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Appendix  Projective Geometry for Machine Vision
, 1992
"... Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a boo ..."
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Cited by 27 (0 self)
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Introduction The idea for this Appendix arose from our perception of a frustrating situation faced by vision researchers. For example, one is interested in some aspect of the theory of perspective image formation such as the epipolar line. The interested party goes to the library to check out a book on projective geometry filled with hope that the necessary mathematical machinery will be directly at hand. These expectations are quickly dashed. Upon opening the book, the expectant reader finds the presentation dominated by endless observations about harmonic relations and a few chapters which explore the minutiae of Pappus' theorem. Finally, as a last cruel twist of irony, the book ends in triumph with a rather exhilarating discourse on the conic pencil. All of the material is presented in the form of theorems defined on points, lines and conics without the use of coordinates, except perhaps for a quick pause to define barycentric coordinates just to taunt the reader. Dejected, the vis
A Theory of the Motion Fields of Curves
 International Journal of Computer Vision
, 1993
"... This paper is a study of the motion field generated by moving 3D curves which are observed by a camera. We first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend. We then go ahead to ..."
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Cited by 16 (1 self)
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This paper is a study of the motion field generated by moving 3D curves which are observed by a camera. We first discuss the relationship between optical flow and motion field and show that the assumptions made in the computation of the optical flow are a bit difficult to defend. We then go ahead to study the motion field of a general curve. We first study the general case of a curve moving nonrigidly and introduce the notion of isometric motion. In order to do this, we introduce the notion of spatiotemporal surface and study its differential properties up to the second order. We show that, contrarily to what is commonly believed, the full motion field of the curve (i.e the component tangent to the curve) cannot be recovered from this surface. We also give the equations that characterize the spatiotemporal surface completely up to a rigid transformation. Those equations are the expressions of the first and second fundamental forms and the Gauss and CodazziMainardi equations. We then...
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"... Geometric invariants are shape desrriptors that remain unchanged under geometric transformations such as projection, or change of the viewpoint. In [,?I we devc’loped a new method of obtaining local projertive antl affine invariants for a general curve without any correspondences. Being local, the ..."
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Geometric invariants are shape desrriptors that remain unchanged under geometric transformations such as projection, or change of the viewpoint. In [,?I we devc’loped a new method of obtaining local projertive antl affine invariants for a general curve without any correspondences. Being local, the invariants are much Ii ss sensitive to occlusion than global invariants. The iniiariants computation is based on a cunonicul method This consists of defining a canonical coo7.dinate system using intrinsic properties of the shapc, independently of the given coordinate system. Since this canonical system is indeptndent of the oriyinul one, it is invariant and all quantities defined in it arc invariant. Here we present a furth.er developnient of the method to obtain local semiinvariants, thud is loid rnvtrriunts for curves with known correspondencrs. Several conjigurations are treated: curves with knoion correspondences of one or two feature points or lines. 1