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CHAPTER 13 CURVES AND SURFACES
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pairs {(PI, ql), (p2, q2), , (Pn, qn)} Solve the absolute orientation problem using this set of conjugate pairs Even though the point sets are not true conjugate pairs, an approximate solution to the transformation between the views will be obtained Apply the transformation to the point set P and recompute the set of closest points Q Now the point set is closer to the model and the pairs of points in P and Q will be closer to correct conjugate pairs The absolute orientation problem can be solved again and the new transformation applied to point set P This procedure is repeated until the sum of the squared distances between closest points (the root-mean-square closest point distance) is below a threshold In the early iterations, the closest point pairs may be very poor approximations to true conjugate pairs, but applying the rigid body transformation obtained by solving the absolute orientation problem with these approximate conjugate pairs brings the point set closer to the model As the iterations continue, the closest point pairs become more like valid conjugate pairs For example, if the point set is initially very far from the model, then all points may be mapped to the same point on the model Clearly, such a many-to-one mapping cannot be a valid set of conjugate pairs, but the first iteration pulls the point set to the model so that the points are centered on the matching model point and subsequent iterations align the center of the point set with the center of the model The final iterations rotate the point set and adjust its position so that the point set is aligned with the model The steps of iterative closest point registration are listed in Algorithm 134
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Algorithm 134 Iterative Closest Point Registration Register a set of points to a surface modeled as a polygonal mesh
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1 Compute the set of closest points 2 Compute the registration between the point sets 3 Apply the transform to register the point sets
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Return to step 1 unless the registration error is within tolerance
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An excellent introduction to the differential geometry of curves and surfaces is the book by do Carmo [67] Bartels, Beatty, and Barsky [22] have written an excellent introduction to spline curves and surfaces The series of books, called Graphics Gems, contains much useful information on geometry, curves, and surfaces [89] The most recent volume includes a disk with code for all of the algorithms presented in the series Surface reconstruction is a common problem in computer vision [92, 232, 233, 35] Surface reconstruction is required for fitting a surface to sparse depth values from binocular stereo and range sensors [28, 92, 232] Sinha and Schunck [223] have proposed a two-stage algorithm for surface reconstruction The first stage removes outliers from the depth measurements and interpolates the sparse depth values onto a uniform grid The second stage fits weighted bicubic splines to the output from the second stage The section on surface segmentation is based on the work of Besl and Jain [28J The iterative closest point algorithm was developed by Besl and McKay [29]
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Exercises
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Define implicit, explicit, and parametric representations of a surface When we consider an image as a two-dimensional mathematical function, can we consider the image as a surface If so, which of the three representations above corresponds to an image Given four points in space, how will you determine whether they are coplanar Define principal curvature, Gaussian curvature, and mean curvature for a surface What information about the nature of a surface do you get by knowing these curvature values at a point Do you get local information or global information about the surface from these curvature values What is a spline curve Why are cubic splines frequently used to represent curves
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