CN104123696A - Focus and context visualization method based on multiresolution - Google Patents

Abstract

The invention relates to a focus and context visualization method based on multiresolution, and belongs to the field of computer graphics and visualization. According to the method, a two dimensional model is deformed on basis of focus and context visualization with minimal deformation. The method comprises firstly dividing the model into uniform grids with small resolution ratio, and judging which grids need to deform; subdividing the model into irregular grids with different resolution ratios according to intensity of model peaks; adding position constraint and boundary constraint; performing iteration solution on a linear system by using a Taucs function library; at least rebuilding road network data. Compared with prior art, the focus and context visualization method based on multiresolution can achieve the effect of rapid and small deformation when focus and context visualization is performed on the model.

Description

A kind of focus context method for visualizing based on multiresolution

Technical field

The present invention relates to a kind of focus context (Focus+Context, F+C) method for visualizing based on multiresolution, belong to computer graphics, visual field.

Background technology

Present a complicated model on screen in, resolution is limited all the time, and operating personnel can not see all details that need region-of-interest.If directly amplify in focusing region, can make model exceed screen border, can't see again an overall information.Therefore just draw focus context (Focus+Context) technology,, in finite baffle, can not only show the details of user's region-of-interest, can ensure that again user is to model entirety macroscopical perception.

Along with the development of Focus+Context technology, substantially can be divided into 2 kinds of methods: the deformation algorithm of the deformation algorithm based on mapping function and based target function optimization.Deformation algorithm based on mapping function has again two kinds of classical methods--radially amplification method of fish eye lens method and Gauss; The deformation algorithm of based target function optimization also has two classical methods--quadratic programming deformation method and deformation quantity Method for minimization.

Furnas in 1986 borrows and signs fish-eye view (Fish-eye view) effect, propose to have the flake algorithm of material impact power, its thought is the amplification factor amplification that user specifies of pressing to being positioned at focus area, utilizes the position of the extra-regional point of predefined mapping function focusing to shine upon simultaneously.

Existing mapping function can be divided into two classes: (1) global map function, and to being positioned at shining upon a little outside focus area.The mapping function that the people such as Harrie use, the magnification ratio of grid is along with reducing with the increase of focus area centre distance; (2) local mapping function, shines upon the point closing in the fixed range of focus area, and the position of other region point remains unchanged.The mapping function that Yamamoto etc. use, the position that is positioned at deformed region point obtains by Bezier (Bezier) interpolating function, makes it away from focus area.Neither belonging to the part that focus area do not belong to deformed region is yet context area, and the point that is positioned at context area keeps its invariant position.

The deformation algorithm based on mapping function of getting up early, everybody finds a very serious problem, the distortion that is exactly model is too serious.User's identification is caused to very large problem, and neither one judgment criteria to distortion result carry out qualitative assessment, realize Focus+Context technology so researcher has proposed the deformation algorithm of based target function.

The common technology that solves optimization problem is mathematical programming (Mathematical Programming).First practical problems is carried out to abstractdesription with some variablees, the function that is then these variablees by the object definition of optimization, i.e. objective function, need to satisfied constraint condition be defined as optimizing process equation or the inequality of these variablees.Researchist has carried out a large amount of research to mathematical programming problem, and the efficient solution that the problem of some particular types is provided, for example, LPs (Linear Pragrams) problem, CQP (Convex Quadratic Programs) problem.So the deformation algorithm of research based target function optimization, how key is defining variable, objective function reasonable in design and constraint condition, thus reach deformation effect both fast and sound.

2008, it is visual that the people such as Yu-Shuen Wang propose to utilize the minimum thought of deformation to realize Focus+Context to three-dimensional model.Wen Zhongxian, to square closure of modelling, divides uniform grid to this closure.The distortion of model points is converted into the distortion to grid.Mainly be divided into following step:

(1) uniform grid based on three-dimensional model is divided;

(2) divide Princple grid and Trivial grid;

(3) definition of based target function;

(4) add edge-restraint condition;

(5) objective function is carried out to iterative;

(6) road net data is rebuild;

The shortcoming of this method is that calculated amount is large.

