CN103559400A - Method for calculating distance from point to implicit curve based on inferior arc evolution - Google Patents

Abstract

The invention provides a method for calculating a numerical value of a distance from a point to an implicit curve in a plane. The method has the advantages of accuracy, robustness, high algorithm efficiency, low requirements on the smoothness of an implicit function, and the like. The method specifically comprises the following steps of continuously reducing radiuses from a circle which employs the point as a circle center and is intersected with the implicit curve until a certain concentric circle is not intersected with the implicit curve, continuously bisecting the radiuses of concentric circles until un-intersected inner circles are infinitely close to intersected outer circles, and finally calculating the distance, wherein an inferior arc evolution algorithm is adopted for judging intersection between the circles and the implicit curve, namely whether function values of dense equal division points of circumferences are opposite signs of the circle center or not is judged according to the sequence of binary coding inverse codes of the dense equal division points, the circles are determined to be intersected with the implicit curve if the function values are the opposite signs of the circle center, otherwise circular arcs in an inferior arc set are subjected to evolution processing such as deletion, reduction or fracturing according to the function values to reduce the coverage of the inferior arc set until the inferior arc set is null, and the circles are determined to be not intersected with the implicit curve.

Description

The point developing based on minor arc is to Implicit Curves distance calculating method

Technical field

The present invention relates to numerical evaluation field, specifically a kind ofly provide some the numerical computation method of Implicit Curves distance for robot calculator.

Background technology

Point has significant application value to the distance calculating method of Implicit Curves in fields such as pattern-recognition, Geometric Modeling, computer vision, medical image processing, is also the important research basis of the subjects such as numerical analysis, computer science, reverse-engineering.Because the analytic expression of point to the distance of Implicit Curves is generally difficult to obtain even may not obtain, so the calculating of distance numerical evaluation normally.

If required distance is a p ₀to the distance of Implicit Curves f (x, y)=0, point to the numerical evaluation of Implicit Curves distance conventionally have local method and global approach minute.Local method contains an iterative process, and as Newton iteration method, it is first obtained on Implicit Curves and p with iterative manner ₀reach the point (being called foot point) of minor increment, namely first solve the following equation root p about p ^*:

$\left\{\begin{array}{c}(p-p_0)\×\▿f\left(p\right)=0\\ f\left(p\right)=0\end{array}\right.$

Wherein,

represent gradient operator, * be vector multiplication cross.So required distance is exactly || p*-p ₀||.Convergence rate in iterative methods is very fast, but is often subject to the impact of singular point and Initial value choice, exists and does not restrain, is absorbed in the shortcomings such as Local Extremum, and the slickness of f (x, y) is had relatively high expectations.Overall situation rule is according to implicit expression function j (x, y) certain global property carries out two-dimensional search to foot point, as document " Geometric constraint solver using multivariate rational spline functions " (Elber, G.and Kim, M.-S., In:Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, ACM, 2001:pp.1-10) and document " Computation of the solutions of nonlinear polynomial systems " (Sherbrooke, E.C.and Patrikalakis, N.M., Computer Aided Geometric Design, v10, 1993:pp.379-405) described method, according to the convex closure character of basis function, adopt B é zier method of cutting out constantly to get rid of non-foot point region, obtain candidate's foot point, finally in these candidate's foot points, choose and p ₀the point that distance is minimum.Global approach can guarantee that robust calculates, but calculated amount is larger, and implicit expression function is had to specific (special) requirements (as convexity).

In addition, also has the method as the compromise of global approach and partial approach, basic framework is still process of iteration, but every step iteration is searched for the estimated position of next foot point according to radial derivative, referring to document " Robust Computation of Foot Points on Implicitly Defined Curves " (Aigner, M.and Jittler, B., In:Mathematical Methods for Curves and Surfaces:

2004, Nashboro Press, 2005:pp.1-10).

Certainly, because speed of convergence and robust calculate, be difficult to satisfactory to both partiesly, at many calculation levels that need, to the application scenario of Implicit Curves distance, usually adopt estimation algorithm, as the second order based on curvature is estimated, simplex is estimated.Specifically can be referring to document " Fitting B-spline curves to point clouds by curvature-based squared distance minimization " (Wang, Wenping; Pottmann, H.; Et a1., ACM Transactions on Graphics, 25 (2), 2006:214-238) and document " Implicit polynomial representation through a fast fitting error estimation " (Rouhani, M.; Sappa, A.D., IEEE Transactions on Image Processing, 21 (4), 2012:2089-2098).The estimation of distance is not design object of the present invention, and target of the present invention is accurate Calculation and robustness.

