CN107392966A - Relative orientation optimization method based on bound constrained function - Google Patents

Relative orientation optimization method based on bound constrained function Download PDF

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CN107392966A
CN107392966A CN201710728719.0A CN201710728719A CN107392966A CN 107392966 A CN107392966 A CN 107392966A CN 201710728719 A CN201710728719 A CN 201710728719A CN 107392966 A CN107392966 A CN 107392966A
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CN107392966B (en
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张征宇
刘明敏
王学源
邹易峰
周润
范金磊
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High Speed Aerodynamics Research Institute of China Aerodynamics Research and Development Center
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T7/00—Image analysis
    • G06T7/80—Analysis of captured images to determine intrinsic or extrinsic camera parameters, i.e. camera calibration
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T2207/00—Indexing scheme for image analysis or image enhancement
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Abstract

The invention discloses a kind of relative orientation optimization method based on bound constrained function, comprise the following steps：Construct bound constrained function；For n to corresponding image points, the relative orientation optimization method containing bounded is established；Relative orientation optimization method containing bounded is converted into unconstrained minimization equation；Solve to obtain the approximate optimal solution of unconstrained minimization equation using general reversion of least square；When meeting stop criterion, iteration ends, the approximate optimal solution using the elements of relative orientation tried to achieve for the last time as the relative orientation optimization method containing bounded.The present invention by constructing bound constrained function, the optimal solution of elements of relative orientation will be stably limited in feasible zone, avoid iteration point surmount obstacles depart from feasible zone caused by Optimization Solution failure.On the other hand, soaring speed of the bound constrained function outside feasible zone is controlled by sharpness factor；The speed to be diminished by index control penalty factor, and then the optimal solution for making elements of relative orientation quickly and reliably converge to.

Description

Relative orientation optimization method based on bound constrained function
Technical field
The invention belongs to machine vision and photogrammetric technology field, and in particular to a kind of based on the relative of bound constrained function Orientation optimization process.
Background technology
With the continuous lifting of slr camera performance, slr camera is used with free camera style, without processing and safeguarding The very high precision calibration plate of cost and high-precision camera motion platform, the self-calibration of camera is realized, it is quick, flexible, at low cost Obtain the three-dimensional data of tested industry spot parts, it has also become one of trend of coordinate measuring technology development.
By solving the coplanar condition equation of corresponding image points, elements of relative orientation is obtained, is the three-dimensional based on slr camera Measurement and the basis of Camera Self-Calibration, and the core of such as 5 points, 7 points and 8 scheduling algorithms of n points perspective problem (PnP).Such as figure Shown in 1, Liang Ge photo centres are respectively S1、S2, object space point PiIn image plane IA、IBOn imaging point be respectively Pi'、Pi", it is of the same name Core line is respectively l, l'.I-th pair corresponding image points Pi' and Pi" corresponding to coplanarity equation be
β=(β0,β1,β2,β3,β4,β5)； (1)
B in formulax,by,bzAnd φ, ω, κ are respectively vertical element and the angle element of elements of relative orientation, phase hereinafter Orientation element is represented with β.Image space coordinate (u 'i,v′i,w′i) and (u "i,v″i,w″i) conversion formula
P′iWith P "iImaging point on the photograph of left and right is respectively (x 'i,y′i,-f),(x″i,y″i,-f), f is photo centre To the vertical range (main away from) of image plane.
But photogrammetric from traditional (aviation) is to be similar to that vertical camera style is different, and the randomness freely photographed can not Produce the big overlaid numeric image of wide-angle with avoiding, if directly using ripe low-angle coplanar condition equation inearized model, Be difficult to accurately and reliably solve elements of relative orientation, it is necessary to more consider coplanar condition equation nonlinear characteristic, can pass through Scale realizes absolute orientation, and high accuracy obtains the three-dimensional coordinate of measured point.
And the eigenmatrix E estimation sides based on singular value decomposition (Singular Value Decomposition, SVD) Method, although relative orientation can be realized, as Hartley propose linear 8 algorithms of standardization, after trying to achieve E, can therefrom separate Go out translation and spin matrix, but translate, the combination of spin matrix has 4 kinds, it is necessary to exclude to misread, due to resolving spin matrix and Translation vector precision is relatively low, subsequently must also carry out nonlinear optimization to improve precision.
