tricubic

Tricubic interpolation is a method for estimating values on a three-dimensional grid by fitting a cubic polynomial to data in the vicinity of the target point. A common approach uses a 4 by 4 by 4 neighborhood of grid points and constructs a cubic polynomial in x, y, and z, so the interpolated value is a weighted combination of the 64 surrounding samples. The result is a smooth function that can provide continuous first and second derivatives with appropriate coefficient schemes.

Implementation variants exist. In separable cubic interpolation, one performs cubic interpolation along x for each of the 4 by 4 slices, then along y, and finally along z. This effectively uses a 1D cubic kernel in each axis. Other approaches determine the full set of 64 coefficients to match function values (and possibly derivatives) at the grid points, yielding tricubic Hermite interpolation or tricubic convolution based on kernel methods such as Keys’ cubic convolution.

Compared with trilinear interpolation, tricubic interpolation offers smoother results and can achieve higher continuity, at the cost of greater computational effort and memory usage due to the larger stencil and more complex coefficient calculations. The choice of variant influences smoothness and accuracy, with some schemes prioritizing exact derivative matches and others prioritizing simple, fast kernels.

Applications of tricubic interpolation include volume rendering, medical and scientific imaging, 3D texture filtering in computer graphics, and any scenario requiring smooth resampling of three-dimensional data on a regular grid. It is one of several techniques for interpolating multidimensional data, with alternatives such as trilinear and spline-based methods.