Showing papers on "Fractal dimension published in 1996"
TL;DR: In this article, the authors showed that fractal surfaces can be super water repellent (superwettable) when the surfaces are composed of hydrophobic (hydrophilic) materials.
Abstract: Wettability of fractal surfaces has been studied both theoretically and experimentally. The contact angle of a liquid droplet placed on a fractal surface is expressed as a function of the fractal dimension, the range of fractal behavior, and the contacting ratio of the surface. The result shows that fractal surfaces can be super water repellent (superwettable) when the surfaces are composed of hydrophobic (hydrophilic) materials. We also demonstrate a super-water-repellent fractal surface made of alkylketene dimer; a water droplet on this surface has a contact angle as large as 174°.
1,500 citations
TL;DR: In this article, asymptotic expansions of these integral forms are used to constrain a unified equation for small-angle scattering, and this approach is extended to describe structural limits to mass-fractal scaling at the persistence length.
Abstract: The Debye equation for polymer coils describes scattering from a polymer chain that displays Gaussian statistics. Such a chain is a mass fractal of dimension 2 as evidenced by a power-law decay of −2 in the scattering at intermediate q. At low q, near q ≃ 2π/Rg, the Debye equation describes an exponential decay. For polymer chains that are swollen or slightly collapsed, such as is due to good and poor solvent conditions, deviations from a mass-fractal dimension of 2 are expected. A simple description of scattering from such systems is not possible using the approach of Debye. Integral descriptions have been derived. In this paper, asymptotic expansions of these integral forms are used to describe scattering in the power-law regime. These approximations are used to constrain a unified equation for small-angle scattering. A function suitable for data fitting is obtained that describes polymeric mass fractals of arbitrary mass-fractal dimension. Moreover, this approach is extended to describe structural limits to mass-fractal scaling at the persistence length. The unified equation can be substituted for the Debye equation in the RPA (random phase approximation) description of polymer blends when the mass-fractal dimension of a polymer coil deviates from 2. It is also used to gain new insight into materials not conventionally thought of as polymers, such as nanoporous silica aerogels.
884 citations
TL;DR: In this article, the fractal dimension of the solid AKD surface was determined to be D ≈ 2.3 applying the box-counting method to the SEM images of the AKD cross section.
Abstract: Super water-repellent surfaces showing a contact angle of 174° for water droplets have been made of alkylketene dimer (AKD). Water droplets roll around without attachment on the super water-repellent surfaces when tilted slightly. The AKD is a kind of wax and forms spontaneously a fractal structure in its surfaces by solidification from the melt. The fractal surfaces of AKD repel a water droplet completely and show a contact angle larger than 170° without any fluorination treatments. Theoretical prediction of the wettability of the fractal surfaces has been given in the previous paper.3 The relationship between the contact angle of the flat surface θ and that of the fractal surface θf is expressed by the equation cos θf = (L/l)D-2 cos θ where (L/l)D-2 is the surface area magnification factor. The fractal dimension of the solid AKD surface was determined to be D ≈ 2.3 applying the box-counting method to the SEM images of the AKD cross section. L and l, which are the largest and the smallest size limits of ...
786 citations
TL;DR: In this article, the authors present a method for the determination of Fractal Dimension in Point Field Models, based on the Hausdorff measure and dimension of a point field.
Abstract: FRACTALS AND METHODS FOR THE DETERMINATION OF FRACTAL DIMENSIONS. Hausdorff Measure and Dimension. Deterministic Fractals. Random Fractals. Methods for the Empirical Determination of Fractal Dimension. THE STATISTICS OF SHAPES AND FORMS. Fundamental Concepts. Representation of Contours. Set Theoretic Analysis. Point Description of Figures. Examples. POINT FIELD STATISTICS. Fundamentals. Finite Point Fields. Poisson Point Fields. Fundamentals of the Theory of Point Fields. Statistics for Homogeneous Point Fields. Point Field Models. Appendices. References. Index.
577 citations
TL;DR: This paper defines the concept of the fractal dimension, D, as a measure of complexity and illustrates the two different general ways of quantitatively measuring D by length-related and mass-related methods.
554 citations
TL;DR: In this paper, the fractal and scale-free nature of interstellar CO clouds with power indices that are independent of distance has been investigated and the results are αL = 1 + D and αM = 1+ D/κ for interstellar fractal dimension D = 2.3 ± 0.3 and a value of κ in the range 2.4-3.7.
