By Yi Ma, Stefano Soatto, Jana Kosecká, S. Shankar Sastry
This e-book introduces the geometry of 3-D imaginative and prescient, that's, the reconstruction of 3D types of items from a suite of 2-D photographs. It info the vintage concept of 2 view geometry and exhibits extra right software for learning the geometry of a number of perspectives is the so-called rank attention of the a number of view matrix. It additionally develops functional reconstruction algorithms and discusses attainable extensions of the idea.
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Additional resources for An Invitation to 3-D Vision: From Images to Geometric Models
5 illustrates a moving rigid object with a coordinate frame C attached to it. 4. 5. A rigid-body motion between a moving frame C and a world frame W. the translation T w eE ]R3 of the origin of the frame C relative to that of the frame W and the vector X c but expressed relative to the frame W . Since X c are the coordinates of the point p relative to the frame C, with respect to the world frame W , it becomes Rwc X c, where Rwc E SO(3) is the relative rotation between the two frames. 17) Usually, we denote the full rigid-body motion by gwe = (R 111C ' Twc ), or simply 9 = (R, T) if the frames involved are clear from the context.
Then, it is immediate to verify that powers of 0 can be reduced by the following two formulae 02 = ww T I, - 03 = -0. 13) can be simplified as ~ ewt = I + (t33! + 5Tt - . . t - 5 ) 0 + (t22! - t4 4! + t6! 6 - ... ) 0 2. 28 Chapter 2. Representation of a Three-Dimensional Moving Scene The two sets of parentheses contain the Taylor series for sin(t) and (1 - cos(t)), respectively. Thus, we have e iA = 1+ wsin(t) + w2 (1 - cos(t)). 0 Using Rodrigues' formula, it is immediate to see that if IIwll have = 1, t = 2k7r, we for all k E Z.
In particular, Yg 1 , g2 16We say that two vectors u, v are related by a linear transformation if u = Av for some matrix A , and by an affine transformation if u = A v + b for some matrix A and vector b. See Appendix A. 4. Rigid-body motion and its representations 31 and 9 E SE(3), we have _ _ _ [Rl glg2 0 T1] [R2 1 0 T2] 1 and [ ROT -RITT] E SE(3). 2 (but given formally in Appendix A). In the homogeneous representation, the action of a rigid-body motion 9 E SE(3) on a vector v = X(q) - X(p) E R3 becomes g*(ii) = gX(q) - gX(p) = gii.