Simple fractal method of assessment of histological images for application in medical diagnostics

We propose new method of assessment of histological images for medical diagnostics. 2-D image is preprocessed to form 1-D landscapes or 1-D signature of the image contour and then their complexity is analyzed using Higuchi's fractal dimension method. The method may have broad medical application, from choosing implant materials to differentiation between benign masses and malignant breast tumors.     

Fractal analysis of the uterine contractions.

The fractal dimension D may be calculated in many ways, since its strict definition, the Hausdorff definition is too complicated for practical estimation. In this paper we perform a comparative study often methods of fractal analysis of time series. In Benoit, a commercial program for fractal analysis, five methods of computing fractal dimension of time series (rescaled range analysis, power spectral analysis, roughness-length, variogram methods and wavelet method) are available. We have implemented some other algorithms for calculating D: Higuchi's fractal dimension, relative dispersion analysis, running fractal dimension, method based on mathematical morphology and method based on intensity differences. For biomedical signals results obtained by means of different algorithms are different, but consistent.     

Curvelet Based Offline Analysis of SEM Images

Manual offline analysis, of a scanning electron microscopy (SEM) image, is a time consuming process and requires continuous human intervention and efforts. This paper presents an image processing based method for automated offline analyses of SEM images. To this end, our strategy relies on a two-stage process, viz. texture analysis and quantification. The method involves a preprocessing step, aimed at the noise removal, in order to avoid false edges. For texture analysis, the proposed method employs a state of the art Curvelet transform followed by segmentation through a combination of entropy filtering, thresholding and mathematical morphology (MM). The quantification is carried out by the application of a box-counting algorithm, for fractal dimension (FD) calculations, with the ultimate goal of measuring the parameters, like surface area and perimeter. The perimeter is estimated indirectly by counting the boundary boxes of the filled shapes. The proposed method, when applied to a representative set of SEM images, not only showed better results in image segmentation but also exhibited a good accuracy in the calculation of surface area and perimeter. The proposed method outperforms the well-known Watershed segmentation algorithm.     

A Fast Image Alignment Approach for 2D Classification of Cryo-EM Images Using Spectral Clustering

Three-dimensional (3D) reconstruction in single-particle cryo-electron microscopy (cryo-EM) is a significant technique for recovering the 3D structure of proteins or other biological macromolecules from their two-dimensional (2D) noisy projection images taken from unknown random directions. Class averaging in single-particle cryo-EM is an important procedure for producing high-quality initial 3D structures, where image alignment is a fundamental step. In this paper, an efficient image alignment algorithm using 2D interpolation in the frequency domain of images is proposed to improve the estimation accuracy of alignment parameters of rotation angles and translational shifts between the two projection images, which can obtain subpixel and subangle accuracy. The proposed algorithm firstly uses the Fourier transform of two projection images to calculate a discrete cross-correlation matrix and then performs the 2D interpolation around the maximum value in the cross-correlation matrix. The alignment parameters are directly determined according to the position of the maximum value in the cross-correlation matrix after interpolation. Furthermore, the proposed image alignment algorithm and a spectral clustering algorithm are used to compute class averages for single-particle 3D reconstruction. The proposed image alignment algorithm is firstly tested on a Lena image and two cryo-EM datasets. Results show that the proposed image alignment algorithm can estimate the alignment parameters accurately and efficiently. The proposed method is also used to reconstruct preliminary 3D structures from a simulated cryo-EM dataset and a real cryo-EM dataset and to compare them with RELION. Experimental results show that the proposed method can obtain more high-quality class averages than RELION and can obtain higher reconstruction resolution than RELION even without iteration.     

Fractal dimension in human cerebellum measured by magnetic resonance imaging

Fractal dimension has been used to quantify the structures of a wide range of objects in biology and medicine. We measured fractal dimension of human cerebellum (CB) in magnetic resonance images of 24 healthy young subjects (12 men and 12 women). CB images were resampled to a series of image sets with different 3D resolutions. At each resolution, the skeleton of the CB white matter was obtained and the number of pixels belonging to the skeleton was determined. Fractal dimension of the CB skeleton was calculated using the box-counting method. The results indicated that the CB skeleton is a highly fractal structure, with a fractal dimension of 2.57 +/- 0.01. No significant difference in the CB fractal dimension was observed between men and women.     