Abstract: Interstellar molecular clouds have power-law size L and mass M distributions of the form n(L) dL = L−αL dL and n(M) dM = M−αM dM, where M Lk is also a power law. These relations are shown to result from the fractal and scale-free nature of interstellar gas with power indices that are independent of distance. The results are αL = 1 + D and αM = 1 + D/κ for interstellar fractal dimension D = 2.3 ± 0.3 and a value of κ in the range 2.4-3.7, as determined from cloud surveys in the literature. The same fractal dimension also results from the expected relation D = κ when the M(L) correlation includes many different surveys, spanning a range of 1010 in mass. These results imply that interstellar CO clouds are the unresolved parts of a pervasive fractal structure in the interstellar gas. The similarity between n(M) for interstellar clouds and n(M) for globular clusters suggests that the clusters formed inside fractal progenitor clouds at a nearly constant efficiency.
479 citations
TL;DR: In this article, the Gauss sums of the Talbot image are given explicitly in closed form, and the phases of these individual images are given in closed-form; this simplifies calculations of Talbot images.
Abstract: Self-images of a grating with period a, illuminated by light of wavelength λ, are produced at distances z that are rational multiples p/q of the Talbot distance z T = a 2/λ; each unit cell of a Talbot image consists of q superposed images of the grating. The phases of these individual images depend on the Gauss sums studied in number theory and are given explicitly in closed form; this simplifies calculations of the Talbot images. In ‘transverse’ planes, perpendicular to the incident light, and with ζ = z/z T irrational, the intensity in the Talbot images is a fractal whose graph has dimension . In ‘longitudinal’ planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension . In certain special diagonal planes, the fractal dimension is . Talbot images are sharp only in the paraxial approximation λ/a → O and when the number N of illuminated slits tends to infinity. The universal form of the post-paraxial smoothing of the edge of the sli...
376 citations
TL;DR: In this paper, the authors demonstrate that the settling velocity models based on impermeable spheres do not accurately relate aggregate size, porosity and settling velocity for highly porous fractal aggregates.
Abstract: Aggregates generated in water and wastewater treatment systems and those found in natural systems are fractal and therefore have different scaling properties than assumed in settling velocity calculations using Stokes' law. In order to demonstrate that settling velocity models based on impermeable spheres do not accurately relate aggregate size, porosity and settling velocity for highly porous fractal aggregates, we generated fractal aggregates by coagulation of latex microspheres in paddle mixers and analyzed each aggregate individually for its size, porosity, and settling velocity. Settling velocities of these aggregates were on average 4−8.3 times higher than those predicted using either an impermeable sphere model (Stokes' law) or a permeable sphere model that specified aggregate permeability for a homogeneous distribution of particles within an aggregate. Fractal dimensions (D) derived from size−porosity relationships for the three batches of aggregates were 1.78 ± 0.10, 2.19 ± 0.12 and 2.25 ± 0.10. ...
320 citations
15 Jan 1996
TL;DR: It is shown that a reasonable estimate of the fluid drag force on a fractal aggregate may be obtained by assigning a constant volume-averaged porosity to the aggregate and using any of the expressions available in the literature for aggregates with uniform permeability.
Abstract: Hydrodynamic properties of fractal aggregates with radially varying permeability are investigated in terms of two parameters: radius of equivalent solid sphere experiencing the same drag as the aggregate (Ω*radius of aggregate) and fluid collection efficiency (η) of the aggregate. Resistance to the fluid flow through the aggregate is predicted to increase with increasing fractal dimension, while the fluid collection efficiency is expected to decrease. It is shown that a reasonable estimate of the fluid drag force on a fractal aggregate may be obtained by assigning a constant volume-averaged porosity to the aggregate and using any of the expressions available in the literature for aggregates with uniform permeability. The two hydrodynamic parameters, Ω and η, are used to modify the existing expressions for interactions between solid spheres to account for the porous nature of aggregates and thus calculate the collision rate kernels for interacting aggregates. The ratio of hydrodynamic radius of an aggregate to its radius of gyration predicted by the proposed model was in reasonable agreement with an experimental value reported in the literature.
226 citations
TL;DR: The results suggested that the RDG-FA theory is a reasonable approximation for optics of a wide range of fractal aggregates, considerably extending its domain of applicability.
