CN107367235A - A kind of axial system error scaling method of infrared surface battle array scanning system - Google Patents

A kind of axial system error scaling method of infrared surface battle array scanning system Download PDF

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CN107367235A
CN107367235A CN201610316710.4A CN201610316710A CN107367235A CN 107367235 A CN107367235 A CN 107367235A CN 201610316710 A CN201610316710 A CN 201610316710A CN 107367235 A CN107367235 A CN 107367235A
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CN107367235B (en
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唐彦琴
顾国华
费小亮
钱惟贤
陈钱
隋修宝
何伟基
任侃
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Nanjing University of Science and Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
    • G01B11/00—Measuring arrangements characterised by the use of optical techniques
    • G01B11/02—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness
    • G01B11/03—Measuring arrangements characterised by the use of optical techniques for measuring length, width or thickness by measuring coordinates of points

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Abstract

The present invention proposes a kind of axial system error scaling method of infrared surface battle array scanning system, the horizontal system of coordinates is established to the transformed error update equation between image coordinate system, then the unknown parameter in error correction equation is estimated with nonlinear least square fitting, update equation is brought into by the unknown parameter estimated is counter, so as to obtain revised azimuth and the angle of pitch.The present invention can make infrared surface battle array scanning system code wheel reading more accurate, can be widely applied in the demarcation of planar array scanning system shafting.

