CN106570864B - Conic fitting method in image based on geometric error optimization - Google Patents
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Abstract
The present invention relates to a kind of conic fitting methods in image based on geometric error optimization, comprising: extracts the point m of the edge image of specific conic section M in imagei；Calculate point miTo the geometric distance d of conic section Mfa(mi,C)；To geometric distance dfa(mi, C) and linear weighted function iteration is carried out, Matrix C relevant to C is obtained with singular value decomposition method1；Using based on most short geometric distance d (mi, C) building objective function, calculating work as C=C1When small quantity Δ u when making the minimization of object functioni,ΔviAnd scale parameter λiValue, and be denoted as Δ u respectivelyi1,Δvi1And λi1；With Δ ui1,Δvi1And λi1For initial value, nonlinear optimization solution is carried out to the objective function, obtains the coefficient matrix C of conic section M2, according to coefficient matrix C2Generate the conic section M in described image after the optimization of specific conic section M2.The present invention, which realizes, combines the high efficiency of conic fitting method and high-precision ideal effect in image, and improves the precision of conic fitting in image.
Description
Technical field
The invention belongs to computer vision fields, and in particular to a kind of conic fitting based on geometric error optimization
Method.
Background technique
In various cultures and natural landscape, conic section is seen everywhere.Image conic fitting is as many
The initial processing step of application all receives attention in fields such as computer vision, commercial measurement, computer graphics.Image
Conic fitting robot navigation, virtual reality, in terms of have critically important application value.
By perspective projection, perspective camera is by a scene change at piece image.A specific secondary song in scene
Line is still a conic section in the image of projection, and still, the conic section in image may be round, oval, parabola,
The straight line of even one degeneration.If we do not know about the priori of scene before detecting a Classification of Quadratic
Know, then computer is difficult to learn the concrete type of conic section in image automatically.Therefore, it is (secondary to study general conic section
Curve type is unknown) fitting problems be very it is necessary to.When conic fitting and then to the type of conic section into
Row identification, then can automate and become easy.Conic fitting in the present invention, if referred to general without specified otherwise
Conic fitting.
A kind of naturally roadmap of conic fitting problem is using linear least square.But in reality
In the task of border, due to blocking, noise, factors, the precision of this method such as fuzzy be difficult to meet our requirement.Therefore, occur
A variety of different optimization algorithms are to improve the precision of conic fitting.At present have Statistics-Based Method, based on algebra away from
From the different precision for being used to improve conic fitting of method and method three classes based on geometric distance optimization algorithm.Base
It often assumes that picture noise obeys certain distribution in the method for statistics, and is based on first order Taylor series expansion, but due to usual
The noise profile of image is difficult accurately to obey the distribution assumed, and when noise is bigger, Taylor's single order series approximation is lost
Effect, so statistical method only can just have preferable fitting effect noise is smaller；Method based on algebraic distance, is established
Objective function is without geometry and physical meanings, to be changed with geometric transformation, therefore based on algebraic distance error in image
After transformation, error may then be become much larger；Method based on geometric distance is the method for orthogonal distance, but this method is each time
Each of iteration picture point requires to solve 4 equation of n th order n about an argument, thus not only complexity it is high and
And it is very sensitive to noise.Although there is the Sampson method of first approximation later, due to only it is a kind of approximately away from
From, therefore precision is not high.In addition, there are also one is the methods based on circle transformation, wherein when optimizing, in addition to optimization conic section
Outside parameter, each picture point needs separately set a parameter in iteration each time, and the parameter for needing to solve is extremely more, simplifies orthogonal
The solution of 4 equation of n th order n of distance, but the number of parameters of solution is significantly increased, therefore complexity is very high.In conclusion at present
High efficiency and high-precision can not be combined to the method for conic fitting in image.
The present invention is directed to develop one kind to meet conic fitting side in two dimensional image high-efficient, with high accuracy simultaneously
Method.
Summary of the invention
In order to solve the above-mentioned technical problem, i.e., the method for conic fitting in image can not be combined efficiently at present
Rate and high-precision problem, the invention proposes a kind of conic fitting methods in image based on geometric error optimization, can
To combine the high efficiency and high-precision of conic fitting in image.