Therefore, the relative orientation optimization method of reliable, the accurate big overlaid numeric image of wide-angle, to ensuring camera from mark Determine precision and measured object three-dimensional measurement precision is most important.
Traditional coplanarity equation group optimization is by Taylor expansion, takes first two to be used as approximate equation, then iterative, Initial value required precision to five elements of relative orientation is high, and reason is：When the Taylor expansion near optimal solution, secondary and high-order Event is smaller, and truncated error is smaller, and convergence can quickly；Otherwise, after casting out secondary and high-order event, truncated error is larger, iteration Caused point range is difficult to reliable conveyance to optimal solution；Though genetic algorithm does not need previously given elements of relative orientation initial value, Phenomena such as low efficiency, Premature Convergence being present and stagnating, is readily obtained locally optimal solution, the elements of relative orientation precision solved is not The problems such as sufficient.
Traditional nonlinear optimization method, then need to know the convex set of elements of relative orientation, but to find five simultaneously The convex set of elements of relative orientation is still difficult at present, therefore, it is more at present by giving elements of relative orientation addition constraint section, so as to Initial value nearby searches out optimal solution, and the common method for solving the problems, such as constrained optimization is penalty function method.Wherein, exterior penalty function method is fitted Close and solve the problem of optimal solution is on restrained boundary, but be difficult to ensure that approximate optimal solution is feasible solution, and in actual measurement In, it is desirable to the feasible solution of former problem can be obtained when stopping iteration.Method of inner penalty function is then given to the point for attempting to depart from feasible zone Punishment, be provided with obstacle equivalent on the border of feasible zone, if but step size controlling it is improper, can make iteration point disengaging feasible zone, lead Cause Optimization Solution failure.
The content of the invention
In order to overcome the disadvantages mentioned above of prior art, it is excellent that the present invention proposes a kind of relative orientation based on bound constrained function Change method, constraint section is determined according to the initial value precision of elements of relative orientation, obtains the feasible zone of coplanarity equation solution；The opposing party Face, the bound constrained function of uniqueness is constructed, optimization solution is limited in feasible zone, it is ensured that in the case where initial value precision is not high, still It can reliably, rapidly converge to the optimal solution of elements of relative orientation.
The technical solution adopted for the present invention to solve the technical problems is：A kind of relative orientation based on bound constrained function is excellent Change method, comprises the following steps：
Step 1: construction bound constrained function：
β represents elements of relative orientation, β in formula0For the initial value of elements of relative orientation；cjRepresent the skew on the basis of initial value Amount；When β is vertical element bxWhen, cjIt is taken as c0；When β is angle element φ, when ω, κ, cjIt is taken as c1；δnFor penalty factor, μ is steep The high and steep factor, τ are the index of penalty factor；
Step 2: the relative orientation optimization method containing bounded is established to corresponding image points for n：
β=(β0,β1,β2,β3,β4,β5)；
I=0,1 ..., n-1
Step 3: the relative orientation optimization method containing bounded is converted into unconstrained minimization equation：
qi(β)=0, i=0,1 ..., n-1 ..., n+4, n >=5
Step 4: solve to obtain the approximate optimal solution of unconstrained minimization equation using general reversion of least square；
Step 5: when meeting stop criterion, iteration ends, using the elements of relative orientation tried to achieve for the last time as containing The approximate optimal solution of the relative orientation optimization method of bound constrained.
Compared with prior art, the positive effect of the present invention is：It is different from existing penalty function method optimization method, this hair It is bright by constructing bound constrained function, the optimal solution of elements of relative orientation will be stably limited in feasible zone, avoids iteration point from getting over The Optimization Solution crossed caused by obstacle departs from feasible zone fails.On the other hand, by sharpness factor μ, control bound constrained function is can The overseas soaring speed of row, μ is bigger, and the speed for tending to infinity in feasible zone external constraint function absolute value is faster (i.e. precipitous Degree is bigger)；Penalty factor δ is controlled by index τnThe speed to diminish, the absolute value of lifting bound constrained function tend to outside feasible zone Infinitely great speed, and then the optimal solution for making elements of relative orientation quickly and reliably converge to.
Brief description of the drawings
Examples of the present invention will be described by way of reference to the accompanying drawings, wherein：
Fig. 1 is coplanar model schematic；
Fig. 2 is δnInfluence to bound constrained function；
Fig. 3 is influences of the μ to bound constrained function；
Fig. 4 is influence of the τ value to bound constrained function；
Fig. 5 is traditional penalty function and bound constrained function effect contrast figure；
Fig. 6 is the flow chart of the inventive method.
