CN104992469A - Automatic preparation body neck-edge line extraction method based on MS complex - Google Patents

Abstract

The invention discloses an automatic preparation body neck-edge line extraction method based on MS complex. The method comprises the following steps: S1) establishing an MS complex structure; S2) carrying out noise reduction processing; S3) carrying out MS complex simplification; S4) screening feature lines; and S5) evolving the feature lines into neck-edge lines. The method is high in efficiency and insensitive to noise, can automatically extract more accurate neck-edge lines, and can be widely applied to the mouth rehabilitation field.

Description

Based on the teeth preparation line extraction method of MS complex

Technical field

The present invention relates to a kind of neck edge line extraction method based on MS complex, belong to oral restoration field.

Background technology

Teeth preparation line extractive technique is the gordian technique that oral cavity CAD/CAM repairs.At present, external advanced oral restoration CAD/CAM system generally all has neck edge line automatically or interactively pick-up function, and its algorithm is mostly unknown.Domesticly also there are some neck edge line drawing algorithms, but more or less there is following shortcoming: (1) extraction accuracy is not high; (2) robustness is bad, comparatively responsive to noise; (3) automaticity is not high, needs an artificial pickup initial point or some unique points.

Relative to neck edge line drawing technology, large quantity research is had based on the feature line extraction technology of a cloud or grid model in reverse-engineering field, its handling object comprises engineering goods, human body, animals and plants, historical relic, the artwork, topography and geomorphology etc., and this provides reference and reference for neck edge line drawing technology to a certain extent.General features line can be divided into viewpoint change with not with the characteristic curve of viewpoint change, wherein do not comprise again the rip out stitches of model surface, boundary line and a few class of paddy crestal line with the characteristic curve of viewpoint change, the difference of teeth preparation line and general features line is: (1) neck edge line can be classified as crestal line, but does not comprise valley line, boundary line and rip out stitches; (2) neck edge line is a closed annular feature line.Therefore, general feature line extraction method can not be directly used in neck edge line drawing.

Qiu Yanjie etc. propose a kind of feature line extraction algorithm based on Morse-Smale (MS) complex in " the triangle gridding feature line extraction based on Morse-Smale complex ", this algorithm is with Moving Least Squares (Moving-Least Square, MLS) Surface Method calculates maximum, the minimum principal curvatures on summit, correspondingly set up MS complex using curvature (Curvedness) as target function, simplify and aftertreatment through complex, automatically obtain clear, characteristic curve accurately, and have that counting yield is high, advantage to insensitive for noise.But the characteristic curve that the method is extracted comprises valley line and boundary line, can not be directly used in neck edge line drawing, and the method also found no at present based on MS complex is for oral restoration, especially neck edge line drawing aspect.

Summary of the invention

For overcoming above-mentioned prior art problem, the present invention proposes a kind of neck edge line extraction method based on MS complex, and the method efficiency is high, to insensitive for noise, automatically can extract neck edge line comparatively accurately.

The technical solution adopted for the present invention to solve the technical problems is:

Based on a neck edge line extraction method for MS complex, it is characterized in that, comprising:

S1, set up MS complex structure;

S2, noise reduction process;

S3, MS complex simplifies;

S4, screening characteristic curve;

S5, characteristic curve is evolved into neck edge line.

Further, described step S1, comprising:

S11, structure target function: calculate each summit mean curvature K with moving least squares surfaces method _h(p _i), and with K _h(p _i) as the target function on each summit;

S12, differentiation critical point: with target function described in S11 for the critical point according to differentiation MS complex, comprise minimum point, maximum point and saddle point;

S13, build ascending, descending arc: from saddle point for, respectively along target function gradient direction and search in the other direction, obtain ascending, descending arc successively; For any summit p of triangle gridding _i, its gradient direction is then

$q:=\max \frac{K_H\left(p_i\right)-K_H\left(q_i\right)}{\left|\right|p_i-q_i\left|\right|},(j=\mathrm{1,2},...,m_i)$

Wherein, q _ifor p _i1-neighborhood point; m _ifor p _i1-neighborhood point number.

Further, described step S4, comprising:

S41, employing algorithm of region growing, to rise arc for carrying out Region dividing to grid model in border;

S42, remove branch, isolated island and the peninsula by area judging.

