Fractal dimension of color fractal images

Fractal contours: Order, chaos, and art

Over the recent decades, a variety of indices, such as the fractal dimension, Hurst exponent, or Betti numbers, have been used to characterize structural or topological properties of art via a singular parameter, which could then help to classify artworks. A single fractal dimension, in particular, has been commonly interpreted as characteristic of the entire image, such as an abstract painting, whether binary, gray-scale, or in color, and whether self-similar or not. There is now ample evidence, however, that fractal exponents obtained using the standard box-counting are strongly dependent on the details of the method adopted, and on fitting straight lines to the entire scaling plots, which are typically nonlinear. Here, we propose a more discriminating approach with the aim of obtaining robust scaling plots and extracting relevant information encoded in them without any fitting routines. To this goal, we carefully average over all possible grid locations at each scale, rendering scaling plots independent of any particular choice of grids and, crucially, of the orientation of images. We then calculate the derivatives of the scaling plots, so that an image is described by a continuous function, its fractal contour, rather than a single scaling exponent valid over a limited range of scales. We test this method on synthetic examples, ordered and random, then on images of algorithmically defined fractals, and finally, examine selected abstract paintings and prints by acknowledged masters of modern art.

Classification of Multiple H&E Images via an Ensemble Computational Scheme

In this work, a computational scheme is proposed to identify the main combinations of handcrafted descriptors and deep-learned features capable of classifying histological images stained with hematoxylin and eosin. The handcrafted descriptors were those representatives of multiscale and multidimensional fractal techniques (fractal dimension, lacunarity and percolation) applied to quantify the histological images with the corresponding representations via explainable artificial intelligence (xAI) approaches. The deep-learned features were obtained from different convolutional neural networks (DenseNet-121, EfficientNet-b2, Inception-V3, ResNet-50 and VGG-19). The descriptors were investigated through different associations. The most relevant combinations, defined through a ranking algorithm, were analyzed via a heterogeneous ensemble of classifiers with the support vector machine, naive Bayes, random forest and K-nearest neighbors algorithms. The proposed scheme was applied to histological samples representative of breast cancer, colorectal cancer, oral dysplasia and liver tissue. The best results were accuracy rates of 94.83% to 100%, with the identification of pattern ensembles for classifying multiple histological images. The computational scheme indicated solutions exploring a reduced number of features (a maximum of 25 descriptors) and with better performance values than those observed in the literature. The presented information in this study is useful to complement and improve the development of computer-aided diagnosis focused on histological images.

On the Quantification of Visual Texture Complexity

Complexity is one of the major attributes of the visual perception of texture. However, very little is known about how humans visually interpret texture complexity. A psychophysical experiment was conducted to visually quantify the seven texture attributes of a series of textile fabrics: complexity, color variation, randomness, strongness, regularity, repetitiveness, and homogeneity. It was found that the observers could discriminate between the textures with low and high complexity using some high-level visual cues such as randomness, color variation, strongness, etc. The results of principal component analysis (PCA) on the visual scores of the above attributes suggest that complexity and homogeneity could be essentially the underlying attributes of the same visual texture dimension, with complexity at the negative extreme and homogeneity at the positive extreme of this dimension. We chose to call this dimension visual texture complexity. Several texture measures including the first-order image statistics, co-occurrence matrix, local binary pattern, and Gabor features were computed for images of the textiles in sRGB, and four luminance-chrominance color spaces (i.e., HSV, YC_bC_r, Ohta’s I₁I₂I₃, and CIELAB). The relationships between the visually quantified texture complexity of the textiles and the corresponding texture measures of the images were investigated. Analyzing the relationships showed that simple standard deviation of the image luminance channel had a strong correlation with the corresponding visual ratings of texture complexity in all five color spaces. Standard deviation of the energy of the image after convolving with an appropriate Gabor filter and entropy of the co-occurrence matrix, both computed for the image luminance channel, also showed high correlations with the visual data. In this comparison, sRGB, YC_bC_r, and HSV always outperformed the I₁I₂I₃ and CIELAB color spaces. The highest correlations between the visual data and the corresponding image texture features in the luminance-chrominance color spaces were always obtained for the luminance channel of the images, and one of the two chrominance channels always performed better than the other. This result indicates that the arrangement of the image texture elements that impacts the observer’s perception of visual texture complexity cannot be represented properly by the chrominance channels. This must be carefully considered when choosing an image channel to quantify the visual texture complexity. Additionally, the good performance of the luminance channel in the five studied color spaces proves that variations in the luminance of the texture, or as one could call the luminance contrast, plays a crucial role in creating visual texture complexity.

