Editing Bicubic interpolation (section)

==Bicubic convolution algorithm==

Bicubic spline interpolation requires the solution of the linear system described above for each grid cell. An interpolator with similar properties can be obtained by applying a [[convolution]] with the following kernel in both dimensions:
<math display="block">W(x) = 
\begin{cases}
 (a+2)|x|^3-(a+3)|x|^2+1 & \text{for } |x| \leq 1, \\
 a|x|^3-5a|x|^2+8a|x|-4a & \text{for } 1 < |x| < 2, \\
 0                       & \text{otherwise},
\end{cases}
</math>
where <math>a</math> is usually set to −0.5 or −0.75. Note that <math>W(0)=1</math> and <math>W(n)=0</math> for all nonzero integers <math>n</math>.

This approach was proposed by Keys, who showed that <math>a=-0.5</math> produces third-order convergence with respect to the sampling interval of the original function.<ref name=Keys>{{cite journal
 | author = R. Keys
 | year = 1981
 | title = Cubic convolution interpolation for digital image processing
 | journal = IEEE Transactions on Acoustics, Speech, and Signal Processing
 | doi = 10.1109/TASSP.1981.1163711
 | volume = 29
 | pages = 1153–1160
 | issue = 6
| bibcode = 1981ITASS..29.1153K
 | citeseerx = 10.1.1.320.776
 }}</ref>

If we use the matrix notation for the common case <math>a = -0.5</math>, we can express the equation in a more friendly manner:
<math display="block">p(t) =
\tfrac{1}{2}
\begin{bmatrix}
1 & t & t^2 & t^3
\end{bmatrix}

\begin{bmatrix}
0 & 2 & 0 & 0 \\
-1 & 0 & 1 & 0 \\
2 & -5 & 4 & -1 \\
-1 & 3 & -3 & 1
\end{bmatrix}

\begin{bmatrix}
f_{-1} \\
f_0 \\
f_1 \\
f_2
\end{bmatrix}
</math>
for <math>t</math> between 0 and 1 for one dimension. Note that for 1-dimensional cubic convolution interpolation 4 sample points are required. For each inquiry two samples are located on its left and two samples on the right. These points are indexed from −1 to 2 in this text. The distance from the point indexed with 0 to the inquiry point is denoted by <math>t</math> here.

For two dimensions first applied once in <math>x</math> and again in <math>y</math>:
<math display="block">\begin{align}
b_{-1}&= p(t_x, f_{(-1,-1)}, f_{(0,-1)}, f_{(1,-1)}, f_{(2,-1)}), \\[1ex]
b_{0} &= p(t_x, f_{(-1,0)}, f_{(0,0)}, f_{(1,0)}, f_{(2,0)}), \\[1ex]
b_{1} &= p(t_x, f_{(-1,1)}, f_{(0,1)}, f_{(1,1)}, f_{(2,1)}), \\[1ex]
b_{2} &= p(t_x, f_{(-1,2)}, f_{(0,2)}, f_{(1,2)}, f_{(2,2)}),
\end{align}</math>
<math display="block">p(x,y) = p(t_y, b_{-1}, b_{0}, b_{1}, b_{2}).</math>