regrid(x,xgrid,ygrid,sx,sy) and
regrid3(x,xgrid,ygrid,sx,sy) work as described below. The mapped
image values are interpolated from the source image x on a double
grid. The coordinates of the points of the coarse grid, measured in the
coordinate system of x, are given by xgrid and
ygrid. A fine grid of sx by sy points is spread
evenly over each cell of the coarse grid, and each point of the fine
grid corresponds to a single element of the mapped image. The image
value at each point of the fine grid is determined through
nearest-neighbor (regrid) or bicubic (regrid3)
interpolation.
For example, the x coordinates of the points on the fine grid in the
cell identified by cell indices i and j are given by
(xgrid(i+1,j) - xgrid(i,j))*ix/sx + (xgrid(i,j+1) -
xgrid(i,j))*iy/sy + (xgrid(i+1,j+1) + xgrid(i,j) - xgrid(i+1,j) -
xgrid(i,j + 1))*ix*iy/(sx*sy) for integer ix between 0 and
sx - 1 and integer iy between 0 and sy - 1.
For linear transformations, such as combinations of rotations, scalings, and translations, only one grid cell (two by two coordinates) is needed.
The bicubic interpolation (for regrid3) is based on the sequence
i = floor(x) b = x - i a = 1 - b w1 = -0.5*b*a*a w2 = 1 + b*b*(1.5*b - 2.5) w3 = 1 + a*a*(1.5*a - 2.5) w4 = -0.5*a*b*b y_interp = y(i-1)*w1 + y(i)*w2 + y(i+1)*w3 + y(i+2)*w4
which yields y_interp at ordinate x, interpolated from
y. Application in two dimensions yields bicubic interpolation.
This particular form of bicubic interpolation is due to s.k. Park and
r.a. Schowengerdt (1983: Comp. Vis. & Im. Proc., Vol. 23, p. 258; with
alpha = -0.5).