It has been ascertained that the human brain is a complex system studied at multiple scales, from neurons and microcircuits to macronetworks The brain is characterized by a hierarchical organization that gives rise to its highly topological and functional complexity Over the last decades, fractal geometry has been shown as a universal tool for the analysis and quantification of the geometric complexity of natural objects, including the brain The fractal dimension has been identified as a quantitative parameter for the evaluation of the roughness of neural structures, the estimation of time series, and the description of patterns, thus able to discriminate different states of the brain in its entire physiopathological spectrum Fractal-based computational analyses have been applied to the neurosciences, particularly in the field of clinical neurosciences including neuroimaging and neuroradiology, neurology and neurosurgery, psychiatry and psychology, and neuro-oncology and neuropathology After a review of the basic concepts of fractal analysis and its main applications to the basic neurosciences in part I of this series, here, we review the main applications of fractals to the clinical neurosciences for a holistic approach towards a fractal geometry model of the brain

Fractals in the Neurosciences, Part II Clinical Applications and Future Perspectives

The purpose of this study is to discuss existing fractal-based algorithms and propose novel improvements of these algorithms to identify tumors in brain magnetic-response (MR) images. Considerable research has been pursued on fractal geometry in various aspects of image analysis and pattern recognition. Magnetic-resonance images typically have a degree of noise and randomness associated with the natural random nature of structure. Thus, fractal analysis is appropriate for MR image analysis. For tumor detection, we describe existing fractal-based techniques and propose three modified algorithms using fractal analysis models. For each new method, the brain MR images are divided into a number of pieces. The first method involves thresholding the pixel intensity values; hence, we call the technique piecewise-threshold-box-counting (PTBC) method. For the subsequent methods, the intensity is treated as the third dimension. We implement the improved piecewise-modified-box-counting (PMBC) and piecewise-triangular-prism-surface-area (PTPSA) methods, respectively. With the PTBC method, we find the differences in intensity histogram and fractal dimension between normal and tumor images. Using the PMBC and PTPSA methods, we may detect and locate the tumor in the brain MR images more accurately. Thus, the novel techniques proposed herein offer satisfactory tumor identification.

http://fractal.org/Life-Science-Technology/Publications/Fractal-analysis-brains.pdf

Fractal analysis of tumor in brain MR images

The present paper places binary morphological filtering into the framework of statistical estimation, the intent being to develop the theory of mean-square (MS) optimization. Classical binary morphological operations are interpreted as numerical functionals on binary N-vectors, so that in the random setting they can be treated as estimators dependent on N binary observation random variables. For single-erosion filters, optimization is achieved by finding the structuring element that minimizes MS error. Using the Matheron representation as a guide, we generalize the analysis to morphological filters given by unions of multiple erosions and optimize by minimizing MS error over all collections of erosions, or over a prefixed number of erosions. In all cases, MS error is relative to the estimation of an unobserved variable by a morphological function of observed variables. A key element in the method is use of the basis form of the Matheron expansion to reduce significantly the structuring-element search. The technique is adapted to special morphological filters by constraining the basis representation in accordance with the class of interest. It is demonstrated that optimization in terms of erosions is equivalent to optimization in terms of dilations.

Optimal mean-square N -observation digital morphological filters: i. optimal binary filters

For both the binary and gray scales, mean square optimal digital morphological filters have been characterized previously in terms of the Matheron erosion representation for increasing, translation-invariant mappings. Included in the characterization is the minimal search strategy for the optimal filter basis; however, without prior statistical information or an adequate image-noise model, even in the binary setting, filter design is computationally intractable for moderately sized observation windows. The mitigation of the computational burden via design constraints is the focus here. Although the resulting filter will be suboptimal, if the constraints are imposed in a suitable manner, little loss of filter performance occurs in return for design tractability. Three approaches are considered: limiting the number of terms in the Matheron expansion, constraining the observation window, and employing structuring element libraries. In the latter methodology various sublibraries are formed and a suboptimal filter is derived from image-noise statistics in conjunction with a basis search restricted to relevant sublibraries. This study analyzes two techniques for library construction: the expert approach involves prior sublibrary formation based on knowledge of important filter bases and the first-order approach employs single-erosion statistical information to limit the basis search to likely useful candidates.

