Shewchuk, Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation, In SCG '03: Proceedings of the Nineteenth Annual Symposium on Computational Geometry, pp. 191-200, 2003.

These techniques give good results for the taking into account of the anisotropy, but the dipping in dimensions d superior gives rise to a very important increase of the computation volume necessary.

Indeed, the computation of a Delaunay triangulation has a complexity proportional to d !, where d is the dimension of the computing space. Generally, the dipping space has a dimension of 6D or even 10D, which makes the calculation volume prohibitive. The present invention aims at providing a solution for reducing the computing load in a space of dimension d.

For this purpose, according to a first aspect, the invention proposes a data processing method of the aforementioned type, characterized in that in step b /, the result of the intersection between said element and a determined Voronoi cell is determined. from the point which associates it x ,, by implementing an iterative processing according to which, at a current iteration step k, an additional point x _jk of the set of points other than the point x is selected, and a result is calculated an intersection of one half of the space E delimited by the mediating hyperplane of the segment (x ,, x _Jk ) and containing x ,, and the result of intersection calculated at the previous iteration step,

and wherein the selection of at least one additional point in the iteration steps is a function of a comparison between the distance between the associated point x and said additional point and twice the maximum distance between the associated point x , and a point of an intersection result calculated during iterative processing.

The invention makes it possible to reduce the volume of calculations necessary for calculating the intersections of Voronoi cells with the element under consideration, by using a test that makes it possible to avoid unnecessary calculations. It also makes it possible to parallelize the treatments for distinct Voronoi cells.

In embodiments, the method according to the invention further comprises one or more of the following features:

in step b /, the n-1 other points of the game arranged in order of increasing distance with respect to the point x ,, are denoted _y , - - - x _{Jn 1} , and the iterative processing is implemented by selecting at the current iteration step k, the point x _jk of the succeeding set of points in said increasing distance order the point x _{Jk 1} selected at the iteration step k-1;

said iterative processing is stopped according to a comparison between the distance between the associated point x, and the point _{ik + i} and the double of the maximum distance existing between the associated point x, and a point of the calculated intersection result. at the iteration step k;

in an initialization step, the index k is set to 1 and the intersection result is set equal to the element;

repeating all the steps a1 and b / defining in each new step a1 a set of n points drawn from at least the previous game;

the steps a 1 and b 2 are repeated; during the step of calculating an updated intersection result equal to the intersection between on the one hand half the space E delimited by a mediating hyperplane of a segment (χ ,, x _jk ) and which contains χ ,, and on the other hand the result of intersection calculated at the preceding iteration step, one implements the following steps:

determining whether a vertex q of the updated intersection result is the intersection between the mediating hyperplane of [x ,, x _p ] and an edge [qi, q ₂ ] between two vertices q1, q2 of the element, where x _p is a point in the set of n points;

and if said vertex q is determined to be the intersection between the mediating hyperplane of [x ,, x _p ] and an edge [qi, q ₂ ] between two vertices q1, q2 of the element, the coordinates of q are determined using the following formulas:

q = A ^ _! + A ₂ q ₂ , with Κ = Δ ^" (- || χ _ρ || ² + 2 <q ₂ , x _p >)

λ ₂ = Δ ¹ (|| χ _ρ || ² - 2 <q _l5 x _p >)

where Δ = -2 <qi, x _p > + 2 <q ₂ , x _p >, and

<.,.> represents the scalar product function;

during the step of calculating an updated intersection result equal to the intersection between on the one hand half of the space E delimited by a mediating hyperplane of a segment (x ,, x _jk ) and which contains x ,, and on the other hand the intersection result calculated at the preceding iteration step, the following steps are implemented:

it is determined whether a vertex q of the updated intersection result is the intersection between the mediating hyperplane of [x ,, x _p ], the mediating hyperplane of [χ ,, x,], and a triangle of vertices ç q ₂ , q ₃ of the element, where x _p and x, are two points of the set of n points;

- and if q is determined to be the intersection between the mediating hyperplane of [x ,, x _p ], the mediating hyperplane of [x ,, X |], and a triangle of vertices qi, q ₂ , q ₃ of the element, the coordinates of q are determined using the following formulas: q = q ₁ + λ ₂ q ₂ + λ ₃ q ₃ , with

Κ = A ^" [(a ₂₃ -a ₂₂ ) || x _p || ² + (a ₁₂ -a ₁₃ ) || xi || ² + c ₃₁ ];

