Lacunarity analysis: A general technique for the analysis of spatial patterns

A texture-based classification of crackles and squawks using lacunarity

An automatic classification method to efficiently discriminate the types of discontinuous breath sounds (DBSs), i.e., fine crackles (FCs), coarse crackles (CC), and squawks (SQ), is presented in this paper. Using the lacunarity of the acquired DBS, the proposed classification method, namely LAC, introduces a texture-based approach that captures the differences in the distribution of FC, CC, and SQ across the breathing cycle, which may lead to more accurate characterization of the pulmonary acoustical changes due to the related pathology. Prior to the lacunarity analysis, wavelet-based denoising of DBS is employed to eliminate effects of the vesicular sound (background noise) to DBS oscillatory pattern. LAC analysis builds its classification power both upon the use of lacunarity at an optimum scale and the approximation of its trajectory across an optimum range of scales using a three-parameter hyperbola model. LAC is applied to 363 DBS corresponding to 25 cases included in four lung sound databases. Results show that LAC efficiently classifies the three DBS categories in the comparison groups of FC-CC, FC-SQ (both with mean accuracy of 100%), CC-SQ (mean accuracy of 99.62%-100%), and FC-CC-SQ (mean accuracy of 99.75%-100%). When compared to other classification tools, LAC seems quite attractive, since, without employing high computational complexity, it results in high classification accuracy. Moreover, LAC introduces a "texture" concept in the analysis of breath sounds, something that strongly relates to the perception of the bioacoustic signals by the physician. Due to its simplicity, LAC could be implemented in a real-time context and be used in clinical medicine as a module of an integrated intelligent patient evaluation system.

Characterization and classification of tumor lesions using computerized fractal-based texture analysis and support vector machines in digital mammograms

OBJECTIVE

This paper presents a detailed study of fractal-based methods for texture characterization of mammographic mass lesions and architectural distortion. The purpose of this study is to explore the use of fractal and lacunarity analysis for the characterization and classification of both tumor lesions and normal breast parenchyma in mammography.

MATERIALS AND METHODS

We conducted comparative evaluations of five popular fractal dimension estimation methods for the characterization of the texture of mass lesions and architectural distortion. We applied the concept of lacunarity to the description of the spatial distribution of the pixel intensities in mammographic images. These methods were tested with a set of 57 breast masses and 60 normal breast parenchyma (dataset1), and with another set of 19 architectural distortions and 41 normal breast parenchyma (dataset2). Support vector machines (SVM) were used as a pattern classification method for tumor classification.

RESULTS

Experimental results showed that the fractal dimension of region of interest (ROIs) depicting mass lesions and architectural distortion was statistically significantly lower than that of normal breast parenchyma for all five methods. Receiver operating characteristic (ROC) analysis showed that fractional Brownian motion (FBM) method generated the highest area under ROC curve (A ( z ) = 0.839 for dataset1, 0.828 for dataset2, respectively) among five methods for both datasets. Lacunarity analysis showed that the ROIs depicting mass lesions and architectural distortion had higher lacunarities than those of ROIs depicting normal breast parenchyma. The combination of FBM fractal dimension and lacunarity yielded the highest A ( z ) value (0.903 and 0.875, respectively) than those based on single feature alone for both given datasets. The application of the SVM improved the performance of the fractal-based features in differentiating tumor lesions from normal breast parenchyma by generating higher A ( z ) value.

CONCLUSION

FBM texture model is the most appropriate model for characterizing mammographic images due to self-affinity assumption of the method being a better approximation. Lacunarity is an effective counterpart measure of the fractal dimension in texture feature extraction in mammographic images. The classification results obtained in this work suggest that the SVM is an effective method with great potential for classification in mammographic image analysis.

