Deblending using a space-varying median filter

Median filter

Conventional MF is based on a scalar-value sorting process. When a set of scalars is sorted to form an ascending or descending sequence, the middle value is chosen as standard for this sequence. In signal-processing or geophysical data analysis fields, this filter is commonly used to remove spiky noise. The more general mathematical formulation of a MF is given as:

$$\hat{u}_{i,j}=\arg\min_{u_m\in U_{i,j}}\sum_{l=1}^{L}\Arrowvert u_m-u_l \Arrowvert_p,$$ (1)

where $\hat{u}_{i,j}$ is the output value for location $x_{i,j}$ ; $U_{i,j}=\{u_1,u_2,\cdots,u_L\}$ , $i$ and $j$ are the position indices in a 2-D profile; and $l$ and $m$ are both indices in the filtering window. $L$ is the length of the filtering window and $p$ denotes $L_p$ norm. Commonly, $p=1$ corresponds to a standard MF.