Embodiment
A kind of relative orientation optimization method based on bound constrained function, comprises the following steps：
1) bound constrained function designs
The bound constrained function that the present invention constructs is as follows
β represents the b in elements of relative orientation in formulax,by,bz, φ, ω, κ, β0For the initial value of elements of relative orientation；cjRepresent Offset on the basis of initial value, you can row domain radius, when β is vertical element β0When cjIt is taken as c0, when β is angle element β3,β4,β5 When cjIt is taken as c1；δnFor penalty factor, μ is sharpness factor, and τ is the index of penalty factor, and control penalty factor diminishes speed.
δnValue it is larger when, bound constrained function has one section of fluctuating in feasible zone boundary, and borderline functional value is very big, so It can diminish across border to functional value during non-feasible zone afterwards, and constantly become big, functional digraph also becomes more and more precipitous.δn Value it is smaller when, steepness of the bound constrained function outside feasible zone is larger, in the borderline fluctuating unobvious of feasible zone.Such as Fig. 2 It is shown, with penalty factor δnReduction, functional value of the bound constrained function outside feasible zone become more precipitous.
Sharpness factor μ is bigger, tends to infinitely great faster (the i.e. steepness of speed in feasible zone external constraint function absolute value It is bigger), but not as obvious when penalty factor changes, Fig. 3 is δnChange the bound constrained that sharpness factor μ is obtained when being fixed as 1 Functional image.
Index τ can control penalty factor δnThe speed to diminish, while it is precipitous outside feasible zone to improve bound constrained function Degree.As shown in figure 4, because penalty factor successively decreases in an iterative process, during squared and the above power, after the step of iteration two or three, The absolute value of bound constrained function tends to be infinitely great quickly outside feasible zone.
2) the relative orientation optimization calculating formula containing bounded
It is as follows to corresponding image points, its relative orientation optimizing expression containing bounded for n：
β=(β0,β1,β2,β3,β4,β5)；
I=0,1 ..., n-1
3) the relative orientation Optimization Steps containing bounded
Step1:Give penalty factor δnTax initial value is 1, τ 2, δnFor the decreasing sequence of numbers more than zero, once all multiply per iteration With a coefficient of reduction, general δnCoefficient of reduction take 1/2, μ to take 100 steepness for ensuring that bound constrained function.Can be with Using document [5 relative orientation [J] Acta Opticas .2015 (01) based on forward intersection:231-238] in method to phase Initial value is assigned to orientation element, also the simulation stereoplotting of photogrammetry can be used to assign initial value to elements of relative orientation.
Step2:Formula (4) expression is the relative orientation optimization problem containing bounded, changes into unconstrained minimization and asks Topic is such as formula (5).Method for transformation is：Constraint equation and all coplanarity equations get up side by side in wushu (4), are merged into nonlinear equation Group, wherein, the bound constrained function elements of relative orientation to be constrained has 4, then there is 4 bound constrained equations.Along with 1 directly Line element constraint equation (relates only to elements of relative orientation model scale, actual is straight in view of one of vertical element Line element only has two independent parameters, is then 1 as one of constraints using baseline length) and n coplanarity equation fi (β)=0, the equation group for then combining to obtain share n+5 formula, and each equation is represented with q.β is elements of relative orientation.
qi(β)=0, i=0,1 ..., n-1 ..., n+4, n >=5 (5)
Step3:[Zhou Changfa .C# numerical computation algorithms programming [M] electronics industries are used to the Nonlinear System of Equations of formula (5) Publishing house, 2007.241-245] method in book, solved with general reversion of least square.Its Jacobian matrix is：
The iterative formula of least square solution for calculating above Nonlinear System of Equations is：
β(k+1)=β(k)-αkZ(k) (7)
Wherein Z(k)For linear algebraic equation systems A(k)Z(k)=Q(k)Linear least-squares solution, i.e. Z(k)=(A(k))-1Q(k), A in formula(k)For k iterative value β(k)Jacobian matrix；Q(k)For the left end functional value of k iterative value, i.e.,
αkTo make α function of a single variableReach the point of minimum, calculated in algorithm with reasonable extremum method αk。
The specific solution that reasonable extremum method finds a function certain extreme points of the y=f (x) in section [a, b] has 3 steps：
(1) two initial experiment point x of selection in section [a, b]0,x1。
(2) △ x are selected, measure f (x first0+△x),f(x0-△x),f(x1+△x),f(x1The values of-△ x), then calculate Go out y '0, y '1.Wherein With resulting point (y′0, x0), (y '1,x1) the coefficient b in formula (9) tried to achieve according to Thiele interpolation coefficient tablesi.New experiment is tried to achieve by formula (9) again Point
(3) in point (x2- △ is x) and point (x2X) place obtains y ' to+△2.If | y '2|≤ε, then x2For extreme point, wherein, ε is real Test precision.Otherwise, continue that new testing site is calculatedIterate, until what is tried to achieve | y 'i|≤ε, Now xiFor required extreme point.