A ₂ = A ^" [(a ₂₁ - ¾ 3) || xp || ² + (ai3-aii) || xi || ² + c ₃ 2];

λ ₃ = Δ ^"1 [(a ₂₂ - a ₂₁ ) II x _p II ² + (ai 1 - a ₁₂ ) || x, || ² + c ₃₃ ] where an = - 2 <q _1; x _p > a ₁₂ = - 2 <q ₂ , x _p >; a ₁₃ = - 2 <q ₃ , x _p >

a ₂₁ = - 2 <q _1; X |>; a ₂₂ = - 2 <q ₂ , x,>; a ₂₃ = - 2 <q ₃ , x,>.

c ₃ i = a ₂₃ ai ₂ - a ₂₂ ai ₃ ; c ₃₂ = a ₂ i has ₃ - ₂₃ years; c ₃₃ = a ₂₂ year - a ₂ i have ₂

<.,.> represents the scalar product function; during the step of calculating an updated intersection result equal to the intersection between on the one hand half the space E delimited by a mediating hyperplane of a segment (χ ,, x _jk ) and which contains χ ,, and on the other hand the intersection result calculated at the previous iteration step, comprising the following steps:

determining whether a vertex q of the updated intersection result is a vertex of the element;

if said vertex q is determined as a vertex of the element, the presence of said vertex q in the half of the space E delimited by the mediating hyperplane of the segment (x ,, x _Jk ) and which contains x, is determined according to the sign of the orient function by: orient (Y [ ⁺ (i, j _k ), q) = sign (|| x _Jk || ² -2 <q, x _Jk >), where

<.,.> represents the scalar product function and sign (x) is the function supplying the sign of the variable x;

during the step of calculating an updated intersection result equal to the intersection between on the one hand half of the space E delimited by a mediating hyperplane of a segment (x ,, x _jk ) and which contains x ,, and on the other hand the intersection result calculated at the previous iteration step:

if q is determined as the intersection between the mediating hyperplane of [x ,, x _k ] and an edge [qi, q ₂ ] between two vertices q1, q2 of the element, the presence of said vertex q in the half of the space E delimited by the mediating hyperplane of the segment Xi, x _jk ) and which contains x, is determined according to the sign of the orient function orient (f [ ⁺ (/, yJ, q) = sign (Δ || x _jk || ² -2 <A q, x _jk> ) sign (Δ), with Δ = -2 <qi, x _p > + 2 <q ₂ , x _p >, where <.,.> Represents the scalar product function and sign (x) is the function supplying the sign of the variable x;

during the step of calculating an updated intersection result equal to the intersection between on the one hand half of the space E delimited by a mediating hyperplane of a segment (x ,, x _Jk ) and which contains x ,, and on the other hand the intersection result calculated at the previous iteration step: it is determined whether a vertex q of the updated intersection result is the intersection between the mediating hyperplane of [χ ,, x _p ], the mediating hyperplane of [χ ,, X |], and a triangle of vertices qi , q ₂ , q3 of the element, where x _p and X | are two points in the game of n points;

if said vertex q of the updated intersection result is determined as the intersection between the mediating hyperplane of [x ,, x _p ], the mediating hyperplane of [x ,, X |], and a triangle of vertices qi , q ₂ , q3 of the element, the presence of said vertex q in the half of the space E delimited by the mediating hyperplane of the segment (x ,, x _h -) and which contains x, is determined according to the sign of the orient function

(Y [ ⁺ (i _k ) .q) defined by: orient (Y [ ⁺ (i, j _k ), q) = sign (Δ || x _k || ² -2 <Δq, x _k >). sign (Δ), where

Δ = C31 + C32 + C33;

C31 = a ₂ 3 3i2 - a ₂ 2 3i 3; C32 = 821.313 "¾3 3i1; C33 = 322 an - 321 3i2;

3i i = - 2 <qi, x _p >; 3i2 = - 2 <q ₂ , x _P >; 3i ₃ = - 2 <q ₃ , x _P >

3 ₂ i = - <qi, X |>; 3 ₂₂ = - 2 <q ₂ , Xi>; 3 ₂₃ = - 2 <q ₃ , Xi> and where

<.,.> represents the scalar product function and sign (x) is the function providing the sign of the variable x.

According to a second aspect, the present invention provides a data processing computer program defining an element in a space E of dimensions d, said program comprising instructions for implementing the steps of a method according to the first aspect of the present invention. invention when executing the program by processing means.

