CN107728106A - Orientation method of micro-array polarized light compass - Google Patents

Orientation method of micro-array polarized light compass Download PDF

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CN107728106A
CN107728106A CN201710914696.2A CN201710914696A CN107728106A CN 107728106 A CN107728106 A CN 107728106A CN 201710914696 A CN201710914696 A CN 201710914696A CN 107728106 A CN107728106 A CN 107728106A
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CN107728106B (en
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胡小平
韩国良
练军想
何晓峰
张礼廉
王玉杰
范晨
吴雪松
蔡宇
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National University of Defense Technology
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S3/00—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received
    • G01S3/78—Direction-finders for determining the direction from which infrasonic, sonic, ultrasonic, or electromagnetic waves, or particle emission, not having a directional significance, are being received using electromagnetic waves other than radio waves
    • G01S3/782—Systems for determining direction or deviation from predetermined direction

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Abstract

The invention discloses an orientation method of a micro-array polarized light compass. The invention can effectively improve the precision of the micro-array polarized light compass for measuring the solar azimuth, has the advantages of simple principle, simple and convenient operation, wide application range and the like, and realizes the polarized light orientation based on the sky polarization mode.

Description

A kind of orientation method of microarray formula polarised light compass
Technical field：
The present invention relates to polarized light sensor fields of measurement, more particularly to a kind of orientation side of microarray formula polarised light compass Method.
Background technology：
Animal possesses outstanding homing capability, and the animal such as desert ant and honeybee is proved to possess the energy for perceiving polarised light Power, they utilize unique polarization vision structure, perceive the natural polarization characteristic of air, extract the azimuth information of the sun, so as to Carry out reliable navigator fix.Bionical polarized light sensor has used for reference the extremely sensitive visually-perceptible system of animal, inclined with air The acquisition of carrier course information is realized based on pattern of shaking, has strong interference immunity, error not with time integral, applied widely Etc. advantage.
Document (" the autonomous navigation method research based on topological node recursion ", National University of Defense Technology's doctorate opinion Text, 2016, first control text) polarized light sensor of the design based on photodiode, it can effectively measure the aerial a direction in day The polarization information of incident light, but measured zone is limited, easily by blocking and weather is influenceed.(" pixel polarizes chip arrays system to document Application standby and its in polarization image enhancing ",《Acta Physica Sinica》, in April, 2014, the 2nd phase, Zhang Zhigang etc. of volume 63) propose The technique for preparing metal nano grating, and array is polarized into camera applications strengthened in polarization image, but do not provide it Method applied to navigation.Document (" Design and Calibration of a Novel Camera-Based Bio- (2016) 3640- of Inspired Polarization Navigation Sensor ", IEEE Sensors Journal 16 3648, Chen Fan) image-type polarized light sensor of the theory deduction based on four mesh cameras error model and its demarcation side Method, and sensor application is measured in atmospheric polarization type, but its error model is not particularly suited for array polarised light compass.
Compared with the image-type polarised light compass based on more mesh cameras, microarray formula polarised light compass have it is simple in construction, The features such as integrated level high and low power consumption.In presently disclosed data, microarray formula polarized light sensor pooled applications increase in image By force, the orientation method research to microarray formula polarised light compass is very few.Therefore, the measurement side of microarray formula polarised light compass is established Journey and calibrated error parameter have to accurate acquisition solar azimuth information to be of great significance.
The content of the invention：
The technical problem to be solved in the present invention is：Establish the measurement equation and calibrated error of microarray formula polarised light compass Parameter, solves a kind of orientation problem of microarray formula polarised light compass.