Lacunarity as a novel measure of cancer cells behavior

An important goal in many branches of science, especially in molecular biology and medicine is the quantitative analysis of the structures and their morphology. The morphology can be analyzed in many ways, in particular by the fractal analysis. Apart from the fractal dimension, an important part of the fractal analysis is the lacunarity measurement which, roughly speaking, characterizes the distribution of gaps in the fractal: a fractal with high lacunarity has large gaps. In this paper, we present an extension of the lacunarity measure to objects with nonregular shapes that enables us to provide a successful discrimination of cancer cell lines. The cell lines differ in the shape of vacuole (the gaps in their body) which is perfectly suited for the lacunarity analysis.

Statistical geometric affinity in human brain electric activity

The representation of the human electroencephalogram (EEG) records by neurophysiologists demands standardized time-amplitude scales for their correct conventional interpretation. In a suite of graphical experiments involving scaling affine transformations we have been able to convert electroencephalogram samples corresponding to any particular sleep phase and relaxed wakefulness into each other. We propound a statistical explanation for that finding in terms of data collapse. As a sequel, we determine characteristic time and amplitude scales and outline a possible physical interpretation. An analysis for characteristic times based on lacunarity is also carried out as well as a study of the synchrony between left and right EEG channels.

A simple digital image processing system to aid in melanoma diagnosis in an everyday melanocytic skin lesion unit: a preliminary report

BACKGROUND

For early melanoma diagnosis, experienced dermatologists have an accuracy of 64-80% using clinical diagnostic criteria, usually the ABCD rule, while automated melanoma diagnosis systems are still considered to be experimental and serve as adjuncts to the naked-eye expert prediction. In an attempt to aid in early melanoma diagnosis, we developed an image processing program with the aim to discriminate melanoma from melanocytic nevi, establishing a mathematical model to come up with a melanoma probability.

METHODS

Digital images of 132 melanocytic skin lesions (23 melanomas and 109 melanocytic nevi) were studied in features of geometry, color, and color texture. A total of 43 variables were studied for all lesions, e.g., geometry, color texture, sharpness of border, and color variables. Univariate logistic regression analysis followed by "-2 log likelihood" test and Spearman's rank correlation coefficient were used to eliminate inappropriate variables, as the presence of multi-collinearity among variables could cause severe problems in any stepwise variable selection method. Initially, "-2 log likelihood" and nonparametric Spearman's rho picked five variables to be included in a multivariate model of prediction. The five-variable model was then reduced to three variables and the performance of each model was tested. The "jackknife" method was performed in order to validate the model with the three variables and its accuracy was weighed vs. the five-variable model by receiver-operating characteristics (ROC) curve plotting. It was concluded that the reduced model did not compromise discriminatory power.

RESULTS

Not all variables contributed much to the model, therefore they were progressively eliminated and the model was finally reduced to three covariates of significance. A predictive equation was calculated, incorporating parameters of geometry, color, and color texture as independent covariates for the prediction of melanoma. The proposed model provides melanoma probability with a 60.9% sensitivity and 95.4% specificity of prediction, an overall accuracy of 89.4% (probability level 0.5), and 8% false-negative results.

CONCLUSIONS

Through a digital image processing system and the development of a mathematical model of prediction, discrimination between melanomas and melanocytic nevi seems feasible with a high rate of accuracy using multivariate logistic regression analysis. The proposed model is an alternative method to aid in early melanoma diagnosis. Expensive and sophisticated equipment is not required and it can be easily implemented in a reasonably priced portable programmable computer, in order to predict previously undiagnosed skin melanoma before histopathology results confirm diagnosis.

MR imaging and osteoporosis: fractal lacunarity analysis of trabecular bone

We develop a method of magnetic resonance (MR) image analysis able to provide parameter(s) sensitive to bone microarchitecture changes in aging, and to osteoporosis onset and progression. The method has been built taking into account fractal properties of many anatomic and physiologic structures. Fractal lacunarity analysis has been used to determine relevant parameter(s) to differentiate among three types of trabecular bone structure (healthy young, healthy perimenopausal, and osteoporotic patients) from lumbar vertebra MR images. In particular, we propose to approximate the lacunarity function by a hyperbola model function that depends on three coefficients, alpha, beta, and gamma, and to compute these coefficients as the solution of a least squares problem. This triplet of coefficients provides a model function that better represents the variation of mass density of pixels in the image considered. Clinical application of this preliminary version of our method suggests that one of the three coefficients, beta, may represent a standard for the evaluation of trabecular bone architecture and a potentially useful parametric index for the early diagnosis of osteoporosis.