These features and advantages of the invention will appear on reading the description which follows, given solely by way of example, and with reference to the appended drawings, in which:

- Figure 1 shows in part A a field of anisotropy prescribed on a 2D surface, in part B a Voronoi diagram resulting and in part C the result of the corresponding triangulation Delaunay;

Figure 2 shows in part A an isotropic surface D, in part B a Voronoi diagram resulting and in part C the result of the corresponding triangulation of Delaunay;

FIG. 3 partially shows a Voronoi diagram of a set of points, in part B the configuration obtained after one iteration of a Lloyd relaxation algorithm, in part C the configuration obtained after 100. iterations of a Lloyd relaxation algorithm and in part D the result of the Delaunay triangulation resulting from the configuration represented in part C;

FIG. 4 is a view of a data processing device in one embodiment of the invention;

FIG. 5 represents steps of a method in one embodiment of the invention;

FIG. 6 represents steps of a method in one embodiment of the invention;

FIG. 7 represents steps of a method in one embodiment of the invention.

Voronoi diagram and Delaunay triangulation:

First, the definitions of a Voronoi diagram and the Delaunay triangulation are recalled.

Given a set X of points such that X = {xi, x ₂ , ..., x _n } with x, 6 ¾ ^d , where 9I is the real space and d the dimension considered, the Voronoi Vor diagram ( X) is the collection of Voronoi Vor cells (x,), i = 1 to n.

The Voronoi Vor cell (x,) associated with the point x, is defined by:

Vor (Xi) = {x 6 ¾ ^d / d (x, Xi) <d (x, Xj), V j = 1 to n} where d (.,.) Indicates the Euclidean distance ¾ ^d .

The dual of the Voronoi Vor (X) diagram, i.e. the abstract simplicial complex derived from the combinatorial Vor (X), is called the Delaunay triangulation.

Each pair of Voronoi Vor cells (x,), Vor (Xj), which has a non-zero intersection, defines a side (x ,, Xj) in the Delaunay triangulation, and each triplet of Voronoi Vor cells (x, ), Vor (Xj), Vor (x _k ) which has a non-zero intersection defines a triangle (x ,, Xj, Xk) -

Delaunay's triangulation has several interesting geometric properties and is used in many applications, for example, but not only, mesh generation treatments (see Jean-Daniel Boissonnat and Mariette Yvinec, Algorithmic Geometry, Cambridge University Press, 1998). .

The centroidal Voronoï Tessallation (CVT) diagram is a Voronoï diagram in which the point x ,, for i = 1 to n, is the center of the Voronoï cell associated with x ,. A CVT can be realized by an algorithm, called Lloyd's Relaxation, which iteratively moves each point x, to the center of Vor (Xi), and provides isotropic triangles (see Stuart P. Lloyd, Least Squares Quantization in PCM, IEEE). Transactions on Information Theory, 28 (2): 129-137, 1982).

With reference to FIG. 3, a 2D Voronoi diagram associated with the points x,

(corresponding to non fat points) is shown in part A. The centers of Voronoi cells are in bold points. In part B of FIG. 3, the configuration obtained after an iteration of the Lloyd relaxation algorithm is represented. In part C of FIG. 3, the configuration obtained after 100 iterations of the Lloyd relaxation algorithm is represented. In part D of FIG. 3, the Delaunay triangulation deduced from the configuration of part C of FIG. 3 is represented.

At present, the concepts of restricted Voronoi diagram and restricted Delaunay triangulation are defined. Let S be a domain included in ¾ ^d .

For example, S is a surface or a volume, or any other domain.

The Voronoi diagram of the set X restricted to S, labeled Vor (X) | _s , is the set of Voronoi Vor cells (x,), i = 1 to n, restricted to S, denoted Vor (Xi) | _s .

The Voronoi cell restricted to S, Vor (Xi) | _s , is equal to Vor (Xi) | _s = Vor (x,) Π S. The dual of the Voronoi diagram Vor (X), ie the abstract simplicial complex deduced from the combinatorial of Vor (X) | _s , is called the restricted Delaunay triangulation. Each triangle of a restricted Delaunay triangulation corresponds to three restricted Voronoi cells with a non-empty intersection.

As a direct consequence of their definition, restricted Voronoi cells can also be defined by Vor (Xi) | _s = S Π Q Π (', /),

y = 1 to n, j ≠ i where] ^ ⁺ (/, y) = {x | d (x, x,) <c / (x, x _y )} is the half-space in 9i ^d limited by l mediator hyperplane of the segment [x ,, Xj] and which contains x ,.

