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not tempt fate by adding outliers to the normally distributed errors Section 1213 includes some discussion on using robust regression for handling mismatched feature points In some applications, it is necessary to match two sets of points that are not conjugate pairs or to match two curves or surfaces that do not have discrete features This is the registration problem, discussed in Section 139
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The image processing operations in machine vision are usually done in the coordinate system of the image array, with the origin at the upper left pixel The rows and columns correspond to integer coordinates for the pixel grid Subpixel measurements add a fractional part to the image array (pixel) coordinate system, leading to pixel coordinates being represented as floating-point numbers; however, subpixel resolution does not change the geometry of the image array coordinate system We can convert from pixel coordinates to image plane coordinates using some assumptions about the camera geometry In Section 129, we will show how the camera parameters can be calibrated so that the mapping from image array (pixel) coordinates to image plane coordinates uses the actual geometry of the camera, including accounting for the effects of lens distortions The approximate transformation from pixel coordinates to image coordinates assumes that the principal axis intersects the image plane in the center of the image array If the image array has n rows and m columns, then the center of the image array is
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m-1 (121) 2 n-1 -(122) cy 2 We use the hat notation to stress that these are estimates for the location of the principal point The x axis is in the direction of increasing column index, but the direction of increasing row index and the y axis point in opposite directions The transformation from pixel coordinates [i, j] to image coordinates (x', y/) is
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CHAPTER 12 CALIBRATION
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This transformation assumes that the spacing between rows and columns in the image array is the same and that image plane coordinates should be expressed in this system of units Let the spacing between columns be Sx and the spacing between rows be Sy We can add these conversion factors to the transformation from pixel to image coordinates:
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If the image sensor has square pixels, then the conversion factors are identical and can be omitted This simplifies the image processing algorithms The image measurements (such as distances) can be converted to real units later, if necessary If the image sensor has nonsquare pixels, then Sx #- Sy and it may be necessary to convert pixel coordinates to image coordinates, using the formulas above, before performing measurements However, some measurements are not affected by nonsquare pixels For example, the centroid can be computed using pixel coordinates and converted to image plane coordinates later, but distances and angles are not invariant to unequal scale factors and should be computed from point locations in image coordinates For example, the centroids of two regions must be converted to image plane coordinates before calculating the distance between the centroids Because of these problems, it is very common to require cameras with square pixels in machine vision applications If an application uses a camera with nonsquare pixels, then you must carefully consider how the measurements will be affected by nonsquare pixels When in doubt, convert to image plane coordinates before making any measurements The image plane coordinate system is part of the camera coordinate system, a viewer-centered coordinate system located at the center of projection with x and y axes parallel to the x' and y' axes in the image plane and a z axis for depth The camera coordinate system is positioned and oriented relative to the coordinate system for the scene, and this relationship is determined
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