Enhanced BC Algorithm Incorporating a Novel Sampling Step and a Fractional Box Count

The Box-Counting (BC) method is one of the most commonly used algorithms for fractal dimension calculation of binary images in the fields of Engineering, Science, Medical Science, Geology and so on due to its simplicity and reliability. One of the issues related to fractal dimension is data sampling that involves a process where a certain size of box is taken from a given image and it has a direct effect on the precision of the fractal dimension estimation. The Geometric Step (GS) method, arithmetic step method, and divisor step method are the representative methods. The GS method is mainly used because of its efficiency. However, the GS method has some drawbacks in nature. If the image size is large, it provides insufficient data for regression analysis. It can be applied to the image of pixel size for 100 [%] pixel utilization. Application of the GS method to an image of may waste pixels in the calculation and degrade the estimation accuracy. In this thesis, a novel sampling method is proposed in order to resolve the shortcomings of the GS method on the basis of the intuitive observation that an estimate may have a higher degree of precision if more pixels are utilized in each step and a sufficiently large number of fitting data are guaranteed. The proposed sampling method is an improved version of the conventional GS method, called the modified GS (MGS) method. The MGS method selects some additional step sizes with higher pixel utilization rate among the middle values between the integer powers of 2 to constitute the overall step set with the GS method. Not all sampling methods including the MGS method can guarantee 100 [%] pixel utilization when the BC method is applied to images of an arbitrary size. This study suggests a novel fractional counting method to resolve the problem of pixel waste. The proposed counting method counts pixels of fractal within a discarded box (not of size) and adds its fractional count normalized by both the average pixel number of all boxes with size and step size to integer count. The performance of the enhanced BC method incorporating the MGS method and fractional counting method is verified on a set of deterministic fractal images whose theoretical dimensions are well known and compared it with those of the existing BC methods. The experimental results show that the proposed method outperforms the conventional BC method and triangle BC method.Contents List of Tables ⅲ List of Figures ⅳ Abstract ⅵ Chapter 1. Introduction 1.1 Motivation 1 1.2 Research objectives 3 1.3 Organization of the thesis 3 Chapter 2. Overview of Fractal Theory 2.1 Definition of fractal 5 2.2 Fractal dimension 7 2.3 Fractal geometry 9 2.3.1 Mandelbrot set and Julia set 10 2.3.2 Koch snowflake (Opened) 11 2.3.3 Apollonian gasket 12 2.3.4 Vicsek fractal 13 2.3.5 Sierpinski triangle 14 2.3.6 Rand cantor 15 2.3.7 Koch curve 85° 16 2.3.8 Sierpinski carpet 17 2.3.9 Hilbert curve 18 Chapter 3. Existing Box-Counting Methods 3.1 Conventional BC method 20 3.2 Triangle BC method 25 Chapter 4. Enhanced BC method 4.1 Existing sampling methods and their drawbacks 27 4.1.1 Sampling methods 27 4.1.2 Pixel utilization 30 4.1.3 Drawbacks of existing sampling methods 30 4.2 New sampling method 32 4.3 Fractional box count 35 4.4 Procedure of the enhanced BC method 38 Chapter 5. Experiments and Review 5.1 Experiments on deterministic fractal image 41 5.1.1 test image 41 5.1.2 Determination of 43 5.1.3 Experiment with images of pixels 44 5.1.4 Experiments on rotated image 45 5.1.5 Experiment with images of pixels 46 5.2 Experiments on non-deterministic fractal images 51 5.2.1 Converting color images to binary images 51 5.2.2 Coastline images 52 Chapter 6. Conclusion 56 References 58 Appendix 61Maste