Abstract: The range of validity of the Rayleigh–Debye–Gans approximation for the optical cross sections of fractal aggregates (RDG-FA) that are formed by uniform small particles was evaluated in comparison with the integral equation formulation for scattering (IEFS), which accounts for the effects of multiple scattering and self-interaction. Numerical simulations were performed to create aggregates that exhibit mass fractallike characteristics with a wide range of particle and aggregate sizes and morphologies, including xp = 0.01–1.0, |m − 1| = 0.1–2.0, N = 16–256, and Df = 1.0–3.0. The percent differences between both scattering theories were presented as error contour charts in the |m − 1|xp domains for various size aggregates, emphasizing fractal properties representative of diffusion-limited cluster–cluster aggregation. These charts conveniently identified the regions in which the differences were less than 10%, between 10% and 30%, and more than 30% for easy to use general guidelines for suitability of the RDG-FA theory in any scattering applications of interest, such as laser-based particulate diagnostics. Various types of aggregate geometry ranging from straight chains (Df ≈ 1.0) to compact clusters (Df ≈ 3.0) were also considered for generalization of the findings. For the present computational conditions, the RDG-FA theory yielded accurate predictions to within 10% for |m − 1| to approximately 1 or more as long as the primary particles in aggregates were within the Rayleigh scattering limit (xp ≤ 0.3). Additionally, the effect of fractal dimension on the performance of the RDG-FA was generally found to be insignificant. The results suggested that the RDG-FA theory is a reasonable approximation for optics of a wide range of fractal aggregates, considerably extending its domain of applicability.
214 citations
TL;DR: The theoretical predictions suggest that the exponent of the power law is directly related to a suitable fractal dimension of the boundaries, to the elongation of the basin, and to the scaling exponent of mainstream lengths.
Abstract: Seemingly unrelated empirical hydrologic laws and several experimental facts related to the fractal geometry of the river basin are shown to find a natural explanation into a simple finite-size scaling ansatz for the power laws exhibited by cumulative distributions of river basin areas. Our theoretical predictions suggest that the exponent of the power law is directly related to a suitable fractal dimension of the boundaries, to the elongation of the basin, and to the scaling exponent of mainstream lengths. Observational evidence from digital elevation maps of natural basins and numerical simulations for optimal channel networks are found to be in good agreement with the theoretical predictions. Analytical results for Scheidegger's trees are exactly reproduced.
TL;DR: In this paper, the authors examine the assumption that the estimated fractal dimension is constant over some biologically relevant range of spatial scales and conclude that ecologists should avoid calculating and using the fractal dimensions of movement paths unless self-similarity (a constant fractal degree) for some range of scales is demonstrated.
Abstract: Several recent papers developed and applied a novel approach for the analysis of animal movement paths, based on calculating the paths' fractal dimensions. The estimated fractal dimension is used to describe the pattern of the interaction between animal movement and landscape heterogeneity, and possibly to extrapolate movement patterns of organisms across spatial scales. Here, I critically examine the key assumption of the fractal approach: that the estimated fractal dimension is constant over some biologically relevant range of spatial scales. Use of a correlated random walk as a null hypothesis for movement suggests that the fractal dimension should grade smoothly from near 1 at very small spatial scales to near 2 at very large spatial scales. Several empirical data sets exhibit a qualitative pattern in agreement with this prediction. I conclude that ecologists should avoid calculating and using the fractal dimension of movement paths, unless self—similarity (a constant fractal dimension) for some range of spatial scales is demonstrated. An alternative approach employing random—walk models provides a more powerful framework for translating individual movements in heterogeneous space into spatial dynamics of populations.
TL;DR: In this article, the Sierpinsky gasket is described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor, and a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of different terms in the discrete renormalization group formalism introduced to tackle this problem.
Abstract: We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time-dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log-periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.
TL;DR: Comparison of the two polymeric flocculants revealed that both the fractal dimension and the tendency for ruptured fragments to reflocculate were sensitive to the polymer used.
TL;DR: In this paper, a quantum wave with probability density, confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state.
Abstract: A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all positions , the graph of the evolution of P is a fractal curve with dimension . For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension . When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension , simple arguments suggest that the dimension of the time fractal is , and that of the space fractal is .
TL;DR: In this article, a new algorithm is proposed for the automatic picking of seismic first arrivals that detects the presence of a signal by analyzing the variation in fractal dimension along the trace.