Summary of the invention

For the distance of the point in accurate Calculation plane to Implicit Curves, reach given arbitrarily small allowable error in advance, and be not subject to the impact of curve geometry, avoid falling into local extremum, rely on the problems such as Initial value choice, the present invention adopts following technical scheme.

A point p in given two dimensional surface ₀and an Implicit Curves, wherein Implicit Curves is for having the binary function f (x, y) of definition and directional derivative bounded according to the definite single branch plane curve of its null value collection, of the present invention some p on certain bounded domain Ω ₀numerical computation method to described Implicit Curves distance, comprises following steps:

Find a functional value f (p) and f (p ₀) contrary sign or null initial point p, now with p ₀for the center of circle, with || p ₀-p|| is that the circle of radius must intersect with Implicit Curves; Putting inside radius is 0, and external radius is || p ₀-p||; Get again the mean value that median radius is inside radius and external radius, then judge with p ₀whether the circle and the Implicit Curves that for the center of circle, the median radius of take, are radius intersect, if intersect, under external radius, are adjusted to median radius, if non-intersect, are adjusted to median radius on inside radius; Get again median radius, judgement Intersection, adjust inside radius or external radius, so repeatedly, until the difference of external radius and inside radius is less than allowable error, return to the mean value of inside radius and external radius as a p ₀distance to described Implicit Curves.

In above-mentioned numerical procedure, can produce a series of concentric circles.Cylindrical and Implicit Curves intersect at the beginning, inner circle is the point of a degeneration, median radius is 1/2 of external radius, with this radius, make new concentric circles, if this concentric circles and Implicit Curves intersect, get again 1/4 of external radius, if also intersect, get again 1/8 ... so, have at the beginning times compression process of a circle.

The whole process of above-mentioned numerical evaluation remains that cylindrical and Implicit Curves intersect, inner circle and the disjoint state of Implicit Curves.Whole cyclic process is exactly by constantly insert the concentric circles of two minutes between cylindrical and inner circle, allows inner circle and cylindrical infinite approach keeping that cylindrical and Implicit Curves are crossing always, under inner circle and the disjoint prerequisite of Implicit Curves.

More formal step is described below:

(a1) get the functional value f (p) and f (p of f ₀) the some p of contrary sign or null value;

(a2) getting initial inside radius is r ₁=0, initial external radius is r ₂=|| p ₀-p||;

(a3)r＝(r ₁+r ₂)/2；

(a4) judgement is with a p ₀whether the circle and the Implicit Curves that for the center of circle, r, are radius intersect;

(a5) if intersect, put r ₂=r, otherwise, r put ₁=r;

(a6) if r ₂-r ₁< ε (ε is allowable error), returns to (r ₁+ r ₂)/2 are as putting p ₀to the distance of described Implicit Curves, algorithm finishes, otherwise, go to step (a3).

In the embodiment of the numerical computation method to Implicit Curves distance of the present invention, the whether crossing method of judgement circle and Implicit Curves comprises following steps:

(b1) get minimum positive integer n, make circumference 2 ⁿafter decile, every section of arc length is less than allowable error ε, and the Along ent distributing successively after circumference equal dividing is ${\mathrm{<figure-callout\; id="0"\; label="q"\; filenames="HSA0000097231850000011.png,HSA0000097231850000021.png"\; state="\{\{state\}\}">q</figure-callout>}}_0,{\mathrm{<figure-callout\; id="1"\; label="q"\; filenames="HSA0000097231850000021.png,HSA0000097231850000031.png"\; state="\{\{state\}\}">q</figure-callout>}}_1,......,q_{2^n-1},q_{2^n}={\mathrm{<figure-callout\; id="0"\; label="q"\; filenames="HSA0000097231850000011.png,HSA0000097231850000021.png"\; state="\{\{state\}\}">q</figure-callout>}}_0.$

(b2) put initial value i=0, initial minor arc collection A={[0,2 ⁿ], minor arc collection be circle superior function value may with f (p ₀) set of circular arc of contrary sign, its element shape is as [s, t], and wherein s, t are respectively the Along ent labels at circular arc two ends, and [s, t] represents by Along ent q _s, q _s+1..., q _tthe circular arc being linked to be, initial minor arc collection only has one from q ₀link

single circular arc, be whole circumference in fact.

