Deblending using a space-varying median filter

Space-varying median filter

For SVMF, $L$ becomes $L_{i,j}$ , varying with respect to location $x_{i,j}$ . The new filtering expression is:

$$\hat{v}_{i,j} = \arg\min_{v_{m}\in U_{i,j} }\sum_{l=1}^{L_{i,j}} \Arrowvert v_{m} -v_l \Arrowvert_p,$$ (2)

where $\hat{v}_{i,j}$ is the output value for location $x_{i,j}$ after applying a SVMF, $U_{i,j}=\{v_1,v_2,\cdots,v_{L_{i,j}}\}$ . The filter length $L_{i,j}$ can be chosen through the following empirical criteria:

$$L_{i,j}=\left\{\begin{array}{ll} L+l_1,\quad 0\quad \le \vert s^L... ... L-l_4,\quad 0.85s_{max}\le\vert s^L_{i,j}\vert \le s_{max} \end{array}\right.,$$ (3)

where $l_1$ ,$l_2$ ,$l_3$ ,$l_4$ are predefined parameters corresponding to the increments or decrements for the length of filter window and are generally chosen as 4,2,2,4 in default, respectively; $s^L_{i,j}$ is the signal reliability (SR), which can be defined as the local similarity (Fomel, 2007) between the initially filtered data $u^L_{i,j}$ with a window length $L$ and the original data $u_{i,j}$ for point $x_{i,j}$ :

$$s^L_{i,j} = \mathbf{S} [u^L_{i,j}, u_{i,j}]$$ (4)

Here, $\mathbf{S} [\mathbf{x},\mathbf{y}]$ denotes the local similarity between $\mathbf{x}$ and $\mathbf{y}$ , and $s_{max}$ denotes the maximum value of the similarity map. Appendix A gives a short review of local similarity.

Deblending using a space-varying median filter