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

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

This ebook introduces the geometry of 3D imaginative and prescient, that's, the reconstruction of 3-D versions of items from a set of 2-D photographs. It information the vintage thought of 2 view geometry and exhibits extra right device 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 sensible reconstruction algorithms and discusses attainable extensions of the idea.

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A camera) from one place to another, we can certainly reverse the action and put it back to its original position. Similarly, we can combine several motions to generate a new one. Roughly speaking, this property of invertibility and composition can be mathematically characterized by the notion of "group" (Appendix A). As we will soon see, the set of rigid-body motions is indeed a group, the so-called special Euclidean group. However, the abstract notion of group is not useful until we can give it an explicit representation and use it for computation.

Given that computers today can easily handle tens of images with hundreds and even thousands of features at a time, the main problem becomes how to discover all the information that is available from all the images and how to efficiently extract it through computation. The attempts to develop algorithms that efficiently utilize a large number of images are marked by the introduction of factorization techniques for the simplified case of orthographic projection [Tomasi and Kanade, 1992]. In spite of the restrictive assumption about the projection model, the practical implications of such factorization techniques were striking.

How do these properties help us describe a rigid-body motion concisely? The fact that distances and orientations are preserved by a rigid-body motion means that individual points cannot move relative to each other. As a consequence, a rigid-body motion can be described by the motion of a chosen point on the body and the rotation of a coordinate frame attached to that point. In order to see this, we represent the configuration of a rigid body by attaching a Cartesian coordinate frame to some point on the rigid body, and we will keep track of the motion of this coordinate frame relative to a fixed world (reference) frame.

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