LAPLACE OPERATOR.ppt [Lecture seule]

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Laplace operator (spatial domain)The Laplace operator is defined by:In the discrete case:wherexy2 2 2xy∂ff2 22∇ f ≡ +2 2∂x∇ f(i,j) ≡∆ f(i,j) +∆ f(i,j)∆ f(i,j) ≡ f(i,j) −f(i−1,j)∆ f(i,j) ≡ f(i,j) −f(i,j−1)∆ f(i,j) ≡ ∆ f(i+ 1,j) −∆ f(i,j)2x x x2y∂∂y[ f(i 1,j) f(i,j) ] [ f(i,j) f(i 1,j) ]≡ + − − − −≡ f(i+ 1,j) + f(i−1,j) −2f(i,j)∆ f(i,j) ≡ f(i,j+ 1) + f(i,j−1) −2f(i,j)
It follows that:Laplace operator[ ]∇ 2 f ≡ f(i+ 1,j) + f(i− 1,j) + f(i,j+ 1) + f(i,j−1) −4f(i,j)Notice that this result is proportional to:f(i,j) − 1 f(i+ 1,j) + f(i− 1,j) + f(i,j) + f(i,j+ 1) + f(i,j−1)5[ ]Hence, the discrete Laplace operator can be replaced by theoriginal function subtracted by an average of this function in asmall neighborhood:∇ 2 f = f(i,j) −f(i,j)
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LAPLACE OPERATOR.ppt [Lecture seule]

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