2011, it is visual that Jan-Henrik Haunet carries out Focus+Context to road network.He is the distortion to road network, so think that the less effect of distortion on road is better.This algorithm is regarded road network as a figure, each section of road is exactly a limit in figure, adopt CQP (Convex Quadratic Program) method, objective function is defined as to the distortion that as far as possible reduces limit in figure, and avoid intersecting between limit and limit by adding constraint condition, thereby the layout on limit in road network structure figure is optimized.Although effect is fine, the time is slow especially, and particularly, in the time that model vertices is more, the time is how much multiples and increases.The shortcoming of the method is that deformation quantity is large, computing time is long.Deformation quantity refers to after model is out of shape, a tolerance of the model after distortion and master pattern otherness.

Summary of the invention

The object of the invention is the defect in order to overcome above-mentioned prior art, solve the model deformation occurring visual to model Focus+Context time large and the problem such as grow computing time, propose a kind of Focus+Context visualization technique based on multi-resolution grid.

The object of the invention is to be achieved through the following technical solutions.

A kind of Focus+Context method for visualizing based on multi-resolution grid of the present invention, its concrete steps are as follows:

Step 1, obtain the original point cloud data of destination object.

Described original point cloud data is two-dimentional cloud data, comprises abscissa value (x represents with symbol) and ordinate value (y represents with symbol) a little.

Step 2, generation uniform grid.

On the basis of the original point cloud data of the destination object obtaining in step 1, generate uniform grid.Concrete operation step is:

Step 2.1: by formula (1), the original point cloud data of destination object is carried out to unit processing, the unitization of obtaining cloud data.

$\left\{\begin{array}{c}{x}^{\′}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}\\ y^{\′}=\frac{y-y_{\min }}{y_{\max }-y_{\min }}\end{array}\right.---\left(1\right)$

Wherein, x ' is each some unitization abscissa value after treatment in original point cloud data; x _maxit is the horizontal ordinate maximal value of original point cloud data mid point; x _minit is the horizontal ordinate minimum value of original point cloud data mid point; Y ' is each some unitization ordinate value after treatment in original point cloud data; y _maxit is the ordinate maximal value of original point cloud data mid point; y _minit is the ordinate minimum value of original point cloud data mid point.

Step 2.2: the unit space of the two-dimensional coordinate system at unit cloud data place is divided into n × n uniform grid, wherein, n=2 ^m, m is positive integer, m value is by artificially determining.Obtain the grid vertex coordinate of uniform grid simultaneously, and uniform grid is numbered in order.Four apex coordinates of each grid are used respectively (x _v1, y _v1), (x _v2, y _v2), (x _v3, y _v3) and (x _v4, y _v4) represent.Described unit space is to be 0 and 1 by abscissa value, and ordinate value is the rectangle that four straight lines of 0 and 1 surround.

Through the operation of step 2, obtain uniform grid.

Step 3, unit cloud data is carried out to pretreatment operation.

On the basis of step 2 operation, by formula (2), unit cloud data is carried out to pretreatment operation, the mapping relations in the unitization of obtaining cloud data between each point and its place grid.

$\left\{\begin{array}{c}{x}^{\′}=w_1{x}_{v1}+w_2{x}_{v2}+w_3{x}_{v3}+w_4{x}_{v4}\\ y^{\′}=w_1{y}_{v1}+w_2{y}_{v2}+w_3{y}_{v3}+w_4{y}_{v4}\\ 1=w_1+w_2+w_3+w_4\end{array}\right.---\left(2\right)$

Wherein, w _ifor the grid vertex (x of current unit cloud data point (x ', y ') and point (x ', y ') place grid _vi, y _vi) related coefficient; I is positive integer, i ∈ [Isosorbide-5-Nitrae].

By the operation of step 3, can the unitization of obtaining cloud data point the grid vertex (x of (x ', y ') and point (x ', y ') place grid _vi, y _vi) related coefficient w _i.

Step 4, obtain multi-resolution grid.

On the basis of step 3 operation, obtain multi-resolution grid.Concrete steps are:

Step 4.1: determine the threshold value (σ represents with symbol) of counting, sum is the quantity of original point cloud data mid point.

The described method of determining the threshold value σ that counts is: statistic procedure three obtains the quantity of unit cloud data point in each uniform grid, and therefrom maximizing, using this maximal value as the threshold value σ that counts.

Step 4.2: the unit space of two-dimensional coordinate system is divided into 2 × 2 grid, obtains segmenting the length of side of grid, l represents with symbol.

