## Generalized Stochastic Subdivision (1987)

Venue: | ACM Transactions on Graphics |

Citations: | 35 - 2 self |

### BibTeX

@ARTICLE{Lewis87generalizedstochastic,

author = {J. P. Lewis},

title = {Generalized Stochastic Subdivision},

journal = {ACM Transactions on Graphics},

year = {1987},

volume = {6},

pages = {167--190}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functions. The generalized construction is suitable for generating a variety of perceptually distinct high-quality random functions, including those with non-fractal spectra and directional or oscillatory characteristics. It is argued that a spectral modeling approach provides a more powerful and somewhat more intuitive perceptual characterization of random processes than does the fractal model. Synthetic textures and terrains are presented as a means of visually evaluating the generalized subdivision technique. Categories and Subject Descriptors: I.3.3 [Computer Graphics]: Picture/Image Generation; I.3.7 [Computer Graphics]: Three Dimensional Graphics and Realism -<F11.

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Citation Context ... gray levels (typically eight or fewer) and joint probabilities defined over fairly small neighborhoods. The textures produced under these restrictions are often fairly abstract or artificial looking =-=[32]-=-, and published comparisons of textures differing in the second-order statistics but having pairwise-identical first- and second-order moments have not clearly indicated the extent to which high-resol... |

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Citation Context ...ptually or otherwise determine the noise in the non-Gaussian cases. While stochastic texture synthesis using the full second-order statistics is becoming computationally tractable with recent methods =-=[11, 12], it -=-is expensive since it naively involves on the order of N · L 2 joint probabilities if L is the number of gray levels and N is the Markov neighborhood size or “memory” of the texture. To date, mos... |

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Citation Context ...n (e.g. erosion) to (16)sFigure 6: Gallery of simulated landscapes created using generalized stochastic subdivision. (6d) Figure 7: Fractal surface without power modification.shave similar statistics =-=[37, 39]-=-. It also appears that the nominal 1/f noises in electrical circuits may only approximate fractal behavior [26]). Theoretical models predicting the fractal spectrum have not been forthcoming in most c... |

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Citation Context ...rations mentioned in this section should be kept in mind however, since any process whose spectrum falls off faster than f −1 will be numerically singular at small scales. Creases Haruyama and Barsk=-=y [13]-=- have controlled the displacement noise variances in a stochastic subdivision procedure to obtain approximate spectral control of the noise. In some cases, however, changing the noise variances can pr... |

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Citation Context ... at multiples of that lag. Experience in spectral modeling indicates that a wide and useful variety of phenomena may be described using this approach; some techniques and applications are surveyed in =-=[2, 1, 27, 41]-=-. Spectral models have been used to generate reasonably good synthetic approximations of random textures such as sand, cork, etc. (e.g., [5]).sThe analytic and computational techniques for spectral mo... |

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Citation Context ...n process. They are equivalent in efficiency to convolution. The superposition of the function may be visible in the synthesized noise if the Poisson process is too sparse. Interpolated noise lattice =-=[30, 31]-=-. In this method an uncorrelated noise defined on a lattice is interpolated to generate an approximately bandlimited noise. This is employed as a spectral basis, with a desired spectrum being approxim... |

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Citation Context ...(differing in the higher-order statistics) cannot be visually discriminated. While counter-examples to the conjecture have been demonstrated, it is nevertheless regarded as being essentially accurate =-=[33]-=-. It seems likely that similar bounds may apply to the perceptual discrimination of statistical characteristics of (for example) terrains or fluid motion, but these have not been established to the au... |

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Citation Context ... noise and so should be used with care when a particular distribution and spectrum are required for theoretical reasons. The use of this technique its effects on the spectrum are briefly described in =-=[16]-=-. The disadvantages of the generalized subdivision technique include the increased computational cost of considering larger neighborhoods in the estimation procedure, and the need to determine or inve... |