In order to solve the above technical problems, solution proposed by the present invention is：
A kind of orientation method of microarray formula polarised light compass, this method comprise the following steps：
Step 1. establishes the measurement equation of microarray formula polarised light compass：
Microarray formula polarised light compass mainly polarizes chip arrays and CCD (Charge-Coupled by wide-angle lens, pixel Device) camera forms；Pixel polarizes the pixel of chip arrays and 4 CCD pixel structures that CCD pixel size is essentially equal, adjacent Into a polarimetry unit, the polarization state resolving to some view directions incident light is completed；Measured for random polarization single Member, the measurement equation of microarray formula polarised light compass are：
Wherein, j be 4 adjacent CCD pixels numbering, fjFor the measured value of j-th of pixel, the total light intensity of incident light is I, degree of polarization d, φ are the angle of polarization of the angle, i.e. incident light of incident light polarization direction and reference direction, βjFor j-th of polarization The polarised direction of piece and the angle of reference direction, β1、β2、β3And β4Respectively 0 °, 90 °, 45 °, 135 °, KjAnd bjTo be to be calibrated CCD camera linearity error parameter, kjAnd εjFor polarizer alignment error parameter to be calibrated, njFor measurement noise；
Step 2.CCD camera linearity error parameter calibrations：
Using the natural light of unbiased as input, and the response of each CCD pixel is recorded under the input of different light intensity, it is minimum Change following object functions to estimate the linearity error parameter K of CCD camerajAnd bj：
Wherein, i is the numbering of measurement, and N is pendulous frequency,Input light intensity in being measured for ith,Measured for ith Measured value；
KjAnd bjIt can be calculated by following formula：
Step 3. polarizer alignment error parameter calibration：
3.1) angle of polarization and degree of polarization of incident light corresponding to polarimetry unit are calculated
(1) formula in step 1 is rewritten as matrix form, obtains the measurement equation of each polarimetry unit：
HX=P (4)
Wherein,
Wherein, k1、k2、k3、k4、ε1、ε2、ε3And ε4For polarizer alignment error parameter to be calibrated, K1、K2、K3、K4、b1、 b2、b3And b4To pass through step 2 CCD camera linearity error parameter obtained by calibrating；
Using Least Square Method sky angle of polarization pattern, then X least-squares estimation is given by：
Wherein,It is X least-squares estimation；
The angle of polarization of incident light and degree of polarization are respectively corresponding to each polarimetry unit：
3.2) polarizer alignment error parameter to be calibrated is solved with interative least square method
It is defined as follows object function：
Wherein,It is the measured value of the angle of polarization in the m times measurement,WithThe first measurement of the angle of polarization is represented respectively Measured value and its measurement error, Δ φmFor the actual value of angle of polarization change, rmFor the measurement error in the m times measurement；
Iterative equation is：
Wherein, x0For iterative initial value, xnAnd xn+1Respectively non trivial solution after n-th and (n+1)th iteration, Jr(x) to be refined Than matrix；
When | | xn+1-xn| | when≤0.01, iterative equation converges to optimal solution；
Step 4. is based on sky angle of polarization model estimation solar azimuth：
4.1) the E direction vectors of incident light are solved
The E direction vectors of incident light are corresponding to each polarimetry unit：
Wherein, e is the E direction vectors of incident light,For the transformation matrix of incident light coordinate system to camera coordinates system；
4.2) equation of the solar direction vector on the E direction vectors of incident light is established
According to Rayleigh scattering model, the E direction vectors of incident light are perpendicular to scattering surface, i.e.,
eTS=0 (10)
Wherein, s is solar direction vector；
Solar direction vector can be estimated to obtain by two incoherent E vectors, define E=[e1 … eN], wherein eN For the E direction vectors of n-th polarimetry unit, can obtain：
ETS=0 (11)
4.3) optimization object function estimation solar direction vector is established
Solar direction vector s optimal estimation can be obtained by solving following optimization problem：
The optimal estimation of solar direction vector is and EETThe corresponding characteristic vector of minimal eigenvalue, solar direction vector It is the orientation of the sun in the projection of horizontal plane.
Compared with prior art, the present invention has advantages below：
1) microarray formula polarised light compass is the image-type polarized light sensor based on monocular camera, have it is simple in construction, The advantages such as the high and low power consumption of integrated level.