(b3) the n bit of i is represented to b ₁b ₂b _nbackward, obtains binary number b _nb ₂b ₁, the integer of its expression is j.

(b4) if q _jdo not drop in arbitrary circular arc of minor arc collection A, skip to step (b7).

(b5) calculate f (q _j), if this functional value and f (p ₀) contrary sign, finish judgement, return results as " circle intersects with Implicit Curves ", otherwise, carry out minor arc evolution, according to f (q _j) be worth with directional derivative upper bound M and get rid of f (q) certainly and f (p ₀) the Along ent q of jack per line, the circular arc in A is deleted, shunk and divides processing, dwindle the coverage of minor arc collection.

(b6) if minor arc collection

finish evaluation algorithm, return results as " circle is non-intersect with Implicit Curves ".

(b7) putting i is i+1, if i=2 ⁿ, finish evaluation algorithm, return results as " circle is non-intersect with Implicit Curves ", otherwise go to step (b3).

Wherein the evolution of the minor arc in step (b5) can be more specific, specifically comprises following steps:

(c1) if | f (q _j) | > 2Mr, puts

finish;

Wherein:

R is radius of a circle;

M is the upper bound of the directional derivative of f (x, y),

$M=\underset{\mathrm{\&Omega;}}{\text{max}}\left|\right|\&dtri;f(x,y)\left|\right|=\underset{\mathrm{\&Omega;}}{\max }\left|\right|(\frac{\&PartialD;f(x,y)}{\&PartialD;x},\frac{\&PartialD;f(x,y)}{\&PartialD;y})\left|\right|<+\&infin;.$

(c2) calculate $a=2\arcsin \frac{\left|f\left(q_j\right)\right|}{2\mathrm{Mr}}.$

(c3) calculate

wherein

represent to take off integer, now f (q _i) must with f (p ₀) jack per line (j-k≤i≤j+k), obtain 2k+1 jack per line Along ent.

(c4) if j=0, directly by circular arc [0,2 ⁿ] be punctured into [k+1,2 ⁿ-k-1], otherwise to each circular arc of minor arc collection A [s, t], minute following five kinds of situations are done respectively to develop and are processed:

(c4.1) j-k > t or j+k < s: keep [s, t] constant;

(c4.2) j+k >=t >=j-k > s:[s, t] be punctured into [s, j-k-1];

(c4.3) t > j+k >=s >=j-k:[s, t] be punctured into [j+k+1, t];

(c4.4) j+k >=t >=s >=j-k: [s, t] deleted from A;

(c4.5) t > j+k >=j-k > s:[s, t] be split into [s, j-k-1] and [j+k+1, t] two circular arcs.

(c5) finish.

Because the only element of initial minor arc collection is a circumference, after develop first (i=0), become real circular arc, but be likely major arc (central angle is greater than π), might not be minor arc.But when developing (i=1) for the second time, due to the special j value calculating method of the present invention, by i=1=(00...001) ₂, obtain j=(100...00) ₂=2 ⁿthe Along ent q of the/2,2nd processing _jmust be the 1st processing Along ent q ₀antipodal point.If this point is jack per line point, algorithm finishes, otherwise carry out the 2nd time, develops, and two circular arcs (if yes) after evolution must be minor arc.This explanation has all only had real minor arc in minor arc collection after developing for the 2nd time always, and these minor arcs constantly shrink, divides or disappear, and always coverage is constantly dwindled, therefore claim that this algorithm is minor arc evolution algorithmic.

Technique effect of the present invention:

1. accuracy and robustness.Numerical procedure of the present invention can reach in advance given arbitrarily small allowable error, and is not subject to the impact of curve geometry, does not exist and falls into local extremum, relies on Initial value choice, the problem such as do not restrain.