Further, described step S41, comprising:

S411, saddle point, maximum point using in MS complex and rise the frontier point of the point on arc as region growing;

S412, select be not divided and do not belong to growth frontier point grid vertex as Seed Points, use v _seedrepresent, if new subdomain is R, make R=v _seed;

The point that S413, searching and subdomain R are adjacent, for each abutment points, judges whether it is growth frontier point, if not, then joins in subdomain R by this point;

S414, repeat S413 until number of vertices no longer increases in subdomain R, namely adjacent with R point is growth frontier point, and constructs the border chained list of R according to the annexation that correspondence rises arc;

S415, repetition S412 ~ S414, until all summits are all divided into subdomain internal point or frontier point;

S416, according to the Region dividing result on summit, the triangular plate of correspondence to be divided: if any one summit is the internal point of a certain subdomain in triangular plate, then this triangular plate is divided in this subdomain; If three summits are frontier point, then first extract not limitrophe limit in Atria bar limit, judge this limit the affiliated subdomain of adjacent triangular plate, then triangular plate undetermined is divided in this subdomain.

Further, the saddle point in MS complex described in described step S411, maximum point and the point risen on arc do not comprise model boundary point.

Further, described step S42, comprising:

S421, removal characteristic curve branch: to any point on characteristic curve, if the non-feature adjoint point of this point all belongs to the same area, then judge that this point is positioned in branch, this point is incorporated in affiliated area, become intra-zone point;

S422, removal isolated island: if region A is surrounded completely by another region B, then judge that A is as isolated island region, is directly incorporated in B by A;

S423, remove the peninsula: if region A, B are adjacent and B is less than A, then judge the peninsula region of B as A, if A, B public boundary is less than the not common border of B, be then incorporated in the region outside A on the not common border of B.

Further, described step S5 first to the smoothing process of characteristic curve of extracting, then selects the point on characteristic curve to carry out curve fitting.

Further, described step S5 first selects the point on characteristic curve to carry out curve fitting, then carries out fairing processing to curve.

The invention has the beneficial effects as follows: the teeth preparation line extraction method that the present invention is based on MS complex, adopt moving least squares surfaces method to calculate mean curvature and using as target function, both reduced the impact of noise on curvature result, and in turn ensure that a MS complex extracting ridges and avoid proposition to get valley line; Again through noise reduction process, and screen characteristic curve on the basis of extracting characteristic curve with MS complex, obtain a closed annular feature line; Finally obtain required neck edge line in conjunction with characteristic curve smoothing processing and curve; This method efficiency is high, to insensitive for noise, automatically can extract neck edge line comparatively accurately.

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described.

Fig. 1 is the overall flow figure of the teeth preparation line extraction method that the present invention is based on MS complex;

Fig. 2 is the process flow diagram of step S1 of the present invention;

Fig. 3 is the process flow diagram of step S4 of the present invention;

Fig. 4 is the process flow diagram of step S41 of the present invention;

Fig. 5 is the process flow diagram of step S42 of the present invention;

Fig. 6 is one embodiment of the present of invention overall flow figure;

Fig. 7 is the schematic diagram of model boundary of the present invention;

Fig. 8 is the embodiment design sketch that the present invention removes characteristic curve branch;

Fig. 9 is the embodiment design sketch that the present invention removes isolated island;

Figure 10 is the embodiment design sketch that the present invention removes the peninsula;

Figure 11 is the neck edge line design sketch that the present invention finally extracts.

Embodiment

With reference to Fig. 1, based on the teeth preparation line extraction method of MS complex, comprising:

S1, set up MS complex structure;

S2, noise reduction process;

S3, MS complex simplifies;

S4, screening characteristic curve;

S5, characteristic curve is evolved into neck edge line.

With reference to Fig. 2, be further used as preferred embodiment, described step S1, comprising:

S11, structure target function: calculate each summit mean curvature K with moving least squares surfaces method _h(p _i), and with K _h(p _i) as the target function on each summit;

S12, differentiation critical point: with target function described in S11 for the critical point according to differentiation MS complex, comprise minimum point, maximum point and saddle point;

S13, build ascending, descending arc: from saddle point for, respectively along target function gradient direction and search in the other direction, obtain ascending, descending arc successively; For any summit p of triangle gridding _i, its gradient direction is then

$q:=\max \frac{K_H\left(p_i\right)-K_H\left(q_i\right)}{\left|\right|p_i-q_i\left|\right|},(j=\mathrm{1,2},...,m_i)$

Wherein, q _ifor p _i1-neighborhood point; m _ifor p _i1-neighborhood point number.

