Appendix B: local similarity

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

$\displaystyle \mathbf{c}=\sqrt{\mathbf{c}_1\circ\mathbf{c}_2}$ (26)

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

$\displaystyle \mathbf{c}_1$ $\displaystyle =\arg\min_{\mathbf{c}_1}\Arrowvert \mathbf{a}-\mathbf{B} \mathbf{c}_1 \Arrowvert_2^2$ (27)
$\displaystyle \mathbf{c}_2$ $\displaystyle =\arg\min_{\mathbf{c}_2}\Arrowvert \mathbf{b}-\mathbf{A} \mathbf{c}_2 \Arrowvert_2^2$ (28)

where $\mathbf{A}$ 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 26-28, $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ denote vectorized 2D matrices. Equations 27 and 28 can be solved using shaping regularization with a local-smoothness constraint:

$\displaystyle \mathbf{c}_1$ $\displaystyle = [\lambda_1^2\mathbf{I} + \mathbf{T}(\mathbf{B}^T\mathbf{B}-\lambda_1^2\mathbf{I})]^{-1}\mathbf{TB}^T\mathbf{b},$ (29)
$\displaystyle \mathbf{c}_2$ $\displaystyle = [\lambda_2^2\mathbf{I} + \mathbf{T}(\mathbf{A}^T\mathbf{A}-\lambda_2^2\mathbf{I})]^{-1}\mathbf{TA}^T\mathbf{a},$ (30)

where $\mathbf{T}$ is a smoothing operator and $\lambda_1$ and $\lambda_2$ are two parameters controlling the physical dimensionality and enabling fast convergence when inversion is implemented iteratively. These two parameters can be chosen as $\lambda_1 = \Arrowvert\mathbf{B}^T\mathbf{B}\Arrowvert_2$ and $\lambda_2 = \Arrowvert\mathbf{A}^T\mathbf{A}\Arrowvert_2$.


2020-03-27