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PubMed1991

Visual discrimination of fractal borders.

The ideas of fractals and fractal dimension are here translated into the realm of visual psychophysics. Borders between two fields of different luminance were used. Because of the finite grain of the visual system, fractal dimension need be defined only within a certain size range. For a fractal dimension of 1.15, the just-detectable difference in fractal dimension was found to be about 0.0085, rising to about 0.015 for a fractal dimension of 1.25. Reducing exposure duration from 1 s to 0.33 s decreases sensitivity to differences in fractal dimension, but there was no gain in increasing the exposure duration. Good visual observers who are naive to the task require some training before reaching optimal performance. The ability to discriminate fractal dimension differs between fractal edges of the same fractal dimension that were generated with differing statistical programs. Even after considerable training, an observer makes 29% errors when asked to distinguish a fractal edge generated with a Gaussian random walk from one with a rectangular random walk. Gaussian random walk fractals can be more easily distinguished from Poissonian and Cauchy ones.

Differential Threshold↗

PubMed1994

Measuring fractal dimensions. Sensitivity to edge-processing functions.

The fractal dimension is a useful tool in quantitative histology and cytology, and its measurement is easily implemented on computerized image analysis systems. However, the optimal conditions for capture of images and the effect of image-processing functions on the measurement of the fractal dimension have not been reported. Edge-processing functions were applied to images of Euclidean (square) and fractal (Koch island, renal angiogram) objects. The fractal dimension of processed images was measured using implementation of the box-counting method, and the area of thresholded image was also recorded. The method was shown to be accurate, with errors of < 1.5% for objects with known fractal dimensions, and highly reproducible, with a reliability coefficient of 0.972 (95% confidence limits of 0.868-0.987). The fractal dimension of the fractal images showed a marked (> 15%) reduction when a binary noise reduction function was applied with the minimum neighbors limit set above 3. In contrast, the fractal dimension of the Euclidean square was unchanged by this function. The reduction in fractal dimension was due to the erosion of complex convolutions at the edge of the fractal objects. Edge-processing functions should be avoided when manipulating images of fractal objects.

Angiography↗

PubMed1997

Influence of Different Parameters on a Dual-Fractal Analysis for Antigen-Antibody Binding Kinetics for Biosensor Applications

The diffusion-limited binding kinetics of antigen (or antibody) in solution to antibody (or antigen) immobilized on a biosensor surface is analyzed within a fractal framework. The fit obtained by a dual-fractal analysis is compared with that obtained from a single-fractal analysis. In some cases, the dual-fractal analysis provides an improved fit when compared with a single-fractal analysis. This was indicated by the regression analysis provided by Sigmaplot (46). It is of interest to note that the state of disorder (or the fractal dimension) and the binding rate coefficient both increase as the reaction progresses on the biosensor surface. For example, for the binding of HIV-1 p24 in solution to monoclonal antibody (MAb) 18 covalently attached to a biosensor surface (49), an increase in the fractal dimension by 59% from a value of Df1 equal to 1.91 to Df2 equal to 2.95 leads to an increase in the binding rate coefficient by a factor of 15 from k1 equal to 21.1 to k2 equal to 339. Also, the binding of MAb 6301 and 6303 in solution to insulin growth factor binding protein-1 (IGFBP-1) covalently attached to the sensor surface is adequately described by a single-fractal analysis (48). The binding of MAb 6302 to IGFBP-1, however, requires dual fractals. This indicates a difference in the binding mechanisms of these MAbs. The different examples analyzed and presented together provide a means by which the antigen-antibody reactions may be better controlled by noting the magnitude of the changes in the fractal dimension and in the binding rate coefficient as the reaction progresses on the biosensor surface. Also, the magnitude of the changes in the binding rate coefficients (k1 and k2) and in the fractal dimensions (Df1 and Df2) as different parameters are changed for the different biosensor applications are of particular value.

Journal Article↗

PubMedDate not supplied

Application of fractal geometry techniques to the study of trabecular bone.

