CN104268934A - Method for reconstructing three-dimensional curve face through point cloud - Google Patents

Abstract

The invention discloses a method for reconstructing a three-dimensional curve face through point cloud. The method comprises the steps of inputting point cloud data P possibly carrying noise and abnormal values and the number m of peaks of the curve face needed to be reconstructed; initializing a dictionary matrix V and a connection matrix B; updating the dictionary matrix V and the connection matrix B in an iteration mode till convergence; outputting a reconstructed triangular net to complete reconstruction of the three-dimensional curve face. By means of the method, the triangular net is directly reconstructed through the point cloud, the noise input into the point cloud can be removed well, non-uniform sampling is processed in a robust mode, and the characteristics in the point cloud can be recovered well. Compared with a conceal method, the triangular net is directly reconstructed through the point cloud so as to avoid accumulated errors caused by multi-step optimization in the conceal method. Compared with a combination method, the reconstruction errors serve as a target function, so that the reconstruction errors of the curve face reconstructed through the method are usually smaller than those reconstructed by the existing combination method.

Description

A kind of method by the direct reconstruction of three-dimensional curved surface of a cloud

Technical field

The invention relates to a kind of method by the direct reconstruction of three-dimensional curved surface of a cloud, the method can according to input point cloud and some cloud on may with normal direction information directly reconstruct high-quality three-dimension curved surface, in three-dimensional scenic modeling, intelligent city, reverse-engineering and machine-building processing and other fields, there is great using value.

Background technology

Curve reestablishing is that recent two decades is to appear at a hot issue in the fields such as computer-aided design (CAD), medical imaging, computer graphics and intelligent city, its object is to find certain mathematical description or model, accurately, represent that the cloud data inputted (passes through laser scanner compactly, or the feature detection acquisition etc. of medical image), and based on this curve and surface at cloud data place is analyzed, revises and drawn.Curve reestablishing technology has a wide range of applications, and wherein reverse-engineering is one of them important application.In background introduction, will take reverse-engineering as the discussion that application background launches to curve reestablishing technology.The subject that reverse Engineering Technology one of being development and ripe and improving of DATA REASONING technology along with computer aided design and manufacture technology and developing rapidly is emerging and technology.Its appearance, changes the Top-Down Design pattern in original CAD/CAM system from drawing to material object, for product rapid research and development and rapid prototypingly provide brand-new approach.In reverse-engineering system (see Fig. 1), the reconstruction of curved surface is undoubtedly one of most important and the most difficult problem.

In whole reverse-engineering, core support is played a part according to the cloud data reconstruction curved surface that scanning device collects, its object is to find certain mathematical description or model, accurately and represent the scan-data (also known as a cloud) inputted compactly, and based on this curved surface at scan-data place is analyzed, revises and drawn, and require that the mathematical notation of original entity or curved surface in the expression of this mathematics and CAD/CAM system is consistent as far as possible.Generally speaking, the problem of curve reestablishing simply can be divided into two stages: first stage reconstructs high-quality polygonal mesh (triangle gridding is in the majority) from a cloud; Subordinate phase, reconstructs non-unified Rational B-splines NURBS (Non-Uniform Rational B-Spline) curved surface from polygonal mesh; The burst matching etc. of parametrization, the burst of polygonal mesh, the feature extraction of polygonal mesh and polygonal mesh is comprised at second stage.

As shown in Figure 2, provide one group of some cloud in space, their samplings are from a space curved surface (two dimension that two dimension is communicated with compact manifold or band border is communicated with compact manifold), except knowing their three-dimensional coordinate, all the other information are known nothing, how to construct a triangle mesh curved surface to carry out interpolation or approach this group point cloud? up to the present, the theoretical research and the algorithm that relate to this problem are existing a lot, but are broadly divided into following two types.

From the angle of matching cloud data, carry out with an implicit surface cloud data that matching inputs.Because implicit surface does not need parametrization, it can represent the curved surface of complicated shape in addition, at curve reestablishing, the aspect of surface-rendering it have more advantage than explicit curved surface.The general thinking of implicit surface matching is such: first construct an Implicitly function according to cloud data, f (p): R ³-R, approaches the curved surface representated by cloud data with zero contour surface of this function, then obtains grid surface by MC (marching cube) method (or other similar methods).The benefit carrying out matching cloud data with implicit function has: the cloud data that can process noise, can fill up the cavity because undersampling causes, have implicit function in addition, can carry out CSG operation etc. to cloud data.Its weak point: the equipotential surface of implicit function may produce unnecessary curved surface, these class methods, in process complex boundary, can be had any problem when itself having the cloud data in cavity.