We thus see that a restricted Voronoi cell Vor (Xi) | _s can be obtained starting from space 3i ^d and iteratively making cuts using hyperplanes Π ⁺ ('> /) for j = 1 to n and j ≠ i.

To apply such an iteration in practice is of no interest because the resulting algorithm would have a super-practical complexity. That's why algorithms existing ones are based on other considerations. For example, methods for calculating Voronoi diagrams based on the properties of the Delaunay triangulation described in the following reference work: "Algorithmic Geometry", Boissonnat and Yvinec, ISBN-13: 978-0521565295 are known. These classical algorithms directly calculate the Delaunay triangulation, inserting the points one by one and iteratively correcting the mesh so as to check the so-called "empty sphere" property.

Principle of the safety radius:

One of the aspects of the invention is to make it possible to determine efficiently, among the mediating hyperplanes defined by the segments [x ,, x,], which are contributors to the Voronoi Vor cell determination (x,) and which are non-contributors. , ie Vor (x,)

Consider x _h -, ... x _{Jn 1} the n-1 points of the set X other than x, and ranked in increasing order of distance to x ,.

Let V _k (Xi) be the intersection of the first k mediating hyperplanes between x, and each of these k first points, and R _k its radius centered on x ,, ie:

V _k (Xi) = ΠΐΓ ( ^{/ '} , 7 /) ^{and Rk = max} t ^d ( ^Xi ' ^x ) ^{1 6 ν} * ()> ^■

/ = 1

The safety radius theorem according to the invention is as follows:

For every j such that d (x "Xj)> 2.R _k , the hyperplane j) is no contributor to the construction of the Voronoi Vor cell (x,), ie V _k (x,) c ·

Proof: considering x 6 V _k (x,) and Xj such that d (Xi, x _jy )> 2 R _k ,

by definition of R _k , d (x, x,) <R _k .

We have ad (x, x) + d (x, Xj)> d (x _i; Xj) (triangular inequality)

therefore, d (x, Xj)> R _k > d (x, x,) and x C] ^ [ ⁺ (/ ^' , y).

It should be noted that this theorem has advantageous practical applications when applied to bounded Voronoi cells, and therefore to restricted Voronoi cells.

Direct consequence of the safety radius theorem:

if d (Xj, x _{jk + 1} )> 2.R _k then V _k (Xi) = Vor (Xi). We call security radius the first value of Rk (ie the largest) encountered, by traversing the x _h -, ... χ _{ί ni} in this order, which satisfies this condition

"D (x _h x _{jk + 1} )> 2.R _k ". Example of application of the principle of safety radius

In the following, we present an example of application of the safety radius theorem to a device and a data processing method, these data defining a domain S0.

In the case considered, S0 is a surface extending in a space of dimension dO, for example dO = 3.

In other cases, S0 may be a volume delimited by a surface or any other type of domain.

In this case, the goal of the treatment is to generate an anisotropic mesh of the S0 domain according to a prescribed anisotropy field.

For this purpose, a data processing device 10 shown in FIG. 4 is considered.

Such a processing device 10 comprises a memory 1 1, a microcomputer

12 and a human-machine interface 13, including a display screen on which to display a mesh generated for the domain S0.

The memory 10 notably comprises digital data defining the domain S0 and a computer program P.

The program P includes software instructions, which when executed by the microcomputer 12, implement the steps indicated below with reference to FIGS. 5-7.

Let the domain S0 of dimension dO, for which an anisotropic mesh governed by a given anisotropy metric must be generated.

In a step 100, a domain S of dimensions d> d0 is made to correspond to domain S0 of dimension d0 in accordance with the documents named references_1 above. For example, d = 6, or 10.

For dO = 3 and d = 6, at any point [x, y, z] of S0, for example, the point [x, y, z, sN _x , sN _y , sN _z ] of S, where the factor s indicates the desired degree of anisotropy

(a small value of s generates an isotropic mesh and a large value of s, an anisotropic mesh). N _x , N _y , N _z are the normal unit vectors at the surface S0 at the point

[X, y,

Then an iterative processing 101 is implemented to determine an isotropic mesh of the S domain. Thus, in a step 102, the value n of the number of Voronoi cells being fixed, a set X of n points of the space ¾ ^d , {xi, x ₂ , ..., x _n } with x, 6 ¾ ^d , is determined for the current iteration.

During the first iteration, the set X of points is for example chosen randomly.