It is well recognized that both trabecular bone density and structure affect the overall bone quality and strength. In this study the aim is to quantify variations in the structural network of trabeculae using the concepts of fractal geometry. Fractal objects are objects that appear statistically similar over a range of scales. Typically fractals do not have smooth surfaces, but instead have rather complex structures with highly convoluted surfaces. Associated with every fractal is a characteristic dimension, called the fractal dimension. In this study, techniques of fractal analysis were used to characterize the trabecular bone matrix on digital images acquired by quantitative computed tomography (QCT) of dried excised human vertebral bodies (density ranging from 76-220 mg/cc) and photomicrography of transiliac crest biopsies. An automatic boundary tracking algorithm was used to identify the trabecular-bone and bone marrow interface, and a box-counting algorithm was used to estimate the fractal dimension of the trabecular boundary. Using this technique for fractal objects, if the boundary being analyzed is covered with boxes of differing sizes, epsilon, then the number of boxes N required to cover the surface increases indefinitely according to the relation N = epsilon-D, where D is the fractal dimension. Using this relationship in a preliminary study on five specimens we have found that the trabecular-bone boundary is fractal in nature. Using photomicrographs of iliac crest biopsies, it is found that the fractal dimension changes with the fractional trabecular bone content.(ABSTRACT TRUNCATED AT 250 WORDS)

Bone Density↗

PubMed2001

Link between truncated fractals and coupled oscillators in biological systems.

This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.

Animals↗

PubMed1995

Fractal correlation in human H-reflex.

The amplitude of the H-reflex has been known to have considerable variability even if the intensity of the stimulation is held constant. However, previous studies largely ignored the time-dependent profile of this variability. Recent mathematical analyses have shown that some seemingly irregular biological signals have fractal properties. A fractal time series is characterized by the property of self-similarity (self-affinity), and has long-range time correlation. The aim of this study was to investigate the question of whether the fluctuation of H-reflex was fractal with strong time-correlation. Soleus H-reflexes were evoked in five healthy subjects at two levels of stimulation intensity [1.2 MT (motor threshold) and 0.9 MT] every 1 s and 1050 successive amplitudes of H-wave and M-wave were recorded twice. The sequences of the H-wave and the M-wave amplitudes were analyzed by "coarse graining spectral analysis" to calculate the percentage of random fractal components in the sequences (%Fractal) and the spectral exponent beta. The %Fractal values of both sequences were above 90% [H-wave: 93.3 +/- 2.3% (1.2 MT), 91.6 +/- 3.1% (0.9 MT); M-wave: 94.3 +/- 3.3%; mean +/- SD]. Nonflat power spectra of the fractal components were observed for the H-wave sequences regardless of the stimulation intensity [beta = 0.75 +/- 0.26 (1.2 MT), 0.80 +/- 0.39 (0.9 MT)], indicating that the sequences were strongly time correlated. On the other hand, the M-wave sequences had a flatter spectrum (beta = 0.26 +/- 0.14) which was close to uncorrelated white noise. These results indicated that: (1) the fractal correlation found in the H-wave sequences was caused neither by the conduction through nerve fibers nor by the transmission at the neuromuscular junction, because the M-wave sequence had a significantly weaker time correlation, and (2) antidromic impulses in a motor nerve induced by the stimulation made a minor contribution to the generation of fractal correlation in the H-wave sequences, because it was preserved when the stimulation intensity was below MT. It was suggested that the fractal correlation in human H-reflex was generated at the synaptic connections to alpha-motoneurons in the spinal cord.

Adult↗

PubMed2026

Complexity in disguise: a systematic review of fractal analysis in psychiatric neuroimaging.