Hybrid Power Spectral and Wavelet Image Roughness Analysis

The Two-Dimensional Wavelet Transform Modulus Maxima (2D WTMM) sliding window methodology has proven to be a robust approach, in particular for the extraction of the Hurst (H) roughness exponent from grayscale mammograms. The power spectrum is a computational analysis based on the Fourier transform that can be used to estimate the roughness of a scale-invariant image or region via the calculation of H. We aim to examine how the calculation of H in fractional Brownian motion (fBm) images and mammograms can be improved. fBm images are generated for H ∈ [0.00,1.00] for testing through the previous 2D WTMM sliding window analysis using the Gaussian smoothing function, the second-order derivative of the Gaussian smoothing function, the Mexican hat, and the power spectrum analysis. The power spectrum is shown to provide a more accurate calculation of H for Htheo \u3c 0.45 (RMSE = 0.01), while the 2D WTMM analysis with the Mexican hat smoothing function provides this for H ≥ 0.45 (RMSE = 0.058) in fBm images. Through the previous implementation of the 2D WTMM sliding window analysis, we have categorized mammographic subregions into three categories: Fatty (H \u3c 0.45), risky dense (0.45 ≤ H ≤ 0.55), and healthy dense mammographic tissue (H \u3e 0.55). The power spectrum and the 2D WTMM analysis are further tested on the CompuMAINE Laboratory’s acquired de-identified Perm and Maine mammographic datasets. From this analysis, it can be concluded that the power spectrum analysis cannot accurately distinguish fatty from dense tissue in grayscale mammograms. The implementation of the Mexican hat smoothing function provides a decrease in the number of mammographic subregions rejected during our analysis. In addition, the Mexican hat smoothing function indicates a greater difference in risky dense mammographic tissue between cancerous and normal patients compared to the previously adapted 2D WTMM analysis with the Gaussian smoothing function. The presence of noise in the Perm mammographic dataset indicates a larger minimum size for the range of wavelet scales a (MinADelta = 3.0) should be used in the calculation of H using the Mexican hat smoothing function in the 2D WTMM sliding window analysis. Higher quality (16-bit) mammograms in the Maine mammographic dataset indicate a similar minimum range of wavelet scales used in previous studies (MinADelta = 1.0) should be used to calculate H with the Mexican hat smoothing function. Through extensive calibration and testing of the power spectrum and 2D WTMM methodologies, we conclude the implementation of the 2D WTMM methodology with the Mexican hat smoothing function provides the most accurate calculation of H ∈ [0.00,1.00] in fBm and mammographic images

Quantitative Imaging in Electron and Confocal Microscopies for Applications in Biology

Among the large number of topics related to the quantification of images in electron and confocal microscopies for applications in biology, we selected four subjects that we consider to be representative of some recent tendencies. The first is the quantification of three-dimensional data sets recorded routinely in scanning confocal microscopy. The second is the quantification of the textural and fractal appearance of images. The two other topics are related to image series, which are more and more often provided by imaging instruments. The first kind of series concerns electron energy-filtered images. We show that the parametric (modelling) approach can be complemented by non-parametric approaches (e.g., different variants of multivariate statistical techniques). The other kind of series consists of multiple mappings of a specimen. We describe several new tools for the study and quantification of the co-location, with potential application to multiple mappings in microanalysis or in fluorescence microscopy

Fractal complexity of Escherichia coli nutrient transport channels is influenced by cell shape and growth environment

Recent mesoscopic characterisation of nutrient-transporting channels in E. coli has allowed the identification and measurement of individual channels in whole mature biofilms. However, their complexity under different physiological and environmental conditions remains unknown. Analysis of confocal micrographs of biofilms formed by cell shape mutants of E. coli shows that channels have a high fractal complexity, regardless of cell phenotype or growth medium. In particular, biofilms formed by the mutant strain ΔompR, which has a wide-cell phenotype, have a higher fractal dimension when grown on rich medium than when grown on minimal medium, with channel complexity affected by glucose and agar concentration in the medium. Osmotic stress leads to a dramatic reduction in ΔompR cell size, but has a limited effect on channel morphology. This work shows that fractal image analysis is a powerful tool to quantify the effect of phenotypic mutations and growth environment on the morphological complexity of internal E. coli biofilm structures. If applied to a wider range of mutant strains, this approach could help elucidate the genetic determinants of channel formation in E. coli biofilms

A Genetic Bayesian Approach for Texture-Aided Urban Land-Use/Land-Cover Classification

