10 Min Talk SOUNDARARAJAN EZEKIEL Department of Computer Science IUP

Fractal Dimension Madelbrot (1982) offered the following tentative def of a fractal A 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-similarity Dimension The 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

Koch Curve Begin with st.line-divide 3 equal parts Replace middle part by 2 sides of an equilateral triangle of length same length as removed part Continue this process

Koch Curve

N=4, F=3 then Capacity Dimension determine the FD of irregular shapes N[r]r N[r], min # of of balls of size r needed to cover the object

Box Dimension(BD) Take ball = contiguous non- overlapping boxes gives BD Kolmogorov entropy, entropy dimension, metric dimension, logarithmic density N[r]=b r -D Area A = N[r] * r 2 = b r 2-D N[r]=b r -D Area A = N[r] * r 2 = b r 2-D Plot Log A versus Log r then BD= 2-s where s is the slope of regression line in the plot

Types of Fractal Dimensions Compass or Ruler dimension:- ( compute fractal dimension of natural objects ex: coastline) Correlation Dimension Correlation 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 Measures Methods for assessing the fractal characteristics of time-varying signals like heart rate, respiratory rate, seismology signal, stock price and so on. such signals which vary, apparently irregularly, have been considered to be driven by external influences which are random, that is to say, just ''noise''. Methods for assessing the fractal characteristics of time-varying signals like heart rate, respiratory rate, seismology signal, stock price and so on. such signals which vary, apparently irregularly, have been considered to be driven by external influences which are random, that is to say, just ''noise''. FD produces a single numeric value that summarizes the irregularity of “roughness” of feature boundary. FD produces a single numeric value that summarizes the irregularity of “roughness” of feature boundary. It describes the “roughness” of images as natural, the way we perceive roughness. It describes the “roughness” of images as natural, the way we perceive roughness.

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 's Wavelet-- the scale we use to see data plays an important role FT non local -- very poor job on sharp spikes Sine wave Waveletdb10 Wavelet db10

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