CN107270866A - Method for estimating distance based on telephoto lens Binocular Stereo Vision System - Google Patents

Method for estimating distance based on telephoto lens Binocular Stereo Vision System Download PDF

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CN107270866A
CN107270866A CN201710328530.2A CN201710328530A CN107270866A CN 107270866 A CN107270866 A CN 107270866A CN 201710328530 A CN201710328530 A CN 201710328530A CN 107270866 A CN107270866 A CN 107270866A
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黄靖
黄椰
姜文
周高景
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Wuhan University of Technology WUT
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Wuhan University of Technology WUT
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01CMEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C3/00Measuring distances in line of sight; Optical rangefinders

Abstract

The invention discloses a kind of method for estimating distance based on telephoto lens Binocular Stereo Vision System, comprise the following steps:1) disparity estimation, accurate parallax value;2) calculating of telephoto lens focal length;3) range measurement:After accurate parallax value k and focal length value f is obtained, camera is calculated to the distance of object.The present invention proposes the focal length that a kind of new method calculates telephoto lens, on this basis, accurate parallax value is obtained with NCC and APF method, compared to the research done before, this paper measurement distance scope is wider, and measurement accuracy is higher.

Description

Method for estimating distance based on telephoto lens Binocular Stereo Vision System
Technical field
The present invention relates to image processing techniques, more particularly to a kind of distance based on telephoto lens Binocular Stereo Vision System Method of estimation.
Background technology
Traditional distance-finding method based on binocular vision is broadly divided into two parts:1) regarding for left and right two images is calculated Difference;2) focal length of camera lens is demarcated, according to obtained parallax and focal length data, distance is calculated with ranging formula.
In the research for calculating left images parallax value, Chen et al. proposes standing based on adaptive ground control point The method of body matching, the method that Qing et al. is filtered using non-local mean is used for parallax, the liter sampling of depth and optimized, Lin Et al. propose a kind of solid matching method based on Block- matching super-pixel segmentation, Mikhail et al. uses two step energy minimizations Algorithm realizes Stereo matching, and it is inconsistent that Jian et al. proposes that the parallax optimization method of coloured image goes to remove border in disparity map Region.
In the research of telephoto lens calibrated focal length, Stamatopoulos proposes the self-calibrating method of telephoto lens, most Long focal length can reach 400mm, but need the network structure configuration of 13 websites and 39 width images, and the method for demarcation is more complicated; Cold et al. the level of detail merged for landform of Xie Wen proposes a kind of dotted line fusion method based on end point, most long demarcation Focal length reaches 150mm, but is not particularly suited for conventional measuring environment;Ian et al. utilizes the color geometrical constraint of a width coloured image To estimate the focal length of camera, the maximum focal length of measurement has 5% error when being 300mm, it is impossible to ensure higher range accuracy.
In newest binocular ranging research, the maximum distance of Liang et al. binoculars measurement is 30m;Sergiu et al. is most The vehicle distances far measured are 95m, but have 2m or so error;The method that Peter et al. compares different Stereo matchings, is tested Card can farthest measure 200m distance using the method for stereoscopic vision, the error for having 4 meters or so.
Present Research for more than, the problem of current binocular ranging is present be --- calculate the parallax of left and right two images Error is big, causes the precision of ranging low;And most of is that short-focus lens are demarcated, the distance that can be measured is short.
The content of the invention
The technical problem to be solved in the present invention is double based on telephoto lens there is provided one kind for defect of the prior art The method for estimating distance of item stereo vision system.
The technical solution adopted for the present invention to solve the technical problems is:One kind is based on telephoto lens binocular stereo vision system The method for estimating distance of system, comprises the following steps:
1) disparity estimation;
1.1) one group of obtained picture is shot by left and right camera by same target;
1.2) determine that same target point is imaged on each self-corresponding pixel coordinate position on left images, obtain their positions Pixel difference be binocular parallax;
1.3) solved using the method for normalized crosscorrelation (NCC) and obtain the parallax k of integer, on this basis, using pair The Parabolic Fit (SPF) of title and the method for asymmetrical Parabolic Fit (APF), which are solved, obtains τ1And τ2, corresponding parallax k1 =k+t1,k2=k+t2
2) calculating of telephoto lens focal length;
2.1) linear regression model (LRM) on parallax is set up:
fi01kii (1)
Wherein, kiThe value of independent variable parallax, f when being observed for ithiFor dependent variable focal length f observation, it is assumed that εi(i= 1,2 ..., n) separate, and obey same normal distribution N (0, σ2);
