lecnotes1 - Lecture Notes 1 An Invitation to 3-D Vision...

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Lecture Notes 1 . An Invitation to 3-D Vision: From Images to Models (in preparation) Y. Ma, J. Koˇ seck´a, S. Soatto and S. Sastry. c ± Yi Ma et. al. Representation of a three dimensional moving scene The study of geometric relationships between a three dimensional scene and its multiple images taken by a moving camera is in fact a study of the interplay between two fundamental transforma- tions: the rigid body motion that models how the camera moves, and the perspective projection which describes the image formation process. Long before these two transformations were brought together, the theories for each have been developed independently. The studies of the principles of motion of a material bodies have a long history belonging to the foundations of physics. By now several classical fields of evolved focusing on the different aspects of the problem. For our purpose the more recent noteworthy insights to the understanding of the motion of rigid body have been made by Chasles and Poinsot in early 1800s. Their findings led to current treatment of the subject which has been since widely adopted in robotics and control. The notion of the screw motion and its infinitesimal version called twist play a central role in our formulation. The screw motion represents the discrete case, characterizing the displacement or configuration of the rigid body, while the twist describes the differential case characterized by the instanteneous velocity of the rigid body. There are several advantages of the representation we present. First it enables us to treat the discrete and differential case in a unified way and provides a clear geometric intuition behind both cases. From the computational standpoint the formulation enables global parametrization of the rigid body motion which does not suffer from the singularities present in other representations which use local coordinates. The formulation sets the stage for variety of linear algebric tools, which enable us to systematically study the problems outlined in the later chapters. We start in this chapter with the introduction of a three dimensional Euclidean space as well as the rigid body transformation acting on the space. The next chapter will then focus on the perspective projection model of the camera. 0.1 A three dimensional Euclidean space We will use E 3 to denote a three dimensional Euclidean space. A Euclidean space, as suggested by its name, is a space which satisfies the five Euclid Axioms. Nonetheless, there is also an analytical way to describe a Euclidean space which serves better for our purposes. A three dimensional Euclidean space E 3 can be described as a space to which we may assign a (global) Cartesian frame XY Z . Every point p E 3 can then be identified with a point in R 3 by its three coordinates [ X 1 ,X 2 3 ] T .
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This note was uploaded on 04/07/2010 for the course CS 685 taught by Professor Luke,s during the Fall '08 term at George Mason.

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lecnotes1 - Lecture Notes 1 An Invitation to 3-D Vision...

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