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Mathematical Form of Microsphere Projection


Mathematical Form of Microsphere Projection


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Reduced Mathemaical Form

 
Mathematical Formula for Microsphere Projection
 


LaTeX Formula

w_i=\max\left\{\left\|\ell_j-I\right\|^{-p}\cos\left(s_i,\ell_j-I\right):j\in\left\{1,2,3,\dots,N\right\}\right\}
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m_i=\mbox{any }v_j:\left[\left(\left\|\ell_j-I\right\|^{-p}\cos\left(s_i,\ell_j-I\right)=w_i\right)\land j\in\left\{1,2,3,\dots,N\right\}\right]
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f(I)=\begin{cases}
v_i\mbox{ if }\exists{i}\in\left\{1,2,3,\dots,N\right\}\left(I=\ell_i\right)\\
\displaystyle\frac{\displaystyle\sum_{i=1}^{P}m_iw_i} {\displaystyle\sum_{i=1}^{P}w_i}\mbox{ otherwise }
\end{cases}
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I = \mbox{Location of interpolation} \\
p = \mbox{Propagation of influence power, } p>0 \\
v_i = \mbox{Value of sample }i,~i\in\left\{1,2,3,\dots,N\right\} \\
\ell_i = \mbox{Location of sample }i,~i\in\left\{1,2,3,\dots,N\right\} \\
N = \mbox{Number of samples} \\
s_i = \mbox{Evenly spaced unit vector on surface of sphere,}~i\in\left\{1,2,3,\dots,P\right\} \\
P = \mbox{Precision (number of unit vectors on sphere),}~P\gg2d \\
d = \mbox{Dimensionality of data (}d=2\mbox{ is planar)}


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