2.2) model obtains factor beta after calculating obtains the parallax of two images0, β1
2.3) according to the regression model set up, parallax value k is substituted into, you can obtain focal length value f;
3) range measurement
After accurate parallax value k and focal length value f is obtained, camera can be calculated by formula (2) to the distance of object:
Wherein B and f represent the baseline and focal length of two cameras, d respectivelypThe size of pixel is represented, K represents parallax, K=k+ τ2
By such scheme, the step 1.3) find optimal k values first by NCC method:
Assuming that the corresponding optimum point P of left imagesLAnd PRMeet:
IL(x, y)=IR(x-k-τ,y)
Wherein,For relational operator, when NCC functions reach maximum, you can obtain corresponding k values.
By such scheme, τ 1 and τ 2 is calculated with SPF and APF respectively;
First τ 1 is calculated using the method for symmetrical Parabolic Fit (SPF):
Then the parallax obtained by the calculating of SPF methods is:
K1=k+ τ1 (5)
Assume that NCC functions are distributed as symmetrical parabola in above formula, but the NCC functions of most of images are asymmetric ,
τ 2 can be calculated using asymmetrical Parabolic Fit;
The curve of left and right two is expressed as:
Yl=a1xl 2+b1xl+c1
Yr=a2xr 2+b2xr+c2 (6)
To calculate two curves in left and right, the value of curve corresponding coefficient need to be determined.Wherein a1And a2Can returning by left image The method of one change cross-correlation, which is calculated, to be obtained.
klAnd krFor τ1The integer value of both sides, general klTake -1 or 0, krTake 0 or 1, b1,c1,b2,c2Calculated by following formula:
So far a1, b1, c1, a2, b2, c2 have been calculated and obtained, can calculated using asymmetrical Parabolic Fit and obtain τ 2 are:
Then the parallax obtained by the calculating of APF methods is:
k2=k+ τ2 (10)
The final parallax K calculated that substitutes into is k2
The beneficial effect comprise that:Propose the focal length that a kind of new method calculates telephoto lens.It is basic herein On, accurate parallax value is obtained with NCC and APF method, compared to the research done before, this paper measurement distance scope Wider, measurement accuracy is higher.
Brief description of the drawings
Below in conjunction with drawings and Examples, the invention will be further described, in accompanying drawing:
Fig. 1 is one group of image of the binocular camera shooting of the embodiment of the present invention;
Fig. 2 is the NCC and symmetrical and asymmetrical Parabolic Fit (SPF) method schematic diagram of the embodiment of the present invention;
Fig. 3 is that the focal length of the embodiment of the present invention calculates schematic diagram;
Fig. 4 is the binocular camera schematic diagram of the embodiment of the present invention;
Fig. 5 is the RTK rangefinder schematic diagrames of the embodiment of the present invention;
Fig. 6 is the shooting image schematic diagram of the embodiment of the present invention;
Fig. 7 is the mcintosh and Overwater-floating ball schematic diagram of the embodiment of the present invention;
Fig. 8 is the method flow diagram of the embodiment of the present invention.
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, with reference to embodiments, to the present invention It is further elaborated.It should be appreciated that specific embodiment described herein is not used to limit only to explain the present invention The fixed present invention.
The inventive method shoots one group of obtained picture as shown in figure 8, Fig. 1 is left and right camera, and Chinese character " text " is in left and right two The coordinate of width image is P respectivelyL(x, y) and PR(x-k- τ, y), wherein parallax is that k+ τ, k and τ represent integer and fractional part respectively Point.Assuming that the corresponding optimum point P of left imagesLAnd PRMeet:
IL(x, y)=IR(x-k-τ,y) (1)
Optimal k values are found using NCC method:
WhereinFor relational operator, when NCC values reach maximum, you can obtain k values.To calculate τ, respectively with SPF and APF Calculate τ1And τ2, wherein SPF and APF are as shown in Figure 2.
First τ 1 is calculated using the method for symmetrical Parabolic Fit (SPF):
Then SPF methods calculate obtained by parallax k1For:
k1=k+ τ1(5)
Assume that NCC functions are distributed as symmetrical parabola in above formula, but the NCC functions of most of images are asymmetric , therefore in order to calculate τ in Fig. 2 right figures2Size, now two curves of Fig. 2 right figures or so are expressed as:
Yl=a1xl 2+b1xl+c1
Yr=a2xr 2+b2xr+c2 (6)
klAnd krFor τ1The value of both sides, general klTake -1 or 0, kr0 or 1 is taken, then b1, c1, b2, c2 can be calculated by following formula :
So far a1, b1, c1, a2, b2, c2Calculate and obtained, can have been calculated using asymmetrical Parabolic Fit and obtain τ2 For:
Then the parallax obtained by the calculating of APF methods is:
k2=k+ τ2 (10)
2) calculating of telephoto lens focal length
Ensure the range accuracy requirement of binocular range-measurement system, it is important to ensure that the inside and outside parameter of binocular camera and binocular are surveyed Accuracy away from systematic parameter in system, so that it is guaranteed that these relevant parameters are updated to the essence of distance measurement result during apart from calculation formula The problem of calibrating of true property, i.e. binocular range-measurement system.Telephoto lens causes mutually confidential away from object because its focal length is big, the angle of visual field is small It is remote enough, find to be difficult to realize when carrying out telephoto lens demarcation with traditional scaling method in an experiment and result is wrong.