Fractal geometric analysis of lung cancer angiogenic patterns

For medical images, the fractal dimension D may be used as an index of irregularity. The angiogenesis patterns of lung cancer were analysed by means of the perimeter-area and box counting algorithms. The fractal nature of all images in the sense of the perimeter-area method and of 68% images in the sense of the box-counting method suggest the possibility to use the fractal dimension as a new non-morphometric parameter evaluating angiogenic processes in neoplasms.     

Fractal Analysis of Cervical Intraepithelial Neoplasia

Introduction

Cervical intraepithelial neoplasias (CIN) represent precursor lesions of cervical cancer. These neoplastic lesions are traditionally subdivided into three categories CIN 1, CIN 2, and CIN 3, using microscopical criteria. The relation between grades of cervical intraepithelial neoplasia (CIN) and its fractal dimension was investigated to establish a basis for an objective diagnosis using the method proposed.

Methods

Classical evaluation of the tissue samples was performed by an experienced gynecologic pathologist. Tissue samples were scanned and saved as digital images using Aperio scanner and software. After image segmentation the box counting method as well as multifractal methods were applied to determine the relation between fractal dimension and grades of CIN. A total of 46 images were used to compare the pathologist''s neoplasia grades with the predicted groups obtained by fractal methods.

Results

Significant or highly significant differences between all grades of CIN could be found. The confusion matrix, comparing between pathologist''s grading and predicted group by fractal methods showed a match of 87.1%. Multifractal spectra were able to differentiate between normal epithelium and low grade as well as high grade neoplasia.

Conclusion

Fractal dimension can be considered to be an objective parameter to grade cervical intraepithelial neoplasia.     

Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.Here we present a new technique for quantitatively determining the scales over which real-world structures show statistical self-similarity. The new technique uses a combination of curve-fitting and tests of curvilinearity of residuals to identify the largest range of contiguous scales that exhibit statistical self-similarity. Consequently, we estimate D only over the statistically identified region of self-similarity and introduce the finite scale- corrected dimension (FSCD). We demonstrate the use of this method in two steps. First, using mathematical fractal curves with known but variable spatial scales of self-similarity (achieved by varying the iteration level used for creating the curves), we demonstrate that our method can reliably quantify the spatial scales of self-similarity. This technique therefore allows accurate empirical quantification of theoretical Ds. Secondly, we apply the technique to digital images of the rhizome systems of goldenrod (Solidago altissima). The technique significantly reduced variations in estimated fractal dimensions arising from variations in the method of preparing digital images. Overall, the revised method has the potential to significantly improve repeatability and reliability for deriving fractal dimensions of real-world branching structures.     

Reconstruction for 3D PET Based on Total Variation Constrained Direct Fourier Method

This paper presents a total variation (TV) regularized reconstruction algorithm for 3D positron emission tomography (PET). The proposed method first employs the Fourier rebinning algorithm (FORE), rebinning the 3D data into a stack of ordinary 2D data sets as sinogram data. Then, the resulted 2D sinogram are ready to be reconstructed by conventional 2D reconstruction algorithms. Given the locally piece-wise constant nature of PET images, we introduce the total variation (TV) based reconstruction schemes. More specifically, we formulate the 2D PET reconstruction problem as an optimization problem, whose objective function consists of TV norm of the reconstructed image and the data fidelity term measuring the consistency between the reconstructed image and sinogram. To solve the resulting minimization problem, we apply an efficient methods called the Bregman operator splitting algorithm with variable step size (BOSVS). Experiments based on Monte Carlo simulated data and real data are conducted as validations. The experiment results show that the proposed method produces higher accuracy than conventional direct Fourier (DF) (bias in BOSVS is 70% of ones in DF, variance of BOSVS is 80% of ones in DF).     

Extracting complexity waveforms from one-dimensional signals

Background  

Nonlinear methods provide a direct way of estimating complexity of one-dimensional sampled signals through calculation of Higuchi's fractal dimension (1<FD<2). In most cases the signal is treated as being characterized by one value of FD and consequently analyzed as one epoch or, if divided into more epochs, often only mean and standard deviation of epoch FD are calculated. If its complexity variation (or running fractal dimension), FD(t), is to be extracted, a moving window (epoch) approach is needed. However, due to low-pass filtering properties of moving windows, short epochs are preferred. Since Higuchi's method is based on consecutive reduction of signal sampling frequency, it is not suitable for estimating FD of very short epochs (N < 100 samples).     

