10 Min Talk10 Min Talk10 Min Talk10 Min Talk

SOUNDARARAJAN SOUNDARARAJAN EZEKIELEZEKIEL

Department of Computer ScienceDepartment of Computer Science

IUPIUP

Fractal DimensionFractal DimensionFractal DimensionFractal DimensionMadelbrot (1982) offered the following tentative

def of a fractalA fractal is by definition a set for which the

Hausdorff-Besicovitch dimension strictly exceeds the Topological dimension

A fractal is a shape made of parts similar to the whole. This definition uses the concept of self-similarity. A set is called strictly self-similar if it can be broken into arbitrary small pieces, each of which is a small replica of the entire set.

Self-similaritySelf-similarity DimensionDimensionSelf-similaritySelf-similarity DimensionDimensionThe self-similarity dimension describes

how many new pieces geometrically similar to the whole object are observed as the resolution is made finer.

If we change the scale by a factor F, and we find that there are N pieces similar to the original, the self-similarity dimension

log

logself

ND

F

Koch CurveKoch CurveKoch CurveKoch CurveBegin with st.line-divide 3 equal partsReplace middle part by 2 sides of an

equilateral triangle of length same length as removed part

Continue this process

Koch CurveKoch Curve

N=4 , F=3 then

Capacity Dimensiondetermine the FD of irregular shapesN[r]N[r] , min # of of balls of size r r needed to

cover the object

41.26185

3self

LogD

Log

[ ](1/ )0

lim LogN rcap Log rrD

Box Dimension(BD)Box Dimension(BD)Box Dimension(BD)Box Dimension(BD)Take ball = contiguous non-overlapping

boxes gives BD

Kolmogorov entropy, entropy dimension, metric dimension, logarithmic density

N[r]=b N[r]=b r r --DD Area A = N[r] * r Area A = N[r] * r22 = b = b r r 2-2-DD Plot Log A versus Log r then BD= 2-s where

s is the slope of regression line in the plot

Types of Fractal DimensionsTypes of Fractal DimensionsTypes of Fractal DimensionsTypes of Fractal DimensionsCompass or Ruler dimension:- ( compute fractal

dimension of natural objects ex: coastline)Correlation DimensionCorrelation Dimension:- weightings that

measures the correlation, among the points Information Dimension:- based on a weighting of

the points of the set within a box that measure the rate at which information changes .

Fractal MeasuresFractal MeasuresFractal MeasuresFractal Measures Methods for assessing the fractal characteristics of time-Methods for assessing the fractal characteristics of time-

varying signals like heart rate, respiratory rate, seismology varying signals like heart rate, respiratory rate, seismology signal, stock price and so on. such signals which vary, signal, stock price and so on. such signals which vary, apparently irregularly, have been considered to be driven apparently irregularly, have been considered to be driven by external influences which are random, that is to say, by external influences which are random, that is to say, just ''noise''just ''noise''..

FD produces a single numeric value that summarizes the FD produces a single numeric value that summarizes the irregularity of “roughness” of feature boundary.irregularity of “roughness” of feature boundary.

It describes the “roughness” of images as natural, It describes the “roughness” of images as natural, the way we perceive roughness.the way we perceive roughness.

What is Wavelet? ( Wavelet Analysis)What is Wavelet? ( Wavelet Analysis) Wavelets are functions that satisfy certain mathematical

requirements and are used to represent data or other functions Idea is not new--- Joseph Fourier--- 1800's Wavelet-- the scale we use to see data plays an important role FT non local -- very poor job on sharp spikes

Sine waveSine wave

WaveletWavelet db10db10

Result2Result2

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