2) by establishing the measurement equation and calibrated error parameter of microarray formula polarised light compass, it is inclined to improve microarray formula Shake the orientation accuracy of light compass.
Brief description of the drawings：
1. Fig. 1 is the schematic flow sheet of the inventive method；
2. Fig. 2 is the schematic diagram of microarray formula polarised light compass；
3. Fig. 3 is the schematic diagram of CCD camera response incident light；
4. Fig. 4 is the schematic diagram of Rayleigh scattering model.
Embodiment：
The present invention is described in further detail below with reference to Figure of description and specific embodiment.
As shown in figure 1, a kind of orientation method of microarray formula polarised light compass of the present invention, this method basis first The structure of microarray formula polarised light compass, the measurement equation of microarray formula polarised light compass is established, it is linear then to demarcate CCD camera Error parameter and polarizer alignment error parameter, it is finally based on sky angle of polarization pattern acquiring solar azimuth.
The particular content of the present invention includes：
Step 1. establishes the measurement equation of microarray formula polarised light compass：
The system of microarray formula polarised light compass is formed as shown in Fig. 2 mainly by wide-angle lens 100, pixel polarizer battle array Row 101 and CCD (Charge-Coupled Device) camera 102 form；The pixel and CCD pixel chi of pixel polarization chip arrays It is very little essentially equal, complete the polarization state resolving to some view directions incident light；
As shown in figure 3, adjacent 4 CCD pixels 1,2,3,4 form a polarimetry unit, a certain polarimetry list CCD pixel is to the response of emergent light through polarizer in member：
fj=KjIj+bj+nj(j=1,2,3,4) (13)
Wherein, j be 4 adjacent CCD pixels numbering, fjFor the measured value of j-th of pixel, IjFor through polarizer The light intensity of emergent light, KjAnd bjFor CCD camera linearity error parameter to be calibrated, njFor measurement noise；
When incident light is partial polarization light, the light intensity through polarizer is：
Wherein, the total light intensity of incident light is I, and degree of polarization d, φ are the angle of incident light polarization direction and reference direction, That is the angle of polarization of incident light, βjThe angle of polarised direction and reference direction for j-th polarizer, β1、β2、β3And β4Respectively 0 °, 90 °, 45 °, 135 °, kjAnd εjFor polarizer alignment error parameter to be calibrated, njFor measurement noise；
Formula (14) is substituted into formula (13), the measurement equation for obtaining microarray formula polarised light compass is：
Step 2.CCD camera linearity error parameter calibrations：
Using the natural light of unbiased as input, and the response of each CCD pixel is recorded under the input of different light intensity, it is minimum Change following object functions to estimate the linearity error parameter K of CCD camerajAnd bj：
Wherein, i is the numbering of measurement, and N is pendulous frequency,Input light intensity in being measured for ith,Measured for ith Measured value；
KjAnd bjIt can be calculated by following formula：
Step 3. polarizer alignment error parameter calibration：
3.1) angle of polarization and degree of polarization of incident light corresponding to polarimetry unit are calculated
(15) formula in step 1 is rewritten as matrix form, obtains the measurement equation of each polarimetry unit：
HX=P (18)
Wherein
Wherein, k1、k2、k3、k4、ε1、ε2、ε3And ε4For polarizer alignment error parameter to be calibrated, K1、K2、K3、K4、b1、 b2、b3And b4To pass through step 2 CCD camera linearity error parameter obtained by calibrating；
Using Least Square Method sky angle of polarization pattern, then X least-squares estimation is given by：
Wherein,It is X least-squares estimation；
The angle of polarization of incident light and degree of polarization are respectively corresponding to each polarimetry unit：
3.2) polarizer alignment error parameter to be calibrated is solved with interative least square method
The rotational angle of two neighboring position is obtained from precise rotating platform, obtains the measured value in N+1 orientation altogether, can be with It is expressed as form：
Wherein,It is the measured value of the angle of polarization in the m times measurement；
It is defined as follows object function：
Wherein,WithThe measured value and its measurement error of the first measurement of the angle of polarization, Δ φ are represented respectivelymFor polarization The actual value of angle change, rmFor the measurement error in the m times measurement；