Then, during the following iterations, the set X is determined according to the results of the last iteration performed for the processing 101.

In a step 103, a step of determining the Voronoi diagram restricted to S, ie Vor (Xj) | _s , is achieved.

In a step 104, a loop output condition is tested. This loop output condition includes for example:

the comparison between the current value k of iteration and a fixed maximum threshold; and or

- the gradient standard of an "objective" function is below a certain threshold. For example, the "objective" function represents the noise power of the sampling

("Quantization noise power" in English), namely the sum of the moments of inertia of the Voronoi cells.

If the loop output condition 105 is not satisfied, the value of k is increased by 1 and an additional iteration of the processing 101 is implemented.

If the loop output condition 104 is satisfied, iterative processing 101 is stopped. A restricted step Delaunay triangulation step 105 is then implemented, to determine the dual of the Voronoi diagram provided at the output of the iterative processing 101, by direct deduction of the combinatorial of the restricted Voronoi diagram, namely for each vertex of the Voronoi restricted, one generates the corresponding Delaunay triangle.

Various operations can then be implemented, in particular the projection on the initial dO dimension space of the result of the Delaunay triangulation obtained in the dimension space d after the iterative processing 101, so as to obtain the desired anisotropic mesh of domain S0.

In the embodiment considered, the step 103 for determining the Vor (Xj) | _s , is carried out in the manner indicated below with reference to FIG.

In a step 103_1, for each x ,, with i = 1 to n, the other points x ,, j = 1 to n and j ≠ i, are sorted in increasing order of distance to x, which results in an ordered list Xj,. . . Xj ^ ■ To do this, the ANN tool is used for example (David M. Mount and Sunil

Arya, ANN: A library for getting close nearest neighbor searching. In Proceedings CGC Workshop on Computational Geometry, pages 33-40, 1997).

In an initialization step 103_2, the variable m is initialized to the value 0, and each of the point coordinates g, ¾ ^d , i = 1 to n.

In a step 103_3, a division of the domain S into subdomains is performed, in order to be able to parallelize the steps performed on distinct subdomains (triangles, tetrahedra, etc.).

In this case, the subdomains f of the surface S are triangles.

In a step 103_4, for each point x ,, with i = 1 to n, and for each subdomain f such that the intersection between the subdomain f and the Voronoi cell associated with X, is nonempty, Vor (Xi) | _f , the Voronoi cell associated with x, and restricted to the subdomain f, is determined, as indicated below with reference to FIG. 7.

Then we assign to a variable m the value of the "mass" of Vor (Xj) | _f , we add to the current value of m, the value of m, and update the value of the coordinates of the point g, according to the formula: g, = g, + m.centre Vor (Xi) | _f , where center Vor (Xi) |, is the center of gravity of Vor (Xj) | _f .

The "mass" of Vor (Xi) | is equal to. J cx. So the "mass" of Vor (Xi) |, is xeVor

1 r

The center of gravity of Vor (Xi) |, is 1- x.dx.

mass of Vor ( _Xi ) \ _{f Vor} _{lx) lf}

In one embodiment, the determination of the set of points X in a step 102 for the iteration k + 1 of the processing 101, is for example such that the point x, = g ,. 1 / ri, where i = 1 to n and g, is the point obtained at the end of step 103_4 implemented for the iteration k of the treatment 101.

According to the invention, the step 103_4 for determining the Voronoi Vor (Xi) | cell, associated with x, and restricted to the subdomain f, is implemented using the safety radius theorem, and considering the points x _h x _{jn 1} ordered according to a distance X, increasing.

In an exemplary embodiment, with reference to FIG. 7, step 103_4 thus comprises the following operations. In an initialization step 103_41, consider a domain V equal to the subdomain f considered, a value t equal to 1 and a value R equal to max {d (x ,, x) / x 6V}.

In a step 103_42, the following steps are iterated as long as d (Xi, _{i (} ) <2R and that t <n:

v = vn [] ⁺ (/, i ₍ );

t = t + 1.

When d (x, _y ) ≥ 2R or t = n, step 103_42 is stopped.

And the current value of V is then equal to the Voronoi cell Vor (Xi) |, associated with x, and restricted to the subdomain f (to continue the iterations of the step 103_42 is no longer useful since the non-mediating hyperplanes still taken into account are not contributors).

For the effective calculation of the intersection V y ^' _f ), Sutherland &Hodgman's reentrant fenestration algorithm could be used, for example (Ivan Sutherland and Gary W. Hodgman, Reentrant Polygon Clipping, Communications of the ACM, 17: 32-42, 1974).

