Random noise attenuation using local signal-and-noise orthogonalization

Appendix B: Review of local similarity

Fomel (2007a) defined local similarity between vectors $\mathbf{a}$ and $\mathbf{b}$ as:

$$\mathbf{c}=\sqrt{\mathbf{c}_1^T\mathbf{c}_2}$$ (14)

where $\mathbf{c}_1$ and $\mathbf{c}_2$ come from two least-squares minimization problems:

$$\mathbf{c}_1 =\arg\min_{\mathbf{c}_1}\Arrowvert \mathbf{a}-\mathbf{B}\mathbf{c}_1 \Arrowvert_2^2,$$ (15)

$$\mathbf{c}_2 =\arg\min_{\mathbf{c}_2}\Arrowvert \mathbf{b}-\mathbf{A}\mathbf{c}_2 \Arrowvert_2^2,$$ (16)

where $\mathbf{A}$ is a diagonal operator composed from the elements of $\mathbf{a}$ : $\mathbf{A}=diag(\mathbf{a})$ and $\mathbf{B}$ is a diagonal operator composed from the elements of $\mathbf{b}$ : $\mathbf{B}=diag(\mathbf{b})$ . Least-squares problems B-2 and B-3 can be solved with the help of shaping regularization with a smoothness constraint:

$$\mathbf{c}_1 = [\lambda_1^2\mathbf{I} + \mathcal{T}(\mathbf{B}^T\mathbf{B}-\lambda_1^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{B}^T\mathbf{a},$$ (17)

$$\mathbf{c}_2 = [\lambda_2^2\mathbf{I} + \mathcal{T}(\mathbf{A}^T\mathbf{A}-\lambda_2^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{A}^T\mathbf{b},$$ (18)

where $\mathbf{\mathcal{T}}$ 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 $\lambda_2 = \Arrowvert\mathbf{A}^T\mathbf{A}\Arrowvert_2$ (Fomel, 2007a). The definition of $\mathbf{c}_1$ and $\mathbf{c}_2$ are equivalent to definition of LOW in this paper.

Random noise attenuation using local signal-and-noise orthogonalization