Iterative equation is：
Wherein, x0For iterative initial value, xnAnd xn+1Respectively non trivial solution after n-th and (n+1)th iteration, Jr(x) to be refined Than matrix；
When | | xn+1-xn| | when≤0.01, iterative equation converges to optimal solution；
Step 4. is based on sky angle of polarization model estimation solar azimuth：
4.1) the E direction vectors of incident light are solved
The E direction vectors of incident light are corresponding to each polarimetry unit：
Wherein, e is the E direction vectors of incident light,For the transformation matrix of incident light coordinate system to camera coordinates system；
As shown in figure 4, O represents the position of observer, S is position of the sun on celestial sphere, and P is observation station in the position of celestial sphere Put, OP is observed direction, incident light coordinate system (OiXiYiZi) arrive camera coordinates system (OXlYlZl) transformation matrixCalculate such as Under：
Wherein, γ and α represents zenith angle and the azimuth of observation station, f respectivelycRepresent focal length of camera, (xc,yc) it is figure Inconocenter, each pixel (x in imagep,yp) corresponding with the observation station P that it is aerial；
4.2) equation of the solar direction vector on the E direction vectors of incident light is established
According to Rayleigh scattering model, the E direction vectors of incident light are perpendicular to scattering surface, i.e.,
eTS=0 (26)
Wherein, s is solar direction vector；
Solar direction vector can be estimated to obtain by two incoherent E vectors, define E=[e1 … eN], wherein eN For the E direction vectors of n-th polarimetry unit, can obtain：
ETS=0 (27)
4.3) optimization object function estimation solar direction vector is established
Solar direction vector s optimal estimation can be obtained by solving following optimization problem：
The optimal estimation of solar direction vector is and EETThe corresponding characteristic vector of minimal eigenvalue, solar direction vector It is the orientation of the sun in the projection of horizontal plane.
Described above is only the preferred embodiment of the present invention, and protection scope of the present invention is not limited merely to above-mentioned implementation Example, all technical schemes belonged under thinking of the present invention belong to protection scope of the present invention.It should be pointed out that for the art Those of ordinary skill for, some improvements and modifications without departing from the principles of the present invention, these improvements and modifications It should be regarded as protection scope of the present invention.

Claims (1)

1. a kind of orientation method of microarray formula polarised light compass, it is characterised in that comprise the following steps：
Step 1. establishes the measurement equation of microarray formula polarised light compass：
Microarray formula polarised light compass mainly polarizes chip arrays and CCD (Charge-Coupled by wide-angle lens, pixel Device) camera forms；Pixel polarizes the pixel of chip arrays and 4 CCD pixel structures that CCD pixel size is essentially equal, adjacent Into a polarimetry unit, the polarization state resolving to some view directions incident light is completed；Measured for random polarization single Member, the measurement equation of microarray formula polarised light compass are：
<mrow> <msub> <mi>f</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <msub> <mi>K</mi> <mi>j</mi> </msub> <mn>2</mn> </mfrac> <mi>I</mi> <mo>&amp;lsqb;</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>k</mi> <mi>j</mi> </msub> <mi>d</mi> <mrow> <mo>(</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mo>(</mo> <mrow> <mn>2</mn> <mi>&amp;phi;</mi> <mo>-</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <msub> <mi>&amp;beta;</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mi>j</mi> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> <mo>+</mo> <msub> <mi>b</mi> <mi>j</mi> </msub> <mo>+</mo> <msub> <mi>n</mi> <mi>j</mi> </msub> <mo>,</mo> <mrow> <mo>(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein, j be 4 adjacent CCD pixels numbering, fjFor the measured value of j-th of pixel, the total light intensity of incident light is I, partially Degree of shaking is d, and φ is the angle of polarization of the angle, i.e. incident light of incident light polarization direction and reference direction, βjFor j-th polarizer The angle of polarised direction and reference direction, β1、β2、β3And β4Respectively 0 °, 90 °, 45 °, 135 °, KjAnd bjFor CCD to be calibrated Camera linearity error parameter, kjAnd εjFor polarizer alignment error parameter to be calibrated, njFor measurement noise；
