高斯拉普拉斯算子

高斯拉普拉斯算子（Laplacian of Gaussian，LoG）

高斯拉普拉斯算子（Laplacian of Gaussian，LoG）提取图像 $f(x, y)$边缘：

1. 图像平滑去噪，高斯低通滤波器（a convolution with a Gaussian kernel of width $\sigma$）

$G_{\sigma}(x, y) = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

2. 边缘检测，拉普拉斯算子（Laplace operator）

$\bigtriangleup (G_{\sigma}(x, y) * f(x, y)) = (\bigtriangleup G_{\sigma}(x, y)) * f(x, y) = \text{LoG} * f(x, y)$

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卷积性质：

$\frac{d}{dt} (h(t) * f(t)) = \frac{d}{dt} \int f(\tau) h(t - \tau) d \tau = \int f(\tau) \frac{d}{dt} h(t - \tau) d \tau = f(t) * \frac{d}{dt} h(t)$

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高斯拉普拉斯算子 $\bigtriangleup G_{\sigma}(x,y)$：

$\frac{\partial^{2}}{\partial x^{2}} G_{\sigma}(x, y) = \frac{1}{\sqrt{2 \pi} \sigma} \frac{x^{2} - \sigma^{2}}{\sigma^{4}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

$\frac{\partial^{2}}{\partial y^{2}} G_{\sigma}(x, y) = \frac{1}{\sqrt{2 \pi} \sigma} \frac{y^{2} - \sigma^{2}}{\sigma^{4}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

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$\frac{\partial}{\partial x} G_{\sigma}(x, y) = \frac{1}{\sqrt{2 \pi} \sigma} \frac{\partial }{\partial x} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right) = - \frac{x}{\sqrt{2 \pi} \sigma^{3}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

$\frac{\partial^{2}}{\partial x^{2}} G_{\sigma}(x, y) = - \frac{1}{\sqrt{2 \pi} \sigma^{3}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right) + \frac{x^{2}}{\sqrt{2 \pi} \sigma^{5}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right) = \frac{1}{\sqrt{2 \pi} \sigma} \frac{x^{2} - \sigma^{2}}{\sigma^{4}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

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$\text{LoG}$定义为：

$\text{LoG} \triangleq \frac{\partial^{2}}{\partial x^{2}} G_{\sigma}(x, y) + \frac{\partial^{2}}{\partial y^{2}} G_{\sigma}(x, y) = \frac{1}{\sqrt{2 \pi} \sigma} \frac{x^{2} + y^{2} - 2 \sigma^{2}}{\sigma^{4}} \exp \left( - \frac{x^{2} + y^{2}}{2 \sigma^{2}} \right)$

二维 $5 \times 5$ $\text{LoG}$算子：

$\begin{bmatrix} 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 1 & 2 & -16 & 2 & 1 \\ 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ \end{bmatrix}$

核矩阵各元素之和必须为零（make sure that the sum (or average) of all elements of the kernel has to be zero）。

边缘检测步骤：

1. LoG滤波（applying LoG to the image）

2. 过零检测（detection of zero-crossings in the image）

3. 门限判决（threshold the zero-crossings to keep only those strong ones (large difference between the positive maximum and the negative minimum)）