box counting method

7/25/2019 Box Counting Method

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Box-Counting Method


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Self-similarity

Example:One of the basic properties of fractal images is the notion of self-similarity. This idea is

easy to explain using the Sierpinski triangle. Note that S may be decomposed into 3

congruent figures, each of which is exactly 1/2 the size of S! See above figure. That is

to say, if we magnify any of the 3 pieces of S shown in figure by a factor of 2, we obtain

an exact replica of S. That is, S consists of 3 self-similar copies of itself, each withmagnification factor 2.

Self similarity means that parts of the object are self similar (they contain parts of

themself). Self similar structures contain parts which are similar to themselves. It iseven possible that the whole structure can be found within the object (i.e.

Mandelbrot Set). Iteration and recursion describe the process of repeating steps.

Self-similarity is a property of the object, not of the steps used to build the object.


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FractalsFractals are infinitely complex patterns that are self-similar across

different scales. They are created by repeating a simple process

over and over in an ongoing feedback loop. So, fractals are natural

or artificial structures which offer a high level of scale invariance or

self-similarity.

Scale invariance relating to fractals means that the scale is notimportant for the result. Structures which are scale invariant can be

observed from different distances but the object always looks the

same.

many natural phenomena are better described using a dimension

between two whole numbers. So while a straight line has a

dimension of one, a fractal curve will have a dimension between

one and two, depending on how much space it takes up as it twists

and curves.


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DimensionsBy defining dimension in terms of the scaling properties of a shape, we

will come up with a quantitative way of describing fractals. This definition

of dimension is known as the self-similarity dimension, because it tells ushow many small self similar pieces of an object fit inside a large piece.

So what does a non-integer dimension mean, anyway?

There are several ways to think about this. First, a dimension between 1

and 2 means that the shape has some qualities of two-dimensionalobjects and some of one dimensional objects. The snowflake is two-

dimensional in the sense that it resides in two-dimensional space. It rests

on the surface of a piece of paper, which is two-dimensional. However, as

the process of building the snowflake proceeds, the shape becomes more

and more edgy,in the sense that its perimeter grows and grows.A fractal dimension is a ratio providing a statistical index

of complexity comparing how detail in a pattern (strictly speaking,

a fractal pattern) changes with the scale at which it is measured. It

has also been characterized as a measure of the space-

filling capacity of a pattern


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Fractal Dimension

There are two definitions of fractal dimension:

Fractal dimension is a ratio providing a statistical index of

complexity comparing how detail in a pattern changes with the

scale at which it is measured.

Fractal dimension is a measure of the space-filling capacity of apattern that tells how a fractal scales differently than the space

it is embedded in.

A fractal dimension does not have to be

an integer.


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Definition of Box-Counting DimensionFor different side lengths r we count N(r), the smallest number of boxes of side length r

needed to cover the shape. But, how does N(r) depend on r?

If the shape is 1-dimensional, such as the line segment, we have seen N(r) = 1/r If the shape is 2-dimensional, such as the (filled-in) unit square, we have seen N(r) =

(1/r)2

If the shape is 3-dimensional, such as the (filled-in) unit cube, we expect N(r) = (1/r)3.

For more complicated shapes, the relation between N(r) and 1/r may not be so clear.

If we suspect that N(r) is approximately k

(1/r)d, a power law relation, how can we find

d?

Taking Log of both sides of N(r) = k(1/r)d, we obtain

Log(N(r)) = Log(k) + Log((1/r)d) = dLog(1/r) + Log(k)

With the expectation that the approximation becomes better for smaller r. Solving for

d and taking the limit as r 0gives db= limr 0Log(N(r))/Log(1/r) (Note as r 0we

have 1/r , soLog(1/r) and Log(k)/Log(1/r) 0.) If the limit exists, it is called the box-counting dimension, db, of the shape.

This limit may be slow to converge; an alternate approach is to notice

Log(N(r)) = dLog(1/r) + Log(k) is the equation of a straight line with slope d and y-

intercept Log(k).

So plotting Log(N(r)) vs Log(1/r), the points should lie approximately on a straight line

with slope db. This is the log-log approach to finding the box-counting dimension.


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Box-Counting Dimension of a Gasket


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Box-Counting Dimension Practice Problems


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Box-Counting Dimension Practice Problems


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Sierpinski carpet

First stage

Calculate next stages similarly


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Koch curveOption 1

Option 2

1. 2.

3.


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Sierpinski Triangle

Stage 2

Stage 3

Stage 4

Stage 5

Calculate D for all stages one by one:


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Stage 1-2 The shape can be decomposed into

N = 3 pieces, each scaled by a

factor of r = 1/2,

so ds= Log(3)/Log(2)

Stage 1 Stage 2

Similar Dimension

box counting method

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