# 斑点检测

## 高斯拉普拉斯算子 (LOG)

${\displaystyle G(x,y;\sigma )={1 \over {\sqrt {2\pi \sigma ^{2}}}}\exp(-{\frac {x^{2}+y^{2}}{2\sigma ^{2}}})}$

，在這裡${\displaystyle \sigma }$標準差，利用其與影像作卷積得到${\displaystyle L}$

${\displaystyle L(x,y;\sigma )=f(x,y)*G(x,y;\sigma )}$

${\displaystyle \nabla ^{2}={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}}$
${\displaystyle \nabla ^{2}L=L_{xx}+L_{yy}}$

${\displaystyle \nabla _{\mathrm {norm} }^{2}L=t\,(L_{xx}+L_{yy})}$

，再藉此找出極值與斑點

${\displaystyle ({\hat {x}},{\hat {y}};{\hat {t}})=\operatorname {argmaxminlocal} _{(x,y;t)}((\nabla _{\mathrm {norm} }^{2}L)(x,y;t))}$

## 高斯差算子（DOG）

${\displaystyle \partial _{t}L={\frac {1}{2}}\nabla ^{2}L}$

${\displaystyle \nabla _{\mathrm {norm} }^{2}L(x,y;t)\approx {\frac {t}{\Delta t}}\left(L(x,y;t+\Delta t)-L(x,y;t)\right)}$.

## 黑塞矩陣行列式（DOH）

${\displaystyle \det H_{\mathrm {norm} }L=t^{2}\left(L_{xx}L_{yy}-L_{xy}^{2}\right)}$

，在這裡HL指的是黑塞矩陣。接著，藉由偵測${\displaystyle \det H_{norm}L}$的局部極值，此方法也可進行自動斑點檢測。[3][5]

${\displaystyle ({\hat {x}},{\hat {y}};{\hat {t}})=\operatorname {argmaxlocal} _{(x,y;t)}((\det H_{\mathrm {norm} }L)(x,y;t))}$.

## 拉普拉斯算子與黑塞矩陣行列式的混合（Hessian-Laplace）

${\displaystyle ({\hat {x}},{\hat {y}})=\operatorname {argmaxlocal} _{(x,y)}((\det HL)(x,y;t))}$
${\displaystyle {\hat {t}}=\operatorname {argmaxminlocal} _{t}((\nabla _{\mathrm {norm} }^{2}L)({\hat {x}},{\hat {y}};t))}$

## References

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