On Geometric and Algebraic Aspects of 3D Affine and Projective Structures from Perspective 2D Views

Abstract
Part I of this paper investigates the differences conceptually and algorithmically - between affine and projective frameworks for the tasks of visual recognition and reconstruction from perspective views. It is shown that an affine invariant exists between any view and a fixed view chosen as a reference view. This implies that for tasks for which a reference view can be chosen, such as in alignment schemes for visual recognition, projective invariants are not really necessary. The projective extension is then derived, showing that it is necessary only for tasks for which a reference view is not available - such as happens when updating scene structure from a moving stereo rig. The geometric difference between the two proposed invariants are that the affine invariant measures the relative deviation from a single reference plane, whereas the projective invariant measures the relative deviation from two reference planes. The affine invariant can be computed from three corresponding points and a fourth point for setting a scale; the projective invariant can be computed from four corresponding points and a fifth point for setting a scale. Both the affine and projective invariants are shown to be recovered by remarkably simple and linear methods. Visual recognition, Structure from motion, 3D Reconstruction, Projective geometry

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