Let S be the set 0,1,��, 255, then, the function FM : S �� S �� [0,1] given byFMS(p,q)=(min(p,q)+bmax(p,q)+b)a(1)where b is a small positive value for preventing max(p, q) = 0 singularity. As the difference between the components p and q become bigger, the value of FMs falls quickly. Thus, we assume FMs(p, q) is the fuzzy distance between the image components p and q. Clearly, FM is F-founded and it meets0��bImax+b��FMS(p,q)��1(2)for all p,q S, Imax is maximum pixel intensity, and Imax = 255 in this paper.2.2. Deinterlacing ImplementationThe proposed filter consists of three steps: (1) pre-processing step, (2) FM-based weight assignation step, and (3) rank-ordered marginal filter step.

To begin with, we conduct interpolation with three missing pixels at location (?1, 0), (0, 0), and (1, 0), with vertical six-tap filters.

After that, we evaluate FM degree using the introduced FM equation. The obtained FM degree is used for assigning weights. Finally, the missing pixel is calculated using the rank-ordered marginal filtering (ROMF) scheme.Let us assume that I is an image and I(c,r) is the pixel intensity at a position of (c, r), c is column number and r is raw number, and I(0,0) is the centered missing pixel to be processed. We denote W as a filtering window centered on the pixel under processing of size N��N,N = 3,5,7,��, which contains n = N2 pixels. The pixels in W are symbolized as I(c,r), and c, r = ?1, 0,1 for N = 3 case.The first step of the ROMF method is vertical six-tap filter (STF).

This fixed coefficient six-tap Wiener filter is widely used to estimate the sub-pixels in video codec, such as MPEG-4, H.

264/AVC, and some deinterlacing methods [16]. The coefficients of this filter can be different such as h = [1, ?5, 20, 20, ?5, 1]/32 or h = [3, ?17, 78, 78, ?17, 3]/128. In this paper, we chose the previous one for our system under the assumption that h can calculate missing lines in the sub-pixel position properly. The missing pixels at (c, 0) position, c = ?1,0,1, are estimated using the adjacent pixels at (c, GSK-3 ?5), (c, ?3), (c, ?1), (c, 1), (c, 3), and (c,5), and we denote them as I(c,?5), I(c,?3), I(c,?1), I(c,1), I(c,3), and I(c,5), respectively.

To interpolate the pixel AV-951 more precisely, we must adapt the filter to accommodate the new interpolation condition. Now, three pixels in the missing line I(?1,0)STF, I(0,0)STF and I(1,0)STF are approximately deinterlaced applying Equation (3); however, they are not the same with the original missing pixel. Figure 1 shows the pixel positions with filter coefficients.I(c,0)STF=h(1)I(c,?5)+h(2)I(c,?3)+h(3)I(c,?1)+h(4)I(c,1)+h(5)I(c,3)+h(6)I(c,5)(3)Figure 1.The pixel positions with filter coefficients.