Step 2.CCD camera linearity error parameter calibrations：
Using the natural light of unbiased as input, and the response of each CCD pixel is recorded under the input of different light intensity, under minimum Object function is stated to estimate the linearity error parameter K of CCD camerajAnd bj：
<mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>(</mo> <msubsup> <mi>f</mi> <mi>j</mi> <mi>i</mi> </msubsup> <mo>-</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> <mo>-</mo> <msub> <mi>b</mi> <mi>j</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
Wherein, i is the numbering of measurement, and N is pendulous frequency,Input light intensity in being measured for ith,For the survey of ith measurement Value；
KjAnd bjIt can be calculated by following formula：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>K</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> <msubsup> <mi>f</mi> <mi>j</mi> <mi>i</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <msup> <mi>N</mi> <mn>2</mn> </msup> </mfrac> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>f</mi> <mi>j</mi> <mi>i</mi> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <mrow> <mo>(</mo> <mrow> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mn>1</mn> <msup> <mi>N</mi> <mn>2</mn> </msup> </mfrac> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>b</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>f</mi> <mi>j</mi> <mi>i</mi> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> <mrow> <mo>(</mo> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <msub> <mi>I</mi> <mi>j</mi> </msub> <mi>i</mi> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
Step 3. polarizer alignment error parameter calibration：
3.1) angle of polarization and degree of polarization of incident light corresponding to polarimetry unit are calculated
(1) formula in step 1 is rewritten as matrix form, obtains the measurement equation of each polarimetry unit：
HX=P (4)
Wherein,
<mrow> <mi>H</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mi>sin</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mn>2</mn> </msub> <mi>sin</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>3</mn> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mn>3</mn> </msub> <mi>sin</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>4</mn> </msub> <mi>cos</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>4</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mn>4</mn> </msub> <mi>sin</mi> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>&amp;beta;</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>&amp;epsiv;</mi> <mn>4</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>X</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>I</mi> <mi>d</mi> <mi> </mi> <mi>cos</mi> <mn>2</mn> <mi>&amp;phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>I</mi> <mi>d</mi> <mi> </mi> <mi>sin</mi> <mn>2</mn> <mi>&amp;phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>I</mi> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>P</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>)</mo> <mo>/</mo> <msub> <mi>K</mi> <mn>1</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>)</mo> <mo>/</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>3</mn> </msub> <mo>)</mo> <mo>/</mo> <msub> <mi>K</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>4</mn> </msub> <mo>-</mo> <msub> <mi>b</mi> <mn>4</mn> </msub> <mo>)</mo> <mo>/</mo> <msub> <mi>K</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
Wherein, k1、k2、k3、k4、ε1、ε2、ε3And ε4For polarizer alignment error parameter to be calibrated, K1、K2、K3、K4、b1、b2、b3 And b4To pass through step 2 CCD camera linearity error parameter obtained by calibrating；
Using Least Square Method sky angle of polarization pattern, then X least-squares estimation is given by：
<mrow> <mover> <mi>X</mi> <mo>^</mo> </mover> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>H</mi> <mi>T</mi> </msup> <mi>H</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>H</mi> <mi>T</mi> </msup> <mi>P</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
Wherein,It is X least-squares estimation；
The angle of polarization of incident light and degree of polarization are respectively corresponding to each polarimetry unit：
