The Variation Method: A Technique To Estimate The Fractal Dimension Of Surfaces

Dubuc, B.; Roques‐Carmes, C.; Tricot, Claude; Zucker, Steven W.

doi:10.1117/12.976511

Cited by 38 publications

(21 citation statements)

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“…Hence, its value may be quite different from Hausdorff measure. As an aside, one consequence of the above proposition is the known result that the Minkowski dimension of a set is greater than or equal to its Hausdorff dimension [9,13].…”

Section: K-0oomentioning

confidence: 94%

An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

Kulkarni¹,

Mitter²,

Richardon³

1990

Signal Processing

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A variational method for the reconstruction and segmentation of images was recently proposed by Mumford and Shah [15]. In this paper we treat two aspects of the problem. The first concerns existence of solutions to the problem; the second concerns representations suitable for computation. Discrete versions of this problem have been proposed and studied in [5,12,14,15]. However, it seems that these discrete versions do not properly approximate the continuous problem in the sense that their solutions may not converge to a solution of the continuous problem as the lattice spacing tends to zero.Here we consider the use of an alternate lattice approximation for the boundaries of the image and Minkowski content as a cost term for the boundaries. Several properties of Minkowski content are derived. These are used to show that partially discrete versions of the variational problem possess some desirable convergence properties. Specifically, under suitable conditions, solutions to the discrete problem converge in the continuum limit to a solution of the continuous problem.The existence result included here is applicable to both discrete and continuous versions of the problem.

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Section: K-0oomentioning

confidence: 94%

An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

Kulkarni¹,

Mitter²,

Richardon³

1990

Signal Processing

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“…Furthermore, the popular Brownian model is actually one dimensional (l-D) , and when applied to 2-D images, l-D profiles (cross-sections) are used. The reported accuracy of some of the most commonly used techniques includes errors ranging from 5% to 10% [ 11] . In addition , the computation involved is usually extensive.…”

Section: The Variation Methodsmentioning

confidence: 99%

“…To compute the local fractal dimension we use a recently proposed approach , named the variation method [ 11] . This method is significantly more accurate than other techniques, and it is also computationally efficient.…”

Section: Introductionmentioning

confidence: 99%

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Fractal-based multifeature texture description

et al. 1991

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Fractal geometry receives recently increasing attention in analyzing natural scenes. The fractal diiaension of a surface image has been used as segmentation feature, but since it is not sufficient to characterize important configurative texture characteristics, additional features are necessary. In this work we use a combination of fractal and non-fractal features to segment and classify natural textures. INTRODUCTIONTextural features are important pattern elements in both human and machine vision. The problem of segmentation and classification of textured images has been widely studied and several approaches based on either statistical or structural representations have been proposed [1]. During the recent years fractal geometry has been used in an effort for a unified image model. The fractal model introduced by Mandelbrot [2] has been quite successful in modeling a wide variety of physical phenomena. Another area where fractals have been used is computer graphics. Synthesized fractal sets have striking resemblance to natural objects (clouds, trees, mountains, etc.), and this suggested that fractals might also be successfully applied to image analysis. As a result, the fractal model received considerable attention in describing natural surfaces and also in texture discrimination.A basic characteristic of a fractal set is the fractal dimension D. From an intuitive point of view, D corresponds to the concept of roughness; i.e. for a plane D is 2 (equal to the topological dimension) , while for highly irregular (space filling) surfaces D approaches 3. Fractals can be either deterministic or statistical. A fundamental property of deterministic fractals is selfsimilarity; i.e. the set looks the same at all scales. As a consequence, the fractal dimension remains invariant to scale variations. Statistical fractals maintain self-similarity in a statistical sense. Physical fractals have the self-similarity property over some limited range of scale. Among the several segmentation algorithms based on fractal dimension alone that have been proposed [3][4][5][6][7], their basic difference lays in the approach used to estimate the local D parameter of the textured surface. Their performance, however, is limited because, as it was reported by several researchers, the fractal dimension alone is not sufficient to characterize natural textures [8][9][10]. Furthermore, segmentation 46 / SPIE Vol. 1521 Image Understanding for Aerospace Applications(1991) Downloaded From: http://proceedings.spiedigitallibrary.org/ on 06/16/2016 Terms of Use: http://spiedigitallibrary.org/ss/TermsOfUse.aspx

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“…25,27 An additional complication with model- Rolland, Goon, and Yu: Synthesis of textured complex backgrounds ing biological tissue as a fractal is that it is difficult to accurately estimate a fractal dimension from digitized data. 28 The indication that the power spectra of mammograms and various natural images follow some power law may be significant. It is however important to note it has been demonstrated that the power spectrum of a statistical complex background is not a complete descriptor of the required background statistics to predict human observer performance in various detection tasks.…”

Section: A Two-component Model Synthesis Frameworkmentioning

confidence: 99%

Synthesis of textured complex backgrounds

Rolland

Goon

1998

Opt. Eng

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The statistical characterization of realistic, complex backgrounds where targets may be embedded is essential in the optimization of methods for target acquisition. A modeling framework for complex backgrounds that yields perceptually realistic images may provide a path towards such essential characterizations. To this end, we developed a framework for the synthesis of statistical textured backgrounds. The results of the syntheses of three rock-like structures and that of grass indicate some of the capabilities of the framework. We further extended the methods to synthesize biological tissue samples which present two forms of background complexity: a slowly, spatially varying meanbackground and a residual texture image. The extended framework allows synthesis of each component independently. A mathematical phantom for modeling inhomogeneous backgrounds is then proposed. First and second order statistics of various textures are then presented and a measure of distance between two images is proposed. Finally we discuss how such a framework may lead to effective statistical descriptions of such complex backgrounds for target acquisition. © 1998 Society of Photo-Optical Instrumentation Engineers. [S0091-3286(98)01807-8] Subject terms: textured backgrounds; complex backgrounds; texture synthesis; first and second order statistics; detection.Paper TAM-18

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The Variation Method: A Technique To Estimate The Fractal Dimension Of Surfaces

Cited by 38 publications

References 0 publications

An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

An Existence Theorem and Lattice Approximations for a Variational Problem Arising in Computer Vision

Fractal-based multifeature texture description

Synthesis of textured complex backgrounds

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