Abstract: A new algorithm is proposed for the automatic picking of seismic first arrivals that detects the presence of a signal by analyzing the variation in fractal dimension along the trace. The “divider-method” is found to be the most suitable method for calculating the fractal dimension. A change in dimension is found to occur close to the transition from noise to signal plus noise, that is the first arrival. The nature of this change varies from trace to trace, but a detectable change is always found to occur. The algorithm has been tested on real data sets with varying S/N ratios and the results compared to those obtained using previously published algorithms. With an appropriate tuning of its parameters, the fractal-based algorithm proved more accurate than all these other algorithms, especially in the presence of significant noise. The fractal method proved able to tolerate noise up to 80% of the average signal amplitude. However, the fractal-based algorithm is considerably slower than the other methods and hence is intended for use only on data sets with low S/N ratios.
TL;DR: In this article, the authors introduce some of the basic concepts of chaos theory, then details a method for quantifying a fractal dimension from a time series, the correlation dimension.
TL;DR: It is argued that the geometry of urban residential development is fractal, and two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected are introduced.
Abstract: In this paper, we argue that the geometry of urban residential development is fractal. Both the degree to which space is filled and the rate at which it is filled follow scaling laws which imply invariance of function, and self-similarity of urban form across scale. These characteristics are captured in population density functions based on inverse power laws whose parameters are fractal dimensions. First we outline the relevant elements of the theory in terms of scaling relations and then we introduce two methods for estimating fractal dimension based on varying the size of cities and the scale at which their form is detected. Exact and statistical estimation techniques are applied to each method respectively generating dimensions which measure the extent and the rate of space filling. These methods are then applied to residential development patterns in six industrial cities in the northeastern United States, with an innovative data source from the TIGER/Line files. The results support the theory of the...
TL;DR: A renormalized capillary length and density of the solid phase is introduced and scaling exponents for the growth velocity and the different length scales are expressed in terms of the fractal dimensions for surface and bulk of these fractal structures.
Abstract: The morphology diagram of possible structures in two-dimensional diffusional growth is given in the parameter space of undercooling \ensuremath{\Delta} versus anisotropy of surface tension \ensuremath{\epsilon}. The building block of the dendritic structure is a dendrite with parabolic tip, and the basic element of the seaweed structure is a doublon. The transition between these structures shows a jump in the growth velocity. We also describe the structures and velocities of fractal dendrites and doublons destroyed by noise. We introduce a renormalized capillary length and density of the solid phase and use scaling arguments to describe the fractal dendrites and doublons. The resulting scaling exponents for the growth velocity and the different length scales are expressed in terms of the fractal dimensions for surface and bulk of these fractal structures. All the considered structures are compact on length scales larger than the diffusion length and they show fractal behavior on intermediate length scales between the diffusion length and a small size cutoff which depends on the strength of noise. \textcopyright{} 1996 The American Physical Society.
TL;DR: Strengths of this new estimator are that it can estimate fractal d at different spatial scales, give an estimate of variation, and combine data from many separate path segments which have been gathered at various spatial scales.
Abstract: Fractal measurements of animal movement paths have been used to analyze how animals view habitats at different spatial scales. One problem has been the absence of error estimates for fractal d estimators. To address this weakness, I present and test 4 new estimators for measuring fractal dimension at different spatial scales, along with estimates of their variation. The estimators are based on dividing the movement path into pairs of steps, forming V's, and then estimating various statistics from each V.
TL;DR: In this paper, a general model of the water retention curve is derived for soils whose pore size distribution is fractal in the sense of the Mandelbrot number-size distribution.
Abstract: The relationship between water content and water potential for a soil is termed its water retention curve. This basic hydraulic property is closely related to the soil pore size distribution, for which it serves as a conventional method of measurement. In this paper a general model of the water retention curve is derived for soils whose pore size distribution is fractal in the sense of the Mandelbrot number-size distribution. This model, which contains two adjustable parameters (the fractal dimension and the upper limiting value of the fractal porosity) is shown to include other fractal approaches to the water retention curve as special cases. Application of the general model to a number of published data sets covering a broad range of soil texture indicated that unique, independent values of the two adjustable parameters may be difficult to obtain by statistical analysis of water retention data for a given soil. Discrimination among different fractal approaches thus will require water retention data of high density and precision.
01 Jan 1996
TL;DR: In this paper, the authors demonstrate that concepts derived from fractal and chaos theory are fundamental to the description and modelling of scalerelated phenomena in biology, from the molecular to ecosystem levels of organization.