From the angle of interpolation cloud data, triangulation is carried out to cloud data, directly set up topological connection relation between points, thus obtain a grid surface.In these class methods, modal algorithm has dough sheet to reject algorithm and region growing algorithm.So-called dough sheet is rejected algorithm and is referred to, first triangulation (general employing Delaunay triangulation) is carried out to cloud data, obtain the set of a triangular plate, reject undesirable triangular plate according to certain rule, remaining triangular plate obtains the reconstruction grid of a cloud through suitable post-processed.This kind of interpolation algorithm be applicable to not having noise or noise smaller, have the cloud data of complex boundary.The cloud data reconstruction effect of method disadvantage to band noise of interpolation type is undesirable, and in addition for nonuniform sampling, and the some cloud gathered has the situation of disappearance not process very well.

Summary of the invention

(1) technical matters that will solve

In the application such as reverse-engineering and 3 D scene rebuilding, often need the three-dimension curved surface recovering scene or model, and the data that existing collecting device collects are with the form of a cloud, this just needs reconstruction algorithm to reconstruct three-dimension curved surface using a cloud as input, and three-dimension curved surface represents with the form of triangular mesh often.Rebuilding triangular mesh by a cloud is an inverse problem, and the noise in cloud data, collection density and sampling uniformity coefficient etc. all have larger impact to last reconstruction; In numerous applications, the feature on the good reserving model of the next energy of the three-dimension curved surface of reconstruction is often needed; It is closed model that existing many methods only can process topological structure.

The present invention preferably resolves by the following technical matters in a cloud reconstruction triangular mesh problem:

1, the present invention can process the cloud data with error, nonuniform sampling.

2, the present invention's triangular mesh of rebuilding out can the feature of good reserving model.

3, the approximate error of the present invention's triangular mesh of rebuilding out and model is little

4, the topological structure of the present invention to handled model does not limit.

Given this, fundamental purpose of the present invention be to provide a kind of robustness based on dictionary learning by the method for the direct reconstruction of three-dimensional curved surface of a cloud, according to the method, user from the three dimensional point cloud with noise and exceptional value, can reconstruct the curved surface possessing original geometry characteristic sum details.

(2) technical scheme

For achieving the above object, the invention provides a kind of method by the direct reconstruction of three-dimensional curved surface of a cloud, comprising:

Steps A: input with the cloud data P of noise and exceptional value and may need the number of vertices m rebuilding curved surface;

Step B: initialization dictionary matrix V and initialization connection matrix B;

Step C: upgrade dictionary matrix V and connection matrix B iteratively, until convergence;

Step D: export the triangular mesh rebuild, complete the reconstruction of three-dimension curved surface.

In such scheme, the dictionary of initialization described in step B matrix V is, with existing Points Sample method, input cloud data P is sampled as m point, specifically comprise: carry out resampling to the cloud data P of input, putting cloud number after making to sample is m, in this, as initial dictionary matrix V.Existing Points Sample method described in step B is the some cloud method for resampling of Poisson disk sampling method, variable density disk sampling method, stochastic sampling method or limit perception.

In such scheme, the connection matrix of initialization described in step B B is according to the projection energy criterion defined and stream shape constraint criterion, constructs triangle gridding, in this, as initial connection matrix B with the cloud data P after resampling, specifically comprise: initialization grid M, for cloud data P mid point p _i, in dictionary matrix V, find the k nearest apart from it point construct a triangle sets T (p _i), wherein 10≤k≤15, this triangle sets T (p _i) contain at most individual triangle; By this triangle sets T (p _i) in there is minimum projection's energy triangle add in this grid M, if the grid M after upgrading remains stream shape, then this triangle is exactly sampled point p _iinitial corresponding triangle, otherwise continue to choose the triangle with least energy from remaining triangle, repeat this operation until this triangle sets is grid M that is empty or that upgrade remain stream shape; Once choose triangle, the column vector b of connection matrix B _iwill by solving as shown in the formula carrying out sub-initialization:

$d(p_i,f)=\left|\right|p_i-p_i^{\′}\left|\right|=\underset{\mathrm{\α},\mathrm{\β},\mathrm{\γ}\≥0}{\underset{\mathrm{\α}+\mathrm{\β}+\mathrm{\γ}=1}{\min }}\left|\right|p_i-(\mathrm{\α}{v}_r+\mathrm{\β}{v}_s+\mathrm{\γ}{v}_t)\left|\right|$

Here p ' _i=α ^*v _r+ β ^*v _s+ γ ^*v _tp _ito the subpoint of triangle f, (α ^*, β ^*, γ ^*) be the barycentric coordinates corresponding to f.

In such scheme, the projection energy criterion of described definition is original sample point p _ito the projection energy E (p of triangle f _i, f), described by following formula:

E(p _i，f)＝E _appr(p _i，f)+ω _eE _edge(f)+ω _nE _normal(f)，

Original sample point p in formula _ito the distance E of the triangle f nearest apart from it _appr(p _i, f)=| d (p _i, f) | ^qweigh the l of input point cloud to the distance of reconstruction curved surface _{2, q}mould, be regular terms, make triangle f as far as possible close to equilateral triangle, thus make triangle gridding quality high as far as possible, normal direction regular terms, with making f normal direction and input point cloud p when normal direction information on the some cloud of input _inormal direction consistent as far as possible, wherein e _ithree limits of triangle f, np _ioriginal sample point p _inormal vector.

In such scheme, described step C comprises:

Step C1: fixing dictionary matrix V, calculates the limit e in current triangle gridding _ithe sampled point projection energy sum E (e that corresponding triangle comprises _i), construct a Priority Queues Q, deposit all pairing element (e _i, E (e _i));

Step C2: if queue Q non-NULL, selects E (e in Q _i) maximum element, carry out limit recurrence update algorithm;

Step C3: when queue Q is empty set, carries out triangle and deletes detection, under the condition not destroying stream shape character, deletes the triangle not having corresponding sampled point;

Step C4: be fixedly connected with matrix B, upgrades dictionary element location matrix V, utilizes alternating direction multiplier method, the optimization problem of correspondence is resolved into two sub-problem solvings.

In such scheme, the limit recurrence update algorithm in described step C2 also comprises:

If e _ibe internal edges, edge flip detection is carried out to it, if E (e after exchanging _i) value reduces, and this operation carry out after this grid remain manifold structure and then carry out swap operation;

If e _ibe boundary edge, virtual triangle carried out to it and adds detection, if E (e after inserting triangle _i) value reduction, then carry out adding triangle operation;

Above-mentioned two kinds of operations once occur, then upgrade corresponding projection energy, and to e _iall neighbours carry out while recurrence update algorithm, otherwise termination algorithm.

(3) beneficial effect

Kind provided by the invention, by the method for the direct reconstruction of three-dimensional curved surface of a cloud, directly rebuilds triangular mesh by a cloud, well can remove the noise in input point cloud, to nonuniform sampling robust and the feature that can well recover in a cloud simultaneously.

Compare implicit method, the present invention has the following advantages:

1, the present invention directly rebuilds triangular mesh by a cloud, avoids the cumulative errors that the multistep optimization problem in implicit method is brought.

2, the inventive method point cloud normal direction information is not required, and in implicit method with towards normal direction information be necessary, but to band towards a normal estimation inherently very difficult problem, and in the place that feature is larger, the estimation of normal direction often has error, and this brings larger error just may to best curve reestablishing.

Compare combined method, the present invention has the following advantages:

1, the present invention is using reconstruction error as objective function, and the reconstruction error of the curved surface making the present invention rebuild out often can be little than existing combined method.

2, the cloud data collected is often with larger noise, and sampled point non-uniform Distribution, and the present invention well can process these class data, and existing combined method is very responsive to this kind of data.