Use of color texture in determining the nature of melanocytic skin lesions—a qualitative and quantitative approach

Melanocytic nevi are recognized as precursors of melanoma. Aiding in early recognition of melanoma, we estimated color texture parameters, fractal dimension and lacunarity of melanoma and other melanocytic nevi. Digital images of the lesions were processed. Graphic three-dimensional pseudoelevation images of the lesions and surrounding skin were produced to identify irregularities in color texture within the lesions. Estimation of lacunarity and fractal dimension followed in order to produce a numerical estimate of the coarseness of color texture. Clinicians readily perceive the resulting "geographical" images. Irregularity in the anaglyph, which might veil malignancy, is effortlessly identified through these images, and therefore an early excision of a suspect lesion is indicated.

Influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media

In recent years, experience has demonstrated that the classical fractal dimensions are not sufficient to describe uniquely the interstitial geometry of porous media. At least one additional index or dimension is necessary. Lacunarity, a measure of the degree to which a data set is translationally invariant, is a possible candidate. Unfortunately, several approaches exist to evaluate it on the basis of binary images of the object under study, and it is unclear to what extent the lacunarity estimates that these methods produce are dependent on the resolution of the images used. In the present work, the gliding-box algorithm of Allain and Cloitre [Phys. Rev. A 44, 3552 (1991)] and two variants of the sandbox algorithm of Chappard et al. [J. Pathol. 195, 515 (2001)], along with three additional algorithms, are used to evaluate the lacunarity of images of a textbook fractal, the Sierpinski carpet, of scanning electron micrographs of a thin section of a European soil, and of light transmission photographs of a Togolese soil. The results suggest that lacunarity estimates, as well as the ranking of the three tested systems according to their lacunarity, are affected strongly by the algorithm used, by the resolution of the images to which these algorithms are applied, and, at least for three of the algorithms (producing scale-dependent lacunarity estimates), by the scale at which the images are observed. Depending on the conditions under which the estimation of the lacunarity is carried out, lacunarity values range from 1.02 to 2.14 for the three systems tested, and all three of the systems used can be viewed alternatively as the most or the least "lacunar." Some of this indeterminacy and dependence on image resolution is alleviated in the averaged lacunarity estimates yielded by Chappard et al.'s algorithm. Further research will be needed to determine if these lacunarity estimates allow an improved, unique characterization of porous media.

Self-referred approach to lacunarity

This paper describes an approach to lacunarity which adopts the pattern under analysis as the reference for the sliding window procedure. The superiority of such a scheme with respect to more traditional methodologies, especially when dealing with finite-size objects, is established and illustrated through applications to diffusion limited aggregation pattern characterization. It is also shown that, given the enhanced accuracy and sensitivity of this scheme, the shape of the window becomes an important parameter, with advantage for circular windows.

Characterizing width uniformity by wave propagation

This work describes a different image analysis approach to characterize the uniformity of objects in agglomerates by using the propagation of normal wave fronts. The problem of width uniformity is discussed and its importance for the characterization of composite structures normally found in physics and biology highlighted. The methodology involves identifying each cluster (i.e., connected component) of interest, which can correspond to objects or voids, and estimating the respective medial axes by using a recently proposed wave front propagation approach, which is briefly reviewed. The distance values along such axes are identified and their mean and standard deviation values obtained. As illustrated with respect to synthetic and real objects (in vitro cultures of neuronal cells), the combined use of these two features provides a powerful description of the uniformity of the separation between the objects, presenting potential for several applications in material sciences and biology.

