Fused Multi-Modal Electron Microscopy

Jason Manassa; Miti Shah; Min Gee Cho; Zichao Wendy Di; Yi Jiang; Jeffrey A. Fessler; Yu-Tsun Shao; Mary C. Scott; Jonathan Schwartz; Robert Hovden

doi:10.69761/MXVR4353

Computational Article

Elemental Microscopy

Contents

Algorithms¶

Algorithm 6.1 (Multi-Modal 2D Data Fusion)

Objective Function:
$\Psi(x) = \frac{1}{2}\left\|\sum_i x_i - b_H\right\|^2 + \lambda_{Chem}\sum_i\left(1^T x_i - b_{i}^T\log(x_i + \epsilon)\right) + \lambda_{TV}\sum_i \left\|x_i\right\|_{TV}$

Inputs:
$b_H \in \mathbb{R}^{n_x \times n_y} \rightarrow$ HAADF Micrograph (Image Dimensions: $n_x \times n_y$ )
$b_C \in \mathbb{R}^{n_e \times n_x \times n_y} \rightarrow$ Raw Chemical Maps (Total # of Elements: $n_e$ )
$N_{iter} = 30 \rightarrow$ Number of Main Cost Function Iterations
$N_{iter TV} = 3 \rightarrow$ Number of Total Variation (TV) Iterations
$\epsilon = 0.2 \rightarrow$ Background Noise Threshold
$\lambda_{HAADF} = \frac{1}{n_e} \rightarrow$ HAADF Weight
$\lambda_{Chem} = 0.08 \rightarrow$ Data Consistency Weight, Ranges from 0 to 1
$\lambda_{TV} = 0.15 \rightarrow$ Regularization Weight, Ranges from 0 to 1
$γ = 1.6 \rightarrow$ Incoherent Linear Imaging factor

Output:
$x \in \mathbb{R}^{n_e \times n_x \times n_y} \rightarrow$ Concatenated Vector of Recovered Maps

Main Function:

Initialize first iterate as raw elemental maps $x_i^0 = b_C$
For $k = 1$ to $N_{iter}$ :
1. $x^k = x^{k-1} - (\lambda_{HAADF}\nabla \Psi_1(x^k) + \lambda_{Chem} \nabla \Psi_2(x^k))$
2. For $i = 1$ to $n_e$ :
  1. $x_i^k = \text{TV\_FGP\_2D}(x_i^k, \lambda_{TV}, N_{iter TV}) \text{ or TV\_GP\_2D}(x_i^k, \lambda_{TV}, N_{iter TV})$
3. End for
End for
Return $x^{N_{iter}}$

Definitions:
$\nabla \Psi_1(x) = -\gamma\text{diag}(x^{-1})A^T(b_H - Ax^{\gamma})$
$\nabla \Psi_2(x) = 1 - b \oslash (x + \epsilon)$
$\oslash$ is element wise division

Algorithm 6.2 (2D Total Variation Fast Gradient Projection Method)

Inputs:
$b \in \mathbb{R}^{n_x \times n_y} \rightarrow$ 2D Image, $\lambda \rightarrow$ Regularization Parameter, $N_{iter TV} \rightarrow$ Number of Iterations

Output:
$x \in \mathbb{R}^{n_e \times n_x \times n_y} \rightarrow$ Concatenated Vector of Recovered Maps

TV_FGP_2D Function:

$({r}_x, {r}_y) = ({p}_x, {p}_y) = \left({0}_{(m-1) \times n}, {0}_{m \times (n-1)}\right), \quad t_1 = 1$
For $k = 1, \ldots, N_{iter TV}$ :
1. $({p}_x, {p}_y) = {P}_p \left[ ({r}_x, {r}_y) + \frac{1}{8\lambda} \mathcal{L}^T\left({P}_c\left[b - \lambda \mathcal{L}({r}_x, {r}_y)\right]\right) \right]$
2. $t_{k+1} = \frac{1 + \sqrt{1 + 4t_k^2}}{2}$
3. $({r}_x^{k+1}, {r}_y^{k+1}) = ({p}_x, {p}_y) + \frac{t_k - 1}{t_{k+1}}\left(({p}_x - {p}_x^{k-1}, {p}_y - {p}_y^{k-1})\right)$
End for
Return $x^* = {P}_c \left[b - \lambda \mathcal{L}({p}_x^N, {p}_y^N)\right]$

Algorithm 6.3 (2D Total Variation Gradient Projection Method)

Inputs:
$b \in \mathbb{R}^{n_x \times n_y} \rightarrow$ 2D Image, $\lambda \rightarrow$ Regularization Parameter, $N_{iter TV} \rightarrow$ Number of Iterations

Output:
$x \in \mathbb{R}^{n_e \times n_x \times n_y} \rightarrow$ Concatenated Vector of Recovered Maps

TV_GP_2D Function:

$p_x^0 = \mathbf{0} \in \mathbb{R}^{(m-1) \times n}$ , $p_y^0 = \mathbf{0} \in \mathbb{R}^{m \times (n-1)}$
For $k = 1, \ldots, N_{iter TV}$ :
1. $(p_x^k, p_y^k) = P_p(p_x^{k-1}, p_y^{k-1}) + \frac{1}{8}\mathcal{L}^T(P_c(b - \lambda \mathcal{L}(p_x^{k-1}, p_y^{k-1})))$
End for
Return $x^* = P_c(b - \lambda \mathcal{L}(p_x^{N_{iter TV}}, p_y^{N_{iter TV}}))$