New Methods for Estimating the Dimension Fractal Introducing the Artificial Intelligence
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New Methods for Estimating the Dimension Fractal Introducing the Artificial Intelligence A. Zerroug · D. Schoëvaërt-Brossault · S. Rebiai
Received: 17 October 2008 / Accepted: 30 October 2008 / Published online: 13 November 2008 © Springer Science+Business Media B.V. 2008
Abstract Methods provided actually for estimating the fractal dimension (especially methods of Richardson, Minkowski, Weibel and Flook) differ only by the way which is how to recover the fractal subject previously studied. The concept of optimal recovery is not found by these methods but in fact, the more we minimize this tool the more it gets close to the theoretical dimension. Therefore, we develop a new concept which incorporates artificial intelligence, consequentially the concept of optimality is put forward. Keywords Adjustment tests · Markov fields · Fractal dimension · Stereological simulation
1 Introduction One of the cancer tumor characteristics is its utmost irregularity of its borders between the surrounding tissue. In fact, this texture is an important indicator of the tumour invasion. Unfortunately, it is too difficult to classify this criteria of similarity within the tumours limits. A pathologist is able to compare efficiently the irregular forms or the nucleus of cells by visual inspection, but this way is less effective to estimate the degree of the tissue irregularity which is aimed to be used to study and classify the type of the tumour according to its geometry and its dynamism [4, 6]. The surrounding of the tumour is an important indication of the tumour dynamism and behaviour introduced by Mandelbrot [3], the fractal dimension is an important mathematic mean which enables to quantify the tumour texture A. Zerroug () Department of Mathematics, University Med Khid, Biskra, Algeria e-mail: [email protected] D. Schoëvaërt-Brossault Laboratoire Image Analysis in Cellular Pathology Institute of Hematology, Hôpital Saint Louis, Paris, France S. Rebiai Department of Mathematics, University of Batna, Batna, Algeria
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(Flook et al. [2]), the fractal dimension of tumors was measured to define its malignant nature [5, 8]. Consequently new estimating methods are provided for the fractal dimension. These methods are more and more closed the best theoretical dimension because the existing methods (such as stereological methods of Richardson, Minkowski, Weibel [9], Flook [1]) are different between each other only by the recovery tool in use. That is to say the most important thing is not the tool itself but the tool should be the minimum, because the more were closed to the fractal dimension theory, the more the cancerologist is owning an important tool of diagnosis; for this we introduce the artificial intelligence.
2 Estimating of the Fractal Dimension 2.1 Proposed Method of Recovery We assume A(M, N ) the matrix representing the curve as below: M: the columns number. N : the rows number. With j ∈ [0, N ] and i ∈ [0, M]. We fix j and we vary I from 0 to M every time that A(i, j ) = 0 we pass to i = i + 1 un
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