With reference to Fig. 3, be further used as preferred embodiment, described step S4, comprising:

S41, employing algorithm of region growing, to rise arc for carrying out Region dividing to grid model in border;

S42, remove branch, isolated island and the peninsula by area judging.

With reference to Fig. 4, be further used as preferred embodiment, described step S41, comprising:

S411, saddle point, maximum point using in MS complex and rise the frontier point of the point on arc as region growing;

S412, select be not divided and do not belong to growth frontier point grid vertex as Seed Points, use v _seedrepresent, if new subdomain is R, make R=v _seed;

The point that S413, searching and subdomain R are adjacent, for each abutment points, judges whether it is growth frontier point, if not, then joins in subdomain R by this point;

S414, repeat S413 until number of vertices no longer increases in subdomain R, namely adjacent with R point is growth frontier point, and constructs the border chained list of R according to the annexation that correspondence rises arc;

S415, repetition S412 ~ S414, until all summits are all divided into subdomain internal point or frontier point;

S416, according to the Region dividing result on summit, the triangular plate of correspondence to be divided: if any one summit is the internal point of a certain subdomain in triangular plate, then this triangular plate is divided in this subdomain; If three summits are frontier point, then first extract not limitrophe limit in Atria bar limit, judge this limit the affiliated subdomain of adjacent triangular plate, then triangular plate undetermined is divided in this subdomain.

Be further used as preferred embodiment, the saddle point in MS complex described in described step S411, maximum point and the point risen on arc do not comprise model boundary point.

With reference to Fig. 5, be further used as preferred embodiment, described step S42, comprising:

S421, removal characteristic curve branch: to any point on characteristic curve, if the non-feature adjoint point of this point all belongs to the same area, then judge that this point is positioned in branch, this point is incorporated in affiliated area, become intra-zone point;

S422, removal isolated island: if region A is surrounded completely by another region B, then judge that A is as isolated island region, is directly incorporated in B by A;

S423, remove the peninsula: if region A, B are adjacent and B is less than A, then judge the peninsula region of B as A, if A, B public boundary is less than the not common border of B, be then incorporated in the region outside A on the not common border of B.

Be further used as preferred embodiment, described step S5 first to the smoothing process of characteristic curve of extracting, then selects the point on characteristic curve to carry out curve fitting.

Be further used as preferred embodiment, described step S5 first selects the point on characteristic curve to carry out curve fitting, then carries out fairing processing to curve.

Below in conjunction with Figure of description and specific embodiment, the present invention is described in further details.

With reference to Fig. 6, the embodiment that the present invention is based on the teeth preparation line extraction method of MS complex comprises the following steps:

Step one: reading model

In the present invention, the applicable object of neck edge line drawing is Tooth preparation triangle grid model, and the triangle grid model that the cloud data being scanned gained by Tooth preparation generates through process belongs to applicable object of the present invention together.

Step 2: set up MS complex structure

The present invention adopt moving least squares surfaces method calculate mean curvature and using as target function, set up MS complex structure on this basis, its concrete computation process is:

(1) target function is built

1) each summit mean curvature K is calculated with moving least squares surfaces method _h(p _i): first the moving least squares surfaces S on Modling model summit, S are that (y, a) along the local minimum in vector field n (x) direction, its implicit equation is energy function e

$S=\left\{x\right|g\left(x\right)=n{\left(x\right)}^T\left(\frac{\∂e(y,n\left(x\right))}{\∂y}{|}_{y=x}\right)=0,x\∈R^3\}$

In formula:

$n\left(x\right)=\frac{{\mathrm{\Σ}}_{p_i\∈Q}{v}_i\mathrm{\θ}(x,p_i)}{\left|\right|{\mathrm{\Σ}}_{p_i\∈Q}{v}_i\mathrm{\θ}(x,p_i)\left|\right|}$

$e(y,a)={\mathrm{\Σ}}_{p_i\∈Q}{\left({(y-p_i)}^{T}a\right)}^2\mathrm{\θ}(y,p_i)$

Wherein, θ is Gauss's weight function, and y, a are position vector and direction vector respectively; p _ifor grid model summit; v _ifor summit p _inormal vector.