Step4：The control accuracy of the least square solution of non-linear least square generalized inverse is eps1=0.00001, very The control accuracy eps2=0.00001 that different value is decomposed, algorithmic statement obtain the approximate optimal solution β of unconstrained problem (5)(m), m is General reversion of least square access times.
Step5：Judge whether to meet stop criterion, if | β(m)-β(m-1)|<ε, then iteration ends, wherein, it is allowed to error ε= 0.00000001, β(m)For the m times elements of relative orientation tried to achieve with general reversion of least square, β now(m)As constraint is asked Inscribe the approximate optimal solution of (4)；Otherwise makeδn (m)For the m times iteration penalty factor value, Step 3 is returned.
It is as shown in Figure 6 to solve flow chart：
Existing exterior penalty function method is adapted to solve the problem of optimal solution is on restrained boundary, it is difficult to ensures that approximate optimal solution is Feasible solution；And existing SUMT interior point method is given to the point for attempting disengaging feasible zone and punished, set equivalent to the border in feasible zone Put obstacle, if but step size controlling it is improper, can make iteration point cross boundary obstacles depart from feasible zone, cause Optimization Solution to fail.
The present invention is by creating the bound constrained function as shown in formula (3), by the optimal solution stability line of elements of relative orientation It is scheduled in feasible zone, and by sharpness factor μ and penalty factor index τ, for outside feasible zone, controlling the absolute of bound constrained function Value tends to infinitely great speed, it is ensured that the optimal solution that elements of relative orientation quickly and reliably converges to.
As shown in figure 5, for example traditional penalty function reciprocal, for given feasible zone [0.12,1.88], iteration point is feasible Penalty function value very little when being moved in domain, once close to border, penalty function value suddenly increases to maximum respectively at left and right sides of section It is worth for 3.825 and 4.662, but iteration point once crosses feasible zone border, penalty function value is too small can not to produce punishment effect, and then Optimization is caused to fail.And for same constraint section, when iteration point is in the range of feasible zone, bound constrained functional value is equal to 0；When When iteration point is outside feasible zone, bound constrained function absolute value minimum value on the left of section is 9.765, right side minimum 9.513.
Therefore, bound constrained function proposed by the present invention, the iteration point of optimization process can be limited in feasible zone, makes its fast The optimal solution of elements of relative orientation is converged to fastly.Optimization efficiency is high and measured value is more accurate, has practical value.