Wavelet packet fractal analysis of neuronal morphology

An image analysis method called two-dimensional wavelet packet analysis (2D WPA) is introduced to quantify branching complexity of neurons. Both binary silhouettes and contour profiles of neurons were analyzed to determine accuracy and precision of the fractal dimension in cell classification tasks. Two-dimensional WPA plotted the slope of decay for a sorted list of discrete wavelet packet coefficients belonging to the adapted wavelet best basis to obtain the fractal dimension for test images and binary representations of neurons. Two-dimensional WPA was compared with box counting and mass-radius algorithms. The results for 2D WPA showed that it could differentiate between neural branching complexity in cells of different type in agreement with accepted methods. The importance of the 2D WPA method is that it performs multiresolution decomposition in the horizontal, vertical, and diagonal orientations.     

Low-pass filtering, a new method of fractal analysis: application to PET images of pulmonary blood flow.

The pattern of a spatial structure that repeats itself independently of the scale of magnification or resolution is often characterized by a fractal dimension (D). Two-dimensional low-pass filtering, which may serve as a method to assess D, was applied to functional images of pulmonary perfusion measured by positron emission tomography. The corner frequency of a low-pass filter is inversely proportional to the resolution scale. The method was applied to three types of images: random noise images, synthetic fractal images, and positron emission tomographic images of pulmonary perfusion. Images were processed with two-dimensional low-pass filters of decreasing corner frequencies, and a spatial heterogeneity index, the coefficient of variation, was calculated for each low-pass-filtered image. The natural logarithm of the coefficient of variation scaled linearly with the natural logarithm of the resolution scale for the PET images studied (average R(2) = 0.99). D ranged from 1.25 to 1.36 for the residual distribution of pulmonary perfusion after vertical gradients were removed by linear regression. D of the same data without removal of vertical gradients ranged from 1.11 to 1.14, but the fractal plots had systematic deviations from linearity and a lower linear correlation coefficient (R(2) = 0. 96). The method includes all data in the lung field and is insensitive to the effects of misregistration. We conclude that low-pass filtering offers new insights into the interpretation of D of two-dimensional functional images as a measure of the frequency content of spatial heterogeneity.     

Fractal dimension and histogram method: Algorithm and some preliminary results of noise-like time series analysis

In the present work a methodological background for the histogram method of time series analysis is developed. Connection between shapes of smoothed histograms constructed on the basis of short segments of time series of fluctuations and the fractal dimension of the segments is studied. It is shown that the fractal dimension possesses all main properties of the histogram method. Based on it a further development of fractal dimension determination algorithm is proposed. This algorithm allows more precision determination of the fractal dimension by using the “all possible combination” method. The application of the method to noise-like time series analysis leads to results, which could be obtained earlier only by means of the histogram method based on human expert comparisons of histograms shapes.     

Microstructural characterization of chitosan and alginate films by microscopy techniques and texture image analysis

The aim of this work is to characterize the microstructure of chitosan and alginate edible films by microscopy techniques and texture image analysis. Edible films were obtained by solution casting and solvent evaporation. The microscopy techniques used in this work were: light, environmental scanning electron and atomic force microscopy. Textural features and fractal dimension were extracted from the images. Entropy and fractal dimension were more useful to evaluate the complexity and roughness of films. The highest values of entropy and fractal dimension corresponded to alginate/chitosan, followed of alginate and chitosan films. An entropy/fractal dimension ratio, proposed here, was useful to characterize the degree of image complexity and roughness of edible films at different magnifications. It was possible to postulate that microscopy techniques combined with texture image analysis are efficient tools to quantitatively evaluate the surface morphology of edible films made of chitosan and alginate.     

The Quasi-Fractal Structure of Fish Brain Neurons

The box-counting method for calculating the fractal dimension (D) with the ImageJ 1.20s software is used as a tool for quantitative analysis of the neuronal morphology in the fish brain. The fractal dimension was determined for several types of neurons in the brain of two teleost species, Pholidapus dybowskii and Oncorhynchus keta. These results were compared with those obtained for some neurons of the human brain. The fractal (fractional) dimension (D), as a quantitative index of filling of two-dimensional space by the black and white image of a cell, is shown to vary from 1.22 to 1.72 depending on the type of neuron. The fractal dimension reaches its maximum in less specialized neurons that carry out a number of different functions. On the other hand, highly specialized neurons display a relatively low fractal dimension. Thus, the fractal dimension serves as a numerical measure of the spatial complexity of the neuron and correlates with the morphofunctional organization of the cell.     

