CN103871107B - A kind of method for quantitative measuring of isosurface extraction precision - Google Patents
A kind of method for quantitative measuring of isosurface extraction precision Download PDFInfo
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- CN103871107B CN103871107B CN201410067145.3A CN201410067145A CN103871107B CN 103871107 B CN103871107 B CN 103871107B CN 201410067145 A CN201410067145 A CN 201410067145A CN 103871107 B CN103871107 B CN 103871107B
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Abstract
The invention discloses the method for quantitative measuring of a kind of isosurface extraction precision, by choosing several sampled point on tri patch in this method for quantitative measuring, calculate sampled point to the Euclidean distance of corresponding point in true curved surface, then according to the distance of sampled point Yu corresponding point, respectively using all sampled points in all tri patchs to the Euclidean distance maximum of corresponding corresponding point as maximum deviation degree, meansigma methods is as average departure degree, mean square deviation is as deviation fluctuating margin, thus accurate and multi-angle measurement obtains isosurface extraction precision.The method for quantitative measuring of the present invention efficiently solves the metric question of the deviation of isosurface extraction result, realizes quantitative measurement from qualitative analysis, facilitates user to select suitable isosurface extraction method according to described application scenario.And method for quantitative measuring realization itself with use isosurface extraction method, applied widely, can have the extraction accuracy of effective quantitative measurement all isosurface extraction method.
Description
Technical field
The present invention relates to computer graphics isosurface extraction field, be specifically related to a kind of isosurface extraction precision
Method for quantitative measuring.
Background technology
For Three-dimensional calibration volume data or implicit function, contour surface refers to that its scalar value or functional value are a certain
The set of the point of constant.By selecting different scalar value or functional value, different contour surfaces can be obtained,
Thus contribute to visualizing and analyzing relevant volume data and implicit surface.Therefore, isosurface extraction is three
Underlying issue in the application such as dimension scalar field data visualization, implicit surface show, three-dimension curved surface reconstruction.
The main thought of isosurface extraction algorithm is: for Three-dimensional calibration volume data, it is common that by discrete voxel list
Unit's set description；Implicit function is carried out spatial sampling, generates basic solid primitives unit；Then at basic solid primitives list
Contour surface is approached with Linear Triangular dough sheet in Yuan.Marching tetrahedra algorithm (mobile tetrahedron,
Mt algorithm) and marching cubes algorithm (marching cubes algorithm, mc algorithm) be two typical
Isosurface extraction algorithm.
In mc algorithm, basic processing unit is cube.The basic thought of algorithm is in traversal volume mesh
All voxels, judge the position relationship between 8 angle points of each voxel and true curved surface successively, then
Connected into tri patch by finding intersection the connected mode according in the most given look-up table, thus obtain
Expression is approached in this voxel in value face.Look-up table has 256 (=28) the most corresponding 256 kinds of situations of individual entry.
The handling process of mt algorithm is similar with mc algorithm flow, first determines whether the position of 4 angle points and true curved surface
Relation, utilizes linear interpolation or high order approximation method to obtain the intersection point of contour surface and voxel limit, then according to
The intersection point tried to achieve is connected into tri patch by the look-up table of mt algorithm.The entry of the look-up table of mt algorithm has
Article 16, (=24)。
Isosurface extraction be Three-dimensional calibration data visualization, implicit surface show, three-dimension curved surface reconstruction etc.
Underlying issue in application.Substantially, discrete cube volume data is to initial body data or implicit function
Three linear approximation represent, now the contour surface in each three linear cube is cubic algebra curved surface, and
Mc algorithm acquired results is the linear approximation for this cubic algebra curved surface, therefore exists in mc algorithm
The process of twice approximation；Discrete tetrahedron volume data is the linear approximation to initial body data or implicit function,
Under the supposed premise of linear approximation, mt algorithm acquired results is accurate, therefore exists in mt algorithm
Once approach.Obviously, in mc and mt algorithm, the sampling density of volume data or implicit function is the biggest, etc.
The approximation accuracy in value face is the highest, but the triangular plate number produced is the most, the space of algorithm and time loss
Also can increase accordingly；Vice versa.Therefore, the approximation accuracy of quantitative measurement both representativeness algorithms for
Relevant science and engineer applied have important reference value.