Urban land-use/land-cover classification is entering a new era with the increased availability of high-resolution satellite imagery and new methods such as texture analysis and artificial intelligence classifiers. Recent research demonstrated exciting improvements of using fractal dimension, lacunarity, and Moran’s I in classification but the integration of these spatial metrics has seldom been investigated. Also, previous research focuses more on developing new classifiers than improving the robust, simple, and fast maximum likelihood classifier. The goal of this dissertation research is to develop a new approach that utilizes a texture vector (fractal dimension, lacunarity, and Moran’s I), combined with a new genetic Bayesian classifier, to improve urban land-use/land-cover classification accuracy. Examples of different land-use/land-covers using post-Katrina IKONOS imagery of New Orleans were demonstrated. Because previous geometric-step and arithmetic-step implementations of the triangular prism algorithm can result in significant unutilized pixels when measuring local fractal dimension, the divisor-step method was developed and found to yield more accurate estimation. In addition, a new lacunarity estimator based on the triangular prism method and the gliding-box algorithm was developed and found better than existing gray-scale estimators for classifying land-use/land-cover from IKONOS imagery. The accuracy of fractal dimension-aided classification was less sensitive to window size than lacunarity and Moran’s I. In general, the optimal window size for the texture vector-aided approach is 27x27 to 37x37 pixels (i.e., 108x108 to 148x148 meters). As expected, a texture vector-aided approach yielded 2-16% better accuracy than individual textural index-aided approach. Compared to the per-pixel maximum likelihood classification, the proposed genetic Bayesian classifier yielded 12% accuracy improvement by optimizing prior probabilities with the genetic algorithm; whereas the integrated approach with a texture vector and the genetic Bayesian classifier significantly improved classification accuracy by 17-21%. Compared to the neural network classifier and genetic algorithm-support vector machines, the genetic Bayesian classifier was slightly less accurate but more computationally efficient and required less human supervision. This research not only develops a new approach of integrating texture analysis with artificial intelligence for classification, but also reveals a promising avenue of using advanced texture analysis and classification methods to associate socioeconomic statuses with remote sensing image textures

Multifractal techniques for analysis and classification of emphysema images

This thesis proposes, develops and evaluates different multifractal methods for detection, segmentation and classification of medical images. This is achieved by studying the structures of the image and extracting the statistical self-similarity measures characterized by the Holder exponent, and using them to develop texture features for segmentation and classification. The theoretical framework for fulfilling these goals is based on the efficient computation of fractal dimension, which has been explored and extended in this work. This thesis investigates different ways of computing the fractal dimension of digital images and validates the accuracy of each method with fractal images with predefined fractal dimension. The box counting and the Higuchi methods are used for the estimation of fractal dimensions. A prototype system of the Higuchi fractal dimension of the computed tomography (CT) image is used to identify and detect some of the regions of the image with the presence of emphysema. The box counting method is also used for the development of the multifractal spectrum and applied to detect and identify the emphysema patterns. We propose a multifractal based approach for the classification of emphysema patterns by calculating the local singularity coefficients of an image using four multifractal intensity measures. One of the primary statistical measures of self-similarity used in the processing of tissue images is the Holder exponent (α-value) that represents the power law, which the intensity distribution satisfies in the local pixel neighbourhoods. The fractal dimension corresponding to each α-value gives a multifractal spectrum f(α) that was used as a feature descriptor for classification. A feature selection technique is introduced and implemented to extract some of the important features that could increase the discriminating capability of the descriptors and generate the maximum classification accuracy of the emphysema patterns. We propose to further improve the classification accuracy of emphysema CT patterns by combining the features extracted from the alpha-histograms and the multifractal descriptors to generate a new descriptor. The performances of the classifiers are measured by using the error matrix and the area under the receiver operating characteristic curve (AUC). The results at this stage demonstrated the proposed cascaded approach significantly improves the classification accuracy. Another multifractal based approach using a direct determination approach is investigated to demonstrate how multifractal characteristic parameters could be used for the identification of emphysema patterns in HRCT images. This further analysis reveals the multi-scale structures and characteristic properties of the emphysema images through the generalized dimensions. The results obtained confirm that this approach can also be effectively used for detecting and identifying emphysema patterns in CT images. Two new descriptors are proposed for accurate classification of emphysema patterns by hybrid concatenation of the local features extracted from the local binary patterns (LBP) and the global features obtained from the multifractal images. The proposed combined feature descriptors of the LBP and f(α) produced a very good performance with an overall classification accuracy of 98%. These performances outperform other state-of-the-art methods for emphysema pattern classification and demonstrate the discriminating power and robustness of the combined features for accurate classification of emphysema CT images. Overall, experimental results have shown that the multifractal could be effectively used for the classifications and detections of emphysema patterns in HRCT images

A practical method for estimating fractal dimension

Pattern Recognition Letters165457-46