3 D Graphics

Download An Invitation to 3-D Vision: From Images to Geometric Models by Yi Ma, Stefano Soatto, Jana Kosecká, S. Shankar Sastry PDF

By Yi Ma, Stefano Soatto, Jana Kosecká, S. Shankar Sastry

Endowing machines with a feeling of imaginative and prescient has been a dream of scientists and engineers alike for over part a century. basically long ago decade, although, has the geometry of imaginative and prescient been understood to the purpose the place this dream turns into possible, thank you additionally to the outstanding growth in imaging and computing undefined.

This booklet addresses a relevant challenge in computing device imaginative and prescient -- how you can get better 3D constitution and movement from a set of 2-D photographs -- utilizing innovations drawn more often than not from linear algebra and matrix idea. the tension is on constructing a unified framework for learning the geometry of a number of pictures of a 3-D scene and reconstructing geometric types from these pictures. The ebook additionally covers suitable points of photograph formation, simple photo processing, and have extraction. The authors bridge the space among thought and perform by way of delivering step by step directions for the implementation of operating imaginative and prescient algorithms and structures.

Written basically as a textbook, the purpose of this e-book is to provide senior undergraduate and starting graduate scholars in computing device imaginative and prescient, robotics, and special effects a high-quality theoretical and algorithmic starting place for destiny study during this burgeoning box. it truly is completely self-contained with useful historical past fabric coated at the start chapters and appendices, and many routines, examples, and illustrations given through the text.

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Additional info for An Invitation to 3-D Vision: From Images to Geometric Models

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1993b]. In this book, these classic results, together with their extension to the continuous motion and multiple-body cases, are presented in Chapters 5, 6, and 7. Extensions of the reconstruction techniques from point features to lines initiated the study of the relationships among three views, since in the line case there is no effective constraint for two views that can be used for motion and structure recovery, as we will explain in Chapter 8. , 1990]. This result was later unified in the two papers of [Spetsakis and Aloimonos, 1990a, Spetsakis and Aloimonos, 1990b] for both the point and line cases, together with unified algorithms.

Egory of local parameterizations. This kind of parameterization is good for only a portion of SO(3), but not for the entire space. e. the three vectors WI, W2, W3 are linearly independent. Define a mapping (a parameterization) from lR,3 to SO(3) as a: (al , a2,a3) f-+ exp(awl+a2w2+a3w3). The coordinates (aI, a2, (3) are called the Lie-Cartan coordinates of the first kind relative to the basis (Wl,W2,W3). Another way to parameterize the group SO(3) using the same basis is to define another mapping from 1R3 to SO(3) by f3: (f31, f32, (33) f-+ exp(f31 WI) exp(f32W2) exp(f33 w 3) ' The coordinates (f31 , f32, (33) are called the Lie-Cartan coordinates of the second kind.

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. Hence, for a given rotation matrix R E SO(3), there are infinitely many exponential coordinates w E JR3 such that eW = R. The exponential map exp : 80(3) - t SO(3) is therefore not one-to-one.

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