2. efficiency of algorithm is high.Algorithm of the present invention is from coil to coil search, point by point scanning " quite force method " outwardly, but actual algorithm speed is not slow.Circle is the main determining factor of efficiency of algorithm with the crossing judgement of Implicit Curves, no matter justifies to intersect with Implicit Curves or non-intersect, all can judge fast.Specifically, in the situation that circle and Implicit Curves are crossing, owing to adopting binary inverse order code (as step (b3)) technology, the sequential processes that algorithm can evenly be encrypted gradually by Along ent, can prove and can at O, run into contrary sign point and finish from step (b5) in (1) time; In circle and the disjoint situation of Implicit Curves, algorithm can be along with the carrying out of minor arc evolution, and minor arc collection becomes fast empty and finishes from step (b6).So algorithm can finish Intersection judgement within very short time, thereby try to achieve fast the distance that a little arrives Implicit Curves.

3. less demanding to the slickness of the function f (x, y) of Implicit Curves.Newton iteration method requires f (x, y) single order differential exists and is continuous, other global approach requires f (x, y) possesses certain global nature, and this algorithm is only required and can be led, and directional derivative bounded, the upper bound is M, in fact the applicable condition of this algorithm can be more loose, only need meet the Lipschitz condition of continuity: | f (p)-f (q) |≤M||p-q||.

4. because algorithm of the present invention is always from contrary sign point, first carry out times compression process of " constantly dwindling; until run into non-intersect concentric circles ", so expanding, the radius of circle that can not exist the method for " constantly expanding, until run into crossing concentric circles " to exist is twice " failing to get or achieve what one wants " phenomenon of just crossing whole branch of a curve.

Accompanying drawing explanation

Fig. 1 is the algorithm general flow chart of preferred embodiment of the present invention.

Fig. 2 is that circle and the Implicit Curves in preferred embodiment of the present invention intersects evaluation algorithm process flow diagram.

Fig. 3 is the minor arc evolution process flow diagram in preferred embodiment of the present invention.

Fig. 4 is the minor arc evolution schematic diagram in preferred embodiment of the present invention.

Fig. 5 is the minor arc evolutionary process instance graph of non-intersect situation in preferred embodiment of the present invention.

Fig. 6 intersects the minor arc evolutionary process instance graph of situation in preferred embodiment of the present invention.

Embodiment

A point p in given two dimensional surface ₀=(x ₀, y ₀) and a bounded domain Ω on have Implicit Curves f (x, y)=0 of definition, some p ₀general steps to the numerical evaluation of Implicit Curves distance is as follows:

(1) get the functional value f (p) and f (p of f ₀) the some p of contrary sign or null value;

(2) getting initial inside radius is r ₁=0, initial external radius is r ₂=|| p ₀-p||;

(3)r＝(r ₁+r ₂)/2；

(4) judgement is with a p ₀whether the circle and the Implicit Curves that for the center of circle, r, are radius intersect;

(5) if intersect, put r ₂=r, otherwise, r put ₁=r;

(6) if r ₂-r ₁< ε (ε is allowable error), returns to (r ₁+ r ₂)/2 are as putting p ₀to the distance of described Implicit Curves, algorithm finishes, otherwise, go to step (3).

Referring to Fig. 1, it is for the algorithm general flow chart of preferred embodiment of the present invention.For asking a p ₀with the distance of Implicit Curves f (x, y)=0, first, perform step 101, find a f (p) and f (p ₀) contrary sign or null some p, even f (p ₀) > 0, get f (p)≤0, if f is (p ₀) < 0, get f (p)>=0 (f (p ₀the situation of)=0 needn't consider, this time point is on Implicit Curves, and required distance is 0).Finding contrary sign point methods has random cultellation, directly observes the methods such as specified point of getting, after be otherwise noted.Then perform step 102, give inside radius r ₁with external radius r ₂initialize is 0 He respectively || p ₀-p||, with p ₀for the center of circle, r ₁for the circle (now deteriorating to a little) of radius and Implicit Curves non-intersect, and with r ₂for the concentric circles of radius and Implicit Curves intersect.Then enter a cyclic process, first perform step 103, get median radius r, then by step 104 judgement, take the circle that r is radius and whether intersect with Implicit Curves, if crossing, perform step 105, replacement external radius r ₂for r, inside radius r ₁remain unchanged, otherwise, perform step 106, replacement inside radius r ₁for r, external radius r ₂remain unchanged.Inside radius or external radius perform step 107 after resetting, judgement r ₂-r ₁whether be less than allowable error ε, if r ₂-r ₁< ε, performs step 108, returns to r ₁with r ₂mean value as required distance, algorithm finishes, otherwise, go to step 103 continuation circulations and carry out.