Accompanying drawing explanation

Fig. 1, the implementation process schematic diagram of reverse-engineering in prior art;

Fig. 2, left figure are input point cloud, and right figure is the three-dimension curved surface after reconstruction;

Fig. 3, the method flow diagram by the direct reconstruction of three-dimensional curved surface of a cloud provided by the invention;

Fig. 4, according to the schematic diagram of the flow process of the curve reestablishing method of the embodiment of the present invention;

Fig. 5, according to the schematic diagram of the sparse coding algorithm flow of the embodiment of the present invention;

Fig. 6, according to the schematic diagram of the sparse coding algorithm flow of the embodiment of the present invention;

Fig. 7, the schematic diagram that the edge flip according to the embodiment of the present invention operates;

Fig. 8, the process of boundary edge: left figure is leg-of-mutton situation for needs insert, right figure does not need to insert leg-of-mutton situation;

Fig. 9, the sparse coding algorithm according to the embodiment of the present invention acts on the example schematic on square model;

Figure 10, according to the schematic diagram of the dictionary updating flow process based on alternating direction multiplier method of the embodiment of the present invention;

Figure 11, what the curve reestablishing method according to the embodiment of the present invention was shown for faceform increases along with iterations, and reconstruction quality improves gradually;

Figure 12, according to the experimental result of the curve reestablishing method of the embodiment of the present invention and the contrast schematic diagram with screened poisson method;

Figure 13, according to the experimental result of the curve reestablishing method of the embodiment of the present invention and the contrast schematic diagram with additive method;

Figure 14, according to the experimental result of the curve reestablishing method of the embodiment of the present invention and the contrast schematic diagram with additive method;

Figure 15, according to the experimental result of the curve reestablishing method of the embodiment of the present invention and the contrast schematic diagram with additive method.

Embodiment

For making the object, technical solutions and advantages of the present invention clearly understand, below in conjunction with specific embodiment, and with reference to accompanying drawing, the present invention is described in more detail.

Technical thought of the present invention is such: input needs the cloud data (may with noise and exceptional value) rebuild and the number of vertices m rebuilding curved surface, utilize existing some cloud method for resampling, resampling is carried out to the some cloud of input, the point cloud number after sampling is made to be m, in this, as initial dictionary matrix V, according to the criterion of definition, with the some cloud structure triangle gridding after resampling, in this, as initial connection matrix B.

The error of rebuilding curved surface and original input point cloud is defined as by the present invention: original point cloud is to the leg-of-mutton minimum projection distance that 3 dictionary element are formed, in order to make this error minimum, the present invention proposes the Optimized model based on dictionary learning, upgrade dictionary element (rebuilding curved surface vertex position) and annexation (triangle gridding) iteratively, simultaneously, in order to noise and exceptional value data robust, the present invention adopts l _{2, q}the distance error tolerance of (0 < q < 1) mould.Finally, in order to the triangle gridding of outputting high quality, the present invention adds related constraint in Optimized model.In renewal annexation process, the present inventor introduces a kind of sparse coding algorithm upgraded based on limit, mainly through three kinds of operations: exchange limit, delete triangle, add triangle, upgrade annexation, ensure that the triangle gridding exported is manifold structure simultaneously.

As shown in Figure 3, Fig. 3 is the method flow diagram by the direct reconstruction of three-dimensional curved surface of a cloud provided by the invention, and the method comprises the following steps:

Step 31: input with the cloud data P of noise and exceptional value and may need the number of vertices m rebuilding curved surface.

Step 32: initialization dictionary matrix V and initialization connection matrix B;

In this step, utilize existing Points Sample method, resampling is carried out to input point cloud, putting cloud number after making to sample is m, in this, as initial dictionary V, according to projection energy and the stream shape constraint criterion of definition, with the some cloud structure triangle gridding after resampling, in this, as initial connection matrix B.

Step 321: initialization dictionary matrix V, is, with existing Points Sample method, input cloud data P is sampled as m point, specifically comprises: carry out resampling to the cloud data P of input, and putting cloud number after making to sample is m, in this, as initial dictionary matrix V.Existing Points Sample method described in step B is Poisson disk sampling method, variable density disk sampling method, stochastic sampling method, the some cloud method for resampling of limit perception or other method for resampling.

