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Topic: Box-counting dimension

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 Minkowski-Bouligand dimension - Wikipedia, the free encyclopedia
For example, the box dimension of a single point is 0, but the box dimension of the collection of rational numbers in the interval [0, 1] has dimension 1.
For example, the Hausdorff dimension, lower box dimension, and upper box dimension of the Cantor set are all equal to log(2)/log(3).
It is possible to define the box dimensions using balls, with either the covering number or the packing number. /wiki/Minkowski-Bouligand_dimension   (915 words)

 Box Counting Dimension
The box-counting dimension can exceed one for a finite iteration Koch curve, but is never greater than the dimension of the space in which the shape is embedded, in this case a 2d plane.
Figure 6.1 demonstrates how the box counting technique can be used to calculate the dimension of a shape.
boxes could be required to cover the square, which would lead to an unrealistic value for the dimension. /~ted/thesis/node73.html   (965 words)

 Box Dimension Discussion
Counting boxes of different sizes along edges of shapes allows one to measure this fractal dimension.
Box-Counting Dimension is a way to measure the length of fractal edges, such as coastlines, the shape of ferns, and mathematical constructs such as the Koch curve.
The box-counting dimension (your goal) is in purple at the bottom center of the screen. /courseware/boxdim/readers/discussion.html   (315 words)

 Fractal Dimension
Not surprisingly, the box dimensions of ordinary Euclidean objects such as points, curves, surfaces, and solids coincide with their topological dimensions of 0, 1, 2, and 3-- this is, of course, what we would want to happen, and follows from the discussion at the beginning of §5.1.
The dimension of the union of finitely many sets is the largest dimension of any one of them, so if we ``grow hair'' on a plane, the result is still a two-dimensional set.
Thus, the embedding dimension of a plane is 2, the embedding dimension of a sphere is 3, and the embedding dimension of a klein bottle is 4, even though they all have (topological) dimension two. /~scott/Book331/Fractal_Dimension.html   (1303 words)

 Fractal Geometry
The box counting dimension of a shape in the plane can be estimated by the slope of the plot of
Every change in the number of selected boxes alters this estimate, so this number estimates the dimension only when box-counting has been completed.
Box size 8 and 4 are quite small and can safely be avoided when learning the mechanics of box-counting. /Fractals/Labs/CoastlineLab/CBCDimDir.html   (183 words)

 FDC - Fractal Dimension Calculator
Advances in the implementation of the box-counting method of fractal dimension estimation
The estimate of the fractal dimension is the slope of this graph.
FDC estimates the fractal dimension of an object represented as a black and white image where the object to be analysed is assumed to be made up of the black pixels. /~pbourke/fractals/fracdim   (1600 words)

 Jeffery J. Leader's MA 325 HW Solutions Page.
For the Hausdorff dimension, the open sets U (see pg.217) may be taken to have arbitrarily small diameter (though not zero, as they are open) and so the Hausdorff measure is always a countable sum of arbitrarily small values and hence the infimum is zero for all s; so, s=0 is the dimension.
Taking fractal dimension greater than topological dimension as the definition includes all objects which have an unexpected dimensionality, but includes objects which are simply squares, etc., if one disregards their genesis; it considers how the curve got there, not just its final geometry.
These physical objects are curve-like (topological dimension 1), as are the four classical fractals (the outline in the case of the island) but attempt to nearly fill space or the plane. /~leader/ma325soln.html   (1477 words)

 Graph Gallery
The box-counting fractal dimension computed with the package [6] is 1.69.
The box-counting fractal dimension computed with the package [6] is 1.70.
The box-counting fractal dimension computed with the package [6] is 1.64. /~patrigna/graphpictures   (762 words)

 Fractal Dimension - 2D Box Counting
The fractal dimension of the contour is related to the slope of the relationship between the logarithm of the number of boxes through which the contour passes and the logarithm of the size of the boxes.
As an option, a log-log plot of the number of boxes verses the size of the box may be produced on one of the buffers.
The box counts and sizes may be written to the formatted file specified by this field. /user/fdl/people/sd/digimage/hlpfiles/lf.htm   (779 words)