In order to obtain actual focal length value, a kind of computational methods of new telephoto lens are proposed herein.We can With approximate focal length is represented with (11) formula:
Wherein B and f represent the baseline and focal length of two cameras, d respectivelypThe size of pixel is represented, k represents regarding for integer Difference, ZdRepresent distance.
As shown in figure 3, binocular camera is placed in into same horizontal line, remote n meters of object is shot, it is known that camera to object Apart from Zd, parallax k can calculate by the image that obtains and obtained, dpDetermined by picture size, thus according to formula (11) it is available away from From n-1, the corresponding focal length value f of n-2 ... n-9n-1,fn-2…fn-9.Due to needing one meter one meter of calculating focal length in aforementioned manners Value, it is inconvenient for use in actually measurement.Therefore in actual measurement process, the above method is calculated to obtained focal length value and sets up pass In the linear regression model (LRM) of parallax:
fi01kii (12)
Wherein, kiThe value of independent variable parallax, f when being observed for ithiFor dependent variable focal length f observation, it is assumed that εi(i= 1,2 ..., n) separate, and obey same normal distribution N (0, σ2).The regression model can be analyzed with software SPSS Arrive, specifically how to obtain regression model using SPSS study, will be shown in following experiment.Obtained using the model in calculating After the parallax of two images, the focal length value that just can be estimated.
3) range measurement
After accurate parallax value k and focal length value f is obtained, camera can be calculated by formula (13) to the distance of object:
Wherein, B and f represent the baseline and focal length of two cameras respectively, and dp represents the size of pixel, and K is represented with APF meters K+ τ can be used in obtained parallax, formula2Instead of K, wherein τ2For the fractional part of parallax.
It is of the invention with existing method, from focal length, survey maximum distance, the error three of ranging in terms of contrast, as a result As shown in table 1.
Table 1. is of the invention to be contrasted with existing binocular distance-finding method
One instantiation:
In this example, camera uses Germany's deterrence Walimex 500mm F832 telephoto lenses.Actual focal length value In 500mm or so.Binocular camera is as shown in Figure 4.In order to obtain accurate distance value, RTK (carrier phase technology) ranging is used Instrument obtains the accurate distance of camera distance object, as shown in Figure 5.
Mark plate is placed in the distance away from camera 60-69m and 140-149m by us, is respectively shot with left and right camera every one meter Piece image.Obtained a series of images data are as shown in Fig. 6-1 and Fig. 6-2.
Parallax is calculated with NCC method to left and right two images, then formula (11) calculates focal length, obtained parallax 2-1 and table 2-2 are shown in Table with the value of focal length.
The parallax value and focal length of table 2-1 60-69m images
The parallax value and focal length of table 2-2 140-149m images
Set up in the experimental data obtained from above-mentioned experiment on the linear regression model (LRM) between focal length and parallax, 60-69m and 140-149m focal length and the data value of parallax are inputted in SPSS, the regression coefficient for obtaining model is analyzed by SPSS Table, is shown in Table 3.
The regression coefficient table of table 3
From regression coefficient table it can be seen that sig=0, i.e. significance test p value be 0, it can be seen that dependent variable and from Linear relationship between variable is obvious, and focal length f and parallax k linearity of regression model is:
F=-0.513*k+604.015 (14)
Model (14) can accurate expression focal length f and parallax k relation, below by with the model calculate it is actual away from From.In actual experiment, by binocular telephoto lens frame in bank, mcintosh chest and Overwater-floating in prospect map are shot respectively Ball.Mcintosh chest is as shown in fig. 7, Overwater-floating ball is as shown in Figure 7.
The picture that obtained mcintosh chest is shot to left and right camera calculates parallax with NCC methods, can obtain integer Parallax k1.Focal length f can obtain according to the linearity of regression model of parallax and focal length1=542.968m, on this basis with symmetrical The method of Parabolic Fit (SPF), can be calculated τ1=0.10, k2=k11.Because the NCC functions of most of images are non- Symmetrical Parabolic Fit (APF), τ is can be calculated with APF method2=0.2995, k3=k12.Two-phase is measured in experiment The distance between machine b=14.8cm, actual range is obtained for 102.053m with RTK (carrier phase difference technology) measuring instrument, By parallax k1,k2,k3With focal length f1It is updated in formula (13), it is different apart from mcintosh chest that calculating obtains telephoto lens Distance measurement result and range error are shown in Table 4-1, and the result and range error of Overwater-floating ball are shown in Table 4-2.
The distance measurement result of table 4-1. mcintosh case distinct methods
The distance measurement result of table 4-2. Overwater-floating ball distinct methods
It should be appreciated that for those of ordinary skills, can according to the above description be improved or converted, And all these modifications and variations should all belong to the protection domain of appended claims of the present invention.

