Automatic calibration device and method for linear spectrum profile sensor

Abstract

The invention relates to an automatic calibration device and method for a line spectrum profile sensor, and the method comprises the steps: employing the line spectrum profile sensor and a laser interferometer to calibrate the linearity of a displacement direction, and arranging a checkerboard marker on a motion axis to calibrate the position of a profile direction; during calibration, the checkerboard target is controlled to move step by step from the near end of the measuring range to the far end of the measuring range, the X-direction position, the Z-direction position and the light intensity in the original contour of the line spectrum contour sensor at each sampling position are recorded, and the data of the laser interferometer are recorded as the Z-direction position reference; detecting light and shade segmentation points of light intensity on the checkerboard target through an inflection point detection algorithm to obtain an X-direction position reference; after full-scale scanning is completed, two-dimensional contour data at different positions form a three-dimensional surface shape for fitting, and a calibration table is generated for calibration. According to the invention, the positions of the displacement direction and the contour direction can be calibrated at the same time in single scanning, the method is suitable for calibration of line spectrum contour sensors with various measuring ranges, and the calibration precision and the production efficiency can be obviously improved.

Description

Technical Field

[0001] The present invention relates to the field of precision measurement and sensing technology, and in particular to an automatic calibration device and method for a line spectrum profile sensor. Background Art

[0002] Spectral confocal displacement sensors have been widely used in the industrial field in recent years due to their significant advantages such as high accuracy, good stability, and a wide range of measurable materials. They have become one of the indispensable important sensors in industrial production applications such as product shape measurement and defect detection. The line spectral profile sensor is a sensor that converts the confocal point into a confocal line through the design of the optical system based on the point spectral confocal sensor. It can obtain the shape data of a line of the measured sample in a single measurement, thus greatly improving the measurement efficiency. However, while the line scanning spectral confocal system has achieved improved measurement performance, it has also brought great challenges to its design, assembly and calibration processes. The calibration process plays a decisive role in the performance of the final product. Due to the complex structure of such sensors and the wide variety of optical and mechanical components involved, errors are inevitably introduced during the manufacturing and assembly stages, which may lead to significant individual differences between products. Therefore, it is difficult to obtain more significant performance improvements at the application level by optimizing only from the design level of the sensor. In order to meet the needs of industrial mass applications, it is necessary to carry out an automated calibration of the line spectral profile sensor to eliminate and reduce the performance differences caused by manual operations during the product manufacturing and assembly process as much as possible. While improving the environmental adaptability of the product, the various accuracy indicators of the product can also be further optimized through test calibration. Summary of the invention

[0003] The purpose of the present invention is to overcome the problems existing in the prior art and to provide an automatic calibration device and method for a line spectrum profile sensor, which can not only significantly improve production efficiency, but also effectively ensure measurement accuracy, and provide guidance and guarantee for the installation and adjustment of the line spectrum profile sensor.

[0004] In order to achieve the above technical objectives and the above technical effects, the present invention is implemented through the following technical solutions:

[0005] An automatic calibration device for a line spectrum profile sensor, comprising a probe fixing fixture, a line spectrum profile sensor, a motion axis, a checkerboard target, an interferometer reflector, a laser interferometer and a host computer, wherein:

[0006] The corresponding measuring system is fixed on the vibration isolation platform by using the fixing fixture to reduce the influence of environmental vibration on the measurement;

[0007] The checkerboard target and the interferometer reflector are arranged on the motion axis 3;

[0008] The host computer controls the motion axis to achieve displacement in the depth direction of the object under test, and obtains the displacement information of the object under test through a laser interferometer. The displacement information of the laser interferometer is returned to the host computer for calibration fitting calculation. In the line length direction, the corresponding calibration system performs X coordinate calibration mapping through a checkerboard target.

[0009] Further, the measurement beam of the line spectral profile sensor, the outgoing beam and the return beam of the laser interferometer are all parallel to the motion direction of the motion axis.