Then mean curvature is

$K_H=\frac{{\left|\right|\▿g\left(x\right)\left|\right|}^2\·\mathrm{Trace}\left(H\right)-\▿g\left(x\right)\·H\left(g\left(x\right)\right)\·{\▿}^{T}g\left(x\right)}{{\left|\right|\▿g\left(x\right)\left|\right|}^3}$

2) with K _h(p _i) as the target function on each summit;

(2) critical point is differentiated: with target function described in S11 for the critical point according to differentiation MS complex, comprise minimum point, maximum point and saddle point;

(3) build ascending, descending arc: from saddle point for, respectively along target function gradient direction and search in the other direction, obtain ascending, descending arc successively; For any summit p of triangle gridding _i, its gradient direction is then

$q:=\max \frac{K_H\left(p_i\right)-K_H\left(q_i\right)}{\left|\right|p_i-q_i\left|\right|},(j=\mathrm{1,2},...,m_i)$

Wherein, q _ifor p _i1-neighborhood point; m _ifor p _i1-neighborhood point number.

Step 3: noise reduction process

In initial MS complex, due to the existence of curvature estimation error, even if also minor fluctuations can be produced in the region that actual curvature is equal, the generation of redundancy critical point and characteristic curve may be caused.These characteristic curves cause primarily of the error of calculation or artificial perturbation, and significance is very little, can regard topological noise as.The present invention is with the difference of the minimum and maximum value of grid model summit mean curvature for benchmark, and choose this benchmark 2% carries out noise reduction process as threshold value.

Step 4: MS complex simplifies

The detailed process that MS complex of the present invention simplifies is:

(1) defined feature line significance: be first connected with saddle point 2 are risen arc and 2 and fall arc and merge into 1 respectively and rise line R and 1 and fall line r; Then minimum point m is obtained by MS complex concept _irising territory A (m _i) and maximum point M _idecline territory D (M _i); Then adjacent with rising line R rising territory A (m ₁), A (m ₂) significance be

$\mathrm{\σ}(R,A\left(m_i\right))=\frac{{\∫}_R{K}_H\mathrm{dR}}{\mathrm{length}\left(R\right)}-\frac{{\∫}_{A\left(m_i\right)}{K}_H\mathrm{dA}}{\mathrm{area}\left(A\left(m_i\right)\right)},(i=\mathrm{1,2})$

In formula,

${\∫}_R{K}_H\mathrm{dR}=\underset{j}{\mathrm{\Σ}}\frac{K_H\left(p_1^j\right)+K_H\left(p_2^j\right)}{2}\left|\right|p_1^j-p_2^j\left|\right|$

${\∫}_{A\left(m_i\right)}{K}_H\mathrm{dA}=\underset{k}{\mathrm{\Σ}}\frac{K_H\left(p_1^k\right)+K_H\left(p_2^k\right)+K_H\left(p_3^k\right)}{3}{A}^x$

Wherein, for 2 end points of jth bar line segment on R; be respectively A (m _i) middle kth leg-of-mutton 3 summits; A ^kfor a kth triangle area.

The overall significance rising line R is

σ(R)＝min{σ(R，A(m ₁))，σ(R，A(m ₂))}

Similarly, adjacent with falling line r decline territory D (M ₁), D (M ₂) significance be

$\mathrm{\σ}(r,D\left(M_i\right))=\frac{{\∫}_{D\left(M_i\right)}{K}_H\mathrm{dD}}{\mathrm{area}\left(D\left(M_i\right)\right)}-\frac{{\∫}_r{K}_H\mathrm{dr}}{\mathrm{length}\left(r\right)},(i=\mathrm{1,2})$

The overall significance falling line r is

σ(r)＝min{σ(r，D(M ₁))，σ(r，D(M ₂))}

(2) the minimum characteristic curve of significance is deleted successively to realize the simplification to complex: if the remarkable angle value of the entirety of characteristic curve is less than simplification threshold value, then delete this characteristic curve and corresponding critical point; If the critical point of deleting is the very big or minimum point be only connected with a saddle point, then this critical point is deleted together with saddle point and ascending, descending arc thereof; Often complete a delete procedure, all will recalculate the significance of characteristic curve, then continue to perform delete procedure until all significances rising line are all greater than simplification threshold value.