Claims (3)

1. a kind of method for estimating distance based on telephoto lens Binocular Stereo Vision System, comprises the following steps:
1) disparity estimation;
1.1) one group of obtained picture is shot by left and right camera by same target;
1.2) determine that same target point is imaged on each self-corresponding pixel coordinate position on left images, obtain the picture of their positions Plain difference is binocular parallax;
1.3) the parallax k for obtaining integer is solved using the method for normalized crosscorrelation, on this basis, symmetrical parabola is used The method of fitting and asymmetrical Parabolic Fit solves and obtains τ1And τ2, corresponding parallax k1=k+t1,k2=k+t2
2) calculating of telephoto lens focal length;
2.1) linear regression model (LRM) on parallax is set up:
fi01kii (1)
Wherein, kiThe value of independent variable parallax, f when being observed for ithiFor dependent variable focal length f observation, it is assumed that εi(i=1, 2 ..., n) separate, and obey same normal distribution N (0, σ2);
2.2) model obtains factor beta after calculating obtains the parallax of two images0, β1
2.3) according to the regression model set up, parallax value k is substituted into, you can obtain focal length value f;
3) range measurement
After accurate parallax value k and focal length value f is obtained, camera can be calculated by formula (2) to the distance of object:
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Wherein B and f represent the baseline and focal length of two cameras, d respectivelypThe size of pixel is represented, K represents parallax, K=k+ τ2
2. the method for estimating distance according to claim 1 based on telephoto lens Binocular Stereo Vision System, its feature exists In the step 1.3) find optimal k values first by NCC method:
Assuming that the corresponding optimum point P of left imagesLAnd PRMeet:
IL(x, y)=IR(x-k-τ,y)
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Wherein,For relational operator, when NCC functions reach maximum, you can obtain corresponding k values.
3. the method for estimating distance according to claim 1 based on telephoto lens Binocular Stereo Vision System, its feature exists In step 1.3) τ is calculated with SPF and APF respectively when calculating the fractional part of parallax1And τ2
First τ is calculated using symmetrical Parabolic Fit SPF method1
<mrow> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mn>4</mn> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
Then the parallax obtained by the calculating of SPF methods is:
k1=k+ τ1 (5)
Assume that NCC functions are distributed as symmetrical parabola in above formula, however the NCC functions of most of images be it is asymmetrical,
τ 2 can be calculated using asymmetrical Parabolic Fit;
The curve of left and right two is expressed as:
Yl=a1xl 2+b1xl+c1
Yr=a2xr 2+b2xr+c2 (6)
To calculate two curves in left and right, the value of curve corresponding coefficient need to be determined;Wherein a1And a2Can be by the normalization of left image The method of cross-correlation is calculated and obtained;
<mrow> <mi>N</mi> <mi>A</mi> <mi>C</mi> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mi>k</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>W</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> <mo>&amp;CircleTimes;</mo> <msub> <mi>W</mi> <mi>L</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <mi>&amp;Delta;</mi> <mi>k</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </mrow>
<mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mi>A</mi> <mi>C</mi> <mrow> <mo>(</mo> <mo>-</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> <msup> <mrow> <mo>(</mo> <mo>-</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> </mrow>
<mrow> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mi>N</mi> <mi>A</mi> <mi>C</mi> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> <msup> <mrow> <mo>(</mo> <mn>0.5</mn> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
klAnd krFor τ1The integer value of both sides, general klTake -1 or 0, krTake 0 or 1, b1,c1,b2,c2Calculated by following formula:
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>b</mi> <mn>1</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mn>1</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>k</mi> <mi>l</mi> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <msup> <msub> <mi>k</mi> <mi>l</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <msup> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>
<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>b</mi> <mn>2</mn> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mn>2</mn> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>k</mi> <mi>r</mi> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mfenced> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <msub> <mi>k</mi> <mi>r</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msup> <msub> <mi>k</mi> <mi>r</mi> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>N</mi> <mi>C</mi> <mi>C</mi> <mrow> <mo>(</mo> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <msup> <msub> <mi>&amp;tau;</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>
So far a1, b1, c1, a2, b2, c2 have been calculated and obtained, can calculated using asymmetrical Parabolic Fit and obtain τ2For:
<mrow> <msub> <mi>&amp;tau;</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> </mrow> <mrow> <mo>-</mo> <mn>2</mn> <mrow> <mo>(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
Then the parallax obtained by the calculating of APF methods is:
k2=k+ τ2 (10)。
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