Facilitation of optimal binary morphological filter design via structuring element libraries and design constraints

Optimal morphological restoration: The morphological filter mean-absolute-error theorem

Even in the binary case, designing optimal morphological filters involves a time-consuming search procedure that, in practice, can be intractable. The present paper provides an algorithm for filter design that is based upon the relationship between the optimal morphological filter and the conditional expectation. In effect, the algorithm proceeds by changing the conditional expectation into a morphological filter while at the same time increasing the mean-square error a minimal amount. Under many noise environments, the new algorithm is extremely efficient, thereby providing a filter design that can be used online for structuring-element updating.© (1991) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Conditional-expectation-based implementation of the optimal mean-square binary morphological filter

Once designed, implementation of an optimal mean-square binary morphological filter is extremely fast, especially when the erosions are implemented on a suitable parallel processor. On the other hand, optimal filter design involes a computationally burdensome search procedure that can, in practice, be intractable. The present paper provides an algorithm for filter design that is based on the relationship between the optimal morphological filter and the conditional expectation. The algorithm proceeds by changing the conditional expectation into a morphological filter while at the same time increasing the mean-square error by a minimal amount. It does so by switching observations between the 1-set and the 0-set of the conditional expectation. The switching algorithm is extremely efficient in many noise environments, and therefore provides a filter design that can be useful for online structuring-element updating. Owing to the relationship between stack and morphological filters, the algorithm is at once useful for finding optimal binary stack filters.

Efficient derivation of the optimal mean-square binary morphological filter from the conditional expectation via a switching algorithm for discrete power-set lattice

In the present work, distinct structures appearing in biomedical images are modeled as fractals. Within an image, the relevant structures are associated to a fractal dimension. Changes in the dimension values, as a function of time, reflect alterations of structural properties. Accurate and robust estimation of this dimension, leads to a precise characterization of changes undergone by the structure. The Continuous Pyramidal Alternating Sequential Filter method is proposed as a robust and accurate fractal dimension estimator. A study on bedrest data of human subjects was conducted. Bedrest is an accepted model for the study of osteoporosis. Here the spine is modeled as a fractal structure. Fractal model were also applied towards analysis of breast cancer and brain tumors. Results from these different studies confirm that fractals can suitably model a variety of biological structures. These studies also suggest that fractal models can be effectively utilized to detect temporal changes undergone by the structures.

Fractal-based characterization of structural changes in biomedical images

Since their introduction by Mandelbrot, fractals have been employed in perhaps almost all fields of study. This work is aimed towards extending the applicability of fractal models within image analysis. Fractal dimension is a key parameter in fractal models and is frequently used to characterize texture in higher level image processing schemes. Robust texture characterization schemes can significantly improve performance of computer vision or image analysis systems that rely upon these. However, modeling texture of digital images using fractal dimension suffers from various limitations. The finite nature and the fixed resolution of a digital image results in inferior accuracy of the fractal dimension estimate. Most existing fractal dimension estimation techniques tend to have very low accuracy when applied to small images, for instance of size 16 x 16 pixels. Even when large images are considered, without special constraints or optimization the accuracy is not very high. One of the main contributions of this work is to present robust new algorithms for computing accurate fractal dimension estimates over both small and large digital images. Analysis of mammograms for breast cancer detection, MRI and CT images for brain tumor detection and X-ray images for detection of osteoporosis is presented as application of the new fractal dimension estimation techniques. A scheme for early detection of the above mentioned ailments, using fractal dimension as the primary parameter is proposed. The other contribution of this work is the introduction of a new generalized family of translation, rotation and scale invariant parameters, namely, fractal moments. Fractal moments are parameters that are well suited not only for fractals, but for any arbitrary digital image analysis applications. Furthermore, it will be shown that when dealing with fractal images, under some circumstances fractal dimension becomes a special case of the more generalized fractal moments. The theory of fractal moments is applied towards pattern recognition and experimental results indicate that these parameters can robustly characterize textures in digital images. To the author's knowledge this is the first time such a theory is being presented, in particular for image analysis.

Accurate fractal dimension estimation and its application to image analysis

A new family of scale, rotation and affine transformation invariant parameters, denoted as multiresolution moments, is introduced. Basic underlying theory of this generalized and extensible family is described. Experimental results obtained by using a specific subset of this family of new parameters towards texture similarity measurement are included to illustrate the applicability and robustness of this family of parameters.

Vivek Swarnakar

Papers

Conditional-expectation-based implementation of the optimal mean-square binary morphological filter

Efficient derivation of the optimal mean-square binary morphological filter from the conditional expectation via a switching algorithm for discrete power-set lattice

Fractal-based characterization of structural changes in biomedical images

Accurate fractal dimension estimation and its application to image analysis

Texture Analysis Using Mult iresolut ion Moments