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

Superabsorbent polymer (SAP) has been widely used as internal curing agent for cement-based materials, duo to its high internal curing efficiency. Pore structure of affected zone around SAP is important for understanding the internal curing mechanism. This paper presents a study of the pore structure evolution of affected zone around SAP in cement paste with water/cement ratio 0.24 and 0.30 from 3 up to 28 days. Pore structure of affected zone and matrix was tested by mercury intrusion porosimetry and nitrogen adsorption and desorption. Surface fractal dimensions were determined using two different fractal models. Results indicated that SAP had no effect on the pore shape of affected zone, excluding the pore volume and pore diameter at early and late ages. Higher critical pore diameter but lower pore volume were found in affected zone at late ages. All the samples showed a multi-fractal property at micro- and macro-fractal scale, and the surface fractal dimension of affected zone was always larger than the matrix, showing a higher pore complexity. Some new findings on internal curing were achieved from an affected zone perspective. The calculated mechanical strength and permeability of affected zone from the pore structure parameters were found higher than the matrix. These new findings may give insight into the effect of internal curing with SAP from the perspective of affected zone.

Pore structure of affected zone around saturated and large superabsorbent polymers in cement paste

Fractal geometry and mercury porosimetry: Comparison and application of proposed models on building stones

Analysis methods of CT-scan images for the characterization of the bone texture: first results

We apply fractal analysis to study the birefringence textures of lyotropic lamellar liquid crystal system (water/cethylpyridinium chloride/decanol). Birefringence texture morphologies are important as they provide information on the molecular ordering as well as defect structures and therefore has been adopted as a standard method in characterizing different phases of liquid crystals. The system under consideration shows a gradual morphological transition from mosaic to oily streak structures and then to maltese cross texture when the water content is increased. Since these textures are the characteristic fingerprints for the lyotropic lamellar phases, it is necessary to have robust techniques to obtain image quantifiers that can characterize the morphological structure of the textures. For this purpose we employ three different approaches namely the Fourier power spectrum for monofractal analysis, the generalized box-counting method for multifractal analysis and multifractal segmentation technique for estimating the space-varying local Hurst exponents. The relationships between estimated image parameters such as the spectral exponent, the Hurst exponent and the fractal dimension with respect to patterns observed in the birefringence textures are discussed.

Fractal analysis of lyotropic lamellar liquid crystal textures

An image texture measure based on the box counting algorithm is evaluated for its potential to characterize human trabecular bone structure in medical images. Although bone images lack the self-similarity of theoretical fractals, bone images are candidates for characterization using fractal analysis because of their highly complex structure. The fractal based measure, herein called the box counting dimension (BCD), is an effective dimension, and does not imply an underlying fractal geometry. The importance of resolution in quantifying bone characteristics using the BCD is addressed. The relationship of BCD to standard measures of trabecular bone structure is also analyzed. To evaluate the variability of the BCD with change in resolution, the BCD is determined for two sections from each of seven 3D X-ray Tomographic Microscopy (XTM) images of human radius bone specimens, while the resolution is varied using lowpass filtering. An automated method of choosing the range of scales for the fractal analysis curve regression is used. The relationship of BCD to trabecular bone width and spacing is analyzed both for the XTM images and for simulated images representing idealized structures. The range of BCD values is 1.21–1.54. Variation in BCD over a range of resolutions is found to be small compared to the variation in BCD between different bone specimens. Maximum change in BCD over a large range of resolutions (17.60–176 microns per pixel) is 0.08. BCD decreases as space between trabeculae increases. Fractal based texture measures may potentially allow clinical monitoring of changes in bone structure — for example, using Magnetic Resonance Imaging at 150–200 micron resolution.

Fractal Based Image Analysis of Human Trabecular Bone using the Box Counting Algorithm: Impact of Resolution and Relationship to Standard Measures of Trabecular Bone Structure

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Fractal Based Image Analysis of Human Trabecular Bone using the Box Counting Algorithm: Impact of Resolution and Relationship to Standard Measures of Trabecular Bone Structure