经典大气散射模型描述如下:
\[I(x)=J(x)t(x)+A(1-t(x)), \]其中\(I(x)\)为带雾图像,\(J(x)\)为清晰图像,\(t(x)\)为透射率,\(A\)为全局全局背景光。通常定义
\[t(x)=e^{-\beta d(x)}, \]其中\(\beta\)为大气散射系数,\(d(x)\)为相机到物体深度。
我们从体渲染角度来考虑带雾图像模型,简化的体渲染方程为:
\[I(s)=\int_0^sT(t)\sigma(t)C(t)\mathbf{d}t+T(s)I_0\\ T(s)=\exp(-\int_0^s\sigma(t)\mathbf{d}t). \]其中\(\sigma\)为衰减率,\(C\)为辐射强度,\(I_0\)为背景光。
于是我们假定雾的衰减率为\(\sigma_0\),辐射强度为\(C_0\),物体深度为\(d(x)\),原式可以化为
\[\begin{aligned} T(s)&=\exp(-\int_0^{d(x)}\sigma_0\mathbf{d}t)\exp(-\int_{d(x)}^s\sigma(t)\mathbf{d}t) \\ I(x)&=\int_0^{d(x)}T(t)\sigma_0C_0\mathbf{d}t+\int_{d(x)}^fT(t)\sigma(t)C(t)\mathbf{d}t+T(f)I_0 \\ &=\sigma_0C_0\int_0^{d(x)}e^{-t\sigma_0}\mathbf{d}t+e^{-d(x)\sigma_0}\int_{d(x)}^fe^{-\int_{d(x)}^f\sigma(t)\mathbf{d}t}\sigma(t)C(t)\mathbf{d}t+e^{-d(x)\sigma_0}e^{-\int_{d(x)}^f\sigma(t)\mathbf{d}t} \\ &=C_0\left[1-e^{-d(x)\sigma_0}\right]+e^{-d(x)\sigma_0}J(x) \\ &=J(x)t(x)+C_0(1-t(x)) \end{aligned} \]与原式形式基本一致。
标签:mathbf,推导,int,模型,exp,图像,sigma,CV From: https://www.cnblogs.com/edlinf/p/17764330.html