Space Of Light Rays Research Articles

Causality and skies: is non-refocussing necessary?

The causal structure of a strongly causal, null pseudo-convex, space-time M is completely characterized in terms of a partial order on its space of skies defined by means of a class of non-negative Legendrian isotopies called sky isotopies. It is also shown that such partial order is determined by the class of future causal celestial curves, that is, curves in the space of light rays which are tangent to skies and such that they determine non-negative sky isotopies. It will also be proved that the space of skies Σ equipped with Low’s (or reconstructive) topology is homeomorphic and diffeomorphic to M under the only additional assumption that M separates skies, that is, that different events determine different skies. The sky-separating property of M is sharp and the previous result provides an answer to the question about the class of space-times whose causal structure, topological and differentiable structure can be reconstructed from their spaces of light rays and skies. These results can be understood as a Malament–Hawking-like theorem stated in terms of the partial order defined on the space of skies.

The 3D visibility complex

Visibility problems are central to many computer graphics applications. The most common examples include hidden-part removal for view computation, shadow boundaries, mutual visibility of objects for lighting simulation. In this paper, we present a theoretical study of 3D visibility properties for scenes of smooth convex objects. We work in the space of light rays, or more precisely, ofmaximal free segments. We group segments that "see" the same object; this defines the3D visibility complex. The boundaries of these groups of segments correspond to thevisual eventsof the scene (limits of shadows, disappearance of an object when the viewpoint is moved, etc.). We provide a worst case analysis of the complexity of the visibility complex of 3D scenes, as well as a probabilistic study under a simple assumption for "normal" scenes. We extend the visibility complex to handle temporal visibility. We give an output-sensitive construction algorithm and present applications of our approach.

Mixed-norm estimates for a restricted X-ray transform in ℝ4 and ℝ5

The Radon transform found important applications in integral geometry and in the study of PDE’s. R1,d is often called the X-ray transform due to its applications in radiology, we denote it by Xfull. It is well-known [19], [12] that the sharp mixed norm estimates for the full X-ray transform implies the Kakeya conjecture and it is related to some of the main problems in the summability of Fourier transform, Fourier restriction and more generally to oscillatory integrals, non-linear P.D.E’s and number theory [7], [1], [2], [20], [3], [14]. For some mapping properties of Xfull, see, e.g., [6], [5], [19] and [12]. Note that G1,d is a 2d− 2-dimensional manifold, thus Xfull is overdetermined for d ≥ 3, and it is of interest to consider its restrictions to lower dimensional subspaces of G1,d. For the definition of the restricted X-ray transforms as part of a more general class of transformations and some of its properties, see [11]. One particular example is the restriction of Xfull to the space of light rays (lines in R making a 45 degree angle with the plane xd = 0). Recently, Wolff [21] obtained mixed norm estimates for this operator (almost sharp in R) and used this information to prove almost sharp bilinear cone restriction estimates in all dimensions. We are interested in the restriction of Xfull to d dimensional line complexes in R. Let d ≥ 3; the subspace Gd of G1,d we are interested in is defined as follows: Let γd be the curve {γd(t) : γd(t) = (1, t, t, ..., td−1), t ∈ (−1, 1)} in R. Let l(t, x) denote