[0010] A method for automatic calibration of a line spectral profile sensor. This method uses a line spectral profile sensor and a laser interferometer to calibrate the linearity of the displacement direction, and arranges a checkerboard target on the motion axis to calibrate the position in the profile direction. During calibration, control the checkerboard target to perform a step-by-step movement from the near end to the far end of the range, record the X-direction position, Z-direction position and light intensity in the original profile of the line spectral profile sensor at each sampling position, and record the laser interferometer data as the Z-direction position reference. Then, detect the bright-dark segmentation point of the light intensity on the checkerboard target through the inflection point detection algorithm to obtain the X-direction position reference. After completing the full-range scan, fit the three-dimensional surface formed by the two-dimensional profile data at different positions to generate a calibration table for calibration.

[0011] Further, the process of collecting and calibrating the line spectral X-Z coordinate data is as follows: At different positions within the range of the line spectral profile sensor, read the profile data. After obtaining multiple profiles, perform operations to obtain a two-dimensional calibration table. Specifically, collect M groups of data and collect P-point profile data. The single-profile collection and data extraction are carried out through the following steps:

[0012] Step S101: Call the original profile reading function to obtain the original profile X coordinate x u and the original profile Z coordinate z u and the light intensity i of the coordinate point u ;

[0013] Step S102: Pass x u and i u into the light intensity mutation point detection function to obtain the mutation point x s , and obtain the corresponding z s of the mutation point x s according to linear interpolation. Obtain the corresponding d s of x s according to the actual interval of the checkerboard;

[0014] Step S103: Collect the interferometer position z i corresponding to the current position;

[0015] Step S104: For X coordinate fitting, take (x s , z s , ds ) as the (x rx , y rx , z rx ) data of the original three-dimensional coordinates;

[0016] Step S105: For the Z - coordinate fitting, take (x u , z u , z i ) as the (x rz , y rz , z rz ) data of the original three - dimensional coordinates.

[0017] Furthermore, during the calibration process, the X - coordinate data is segmented and extracted through the following specific steps:

[0018] Step S201: Define the zero - point position pixel in the X - direction as zeropixel = P / 2, and the line width as W;

[0019] Step S202: Obtain the set of light - intensity mutation points xs according to the inflection - point segmentation extraction algorithm, and the unit of xs is pixel;

[0020] Step S203: Conduct a linear increasing search in xs to find the index i such that xs i ≤zeropixel and xs i+1 >zeropixel;

[0021] Step S204: The index offset of the zero - point position pixel in xs

[0022] Step S205: For xs i , its corresponding actual position ds i =(i - zerooffset)×W.

[0023] Furthermore, during calibration, x rz , x rx corresponds to the position of the points on the contour in the X - direction, so its range is between [0, P], and y rz , y rx corresponds to the position of the points on the contour in the Z - direction, so its range is between [0, Q]. The three - dimensional data fitting and calibration table generation are carried out through the following specific steps:

[0024] Step S301: Call the original contour reading function to obtain the original data of the P points for Z - coordinate fitting (x rz , y rz , z rz ), and the original data of the X - coordinate (x rx , y rx , z rx) The number of points is determined according to the number of detected mutations;

[0025] Step S302: Repeatedly collect M profiles;

[0026] Step S303: Generate grid coordinates in the X direction from 0:N / 63:N and in the Y direction from 0:Q / 255:Q, with a size of axb, and expand it to obtain (x f , y f );

[0027] Step S304: Pass (x rz , y rz , z rz ) and (x f , y f ) into the surface fitting function F, and its return value z fz is the value of Z in the calibration table;

[0028] Step S305: Pass (x rx , y rx , z rx ) and (x f , y f ) into the surface fitting function F, and its return value z fx is the value of X in the calibration table.

[0029] Furthermore, the bilinear interpolation algorithm is used to complete the calculation of the calibration value.

[0030] Furthermore, when performing verification, the Z-direction verification is completed through the following specific steps:

[0031] Step S401: For each profile, calculate its minimum value xmin and maximum value xmax in the X direction;

[0032] Step S402: The minimum and maximum values in the X direction of N profiles respectively form arrays Xmin = [xmin 1 , xmin 2 , …, xmin N and Xmax = [xmax 1 , xmax 2 , …, xmax N ;

[0033] Step S403: Find the maximum value xminmax of Xmin and the minimum value xmaxmin of Xmax to ensure that corresponding data can be found for all profiles at xminmax and xmaxmin;

[0034] Step S404: Generate an arithmetic progression xinterp of 100 points between xminmax and xmaxmin;

[0035] Step S405: obtaining the Z-direction interpolation position zinterp corresponding to xinterp through interpolation;