In above-mentioned implementation step 101, find contrary sign point and be easy to realize.Have two ways at least, first way is random selecting point, the Area Ratio on the occasion of region and negative territory that might as well be located at f (p) in the Ω of region is u:v (0 < u, v < 1, u+v=1), the probability that n random selecting point obtains the non-negative point of f (p) is 1-v ⁿ, obtaining non-probability is on schedule 1-u ⁿ, both all can increase and be tending towards fast 1 with n.In addition, if fruit dot is for one group of point { p to the calculating of Implicit Curves distance ₀, p ₁, p ₂... calculate, be first p ₀calculate any one nonnegative value point of obtaining in the process of distance and non-can be to other p in group on the occasion of putting ₁, p ₂... as initial contrary sign point, use.If encounter the point of a f (q)=0, such point not only can be used as nonnegative value point but also can be used as non-on the occasion of point.Second way is direct observation, implicit expression function f (x major part to particular form, y), such as binary polynomial, tensor product B-splines, can be according to function f (p)=f (x, y) monotonicity in particular orientation (as coordinate axis), gradually through observing or get particular value, just directly seek nonnegative value point, non-on the occasion of point, even zero point.

Referring to Fig. 2, it,, for the circle in preferred embodiment of the present invention and the crossing evaluation algorithm process flow diagram of Implicit Curves, is the further refinement of Fig. 1 step 104.First perform step 201, get minimum positive integer n, make circumference 2 ⁿevery section of arc length after decile is less than allowable error ε, and n meets

$\frac{2\mathrm{\&pi;r}}{2^n}<\mathrm{\&epsiv;}\&le;\frac{2\mathrm{\&pi;r}}{2^{n-1}}$

Wherein r is described radius of a circle.

The Along ent of getting on circumference is

$q_i=(x_0+\mathrm{<figure-callout\; id="2"\; label="r\; cos"\; filenames="HSA0000097231850000011.png,HSA0000097231850000021.png"\; state="\{\{state\}\}">r</figure-callout>}\cos \frac{2\mathrm{\&pi;i}}{2^n},{\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="HSA0000097231850000011.png,HSA0000097231850000021.png"\; state="\{\{state\}\}">y</figure-callout>}}_0+\mathrm{<figure-callout\; id="2"\; label="r\; sin"\; filenames="HSA0000097231850000011.png,HSA0000097231850000021.png"\; state="\{\{state\}\}">r</figure-callout>}\sin \frac{2\mathrm{\&pi;i}}{2^n}),(i=\mathrm{0,1},...,2^n-1)$

Wherein r and p ₀=(x ₀, y ₀) be respectively described radius of a circle and the center of circle, an array q[0..2 for these Along ents ⁿ-1] represent.Then perform step 202, the initial value of putting i is 0, and the original state of putting minor arc collection A is { [0,2 ⁿ], only have a circular arc, it is " circular arc " of whole circumference in fact.Execution step 203, calculates i as the backward number of n bit, if i=is (b ₁b ₂b _n) ₂, j=(b _nb ₂b ₁) ₂.Then perform step 204, judge Along ent q[j] whether drop in certain circular arc of A, if do not drop on arbitrary circular arc, perform step 211 with step 212:i increasing 1 and judge whether i reaches maximal value 2 ⁿ, otherwise carry out series of steps below.First be step 205, computing function value f (q[j]), is then step 206, judge this functional value whether with f (p ₀) contrary sign.If contrary sign (comprise f (q[j])=0), perform step 207, return to judged result " circle intersects with Implicit Curves ", algorithm finishes; Not so, f (q[j]) and f (p ₀) strict jack per line, perform step 208, carry out minor arc evolution.Minor arc develop be according to f (q[j]) value with directional derivative upper bound M eliminating f (q) sure and f (p ₀) the Along ent q of jack per line, the circular arc in A is deleted, shunk and divides processing, dwindle the coverage of minor arc collection.It is step 209 that minor arc develops afterwards, judges whether minor arc collection is empty, if it is empty, returns to judged result " circle is non-intersect with Implicit Curves ", and algorithm finishes, otherwise, perform step 211 with step 212:i increasing 1 and judge whether i arrives maximal value 2 ⁿ.

If run into after i increases 1 and reach maximal value 2 in step 211 and step 212 ⁿ, illustrate that all Along ents are disposed, do not run into contrary sign point, so perform step 210, return to judged result " circle is non-intersect with Implicit Curves ", algorithm finishes.