A kind of conic fitting method based in geometric error optimization image proposed by the present invention, which is characterized in that
Include the following steps:
Step 1, the point m of the edge image of specific conic section M in image is extractedi, wherein mi=(ui vi1)T, i=
1...N；
Step 2, point m is calculatediTo the geometric distance d of conic section Mfa(mi, C), wherein C is the coefficient square of conic section M
Battle array；
Step 3, to geometric distance dfa(mi, C) and linear weighted function iteration is carried out, it is obtained and C phase with singular value decomposition method
The Matrix C of pass1；
Step 4, using based on most short geometric distance d (mi, C) building objective function, calculating work as C=C1When make target
Small quantity Δ u when function minimizationi,ΔviAnd scale parameter λiValue, and be denoted as Δ u respectivelyi1,Δvi1And λi1；
Step 5, with Δ ui1,Δvi1And λi1For initial value, nonlinear optimization solution is carried out to the objective function, is obtained
The coefficient matrix C of conic section M2, according to coefficient matrix C2It generates secondary after the optimization of specific conic section M in described image
Curve M2。
Preferably, the geometric distance dfa(mi, C) and it is the Sampson distance weighted, calculation method are as follows:
Wherein,A, b, c, d, e, f are respectively the matrix coefficient of conic section M,
Preferably, the most short geometric distance d (mi, C) calculation formula are as follows:
Wherein, Δ ui、ΔviFor small quantity.
The objective function is based on most short geometric distance d (mi, C) and constraint function determination, the solution of objective function
Analyse expression formula are as follows:
Wherein,λiFor scale ginseng
Number.
The constraint function are as follows:
In constraint function,
p+、p-For the intersection point of straight line L' and conic section M, L' is passing point miAnd with polar curve L=CmiOrthogonal straight line.
Preferably, to geometric distance dfa(mi, C) and carry out linear weighted function iteration method particularly includes:
Step 31, the relevant coefficient matrix C of conic section M is calculated(k), k expression the number of iterations；
Step 32, step 31 is repeated, when convergence meets condition, records coefficient matrix C at this time(k), and enable C1=C(k)。
Preferably, the relevant coefficient matrix C of the calculating conic section M(k)Method are as follows:
As k=0, linear system is solved using singular value decomposition methodI=1...N obtains coefficient
Matrix C(0)；
As k > 0, linear system is solved using singular value decomposition methodIt obtains and secondary song
The relevant coefficient matrix C of line M(k)；
WhereinTo enable C=C(k-1), and be calculated by following formula:
Preferably, the condition of convergence described in step 32 are as follows:
Wherein, V (C)=(a b c d e f)T, ε is preset threshold.
Preferably, p in the constraint functioniAnalytical Expression formula are as follows:
Preferably, coefficient matrix C is obtained in step 52Afterwards, pass throughTo C2Standardization, wherein | | C2||FTable
Show C2F norm.
Conic fitting method can be pushed away directly in a kind of image based on geometric error optimization proposed by the invention
Extensively in quadric fitting based on depth image or three-dimensional point.
Conic fitting method in a kind of image based on geometric error optimization proposed by the present invention, overcomes existing figure
The method of conic fitting can not combine high efficiency and defective as in, realize and combine two in image
The high efficiency of secondary curve-fitting method and high-precision ideal effect, and improve the precision of conic fitting in image.
Detailed description of the invention
Fig. 1 is polar curve L schematic diagram of the point m about conic section M；
Fig. 2 is point miTo the geometric distance d (m of conic section Mi,pi) schematic diagram；
Fig. 3 is the schematic diagram between exterior point and conic section M；
Fig. 4 is conic fitting method flow schematic diagram of the invention.
Specific embodiment
The preferred embodiment of the present invention described with reference to the accompanying drawings.It will be apparent to a skilled person that this
A little embodiments are used only for explaining technical principle of the invention, it is not intended that limit the scope of the invention.
Technical solution of the present invention is illustrated in order to clearer, below with reference to theory deduction and specific reality of the invention
Mode is applied technical solution of the present invention is described in detail.
The present invention, which is directed to, can not combine high efficiency and height to the method for conic fitting in image in the prior art
The problem of precision and propose, and the invention proposes a kind of new points to the geometric distance calculation method of conic section, is based on
This devises a kind of high efficient secondary curve-fitting method of linear weighted function, and further approaches most short geometric distance, has higher
Precision.The method of the present invention can be directly generalized in quadric fitting based on depth image or three-dimensional point.
1, the derivation of geometric distance calculation method of the present invention
For the image containing conic section, the edge graph picture point m of conic section M is extractedi=(ui vi 1)T, i=
1...N；
Shown in the coefficient matrix C such as expression formula (1) of conic section M:
In the absence of noise, available:
Point miPolar curve L about conic section M is Cmi, it is denoted as Cmi=L=(l1,l2,l3)T.Then miThe distance between L
Calculation method such as formula (2) shown in:
Wherein | | indicate the absolute value of the numerical value of element between two vertical lines.