Description

A kind of axial system error scaling method of infrared surface battle array scanning system
Technical field
The invention belongs to error calibration technical field, and in particular to a kind of axial system error demarcation side of infrared surface battle array scanning system Method.
Background technology
Altitude azimuth form infrared surface battle array scanning system is mainly shown during target bearing is obtained by the code-disc on turntable Device carries out reading, but the machining that is constrained in turntable production process and the limitation for harmonizing precision, code wheel reading and The real angle of realistic objective may be inconsistent.Therefore, rotary table error analysis how is carried out to obtain correction factor to red It is extremely important in the sweeping of outside battle array scanning system, succeeding target detect and track precision is influenceed very big.
The content of the invention
The present invention provides a kind of axial system error scaling method of infrared surface battle array scanning system, can make infrared surface battle array scanning system Code wheel reading is more accurate, can be widely applied in the demarcation of planar array scanning system shafting.
In order to solve the above-mentioned technical problem, the present invention provides a kind of axial system error scaling method of infrared surface battle array scanning system, The horizontal system of coordinates is established to the transformed error update equation between image coordinate system, is then estimated with nonlinear least square fitting The unknown parameter in error correction equation is counted out, update equation is brought into by the unknown parameter estimated is counter, so as to be corrected Azimuth afterwards and the angle of pitch.
Further, shown in the transformed error update equation such as formula (1) between the horizontal system of coordinates and image coordinate system,
In formula (1), T (X, Y, Z) is the horizontal system of coordinates, and X, Y, Z are respectively three axle coordinates of the horizontal system of coordinates；For image coordinate system,WithRespectively image coordinate；
In formula (1), M (v, Av, A', E') and spin matrix between the horizontal system of coordinates and turntable coordinate system, it is such as formula (2) shown in,
M=Ml(v)Mz(A')Mx(E') (2)
In formula (2), Ml(v) it is the correction matrix M to vertical axis angle vl(v), it is such as formula (3)：
In formula (3), AvFor angle of inclination v incline direction；
In formula (2), Mz(A) it is the correction matrix to anglec of rotation A' in orientation, it is such as formula (4)：
In formula (2), Mx(E') it is the correction matrix to anglec of rotation E' in pitching, it is such as formula (5)：
In formula (1), h (cx,cy,Fx,Fy,Tx,Ty,Tz) transformational relation between turntable coordinate system and image coordinate system is represented, And have：
Wherein, Tx、TyAnd TzRespectively turntable coordinate system triaxial coordinate is in the translational movement to camera coordinates system, cxAnd cy The respectively pixel point coordinates of camera, x, y, z are respectively turntable coordinate system triaxial coordinate, FxFor equivalent Jiao of X-axis Away from FyFor the equivalent focal length of Y-axis；
C is first asked using LM non-linear least square methodsx、cy、Fx、Fy、Tx、TyAnd TzOptimal value, then basis cx、cy、Fx、Fy、Tx、TyAnd TzOptimal value, substitute into formula (1), ask for v, Av, A' and E'.
Compared with prior art, its remarkable advantage is the present invention, and the horizontal system of coordinates exists to the total transfer equation of image coordinate When correcting turntable code wheel reading, it is contemplated that all variables, and the value of specific variable is obtained by Optimized Iterative method, from And can be very good to correct orientation and pitching angle reading, the engineering debugging to reality has reference significance.
Brief description of the drawings
Fig. 1 is turntable measuring coordinate system schematic diagram.
Fig. 2 is LM nonlinear least square method iterativecurve schematic diagrames.
Embodiment
It is readily appreciated that, according to technical scheme, in the case where not changing the connotation of the present invention, this area Those skilled in the art can imagine infrared surface battle array scanning system of the present invention axial system error scaling method a variety of implementations Mode.Therefore, detailed description below and accompanying drawing are only the exemplary illustrations to technical scheme, without answering When the whole for being considered as the present invention or it is considered as limitation or restriction to technical solution of the present invention.
The axial system error scaling method of infrared surface battle array scanning system of the present invention, with regard to the horizontal system of coordinates to turntable coordinate system, The image coordinate system of thermal imaging system is arrived again, the transformed error update equation between coordinate system is proposed, then with a non-linear most young waiter in a wineshop or an inn Multiply fitting and estimate unknown parameters ' value, brought into so as to counter in update equation, so as to obtain revised orientation and the angle of pitch, So that code wheel reading precision improves, specific implementation step is as follows：
Step 1, derive the spin matrix M between horizontal system of coordinates O-XYZ and turntable coordinate system o-xyz；
Step 2, derive the conversion relational expression between the image coordinate system (x~, y~) of turntable coordinate system o-xyz and thermal imaging system；
Step 3, horizontal system of coordinates O-XYZ is established to total transition matrix between image coordinate (x~, y~), uses original-party Position and pitching angular dimensions try to achieve unknown parameter in total transition matrix by LM nonlinear least square methods (v,Av,A',E',Cx,Cy,Fx,Fy,Tx,Ty,Tz) estimate；
Step 4, parameter (v, the A that will be obtainedv,A',E',Cx,Cy,Fx,Fy,Tx,Ty,Tz) estimate counter bring Horizon into Coordinate system O-XYZ obtains revised orientation and the angle of pitch into the total transfer equation of image coordinate (x~, y~), so as to complete Demarcated into axial system error.
For step 1, if the horizontal system of coordinates is O-XYZ, horizontal system of coordinates origin O is the friendship of the axle of turntable coordinate system three Point.Wherein, Z axis and turntable horizontal plane, in the horizontal plane, and Y-axis points to the earth north, three axles for X, Y-axis It is orthogonal and meet right-handed coordinate system.The sensing of the actually each axle of the horizontal system of coordinates is respectively turntable in theoretical zero-bit The sensing of trunnion axis, collimation axis and vertical axis, turntable coordinate system are O-XYZ, and y-axis is the finger of the optical axis after alignment target To z-axis is orthogonal with y-axis and points to zenith, while has x-axis and y, and z-axis is orthogonal, composition right-handed coordinate system, two Coordinate Conversion is shown in Fig. 1.