Step 4.3: judge successively the quantity (s represents with symbol) of unit cloud data point in each segmentation grid, if s >=σ, and l > 1/n, continue this segmentation grid to be divided into 2 × 2 grid.Each segmentation grid repeats the operation of this step, until can not be subdivided.

By the operation of above-mentioned steps, obtain multi-resolution grid.

Step 5, the grid in multi-resolution grid is divided into effectively (Principle) grid and redundancy (Trivial) grid.

On the basis of step 4 operation, the total-grid in multi-resolution grid is divided into effectively (Principle) grid and redundancy (Trivial) grid.Concrete operation method is: the multi-resolution grid that step 4 is obtained travels through, if there is no any point in original point cloud data in current grid, is redundancy grid by this grid mark; Otherwise, be effective grid by this grid mark.

Step 6, objective definition function.

On the basis of step 5 operation, objective definition function.Concrete steps are:

Use symbol c _krepresent any grid in the multi-resolution grid that obtains of step 4, k represents the index of each grid in multi-resolution grid that step 4 obtains, 1≤k≤K, and K represents the number of grid in multi-resolution grid; With symbol c ' _krepresent grid c in multi-resolution grid _kgrid after distortion; C ' _k=s _kc _k, wherein, s _kbe the equal proportion scaled matrix of 2 × 2, represent grid c _kdeformation coefficient.

$s_k=\left(\begin{array}{cc}{s}_{k,x}& \\ & s_{k,y}\end{array}\right)---\left(3\right)$

Wherein, s _k,xand s _k,yrepresent respectively grid c _kthe x on summit, y coordinate zoom factor; s _k,x=s _k,y.

The objective function of distortion of the mesh is defined as:

||C′-SC|| ²＝0 (4)

Wherein, C represents all grids in multi-resolution grid that step 4 obtains; S represents the equal proportion scaled matrix of all grids; Grid in C ' expression multi-resolution grid after all distortions of the mesh.

Because multi-resolution grid is divided into effective grid and redundancy grid, therefore only need to calculate effective grid, objective function is reduced to:

||C′ _P-S _PC _P|| ²＝0 (5)

Wherein, C _pall effective grids in the multi-resolution grid that expression step 5 obtains; S _prepresent the equal proportion scaled matrix of all effective grids; C ' _prepresent the grid after all effective grid distortion in multi-resolution grid.

Use symbol represent that in multi-resolution grid, any one grid in all effective grids (is used symbol c _prepresent) e article of limit, e ∈ 0,1,2,3}, v _p1and v _p2represent respectively termination summit and the initial vertex on e article of limit; In multi-resolution grid, any one grid in all effective grids can be expressed as the set on limit

By grid c _pwith vertex representation be c _p=q _pv _p, wherein, v _prepresent grid c _pvertex matrix; q _pfor the representing matrix on limit, q _pelement value in matrix is as follows:

$q_p^{u,v}=\left\{\begin{array}{ccc}1& \mathrm{if}& u=e,v=p1\\ -1& \mathrm{if}& u=e,v=p2\\ 0& \mathrm{otherwise}& \end{array}\right.---\left(6\right)$

Wherein, represent q _pin matrix, u is capable, the value of the element of v row; { 0,1,2,3}, v is positive integer to u ∈.

Use Q _prepresent the representing matrix on the limit of all effective grids in multi-resolution grid, all effective grid set C in multi-resolution grid _pcan be expressed as C _p=Q _pv _p, the objective function that formula (5) represents like this can use formula (7) to represent.

||Q _PV′ _P-S _PQ _PV _P|| ²＝0 (7)

Wherein, V _prepresent all effective grid C in multi-resolution grid _pvertex matrix; V ' _prepresent all effective grid C in multi-resolution grid _pvertex matrix after distortion.

In the process of being out of shape, the mesh space of the absolute coordinates of finding only to use a summit after to whole distortion positions, and can cause model " shake ", so will be that each grid vertex point of addition retrains." shake " of described model refers to the overall offset of mesh space with respect to original position.

ω||IV′ _P-V _P|| ²＝0 (8)

Wherein, the intensity that ω is position constraint, ω is artificial setting value, ω ∈ [0,1]; I is and V ' _pthe unit matrix that exponent number is identical.

