Sampling and density
How-to build a continuous field from arbitrary sampled particles ?
- Consider arbitrary continuous field \(A(p)\)
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- Def. of convolution: \(A(p) = (A\star\delta)(p)=\int_\Omega A(q)\,\delta(p-q)\,\mathrm{d}q\)
- 1. Consider \(W_h\) a smooth kernel with \(\int_\Omega W_h(p) \, \mathrm{d}p=1\)
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- \(A(p) \simeq (A\star W_h)(p)=\int_\Omega A(q)\,W_h(p-q)\,\mathrm{d}q\)
- Low pass filter applied to A
- 2. Discrete sampling on \(p_j\)
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- \(A(p) = \sum_j A(p_j) W_h(p-p_j) V_j\)
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- \(V_j\): small volume associated to \(p_j\)