Step 322: initialization connection matrix B: be the projection energy criterion according to definition and stream shape constraint criterion, triangle gridding is constructed with the cloud data P after resampling, in this, as initial connection matrix B, specifically comprise: initialization grid M, for cloud data P mid point p _i, in dictionary matrix V, find the k nearest apart from it point construct a triangle sets T (p _i), wherein 10≤k≤15, this triangle sets T (p _i) contain at most individual triangle; By this triangle sets T (p _i) in there is minimum projection's energy triangle add in this grid M, if the grid M after upgrading remains stream shape, then this triangle is exactly sampled point p _iinitial corresponding triangle, otherwise continue to choose the triangle with least energy from remaining triangle, repeat this operation until this triangle sets is grid M that is empty or that upgrade remain stream shape; Once choose triangle, the column vector b of connection matrix B _iwill by solving as shown in the formula carrying out sub-initialization:

$d(p_i,f)=\left|\right|p_i-p_i^{\&prime;}\left|\right|=\underset{\mathrm{\&alpha;},\mathrm{\&beta;},\mathrm{\&gamma;}\&GreaterEqual;0}{\underset{\mathrm{\&alpha;}+\mathrm{\&beta;}+\mathrm{\&gamma;}=1}{\min }}\left|\right|p_i-(\mathrm{\&alpha;}{v}_r+\mathrm{\&beta;}{v}_s+\mathrm{\&gamma;}{v}_t)\left|\right|$

Here p ' _i=α ^*v _r+ β ^*v _s+ γ ^*v _tp _ito the subpoint of triangle f, (α ^*, β ^*, γ ^*) be the barycentric coordinates corresponding to f.

Wherein, the projection energy criterion of described definition is original sample point p _ito the projection energy E (p of triangle f _i, f), described by following formula:

E(p _i，f)＝E _appr(p _i，f)+ω _eE _edge(f)+ω _nE _normal(f)，

Original sample point p in formula _ito the distance E of the triangle f nearest apart from it _appr(p _i, f)=| d (p _i, f) | ^qweigh the l of input point cloud to the distance of reconstruction curved surface _{2, q}mould, be regular terms, make triangle f as far as possible close to equilateral triangle, thus make triangle gridding quality high as far as possible, normal direction regular terms, with making f normal direction and input point cloud p when normal direction information on the some cloud of input _inormal direction consistent as far as possible, wherein e _ithree limits of triangle f, np _ioriginal sample point p _inormal vector.

Step 33: upgrade dictionary matrix V and connection matrix B iteratively, until convergence;

Step 331: fixing dictionary matrix V, calculates the limit e in current triangle gridding _ithe sampled point projection energy sum E (e that corresponding triangle comprises _i), construct a Priority Queues Q, deposit all pairing element (e _i, E (e _i)).

Step 332: if queue Q non-NULL, selects E (e in Q _i) maximum element, carry out limit recurrence update algorithm;

Step 332-1: if e _ibe internal edges, edge flip detection is carried out to it, if E (e after exchanging _i) value reduces, and this operation carry out after this grid remain manifold structure and then carry out swap operation; If e _ibe boundary edge, virtual triangle carried out to it and adds detection, if E (e after inserting triangle _i) value reduction, then carry out adding triangle operation; Above-mentioned two kinds of operations once occur, then upgrade corresponding projection energy, and to e _iall neighbours carry out while recurrence update algorithm, otherwise termination algorithm.

Step 333: when queue Q is empty set, carries out triangle and deletes detection, under the condition not destroying stream shape character, deletes the triangle not having corresponding sampled point.

Step 334: be fixedly connected with matrix B, upgrades dictionary element location matrix V, utilizes alternating direction multiplier method, the optimization problem of correspondence is resolved into two sub-problem solvings.

Step 34: export the triangular mesh rebuild, complete the reconstruction of three-dimension curved surface.

Adopt such scheme of the present invention, the high-quality triangle mesh curved surface of possessing characteristic sum details can be reconstructed by three dimensional point cloud (with noise and exceptional value).

Based on the technical scheme that the invention described above provides, be described in greater detail below in conjunction with specific embodiment.The invention provides a kind of robustness curve reestablishing method based on dictionary learning, according to the method, by the three dimensional point cloud with noise and exceptional value, the curved surface with raw data characteristic sum details can be reconstructed.