By L=CmiIt substitutes into formula (2), then point miDistance to polar curve L may be updated as the expression formula as shown in formula (3):
Wherein,There are following relationships in G: a2+d2≥0,(ba-
d2)2>=0, det (G)=0.
For being fitted shown in the famous Sampson distance definition such as formula (4) of conic section:
Derivation result substitution formula (4) can be obtained into formula (5):
Compare formula (3) and formula (5), it is known that Sampson distance is point miTo the half of polar curve L distance.Such as Fig. 1 (a)
It is shown, point m is shown to polar curve L distance by runic line segment form, when there are noise, the distance calculated by formula (3) is that point m is arrived
The distance of polar curve L is d, and d/2 is Sampson distance；When noise is not present, m is located on conic section M, and L is cutting at m
Line, as shown in Fig. 1 (b), the value of formula (3) and formula (5) is 0 at this time.Due to the molecule of formula (5) typically refer to algebra away from
From therefore, Sampson distance is the algebraic distance of weighting.
Difference m is calculated using formula (3)iDistance value, and take its quadratic sum, or calculate difference m using formula (5)i
Distance value, and take its quadratic sum as fitting conic section cost function.One is differed between the two cost functions admittedly
Calibration amount 4.When therefore finding minimum value C by optimization, the two cost functions are of equal value.
As shown in Fig. 2, M is a conic section, miIt is a point, L=CmiIt is polar curve.Straight line L' passes through miAnd with pole
Line L-orthogonal, the intersection point of two straight lines are qi.Sampson distance is d (mi,qi)/2.With point miLocate the increase of noise, d (mi,qi)
Become increasing, and the error of Sampson distance is also increasing.Under normal conditions, the distribution of noise is not on image
Uniformly, therefore the noise for being put on different images, Sampson distance should be endowed different weighted values, but should not be
One fixed 1/2.
The present invention proposes a kind of new miShown in geometric distance such as formula (6) between M:
d(mi, C) and=min { d (p+,mi),d(p-,mi)} (6)
It enables
Then d (mi, C) and=d (mi,pi)；
Wherein, p+、p-Two o'clock is the intersection point of straight line L' and conic section M.
As shown in Fig. 2, d (mi,pi) can be than d (mi,qi) m is measured more accuratelyiThe distance between M.
In order to calculate d (mi, C), it needs to calculate pi, piIt can be solved and be obtained by formula (7):
Direct solution formula (7) is not easy to, and the present invention provides a kind of very succinct mode and obtains defining for formula (7)
Analytical form, specific as follows:
Q is calculated firsti.L=CmiIt can be expressed as L=(l1,l2,l3)T.L' is with L-orthogonal and by mi=(ui vi1)T,qi
It is the intersection point of straight line L and straight line L '.So available qiCalculation formula, as shown in formula (8):
Result is rewritten, qiCalculation formula can further be transformed to formula (9).
Wherein The last line of Matrix C to be replaced with the matrix after full 0 row.
Due to qiOn the linel, thereforeIn conjunction with L=CmiObtain formula (10):
Two solution p+、p-With qi、miCollinearly, therefore formula (11) can be obtained:
p±=λ1qi+λ2mi (11)
Wherein, formula (11) is indicated using homogeneous coordinates, λ1,λ2It is two scale parameters.
Formula (11) are substituted into first equation of formula (7) and use formula (10), available formula (12):
Under normal conditions, qiAnd miPositioned at the both sides of conic section M.Therefore,
Solution formula (12) available formula (13):
Notice mi=(ui vi 1)TThe last one element be 1, and the q in formula (9)iThe last one element
It is 1, according to formula (11) by p±Formula (14) is normalized to obtain:
So p±The last one element be also 1.So p±To miDistance square according to square formula (15) calculate:
Formula (13) are substituted into formula (15) and select a lesser value, obtain formula (16):
By q in formula (9)iExpression formula substitute into formula (16), it is available apart from expression formula (17):
Wherein,
And denominatorIt is not 0.Compared with formula (5), formula (17) is the Sampson distance of a weighting,
It is also the algebraic distance of a weighting, there is specific physical geometry meaning.
piExpression can also be obtained according to formula (18):
Here it is the solutions of formula (7).And d2(mi,pi)=d2(mi, C), formula (17) can be obtained.
Enable p'iFor range points miFormula (19) can be obtained in point on nearest curve M:
p'i=pi+(Δui,Δvi,0)T (19)
Wherein, Δ ui,ΔviIndicate small quantity.
