Appendix: Local similarity

Local similarity between vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as:

$$\mathbf{c}=\sqrt{\mathbf{c}_1\circ\mathbf{c}_2},$$ (10)

where $\circ$ denotes element-wise product. $\mathbf{c}_1$ and $\mathbf{c}_2$ come from two least-squares minimization problems:

$$\min_{\mathbf{c}_1}\Arrowvert \mathbf{a}-\mathbf{B} \mathbf{c}_1 \Arrowvert_2^2,$$ (11)

$$\min_{\mathbf{c}_2}\Arrowvert \mathbf{b}-\mathbf{A} \mathbf{c}_2 \Arrowvert_2^2.$$ (12)

Here, $\mathbf{A}$ is a diagonal operator composed of the elements of $\mathbf{a}$, $\mathbf{B}$ is a diagonal operator composed of the elements of $\mathbf{b}$. Note that in equations 10-12, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ denote vectorized 2D matrices. Equations 11 and 12 can be solved using shaping regularization with a local-smoothness constraint:

$$\mathbf{c}_1 = [\lambda_1^2\mathbf{I} + \mathbf{T}(\mathbf{B}^T\mathbf{B}-\lambda_1^2\mathbf{I})]^{-1}\mathbf{TB}^T\mathbf{b},$$ (13)

$$\mathbf{c}_2 = [\lambda_2^2\mathbf{I} + \mathbf{T}(\mathbf{A}^T\mathbf{A}-\lambda_2^2\mathbf{I})]^{-1}\mathbf{TA}^T\mathbf{a},$$ (14)

where $\mathbf{T}$ is a smoothing operator, $\lambda_1$ and $\lambda_2$ are two parameters controlling the physical dimensionality and enabling fast convergence when inversions for $\mathbf{c}_1$ and $\mathbf{c}_2$ expressed in equations 13 and 14 are implemented iteratively. $\lambda_1$ and $\lambda_2$ can be chosen as $\lambda_1 = \Arrowvert\mathbf{B}^T\mathbf{B}\Arrowvert_2$ and $\lambda_2 = \Arrowvert\mathbf{A}^T\mathbf{A}\Arrowvert_2$.