[0036] Step S406: Calculate zinterp and the interferometer count z corresponding to the height of each profile i The difference zdiff;

[0037] Step S407: zdiff of all contours constitutes the overall residual zresidual in the Z direction;

[0038] Step S408: Linearity

[0039] Furthermore, during verification, the X-direction verification is completed through the following specific steps:

[0040] Step S501: define the width of the X-direction scribe line as W;

[0041] Step S502: Obtain a light intensity mutation point set xs according to an inflection point segmentation and extraction algorithm, where the unit of xs is mm;

[0042] Step S503: Perform a linear incremental search on xs to find the index i such that xs i ≤0 and xs i+1 >0;

[0043] Step S504: Index offset of zero-position pixel in xs

[0044] Step S505: For xs i , which corresponds to the actual position ds i =(i-zerooffset)×W;

[0045] Step S506: X-direction residual xresidual=xs-ds.

[0046] The beneficial effects of the present invention are:

[0047] The present invention eliminates and reduces performance differences caused by manual operations during product manufacturing and assembly as much as possible, further optimizes various product accuracy indicators through test calibration while improving the environmental adaptability of the product, and can achieve simultaneous calibration of the displacement direction and the profile direction in a single scan. It is suitable for the calibration of line spectrum profile sensors of various ranges, and can significantly improve calibration accuracy and production efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 It is a schematic diagram of an automatic calibration device for a line spectrum profile sensor of the present invention;

[0049] Figure 2 Schematic diagram of the single-profile acquisition and data extraction process of the present invention;

[0050] Figure 3 Schematic diagram of the X-axis coordinate sampling process of the present invention;

[0051] Figure 4 Schematic diagram of the process of dividing and extracting the X coordinates of the line spectrum of the present invention;

[0052] Figure 5 Schematic diagram of the division of the X coordinates of the line spectrum of the present invention;

[0053] Figure 6 Schematic diagram of the three-dimensional data fitting and calibration table generation process of the present invention;

[0054] Figure 7 Schematic diagram of the Z coordinate data fitting of the present invention;

[0055] Figure 8 Schematic diagram of the generation of the Z-direction verification interpolation data of the present invention;

[0056] Figure 9 Schematic diagram of the Z-direction verification process of the present invention;

[0057] Figure 10 Schematic diagram of the X-direction verification process of the present invention.

[0058] Explanation of the reference numerals in the figure: 1. Probe fixing tooling, 2. Line spectrum profile sensor, 3. Moving axis, 4. Checkerboard target, 5. Interferometer mirror, 6. Laser interferometer, 7. Host computer, 8. Measuring beam, 9. Return beam, 10. Output beam. Detailed implementation manners

[0059] The present invention will be described in detail below with reference to the accompanying drawings and in conjunction with embodiments.

[0060] As Figure 1 shown, an automatic calibration device for a line spectrum profile sensor includes a probe fixing tooling 1, a line spectrum profile sensor 2, a moving axis 3, a checkerboard target 4, an interferometer mirror 5, a laser interferometer 6, and a host computer 7, wherein:

[0061] The corresponding measurement system is fixed on the vibration isolation platform by using the fixing tooling 1 to reduce the influence of environmental vibration on the measurement;

[0062] The checkerboard target 4 and the interferometer mirror 5 are arranged on the moving axis 3, and the moving axis 3 is an electric moving stage. The line spectrum profile sensor 2 and the laser interferometer 6 are respectively arranged outside both ends of the electric moving stage;

[0063] The host computer 7 controls the motion axis 3 to achieve displacement in the depth direction of the object under test, and obtains the displacement information of the object under test through the laser interferometer 6. The displacement information of the laser interferometer 6 is returned to the host computer 7 for calibration fitting calculation. In the on-line length direction, the corresponding calibration system performs X coordinate calibration mapping through the checkerboard target 4.

[0064] The measurement beam 8 of the line spectral profile sensor 2, the outgoing beam 10 and the return beam 9 of the laser interferometer 6 are all parallel to the motion direction of the motion axis 3.