OBJECTIVES: Psychiatric diagnosis and fractal studies are complex processes that extend beyond clinical evaluation and require careful methodological considerations in neuroimaging. Over the years, fractals have helped reduce these complexities in research, but they still cannot grant clinical diagnoses. Thus, the main objective was a systematic review exploring the potential applications of fractal analysis in characterizing psychiatric conditions through neuroimaging techniques-including both functional and structural MRI. MATERIALS AND METHODS: A systematic literature review was conducted on PubMed, identifying thirty-nine original studies that met the inclusion criteria. Areas showing statistical significance (p < 0.05) were reported. These studies were categorized according to DSM-V classification and examined for the description of psychiatric conditions through the fractal analysis. RESULTS: The review primarily focuses on young adults with psychiatric conditions compared to control groups. Schizophrenia and Autism Spectrum Disorder are major areas of investigation, and fractal dimension (FD) is the primary analysis method used to reflect brain patterns. Studies that calculated whole-brain FD may have underestimated local abnormalities due to the inclusion of a high percentage of tissue, potentially resulting in overlooked findings. Notably, abnormalities in the frontal cortex represent a common neurobiological feature across several psychiatric conditions. CONCLUSIONS: The findings from this systematic review shed light on the use of fractal analysis to quantify complex brain patterns in both psychiatric patients and healthy individuals. However, it is essential to recognize the need for further research to elucidate a fractal analysis protocol that allows for optimal extraction of psychiatric insights. KEY POINTS: Question Fractal analysis applied to structural and functional MRI help characterize brain alterations across psychiatric conditions. Findings This review shows consistent fractal patterns across multiple psychiatric disorders, especially in frontal regions. Despite heterogeneous methodologies, results highlight shared structural and functional abnormalities. Clinical relevance Fractal analysis may offer complementary characterization of subtle brain organization across psychiatric disorders. Its potential clinical utility-such as improving diagnostic characterization, earlier detection, among others-remains limited by the current absence of a standardized protocol.

Humans↗

PubMed1998

Fractal properties of cancellous bone of the iliac crest in vertebral crush fracture.

Fractal analysis is a method for describing complex shapes, including the cancellous structure of bone. It describes the surface texture and form of individual trabecular profiles and the overall cancellous structure. Sixty-four postmenopausal women with symptoms of back pain were referred for investigation for osteoporosis. The patients were divided into two groups for comparison: vertebral crush fracture (n = 31, mean age 68.58 +/- 6.47 years), and no vertebral crush fracture (n = 33, mean age 63.36 +/- 7.21 years). Cores of cancellous bone, 3 mm in diameter, were taken from the iliac crest and sectioned. A box-counting method implemented on an image analyzer was used to measure the fractal dimension. Three fractal dimensions describing trabecular surface texture (fractal 1), trabecular shape (fractal 2), and trabecular arrangement (fractal 3) were measured, indicating that cancellous bone has sectional self-similarity. Conventional histomorphometry was also performed on the samples. The results show that fractal 2 is significantly lower in the vertebral crush fracture group than in the nonfracture group (1.15 +/- 0.10 < 1.23 +/- 0.090, p < 0.0013). The histomorphometric analysis shows that bone surface total volume (p < 0.0002), trabecular number (p < 0.0001), and osteoid surface bone surface (p < 0.028) are significantly lower in the fracture group than the nonfracture group. Eroded surface/bone surface (p < 0.056) follows this trend, whereas trabecular separation (p < 0.001) is significantly higher in the fracture group than in the nonfracture group. Fractal 1 and fractal 3 were not significantly different between study groups. The fractal dimension detects changes in the cancellous architecture and gives information about iliac bone transformation in postmenopausal women with vertebral fracture.

Aged↗

PubMed2000

Methodological principles for fractal analysis of trabecular bone.

A standardized methodology for the fractal analysis of histological sections of trabecular bone has been established. A modified box counting method has been developed for use on a PC-based image analyser. The effect of image analyser settings, magnification, image orientation and threshold levels was determined. Also, the range of scale over which trabecular bone is effectively fractal was determined and a method formulated to calculate objectively more than one fractal dimension from the modified Richardson plot. The results show that magnification, image orientation and threshold settings have little effect on the estimate of fractal dimension. Trabecular bone has a lower limit below which it is not fractal (lambda < 25 microm) and the upper limit is 4250 microm. There are three distinct fractal dimensions for trabecular bone (sectional fractals), with magnitudes greater than 1.0 and less than 2.0. It has been shown that trabecular bone is effectively fractal over a defined range of scale. Also, within this range, there is more than one fractal dimension, describing spatial structural entities. Fractal analysis is a model-independent method for describing a complex multifaceted structure, which can be adapted for the study of other biological systems. This may be at the cell, tissue or organ level and complements conventional histomorphometric and stereological techniques.

Femur↗

PubMed2002

Fractal characterization of complexity in temporal physiological signals.