<mrow> <mi>&amp;phi;</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> </mfrac> <mo>)</mo> </mrow> <mo>,</mo> <mi>d</mi> <mo>=</mo> <mfrac> <msqrt> <mrow> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>1</mn> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>2</mn> <mn>2</mn> </msubsup> </mrow> </msqrt> <msub> <mover> <mi>x</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
3.2) polarizer alignment error parameter to be calibrated is solved with interative least square method
It is defined as follows object function：
<mrow> <mtable> <mtr> <mtd> <mrow> <mi>min</mi> <mi> </mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>|</mo> <mo>|</mo> <mi>r</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>|</mo> <mo>|</mo> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msup> <msub> <mi>r</mi> <mi>m</mi> </msub> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>r</mi> <mi>m</mi> </msub> <mo>=</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>~</mo> </mover> <mi>m</mi> </msub> <mo>-</mo> <msub> <mover> <mi>&amp;phi;</mi> <mo>~</mo> </mover> <mn>0</mn> </msub> <mo>-</mo> <mi>&amp;delta;</mi> <msub> <mover> <mi>&amp;phi;</mi> <mo>~</mo> </mover> <mn>0</mn> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;&amp;phi;</mi> <mi>m</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>&amp;equiv;</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>&amp;epsiv;</mi> <mn>1</mn> </msub> <mo>&amp;equiv;</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>x</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>&amp;epsiv;</mi> <mn>4</mn> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mn>2</mn> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mn>3</mn> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mn>4</mn> </msub> </mtd> <mtd> <mrow> <mi>&amp;delta;</mi> <msub> <mover> <mi>&amp;phi;</mi> <mo>~</mo> </mover> <mn>0</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
Wherein,It is the measured value of the angle of polarization in the m times measurement,WithThe measured value of the first measurement of the angle of polarization is represented respectively With its measurement error, Δ φmFor the actual value of angle of polarization change, rmFor the measurement error in the m times measurement；
Iterative equation is：
<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msub> <mi>J</mi> <mi>r</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <msub> <mi>J</mi> <mi>r</mi> </msub> <mo>(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>J</mi> <mi>r</mi> </msub> <msup> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mi>T</mi> </msup> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
Wherein, x0For iterative initial value, xnAnd xn+1Respectively non trivial solution after n-th and (n+1)th iteration, Jr(x) it is Jacobi Matrix；
When | | xn+1-xn| | when≤0.01, iterative equation converges to optimal solution；
Step 4. is based on sky angle of polarization model estimation solar azimuth：
4.1) the E direction vectors of incident light are solved
The E direction vectors of incident light are corresponding to each polarimetry unit：
<mrow> <mi>e</mi> <mo>=</mo> <msubsup> <mi>C</mi> <mi>i</mi> <mi>l</mi> </msubsup> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&amp;phi;</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Wherein, e is the E direction vectors of incident light,For the transformation matrix of incident light coordinate system to camera coordinates system；
4.2) equation of the solar direction vector on the E direction vectors of incident light is established
According to Rayleigh scattering model, the E direction vectors of incident light are perpendicular to scattering surface, i.e.,
eTS=0 (10)
Wherein, s is solar direction vector；
Solar direction vector can be estimated to obtain by two incoherent E vectors, define E=[e1 … eN], wherein eNFor The E direction vectors of N number of polarimetry unit, can be obtained：
ETS=0 (11)
4.3) optimization object function estimation solar direction vector is established
Solar direction vector s optimal estimation can be obtained by solving following optimization problem：
<mrow> <mtable> <mtr> <mtd> <mrow> <munder> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> <mi>s</mi> </munder> <mrow> <mo>(</mo> <msup> <mi>s</mi> <mi>T</mi> </msup> <msup> <mi>EE</mi> <mi>T</mi> </msup> <mi>s</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </mtd> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> </mrow> </mtd> <mtd> <mrow> <msup> <mi>s</mi> <mi>T</mi> </msup> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
The optimal estimation of solar direction vector is and EETThe corresponding characteristic vector of minimal eigenvalue, solar direction vector is in level The projection in face is the orientation of the sun.
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