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There exist several methods for calculating the fractal dimension of objects represented as 2D digital images. For example, Box counting, Minkowski dilation or Fourier analysis can be employed. However, there appear to be some limitations. It is not possible to calculate only the fractal dimension of an irregular region of interest in an image or to perform the calculations in a particular direction along a line on an arbitrary angle through the image. The calculations must be made for the whole image. In this paper, a new method to overcome these limitations is proposed. 2D images are appropriately prepared in order to apply 1D signal analyses, originally developed to investigate nonlinear time series. The Higuchi dimension of these 1D signals is calculated using Higuchi's algorithm, and it is shown that both regions of interests and directional dependencies can be evaluated independently of the whole picture. A thorough validation of the proposed technique and a comparison of the new method to the Fourier dimension, a common two dimensional method for digital images, are given. The main result is that Higuchi's algorithm allows a direction dependent as well as direction independent analysis. Actual values for the fractal dimensions are reliable and an effective treatment of regions of interests is possible. Moreover, the proposed method is not restricted to Higuchi's algorithm, as any 1D method of analysis, can be applied.

Physiological processes are regulated by nonlinear dynamical systems. Various nonlinear measures have frequently been used for characterizing the complexity of fractal time signals to detect system features that cannot be derived from linear analyses. We analysed human balance dynamics ranging from simple standing to balancing on one foot with closed eyes to study the inherent methodological problems when applying fractal dimension analysis to real-world signals. Higuchi dimension was used as an example. Choice of measurement and analysis parameters has a distinct influence on the computed dimension. Noise increases the fractional dimension which may be misinterpreted as a higher complexity of the signal. Publications without specifying the parameter setting, or without analysing the noise-sensitivity are not comparable to findings of others and therefore of limited scientific value.

Complex systems such as the global climate, biological organisms, civilisation, technical or social networks exhibit diverse behaviours at various temporal and spatial scales, often characterized by nonlinearity, feedback loops, and emergence. These systems can be characterized by physical quantities such as entropy, information, chaoticity or fractality rather than classical quantities such as time, velocity, energy or temperature. The drawback of these complexity quantities is that their definitions are not always mathematically exact and computational algorithms provide estimates rather than exact values. Typically, evaluations can be cumbersome, necessitating specialized tools. We are therefore introducing ComsystanJ, a novel and user-friendly software suite, providing a comprehensive set of plugins for complex systems analysis, without the need for prior programming knowledge. It is platform independent, end-user friendly and extensible. ComsystanJ combines already known algorithms and newer methods for generalizable analysis of 1D signals, 2D images and 3D volume data including the generation of data sets such as signals and images for testing purposes. It is based on the framework of the open-source image processing software Fiji and ImageJ2. ComsystanJ plugins are macro recordable and are maintained as open-source software. ComsystanJ includes effective surrogate analysis in all dimensions to validate the features calculated by the different algorithms. Future enhancements of the project will include the implementation of parallel computing for image stacks and volumes and the integration of artificial intelligence methods to improve feature recognition and parameter calculation.

Complex systems such as the global climate, biological organisms, civilisation, technical or social networks exhibit diverse behaviours at various temporal and spatial scales, often characterized by nonlinearity, feedback loops, and emergence. These systems can be characterized by physical quantities such as entropy, information, chaoticity or fractality rather than classical quantities such as time, velocity, energy or temperature. The drawback of these complexity quantities is that their definitions are not always mathematically exact and computational algorithms provide estimates rather than exact values. Typically, evaluations can be cumbersome, necessitating specialized tools. We are therefore introducing ComsystanJ, a novel and user-friendly software suite, providing a comprehensive set of plugins for complex systems analysis, without the need for prior programming knowledge. It is platform independent, end-user friendly and extensible. ComsystanJ combines already known algorithms and newer methods for generalizable analysis of 1D signals, 2D images and 3D volume data including the generation of data sets such as signals and images for testing purposes. It is based on the framework of the open-source image processing software Fiji and ImageJ2. ComsystanJ plugins are macro recordable and are maintained as open-source software. ComsystanJ includes effective surrogate analysis in all dimensions to validate the features calculated by the different algorithms. Future enhancements of the project will include the implementation of parallel computing for image stacks and volumes and the integration of artificial intelligence methods to improve feature recognition and parameter calculation.

The ever decreasing area of forests has lead to environmental and economical challenges and has brought with it a renewed interest in developing methodologies that quantify the extent of deforestation and reforestation. In this study we analyzed the deforested areas of the Apuseni Mountains, which has been under economic pressure in recent years and resulted in widespread deforestation as a means of income. Deforested surface dynamics modeling was based on images contained in the Global Forest Database, provided by the Department of Geographical Sciences at Maryland University between 2000 and 2014. The results of the image particle analysis and modelling were based on Total Area (ha), Count of patches and Average Size whereas deforested area distribution was based on the Local Connected Fractal Dimension, Fractal Fragmentation Index and Tug-of-War Lacunarity as indicators of forest fragmentation or heterogeneity. The major findings of the study indicated a reduction of the tree cover area by 3.8%, an increase in fragmentation of 17.7% and an increase in heterogeneity by 29%, while fractal connectivity decreased only by 0.1%. The fractal and particle analysis showed a clustering of forest loss areas with an average increase from 1.1 to 3.0 ha per loss site per year. In conclusion, the fractal and particle analysis provide a relevant methodological framework to further our understanding of the spatial effects of economic pressure on forestry.

