Coherent and Exact Polygon-to-Polygon Visibility

Visibility computation is a classical problem in computer graphics. A wide variety of algorithms provides solutions with a different accuracy. However, the four dimensional nature of the 3D visibility has prevented for a long time from leading to exact from-polygon visibility algorithms. Recently, the two first tractable solutions were presented by Nirenstein, then Bittner. Their works give the opportunity to design exact visibility tools for applications that require a high level of accuracy. This paper presents an approach that takes advantage of both Nirenstein and Bittner methods. On the one hand, it relies on an optimisation of Nirenstein’s algorithm that increases the visibility information coherence and the computation robustness. On the other hand, it provides an exact visibility data structure as Bittner does, but also suited for non-oriented polygon-to-polygon visibility queries. Exact visibility, CSG, Plücker space 1. INTRODUCTION Visibility computation is a recurring problem in computer graphics applications. A wide variety of algorithms exists in the literature but they provide a different accuracy. This has led to a general algorithm classification : • Aggressive : the visibility is underestimated. • Conservative : the visibility is overestimated. • Approximate : both aggressive and conservative. • Exact : the visibility is exactly computed. Solutions for these first three categories are usually fast. Most of them are designed in a context of visibility culling [Coh03a]. In contrast, exact algorithms require a significant computational effort, especially for from-polygon or from-region visibility. This problem has been considered for a long time as intractable due to the four dimensional nature of the visibility in 3D environment. 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It allows an exact computation of the visibility from a polygon. At the same time, Bittner[Bit02c] proposed another solution. The exact visibility is not necessary for most application. However, it has potential to improve high quality rendering of complex scenes or realistic lighting effects. The works of Nirenstein and Bittner give the opportunity to design efficient visibility tools encoding an exact information. This paper presents an exact visibility algorithm that takes advantage of both Nirenstein and Bittner methods. It relies on Nirenstein algorithm but provides in output a structured visibility information as Bittner does. Moreover, it presents an optimisation of Nirenstein algorithm that improves the visibility information coherence. As a consequence, it gets a noticeable property : By reducing the visibility result complexity, the number of performed operations decreases, improving the robustness. The second section explains the mathematical underlying and the general approach used by Nirenstein and Bittner for exact visibility computation. The third one gives an overview of the two existing algorithms and underlines their differences. From this short study, the section four presents our approach emphasising our optimisation that provides a coherent visibility information and improves robustness. At last, results are given in the section five. 2. BACKGROUND The two solutions proposed by Nirenstein and Bittner both rely on the same approach. They solve the visibility problem between polygons by performing CSG operations on polytopes (convex “volume”) in the Plücker space. This section begins with a presentation of the Plücker space where operates the solution. Next, it gives an overview of the approach allowing exact visibility computation from 3D polygons. Plücker Space The Plücker space [Som59a] is a five dimensional projective space P. It provides an elegant parametrisation for dealing with directed lines in R. Each line l passing through the point (px, py, pz) and next through (qx, qy, qz) is defined in P by πl = (π0, π1, π2, π3, π4, π5), with : π0 = qx − px π3 = qzpy − qypz π1 = qy − py π4 = qxpz − qzpx π2 = qz − pz π3 = qypx − qxpy Notice that (π0, π1, π2) is the direction of l and (π3, π4, π5) encodes its location. Next, let us consider the dual mapping within P : Each π ∈ P can be associated with a dual hyperplane hπ defined by : hπ = {x ∈ P 5 | π3x0 + π4x1 + π5x2 +π0x3 + π1x4 + π2x5 = 0}