As shown in Figure 4, input three-dimensional point cloud P (with noise and exceptional value), utilize existing Points Sample method, be m point by P resampling, in this, as initial dictionary V, according to projection energy and the stream shape constraint criterion of the present invention's definition, with the some cloud after sampling, namely initial dictionary V constructs triangle gridding, as initial connection matrix B.

The reconstruction curved surface defined according to the present invention and the approximate error of original point cloud:

$E_{\mathrm{appr}}=\frac{1}{n}{\left|\right|z\left|\right|}_{2,q}=\frac{1}{n}{\left|\right|P-\mathrm{VB}\left|\right|}_{2,q}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}{\left|\right|p_i-{\mathrm{Vb}}_i\left|\right|}_2^q$

It is original point cloud to the leg-of-mutton minimum projection distance that 3 dictionary element are formed, corresponding sparse constraint is:

||b _i|| ₀≤3，||b _i|| ₁＝1，b _i≥0

Wherein b _ifor the column vector of connection matrix B.

B should meet the constraint of stream shape: B ∈ MT

In order to make this error minimum, the present invention proposes the bent Optimized model (formula (1)) of robustness based on dictionary learning:

$\underset{V,B}{\min }E=E_{\mathrm{appr}}+E_{\mathrm{reg}}$

s.t.||b _i|| ₀≤3，||b _i|| ₁＝1，b _i≥0

B∈MT

Wherein be regular terms, obtain high-quality triangle mesh curved surface by regularization triangle, here, e _irepresent the limit of triangle gridding M, 1 is the number on limit, and E is the matrix storing side information.

If the three-dimensional point cloud P of input is with reliable normal direction information, all right addition method of the present invention is to regularization term:

$E_{\mathrm{normal}}(p_i,f)=\frac{1}{3n}\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}\underset{e_i\&Element;f}{\mathrm{\&Sigma;}}{(e_i\&CenterDot;n_{p_i})}^2$

Make triangle f normal direction and corresponding input point cloud p _inormal direction consistent as far as possible.

Based on above Optimized model, the present invention upgrades dictionary element (some cloud position) and annexation (triangle gridding) iteratively until convergence, and the high-quality triangle gridding of raw data characteristic sum details is possessed in finally output.

When dictionary V fixes, the present invention utilizes the sparse coding algorithm based on limit to upgrade corresponding connection matrix B, as shown in Figure 5, with the limit e of current grid M _iand the projection energy of correspondence and value for pairing element, set up queue Q, and deposit all pairing element (e _i, E (e _i)), each iteration, deletes first element with maximal projection energy in Q, according to the limit e of this element _itype, carry out recurrence limit update process, as shown in Figure 4: for e _ithe situation of internal edges, if E is (e _i) reduce after carrying out edge flip, and the grid after edge flip remains manifold structure, then to e _icarry out shown in swap operation (as Fig. 5); For e _ithe situation of boundary edge, by e _itwo boundary edge adjacent with it are connected respectively, obtain two virtual triangles (as shown in Figure 6), if correspondence comprise e _ileg-of-mutton sampled point, can project on virtual triangle, and projection energy reduces, then join in current grid M using the virtual triangle with less projection energy as new triangle, meanwhile, pairing element corresponding to newly-generated limit joins in queue Q.For both of these case, as E (e _i) time, corresponding sampled point, limit and leg-of-mutton projection energy all need to upgrade, limit e _ineighbours limit also carry out upgrading (shown in the left figure of figure and Fig. 8 as left in Fig. 7) according to the descending of new projection energy.

As shown in Figure 9, these operations based on limit can reduce the target energy of Optimized model of the present invention, and by this recursive strategies, these partial operations can be spread to other regions of grid.

When queue Q is empty set, algorithm carries out triangle and deletes detection, under the condition not destroying stream shape character, deletes the triangle that those do not have corresponding sampled point.

The sparse coding algorithm based on limit that the present invention proposes has following character:

1. flow shape.Because edge flip and edge triangles insert the stream shape character that can't affect input grid, so current grid is stream shape in whole algorithmic procedure always.

2. energy reduces.Because the operation in algorithm only can reduce to correspond to the projection energy comprised when leg-of-mutton sampled point in front, and the projection energy of other points remains unchanged, so the total energy of Optimized model (formula (1)) is monotonic decreasing (as Fig. 7).

