Merge pull request #3 from slimgroup/math

start derivations page

docs/make.jl

@@ -5,6 +5,7 @@ makedocs(sitename="Photoacoustic imaging in Julia",
          authors="Rafael Orozco and Mathias Louboutin",
          pages = Any[
              "Home" => "index.md",
+             "Theory" => "derivations.md",
              "Tutorials" => ["LSQR.md", "LearnedPrior.md"]
          ])
 

docs/src/derivations.md

@@ -0,0 +1,76 @@
+
+# PhotoAcoustic.jl Adjoint and Jacobian derivations. 
+To the best of our knowledge, the literature does not contain a derivation for the adjoint of the forward photoacoustic operator using the second order wave equation and its Jacobian with respect to the speed of sound. 
+
+## Forward model
+The forward model of a photoacoustic experiment can be described by the second order wave equation with active source for space $x$ and time $t$:
+
+$$\frac{1}{c(x)^2}\frac{\partial^2}{\partial t^2}u(x,t) - \nabla^2 u(x,t) = 0$$
+
+where the source is instead defined in the initial state:
+```math
+\begin{align}
+u(x,0) &= p_0 \\
+\dot u(x,0) &= 0.
+\end{align}
+```
+
+The spatial distribution $p_0$ is the initial acoustic source caused by the photonic impulse and is the parameter of interest when performing inversion. 
+
+## Adjoint model derivation
+The derivation is based on procedure and notation from [PDE-constrained optimization and the adjoint method](https://cs.stanford.edu/~ambrad/adjoint_tutorial.pdf). We start by defining our optimization problem:
+
+$$\underset{p_0}{\operatorname{min}} \, \, F(u,p_0) \, \,  \mathrm{where} \, \, F(u,p_0) = \int_{0}^{T}f(u,t)dt = \frac{1}{2} \sum_{r=1}^{R} \int_{0}^{T}\left|| u_{r}(t) - d_{r}(t)\right||_{2}^{2}dt$$
+```math
+\begin{align}
+\text{subject to} \, \, \ddot u &= m\nabla^2 u\\
+u(0) &= p_0 \\
+\dot u(0) &= 0. 
+\end{align}
+```
+Where we have set $m=\frac{1}{c(x)^2}$ to be the slowness squared. For ease of derivation of writing the Lagrangian, we will assign functions to each of these three constraints:
+```math
+\begin{align}
+ h(\ddot u,p_0,t) &= m\nabla^2 u\\
+g(u(0),p_0) &= u(0) - p_0 = 0 \\
+k(\dot u(0)) &= \dot u(0) = 0
+\end{align}
+```
+In order to solve the minimization problem, our goal is to obtain the sensitivity of our functional $F$ with respect to the variable $p_0$:
+
+$$d_{p_0}F.$$
+
+We start by writing out the full Lagrangian containing our functional and constraints: [...can add more details here...]. By selecting the correct multipliers [...can add more details here...] we see that the total derivative is:
+
+$$d_{p_0}F = d_{p_0}L = \int_{0}^{T}[\partial_{p_0}f - \lambda^{\top}\partial_{p_0}h]dt - \dot \lambda(0)^{\top}[\partial_{u(0)}g]^{-1}\partial_{p_0}g + \lambda(0)^{\top}[\partial _{\dot u(0)}k]^{-1} \partial_{p_0}k$$
+
+Where the notation $\partial_{p_0}f$ means the Jacobian of $f$ with respect to $p_0$ and $[\cdot]^{-1}$ is the matrix inverse. Lets simplify the expression by noting the following equalities:
+```math
+\begin{align}
+\partial_{p_0}f &= 0 \\
+\partial_{p_0}h &= 0 \\
+[\partial_{u(0)}g]^{-1} &= -I \\
+\partial_{p_0}g  &= I \\
+\partial_{p_0}k &= 0
+\end{align}
+```
+giving the total derivative:
+
+$$d_{p_0}F = - \dot \lambda(0)^{\top}$$
+
+where $\lambda$ is the adjoint wavefield given by the solution of:
+
+$$\ddot \lambda = \partial_{u}h^{\top}\lambda  -\partial_{u}f^{\top}  \, \, \rightarrow \, \, 
+\ddot \lambda -{m}\nabla^2\lambda = - \partial_{u}f^{\top} =  \sum_{r=1}^{R} \int_{0}^{T}(u_{r}(t) - d_{r}(t)dt.$$
+
+Note that this is the same wave equation as the forward model but with a source term given by the residual $-\partial_{u}f^{\top}$and it is solved in reverse time (from $t=T$ to $t=0$) with the "initial" state:
+```math
+\begin{align}
+ \lambda(T)^{\top} &= 0 \\
+\dot \lambda(T)^{\top} &= 0
+\end{align}
+```
+
+## Jacobian derivation
+TBD
+