Color Image Complexity versus Over-Segmentation: A Preliminary Study on the Correlation between Complexity Measures and Number of Segments

It is said that image segmentation is a very difficult or complex task. First of all, we emphasize the subtle difference between the notions of difficulty and complexity. Then, in this article, we focus on the question of how two widely used color image complexity measures correlate with the number of segments resulting in over-segmentation. We study the evolution of both the image complexity measures and number of segments as the image complexity is gradually decreased by means of low-pass filtering. In this way, we tackle the possibility of predicting the difficulty of color image segmentation based on image complexity measures. We analyze the complexity of images from the point of view of color entropy and color fractal dimension and for color fractal images and the Berkeley data set we correlate these two metrics with the segmentation results, more specifically the number of quasi-flat zones and the number of JSEG regions in the resulting segmentation map. We report on our experimental results and draw conclusions.

Novel digital image analysis using fractal dimension for assessment of skin radiance

BACKGROUND

Despite a strong desire to quantify skin radiance in the field of cosmetics, there does not exist a robust method to characterize it. Classical shine that quantifies the specular reflection from skin has been commonly used as the metric to characterize radiance. However, it does not always correlate with the perceived radiance as there are many other parameters that inform radiance perception including spatial distribution of shine and color homogeneity.

MATERIALS AND METHODS

In this work, we propose a novel method using fractal analysis to better characterize radiance by considering the spatial heterogeneity of pixel intensities as well as color evenness. A simulated image library (nine images) from very dull to very bright was created using bare face images of 20 panelists. Product images taken post-product usage were ranked along this library by finding the image in the library that most resembles the product image by our algorithm as well as experts. Additionally, classical shine and color measurements were made as benchmarks.

RESULTS

Our results confirm a strong correlation (R²  = 0.99) between the expert radiance rankings and the rankings by fractal dimension algorithm. The new algorithm offers an improved product differentiation compared with classical shine or color measurements.

CONCLUSION

Fractal dimension calculation offers higher sensitivity and resolution compared with other descriptors such as classical shine or color heterogeneity. In cases where the image rank is dominated by pixel intensities rather than color evenness, the image ranks resulting from calculating the fractal dimension is comparable with use of classical shine as the ranking parameter.

Multidimensional and fuzzy sample entropy (SampEnMF) for quantifying H&E histological images of colorectal cancer

In this study, we propose to use a method based on the combination of sample entropy with multiscale and multidimensional approaches, along with a fuzzy function. The model was applied to quantify and classify H&E histological images of colorectal cancer. The multiscale approach was defined by analysing windows of different sizes and variations in tolerance for determining pattern similarity. The multidimensional strategy was performed by considering each pixel in the colour image as an n-dimensional vector, which was analysed from the Minkowski distance. The fuzzy strategy was a Gaussian function used to verify the pertinence of the distances between windows. The result was a method capable of computing similarities between pixels contained in windows of various sizes, as well as the information present in the colour channels. The power of quantification was tested in a public colorectal image dataset, which was composed of both benign and malignant classes. The results were given as inputs for classifiers of different categories and analysed by applying the k-fold cross-validation and holdout methods. The derived performances indicate that the proposed association was capable of distinguishing the benign and malignant groups, with values that surpassed those results obtained with important techniques available in the Literature. The best performance was an AUC value of 0.983, an important result, mainly when we consider the difficulties of clinical practice for the diagnosis of the colorectal cancer.

Comparative study on fractal analysis of interferometry images with application to tear film surface quality assessment

The purpose of this research was to evaluate the applicability of the fractal dimension (FD) estimators to assess lateral shearing interferometric (LSI) measurements of tear film surface quality. Retrospective recordings of tear film measured with LSI were used: 69 from healthy subjects and 41 from patients diagnosed with dry eye syndrome. Five surface quality descriptors were considered, four based on FD and a previously reported descriptor operating in a spatial frequency domain (M₂), presenting temporal kinetics of post-blink tear film. A set of 12 regression parameters has been extracted and analyzed for classification purposes. The classifiers are assessed in terms of receiver operating characteristics and areas under their curves (AUC). Also, the computational loads are estimated. The maximum AUC of 82.4% was achieved for M₂, closely followed by the binary box-counting (BBC) FD estimator with AUC=78.6%. For all descriptors, statistically significant differences between the subject groups were found (p<0.05). The BBC FD estimator was characterized with the highest empirical computational efficiency that was about 30% faster than that of M₂, while that based on the differential box-counting exhibited the lowest efficiency (4.5 times slower than the best one). Concluding, FD estimators can be utilized for quantitative assessment of tear film kinetics. They provide a viable alternative to previously used spectral counter parameters, and at the same time allow higher computational efficiency.

Is a two-dimensional generalization of the Higuchi algorithm really necessary?