This review first gives an overview on the concept of fractal geometry with definitions and explanations of the most fundamental properties of fractal structures and processes like self-similarity, power law scaling relationship, scale invariance, scaling range and fractal dimensions. Having laid down the grounds of the basics in terminology and mathematical formalism, the authors systematically introduce the concept and methods of monofractal time series analysis. They argue that fractal time series analysis cannot be done in a conscious, reliable manner without having a model capable of capturing the essential features of physiological signals with regard to their fractal analysis. They advocate the use of a simple, yet adequate, dichotomous model of fractional Gaussian noise (fGn) and fractional Brownian motion (fBm). They demonstrate the importance of incorporating a step of signal classification according to the fGn/fBm model prior to fractal analysis by showing that missing out on signal class can result in completely meaningless fractal estimates. Limitation and precision of various fractal tools are thoroughly described and discussed using results of numerical experiments on ideal monofractal signals. Steps of a reliable fractal analysis are explained. Finally, the main applications of fractal time series analysis in biomedical research are reviewed and critically evaluated.

Algorithms↗

PubMed1996

Fractal analysis of mammographic lesions: a feasibility study quantifying the difference between benign and malignant masses.

The increased use of screening mammography has led to increased pressure to differentiate between benign and malignant lesions. Even those lesions considered "suspicious" by qualitative radiologists' interpretations may prove malignant in less than 30% of cases. Fractal analysis is a mathematical technique that quantifies complex shapes. The hypothesis tested is that fractal analysis can quantify the difference between the shapes of benign and malignant lesions as imaged by mammography. Ten mammograms from patients with biopsy-proven invasive ductal carcinoma and 10 mammograms from patients with biopsy-proven benign disease were compared using the box-counting technique of fractal analysis. The fractal dimension of the mediolateral and craniocaudal views were added together to derive the composite fractal dimension. Statistical analysis was done using the Mann-Whitney U test. The median composite fractal dimension for benign lesions was 1.831 (range 1.359-2.009) and for malignant lesions 2.477 (range 2.084-3.158) (P < 0.0001). In addition, all benign lesions had fractal dimensions < or = 2.009, and all malignant lesions had fractal dimensions > or = 2.084. In this sample of 10 mammograms of malignant lesions versus 10 mammograms of benign masses, the composite fractal dimension was perfectly discriminatory. Fractal analysis may be useful to evaluate mammographically discovered breast masses. A blinded, prospective trial will be needed to determine its ultimate usefulness.

Breast Neoplasms↗

PubMed2001

Three-dimensional tumor perfusion reconstruction using fractal interpolation functions.

It has been shown that the perfusion of blood in tumor tissue can be approximated using the relative perfusion index determined from dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool. Also, it was concluded in a previous report that the blood perfusion in a two-dimensional (2-D) tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. In this paper, the three-dimensional (3-D) tumor perfusion is reconstructed from the 2-D slices using the method of fractal interpolation functions (FIF), i.e., the piecewise self-affine fractal interpolation model (PSAFIM) and the piecewise hidden variable fractal interpolation model (PHVFIM). The fractal models are compared to classical interpolation techniques (linear, spline, polynomial) by means of determining the 2-D fractal dimension of the reconstructed slices. Using FIFs instead of classical interpolation techniques better conserves the fractal-like structure of the perfusion data. Among the two FIF methods, PHVFIM conserves the 3-D fractality better due to the cross correlation that exists between the data in the 2-D slices and the data along the reconstructed direction. The 3-D structures resulting from PHVFIM have a fractal dimension within 3%-5% of the one reported in literature for 3-D percolation. It is, thus, concluded that the reconstructed 3-D perfusion has a percolation-like scaling. As the perfusion term from bio-heat equation is possibly better described by reconstruction via fractal interpolation, a more suitable computation of the temperature field induced during hyperthermia treatments is expected.

Animals↗

PubMed2003

A novel approach to diagnose diabetes based on the fractal characteristics of retinal images.