The fractal dimension (FD) can be used as a measure for morphological complexity in biological systems. The aim of this study was to test the usefulness of this quantitative parameter in the context of cerebral vascular complexity. Fractal analysis was applied on ten patients with cerebral arteriovenous malformations (AVM) and ten healthy controls. Maximum intensity projections from Time-of-Flight MRI scans were analyzed using different measurements of FD, the Box-counting dimension, the Minkowski dimension and generalized dimensions evaluated by means of multifractal analysis. The physiological significance of this parameter was investigated by comparing values of FD first, with the maximum slope of contrast media transit obtained from dynamic contrast-enhanced MRI data and second, with the nidus size obtained from X-ray angiography data. We found that for all methods, the Box-counting dimension, the Minkowski dimension and the generalized dimensions FD was significantly higher in the hemisphere with AVM compared to the hemisphere without AVM indicating that FD is a sensitive parameter to capture vascular complexity. Furthermore we found a high correlation between FD and the maximum slope of contrast media transit and between FD and the size of the central nidus pointing out the physiological relevance of FD. The proposed method may therefore serve as an additional objective parameter, which can be assessed automatically and might assist in the complex workup of AVMs.

ContextForest loss and fragmentation pose extreme threats to biodiversity. Their efficient characterization from remotely sensed data therefore has strong practical implications. Data are often separately analyzed for spatial fragmentation and disorder, but no existing metric simultaneously quantifies both the shape and arrangement of fragments.ObjectivesWe present a fractal fragmentation and disorder index (FFDI), which advances a previously developed fractal index by merging it with the Rényi information dimension. The FFDI is designed to work across spatial scales, and to efficiently report both the fragmentation of images and their spatial disorder.MethodsWe validate the FFDI with 12,600 synthetic hierarchically structured random map (HRM) multiscale images, as well as several other categories of fractal and non-fractal test images (4880 images). We then apply the FFDI to satellite imagery of forest cover for 10 distinct regions of the Romanian Carpathian Mountains from 2000–2021.ResultsThe FFDI outperformed its two individual components (fractal fragmentation index and Rényi information dimension) in resolving spatial patterns of disorder and fragmentation when tested on HRM classes and other image types. The FFDI thus offers a clear advantage when compared to the individual use of fractal fragmentation index and the Information Dimension, and provided good classification performance in an application to real data.ConclusionsThis work improves on previous characterizations of landscape patterns. With the FFDI, scientists will be able to better monitor and understand forest fragmentation from satellite imagery. The FFDI may also find wider applicability in biology wherever image analysis is used.

The complexity in the styles of 1200 Byzantine icons painted between 13th and 16th from Greece, Russia and Romania was investigated through the Kolmogorov algorithmic information theory. The aim was to identify specific quantitative patterns which define the key characteristics of the three different painting schools. Our novel approach using the artificial surface images generated with Inverse FFT and the Midpoint Displacement (MD) algorithms, was validated by comparison of results with eight fractal and non-fractal indices. From the analyzes performed, normalized Kolmogorov compression complexity (KC) proved to be the best solution because it had the best complexity pattern differentiations, is not sensitive to the image size and the least affected by noise. We conclude that normalized KC methodology does offer capability to differentiate the icons within a School and amongst the three Schools.

The assessment of forest health and terrain usability is closely tied to slope accessibility. Current methods for evaluating terrain accessibility based solely on slope characteristics often lack precision and fail to capture the combined effects of topography and vegetation. This study introduces succolarity, together with succolarity reservoir and delta (Δ) succolarity, as fractal-based measures for assessing undeveloped land accessibility. The analysis focused on two test areas: the Ceahlău Mountains and the Blaj–Vulpăr Hills. Results revealed lower accessibility values for the Ceahlău Mountains (0.01 to 0.23 for slopes of 0–5° and 0–30°) compared to the Blaj–Vulpăr Hills (0.035 to 0.598 for the same ranges). These significant contrasts demonstrate that terrain fragmentation and compact forests act as decisive constraints, with slope predominating in mountains and vegetation in hilly areas. The findings are valuable for environmental agencies, emergency services, and research groups studying land morphology and mobility. Practical applications include infrastructure planning, sustainable land-use management, and strategic operations in remote terrains. Incorporating additional datasets (e.g., hydrographic networks, seasonal vegetation) and refining methodologies will further enhance succolarity-based assessments, supporting sustainable development in challenging environments.