Fractal dimensions of data series, particularly time series can be estimated very well by using Higuchi's algorithm. Without phase space constructions, the fractal dimension of a one-dimensional data stream is calculated. Higuchi's method is well accepted and widely applied, because it is very reliable and easy to implement. A generalization of the genuine 1D algorithm to two dimensions would be desirable in order to investigate digital images. In this study, we propose several 2D generalization algorithms and evaluate differences between them. Additionally, a comparison to previously published pseudo 2D generalizations, and to the Fourier and the Blanket method are presented. The algorithms were tested on artificially generated grey value and red-green-blue colour images. It turned out that the proposed 2D generalized Higuchi algorithms are very robust, but differences in between the generalizations as well as differences to the pseudo 2D algorithms are astonishingly small.

Computer-Aided Prostate Cancer Diagnosis From Digitized Histopathology: A Review on Texture-Based Systems

Prostate cancer (PCa) is currently diagnosed by microscopic evaluation of biopsy samples. Since tissue assessment heavily relies on the pathologists level of expertise and interpretation criteria, it is still a subjective process with high intra- and interobserver variabilities. Computer-aided diagnosis (CAD) may have a major impact on detection and grading of PCa by reducing the pathologists reading time, and increasing the accuracy and reproducibility of diagnosis outcomes. However, the complexity of the prostatic tissue and the large volumes of data generated by biopsy procedures make the development of CAD systems for PCa a challenging task. The problem of automated diagnosis of prostatic carcinoma from histopathology has received a lot of attention. As a result, a number of CAD systems, have been proposed for quantitative image analysis and classification. This review aims at providing a detailed description of selected literature in the field of CAD of PCa, emphasizing the role of texture analysis methods in tissue description. It includes a review of image analysis tools for image preprocessing, feature extraction, classification, and validation techniques used in PCa detection and grading, as well as future directions in pursuit of better texture-based CAD systems.

Multifractal geometry in analysis and processing of digital retinal photographs for early diagnosis of human diabetic macular edema

OBJECTIVE

The purpose of this paper is to determine a quantitative assessment of the human retinal vascular network architecture for patients with diabetic macular edema (DME). Multifractal geometry and lacunarity parameters are used in this study.

MATERIALS AND METHODS

A set of 10 segmented and skeletonized human retinal images, corresponding to both normal (five images) and DME states of the retina (five images), from the DRIVE database was analyzed using the Image J software. Statistical analyses were performed using Microsoft Office Excel 2003 and GraphPad InStat software.

RESULTS

The human retinal vascular network architecture has a multifractal geometry. The average of generalized dimensions (Dq) for q = 0, 1, 2 of the normal images (segmented versions), is similar to the DME cases (segmented versions). The average of generalized dimensions (Dq) for q = 0, 1 of the normal images (skeletonized versions), is slightly greater than the DME cases (skeletonized versions). However, the average of D2 for the normal images (skeletonized versions) is similar to the DME images. The average of lacunarity parameter, Λ, for the normal images (segmented and skeletonized versions) is slightly lower than the corresponding values for DME images (segmented and skeletonized versions).

CONCLUSION

The multifractal and lacunarity analysis provides a non-invasive predictive complementary tool for an early diagnosis of patients with DME.

Color diffusion model for active contours - an application to skin lesion segmentation

Most of the existing diffusion models are defined for gray-scale images. We propose a diffusion model for color images to be used as external energy for active contours. Our diffusion model is based on the first-order moment of the correlation integral expressed using ΔE distances in the CIE Lab color space. We use a multi-scale approach for active contours, the diffusion being independently computed at various scales. We validate the model on synthetic images, including multi-fractal color textures, as well as medical images representing melanoma. We conclude that the proposed diffusion model is valid for use in skin lesion segmentation in color images using active contours.

Detection of prostate cancer on histopathology using color fractals and Probabilistic Pairwise Markov models

In this paper we present a system for detecting regions of carcinoma of the prostate (CaP) in H&E stained radical prostatectomy specimens using the color fractal dimension. Color textural information is known to be a valuable characteristic to distinguish CaP from benign tissue. In addition to color information, we know that cancer tends to form contiguous regions. Our system leverages the color staining information of histology as well as spatial dependencies. The color and textural information is first captured using color fractal dimension. To incorporate spatial dependencies, we combine the probability map constructed via color fractal dimension with a novel Markov prior called the Probabilistic Pairwise Markov Model (PPMM). To demonstrate the capability of this CaP detection system, we applied the algorithm to 27 radical prostatectomy specimens from 10 patients. A per pixel evaluation was conducted with ground truth provided by an expert pathologist using only the color fractal feature first, yielding an area under the receiver operator characteristic curve (AUC) curve of 0.790. In conjunction with a Markov prior, the resultant color fractal dimension + Markov random field (MRF) classifier yielded an AUC of 0.831.