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The aim of this study is the microcalcification segmentation in digital mammograms. We propose two different methods which are based on the combination of the multifractal analysis with, respectively, the fractal analysis and then with the lacunarity. Our approach consists of two steps. On the first stage, we created the “α_image.” This image was constructed by singularity coefficient deduced from multifractal spectrum of the original image. On the second stage, in order to enhance the visualization of microcalcifications, we create the “f(α)_image” based on global regularity measure of “α_image.” Two different techniques are used: the box counting (BC) used to calculate fractal dimension and the gliding box method used to measure lacunarity. These techniques were applied in order to compare results. Our proposed approaches were tested on mammograms from “MiniMIAS database.” Results demonstrate that microcalcifications were successfully segmented.

Osteoporosis is a disease characterized by low bone mass and deterioration of micro-architectural bone tissue, which provokes an increased risk of fracture. This work treats the texture characterization of trabecular bone radiographs. The goal is to analyse according to clinical research a group of 174 subjects: 87 osteoporotic patients with various bone fracture types and 87 healthy subjects. In order to characterize osteoporosis, a method of lacunarity measurement for grayscale image is used. This approach allowed the discrimination between healthy subjects and patients with osteoporosis. The results show an improved classification rate compared to another work [1].

For texture analysis, several features such as co-occurrence matrices, Gabor filters and the wavelet transform are used. Recently, fractal geometry appeared to be an effective feature to analyze texture. But it is often restricted to 2D images, while 3D information can be very important especially in medical image processing. Moreover applications are limited to the use of fractal dimension. This study focuses on the benefits of fractal geometry in a classification method based on volumic texture analysis. The proposed methods make use of fractal and multifractal features for a 3D texture analysis of a voxel neighborhood. They are validated with synthetic data before being applied on real images. Their efficiencies are proved by comparison to some other texture features in supervised classification processes (AdaBoost and support vector machine classifiers). The results showed that features based on fractal geometry (by combining fractal and multifractal features) contributed to new texture characterization. Information on new features was useful and complementary for a classification method. This study suggests that fractal geometry can provide a new useful information in 3D texture analysis, especially in medical imaging.

In this work we study the some general fractal sums of pulses defined in ℝ by: $$F(t) =\sum^{+\infty}_{n=1}a_nG(\lambda_n^{-1}(t-X_n))$$ where (an), (λn) two positive scalar sequences such that ∑an is divergent, and (λn) is non-increasing to 0, G is an elementary bump and Xn are independent random variables uniformly distributed on a sufficiently large domain Ω. We investigate the Hausdorff dimension of the graph of G and in particular we answer a question given by Tricot in (Courbes et dimensions fractales, Springer, Berlin, 1995).

Let n>m≥1 be two integers. At first we obtain general results for the multifractal analysis of the orthogonal projections on m-dimensional linear subspaces of singular measures μ on ℝn satisfying the multifractal formalism. The results hold for γn,m-almost every such subspace, where γn,m is the uniform measure on the Grassmannian manifold Gn,m. Let μ be such a measure and suppose that its upper Hausdorff dimension is less than or equal to m. Let I stand for the interval over which the singularity spectrum of μ is increasing. We prove that there exists a non-trivial subinterval of I such that for every , for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds at α for μV, the orthogonal projection of μ on V. Moreover, in some cases the result is optimal in the sense that the interval is maximal in I. Also, we determine the Lq-spectrum τμV(q) on the minimal interval J necessary to recover the singularity spectrum of μV over as the Legendre transform of τμV. The interval J and the function τμV(q) do not depend on V, and τμV(q) can differ from τμ on a non-trivial interval. For Gibbs measures and some of their discrete counterparts, we show the stronger uniform result: for γn,m-almost every m-dimensional subspace V, the multifractal formalism holds for μV over the whole interval . As an application, we obtain a part of the singularity spectrum of some self-similar measures on attractors of iterated function systems which do not satisfy the weak separation condition.