Step 5: screening characteristic curve

The present invention screens the characteristic curve after MS complex simplifies, and first carries out Region dividing to grid model, then removes the characteristic curve irrelevant with required neck edge line by area judging, and finally obtain a closed annular feature line, its detailed process is:

(1) algorithm of region growing is adopted, to rise arc for carrying out Region dividing to grid model in border:

1) using the saddle point in MS complex, maximum point and rise the frontier point of the point on arc as region growing, but frontier point herein does not comprise model boundary point, can avoid the boundary line of extracting model like this; It is noted that zone boundary refers to the unique point or characteristic curve surrounding region, and model boundary refers to the border bottom Tooth preparation grid model, model boundary as shown in Figure 7;

2) select be not divided and do not belong to growth frontier point grid vertex as Seed Points, use v _seedrepresent, if new subdomain is R, make R=v _seed;

3) find the point adjacent with subdomain R, for each abutment points, judge whether it is growth frontier point, if not, then joins in subdomain R by this point;

4) repeat S413 until number of vertices no longer increases in subdomain R, namely adjacent with R point is growth frontier point, and rises the border chained list of the annexation structure R of arc according to correspondence;

5) S412 ~ S414 is repeated, until all summits are all divided into subdomain internal point or frontier point;

6) according to the Region dividing result on summit, the triangular plate of correspondence is divided: if any one summit is the internal point of a certain subdomain in triangular plate, then this triangular plate is divided in this subdomain; If three summits are frontier point, then first extract not limitrophe limit in Atria bar limit, judge this limit the affiliated subdomain of adjacent triangular plate, then triangular plate undetermined is divided in this subdomain.

(2) branch, isolated island and the peninsula is removed by area judging.

1) characteristic curve branch is removed: the present invention removes the embodiment of characteristic curve branch as shown in Figure 8, the non-feature adjoint point of the upper any point of characteristic curve a, b all belongs to region A, non-feature adjoint point refers to the neighborhood point not on characteristic curve, then judge that a, b are as the branch in A, a, b are incorporated to A, namely remove a, b;

2) isolated island is removed: the present invention removes the embodiment of isolated island as shown in Figure 9, and region B is surrounded completely by another region A, then judge that B is as isolated island region, is directly incorporated in A by B, namely removes B;

3) peninsula is removed: the present invention removes the embodiment on the peninsula as shown in Figure 10, and region A, B are adjacent and B is less than A, then judge the peninsula region of B as A; Region C and A, B are adjacent, and characteristic curve a, b are the border of A, and characteristic curve b, c are the border of B, and b is the public boundary of A, B, if b is less than c, then known b is optimal path, is incorporated to by c in the C of region, namely removes c.

Step 6: characteristic curve is evolved into neck edge line

The present invention first to the smoothing process of characteristic curve of extracting, then selects the point on characteristic curve to carry out curve fitting.

The embodiment that characteristic curve is evolved into neck edge line by the present invention is: first adopt discrete Laplacian algorithm to the smoothing process of characteristic curve, then with B-spline curves, unique point is fitted to neck edge line, its detailed process is:

(1) adopt discrete Laplacian algorithm to the smoothing process of characteristic curve

1) new node is calculated

$x^{\′}=\frac{1}{2t}\underset{k=1}{\overset{m}{\mathrm{\Σ}}}(x-x_k)$

Wherein, x be on characteristic curve l a bit; x _kfor the adjoint point of x on l; M is adjoint point number; T is smoothing factor; The new node that x ' obtains after Laplace transform for x;

2) line segment that new node is formed by connecting is projected on grid surface, to ensure that the characteristic curve smoothly can not produce relatively large deviation relative to triangle gridding.

(2) with B-spline curves, unique point is fitted to neck edge line.

The neck edge line finally extracted is as shown in heavy black line bar in Figure 11.

Compared with prior art, the present invention adopt moving least squares surfaces method calculate mean curvature and using as target function, both reduced the impact of noise on curvature result, and in turn ensure that a MS complex extracting ridges and avoid proposition to get valley line; Again through noise reduction process, and screen characteristic curve on the basis of extracting characteristic curve with MS complex, obtain a closed annular feature line; Finally obtain required neck edge line in conjunction with characteristic curve smoothing processing and curve; This method efficiency is high, to insensitive for noise, automatically can extract neck edge line comparatively accurately.