After conformity goal function and position constraint condition, the objective function that can be improved, as shown in formula (9).

||Q _PV′ _P-S _PQ _PV _P|| ²+ω||IV′ _P-V _P|| ²＝0 (9)

Add net boundary constraint condition and allow each grid vertex remain in border, obtain formula (10), formula (10) is objective function.

$\left\{\begin{array}{c}{\left|\right|Q_P{V}_P^{\′}-S_P{Q}_P{V}_P\left|\right|}^2+\mathrm{\ω}{\left|\right|{\mathrm{IV}}_P^{\′}-V_P\left|\right|}^2=0^2\\ x_l\≤v_x\≤x_h,y_l\≤v_y\≤y_h\end{array}\right.---\left(10\right)$

Wherein, v _xand v _ythe x coordinate and the y that represent respectively the summit of any grid in all effective grids in multi-resolution grid sit target value; x _land y _lrepresent respectively the minimum value of x coordinate and the minimum value of y coordinate in all summits of multi-resolution grid; x _hand y _hrepresent respectively the maximal value of x coordinate and the maximal value of y coordinate in all summits of multi-resolution grid.

Step 7, objective function is solved.

On the basis of step 6 operation, formula (10) is solved.Concrete operation step is:

Step 7.0: formula (10) is expressed as to formula (11).

AV′ _P＝b(V _P)(11)

Wherein, $A={[Q_P^T,{\mathrm{\ωI}}^T]}^T,$ $b\left(V_P\right)={[{\left(S_P{Q}_P{V}_P\right)}^T,{\mathrm{\ωV}}_P^T]}^T.$

A is multiplied by formula (11) equal sign both sides simultaneously ^t, be converted to and solve linear system:

(A ^TA)V′ _P＝A ^Tb(V _P) (12)

Step 7.1: if iteration, V for the first time _pget initial value, simultaneously according to V _pinitial value, calculate the initial area of each effective grid and (use symbol M _prepresent); Otherwise V is set _p=V ' _p, V ' _pfor the result obtaining in step 7.4, and according to current V _pvalue, calculate the effective grid after each distortion area (with symbol M ' _prepresent).

Step 7.2: all effective grid c in the multi-resolution grid that calculation procedure four obtains _pzoom factor (use symbol f _prepresent).If this effective grid is in focus area, the zoom factor of this effective grid is the artificial enlargement factor of setting (representing λ > 1 by sign of lambda), i.e. f _p=λ.If this effective grid, not in focus area, calculates the zoom factor of this effective grid according to formula (13).Described focus area is by the artificial region of specifying of user, and therefore non-focus area is the region beyond focus area in view.According to all effective grid c in the multi-resolution grid calculating _pzoom factor, draw zoom factor matrix S _p.

$f_p={(M_p^{\′}/M_p)}^{1/2}{I}^{\′}---\left(13\right)$

Wherein, I ' is 2 × 2 unit matrixs.

Step 7.3: the zoom factor matrix S of calculating according to step 7.2 _p, calculate b (V _p), $b\left(V_P\right)={[{\left(S_P{Q}_P{V}_P\right)}^T,{\mathrm{\ωV}}_P^T]}^T.$

Step 7.4: by the 7.3 b (V that calculate _p), be brought into formula (12), calculate V ' _p.If V ' _prestrain or iterations reaches preset value, stopped iteration; Otherwise repeating step 7.1 is to the operation of step 7.4.

Step 8, reconstruction model.

On the basis of step 7 operation, the V ' that uses step 7 to obtain _preconstruction model, obtains the two dimensional model after distortion.

Beneficial effect

Compared with the prior art, the inventive method can, model being carried out to Focus+Context visual time, reach fast and the less effect of deformation quantity.

Brief description of the drawings

Fig. 1 is State of Massachusetts, US Boston (Boston) city road network original graph in the specific embodiment of the invention; Wherein, region A is focus area;

Fig. 2 is the operational flowchart of the Focus+Context method for visualizing based on multiresolution in the specific embodiment of the invention;

Fig. 3 is the inventive method and existing method amplification effect comparison diagram in the embodiment of the present invention; Wherein, Fig. 3 (a) uses the method for Yamamoto to amplify the focus area A in Fig. 1, the amplification effect figure obtaining; Fig. 3 (b) uses the method for Haunert to amplify the focus area A in Fig. 1, the amplification effect figure obtaining; Fig. 3 (c) uses the inventive method to amplify the focus area A in Fig. 1, the amplification effect figure obtaining.