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Preservation of the rescaled adjusted range (2: Simulation studies using Box-Jenkins models
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Citation Context ...pectra will be mentioned. Several researchers have concluded that the f −d spectrum (or equivalently the “fractal dimension”) is not adequate to describe natural phenomena of interest over all s=-=cales [24, 29]-=-. Empirical studies of the possible fractal nature of various phenomena have also found that the fractal dimension is often different at different scales, although ranges of scales described by a cons... |

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Citation Context .... (6d) Figure 7: Fractal surface without power modification.shave similar statistics [37, 39]. It also appears that the nominal 1/f noises in electrical circuits may only approximate fractal behavior =-=[26]).-=- Theoretical models predicting the fractal spectrum have not been forthcoming in most cases (an exception is the well established Kolmogorov −5/3 law in the statistical theory of turbulence [27]). I... |

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Citation Context ...pectra will be mentioned. Several researchers have concluded that the f −d spectrum (or equivalently the “fractal dimension”) is not adequate to describe natural phenomena of interest over all s=-=cales [24, 29]-=-. Empirical studies of the possible fractal nature of various phenomena have also found that the fractal dimension is often different at different scales, although ranges of scales described by a cons... |

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Citation Context ... is used to model Markovian noises, for example, a locally stationary approximation to Brownian motion. However, the construction has been applied to the non-Markovian fractional noises f −d , d �=-== 2 [9]; in-=- these cases disregarding the required autocorrelation produces “creases” (Section 4). The general problem of estimating the value of a stochastic process given knowledge of the process at other p... |

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Citation Context ... obeys this spectrum is nondifferentiable and has infinite characteristics (length, average power). The viability of this aspect of the fractal model was also questioned in the geophysical literature =-=[19, 39]-=-. Although some fractal practitioners state that the fractal model is only intended to apply over a limited range of scales, the global properties of self-similarity and non-differentiability are the ... |

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Citation Context ...ing differentiability and scaledependent features. The nondifferentiability of the fractal models results in terrains that in the author’s opinion appear too rough at small scales (e.g., Figure 7 an=-=d [42]). Most of-=- the terrain figures accompanying this article were produced using two-dimensional variations of a “Markaussian” autocorrelation R(τ) = exp(−|τ| α ) with 1 < α < 2, which produces noises fal... |

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Digital Elevation Models. National Cartographic Information
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Citation Context ...jacent objects. In fact the subdivision can be “seeded” with data obtained from another source, such as the digital elevation models (DEMs) available from the National Cartographic Information Cen=-=ter [7]-=-. In this paper we generalize the stochastic subdivision construction [10] to produce random functions with prescribed correlations and spectra. The generalized subdivision technique produces high-qua... |

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Fractional Brownian relief: an exact local method
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Citation Context ... (oscillatory noises are not possible however). Another variation [25] uses a higher quality interpolant for the point estimation, alleviating the problem with creases. Fractional Brownian simulation =-=[8]-=-. This method applies a standard multivariate simulation technique in which uncorrelated random variables are linearly transformed by a factored form (Choleski decomposition) of the desired correlatio... |

1 |
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Citation Context ...ting process [20]. The Poisson faulting process is a sum of randomly placed step functions with random heights. It produces a Brownian noise (f −2 spectrum). A related technique is sparse convolutio=-=n [16, 17]-=-, in which an arbitrary function is convolved with a Poisson point process (in one dimension the Poisson faulting process is a special case of sparse convolution). The resulting spectrum is the transf... |

1 |
Stochastic models for the Earth’s relief, the shape and dimension of the coastlines, and the number-area rule for islands
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Citation Context ...hod is global and less efficient than FFT synthesis (it is O(N 2 )), though an incremental refinement method that is local in a limited-precision implementation is described. Poisson faulting process =-=[20]. -=-The Poisson faulting process is a sum of randomly placed step functions with random heights. It produces a Brownian noise (f −2 spectrum). A related technique is sparse convolution [16, 17], in whic... |