Bicubic Interpolation - Bicubic Interpolation

Bicubic Interpolation

Suppose the function values and the derivatives, and are known at the four corners, and of the unit square. The interpolated surface can then be written

The interpolation problem consists of determining the 16 coefficients . Matching with the function values yields four equations,

Likewise, eight equations for the derivatives in the -direction and the -direction

And four equations for the cross derivative .

where the expressions above have used the following identities,

.

This procedure yields a surface on the unit square which is continuous and with continuous derivatives. Bicubic interpolation on an arbitrarily sized regular grid can then be accomplished by patching together such bicubic surfaces, ensuring that the derivatives match on the boundaries.

If the derivatives are unknown, they are typically approximated from the function values at points neighbouring the corners of the unit square, e.g. using finite differences.

Grouping the unknown parameters in a vector,

and letting

,

the problem can be reformulated into a linear equation where its inverse is:

.