[0065] A method for automatic calibration of a line spectral profile sensor. This method uses the line spectral profile sensor 2 and the laser interferometer 6 to calibrate the linearity of the displacement direction (i.e., the Z direction), and arranges the checkerboard target 4 on the motion axis 3 to calibrate the position in the profile direction (i.e., the X direction). During calibration, control the checkerboard target 4 to perform a step-by-step movement from the near end of the range to the far end of the range, record the X-direction position, Z-direction position and light intensity in the original profile of the line spectral profile sensor 2 at each sampling position, and record the data of the laser interferometer 6 as the Z-direction position reference. Then, detect the light intensity at the light and dark segmentation points on the checkerboard target 4 through the inflection point detection algorithm to obtain the X-direction position reference. After completing the full-range scan, fit the three-dimensional surface formed by the two-dimensional profile data at different positions to generate a calibration table for calibration.

[0066] As Figure 2 shown, the process of collecting and calibrating the X-Z coordinate data of the line spectrum is as follows: At different positions within the range of the line spectral profile sensor 2, read the profile data. After obtaining multiple profiles, perform calculations to obtain a two-dimensional calibration table. The following process takes collecting 100 groups of data as an example, and the number of profile data points is taken as P points:

[0067] Step S101: Call the original profile reading function to obtain the original profile X coordinate x u of the P points, the original profile Z coordinate z u , and the light intensity i u of the coordinate point;

[0068] Step S102: Pass x u and i u into the light intensity mutation point detection function to obtain the mutation point x s . According to linear interpolation, obtain the z s corresponding to the mutation point x s . Obtain the d s corresponding to x s according to the actual interval of the checkerboard;

[0069] Step S103: Collect the interferometer position z i corresponding to the current position;

[0070] Step S104: For X coordinate fitting, take (xs , z s , d s ) as the (x rx , y rx , z rx ) data of the original three-dimensional coordinates;

[0071] Step S105: For the Z coordinate fitting, take (x u , z u , z i ) as the (x rz , y rz , z rz ) data of the original three-dimensional coordinates.

[0072] As Figure 3 shown, at the same Z-axis position, measure the checkerboard target 4 with alternating light and dark, and the returned light intensity magnitudes are inconsistent. Through the inflection point detection algorithm, locate the point x p where the mutation occurs, and record the original coordinate position x r at this time. Taking x r as the abscissa and x p as the ordinate, we get: ds i = (i - zerooffset) × W.

[0073] As Figure 4 and Figure 5 shown, during the calibration process, the X coordinate data is segmented and extracted through the following specific steps:

[0074] Step S201: Define the zero position pixel in the X direction as zeropixel = P / 2, and the line width as W;

[0075] Step S202: Obtain the set of light intensity mutation points xs according to the inflection point segmentation and extraction algorithm, and the unit of xs is pixel;

[0076] Step S203: Conduct a linear increasing search in xs to find the index i such that xs i ≤ zeropixel and xs i+1 > zeropixel;

[0077] Step S204: The index offset of the zero position pixel in xs

[0078] Step S205: For xs i , its corresponding actual position ds i = (i - zerooffset) × W.

[0079] As Figure 6 and Figure 7As shown, when the online spectral profile sensor 2 performs automatic calibration, x rz , x rx corresponds to the position of the point on the profile in the X direction, so its range is between [0, P], y rz , y rx corresponds to the position of the point on the profile in the Z direction, so its range is between [0, Q]. The three-dimensional data fitting and calibration table generation are carried out through the following specific steps:

[0080] Step S301: Call the original profile reading function to obtain the original data of P points for Z coordinate fitting (x rz , y rz , z rz ). The number of original data points of the X coordinate (x rx , y rx , z rx ) is determined according to the number of detected mutation points;

[0081] Step S302: Repeatedly collect M profiles;

[0082] Step S303: Generate grid coordinates of 0:N / 63:N in the X direction multiplied by 0:Q / 255:Q in the Y direction, with its size being axb, and expand it to obtain (x f , y f );

[0083] Step S304: Pass (x rz , y rz , z rz ) and (x f , y f ) into the surface fitting function F, and its return value z fz is the value of Z in the calibration table;

[0084] Step S305: Pass (x rx , y rx , z rx ) and (x f , y f ) into the surface fitting function F, and its return value z fx is the value of X in the calibration table.

[0085] The calibration value calculation is completed using the bilinear interpolation algorithm.

