deletions | additions       diff --git a/res_pre.md b/res_pre.md  index 6bae35c..cb9c630 100644  --- a/res_pre.md  +++ b/res_pre.md 
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The only preprocessing step used is convolving the image with a \((2n+1)\times(2n+1)\) kernel where the center element is \(1/(n+1)\) and the k-th layer surrounding the center is made up of values \(1/(8k\cdot(n+1))\). For example, when n=1 the kernel would be   \[  \left( \begin{array}{ccc}  1/16 & 1/16 \tfrac{1}{16}  & 1/16\\\\  1/16 \tfrac{1}{16}  & ^1/_2 \tfrac{1}{16}\\\\  \tfrac{1}{16}  & 1/16\\\\  1/16 \tfrac{1}{2}& \tfrac{1}{16}\\\\  \tfrac{1}{16}  & 1/16 \tfrac{1}{16}  & \tfrac{1}{16} \end{array} \right) \]  Convolving the image with this kind of kernel reduces the noise while retaining more of the original signal than simple averaging. Contributions from the i-th layer around the center will have the same weight as the central pixel. In this work we use the kernel with \(n=1\).