Arbitrary 3D Resolution Discrete Ray Tracing of Implicit Surfaces

Abstract

Abstract.A new approach to ray tracing implicit surfaces based on recursive space subdivision is presented in this paper. Interval arithmetic, already used to calculate intersections in ray tracing and ray casting (numerically or subdividing 1D or 2D spaces), is now used here to implement a ray tracing based on reliable rays traversals into a potentially infinite octree-like subdivided space, eliminating explicit intersections. Novel, robust and efficient algorithms for ray voxelization and BSP octant ordering are used to recursively traverse rays through the space. Implicit surfaces are robustly voxelized and hierarchically stored into an octree to a certain given level. During rendering, the subdivision based voxelization of surfaces and rays continues further down until a resolution near the discrete domain of the floating point numbers is acquired. To guarantee robustness of the ray voxelization, interval arithmetic with calculations performed under appropriate rounding modes in Pentium-4 x87 and SSE2 FPUs respectively is applied. The major advantage is that the traversal algorithm is guaranteed to find reliable intersections between the rays and the scene without any explicit intersection calculation, solving a known precision problem of the ray traversal in a previous approach, used here for comparison. The precision of the traversal can be arbitrarily increased within the limitation of the floating point representation.

Point rendering of non-manifold surfaces with features

“…They can also render surfaces that can't be rendered with traditional ray casting techniques due to difficulties in solving the ray-surface intersection equation. Recently however, [Stolte 2005] discussed a recursive space subdivision based ray casting technique that renders implicit surfaces without having to do any ray-surface intersections. His algorithm does this by subdividing space into voxels that are as small as floating point arithmetic allows.…”

Point‐Based Rendering of Non‐Manifold Surfaces

“…They can also render surfaces that can't be rendered with traditional ray casting techniques due to difficulties in solving the ray‐surface intersection equation. Recently however, Stolte [Sto05] discussed a recursive space subdivision based ray casting technique that renders implicit surfaces without having to do any ray‐surface intersections. His algorithm does this by subdividing space into voxels that are as small as floating‐point arithmetic allows.…”

Evaluating Straight-Line Programs over Balls

“…Changing the rounding mode for almost each arithmetic operation seriously slows down computations, and might involve a serious stall of a hundred of cycles by breaking FPU pipelines of some hardware such as Intel x87. To avoid switching rounding modes, the author of [26] uses different independent rounding modes on the x87 and SSE units. Specific solutions to diminish the swap of rounding modes also rely on the opposite trick in internal computations or directly in the representation [4,12,30].…”