Computer rendering of stochastic models

Abstract

A recurrent problem in generating realistic pictures by computers is to represent natural irregular objects and phenomena without undue time or space overhead. We develop a new and powerful solution to this computer graphics problem by modeling objects as sample paths of stochastic processes. Of particular interest are those stochastic processes which previously have been found to be useful models of the natural phenomena to be represented. One such model applicable to the representation of terrains, known as “fractional Brownian motion,” has been developed by Mandelbrot. The value of a new approach to object modeling in computer graphics depends largely on the efficiency of the techniques used to implement the model. We introduce a new algorithm that computes a realistic, visually satisfactory approximation to fractional Brownian motion in faster time than with exact calculations. A major advantage of this technique is that it allows us to compute the surface to arbitrary levels of details without increasing the database. Thus objects with complex appearances can be displayed from a very small database. The character of the surface can be controlled by merely modifying a few parameters. A similar change allows complex motion to be created inexpensively.

Contact between representative rough surfaces

“…Since the mechanical response of the generated surfaces is affected by their height distributions and the PSD, the surface generation procedure has to include both of them into consideration. While the typical PSD of self-affine surfaces is easily obtained using a random midpoint algorithm [13] or by filtering a white noise [26], the normality of the height distribution of simulated surfaces was, to the best of our knowledge, often ignored. We will demonstrate that even if available numerical techniques permit to generate a self-affine surface with a small error on the resulting PSD, the distribution of heights strongly depends on the range of wavelengths [λ s , λ l ] included in the spectrum.…”

“…Many of the numerical investigations are based on synthesized surfaces preserving the aspect of self-affinity. In early studies [6,12] the artificial rough surfaces obeyed fractality which scaled down to the domain discretization [13]. Later it was realized that the surface has to be smooth enough [7,14] to represent correctly the mechanics of contact and to obtain a reliable estimation of the contact area growth comparable with analytical theories.…”

Earthquake Recurrence in Simulated Fault Systems

“…The fractally rough surfaces are generated using the method of random mid-point displacement (FOURNIER et al, 1982) whereby the fault surface is repeatedly divided and the midpoints of the new divisions are randomly displaced by a normally distributed random variable with a standard deviation given by…”

Generation of Roughness Using the Random Midpoint Displacement Method and Its Application to Quantification of Joint Roughness

“…브라운 프로파일을 생성하기위해 빈번하게 사용되는 방법은 랜덤중점변위법(random midpoint displacement) 과 스펙트럼합성법이다 (Fournier et al, 1982;Fox, 1987;Saupe, 1988;Voss, 1988). …”