The Science of Fractal Images

Barnsley, Michael F.; Devaney, Robert L.; Mandelbrot, Benoît B.; Peitgen, Heinz‐Otto; Saupe, Dietmar; Voss, Richard F.

doi:10.1007/978-1-4612-3784-6

Cited by 359 publications

(168 citation statements)

References 44 publications

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“…In other words, assume that we k n o w 4 how p(t) = fthe expected density of extremum points at scale tg (2) varies with t. What we w ant to de ne is a transformation function h such that the e ective scale can be written = h(t). The decay rate requirement can be stated as:…”

Section: De Nition and Derivationmentioning

confidence: 99%

Effective scale: a natural unit for measuring scale-space lifetime

Lindeberg

1993

IEEE Trans. Pattern Anal. Machine Intell.

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We develop how a notion of e ective scale can be i n t r oduced in a formal way. For continuous signals a scaling argument directly gives that a natural unit for measuring scale-space lifetime is in terms of the logarithm of the ordinary scale parameter. That approach is, however, not appropriate for discrete signals, since then an in nite lifetime would be assigned to structures existing in the original signal. Here we show how such an e ective scale parameter can be de ned a s t o g i v e c onsistent results for both discrete and continuous signals. The treatment is based u p on the assumption that the probability that a local extremum disappears during a short scale interval should not vary with scale. As a tool for the analysis we give estimates of how the density of local extrema can be expected to vary with sca l e i n t h e s c ale-space r epresentation of di erent random noise signals, both in the continuous and discrete cases.

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Section: De Nition and Derivationmentioning

confidence: 99%

Effective scale: a natural unit for measuring scale-space lifetime

Lindeberg

1993

IEEE Trans. Pattern Anal. Machine Intell.

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“…The most widely used algorithms in signal and image processing are described in (Barnsley et al, 1988;Jenna ne et al, 1996Jenna ne et al, , 2001). …”

Section: Measure Ment Of 2d and 3d Self-similaritymentioning

confidence: 99%

Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection

Jennane

Harba

Lemineur

et al. 2007

Medical Image Analysis

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It has been shown that the analysis of two dimensional (2D) bone X-ray images based on the fractional Brownian motion (fBm) model is a good indicator for quantifying alterations in the three dimensional (3D) bone micro-architecture. However, this 2D measurement is not a direct assessment of the 3D bone properties. In this paper, we first show that S 3D , the self-similarity parameter of 3D fBm, is linked to S 2D , that of its 2D projection, by S 3D = S 2D -0.5. In the light of this theoretical result, we have experimentally examined whether this relation holds for trabecular bone. Twenty one specimens of trabecular bone were derived from frozen human femoral heads. They were digitized using a high resolution -CT. Their projections were simulated numerically by summing the data in the three orthogonal directions and both 3D and 2D self-similarity parameters were measured. Results show that the self-similarity of the 3D bone volumes and that of their projections are linked by the previous equation. This demonstrates that a simple projection provides 3D information about the bone structure. This information can be a valuable adjunct to the bone mineral density for the early diagnosis of osteoporosis.

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“…As shapes like these are impossible to fully define with Euclidean geometry and cannot be truly modeled by Euclidean constructs (such as PCA), a new type of non-Euclidean geometry is necessary in order to classify them. The measure of a fractal's complexity and space-filling ability is known as a fractal dimension [7,8]. A fractal dimension is not necessarily a whole number, although the concept expressed is similar to that of whole-number (or Euclidean) dimensions; an unbroken line will have a fractal dimension of 1, an unbroken plane will have a fractal dimension of 2, etc.…”

Section: Chaosmentioning

confidence: 99%

Chaos theory in chemistry and chemometrics: a review

Cramer

Booksh

2006

Journal of Chemometrics

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Since the initial development of chaos theory and corollary concepts such as the strange attractor and fractal dimensionality, there has been a nontrivial effort to apply these concepts to the quantification of material and data structures. There exists within chaos theory modeling concepts that allow for thorough complexity analyses not available through alternate paradigms. The purpose of this paper is to review the main body of work that has been pursued in both experimental chemistry research and related data analysis development with respect to chaos theory and derivative ideas.

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The Science of Fractal Images

Cited by 359 publications

References 44 publications

Effective scale: a natural unit for measuring scale-space lifetime

Effective scale: a natural unit for measuring scale-space lifetime

Estimation of the 3D self-similarity parameter of trabecular bone from its 2D projection

Chaos theory in chemistry and chemometrics: a review

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