Implicit surfaces are a powerful shape primitive for computer graphics. This thesis focuses on a shape modelling approach which generates synthetic shapes by the specification of an implicit surface generating function that has random properties. This type of graphic object\ud can be called a stochastic implicit surface because the surface is perceived as the realisation of a stochastic process. The main contributions of this thesis are in the form of new and improved modelling and rendering algorithms to deal with stochastic implicit surfaces that can be complex and feature fractal scaling properties.\ud \ud On the modelling side, a new topological correction algorithm is proposed to detect disconnected surface parts that arise as a consequence of the implicit surface representation. A surface deformation algorithm, based on advection through a vector field, is also presented.\ud \ud On the rendering side, several algorithms are proposed. First, an improved ray casting method is presented that guarantees correct intersections between view rays and the surface. Second, a new progressive refinement rendering algorithm is proposed that provides a dynamic rendering environment where the image quality steadily increases with time. Third, a distributed rendering mechanism is presented to deal with the long computation times involved in the image synthesis of stochastic implicit surfaces.\ud \ud An application of the proposed techniques is given in the context of the procedural modelling of a planet. A procedural planet model must be able to generate synthetic planets showing the correct range of geological scales. The planet is generated as a stochastic implicit surface. This\ud represents an improvement over previous models that generated planets as displacement maps over a sphere. Terrain features that were previously difficult to model can be achieved through the implicit surface approach. This approach is a generalisation over those previous models since displacement maps over the sphere can be recast as implicit surfaces.\u
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