[0086] As Figure 8 and Figure 9 shown, after the calibration is completed, verification is carried out. The Z-direction verification is completed through the following specific steps:

[0087] Step S401: For each profile, calculate its minimum value xmin and maximum value xmax in the X direction;

[0088] Step S402: The minimum and maximum values in the X direction of N contours respectively form arrays Xmin = [xmin 1 , xmin 2 , …, xmin N and Xmax = [xmax 1 , xmax 2 , …, xmax N ;

[0089] Step S403: Find the maximum value xminmax of Xmin and the minimum value xmaxmin of Xmax to ensure that corresponding data can be found for all contours at xminmax and xmaxmin;

[0090] Step S404: Generate an arithmetic progression xinterp of 100 points between xminmax and xmaxmin;

[0091] Step S405: Obtain the interpolation position zinterp in the Z direction corresponding to xinterp through interpolation;

[0092] Step S406: Calculate the difference zdiff between zinterp and the interferometer count z i of the height corresponding to each contour;

[0093] Step S407: The zdiff of all contours forms the overall residual zresidual in the Z direction;

[0094] Step S408: Linearity

[0095] As Figure 10 shown, after completing the verification in the Z direction, the verification in the X direction is completed through the following specific steps:

[0096] Step S501: Define the line width in the X direction as W;

[0097] Step S502: Obtain the set xs of light intensity mutation points according to the inflection point segmentation extraction algorithm, and the unit of xs is mm;

[0098] Step S503: Perform a linear increasing search on xs to find the index i such that xs i ≤0 and xs i+1 >0;

[0099] Step S504: Index offset of the zero position pixel in xs

[0100] Step S505: For xs i , its corresponding actual position ds i=(i - zerooffset) × W;

[0101] Step S506: The X - direction residual xresidual = xs - ds.

[0102] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, various modifications and variations can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. An automatic calibration device for a line spectrum profile sensor, characterized in that: The device comprises a probe fixing tool (1), a line spectrum profile sensor (2), a moving axis (3), a checkerboard target (4), an interferometer reflector (5), a laser interferometer (6) and a host computer (7), wherein: The corresponding measurement system is fixed on the vibration isolation platform using the fixing fixture (1) to reduce the influence of environmental vibration on the measurement; The checkerboard target (4) and the interferometer reflector (5) are arranged on the motion axis (3); The host computer (7) controls the motion axis (3) to achieve displacement in the depth direction of the object to be measured and obtains displacement information of the object to be measured through a laser interferometer (6). The displacement information of the laser interferometer (6) is returned to the host computer (7) for calibration fitting calculation. In the line length direction, the corresponding calibration system performs calibration mapping of the X coordinate through a checkerboard target (4).

2. The automatic calibration device for line spectrum profile sensor according to claim 1, characterized in that: The measuring light beam (8) of the line spectrum profile sensor (2), the light emitting light beam (10) and the returning light beam (9) of the laser interferometer (6) are all parallel to the moving direction of the moving axis (3).

3. A calibration method for the automatic calibration device of a line spectrum profile sensor according to claim 1 or 2, characterized in that: The method uses a line spectrum profile sensor (2) and a laser interferometer (6) to calibrate the linearity of the displacement direction, and arranges a checkerboard target (4) on a motion axis (3) to calibrate the position in the profile direction; during calibration, the checkerboard target (4) is controlled to move stepwise from the near end of the range to the far end of the range, and the X-direction position, Z-direction position and light intensity in the original profile of the line spectrum profile sensor (2) at each sampling position are recorded, and the laser interferometer (6) data is recorded as the Z-direction position reference; then, the light intensity light and dark segmentation points on the checkerboard target (4) are detected by an inflection point detection algorithm to obtain the X-direction position reference; after completing the full range scan, the two-dimensional profile data at different positions are fitted to form a three-dimensional surface shape, and a calibration table is generated for calibration.

