# Frequency-Based Adaptive Interpolation Filter in Intra Prediction
## 3 Proposed Method
### Design of 8-Tap Interpolation Filter
\begin{pmatrix}\frac{16}{32}r[i_{0}]+\frac{112}{32}r[i_{0}+1]+\frac{336}{32}r[ i_{0}+2]\\ +\frac{560}{32}r[i_{0}+3]+\frac{560}{32}r[i_{0}+4]+\frac{336}{32}r[i_{0}+5]\\ +\frac{112}{32}r[i_{0}+6]+\frac{16}{32}r[i_{0}+7]\\ =\begin{pmatrix}2&r[i_{0}]+14&r[i_{0}+1]+42&r[i_{0}+2]\\ +70&r[i_{0}+3]+70&r[i_{0}+4]+42&r[i_{0}+5]\\ +14&r[i_{0}+6]+2&r[i_{0}+7]\end{pmatrix}\end{pmatrix}  \tag{10}\]  
The integer reference samples used to derive the 8-tap SIF coefficients are shown in Figure 3, where eight integer samples \(r[i_{0}]\) to \(r[i_{0}+7]\) to derive the filter coefficients for black \(i_{0}+3+16/32\)-pixel position are colored gray and \(r[i_{0}]\) is the start sample of eight \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Index \(i\)** & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline _0/32-pixel filter[i]_ & 4 & 24 & 60 & 80 & 60 & 24 & 4 & 0 \\ _1/32-pixel filter[i]_ & 4 & 24 & 60 & 80 & 60 & 24 & 4 & 0 \\ _2/32-pixel filter[i]_ & 4 & 23 & 58 & 79 & 61 & 26 & 5 & 0 \\ _3/32-pixel filter[i]_ & 4 & 23 & 58 & 79 & 61 & 26 & 5 & 0 \\ _4/32-pixel filter[i]_ & 3 & 21 & 58 & 78 & 62 & 28 & 6 & 0 \\ _5/32-pixel filter[i]_ & 3 & 21 & 58 & 78 & 62 & 28 & 6 & 0 \\ _6/32-pixel filter[i]_ & 3 & 20 & 53 & 76 & 64 & 31 & 8 & 1 \\ _7/32-pixel filter[i]_ & 3 & 20 & 53 & 76 & 64 & 31 & 8 & 1 \\ _8/32-pixel filter[i]_ & 3 & 19 & 51 & 75 & 65 & 33 & 9 & 1 \\ _9/32-pixel filter[i]_ & 3 & 19 & 51 & 75 & 65 & 33 & 9 & 1 \\ _10/32-pixel filter[i]_ & 3 & 18 & 49 & 74 & 66 & 35 & 10 & 1 \\ _11/32-pixel filter[i]_ & 3 & 18 & 49 & 74 & 66 & 35 & 10 & 1 \\ _12/32-pixel filter[i]_ & 3 & 17 & 47 & 73 & 67 & 37 & 11 & 1 \\ _13/32-pixel filter[i]_ & 3 & 17 & 47 & 73 & 67 & 37 & 11 & 1 \\ _14/32-pixel filter[i]_ & 2 & 15 & 44 & 71 & 69 & 40 & 13 & 2 \\ _15/32-pixel filter[i]_ & 2 & 15 & 44 & 71 & 69 & 40 & 13 & 2 \\ _16/32-pixel filter[i]_ & **2** & **14** & **42** & **70** & **70** & **42** & **14** & **2** \\ \hline \end{tabular}
\end{table} 
Table 5: The 8-tap Smoothing Interpolation Filter (SIF) coefficients for interpolation. Figure 2: Derivation of \(z[n]\) from \(h[n]\). reference samples. Filter coefficients are scaled to an integer implementation. All \(p/32\) interpolation filter coefficients in Table 5 are derived by substituting \(p=0\), 1, 2, \(\ldots\), 15 into Equation (9) in the same way as \(p=16\) in Equation (10). Interpolation filters at \(p=17\), 18, \(\ldots\), 31, which are not shown in Table 5, can be easily obtained with even symmetry on \(i_{0}+3.5\), reflecting the filter coefficients of \(p=15\), 14, \(\ldots\), 1, respectively. Tables 4 and 5 show 8-tap DCT-IF and 8-tap SIF coefficients, where index = 0 and index = 7 correspond to \(r[i_{0}]\) and \(r[i_{0}+7]\) in Figure 3, respectively. In Tables 4 and 5, index \(i\) is the position of the integer reference sample. Figure 4 shows the magnitude response at 16/32-pixel positions of 4-tap DCT-IF, 4-tap SIF, 8-tap DCT-IF, and 8-tap SIF, where the X-axis represents the normalized radian frequency and the Y-axis represents the magnitude response. The 8-tap DCT-IF has better HPF characteristics than the 4-tap DCT-IF and the 8-tap SIF has better LPF characteristics than the 4-tap SIF. Therefore, 8-tap SIF results in better interpolation than 4-tap SIF in low-frequency reference samples and 8-tap DCT-IF provides better interpolation than 4-tap DCT-IF in high-frequency reference samples. Figure 5 shows the average correlation values of the reference samples for various video resolutions and each \(nTbS\) defined in Equation (3), which is determined according to the CU size at each picture resolution. The correlation increases as the CU size is larger and the video resolution is higher as shown in Figure 5, where video resolutions A1, A2, B, C, and D are shown in parentheses. The intra CU size partition in video coding depends on the prediction performance to enhance the coding in terms of bit rate and distortion. The prediction performance depends on the prediction errors between the prediction samples and the samples in the current CU.