Fractal properties of the lattice Lotka-Volterra model

The lattice Lotka-Volterra (LLV) model is studied using mean-field analysis and Monte Carlo simulations. While the mean-field phase portrait consists of a center surrounded by an infinity of closed trajectories, when the process is restricted to a two-dimensional (2D) square lattice, local inhomogeneities/fluctuations appear. Spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures. Reactions take place only at the interfaces between different species and the borders adopt locally fractal structure. Intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations. The box-counting fractal dimension of the LLV dynamics on a 2D support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators. Lacunarity analysis is used to determine the degree of clustering of homologous species. Besides the spontaneous clustering that takes place on a regular 2D lattice, the effects of fractal supports on the dynamics of the LLV are studied. For supports of dimensionality D(s)<2 the lattice can, for certain domains of the reaction constants, adopt a poisoned state where only one of the species survives. By appropriately selecting the fractal dimension of the substrate, it is possible to direct the system into a poisoned or oscillatory steady state at will.

Fractal signature and lacunarity in the measurement of the texture of trabecular bone in clinical CT images

Fractal analysis is a method of characterizing complex shapes such as the trabecular structure of bone. Numerous algorithms for estimating fractal dimension have been described, but the Fourier power spectrum method is particularly applicable to self-affine fractals, and facilitates corrections for the effects of noise and blurring in an image. We found that it provided accurate estimates of fractal dimension for synthesized fractal images. For natural texture images fractality is limited to a range of scales, and the fractal dimension as a function of spatial frequency presents as a fractal signature. We found that the fractal signature was more successful at discriminating between these textures than either the global fractal dimension or other metrics such as the mean width and root-mean-square width of the spectral density plots. Different natural textures were also readily distinguishable using lacunarity plots, which explicitly characterize the average size and spatial organization of structural sub-units within an image. The fractal signatures of small regions of interest (32x32 pixels), computed in the frequency domain after corrections for imaging system noise and MTF, were able to characterize the texture of vertebral trabecular bone in CT images. Even small differences in texture due to acquisition slice thickness resulted in measurably different fractal signatures. These differences were also readily apparent in lacunarity plots, which indicated that a slice thickness of 1 mm or less is necessary if essential architectural information is not to be lost. Since lacunarity measures gap size and is not predicated on fractality, it may be particularly useful for characterizing the texture of trabecular bone.

Fractal characterization of chromatin appearance for diagnosis in breast cytology

This study explores the use of fractal analysis in the numerical description of chromatin appearance in breast cytology. Images of nuclei from fine-needle aspiration biopsies of the breast are characterized in terms of their Minkowski and spectral fractal dimensions, for 19 patients with benign epithelial cell lesions and 22 with invasive ductal carcinomas. Chromatin appearance in breast epithelial cell nuclear images is demonstrated to be fractal, suggesting that the three-dimensional chromatin structure in these cells also has fractal properties. A statistically significant difference between the mean spectral dimensions of the benign and malignant cases is demonstrated. The two fractal dimensions are very weakly correlated. A statistically significant difference between the benign and malignant cases in lacunarity, a fractal property characterizing the size of holes or gaps in a texture, is found over a wide range of scales. These differences are particularly pronounced at the smallest and largest scales, corresponding respectively to fine-scale texture, indicating whether chromatin is clumped or fine, and to large-scale structures like nucleoli. Logistic regression and artificial neural network classification models are developed to classify unknown cases on the basis of fractal measures of chromatin texture. Using leave-one-out cross-validation, the best logistic regression classifier correctly diagnoses 95.1 per cent of the cases. The best neural network model can correctly classify all of the cases, but it is unclear whether this is due to overtraining. Fractal dimensions and lacunarity are useful tools for the quantitative characterization of chromatin appearance, and can potentially be incorporated into image analysis devices to assure the quality and reproducibility of diagnosis by breast fine-needle aspiration biopsy.