A novel diagnostic scheme to develop quantitative indexes of diabetes is introduced in this paper. The fractal dimension of the vascular distribution is estimated because we discovered that the fractal dimension of a severe diabetic patient's retinal vascular distribution appears greater than that of a normal human's. The issue of how to yield an accurate fractal dimension is to use high-quality images. To achieve a better image-processing result, an appropriate image-processing algorithm is adopted in this paper. Another important fractal feature introduced in this paper is the measure of lacunarity, which describes the characteristics of fractals that have the same fractal dimension but different appearances. For those vascular distributions in the same fractal dimension, further classification can be made using the degree of lacunarity. In addition to the image-processing technique, the resolution of original image is also discussed here. In this paper, the influence of the image resolution upon the fractal dimension is explored. We found that a low-resolution image cannot yield an accurate fractal dimension. Therefore, an approach for examining the lower bound of image resolution is also proposed in this paper. As for the classification of diagnosis results, four different approaches are compared to achieve higher accuracy. In this study, the fractal dimension and the measure of lacunarity have shown their significance in the classification of diabetes and are adequate for use as quantitative indexes.

Algorithms↗

PubMed2004

Fractal analysis of surface growth patterns in endometrioid endometrial adenocarcinoma.

OBJECTIVE: A fractal is a shape made of parts similar to the whole in some way. The objective of this study was to determine whether surface growth patterns in endometrioid endometrial adenocarcinoma are fractal, and the mean fractal dimension differs according to histologic grades and depth of myometrial invasion. METHODS: After the images of photographs of 120 resected uteri with endometrial cancers were digitized, the fractal dimensions of surface of tumors were measured using a fractal analysis software. RESULTS: The mean fractal dimensions of surface growth patterns in G1, G2, and G3 adenocarcinoma were 2.318, 2.303, and 2.383, respectively. These values were significantly greater than the topological dimension of a surface (= 2). The value was significantly higher in G3 than in G2 (p = 0.03). And although not statistically significant, the value of G3 was greater than G1 (p = 0.10) and than (G1 and G2) group (p = 0.06). No significant difference nor tendency was found in the fractal dimension of the surface of the tumor according to depth of invasion. CONCLUSION: This study shows that the surface of endometrioid endometrial adenocarcinoma has a fractal structure, and the mean fractal dimension may differ according to histologic grades. Our report proposes a new way of looking at endometrial cancer pathology. We believe that fractal geometry gives insights into tumor morphology and becomes a useful tool for analyzing complex and irregular tumor growth patterns mathematically.

Adult↗

PubMed1997

Fractal properties of subchondral cancellous bone in severe osteoarthritis of the hip.

Primary osteoarthritis of the hip results in changes to the architecture of subchondral cancellous bone. These changes in architecture occur through the action of osteoclasts and osteoblasts in selectively removing and adding bone. The quantitative description of the bone architecture helps in understanding the etiology of primary osteoarthritis. Fractal analysis is a method for describing complex shapes, which is expressed numerically as the fractal dimension. A box counting method was used, where the perimeter of binary profiles of cancellous bone samples was measured for different box sizes. The fractal dimension was the absolute value of the slope of the straight line segments from the plot of the log number of boxes versus the log box size. Cancellous bone samples from two subchondral regions, superior and inferomedial, to the fovea were analyzed from primary severe osteoarthritic specimens taken following total hip replacement surgery (n = 19, aged 51-80 years) and autopsy controls (n = 25, aged 18-90 years). There were three straight line segments identified on the log-log plot, for each subject, indicating a fractal dimension over three different ranges of scale. The results show that in the superior region there is a highly significant difference between the groups (p < 0.0001) for fractal 1 and pivot point 2. The histomorphometry shows significant differences for bone volume/total volume, bone surface/total volume, trabecular separation, and osteoid surface/total volume between groups. In the inferomedial region fractal 1 and fractal 2 are significantly different. For the histomorphometry, trabecular thickness and eroded surface/total volume are significantly different between the groups. The pivot points, i.e., the box size at which the fractal dimension changes, were of similar magnitude to the trabecular thickness and trabecular separation. These data suggest that the fractal geometry analysis of cancellous bone identifies architectural features not easily recognized by conventional bone histomorphometry. The fractal dimension is a descriptor of bone structure which simplifies the description of a complex structure and enables changes in cancellous bone architecture, due to disease, to be identified.

Adolescent↗

PubMed1998

A comparison of 2 patient populations using fractal analysis.