Embodiment

According to technique scheme, below in conjunction with drawings and Examples, the present invention is described in detail.

In the present embodiment, State of Massachusetts, US Boston (Boston) city road network is carried out to modeling, boston city road network original graph as shown in Figure 1.Boston city road net data comprises 3379 roads and 2852 summits.

A kind of Focus+Context visualization technique based on multi-resolution grid that uses the present invention to propose amplifies the focus area of boston city road network original graph, and focus area is as the a-quadrant in Fig. 1, and as shown in Figure 2, concrete operation step is its operating process:

Step 1, obtain the original point cloud data of destination object.

Described original point cloud data is two-dimentional cloud data, comprises abscissa value x and ordinate value y a little.

Step 2, generation uniform grid.

On the basis of the original point cloud data of the destination object obtaining in step 1, generate uniform grid.Concrete operation step is:

Step 2.1: by formula (1), the original point cloud data of destination object is carried out to unit processing, the unitization of obtaining cloud data.In formula (1), x _min=-71.0740, x _max=-71.0490; y _min=42.3505, y _max=42.3710.

Step 2.2: the unit space of the two-dimensional coordinate system at unit cloud data place is divided into n × n uniform grid, and wherein, n=32, obtains the grid vertex coordinate of uniform grid simultaneously, and uniform grid is numbered in order.Four apex coordinates of each grid are used respectively (x _v1, y _v1), (x _v2, y _v2), (x _v3, y _v3) and (x _v4, y _v4) represent.Described unit space is to be 0 and 1 by abscissa value, and ordinate value is the rectangle that four straight lines of 0 and 1 surround.

Through the operation of step 2, obtain uniform grid.

Step 3, unit cloud data is carried out to pretreatment operation.

On the basis of step 2 operation, by formula (2), unit cloud data is carried out to pretreatment operation, the mapping relations in the unitization of obtaining cloud data between each point and its place grid.

By the operation of step 3, can the unitization of obtaining cloud data point the grid vertex (x of (x ', y ') and point (x ', y ') place grid _vi, y _vi) related coefficient w _i.

Step 4, obtain multi-resolution grid.

On the basis of step 3 operation, obtain multi-resolution grid.Concrete steps are:

Step 4.1: determine the threshold value σ that counts, sum is the quantity of original point cloud data mid point, sum=2852, σ=56.

Step 4.2: the unit space of two-dimensional coordinate system is divided into 2 × 2 grid, obtains segmenting the length of side l of grid.

Step 4.3: judge successively the quantity s of unit cloud data point in each segmentation grid, if s >=σ, and l > 1/n, continue this segmentation grid to be divided into 2 × 2 grid.Each segmentation grid repeats the operation of this step, until can not be subdivided.

By the operation of above-mentioned steps, obtain multi-resolution grid.

Step 5, the grid in multi-resolution grid is divided into effective grid and redundancy grid.

On the basis of step 4 operation, the total-grid in multi-resolution grid is divided into effective grid and redundancy grid.

Step 6, objective definition function.

On the basis of step 5 operation, objective definition function, as shown in formula (10), ω=0.001 in formula (10)

Step 7, objective function is solved.

On the basis of step 6 operation, formula (10) is solved.Concrete operation step is:

Step 7.0: formula (10) is expressed as to formula (11), A is multiplied by formula (11) equal sign both sides simultaneously ^t, be converted to and solve linear system, as shown in formula (12).

Step 7.1: if iteration, V for the first time _pget initial value, simultaneously according to V _pinitial value, calculate the initial area M of each effective grid _p; Otherwise V is set _p=V ' _p, V ' _pfor the result obtaining in step 7.4, and according to current V _pbe worth, calculate the area M ' of the effective grid after each distortion _p.

Step 7.2: all effective grid c in the multi-resolution grid that calculation procedure four obtains _pzoom factor f _p.If this effective grid is in focus area, the zoom factor of this effective grid is artificial enlargement factor λ=3 of setting, i.e. f _p=3.If this effective grid, not in focus area, calculates the zoom factor of this effective grid according to formula (13).Described focus area is by the artificial region of specifying of user, and therefore non-focus area is the region beyond focus area in view.According to all effective grid c in the multi-resolution grid calculating _pzoom factor, draw zoom factor matrix S _p.

