沿从(u,v) 到 (x,y) 光线的辐射: $\overline L(x,y,u,v; z)$
$$E(x,y; z) = \frac{1}{z^2} \iint{\overline L(x,y,u,v; z) \cos^4\phi dudv}$$
$$L(x,y,u,v; z) = \overline L(x,y,u,v; z) \cos^4\phi$$
将$L(x,y,u,v; z) $传播到$L(x,y,u,v; z') $ $\alpha=\frac{z'}{z}$,
$$\begin{array}{l} L(x,y,u,v; z') & = L(x,y,u,v; \alpha z)\ & = L(u+\frac{x-u}{\alpha}, v+\frac{y-v}{\alpha}, u, v; z)\ & = L( u \left(1- \frac{1}{\alpha} \right) + \frac{x}{\alpha}, v \left(1- \frac{1}{\alpha} \right) + \frac{y}{\alpha}, u, v; z)\ \end{array}$$
Transform $LF(x,y,u,v; F)$ to intensity at a depth $\alpha z$, $\Delta z = z'-z = (\alpha-1)z$:
$$\begin{array}{l} T \left{ L(x,y,u,v; z); \alpha \right} & = E(x,y; \alpha z)\ & = \frac{1}{\alpha^2 z^2} \iint{ L( u \left(1- \frac{1}{\alpha} \right) + \frac{x}{\alpha}, v \left(1- \frac{1}{\alpha} \right) + \frac{y}{\alpha}, u, v; z) dudv}\ \end{array}$$
Shear matrix:
$$\begin{array}{l}
T(\alpha)=
\begin{pmatrix}
\frac{1}{ \alpha} & 0 & 1- \frac{1}{ \alpha} & 0 \
0 & \frac{1}{ \alpha} & 0 & 1- \frac{1}{ \alpha} \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
\end{pmatrix}
\end{array}$$
$$\begin{array}{l}
T^{-1}(\alpha)=
\begin{pmatrix}
\alpha & 0 & 1- \alpha & 0 \
0 & \alpha & 0 & 1- \alpha \
0 & 0 & 1 & 0 \
0 & 0 & 0 & 1 \
\end{pmatrix}
\end{array}$$
沿从(u,v) 到 (x,y) 光线的辐射: $\overline L(x,y, \tan \theta_x, \tan \theta_y; z)$, where $\tan \theta_x = \frac{u-x}{z}, \tan \theta_y = \frac{v-y}{z}$
$$E(x,y; z) = \frac{1}{z^2} \iint{\overline L(x,y,\tan \theta_x, \tan \theta_y; z) \cos^4\phi d\tan \theta_xd\tan \theta_y}$$
$$L(x,y,\tan \theta_x, \tan \theta_y; z) = \overline L(x,y,\tan \theta_x, \tan \theta_y; z ) \cos^4\phi$$
将$L(x,y,u,v; z) $传播到$L(x,y,u,v; z') $
$\Delta z = z'-z = (\alpha-1)z$
Transform $LF(x,y,u,v; F)$ to intensity at a depth $\alpha z$, $\Delta z = z'-z = (\alpha-1)z$:
$$\begin{array}{l} T \left{ L(x,y,\tan \theta_x, \tan \theta_y; z); \Delta z \right} \ = E(x,y; z')\ = \frac{1}{(z+\Delta z)^2} \iint{ L(x + \Delta z \tan \theta_x, y + \Delta z \tan \theta_y, \tan \theta_x, \tan \theta_y; z) d\tan \theta_xd\tan\theta_y}\ \end{array}$$
Shear matrix:
$$\begin{array}{l}
T(\Delta z)=
\begin{pmatrix}
1 &0 & \Delta z &0 \
0 &1 &0 &\Delta z \
0 &0 &1 &0 \
0 &0 &0 &1 \
\end{pmatrix} \
\end{array}$$
$$\begin{array}{l}
T^{-1}(\Delta z)=
\begin{pmatrix}
1 &0 & -\Delta z &0 \
0 &1 &0 &-\Delta z \
0 &0 &1 &0 \
0 &0 &0 &1 \
\end{pmatrix} \
\end{array}$$
Projection(or refocusing): from LF to intensity
Propagation: through a Lens, with a distance
[[ Camera Simulation| Camera Simulation]]
[[Coded aperture|Coded aperture]]
[[Light field moment imaging|Light field moment imaging]]