In sum, all there is deviation in the extraction result of isosurface extraction algorithm, and deviation is essentially from two sides
Face: (1) is to the linear of voxel cell or three linear hypothesis；(2) tri patch is used to approach in voxel cell
Contour surface.At present, isosurface extraction algorithm is of a great variety, but passes judgment on and extract the achievement of result precision the most very
Few.And only a small amount of extraction accuracy judgment criteria research all concentrates on the process of difference situation, extraction
The qualitative aspects such as result difference on subjective vision.Therefore, industry lack one can quantitative measurement equivalent
The judgment criteria of face extraction accuracy and the computational methods of correspondence.
Summary of the invention
For the deficiencies in the prior art, the present invention proposes the method for quantitative measuring of a kind of isosurface extraction precision.
The method for quantitative measuring of a kind of isosurface extraction precision, described isosurface extraction precision includes average inclined
From degree, the fluctuating margin of deviation and maximum deviation degree, described method for quantitative measuring includes:
(1) on each tri patch obtained by the extraction of isosurface extraction algorithm, several samplings are uniformly taken
Point；
(2) determine all sampled points corresponding point in true curved surface, and each sampled point is to corresponding point
Euclidean distance；
(3) compare each sampled point and obtain the Euclidean of maximum to the size of the Euclidean distance of corresponding corresponding point
Distance, calculates the meansigma methods of all sampled points of all tri patchs and the Euclidean distance of corresponding corresponding point and equal
Variance, and using described meansigma methods as average departure degree, fluctuates width using described mean square deviation as deviation
Degree, using maximum Euclidean distance as maximum deviation degree.
The method for quantitative measuring of the isosurface extraction precision of the present invention utilize the result of isosurface extraction to obtain
Quantitative to contour surface extraction accuracy of tri patch and the distance of true curved surface.Calculating isosurface extraction obtains
The distance of tri patch and true curved surface time, by choosing several sampled point on tri patch, meter
Calculation sampled point is to the distance of corresponding point in true curved surface, then according to the Euclidean of sampled point with corresponding corresponding point
Distance, by comparing the Euclidean distance that obtains maximum, (this distance is all result dough sheets obtained of extraction
Forward direction Hausdorff distance to true curved surface) as maximum deviation degree, and calculate all Euclidean distances
Meansigma methods and mean square deviation, respectively as maximum deviation degree, average departure degree and deviation fluctuating margin, from
And accurately and the measurement of multi-angle obtains isosurface extraction precision, efficiently solve the inclined of isosurface extraction result
The metric question of difference, achieves quantitative measurement from qualitative analysis, facilitates user according to described application scenario
Select suitable isosurface extraction method.And method for quantitative measuring realization itself with use isosurface extraction side
Method, applied widely, can there is the extraction accuracy of effective quantitative measurement all isosurface extraction method.
Hausdorff distance describes the spacing of two compact subsets in metric space, is widely used in such as geometry
The fields such as modeling, Model Matching.Hausdorff distance has directivity, including forward direction Hausdorff distance and
Backward Hausdorff distance.In prior art, extract result and (use the extraction of isosurface extraction algorithm to obtain
The set of all tri patchs) in the forward direction Hausdorff distance of true curved surface according to formula:
Being calculated, wherein S is for extracting result, and D is true curved surface, and p is the point in S,
Represent the maximum in point all of in S to the Euclidean distance of true curved surface.These computational methods need traversal
In extraction result, institute is a little, and each point is required for traveling through institute in D, and a little, therefore amount of calculation is very greatly, imitates
Rate is low.The present invention directly utilizes the maximum approximation conduct in Euclidean distance between sampled point and true curved surface
Three extraction results, to the forward direction Hausdorff distance of true curved surface, greatly reduce amount of calculation, are conducive to improving
Measure efficiency.
In described step (2) corresponding point of each sampled point be in true curved surface with the Euclidean of this sampled point away from
From minimum point.
How described step (2) is realized by optimized algorithm.
As preferably, described step (2) use interior point method determine that each sampled point is in true curved surface
Corresponding point, and each sampled point is to the Euclidean distance of corresponding point.With other optimization method phases such as positive collection methods
Ratio, interior point method is polynomial time algorithm, and solution efficiency is higher.
The sampled point number taken on each tri patch is identical.The sampled point taken in described step (1)
For the intersection point between three summits of tri patch and the bisector on three bases being parallel to tri patch.