The situation that finishes algorithm by step 212 → 210 is undesirable because algorithm process all Along ents, efficiency is lower.In fact, this evaluation algorithm can not finish from step 212 → 210 on very large probability, and by finishing in step 206 → 207 or step 209 → 210, in the situation that circle and Implicit Curves are crossing, can at O, in (1) time, by step 206 → 207, finish, this point is guaranteed by Along ent special processing order of the present invention (seeing step 203); In circle and the disjoint situation of Implicit Curves, can be along with develop the carrying out of (step 208) of minor arc, the coverage of minor arc collection is dwindled fast, finally becomes empty, so finish by step 209 → 210.So the expectation value of judging efficiency is very high.

As shown in Figure 3, it,, for the better implementation step that minor arc develops, is the further refinement of Fig. 2 step 208.Now, f (q[j]) and f (p ₀) strict jack per line, first perform step 301, judgement | f (q[j] | > 2Mr, wherein:

R is radius of a circle;

M is the upper bound of the directional derivative of function f (x, y), that is:

$M=\underset{\mathrm{\&Omega;}}{\text{max}}\left|\right|\&dtri;f(x,y)\left|\right|=\underset{\mathrm{\&Omega;}}{\max }\left|\right|(\frac{\&PartialD;f(x,y)}{\&PartialD;x},\frac{\&PartialD;f(x,y)}{\&PartialD;y})\left|\right|<+\&infin;.$

The binary function that the upper bound M of directional derivative uses various typical curves, be easy to estimate.In fact, z=f (x, y) can regard the space curved surface of an explicit representation as, so to binary polynomial f (x, y), M can be according to Polynomial Coefficient Estimation, to parametric surfaces such as B é zier curved surface, tensor product B-batten, NURBS, M can and control parameter estimation (referring to document < < Computer-aided Geometric Design > > chapter 2 2.3-2.7 according to basis function, kingdom Jin,Wang state is clear, Zheng Jianmin work, Higher Education Publishing House, 2001), the curved surface that matching obtains to estimating according to distance field, more easily estimate, because to have the mould of directional derivative be 1 feature (the Algebraic B-Spline Curve Reconstruction > > referring to document < < based on Signed Distance Field to distance field accurately everywhere, Li Yun sunset, Feng ties green grass or young crops, Jin Xiaogang, < < Journal of Software > > the 18th volume the 9th phase 2306-2317 page, 2007), so M can be estimated as a number that ratio 1 is slightly large, such as 1.5.

Even if run into the situation that is difficult to estimation, M is estimated as

$M=\frac{v_{\max }-v_{\min }}{D}\&times;20$

Wherein:

V _maxfor the maximal value of f (x, y) in the Ω of region;

V _minfor the minimum value of f (x, y) in the Ω of region;

D is the size (diameter of border circular areas, the length and width mean value of rectangular area) of region Ω.

If the maximal value of f (x, y) and minimum value are also difficult to estimation, M gets large several 1000.

About the estimation of 2 explanation: the first, M of the estimation of M, only affect the sampling density of Along ent of the circle of algorithm of the present invention, although underestimate, likely there is deviation in the calculating at the sharp-pointed place of Implicit Curves only.Can imagine, if at all Along ent q _j, functional value f (q _j) and f (p ₀) jack per line, according to the present invention, be mistaken for circle non-intersect with Implicit Curves, and it is in fact crossing, this explanation implicit expression function f (x, y) there is a long and narrow contrary sign partly inner by " stretching into " circle between two adjacent Along ents, so curve exists a sharp features, and such Implicit Curves is unsettled to curve and curve modeling, in the technical research of actual Geometric Modeling, originally just need to avoid.The second, if be the most conservative estimation M (such as M=1000), only have influence on judging efficiency when the present invention justifies and Implicit Curves is non-intersect, the judging efficiency that intersects situation is unaffected.