ByWithIt obtains formula (20):
The expression formula on formula (20) left side is expressed as CONi。
miSquare such as formula (21) of the shortest distance are shown between C:
If (mi TGmi)2< (mi TCmi)(mi TWmi), then it is assumed that miIt is exterior point and deletes them, because these points is separate
Conic section.Fig. 3 (a), Fig. 3 (b), Fig. 3 (c) respectively show this kind of points under oval, hyperbola and parabola, meet this
The point of kind situation is respectively positioned in black region.
The objective function for being based ultimately upon most short geometric distance is established as shown in formula (22):
Formula (20), (21) and (17) are substituted into (22) and obtain objectives function, as shown in formula (23):
WhereinFor scale parameter.
2, in conjunction with above content, the present invention is based on conic fitting method such as Fig. 4 institutes in the image of geometric error optimization
Show, specifically include:
Step 1, the point m of specific conic section M edge image in image is extractedi, wherein mi=(ui vi 1)T, i=
1...N；Exterior point is removed using RANSAC algorithm, then the point on image is fitted to conic section M, the coefficient square of conic section M
Shown in battle array C such as formula (1).
Step 2, point m is calculatediTo the geometric distance d of conic section Mfa(mi,C)。
The geometric distance dfa(mi, C) and it is the Sampson distance weighted, it can be extracted square root and be acquired by formula (17).
The most short geometric distance d (mi, C) and it can be extracted square root and be acquired by formula (21).
The objective function is based on most short geometric distance d (mi, C) and constraint function determination, the solution of the objective function
Analysis representation formula is formula (23)；The constraint function is formula (20)；P in constraint functioniAnalytical Expression formula can be with
For formula (18).
Step 3, to geometric distance dfa(mi, C) and linear weighted function iteration is carried out, it is obtained and C phase with singular value decomposition method
The Matrix C of pass1；
It is described to geometric distance dfa(mi, C) and carry out linear weighted function iteration method particularly includes:
Step 31, the relevant coefficient matrix C of conic section M is calculated(k), k expression the number of iterations；
Step 32, step 31 is repeated, until recording coefficient matrix C at this time when meeting the condition of convergence(k), and enable C1
=C(k)。
The relevant coefficient matrix C of conic section M is calculated in the present embodiment(k)Method are as follows:
As k=0, linear system is solved using singular value decomposition methodI=1...N obtains coefficient
Matrix C(0)；
As k > 0, linear system is solved using singular value decomposition methodIt obtains and secondary song
The relevant coefficient matrix C of line M(k)；
WhereinCalculation method are as follows: enable C=C(k-1), and be calculated by formula (24):
Shown in such as formula of the condition of convergence described in step 32 (25):
Wherein V (C)=(a b c d e f)T, ε is preset threshold.
Step 4, using based on most short geometric distance d (mi, C) building objective function (23), calculating work as C=C1When make
Small quantity Δ u when the minimization of object functioni,ΔviAnd scale parameter λiValue, and be denoted as Δ u respectivelyi1,Δvi1And λi1；
Step 5, with Δ ui1,Δvi1And λi1For initial value, nonlinear optimization solution is carried out to the objective function, is obtained
Coefficient matrix C2, according to coefficient matrix C2Generate the conic section M in described image after the optimization of specific conic section M2.This step
Coefficient matrix C is obtained in rapid2Afterwards, it needs to pass throughTo C2Standardization, wherein | | C2||FIndicate C2F norm.
Nonlinear optimization solution is carried out to the objective function, can be to optimize formula algorithm using Nonlinear Numerical to public affairs
Formula (23) optimizes, and the Nonlinear Numerical optimization formula algorithm can be quasi-Newton method.
The present invention can also be directly generalized in quadric fitting, and specific embodiment party formula and above-mentioned conic section are quasi-
Conjunction method is consistent.
When being fitted to quadratic surface, reference target function (23) constructs the objective function of Quadratic Surface Fitting
(26):
Whereinmi=(ui vi si 1)T,
The adjustable method of the corresponding Quadratic Surface Fitting of formula (24) is formula (27)；
The adjustable method of the corresponding Quadratic Surface Fitting of formula (18) is formula (28):
The adjustable method of the corresponding Quadratic Surface Fitting of formula (19) is formula (29):
p'i=pi+(Δui,Δvi,Δsi 0)T (29)
The adjustable method of the corresponding Quadratic Surface Fitting of formula (17) is formula (30), which is that point arrives quadratic surface
Distance calculation formula.
