deletions | additions       diff --git a/subsection_Approximation_of_probability_distribution__.tex b/subsection_Approximation_of_probability_distribution__.tex  index 1bac2be..464a55c 100644  --- a/subsection_Approximation_of_probability_distribution__.tex  +++ b/subsection_Approximation_of_probability_distribution__.tex 
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\[P_{PW}((x,y),T) = \sum_{(x',y')\in T}K((x,y),(x',y'))\]  where $K$ is a gaussian kernel. In practice we take a discrete gaussian kernel filter for $K$, given by the 3x3 matrix:  \[ M_{K} = \left| \left(  \begin{array}{ccc} 0.0625 & 0.125 & 0.0625 \\  0.125 & 0.25 & 0.125 \\  0.0625 & 0.125 & 0.0625 \end{array} \right|.\] \right).\]  Whereas the simple histogram method places a spike function (i.e. $K = \delta$) at the bin corresponding to $(x,y)$ and update only a single bin, Parzen windowing places a kernel at the bin of $(x,y)$ and updates all bins falling under the kernel with the corresponding kernel value.  As a result using a gaussian filter, the estimated distributions are more smooth and less sparse.