Step 7.3: the zoom factor matrix S of calculating according to step 7.2 _p, calculate b (V _p), $b\left(V_P\right)={[{\left(S_P{Q}_P{V}_P\right)}^T,{\mathrm{\ωV}}_P^T]}^T.$

Step 7.4: by the 7.3 b (S that calculate _p), be brought into formula (12), calculate V ' _p.If V ' _prestrain or iterations reaches preset value, stopped iteration; Otherwise repeating step 7.1 is to the operation of step 7.4.

Step 8, reconstruction model.

On the basis of step 7 operation, the V ' that uses step 7 to obtain _preconstruction model, obtains the two dimensional model after distortion.

Through the operation of above-mentioned steps, the amplification effect figure obtaining is as shown in Fig. 3 (c).

For effect of the present invention is described, use respectively the method for Yamamoto and the method for Haunert to amplify identical focus area, the amplification effect figure obtaining is as shown in Fig. 3 (a) and Fig. 3 (b).The computing time of 3 kinds of methods is as shown in table 1.

Table 1 adopts 3 kinds of distinct methods focusing region A to amplify the comparison of spent time

Method	Time
		Yamamoto method	6.331s
Haunert method	6.70s
		This method	4.28s

By contrast, can find out that the method that the present invention proposes is all minimum aspect deformation quantity and calculated amount.

Although described by reference to the accompanying drawings embodiments of the present invention, to those skilled in the art, under the premise without departing from the principles of the invention, can also make some improvement, these also should be considered as belonging to protection scope of the present invention.

Claims (1)

1. the Focus+Context method for visualizing based on multi-resolution grid, is characterized in that: its concrete steps are as follows:

Step 1, obtain the original point cloud data of destination object;

Described original point cloud data is two-dimentional cloud data, comprises abscissa value x and ordinate value y a little;

Step 2, generation uniform grid;

On the basis of the original point cloud data of the destination object obtaining in step 1, generate uniform grid; Concrete operation step is:

Step 2.1: by formula (1), the original point cloud data of destination object is carried out to unit processing, the unitization of obtaining cloud data;

$\left\{\begin{array}{c}{x}^{\′}=\frac{x-x_{\min }}{x_{\max }-x_{\min }}\\ y^{\′}=\frac{y-y_{\min }}{y_{\max }-y_{\min }}\end{array}\right.---\left(1\right)$

Wherein, x ' is each some unitization abscissa value after treatment in original point cloud data; x _maxit is the horizontal ordinate maximal value of original point cloud data mid point; x _minit is the horizontal ordinate minimum value of original point cloud data mid point; Y ' is each some unitization ordinate value after treatment in original point cloud data; y _maxit is the ordinate maximal value of original point cloud data mid point; y _minit is the ordinate minimum value of original point cloud data mid point;

Step 2.2: the unit space of the two-dimensional coordinate system at unit cloud data place is divided into n × n uniform grid, wherein, n=2 ^m, m is positive integer, m value is by artificially determining; Obtain the grid vertex coordinate of uniform grid simultaneously, and uniform grid is numbered in order; Four apex coordinates of each grid are used respectively (x _v1, y _v1), (x _v2, y _v2), (x _v3, y _v3) and (x _v4, y _v4) represent; Described unit space is to be 0 and 1 by abscissa value, and ordinate value is the rectangle that four straight lines of 0 and 1 surround;

Through the operation of step 2, obtain uniform grid;

Step 3, unit cloud data is carried out to pretreatment operation;

On the basis of step 2 operation, by formula (2), unit cloud data is carried out to pretreatment operation, the mapping relations in the unitization of obtaining cloud data between each point and its place grid;

$\left\{\begin{array}{c}{x}^{\′}=w_1{x}_{v1}+w_2{x}_{v2}+w_3{x}_{v3}+w_4{x}_{v4}\\ y^{\′}=w_1{y}_{v1}+w_2{y}_{v2}+w_3{y}_{v3}+w_4{y}_{v4}\\ 1=w_1+w_2+w_3+w_4\end{array}\right.---\left(2\right)$