4. The automatic calibration method of the line spectrum profile sensor according to claim 3, characterized in that: The process of collecting and calibrating the XZ coordinate data of the line spectrum is as follows: at different positions within the measuring range of the line spectrum profile sensor (2), profile data is read to obtain multiple groups of profiles, and then a two-dimensional calibration table is obtained by calculation. Specifically, M groups of data are collected, and P point profile data is collected. Single profile collection and data extraction are performed through the following steps: Step S101: Call the original contour reading function to obtain the original contour X coordinate x of point P u , Z coordinate of the original contour u , light intensity at coordinate point i u ; Step S102: x u 、i u Enter the light intensity mutation point detection function to get the mutation point x s , according to linear interpolation, the mutation point x is obtained s The corresponding z s Get x according to the actual spacing of the chessboard s The corresponding d s ; Step S103: Collect the interferometer position z corresponding to the current position i ; Step S104: For X coordinate fitting, take (x s , z s , d s ) as the original three-dimensional coordinate (x rx ,y rx , z rx )data; Step S105: For Z coordinate fitting, take (x u , z u , z i ) as the original three-dimensional coordinate (x rz ,y rz , z rz )data.

5. The automatic calibration method of the line spectrum profile sensor according to claim 4, characterized in that: During the calibration process, the X-coordinate data segmentation and extraction is performed through the following specific steps: Step S201: define the zero pixel in the X direction as zeropixel=P / 2, and the width of the scribe line as W; Step S202: Obtain a light intensity mutation point set xs according to an inflection point segmentation and extraction algorithm, where the unit of xs is pixel; Step S203: Perform a linear incremental search on xs to find the index i such that xs i ≤zeropixel and xs i+1 >zeropixel; Step S204: Index offset of zero-position pixel in xs Step S205: For xs i , which corresponds to the actual position ds i =(i-zerooffset)×W.

6. The automatic calibration method of the line spectrum profile sensor according to claim 5, characterized in that: During calibration, x rz , x rx It corresponds to the position of the point on the contour in the X direction, so its range is between [0, P], y rz ,y rx It corresponds to the position of the point on the contour in the Z direction, so its range is between [0, Q]. The following specific steps are used to perform three-dimensional data fitting and calibration table generation: Step S301: Call the original contour reading function to obtain the Z coordinate fitting of point P (x rz ,y rz , z rz ) original data, original data of X coordinate (x rx ,y rx , z rx ) The number of points is determined according to the number of mutation points detected; Step S302: repeatedly collecting M contours; Step S303: Generate a grid coordinate of 0:N / 63:N in the X direction and 0:Q / 255:Q in the Y direction, whose size is axb, and expand it to obtain (x f ,y f ); Step S304: rz ,y rz , z rz ) and (x f ,y f ) Pass in the surface fitting function F, and its return value is z fz This is the value of Z in the calibration table; Step S305: rx ,y rx , z rx ) and (x f ,y f ) Pass in the surface fitting function F, and its return value is z fx This is the value of X in the calibration table.

7. The automatic calibration method of the line spectrum profile sensor according to claim 6, characterized in that: The calibration value calculation is completed using a bilinear interpolation algorithm.

8. The automatic calibration method for line spectrum profile sensor according to claim 7, characterized in that: When performing verification, complete the Z direction verification through the following specific steps: Step S401: for each contour, calculate its minimum value xmin and maximum value xmax in the X direction; Step S402: The minimum and maximum values ​​of the N contours in the X direction are respectively formed into an array Xmin=[xmin1, xmin2, ..., xmin N ] and Xmax=[xmax1, xmax2,…,xmax N ]; Step S403: Find the maximum value xminmax of Xmin and the minimum value xmaxmin of Xmax, ensuring that all contours can find corresponding data at xminmax and xmaxmin; Step S404: Generate a 100-point arithmetic progression xinterp at xminmax and xmaxmin; Step S405: obtaining the Z-direction interpolation position zinterp corresponding to xinterp through interpolation; Step S406: Calculate zinterp and the interferometer count z corresponding to the height of each profile i The difference zdiff; Step S407: zdiff of all contours constitutes the overall residual zresidual in the Z direction; Step S408: Linearity 9. The automatic calibration method of the line spectrum profile sensor according to claim 8, characterized in that: When performing verification, complete the X-direction verification through the following specific steps: Step S501: define the width of the X-direction scribe line as W; Step S502: Obtain a light intensity mutation point set xs according to an inflection point segmentation and extraction algorithm, where the unit of xs is mm; Step S503: Perform a linear incremental search on xs to find the index i such that xs i ≤0 and xs i+1 >0; Step S504: Index offset of zero-position pixel in xs Step S505: For xs i , which corresponds to the actual position ds i =(i-zerooffset)×W; Step S506: X-direction residual xresidual=xs-ds.

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