The use of the safety radius theorem allows us to determine only those sections of the Voronoi cell that are useful for calculating the intersection between this cell and the considered subdomain f, thus greatly reducing the necessary computational volume. which is very appreciable especially when the value of the dimension d of the space considered increases.

Moreover, it allows the calculation in parallel and independently of intersections for respective Voronoi cells. In one embodiment, a method according to the invention implements the steps indicated below, to define the intersection between a subdomain f and a Voronoi cell, in a space of dimensions d, using the products scalars and the linear combination of vectors, the calculation volume implemented being independent of the dimension d.

It should be noted that these provisions can be implemented independently of the use of the so-called safety radius test.

In the step of calculating V y ^' _i ) Using Sutherland &Hodgman's re-entrant fenestration algorithm, it is necessary to determine the coordinates of the new vertices of the domain resulting from this intersection, and also to determine which side a vertex is with respect to a half-space] ^ [ ⁺ (/ ^' , y).

The function of determining on which side a vertex q is located with respect to the half-space Π ⁺ ('> /) ^is named orient (] ^ [ ⁺ (/, y), q). If the result of orient (YY (i, y), q) is positive, the vertex q is located in the half-space Π (/, y).

Otherwise, the vertex q is outside the half-space Π ⁺ ('> /) ■

Summits q of the domain resulting in some cases have already been determined during previous iteration (s) or appear only in the current iteration.

Three different configurations exist for a vertex q:

a- q is a vertex of the subdomain f considered;

b- q is the intersection between the mediating hyperplane of [x ,, x _k ] and an edge [qi, q ₂ ] between two vertices qi, q ₂ of the subdomain f;

c- q is the intersection between two mediating hyperplanes (for example the mediating hyperplane of [x ,, x _k ] and that of [χ ,, x,], and the subdomain f; in this case, the subdomain is the triangle of vertices qi, q ₂ , q ₃ .

Without loss of generality, we suppose x, at the origin of the space considered. It is brought back by applying a translation.

In the case a /, the coordinates of q are known, and

orient (] ^ [ ⁺ (/, y), q) = sign (|| x _j || ² -2 <q, Xj>), where <,> represents the scalar product function.

In the case b /, the barycentric coordinates (λ _1; λ ₂ ) of q with respect to qi, q ₂ are determined: q = qi + λ ₂ q ₂ .

By naming Δ the determinant of the 2x2 matrix above:

Δ = -2 <q _1; x _k > + 2 <q ₂ , x _k >

= Δ ¹ (- II x _k H ² + 2 <q ₂ , x _k >) and A ₂ = A- (|| x _k || ² - 2 <q ₁ , x _k >)

and orient (Π}), q) = sign (Δ || Xj || ² -2 <Δq, Xj>). sign (Δ).

In the case c /, the barycentric coordinates (λι, λ ₂ , λ ₃ ) of q with respect to qi, q ₂ , q ₃ are determined: q = λι qi + λ ₂ q ₂ + λ ₃ q ₃ .

an = - 2 <q _1; x _k >; a ₁₂ = - 2 <q ₂ , x _k >; a ₁₃ = - 2 <q ₃ , x _k >

The following cofactors are calculated:

c ₃ i = a ₂₃ 3i ₂ - a ₂₂ ai ₃ ; c ₃₂ = a ₂ i has ₃ - ₂₃ years; c ₃₃ = a ₂₂ year - a ₂ i have ₂

Δ being the development of the determinant with respect to the last line of the 3x3 matrix: Δ = c ₃ i + c ₃₂ + c ₃₃ .

It follows:

λι = A ^" [(a ₂₃ -a ₂₂ ) || x _k || ² + (a ₁₂ -a ₁₃ ) || xi || ² + c ₃₁ ];

A ₂ = A ^" [(a ₂₁ -a ₂₃ ) || x _k || ² + (a ₁₃ -a ₁₁ ) || xi || ² + c ₃₂ ];

λ ₃ = Δ ^"1 [(a ₂₂ - a ₂₁ ) II x _k II ² + (a! -a ^) || xi || ² + c ₃₃ ]; and orient (Π (', j), q) = sign (Δ || x || ² -2 <Δq, Xj>) sign (Δ) The formulation of q and the orient function is therefore independent of the dimension d, in that only the scalar product <.,.> and linear combinations intervene, and where the dimension of the linear systems is independent of d.

