10 min talk soundararajan ezekiel department of computer science iup
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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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