 Dimensions - Demo 1
The simplest definition of the dimension is the box counting dimension or capacity.
The dangers of extracting a false "dimension" from too small a range of box sizes, squeezed by the practical limitations at large and small box sizes, should be borne in mind.
It should be apparent that as the box size is reduced, finer levels of the "curve" of the Henon attractor are resolved, so that the number of boxes grows faster than linearly with the inverse box size, but not as fast as quadratically (which would give boxes covering the whole plot). /~mcc/Chaos_Course/Lesson9/Demo1.html   (618 words)

 Dimension - Art History Online Reference and Guide
The Krull dimension of a commutative ring, named after Wolfgang Krull (1899 - 1971), is defined to be the maximal number of strict inclusions in an increasing chain of prime ideals in the ring.
In the physical sciences and in engineering, the dimension of a physical quantity is the expression of the class of physical unit that such a quantity is measured against.
Dimension (from Latin "measured out") is, in essence, the number of degrees of freedom available for movement in a space. /art_history/Size   (751 words)

Shows regression lines for the box counting dimension and mass dimensions at the original location, an average box counting dimension, a minimum cover dimension (from the minimum count at each box size over multiple origins), and a slope corrected version that has horizontal intervals removed from the data.
The data are not the same as in a box count for the fractal dimension, because that count is, in effect, over a fixed grid, meaning individual boxes of a certain size do not overlap.
This function finishes preparing data that was gathered during box counting for the final steps of multifractal analysis. /   (5584 words)

Using the `global measure formula' for the Patterson measure and using an estimate on the horoball counting function we show that the Hausdorff dimension of the limit set is equal to both the box-counting dimension and the packing dimension of the limit set.
We determine the box-counting dimension of the limit set of a general geometrically finite Kleinian group.
Thus, by a result of Sullivan, we obtain the result that for a geometrically finite group these three different types of dimension coincide with the exponent of convergence of the group. /~bos/abstractBOX.html   (98 words)

The box counting algorithm was based originally on ImageJ's function and an NIH Image plugin by Dr. Jelinek.
The intent was to develop a practical tool for biological cell morphology analysis, addressing several inherent problems associated with box counting.
dimensions over an image, including colour coding showing how the distribution of complexity changes with the scale at which it is assessed (in addition to multifractal spectra) /   (548 words)

 Heriot-Watt Maths Research Report HWM02-7
We prove existence and smoothness of solutions, existence of an invariant measure for the corresponding Markov process and we define the spatial densities of topological entropy, of measure-theoretic entropy, and of upper box-counting dimension.
J Rougemont, Space-time invariant measures, entropy, and dimension for stochastic Ginzburg-Landau equations, Commun.
The bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution. /maths/deptreps/HWM02-7.html   (143 words)

Notice that the box size can be changed and was to calculate the approximatedimension as the plot below is for a lower grid size than the Boxes = 3000 which gives an approximate box counting dimension of 1.19.
Estimate the box-counting dimension (Do) of this attractor using grids with smaller and smaller sizes.
Boxes where started at 8x8 grid and went to a 3000x3000 grid. /~mary/assign_11.html   (380 words)

 Measuring Fractal Dimension
The Box-counting dimension is much more widely used than the self-similarity dimension since the box-counting dimension can measure pictures that are not self-similar (and most real-life applications are not self-similar).
The box-counting dimension measure is equal to the slope of that line.
It is important to note that not all types of dimension measurement will give the same answer to a single problem. /~ergreen/honors_thesis/dimension.html   (522 words)

Box -counting Dimension If you have a less regular object, such as the coastline of Great Britain, then it cannot be divided up into identical parts.
If we do this for several box sizes, and plot the number of boxes passed through against the number along one side of the square on log-log paper then we may get an approximation to a straight line.
Fractal Dimension of a fractal which may be broken down into a number of smaller copies of the original can be defined by analogy with lines squares and cubes. /homepage/staff/cmspstri/dim.htm   (335 words)

 Hausdorff Measure and Dimension
The dimension of an object describes the way in which it occupies space, and thus how its size might be quantified.
is equal to the Hausdorff dimension of that object.
The size of an object can be measured by counting the number of rulers required to cover it. /~ted/thesis/node72.html   (277 words)