In addition, it lends itself well to arbitrary precision arithmetic implementation and filtering (interval arithmetic or quasi-static filters).

Recentering (translation of x, originally) improves filtering performance.

The algorithm implementing these steps of calculating the coordinates of q and the orient function is parametrized by a geometric kernel, defining the types of points and vectors, the dot product and the linear combinations of vectors. This allows to have a working implementation whatever the dimension d.

In the example described above, it has been considered that the subdomain f is a triangle. However, the formulas indicated above are valid also for any subdomain f, for example an arbitrary polygon or any object of greater size, for example a tetrahedron (these formulas are therefore applicable in the case of a volume mesh).

The use of the safety radius theorem to reduce the computations of a restricted Voronoi diagram or a restricted Delaunay triangulation has been described above as part of an application to the mesh of an element. Of course, she can intervene in other areas implementing such calculations and provides the same benefits.

The same goes for the principles of calculating vertices resulting from intersections independently of the dimension d and the function orient.

These provisions are very advantageous, and for values of d equal to 2, 3, or greater than or equal to 4.

The applications of a technique according to the invention other than the mesh comprise for example:

3D reconstruction applications, in particular the steps of passing from a cloud of points to a surface;

- robotics applications: for example in the representation of configuration spaces, or the management of autonomous robot fleets or trajectory planning;

- astrophysical applications: for example, calculations in phase spaces;

- artificial intelligence applications, such as document searches ("google" type, which also uses large spaces).

Claims

1. A data processing method defining an element in a space E of dimension d, said method comprising the steps of:

a set of n points in said space is defined;

b / calculating the intersection between said element and Voronoi cells each determined from an associated point respectively among said n defined points;

said method being characterized in that in step b /, the result of the intersection between said element and a Voronoi cell determined from the point associated with it is determined by implementing an iterative process according to which

at a current iteration step k, an additional point x _h - of the set of points other than the point x is selected, and an updated intersection result equal to the intersection between on the one hand half of the space E delimited by the mediating hyperplane of the segment (χ ,, x _jk ) and which contains χ ,, and on the other hand the result of intersection calculated at the preceding iteration step,

2. Method according to claim 1, wherein in step b /, the n-1 other points of the game sorted in order of increasing distance from the point x ,, are denoted _y - _{in i} , and the iterative processing is implemented by selecting at the current iteration step k, the point x _h - of the set of points which succeeds in said increasing distance order the point x _h - ₁ selected at the step of iteration k-1.

3. Method according to claim 2, wherein said iterative processing is stopped according to a comparison between the distance between the associated point x, and the point _{ik + i} and twice the maximum distance existing between the associated point x, and a point of the intersection result calculated at the iteration step k.

4. Method according to one of the preceding claims, wherein in an initialization step, the index k is set to 1 and the intersection result is set equal to the element.

5. Method according to one of the preceding claims, wherein it repeats all steps a and b / defining in each new step a / a set of n points from at least the previous game.

6. Method according to one of the preceding claims, wherein it repeats all the steps a and b /.

7. Method according to one of the preceding claims, wherein, during the step of calculating an updated intersection result equal to the intersection between first half of the space E delimited by a hyperplane mediator of a segment (x ,, x _h -) and which contains

Xi, and secondly the intersection result calculated at the preceding iteration step, the following steps are implemented:

q = λι qi + A ₂ q ₂ , with λι = Δ ^" (- || χ _ρ || ² + 2 <q ₂ , x _p >)

λ ₂ = Δ ¹ (II x _P II ² - 2 <q _!, x _p >)

where Δ = -2 <q _; x _p > + 2 <q ₂ , x _p >, and

<.,.> represents the scalar product function.

8. Method according to one of the preceding claims, wherein, during the step of calculating an updated result of intersection equal to the intersection between first half of the space E delimited by a hyperplane mediator of a segment (x ,, x _h -) and which contains

it is determined whether a vertex q of the updated intersection result is the intersection between the mediating hyperplane of [x ,, x _p ], the mediating hyperplane of [x ,, X |], and a triangle of vertices qi , q ₂ , q ₃ of the element, where x _p and X | are two points in the game of n points;

- and if q is determined to be the intersection between the mediating hyperplane of [x ,, x _p ], the mediator hyperplane of [χ ,, x,], and a triangle of vertices qi, q ₂ , q ₃ of the element, the coordinates of q are determined using the following formulas: q = λι qi + λ ₂ q ₂ + λ ₃ q ₃ , with