3. calculated amount validity.Experiment shows, the complexity of the recursive algorithm that limit upgrades is linear with the number on limit, and this is faster than common greedy algorithm.

When annexation B fixes, dictionary updating optimizes vertex position V exactly, is equivalent to and asks following problem (formula (2)):

$\underset{V}{\min }{E}_{\mathrm{appr}}+E_{\mathrm{reg}}$

Direct optimization (2) is very difficult, due to E _apprl containing non-differentiability _{2, q}mould, the present invention matrix Z replaces E _apprin P-VB, dictionary updating becomes (formula (3)):

$\underset{V,Z}{\min }F(V,Z)$

s.t.h(V，Z)＝0

Here, F (V, Z)=E _appr+ E _regand h (V, Z)=Z-P+VB.

The Augmented Lagrangian Functions of formula (3) is:

$L(V,Z,D)=F(V,Z)+\frac{\mathrm{\&gamma;}}{2}({\left|\right|h(V,Z)+\frac{D}{\mathrm{\&gamma;}}\left|\right|}_F^2-{\left|\right|\frac{D}{\mathrm{\&gamma;}}\left|\right|}_F^2)$

Wherein D is Lagrange multiplier.

The problems referred to above can be solved by alternating direction multiplier method, and as shown in Figure 10, wherein Z-subproblem is (formula (4)):

$\underset{z_i}{\min }\underset{i=1}{\overset{n}{\mathrm{\&Sigma;}}}({\left|\right|z_i\left|\right|}_2^q+\frac{\mathrm{n\&gamma;}}{2}{\left|\right|z_i-x_i\left|\right|}_2^2),$

Wherein, z _ii-th row of matrix Z, x _ibe i-th row.(4) can be decomposed into n minor issue, each minor issue can be reduced to a scalar problem:

$\underset{{\mathrm{\&alpha;}}_i}{\min }{\left|\right|{\mathrm{\&alpha;}}_i{x}_i\left|\right|}_2^q+\frac{\mathrm{n\&gamma;}}{2}{\left|\right|{\mathrm{\&alpha;}}_i{x}_i-x_i\left|\right|}_2^2$

Optimal value can be tried to achieve by a few step iteration.

V-subproblem is:

$\underset{V}{\min }\frac{{\mathrm{\&omega;}}_e}{l}{\left|\right|\mathrm{VE}\left|\right|}_F^2+\frac{{\mathrm{\&omega;}}_n}{3n}{\left|\right|\mathrm{VEC}\left|\right|}_F^2+\frac{\mathrm{\&gamma;}}{2}{\left|\right|Z-P+\mathrm{VB}+\frac{D}{\mathrm{\&gamma;}}\left|\right|}_F^2,$

This is a quadratic problem, can pass through solution linear system solution.The method of conjugate gradient that the present invention utilizes incomplete cholesky to decompose solves.

As shown in figure 11, the optimization method of alternating direction that proposes of the present invention can the trigonometric ratio of optimized reconstruction grid and vertex position iteratively.

As shown in Figure 12 to Figure 15, the robustness curve reestablishing method that the present invention proposes, not only to noise and exceptional value robust, and when possessing data characteristics and details, can also export the high-quality triangle gridding less with original point cloud error.

Above-described specific embodiment; object of the present invention, technical scheme and beneficial effect are further described; be understood that; the foregoing is only specific embodiments of the invention; be not limited to the present invention; within the spirit and principles in the present invention all, any amendment made, equivalent replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (7)

1., by a method for the direct reconstruction of three-dimensional curved surface of a cloud, it is characterized in that, comprising:

Steps A: input with the cloud data P of noise and exceptional value and may need the number of vertices m rebuilding curved surface;

Step B: initialization dictionary matrix V and initialization connection matrix B;

Step C: upgrade dictionary matrix V and connection matrix B iteratively, until convergence:

Step D: export the triangular mesh rebuild, complete the reconstruction of three-dimension curved surface.

2. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 1, it is characterized in that, the dictionary of initialization described in step B matrix V is, with existing Points Sample method, input cloud data P is sampled as m point, specifically comprise: resampling is carried out to the cloud data P of input, putting cloud number after making to sample is m, in this, as initial dictionary matrix V.

3. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 2, it is characterized in that, existing Points Sample method described in step B is the some cloud method for resampling of Poisson disk sampling method, variable density disk sampling method, stochastic sampling method or limit perception.

4. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 1, it is characterized in that, the connection matrix of initialization described in step B B is according to the projection energy criterion defined and stream shape constraint criterion, triangle gridding is constructed with the cloud data P after resampling, in this, as initial connection matrix B, specifically comprise:

Initialization grid M, for cloud data P mid point p _i, in dictionary matrix V, find the k nearest apart from it point construct a triangle sets T (p _i), wherein 10≤k≤15, this triangle sets T (p _i) contain at most individual triangle; By this triangle sets T (p _i) in there is minimum projection's energy triangle add in this grid M, if the grid M after upgrading remains stream shape, then this triangle is exactly sampled point p _iinitial corresponding triangle, otherwise continue to choose the triangle with least energy from remaining triangle, repeat this operation until this triangle sets is grid M that is empty or that upgrade remain stream shape; Once choose triangle, the column vector b of connection matrix B _iwill by solving as shown in the formula carrying out sub-initialization:

$d(p_i,f)=\left|\right|p_i-p_i^{\&prime;}\left|\right|=\underset{\mathrm{\&alpha;},\mathrm{\&beta;},\mathrm{\&gamma;}\&GreaterEqual;0}{\underset{\mathrm{\&alpha;}+\mathrm{\&beta;}+\mathrm{\&gamma;}=1}{\min }}\left|\right|p_i-(\mathrm{\&alpha;}{v}_r+\mathrm{\&beta;}{v}_s+\mathrm{\&gamma;}{v}_t)\left|\right|$

Here p ' _i=α ^*v _r+ β ^*v _s+ γ ^*v _tp _ito the subpoint of triangle f, (α ^*, β ^*, γ ^*) be the barycentric coordinates corresponding to f.

5. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 4, it is characterized in that, the projection energy criterion of described definition is original sample point p _ito the projection energy E (p of triangle f _i, f), described by following formula:

E(p _i，f)＝E _appr(p _i，f)+ω _eE _edge(f)+ω _nE _normal(f)，

Original sample point p in formula _ito the distance E of the triangle f nearest apart from it _appr(p _i, f)=| d (p _i, f) | ^qweigh the l of input point cloud to the distance of reconstruction curved surface _{2, q}mould, be regular terms, make triangle f as far as possible close to equilateral triangle, thus make triangle gridding quality high as far as possible, normal direction regular terms, with making f normal direction and input point cloud p when normal direction information on the some cloud of input _inormal direction consistent as far as possible, wherein e _ithree limits of triangle f, np _ioriginal sample point p _inormal vector.

6. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 1, it is characterized in that, described step C comprises:

Step C1: fixing dictionary matrix V, calculates the limit e in current triangle gridding _ithe sampled point projection energy sum E (e that corresponding triangle comprises _i), construct a Priority Queues Q, deposit all pairing element (e _i, E (e _i));

Step C2: if queue Q non-NULL, selects E (e in Q _i) maximum element, carry out limit recurrence update algorithm:

Step C3: when queue Q is empty set, carries out triangle and deletes detection, under the condition not destroying stream shape character, deletes the triangle not having corresponding sampled point;

Step C4: be fixedly connected with matrix B, upgrades dictionary element location matrix V, utilizes alternating direction multiplier method, the optimization problem of correspondence is resolved into two sub-problem solvings.

7. the method by the direct reconstruction of three-dimensional curved surface of a cloud according to claim 6, it is characterized in that, the limit recurrence update algorithm in described step C2 also comprises:

If e _ibe internal edges, edge flip detection is carried out to it, if E (e after exchanging _i) value reduces, and this operation carry out after this grid remain manifold structure and then carry out swap operation;

If e _ibe boundary edge, virtual triangle carried out to it and adds detection, if E (e after inserting triangle _i) value reduction, then carry out adding triangle operation;

Above-mentioned two kinds of operations once occur, then upgrade corresponding projection energy, and to e _iall neighbours carry out while recurrence update algorithm, otherwise termination algorithm.