 HESS - Abstract
The spectrum of generalised fractal dimensions, D(q), and the sequence of mass exponents, τ(q), were obtained using an efficient generalised box-counting algorithm.
Finally, some open mathematical problems related to the assessment of the box-counting dimension are discussed.
The importance of the specific number of digitised points is underlined, in order to accurately define, the geometry of river networks through a direct generalised box-counting measure that is not influenced by their topology. /EGU/hess/hs4/4-105.htm   (185 words)

 Box Dimension Details
All the box-counting (selecting boxes on the picture) fills in the "Box Counts" data table, and that data drives the plot and the calculation of the dimension.
Please see Box-Counting Dimension for a general mathematical discussion, or Discussion for how to use the module.
Note: if a count is zero (no boxes are selected at that size), nothing is plotted for that box size, and it is dropped from all calculations. /courseware/boxdim/readers/underhood.html   (155 words)

 Geometry Of Hyperbolic Julia-Lavaurs Sets (ResearchIndex)
We show that the upper ball-(box) counting dimension and the Hausdorff dimension of J are equal, that the h -dimensional Hausdorff measure of J vanishes and that the h -dimensional packing measure of J is positive and nite.
3 dimension of Julia-Lavaurs sets as a function of the phase (context) - Urba, Zinsmeister et al.
If g is derived from the parabolic quadratic polynomial f(z) = z 2 + 1 4, then the Hausdorff dimension h is a real-analytic function of. /489193.html   (361 words)

 Point Pattern: Box-counting Dimension
(a) A self-similar point pattern, generated using theLévy flight model; (b,c) Shaded boxes are occupied by at least one point. /faculties/science/botany/labs/ecology/fractals/Figures/Fig11.html   (20 words)

 Box-Counting Dimension, Mu-Ency at MROB
You compute the box-counting dimension from a grid that is superimposed on a fractal image and counting how many boxes in the grid contain part of the fractal.
Box-counting dimension is a simple way of estimating the Hausdorff dimension for fractals.
Then you increase the number of boxes in the grid (but covering the same area: the boxes get smaller) and count again. /pub/muency/boxcountingdimension.html   (226 words)

 Analyses supported by IAN (as of October 7th, 2004)
FDB estimates the fractal dimension of the image using the box counting method.
Definition: The calculation of FDB is the log-log regression of box size versus number of boxes required to cover the image.
FDP estimates the fractal dimension of the image using the perimeter/area method as described in [Sugihari 90]. /projects/ian/analyses.htm   (2567 words)

Even though the box-counting dimension of the Weierstrass function is easy to calculate~\cite{Tricot95da}, the proof that its Hausdorff dimension has the same value is still lacking.
It is enough to analyze the dimension of %$P_{xt}(x,t)$: \[ \widetilde{P}(t) := \sum_{c=1}^\infty q^{2c(s-2)} \sin q^c x \sum_{d=1}^c q^{-d(s-2)}\sin q^{c-d}x \cos [(q^{2c}-q^{2(c-d)})t].
Lower bounds on the Hausdorff dimension of the graph were found by Mauldin~\cite{Mauldin86da,Mauldin86db}. /mp_arc/papers/01-293   (2925 words)

 Fractal dimension estimation
The algorithm used in this program was inspired by the paper entitled "A Fast Algorithm To Determine Fractal Dimensions By Box Counting", which was written by Liebovitch and Toth, and which appeared in the journal "Physics Letters A", volume 141, pp 386-390, (1989).
There are "two-point" estimates of dimension for each scale shift (division of cell size by two), plus overall estimates based on fitting a least-squares line to a log-log plot of cell count versus cell size.
FD3 is quite accurate (typically well within 5% when tested on reasonably-sized samples of fractals whose dimension are known exactly) It is quite fast -- O(NlogN) where N is the number of data lines (points) input. /morph/fd3.html   (303 words)

 Chaos and fractals solutions
) = (log(2) + log(4))/log(4) = (log(2)/log(4)) + 1 = dimension of the Cantor set+ dimension of the interval. /research/chaos/Ch3_22Sol.html   (20 words)

 Jackson's fractals
The fractal dimension is essentially a measure of self-similarity (it is sometimes referred to as the similarity dimension).
From this observation, we define the fractal dimension for a self-similar object of N parts, scaled by a ratio r from the whole, to be:
Correspondingly, the researchers found that the fractal dimension of these works increased steadily from a low value close to 1 to a high value of 1.72. /issue9/news/pollock   (840 words)

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