This paper presents a new approach for synthesizing breast tissue images based on a random fractal process, the fractional Brownian motion (fBm). This work deals with modeling Regions of Interest (ROIs) of mammographic images. Diverse synthetic ROIs were generated: healthy ones and others with microcalcifications according to fatty and dense tissue. Microcalcifications were injected in several dispositions in order to model benign and malignant cases. The aim of this study resides in two points: (1) the generation of synthetic images of mammograms for researchers and radiologists in order to test their tools and orient the choice of their parameters to enhance the diagnostic accuracy; and (2) to compare two microcalcification segmentation approaches: ‘Sq-Sq’ approach based on multifractal analysis and the ‘MM’ approach based on Mathematical Morphology. In fact, the results proved that the ‘Sq-Sq’ method can detect microcalcifications with different arrangements for any type of tissue and were evaluated using a qualitative test by an expert and a quantitative one based on the Area Overlap Measure (AOM) and the Dice coefficient. The ‘Sq-Sq’ approach yield a mean of 0.8±0.06 for AOM and 0.8446 for Dice coefficient for all segmented images.

In this paper we intend to generalize the L q -spectrum relatively to two measures and to study its behavior under orthogonal projections. A uniform result is obtained in the case of “Frostman like measures”.

We study the multifractal formalism for Bernoulli products of invertible matrices. Using the Fourier-Laplace transform, we prove the existence of a Frostman measure and so the validity of multifractal formalism. As an application we give an estimation of the spectrum of singularities of a harmonic function defined on the Sierpinski gasket.

In the present paper, we characterize the approximation numbers orbits of conjugation of a bounded operator T in an Hilbert space and there relationship with the eigenvalues of T . As a consequence we obtain that for normal operators | λ n ( T ) | = inf { ρ ( T - L ) : TL = LT and dim R ( L ) n } , where λ n ( T ) is the n-th eigenvalue of T . We illustrate with an example that the equality doesn’t hold in general.

The inner knowledge of volumes from images is an ancient problem. This question be- comes complicated when it concerns quantization, as the case of any measurement and in particular the calculation of fractal dimensions. Trabecular bone tissues have, like many natural elements, an architecture which shows a fractal aspect. Many studies have already been developed according to this approach. The question which arises however is to know to which extent it is possible to get an exact determination of the fractal dimension of a given volume only from the fractal measurement made on the projections or slice images given by medical imaging. This paper gives general results about the Minkowski dimensions and contents of projections and sections of a set. We also show with examples that they depend essentially on the directions of the planes and so there is - in gen- eral case - no relation between 3D and 2D fractal dimensions. This consideration is then illustrated with examples from synthetic models and from CT scan images of wrists. In conclusion, this study reveals that the quantitative characterization of an organic volume (in particular osseous) requires taking into account the whole volume, and not only some of its slices or projections.

The aim of this study is the detection and segmentation of microcalcifications in digital mammograms using multifractal analysis. To detect the suspicious Region Of Interest (ROI), containing anomalies, we propose to decompose the whole image into ROIs and compare the multifractal spectrums based on the q-structure functions of each one. The segmentation of microcalcifications consists of two steps. On the first step, we create an image denoted ‘α_image’. This image is constructed using the singularity coefficient, deduced from multifractal spectrum. Then, in the next step, we enhance the visualization of microcalcifications by creating an image denoted ‘f(α)_image’ based on the global regularity measure of the ‘α_image’ spectrum. We investigated the robustness of our approach using a data set of mammograms from ‘MiniMIAS’ database. Results demonstrate the accuracy of our approach, which successfully detect and segment microcalcifications with irregular form and small size.

Resume En nous placant dans la situation consideree par L. Olsen, nous justifions le formalisme multifractal sous des hypotheses moins restrictives sur la mesure.

Multifractal analysis is known as a useful tool in signal analysis. However, the methods are often used without methodological validation. In this study, we present multidimensional models in order to validate multifractal analysis methods.

International audience; By obtening a new sufficient condition for a valid multifractal formalism, we improve in this paper a result developped by Olsen (1995, Adv. Math.). In particular, we describe a large class of measures satisfying the multifractal formalism and for which the construction of Gibbs measures is not possible. Some of these measures are not unidimensional but have a nontrivial multifractal spectrum and then give a negative answer to a question asked by S.J. Taylor (1995, J. Fourier Anal. Appl., special issue). We also describe a necessary condition of validity for the formalism which is very close to the sufficient one. This necessary condition allows us to describe a measure $\mu$ for which the multifractal packing dimension function $B_\mu(q)$ is a nontrivial real analytic function but the multifractal formalism is nowhere satisfied. This example gives also a solution to a problem posed by Taylor in (cited above).