Return Fig. 3, if step 301 judgment result is that very, illustrate that whole circle is in the jack per line region of f (x, y), perform step 302, put minor arc and integrate A as empty set, finish; Otherwise, perform step 303 and step 304, determine must with f (p ₀) number of Along ent of jack per line, step 303 is to calculate the radian (one-sided) of jack per line arc, step 304 is to calculate the number k (one-sided) of jack per line Along ent, so by q[j] centered by total 2k+1 the jack per line Along ent in both sides (comprise q[j] own), label is from j-k to j+k.Then according to jack per line arc [j-k, j+k] to the processing of developing of minor arc collection, be first to perform step 305, see and whether processing first Along ent, when first Along ent of processing, notice that jack per line arc covers q[0] both sides, so perform step 306, circumference becomes circular arc [k+1,2 after shrinking ⁿ-k-1]; While processing later Along ent, perform step 307, to each circular arc in minor arc collection A [s, t], minute 5 kinds of situations are processed respectively, as step 308～312.5 kinds of situations as shown in Figure 4, have been listed the various evolution schematic diagram of difform jack per line arc to same minor arc, and wherein thick line arc 401 represents current minor arc [s, t], and fine rule arc 402 represents jack per line arc [j-k, j+k].

Situation 1 (as Fig. 4 (a)): j-k > t or j+k < s, illustrate that jack per line arc and current minor arc do not hand over, keep [s, t] constant.

Situation 2 (as Fig. 4 (b)): j+k >=t >=j-k > s, illustrate that jack per line arc and current minor arc afterbody are overlapping, [s, t] is punctured into [s, j-k-1].

Situation 3 (as Fig. 4 (c)): t > j+k >=s >=j-k, illustrate that jack per line arc and current minor arc stem are overlapping, [s, t] is punctured into [j+k+1, t].

Situation 4 (as Fig. 4 (d)): j+k >=t >=s >=j-k, illustrates that jack per line arc has covered whole current minor arc, deletes [s, t] from A.

Situation 5 (as Fig. 4 (e)): t > j+k >=j-k > s, illustrate that jack per line arc drops on current minor arc inner, [s, t] is split into [s, j-k-1] and [j+k+1, t] two minor arcs.

Each minor arc in A is disposed, finishes.

Fig. 5 is the minor arc evolutionary process instance graph of non-intersect situation, shows the substep dynamic changing process that in circle and the non-intersect situation of Implicit Curves, minor arc develops, n=4 wherein, and it is 2n=16 that decile is counted, and supposes j=0,1,2 ..., 15 o'clock jack per line arc [j-k, j+k] in k value (determining according to f functional value) be respectively 3,3,4,3,2,1,0,1,1,0,0,1,1,2,3,3.

(a) i=0, j=0, calculating f (q[j]) obtain k=3, shrink [0,16], obtain A={[4,12] };

(b) i=1, j=8, calculating f (q[j]) obtain k=1, division [5,11], obtains A={[4,6], [10,12] };

(c) i=2, j=4, calculating f (q[j]) obtain k=2, delete [4,6], obtain A={[10,12] };

(d) i=3, j=12, calculating f (q[j]) obtain k=1, shrink [10,12], obtain A={[10,10] };

(e) i=4, j=2, j does not drop in arbitrary minor arc;

(f) i=5, j=10, calculating f (q[j]) obtain k=0, delete [10,10],

return results " non-intersect ", finish.

The method of pointwise judgement needs 16 f functional values to calculate in order, and adopt minor arc to develop, only needs to calculate for 5 times.

Fig. 6, for intersecting the minor arc evolutionary process instance graph of situation, shows circle and intersects with Implicit Curves the substep dynamic changing process that in situation, minor arc develops, n=4 wherein, and it is 2 that decile is counted ⁿ=16, suppose j=0,1,2 ..., the k value in 15 o'clock jack per line arc [j-k, j+k] (determining according to f functional value) is respectively 2,2,1,0,--,--,--,--, 0,--,--,--, 0,1,2,2.(--represent contrary sign)

(a) i=0, j=0, calculating f (q[j]) obtain k=2, shrink [0,16], obtain A={[3,13] };

(b) i=1, j=8, calculating f (q[j]) obtain k=0, division [3,13], obtains A={[3,7], [9,13] };

(c) i=2, j=4, calculating f (q[j]) obtain contrary sign point, return results " intersecting ", finish.

The method of pointwise judgement on average needs 8 f functional values to calculate in order, and adopt minor arc to develop, only needs to calculate for 3 times.