This study was undertaken to demonstrate that the fractal dimensions calculated using digitized non-standardized, clinical radiographs of mandibular alveolar bone from a population of patients diagnosed with periodontitis are statistically different from fractal dimensions calculated from another population diagnosed as having gingivitis or healthy gingiva. The fractal dimension was calculated using a public domain fractal analysis program distributed by the National Institutes of Health (NIH). Fractal dimensions were calculated from digitized clinical radiographs for 29 patients diagnosed with healthy gingiva and/or gingivitis and 32 patients diagnosed with periodontitis and compared. To estimate the reproducibility of the technique, we recalculated the fractal dimension from images of the gingivitis patients 3 months after the original calculations and compared them to the originals. A 2 sample, 2-tailed Student t test showed the gingivitis data group to be different from the periodontitis data group (P = 0.0012). The original gingivitis and repeat gingivitis groups fractal dimension calculation were the same and analysis showed the two data sets were not significantly different (P = 0.99). We found that: 1) fractal dimensions could be used to distinguish between gingivitis and periodontitis patient groups; 2) fractal dimensions could be calculated from non-standardized clinical radiographs; and 3) fractal dimensions for gingivitis patients were reproducible over a 3-month period.

Adult↗

PubMed1995

Retinal neurons and vessels are not fractal but space-filling.

Many branched patterns in nature are hypothesized to be fractal, i.e., statistically self-similar across a range of scales. We tested this hypothesis on the two-dimensional arbors of retinal neurons and blood vessels. First, we measured fractalness on synthetic fractal and nonfractal patterns. The synthetic fractal patterns exhibited self-similarity over a decade of scale, but the nonfractal "controls" showed hardly any self-similarity. Neuronal and vascular patterns showed no greater self-similarity than the controls. Second, we manipulated a synthetic fractal pattern to remove its self-similarity and found this to be reflected in a loss of measured fractalness. The same manipulation of the nonfractal control and also of the neural and vascular patterns did not alter their measured fractalness. Third, we "grew" patterns of branched line segments according to a variety of nonfractal algorithms. These patterns were, if anything slightly more fractal than the neural and vascular patterns. We conclude that the biological patterns studied here are not fractal. Finally, we measured extended versions of these patterns: a contiguous array of homotypic neuron arbors and a vascular pattern with a high degree of total detail. These patterns showed a "fractal dimension" of 2, which implies that down to some cut-off scale they fill space completely. Thus, neural and vascular patterns might best be described as quasi-regular lattices.

Animals↗

PubMed2000

Structural complexity in mussel beds: the fractal geometry of surface topography.

Seafloor topographic complexity is ecologically important because it provides habitat structure and alters boundary-layer flow over the bottom. Despite its importance, there is little agreement on how to define and measure surface complexity. The purpose of this investigation was to utilize fractal geometry of vertical cross-section profiles to characterize the surface topography of the soft-bottom mussel bed (Mytilus edulis L.) at Bob's Cove, ME, USA. Mussels there have been shown previously to have spatially ordered fractal characteristics in the horizontal plane. Two hypotheses were tested. The first was that the bed surface is fractal over the spatial scale of 1.44-200 mm, with fractal dimension less than or equal to 1.26, the value for the Koch curve, our model for bed profiles. The second was that bed surface topography (i.e., in vertical profile) is less complex than the mussel bed spatial pattern (i.e., aerial view in the horizontal plane). Both hypotheses were supported. Cross-sections of plaster casts of the bed produced 88 surface profiles, all of which were fractal over the entire spatial scale of more two orders of magnitude employed in the analysis. Fractal dimension values (D) for individual profiles ranged from 1.031 to 1.310. Fractal dimensions of entire casts ranged up to mean (1.242+/-0.046) and median (1.251) values similar to 1.26, the theoretical value of the Koch curve. The bed surface was less complex than the bed spatial pattern because every profile had D<1.36, the smallest value previously obtained from aerial views of the bed. The investigation demonstrated for the first time that surface topography of a soft-bottom mussel bed was fractal at a spatial scale relevant to hydrodynamic processes and habitat structure important for benthic organisms. The technique of using cross-section profiles from casts of the bed surface avoided possible underestimates of fractal dimension that can result from other profiling methods reported in the literature. The results demonstrate that fractal dimension can be useful in the analysis of habitat space and water flow over any irregular seafloor surface because it incorporates the size, shape, and scale of roughness elements into a simple, numerical metric.

Journal Article↗