Optimal image interpolation using local low-order surfaces

Desirable features of any digital image resolution-enhancement algorithm include exact interpolation (for "distortionless" or "lossless" processing) adjustable resolution, adjustable smoothness, and ease of computation. A given low-order polynomial surface (linear, quadratic, cubic, etc.) optimally fit by least squares to a given local neighborhood of a pixel to be interpolated can enable all of these features. For example, if the surface is cubic, if a pixel and the 5-by-5 pixel array surrounding it are selected, and if interpolation of this pixel must yield a 4-by-4 array of sub-pixels, then the 10 coefficients that define the surface may be determined by the constrained least squares solution of 25 linear equations in 10 unknowns, where each equation sets the surface value at a pixel center equal to the pixel gray value and where the constraint is that the mean of the surface values at the sub-pixel centers equals the gray value of the interpolated pixel. Note that resolution is adjustable because the interpolating surface for each pixel may be subdivided arbitrarily, that smoothness is adjustable (within each pixel) because the polynomial order and number neighboring pixels may be selected, and that the most computationally demanding operation is solving a relatively small number of simultaneous linear equations for each pixel.