Claims (3)

1. the some p in a two dimensional surface ₀numerical computation method to Implicit Curves distance, wherein Implicit Curves is for there being the binary function f (x of definition and directional derivative bounded on certain bounded domain Ω, y) single branch plane curve of determining according to its null value collection, is characterized in that comprising following steps: find a functional value f (p) and f (p ₀) contrary sign or null initial point p, now with p ₀for the center of circle, with || p ₀-p|| is that the circle of radius must intersect with Implicit Curves; Putting inside radius is 0, and external radius is || p ₀-p||; Get again the mean value that median radius is inside radius and external radius, then judge with p ₀whether the circle and the Implicit Curves that for the center of circle, the median radius of take, are radius intersect, if intersect, under external radius, are adjusted to median radius, if non-intersect, are adjusted to median radius on inside radius; Get again median radius, judgement Intersection, adjust inside radius or external radius, so repeatedly, until the difference of external radius and inside radius is less than allowable error, return to the mean value of inside radius and external radius as a p ₀distance to described Implicit Curves.

2. numerical computation method as claimed in claim 1, is characterized in that, wherein the whether crossing method of judgement circle and Implicit Curves comprises following steps:

(1) get minimum positive integer n, make circumference 2 ⁿafter decile, every section of circular arc length is less than allowable error, and the Along ent distributing successively after circumference equal dividing is $q_0,q_1,......,q_{2^n-1},q_{2^n}=q_0;$

(2) put initial value i=0, initial minor arc collection A={[0,2 ⁿ], minor arc collection be circle superior function value may with f (p ₀) set of circular arc of contrary sign, its element shape is as [s, t], and wherein s, t are respectively the Along ent labels at circular arc two ends, and [s, t] represents by Along ent q _s, q _s+1..., q _tthe circular arc being linked to be, initial minor arc collection only has one from q ₀link

single circular arc, be whole circumference in fact;

(3) the n bit of i is represented to b ₁b ₂b _nbackward, obtains binary number b _nb ₂b ₁, the integer of its expression is j;

(4) if q _jdo not drop in arbitrary circular arc of minor arc collection A, skip to step (7);

(5) calculate f (q _j), if this functional value and f (p ₀) contrary sign, finish judgement, return results as " circle intersects with Implicit Curves ", otherwise, carry out minor arc evolution, according to f (q _j) be worth with directional derivative upper bound M and get rid of f (q) certainly and f (p ₀) the Along ent q of jack per line, the circular arc in A is deleted, shunk and divides processing, dwindle the coverage of minor arc collection;

(6) if minor arc collection

finish evaluation algorithm, return results as " circle is non-intersect with Implicit Curves ";

(7) putting i is i+1, if i=2 ⁿ, finish evaluation algorithm, return results as " circle is non-intersect with Implicit Curves ", otherwise go to step (3).

3. crossing evaluation algorithm as claimed in claim 2, is characterized in that, wherein the evolution of the minor arc in step (5) comprises following steps:

If 1. | f (q _j) | > 2Mr, puts

finish;

Wherein:

R is radius of a circle;

M is the upper bound of the directional derivative of f (x, y),

$M=\underset{\mathrm{\&Omega;}}{\text{max}}\left|\right|\&dtri;f(x,y)\left|\right|=\underset{\mathrm{\&Omega;}}{\max }\left|\right|(\frac{\&PartialD;f(x,y)}{\&PartialD;x},\frac{\&PartialD;f(x,y)}{\&PartialD;y})\left|\right|<+\&infin;;$

2. calculate $a=2\arcsin \frac{\left|f\left(q_j\right)\right|}{2\mathrm{Mr}};$

3. calculate wherein

represent to take off integer, now f (q _i) must with f (p ₀) jack per line (j-k≤i≤j+k), obtain 2k+1 jack per line Along ent;

If 4. j=0, directly by circular arc [0,2 ⁿ] be punctured into [k+1,2 ⁿ-k-1], otherwise to each circular arc of minor arc collection A [s, t], minute following five kinds of situations are done respectively to develop and are processed:

(a) j-k > t or j+k < s: keep [s, t] constant;

(b) j+k >=t >=j-k > s:[s, t] be punctured into [s, j-k-1];

(c) t > j+k >=s >=j-k:[s, t] be punctured into [j+k+1, t];

(d) j+k >=t >=s >=j-k: [s, t] deleted from A;

(e) t > j+k >=j-k > s:[s, t] be split into [s, j-k-1] and [j+k+1, t] two circular arcs;

5. finish.

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