Wherein, w _ifor the grid vertex (x of current unit cloud data point (x ', y ') and point (x ', y ') place grid _vi, y _vi) related coefficient; I is positive integer, i ∈ [Isosorbide-5-Nitrae];

By the operation of step 3, can the unitization of obtaining cloud data point the grid vertex (x of (x ', y ') and point (x ', y ') place grid _vi, y _vi) related coefficient w _i;

Step 4, obtain multi-resolution grid;

On the basis of step 3 operation, obtain multi-resolution grid; Concrete steps are:

Step 4.1: determine the threshold value σ that counts, sum is the quantity of original point cloud data mid point;

The described method of determining the threshold value σ that counts is: statistic procedure three obtains the quantity of unit cloud data point in each uniform grid, and therefrom maximizing, using this maximal value as the threshold value σ that counts;

Step 4.2: the unit space of two-dimensional coordinate system is divided into 2 × 2 grid, obtains segmenting the length of side of grid, l represents with symbol;

Step 4.3: judge successively the quantity s of unit cloud data point in each segmentation grid, if s >=σ, and l > 1/n, continue this segmentation grid to be divided into 2 × 2 grid; Each segmentation grid repeats the operation of this step, until can not be subdivided;

By the operation of above-mentioned steps, obtain multi-resolution grid;

Step 5, the grid in multi-resolution grid is divided into effective grid and redundancy grid;

On the basis of step 4 operation, the total-grid in multi-resolution grid is divided into effective grid and redundancy grid; Concrete operation method is: the multi-resolution grid that step 4 is obtained travels through, if there is no any point in original point cloud data in current grid, is redundancy grid by this grid mark; Otherwise, be effective grid by this grid mark;

Step 6, objective definition function;

On the basis of step 5 operation, objective definition function; Concrete steps are:

Use symbol c _krepresent any grid in the multi-resolution grid that obtains of step 4, k represents the index of each grid in multi-resolution grid that step 4 obtains, 1≤k≤K, and K represents the number of grid in multi-resolution grid; With symbol c ' _krepresent grid c in multi-resolution grid _kgrid after distortion; C ' _k=s _kc _k, wherein, s _kbe the equal proportion scaled matrix of 2 × 2, represent grid c _kdeformation coefficient;

$s_k=\left(\begin{array}{cc}{s}_{k,x}& \\ & s_{k,y}\end{array}\right)---\left(3\right)$

Wherein, s _k,xand s _k,yrepresent respectively grid c _kthe x on summit, y coordinate zoom factor; s _k,x=s _k,y;

The objective function of distortion of the mesh is defined as:

||C′-SC|| ²＝0 (4)

Wherein, C represents all grids in multi-resolution grid that step 4 obtains; S represents the equal proportion scaled matrix of all grids; Grid in C ' expression multi-resolution grid after all distortions of the mesh;

Because multi-resolution grid is divided into effective grid and redundancy grid, therefore only need to calculate effective grid, objective function is reduced to:

||C′ _P-S _PC _P|| ²＝0 (5)

Wherein, C _pall effective grids in the multi-resolution grid that expression step 5 obtains; S _prepresent the equal proportion scaled matrix of all effective grids; C ' _prepresent the grid after all effective grid distortion in multi-resolution grid;

Use symbol represent any one grid c in all effective grids in multi-resolution grid _pe article of limit, e ∈ 0,1,2,3}, v _p1and v _p2represent respectively termination summit and the initial vertex on e article of limit; In multi-resolution grid, any one grid in all effective grids can be expressed as the set on limit

By grid c _pwith vertex representation be c _p=q _pv _p, wherein, v _prepresent grid c _pvertex matrix; q _pfor the representing matrix on limit, q _pelement value in matrix is as follows:

$q_p^{u,v}=\left\{\begin{array}{ccc}1& \mathrm{if}& u=e,v=p1\\ -1& \mathrm{if}& u=e,v=p2\\ 0& \mathrm{otherwise}& \end{array}\right.---\left(6\right)$

Wherein, represent q _pin matrix, u is capable, the value of the element of v row; U ∈ 0,1,2,3}, v is positive integer;

Use Q _prepresent the representing matrix on the limit of all effective grids in multi-resolution grid, all effective grid set C in multi-resolution grid _pcan be expressed as C _p=Q _pv _p, the objective function that formula (5) represents like this can use formula (7) to represent;

||Q _PV′ _P-S _PQ _PV _P|| ²＝0 (7)