Sampled point has 3 three summits being tri patch, additionally, a part of intersection point is 2 bisectors
Intersection point, part is the intersection point of 3 bisectors, and wherein the intersection point of 2 bisectors is positioned at the limit of tri patch
On, the intersection point of 3 bisectors falls in tri patch.
Described bisector is n bisector, and wherein n is the natural number being not zero, and selects according to practical situation,
When n=1 i.e. represents not decile, directly with three summits of tri patch as sampled point.Sampled point number is actual
On depend on the value of n.The sampled point number taken on each tri patch is (n+1) × (n+2)/2.
The isodisperse of described bisector is 2～6.Isodisperse is the highest, then the number of sampled point is the most.Generally
The most result of calculation of sampled point is the most accurate, but amount of calculation is relatively big, and efficiency is low, considers generally 2～6.
As preferably, the isodisperse of described bisector is 4.Consider the accurate of efficiency and result of calculation
Degree, during usual triangle intersection point between 4 bisectors that sampled point is three bases being parallel to tri patch,
I.e. one tri patch is sufficient for accuracy computation requirement when taking 15 sampled points.
The method for quantitative measuring of the isosurface extraction precision of the present invention utilize isosurface extraction result with true
The distance of curved surface quantitatively, with multi-angle measure isosurface extraction algorithm precision, efficiently solve contour surface and carry
Take the metric question of the deviation of result, realize quantitative measurement from qualitative analysis, facilitate user according to described
Application scenario selects suitable isosurface extraction method.And method for quantitative measuring realization itself with use equivalence
Face extracting method, applied widely, can there is the extraction essence of effective quantitative measurement all isosurface extraction method
Degree.
Detailed description of the invention
Below in conjunction with detailed description of the invention, the method for quantitative measuring of the isosurface extraction precision of the present invention is entered
Row describes in detail.
The method for quantitative measuring measurement utilizing this contour surface extraction accuracy in the present embodiment utilizes marching cube
The extraction accuracy of the result of the isosurface extraction that algorithm (mc algorithm) completes, extracts the tri patch obtained
Number be 2752.
The method for quantitative measuring medium value face extraction accuracy of the isosurface extraction precision of the present embodiment includes averagely
Departure degree, the fluctuating margin of deviation and maximum deviation degree.
The method for quantitative measuring of the isosurface extraction precision of the present embodiment includes:
The method for quantitative measuring of a kind of isosurface extraction precision includes:
(1) on each tri patch obtained by the extraction of isosurface extraction algorithm, several samplings are uniformly taken
Point, specific as follows:
With between three summits of tri patch and the n bisector on three bases being parallel to tri patch
Intersection point is as the sampled point of this tri patch, and the sampled point number obtained is (n+1) × (n+2)/2, wherein n
For the natural number being not zero.
N=4 in the present embodiment, i.e. with three summits and three bases being parallel to tri patch of tri patch
Quarterline between intersection point as the sampled point of this tri patch.Obtain altogether 15 sampled points, its
In 3 be three summits of tri patch, 9 is that the intersection point of two four bisectors (is positioned at tri patch
Limit on), 3 is the intersection point (being positioned at tri patch) of three bisectors.
(2) determine all sampled points corresponding point in true curved surface, and each sampled point is to corresponding point
Euclidean distance.
The present embodiment utilize interior point method determine this sampled point unique corresponding point in true curved surface respectively, so
Rear calculating Euclidean distance between the two.The nonlinear optimal problem of this step actually belt restraining, its
Solve relevant being provided that
Majorized function is the Euclidean distance formula between sample point and corresponding point, and constraints is true curved surface
Toroidal function, initial point is sample point, the span of optimal solution be centered by sample point with voxel with
Etc. the cubic space of size, the function value that the iterations in exit criteria is 100, optimal solution is corresponding
It is 10-8, iteration ends step-length be 10-8。
(3) compare each sampled point and obtain the Euclidean of maximum to the size of the Euclidean distance of corresponding corresponding point
Distance, calculates the meansigma methods of all sampled points of all tri patchs and the Euclidean distance of corresponding corresponding point and equal
Variance, and using described meansigma methods as average departure degree, fluctuates width using described mean square deviation as deviation
Degree, using maximum Euclidean distance as maximum deviation degree.