Abstract: The importance of spatial and temporal scaling to the study of biological systems and processes has long been recognized. We demonstrate that concepts derived from fractal and chaos theory are fundamental to the description and modelling of scalerelated phenomena in biology, from the molecular to ecosystem levels of organization. Algorithms for estimating the fractal dimension are described, and numerous applications of fractal theory in the biological sciences are summarized.
Journal Article•
TL;DR: This work focuses on the application of the fractal approach to measuring irregular and complex features of pathologic cells and tissues and also on its potential role in the understanding of tumor biology.
TL;DR: The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty and provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
Abstract: The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P < 0.0001), BS/TV (r = 0.74, P < 0.0001), Tb.Th (r = 0.50, P < 0.02), Tb.Sp (r = -0.81, P < 0.0001), and Tb.N (r = 0.76, P < 0.0001). This method for calculating fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
TL;DR: In this article, the fractal component of a computer program, STRUCTURA, was used to measure fractal dimension of soil from images of soil thin sections, which were analyzed in order to obtain a range of fractal dimensions.
Abstract: Aspects of fractal geometry have been used to give quantitative measurements of soil structure. Fractal dimensions measured were the mass fractal dimension (D m ), surface fractal dimension (D s ), and the spectral dimension (d). We investigated the fractal component of a computer program, STRUCTURA, which measures the fractal dimension of soil from images of soil thin sections. Six thin sections, each showing different structural characteristics, were analyzed in order to ohtain a range of fractal dimensions. The dimensions, in particular D m and d, were shown to discriminate the different structures. The values of D m and d ranged from 1.682 to 1.852 and 1.236 to 1.668, respectively. A further objective was to use these results, together with fractal theory, to show the potential fractal geometry has in predicting physical processes such as diffusion within the soil. To assist with the interpretation of fractal dimensions, the dimensions of different soil samples with the same porosity were compared.
TL;DR: A theoretical comparison shows that algorithms invariant to these transformations of the image intensity values are restricted to Flat Structuring Element, Variogram, and Power Spectrum methods.
TL;DR: In this paper, the Weiestrass-Mandelbrot fractal function with known fractal dimensions was used to calculate the fractal dimension of a machined surface profile.
TL;DR: Shih et al. as discussed by the authors used fractal geometric relationships between the elastic moduli (G′) of the fats and their SFC values using the approach of Shih, W.H., WY. Kim, J. Liu, and I.A.
Abstract: The rheological properties of interesterified and noninteresterified butterfat-canola oil blends do not seem to be strongly related to either solid fat content (SFC) or crystal polymorphic behavior, but rather to the microstructure of the fat crystal network. The microstructure of the fats was quantified by using fractal geometric relationships between the elastic moduli (G′) of the fats and their SFC values using the approach of Shih, W.H., W.Y. Shih, S.I. Kim, J. Liu, and I.A. Aksay [Phys. Rev. A 42:4772–4779 (1990))] for weak-link regimes. Chemical interesterification decreased the fractal dimension of the fat crystal network from 2.46 to 2.15. We propose that fat microstructure, as quantified by a fractal dimensionality, could be modified to attain specific rheological properties.
TL;DR: The fractal dimension of the three measures of complexity used declined with stress, both from pregnancy and from parasitic infection, and provides a new and effective way to assess the general state of animals’ health in the field, without the need for capture and handling.
TL;DR: This work estimates the fractal dimension of the white matter surface for each hemisphere and 10 coronal blocks of each hemisphere in 30 normal adult subjects, and finds abnormal fractal dimensions were found in subjects with epilepsy and on routine MR imaging.
Abstract: The convolutions of the cerebral cortex are difficult to describe and delineate. Our understanding of the development of the brain and its associated maldevelopment would be assisted by quantitative analysis of the cortex. Volumetric magnetic resonance (MR) imaging provides high-resolution anatomical data from which we can reconstruct the white matter as a three-dimensional object and extract its surface (the grey/white matter interface). Threedimensional fractal analysis of this surface is a method of quantifying the surface complexity dependent upon the variation of the surface area under different scales of inspection. We estimate the fractal dimension of the white matter surface for each hemisphere and 10 coronal blocks of each hemisphere in 30 normal adult subjects. These values are tightly distributed and have been used to define a normal range of fractal dimensions. Abnormal fractal dimensions were found in 8/16 subjects with epilepsy and a gyral abnormality observed on routine MR imaging; and in 9/23 subjects with epilepsy and normal routine MR imaging. These analytical techniques offer additional information about the structure of the cortex in normal brains and about abnormalities of structure in subjects with suspected but unobserved structural abnormalites.