Wherein, V _prepresent all effective grid C in multi-resolution grid _pvertex matrix; V ' _prepresent all effective grid C in multi-resolution grid _pvertex matrix after distortion;

In the process of being out of shape, the mesh space of the absolute coordinates of finding only to use a summit after to whole distortion positions, and can cause model " shake ", so will be that each grid vertex point of addition retrains; " shake " of described model refers to the overall offset of mesh space with respect to original position;

ω||IV′ _P-V _P|| ²＝0 (8)

Wherein, the intensity that ω is position constraint, ω is artificial setting value, ω ∈ [0,1]; I is and V ' _pthe unit matrix that exponent number is identical;

After conformity goal function and position constraint condition, the objective function that can be improved, as shown in formula (9);

||Q _PV′ _P-S _PQ _PV _P|| ²+ω||IV′ _P-V _P|| ²＝0 (9)

Add net boundary constraint condition and allow each grid vertex remain in border, obtain formula (10), formula (10) is objective function;

$\left\{\begin{array}{c}{\left|\right|Q_P{V}_P^{\′}-S_P{Q}_P{V}_P\left|\right|}^2+\mathrm{\ω}{\left|\right|{\mathrm{IV}}_P^{\′}-V_P\left|\right|}^2=0^2\\ x_l\≤v_x\≤x_h,y_l\≤v_y\≤y_h\end{array}\right.---\left(10\right)$

Wherein, v _xand v _ythe x coordinate and the y that represent respectively the summit of any grid in all effective grids in multi-resolution grid sit target value; x _land y _lrepresent respectively the minimum value of x coordinate and the minimum value of y coordinate in all summits of multi-resolution grid; x _hand y _hrepresent respectively the maximal value of x coordinate and the maximal value of y coordinate in all summits of multi-resolution grid;

Step 7, objective function is solved;

On the basis of step 6 operation, formula (10) is solved; Concrete operation step is:

Step 7.0: formula (10) is expressed as to formula (11);

AV′ _P＝b(V _P) (11)

Wherein, $A={[Q_P^T,{\mathrm{\ωI}}^T]}^T,$ $b\left(V_P\right)={[{\left(S_P{Q}_P{V}_P\right)}^T,{\mathrm{\ωV}}_P^T]}^T;$

A is multiplied by formula (11) equal sign both sides simultaneously ^t, be converted to and solve linear system:

(A ^TA)V′ _P＝A ^Tb(V _P) (12)

Step 7.1: if iteration, V for the first time _pget initial value, simultaneously according to V _pinitial value, calculate the initial area M of each effective grid _p; Otherwise V is set _p=V ' _p, V ' _pfor the result obtaining in step 7.4, and according to current V _pbe worth, calculate the area M ' of the effective grid after each distortion _p;

Step 7.2: all effective grid c in the multi-resolution grid that calculation procedure four obtains _pzoom factor f _p; If this effective grid is in focus area, the zoom factor of this effective grid is the artificial enlargement factor λ setting, λ > 1, i.e. f _p=λ; If this effective grid, not in focus area, calculates the zoom factor of this effective grid according to formula (13); Described focus area is by the artificial region of specifying of user, and therefore non-focus area is the region beyond focus area in view; According to all effective grid c in the multi-resolution grid calculating _pzoom factor, draw zoom factor matrix S _p;

$f_p={(M_p^{\′}/M_p)}^{1/2}{I}^{\′}---\left(13\right)$

Wherein, I ' is 2 × 2 unit matrixs;

Step 7.3: the zoom factor matrix S of calculating according to step 7.2 _p, calculate b (V _p), $b\left(V_P\right)={[{\left(S_P{Q}_P{V}_P\right)}^T,{\mathrm{\ωV}}_P^T]}^T;$

Step 7.4: by the 7.3 b (V that calculate _p), be brought into formula (12), calculate V ' _p; If V ' _prestrain or iterations reaches preset value, stopped iteration; Otherwise repeating step 7.1 is to the operation of step 7.4;

Step 8, reconstruction model;

On the basis of step 7 operation, the V ' that uses step 7 to obtain _preconstruction model, obtains the two dimensional model after distortion.