In no axial system error ideally, turntable points to target from zero-bit and needs to rotate A angles in orientation, bows Rotation E angles are faced upward, horizontal system of coordinates O-XYZ can be regarded as and first turn A angles further around trunnion axis Y around vertical axis X-axis Axle rotates E angles, finally obtains turntable coordinate system o-xyz, i.e. horizontal system of coordinates O-XYZ and turntable coordinate system Between o-xyz shown in transformational relation such as formula (1)：
Wherein, Mz(A) be in horizontal system of coordinates orientation rotate A angles transition matrix
Mx(E) it is that horizontal coordinate tie up in pitching the transition matrix for rotating E angles, and
In fact, because turntable has axial system error during target detection, cause transition matrix can be by each inclination The influence at angle, the conversion between the horizontal system of coordinates and turntable coordinate system can represent by formula (4),
Wherein, spin matrix total between the horizontal system of coordinates and turntable coordinate system M, its mainly by vertical axis angle v, Anglec of rotation E' is determined in anglec of rotation A' and pitching in orientation, anglec of rotation E' in anglec of rotation A' and pitching in orientation It is instrument code-disc true plot.
To vertical axis angle v correction matrix Ml(v) such as formula (5)：
Wherein, AvFor angle of inclination v incline direction.
To the correction matrix M of anglec of rotation A' in orientationz(A) such as formula (6)：
To the correction matrix M of anglec of rotation E' in pitchingx(E') such as formula (7)：
Finally, according to the sequencing of rotation, by Ml(v)、Mz(A')、Mx(E') continuous phase is multiplied arrives the horizontal system of coordinates Total spin matrix M between turntable coordinate system, its as shown in formula (8),
M=Ml(v)Mz(A')Mx(E') (8)
In this way, altitude azimuth form coordinate system O-XYZ obtains turntable coordinate system o-xyz by spin matrix M, wherein, need The parameter to be used has：Anglec of rotation E' in anglec of rotation A' and pitching on from the orientation that code-disc is truly read, vertical axis Angle of inclination v and incline direction Av.But these three parameters need to correct because error be present, therefore by actual code wheel reading It is placed in the unknown parameter of subsequent algorithm, together iteration.
It is sharp first in order to complete the conversion between turntable coordinate system o-xyz and image coordinate (x~, y~) for step 2 With central projection model, coordinate transformation relation of the image coordinate system (x~, y~) between turntable coordinate system o-xyz is obtained. Here it is divided into two parts again, first, turntable coordinate system o-xyz to camera coordinates system K-XkYkZkTransformational relation, Second, camera coordinates system K-XkYkZkWith the transformational relation of image coordinate system (x~, y~).
Turntable coordinate system o-xyz to camera coordinates system K-XkYkZkTransformational relation can be represented by translational movement, It is as shown in formula (9)：
Wherein, T=(Tx,Ty,Tz) it is translation vector, it is coordinate of the turntable coordinate origin in camera coordinates system, Turntable coordinate origin is namely moved on to the translational movement of camera coordinates system origin.
For camera coordinates system K-XkYkZkWith image coordinate systemConversion, according to perspective projection model Fundamental relation, image coordinate can be obtainedWith camera coordinates system K-XkYkZkRelation, it is such as formula (10) institute Show：
Wherein, (cx,cy) be image pixel point coordinates, i.e. optical axis and image planes intersection point image coordinate；ZkFor object point To the distance projection in the direction of the optical axis of photocentre, so Zk≠0；The equivalent focal length F of X-axisxFor focal length f and pixel Transverse and longitudinal size dxThe ratio between；The equivalent focal length F of Y-axisyFor focal length f and the transverse and longitudinal size d of pixelyThe ratio between.
By the conversion of formula (9) and the coordinate system twice of formula (10), most turntable coordinate system o-xyz is converted into figure at last As coordinate systemTransfer equation such as formula (11) shown in,
For step 3, first in view of the unknown parameter in formula, to the horizontal system of coordinates represented by formula (4)
Conversion relational expression can further use formula (12) to represent between O-XYZ and turntable coordinate system o-xyz：
T (X, Y, Z)=M (v, Av,A',E')R(x,y,z) (12)
Wherein, M (v, Av, A', E') turn between function representation horizontal system of coordinates O-XYZ and turntable coordinate system o-xyz Change relational expression, although angle A ', angle E' have reading on code-disc, need to correct, therefore made unknown parameter Processing, M (v, Av, A', E') in share 4 unknown parameters, respectively v, Av,A',E'；T (X, Y, Z) and R (x, y, z) represents the horizontal system of coordinates and turntable coordinate system respectively.
Furthermore for turntable coordinate system o-xyz and image coordinateBetween transformational relation formula (11) can enter One step symbol h (cx,cy,Fx,Fy,Tx,Ty,Tz) represent, such as formula (13)：
Wherein, h (cx,cy,Fx,Fy,Tx,Ty,Tz) there are 7 unknown parameters, respectively cx,cy,Fx,Fy,Tx,Ty,Tz；Image coordinate system and turntable coordinate system are represented respectively with R (x, y, z).
For step 3, combining step one and step 2, total conversion side of the horizontal system of coordinates and image coordinate system can be obtained Journey：
In order to obtain 11 unknown parameters of the above, the present invention is first asked using LM non-linear least square methods (cx,cy,Fx,Fy,Tx,Ty,Tz) optimal value, then according to (cx,cy,Fx,Fy,Tx,Ty,Tz) optimal value, formula (13) is substituted into, Ask v, Av, A', E', obtain correction value.
Setting parameter equationWherein Horizon is sat Mark (X, Y, Z) to obtain by global positioning systems such as GPS, and correspondingly obtain image coordinateTherefore by KnowTo solve unknown parameter (cx,cy,Fx,Fy,Tx,Ty,Tz,v,Av,A',E').So as to above-mentioned solution Problem is converted into the optimized algorithm as shown in formula (15)：
Choose 11 unknown parameter (cx,cy,Fx,Fy,Tx,Ty,Tz,v,Av, A', E') initial point, parameter beta, damping system Number μ, amplification coefficient ν, obtain the final iteration optimization value of 11 unknown parameters, so as to band by iterative algorithm Enter final conversion formulaIn, correct what each turntable obtained Orientation values A' and pitch value v'.
The present invention carries out simulation optimization using MATLAB, obtains specific parameter optimization value, detailed process is as follows：
First construct W=[x；y；z]-M；, M is the obtained coordinate system of matrix rotation, [x；y；Z] initially verified for GPS Point value, initial value X0=[1 1111 1]；Call min LM functions：
X=min LM (W, X0,0.4,2,1.5,[cx cy Fx Fy Tx Ty Tz v Av])
Wherein, iterativecurve is shown in Fig. 2, and when finally probably iterating to 100 times, min tends towards stability, so as to according to x to Measure each interior optimization value cx、cy、Fx、Fy、Tx、Ty、Tz、v、Av。