Claims (5)

  1. A kind of 1. relative orientation optimization method based on bound constrained function, it is characterised in that：Comprise the following steps：
    Step 1: construction bound constrained function：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>p</mi> <mrow> <mo>(</mo> <mi>&amp;beta;</mi> <mo>,</mo> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>&amp;mu;</mi> <mo>,</mo> <mi>&amp;tau;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <mi>&amp;beta;</mi> <mo>-</mo> <msup> <mi>&amp;beta;</mi> <mn>0</mn> </msup> <mo>+</mo> <msub> <mi>c</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <mi>&amp;beta;</mi> <mo>-</mo> <msup> <mi>&amp;beta;</mi> <mn>0</mn> </msup> <mo>+</mo> <msub> <mi>c</mi> <mi>j</mi> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <mi>&amp;beta;</mi> <mo>-</mo> <msup> <mi>&amp;beta;</mi> <mn>0</mn> </msup> <mo>-</mo> <msub> <mi>c</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;mu;</mi> <mfrac> <mrow> <mi>&amp;beta;</mi> <mo>-</mo> <msup> <mi>&amp;beta;</mi> <mn>0</mn> </msup> <mo>-</mo> <msub> <mi>c</mi> <mi>j</mi> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>}</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
    β represents elements of relative orientation, β in formula0For the initial value of elements of relative orientation；cjRepresent the offset on the basis of initial value；When β is vertical element bxWhen, cjIt is taken as c0；When β is angle element φ, when ω, κ, cjIt is taken as c1；δnFor penalty factor, μ for it is precipitous because Son, τ are the index of penalty factor；
    Step 2: the relative orientation optimization method containing bounded is established to corresponding image points for n：
    β=(β0,β1,β2,β3,β4,β5)；
    fiThe β of (β)=00=bx,β1=by,β2=bz,β4=ω, β5=k
    I=0,1 ..., n-1
    <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>0</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>0</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>0</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;mu;</mi> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>0</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;mu;</mi> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>3</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>4</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>4</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>4</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;mu;</mi> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>4</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>5</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>-</mo> <msup> <mi>&amp;mu;</mi> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>5</mn> <mn>0</mn> </msubsup> <mo>+</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </mrow> </msup> <mo>+</mo> <mfrac> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> <mrow> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>5</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;mu;</mi> <mfrac> <mrow> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> <mo>-</mo> <msubsup> <mi>&amp;beta;</mi> <mn>5</mn> <mn>0</mn> </msubsup> <mo>-</mo> <msub> <mi>c</mi> <mn>1</mn> </msub> </mrow> <mrow> <msup> <msub> <mi>&amp;delta;</mi> <mi>n</mi> </msub> <mi>&amp;tau;</mi> </msup> </mrow> </mfrac> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <msub> <mi>&amp;beta;</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>-</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
    Step 3: the relative orientation optimization method containing bounded is converted into unconstrained minimization equation：
    qi(β)=0, i=0,1 ..., n-1 ..., n+4, n >=5
    Step 4: solve to obtain the approximate optimal solution of unconstrained minimization equation using general reversion of least square；
    Step 5: when meeting stop criterion, iteration ends, using the elements of relative orientation tried to achieve for the last time as containing boundary treaty The approximate optimal solution of the relative orientation optimization method of beam.
  2. 2. the relative orientation optimization method according to claim 1 based on bound constrained function, it is characterised in that：Using minimum Two, which multiply the method that generalized inverse is solved to unconstrained minimization equation, is：
    (1) δ is maden=1, τ=2, δnCoefficient of reduction take 1/2, μ=100, and give elements of relative orientation to assign initial value；
    (2) Jacobian matrix is established：
    <mrow> <msub> <mi>A</mi> <mrow> <mo>(</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> <msub> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> </msub> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>0</mn> </msub> </mrow> </mfrac> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> </mrow> </mfrac> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mfrac> <mrow> <mo>&amp;part;</mo> <msub> <mi>q</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> </msub> </mrow> <mrow> <mo>&amp;part;</mo> <msub> <mi>&amp;beta;</mi> <mn>5</mn> </msub> </mrow> </mfrac> </mtd> </mtr> </mtable> </mfenced> </mrow>
    (3) iterative formula of least square solution is determined：
    β(k+1)=β(k)-αkZ(k)
    Wherein, Z(k)=(A(k))-1Q(k), it is linear algebraic equation systems A(k)Z(k)=Q(k)Linear least-squares solution, A in formula(k)For K iterative value β(k)Jacobian matrix；Q(k)For the left end functional value of k iterative value：
    Q(k)=(q0 (k),q1 (k),…,qn+4 (k))T
    qi (k)=qi(β(k)), i=0,1 ..., n+4
    αkTo make α function of a single variableReach the point of minimum；
    (4) approximate optimal solution of unconstrained minimization equation is obtained when algorithmic statement.
  3. 3. the relative orientation optimization method according to claim 2 based on bound constrained function, it is characterised in that：Algorithmic statement The control accuracy for referring to least square solution is eps1=0.00001, and the control accuracy of singular value decomposition is eps2=0.00001.
  4. 4. the relative orientation optimization method according to claim 3 based on bound constrained function, it is characterised in that：Step 5 institute Stop criterion is stated to refer to | β(k)-β(k-1)|<ε, wherein, it is allowed to error ε=0.00000001, β(k)It is relatively fixed to be tried to achieve for kth time To element.
  5. 5. the relative orientation optimization method according to claim 4 based on bound constrained function, it is characterised in that：If it is unsatisfactory for Stop criterion, then makeδn (k)For kth time iterative penalty factor values, unconstrained minimization equation is carried out again Solve.
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