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 booklet introduces the geometry of 3D imaginative and prescient, that's, the reconstruction of 3D types of items from a suite of 2-D photos. It information the vintage idea 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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Extra info for An Invitation to 3-D Vision: From Images to Geometric Models

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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.

We denote such a map by g(t) : JR3 ~ JR3 ; X t--* g(t)(X). If instead of looking at the entire continuous path of the moving object, we concentrate on the map between its initial and final configuration, we have a rigid-body displacement, denoted by g: JR3 -+ JR3; X t--* g(X). Besides transforming the coordinates of points, 9 also induces a transformation on vectors. Suppose that v is a vector defined by two points p and q with coordinates 20 Chapter 2. (v) ~ g(Y) - g(X). (v)11 = Ilvll for all free vectors v E ]R3.

Obviously, X c = [X1c, X 2c , X3c]T are the coordinates of the same point p with respect to the frame C. Therefore, we have Xw = X1crl + X 2cT2 + X3cT3 = RwcXc. In this equation, the matrix Rwc transforms the coordinates X c of a point p relative to the frame C to its coordinates X w relative to the frame W. Since Rwc is a rotation matrix, its inverse is simply its transpose, Xc = R;;~Xw = R~cXw' That is, the inverse transformation of a rotation is also a rotation; we call it Rcw, following an established convention, so that Rcw = R;;~ = R~c' The configuration of a continuously rotating object can then be described as a trajectory R(t) : t r-+ 80(3) in the space 80(3).

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