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Height fields have become an important element of realistic real-time image synthesis to represent surface details. In this paper, we focus on the frequent case of static height-field data, for which we can precompute acceleration structures. While many rendering algorithms exist that impose tradeoffs between speed and accuracy, we show that even accurate rendering can be combined with high performance. A careful analysis of the surface defined by the height values, leads to an efficient and accurate precomputation method. As a result, each texel stores a safety shape inside which a ray cannot cross the surface twice. This property ensures that no intersections are missed during the efficient marching method. Our analysis is general and can even consider visibility constraints that are robustly integrated into the precomputation. Further, we propose a particular instance of safety shapes with little memory overhead, which results in a rendering algorithm that outperforms existing methods, both in terms of accuracy and performance.