A direction‐selective local‐thresholding method,DSLT, in combination with a dye‐based method for automated three‐dimensional segmentation of cells and airspaces in developing leaves

Quantifying the anatomical data acquired from three‐dimensional (3D) images has become increasingly important in recent years. Visualization and image segmentation are essential for acquiring accurate and detailed anatomical data from images; however, plant tissues such as leaves are difficult to image by confocal or multi‐photon laser scanning microscopy because their airspaces generate optical aberrations. To overcome this problem, we established a staining method based on Nile Red in silicone‐oil solution. Our staining method enables color differentiation between lipid bilayer membranes and airspaces, while minimizing any damage to leaf development. By repeated applications of our staining method we performed time‐lapse imaging of a leaf over 5 days. To counteract the drastic decline in signal‐to‐noise ratio at greater tissue depths, we also developed a local thresholding method (direction‐selective local thresholding, DSLT) and an automated iterative segmentation algorithm. The segmentation algorithm uses the DSLT to extract the anatomical structures. Using the proposed methods, we accurately segmented 3D images of intact leaves to single‐cell resolution, and measured the airspace volumes in intact leaves.     

Document image binarisation using a supervised neural network

Advances in digital technologies have allowed us to generate more images than ever. Images of scanned documents are examples of these images that form a vital part in digital libraries and archives. Scanned degraded documents contain background noise and varying contrast and illumination, therefore, document image binarisation must be performed in order to separate foreground from background layers. Image binarisation is performed using either local adaptive thresholding or global thresholding; with local thresholding being generally considered as more successful. This paper presents a novel method to global thresholding, where a neural network is trained using local threshold values of an image in order to determine an optimum global threshold value which is used to binarise the whole image. The proposed method is compared with five local thresholding methods, and the experimental results indicate that our method is computationally cost-effective and capable of binarising scanned degraded documents with superior results.     

Automated image segmentation by fractal grey tone functions

A new image segmentation method is presented. It is based on mathematical derivations stemming from both mathematical morphology and fractal geometry. Structures of different nature are most often characterized by a fractal grey tone function, in a certain range of resolutions, with a particular value of the fractal dimension. Our method allows the isolation of such structures. It only requires a series of dilated and eroded grey tone images.     

Differences Help Recognition: A Probabilistic Interpretation

This paper presents a computational model to address one prominent psychological behavior of human beings to recognize images. The basic pursuit of our method can be concluded as that differences among multiple images help visual recognition. Generally speaking, we propose a statistical framework to distinguish what kind of image features capture sufficient category information and what kind of image features are common ones shared in multiple classes. Mathematically, the whole formulation is subject to a generative probabilistic model. Meanwhile, a discriminative functionality is incorporated into the model to interpret the differences among all kinds of images. The whole Bayesian formulation is solved in an Expectation-Maximization paradigm. After finding those discriminative patterns among different images, we design an image categorization algorithm to interpret how these differences help visual recognition within the bag-of-feature framework. The proposed method is verified on a variety of image categorization tasks including outdoor scene images, indoor scene images as well as the airborne SAR images from different perspectives.     

Fast rotational matching of single-particle images

The presence of noise and absence of contrast in electron micrographs lead to a reduced resolution of the final 3D reconstruction, due to the inherent limitations of single-particle image alignment. The fast rotational matching (FRM) algorithm was introduced recently for an accurate alignment of 2D images under such challenging conditions. Here, we implemented this algorithm for the first time in a standard 3D reconstruction package used in electron microscopy. This allowed us to carry out exhaustive tests of the robustness and reliability in iterative orientation determination, classification, and 3D reconstruction on simulated and experimental image data. A classification test on GroEL chaperonin images demonstrates that FRM assigns up to 13% more images to their correct reference orientation, compared to the classical self-correlation function method. Moreover, at sub-nanometer resolution, GroEL and rice dwarf virus reconstructions exhibit a remarkable resolution gain of 10-20% that is attributed to the novel image alignment kernel.