拉普拉斯算子（Laplace Operator）

拉普拉斯算子（Laplace Operator）为标量算子（a scalar operator）：两个梯度向量算子（gradient vector operator）的内积（dot product，inner product）

$\bigtriangleup = \bigtriangledown \cdot \bigtriangledown = \bigtriangledown^{2} = \begin{bmatrix} \frac{\partial}{\partial x_{1}} & \cdots & \frac{\partial}{\partial x_{N}} \end{bmatrix}\begin{bmatrix} \frac{\partial}{\partial x_{1}} \\ \vdots \\ \frac{\partial}{\partial x_{N}} \end{bmatrix} = \sum_{n = 1}^{N} \frac{\partial^{2}}{\partial x_{n}^{2}}$△=▽⋅▽=▽2=[∂x1​∂​​⋯​∂xN​∂​​]⎣⎢⎡​∂x1​∂​⋮∂xN​∂​​⎦⎥⎤​=n=1∑N​∂xn2​∂2​

$N = 2$N=2维空间中，

$\bigtriangleup = \bigtriangledown \cdot \bigtriangledown = \bigtriangledown^{2} = \begin{bmatrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \end{bmatrix}\begin{bmatrix} \frac{\partial}{\partial x} \\ \frac{\partial}{\partial y} \end{bmatrix} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$△=▽⋅▽=▽2=[∂x∂​​∂y∂​​][∂x∂​∂y∂​​]=∂x2∂2​+∂y2∂2​

将拉普拉斯算子作用于（apply to）二维函数$f(x, y)$f(x,y)，得到标量函数

$\bigtriangleup f(x, y) = \frac{\partial^{2} f}{\partial x^{2}} + \frac{\partial^{2} f}{\partial y^{2}}$△f(x,y)=∂x2∂2f​+∂y2∂2f​

离散情况（discrete case）下，用二阶差分（second order difference）代替二阶微分（second order differentiation）运算。

• 对于一维函数，若一阶差分定义为：

$\bigtriangledown f[n] = f[n + 1] - f[n]$▽f[n]=f[n+1]−f[n]

则二阶差分为：

$\begin{aligned} \bigtriangleup f[n] & = \bigtriangledown (\bigtriangledown f[n]) := \bigtriangledown f[n] - \bigtriangledown f[n - 1] \\ & = (f[n + 1] - f[n]) - (f[n] - f[n - 1]) \\ & = f[n + 1] - 2 f[n] + f[n - 1] \\ \end{aligned}$△f[n]​=▽(▽f[n]):=▽f[n]−▽f[n−1]=(f[n+1]−f[n])−(f[n]−f[n−1])=f[n+1]−2f[n]+f[n−1]​

为保证$\bigtriangleup f[n]$△f[n]以$f[n]$f[n]为中心对称，定义$\bigtriangleup f[n] := \bigtriangledown f[n] - \bigtriangledown f[n - 1]$△f[n]:=▽f[n]−▽f[n−1]。拉普拉斯算子可通过一维卷积实现（1-D convolution with a kernel $[1, -2, 1]$[1,−2,1]）。

• 二维情况：

$\begin{aligned} \bigtriangleup f[m, n] & := \bigtriangleup_{m} f[m, n] + \bigtriangleup_{n} f[m, n] \\ & = f[m + 1, n] - 2 f[m, n] + f[m - 1, n] + f[m, n + 1] - 2 f[m, n] + f[m, n - 1] \\ & = f[m + 1, n] + f[m - 1, n] + f[m, n + 1] + f[m, n - 1] - 4 f[m, n] \\ \end{aligned}$△f[m,n]​:=△m​f[m,n]+△n​f[m,n]=f[m+1,n]−2f[m,n]+f[m−1,n]+f[m,n+1]−2f[m,n]+f[m,n−1]=f[m+1,n]+f[m−1,n]+f[m,n+1]+f[m,n−1]−4f[m,n]​

相应二维卷积核（2-D convolution kernel）为

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \\ \end{bmatrix} \text{or} \begin{bmatrix} 1 & 1 & 1 \\ 1 & -8 & 1 \\ 1 & 1 & 1 \\ \end{bmatrix}$⎣⎡​010​1−41​010​⎦⎤​or⎣⎡​111​1−81​111​⎦⎤​

拉普拉斯算子是一种高通滤波器（high-pass filtering kernels），$f[n]$f[n]的边缘可通过检测$\bigtriangleup f[n]$△f[n]的过零（zero crossing）点实现。

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