λι = A ^"1 [(a ₂₃ - a ₂₂ ) || x _P || ² + (ai2 - a ₁₃ ) || xi || ² + c ₃ i];

A ₂ = A ^~ [(a ₂₁ - a ₂₃ ) || xp || ² + (a ₁ 3 - a _{1 1} ) || xi || ² + c ₃₂ ];

λ ₃ = Δ ^"1 [(a ₂₂ - a ₂₁ ) || x _P || ² + (ai ₁ - a ₁₂ ) || x, || ² + c ₃₃ ] where an = - 2 <qi, x _p > a ₁₂ = - 2 <q ₂ , x _p >; a ₁₃ = - 2 <q ₃ , x _p >

a ₂₁ = - 2 <qi, xi>; a ₂₂ = - 2 <q ₂ , x,>; a ₂₃ = - 2 <q ₃ , x,>.

C31 = ¾3 3ΐ ₂ "¾ ₂ 3.13! C ₃₂ = 3 ₂₁ 3i3" 3 ₂₃ & _{1 1} ^' , C ₃₃ = a ₂₂ year - a ₂ i Ά ₁₂

<.,.> represents the scalar product function.

9. Method according to one of the preceding claims, wherein, during the step of calculating an updated intersection result equal to the intersection between first half of the space E delimited by a hyperplane mediator of a segment (x ,, x _h -) and which contains

if said vertex q is determined as a vertex of the element, the presence of said vertex q in the half of the space E delimited by the mediating hyperplane of the segment (x ,, x _h -) and which contains x, is determined by the sign of the function orient (] ^ [ ⁺ (/, y _fc ), q) defined by: orient (Y [ ⁺ (i, j _k ), q) = sign (|| x _jk || ² -2 <q, x _k >), where

<.,.> represents the scalar product function, and sign (x) is the function providing the sign of the variable x.

10. Method according to one of the preceding claims, wherein, during the step of calculating an updated intersection result equal to the intersection between first half of the space E delimited by a hyperplane mediator of a segment (x ,, x _h -) and which contains x ,, and secondly the intersection result calculated at the previous iteration step, the following steps are implemented:

determining whether a vertex q of the updated intersection result is the intersection between the mediating hyperplane of [x ,, x _p ] and an edge [qi, q ₂ ] between two vertices q1, q2 of the element, where x _p is a point in the set of n points; if q is determined as the intersection between the mediating hyperplane of [χ ,, x _k ] and an edge [qi, q ₂ ] between two vertices q1, q2 of the element, the presence of said vertex q in the half the space E delimited by the mediating hyperplane of the segment (Xi, x _jk ) and which contains x, is determined according to the sign of the function orient (] ^ [ ⁺ (/, y _fc ), q) defined by : orient (f [ ⁺ (/, y _A ), q) = sign (Δ || x _Jk || ² -2 _{<A q,} x _Jk> ). sign (Δ), with Δ = -2 <q _1; x _p > + 2 <q ₂ , x _p >, and where <.,.> represents the scalar product function, and sign (x) is the function supplying the sign of the variable x.

1 1. Method according to one of the preceding claims, wherein during the step of calculating an updated intersection result equal to the intersection between on the one hand half of the space E delimited by a mediator hyperplane. a segment (x ,, x _jk ) and which contains x ,, and secondly the calculated intersection result at the preceding iteration step, the following steps are implemented:

if said vertex q of the updated intersection result is determined as the intersection between the mediating hyperplane of [x ,, x _p ], the mediating hyperplane of [χ ,, x,], and a triangle of vertices qi , q ₂ , c ₃ of the element, the presence of said vertex q in the half of the space E delimited by the mediating hyperplane of the segment (x ,, x _jk ) and which contains x, is determined according to the sign of the orient function

Δ = C31 + C3 ₂ + C33;

C31 = a ₂ 3 a ₂ - a _{22 a} 3; C3 ₂ = a ₂ i a-i3 - a ₂ 3 year; 033 = a ₂₂ year - a ₂ i have ₂ ;

an = - 2 <q _1; x _p >; a ₁₂ = - 2 <q ₂ , x _p >; a ₁₃ = - 2 <q ₃ , x _p >

a ₂₁ = - 2 <q _1; X |>; a ₂₂ = - 2 <q ₂ , x,>; a ₂₃ = - 2 <q ₃ , Xi> and where

12. A data processing computer program defining an element in a space E of dimensions d, said program comprising instructions for implementing the steps of a method according to one of the preceding claims, during an execution of the program by means of treatment.