In the present embodiment, kth tri patch is to the forward direction Hausdorff distance of true curved surfaceAccording to
Formula:
Calculate, wherein:
N is the number of the sampled point on each tri patch, N=15 in the present embodiment,
For kth triangle unilateral on i-th (i=1,2 ... N) individual sampled point, S be extract obtain
The set of all tri patchs, D is true curved surface, k=1,2 ... m, m are number (this of tri patch
M=2752 in embodiment),
For sampled pointEuclidean distance (i.e. sampled point to true curved surface DWith true curved surface D
In the Euclidean distance of corresponding point,It is right to representTake maximum.
The method maximum in all sampled points to the distance of true curved surface D should be used as tri patch mutually
To the forward direction Hausdorff distance of true curved surface, the forward direction Hausdorff distance obtained is approximation.
Utilize formula:
Calculate the meansigma methods of each sampled point and the Euclidean distance of corresponding corresponding point, nsFor all tri patchs
The sum of middle sampled point,Represent that traversal i and k is to allSummation.When isosurface extraction is calculated
When method result dough sheet number is m, each tri patch takes 15 sample points, therefore one has 15 × m
Sampled point, i.e. ns=15×m。
Utilize formula:
Calculate the mean square deviation of each sampled point and the Euclidean distance of corresponding corresponding point,
Represent traversal i and k pairSummation.
The above, the only detailed description of the invention of the present invention, but protection scope of the present invention is not limited to
This, any those familiar with the art, in the technical scope that the invention discloses, can readily occur in
Change or replacement, all should contain within protection scope of the present invention.
Claims (6)
1. the method for quantitative measuring of an isosurface extraction precision, it is characterised in that described contour surface carries
Take precision and include average departure degree, the fluctuating margin of deviation and maximum deviation degree, described quantitative measurement
Method includes:
(1) on each tri patch obtained by the extraction of isosurface extraction algorithm, several samplings are uniformly taken
Point；
(2) determine all sampled points corresponding point in true curved surface, and each sampled point is to corresponding point
Euclidean distance；
(3) compare each sampled point and obtain the Euclidean of maximum to the size of the Euclidean distance of corresponding corresponding point
Distance, calculates the meansigma methods of all sampled points of all tri patchs and the Euclidean distance of corresponding corresponding point and equal
Variance, and using described meansigma methods as average departure degree, fluctuates width using described mean square deviation as deviation
Degree, using maximum Euclidean distance as maximum deviation degree；
In described step (2) corresponding point of each sampled point be in true curved surface with the Euclidean of this sampled point away from
From minimum point.
2. the method for quantitative measuring of isosurface extraction precision as claimed in claim 1, it is characterised in that
Described step (2) use interior point method determine each sampled point corresponding point in true curved surface, and respectively
Individual sampled point is to the Euclidean distance of corresponding point.
3. the method for quantitative measuring of isosurface extraction precision as claimed in claim 2, it is characterised in that
The sampled point number taken on each tri patch is identical.
4. the method for quantitative measuring of isosurface extraction precision as claimed in claim 3, it is characterised in that
Sampled point is tri patch three summits taken in described step (1) and be parallel to the three of tri patch
Intersection point between the bisector on bar base.
5. the method for quantitative measuring of isosurface extraction precision as claimed in claim 4, it is characterised in that
The isodisperse of described bisector is 2～6.
6. the method for quantitative measuring of isosurface extraction precision as claimed in claim 5, it is characterised in that
The isodisperse of described bisector is 4.
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| CN101672637A (en) * | 2009-09-24 | 2010-03-17 | 华东理工大学 | Digitizing detection method of complicated curved face |
| CN101702237A (en) * | 2009-11-10 | 2010-05-05 | 南京大学 | Horizontal surface fitting method based on quadric error metrics (QEM) |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN101672637A (en) * | 2009-09-24 | 2010-03-17 | 华东理工大学 | Digitizing detection method of complicated curved face |
| CN101702237A (en) * | 2009-11-10 | 2010-05-05 | 南京大学 | Horizontal surface fitting method based on quadric error metrics (QEM) |
Non-Patent Citations (2)
| Title |
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| MEASURING ERROR BETWEEN SIMILAR SURFACES;Kestutis Jankauskas 等;《Information Technology & Control》;20081231;第37卷(第3期);第198-204页 * |
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