More than that better enforcement of the present invention is illustrated, but the invention is not limited to described embodiment, those of ordinary skill in the art also can make all equivalent variations or replacement under the prerequisite without prejudice to spirit of the present invention, and these equivalent modification or replacement are all included in the application's claim limited range.

Claims (8)

1., based on the teeth preparation line extraction method of MS complex, it is characterized in that, comprising:

S1, set up MS complex structure;

S2, noise reduction process;

S3, MS complex simplifies;

S4, screening characteristic curve;

S5, characteristic curve is evolved into neck edge line.

2. the teeth preparation line extraction method based on MS complex according to claim 1, it is characterized in that, described step S1, comprising:

S11, structure target function: calculate each summit mean curvature K with moving least squares surfaces method _h(p _i), and with K _h(p _i) as the target function on each summit;

S12, differentiation critical point: with target function described in S11 for the critical point according to differentiation MS complex, comprise minimum point, maximum point and saddle point;

S13, build ascending, descending arc: from saddle point for, respectively along target function gradient direction and search in the other direction, obtain ascending, descending arc successively; For any summit p of triangle gridding _i, its gradient direction is then

$q:=\max \frac{K_H\left(p_i\right)-K_H\left(q_i\right)}{\left|\right|p_i-q_i\left|\right|},(j=\mathrm{1,2},...,m_i)$

Wherein, q _ifor p _i1-neighborhood point; m _ifor p _i1-neighborhood point number.

3. the teeth preparation line extraction method based on MS complex according to claim 1, it is characterized in that, described step S4, comprising:

S41, employing algorithm of region growing, to rise arc for carrying out Region dividing to grid model in border;

S42, remove branch, isolated island and the peninsula by area judging.

4. the teeth preparation line extraction method based on MS complex according to claim 1, it is characterized in that, described step S41, comprising:

S411, saddle point, maximum point using in MS complex and rise the frontier point of the point on arc as region growing;

S412, select be not divided and do not belong to growth frontier point grid vertex as Seed Points, use v _seedrepresent, if new subdomain is R, make R=v _seed;

The point that S413, searching and subdomain R are adjacent, for each abutment points, judges whether it is growth frontier point, if not, then joins in subdomain R by this point;

S414, repeat S413 until number of vertices no longer increases in subdomain R, namely adjacent with R point is growth frontier point, and constructs the border chained list of R according to the annexation that correspondence rises arc;

S415, repetition S412 ~ S414, until all summits are all divided into subdomain internal point or frontier point;

S416, according to the Region dividing result on summit, the triangular plate of correspondence to be divided: if any one summit is the internal point of a certain subdomain in triangular plate, then this triangular plate is divided in this subdomain; If three summits are frontier point, then first extract not limitrophe limit in Atria bar limit, judge this limit the affiliated subdomain of adjacent triangular plate, then triangular plate undetermined is divided in this subdomain.

5. the teeth preparation line extraction method based on MS complex according to claim 1, is characterized in that, the saddle point in MS complex described in described step S411, maximum point and the point risen on arc do not comprise model boundary point.

6. the teeth preparation line extraction method based on MS complex according to claim 1, it is characterized in that, described step S42, comprising:

S421, removal characteristic curve branch: to any point on characteristic curve, if the non-feature adjoint point of this point all belongs to the same area, then judge that this point is positioned in branch, this point is incorporated in affiliated area, become intra-zone point;

S422, removal isolated island: if region A is surrounded completely by another region B, then judge that A is as isolated island region, is directly incorporated in B by A;

S423, remove the peninsula: if region A, B are adjacent and B is less than A, then judge the peninsula region of B as A, if A, B public boundary is less than the not common border of B, be then incorporated in the region outside A on the not common border of B.

7. the teeth preparation line extraction method based on MS complex according to claim 1, is characterized in that, described step S5 first to the smoothing process of characteristic curve of extracting, then selects the point on characteristic curve to carry out curve fitting.

8. the teeth preparation line extraction method based on MS complex according to claim 1, is characterized in that, described step S5 first selects the point on characteristic curve to carry out curve fitting, then carries out fairing processing to curve.