Claims (2)

1. a kind of axial system error scaling method of infrared surface battle array scanning system, it is characterised in that establish the horizontal system of coordinates and arrive Transformed error update equation between image coordinate system, then estimates error correction equation with nonlinear least square fitting In unknown parameter, update equation is brought into by the unknown parameter estimated is counter, so as to obtain revised azimuth and pitching Angle.
2. axial system error scaling method as claimed in claim 1, it is characterised in that the horizontal system of coordinates and image coordinate Shown in transformed error update equation such as formula (1) between system,
<mrow> <mi>T</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>,</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>M</mi> <mrow> <mo>(</mo> <mi>v</mi> <mo>,</mo> <msub> <mi>A</mi> <mi>v</mi> </msub> <msup> <mi>a</mi> <mo>,</mo> </msup> <mo>,</mo> <msup> <mi>E</mi> <mo>,</mo> </msup> <mo>)</mo> </mrow> <mi>h</mi> <mrow> <mo>(</mo> <msub> <mi>c</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>c</mi> <mi>y</mi> </msub> <mo>,</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>y</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> <mi>D</mi> <mrow> <mo>(</mo> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>,</mo> <mover> <mi>y</mi> <mo>~</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In formula (1), T (X, Y, Z) is the horizontal system of coordinates, and X, Y, Z are respectively three axle coordinates of the horizontal system of coordinates；For image coordinate system,WithRespectively image coordinate；
In formula (1), M (v, Av, A', E') and spin matrix between the horizontal system of coordinates and turntable coordinate system, it is such as formula (2) shown in,
M=Ml(v)Mz(A')Mx(E') (2)
In formula (2), Ml(v) it is the correction matrix M to vertical axis angle vl(v), it is such as formula (3)：
<mrow> <msub> <mi>M</mi> <mi>l</mi> </msub> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>sin</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>cos</mi> <mi> </mi> <mi>v</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>sin</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>sin</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> <mi>sin</mi> <mi> </mi> <mi>v</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msub> <mi>A</mi> <mi>v</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
In formula (3), AvFor angle of inclination v incline direction；
In formula (2), Mz(A) it is the correction matrix to anglec of rotation A' in orientation, it is such as formula (4)：
<mrow> <msub> <mi>M</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mo>,</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msup> <mi>A</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msup> <mi>A</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msup> <mi>A</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msup> <mi>A</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
In formula (2), Mx(E') it is the correction matrix to anglec of rotation E' in pitching, it is such as formula (5)：
<mrow> <msub> <mi>M</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>E</mi> <mo>,</mo> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msup> <mi>E</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi> </mi> <msup> <mi>E</mi> <mo>,</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mi>sin</mi> <mi> </mi> <msup> <mi>E</mi> <mo>,</mo> </msup> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi> </mi> <msup> <mi>E</mi> <mo>,</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
In formula (1), h (cx,cy,Fx,Fy,Tx,Ty,Tz) transformational relation between turntable coordinate system and image coordinate system is represented, And have：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>~</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msub> <mi>T</mi> <mi>x</mi> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>c</mi> <mi>x</mi> </msub> <mi>z</mi> <mo>+</mo> <msub> <mi>F</mi> <mi>x</mi> </msub> <msub> <mi>T</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>c</mi> <mi>x</mi> </msub> <msub> <mi>T</mi> <mi>z</mi> </msub> </mrow> <mrow> <mi>Z</mi> <mo>+</mo> <msub> <mi>T</mi> <mi>z</mi> </msub> </mrow> </mfrac> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>~</mo> </mover> <mo>=</mo> <mfrac> <mrow> <msub> <mi>F</mi> <mi>y</mi> </msub> <mi>y</mi> <mo>+</mo> <msub> <mi>c</mi> <mi>y</mi> </msub> <mi>z</mi> <mo>+</mo> <msub> <mi>F</mi> <mi>y</mi> </msub> <msub> <mi>T</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>c</mi> <mi>y</mi> </msub> <msub> <mi>T</mi> <mi>z</mi> </msub> </mrow> <mrow> <mi>Z</mi> <mo>+</mo> <msub> <mi>T</mi> <mi>z</mi> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced>
Wherein, Tx、TyAnd TzRespectively turntable coordinate system triaxial coordinate is in the translational movement to camera coordinates system, cxAnd cy The respectively pixel point coordinates of camera, x, y, z are respectively turntable coordinate system triaxial coordinate, FxFor equivalent Jiao of X-axis Away from FyFor the equivalent focal length of Y-axis；
C is first asked using LM non-linear least square methodsx、cy、Fx、Fy、Tx、TyAnd TzOptimal value, then basis cx、cy、Fx、Fy、Tx、TyAnd TzOptimal value, substitute into formula (1), ask for v, Av, A' and E'.
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