Those skilled in the art should be able to recognize that, mould described in conjunction with the examples disclosed in the embodiments of the present disclosure
Block, unit and method and step, can be realized with electronic hardware, computer software, or a combination of the two, in order to clearly say
The interchangeability of bright electronic hardware and software generally describes each exemplary composition according to function in the above description
And step.These functions are executed actually with electronic hardware or software mode, depending on technical solution specific application and set
Count constraint condition.Those skilled in the art can realize described function using distinct methods to each specific application
Can, but such implementation should not be considered as beyond the scope of the present invention.
So far, it has been combined preferred embodiment shown in the drawings and describes technical solution of the present invention, still, this field
Technical staff is it is easily understood that protection scope of the present invention is expressly not limited to these specific embodiments.Without departing from this
Under the premise of the principle of invention, those skilled in the art can make equivalent change or replacement to the relevant technologies feature, these
Technical solution after change or replacement will fall within the scope of protection of the present invention.
Claims (9)
1. a kind of conic fitting method in image based on geometric error optimization, which comprises the steps of:
Step 1, the point m of specific conic section M edge image in image is extractedi, wherein mi=(ui vi1)T, i=1...N；
Step 2, point m is calculatediTo the geometric distance d of conic section Mfa(mi, C), wherein C is the coefficient matrix of conic section M；
Step 3, to geometric distance dfa(mi, C) and linear weighted function iteration is carried out, it is obtained with singular value decomposition method relevant to C
Matrix C1；
Step 4, using based on most short geometric distance d (mi, C) building objective function, calculating work as C=C1When make objective function
Small quantity Δ u when minimumi,ΔviAnd scale parameter λiValue, and be denoted as Δ u respectivelyi1,Δvi1And λi1；
Step 5, with Δ ui1,Δvi1And λi1For initial value, nonlinear optimization solution is carried out to the objective function, obtains coefficient
Matrix C2, according to coefficient matrix C2Generate the conic section M in described image after the optimization of specific conic section M2。
2. conic fitting method in the image according to claim 1 based on geometric error optimization, the geometry
Distance dfa(mi, C) and it is the Sampson distance weighted, calculation method are as follows:
Wherein,A, b, c, d, e, f are respectively the matrix coefficient of conic section M,
3. conic fitting method in the image according to claim 2 based on geometric error optimization, which is characterized in that
The most short geometric distance d (mi, C) calculation method are as follows:
Wherein, Δ ui、ΔviFor small quantity.
4. conic fitting method in the image according to claim 3 based on geometric error optimization, which is characterized in that
The objective function is based on most short geometric distance d (mi, C) and constraint function determination, the analytic representation public affairs of the objective function
Formula are as follows:
Wherein,λiFor scale parameter；
The constraint function are as follows:
In constraint function,
p+、p-For the friendship of straight line L' and conic section M
Point, L' are passing point miAnd with polar curve L=CmiOrthogonal straight line.
5. conic fitting method in the image according to claim 3 based on geometric error optimization, which is characterized in that
It is described to geometric distance dfa(mi, C) and carry out linear weighted function iteration method particularly includes:
Step 31, the relevant coefficient matrix C of conic section M is calculated(k), k expression the number of iterations；
Step 32, step 31 is repeated, until recording coefficient matrix C at this time when meeting the condition of convergence(k), and enable C1=C(k)。
6. conic fitting method in the image according to claim 5 based on geometric error optimization, which is characterized in that
The relevant coefficient matrix C of the calculating conic section M(k)Method are as follows:
As k=0, linear system is solved using singular value decomposition methodObtain coefficient square
Battle array C(0)；
As k > 0, linear system is solved using singular value decomposition methodIt obtains and conic section M phase
The coefficient matrix C of pass(k)；
WhereinTo enable C=C(k-1), and be calculated by following formula:
7. conic fitting method in the image according to claim 6 based on geometric error optimization, which is characterized in that
The condition of convergence described in step 32 are as follows:
Wherein V (C)=(a b c d e f)T, ε is preset threshold.
8. conic fitting method in the image according to claim 4 based on geometric error optimization, which is characterized in that
P in the constraint functioniAnalytical Expression formula are as follows:
9. conic fitting method in the image according to claim 8 based on geometric error optimization, which is characterized in that
Coefficient matrix C is obtained in step 52Afterwards, pass throughTo C2Standardization, wherein | | C2||FIndicate C2F norm.
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| Conic Fitting Using the Geometric Distance;Peter Sturm, Pau Gargallo;《ACCV 2007, Part II, LNCS 4844》;20071231;784-795 * |
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