To the Graduate Council: I am submitting herewith a thesis written by ...

Chapter 4: Algorithm Overview 58With the help of Figure 4.8 we would like to explain an issue with the Shannon typeentropy measures. As the resolution of the data increases, the number of points in thedensity is also going to increase and ∆ tends towards zero. Using Reiman’s definitionof integrals we can rewrite Equation 4.18 as−∆f( x )log( ∆f( x ) ) = −i− ∞ −∞i∆f( xi)log( f (x ) ) −i∆f( x )log( ∆ ) )i(4.22)f ( x )log f ( x )dx = lim( H Shannon + log( ∆ ))∆→0 (4.23)We see that as the number of points approaches the continuous random variable, thereis a quantum jump in the amount of information measured. We needed a measure thatis normalized and improves with resolution. The measures that we have presented inTable 4.2 have an upper limit that is directly proportional to the number of charactersin a symbol. Since we need to have the shape information quantized and independentof resolution, we have studied different divergence measures such as KL divergence (Equation 4.24), Jenson-Shannon divergence (Equation 4.25) and Chi-Squareddivergence measures before extending Shannon’s definition for our CVM.p( x )H KL = − p( x )logq( x )(4.24)p + q H Shannon(p ) + H Shannon( q )H JS = H Shannon() −22where p is the density of the object of interest and q is the density of thereference. (4.25)We have discussed the supporting theory for the proposed CVM algorithm. In the nextchapter we discuss implementation decisions for the algorithm and present theexperimental results of our algorithm on different datasets.Figure 4.8: Resolution issue with Shannon type measures.