My question arises from this reading link (In this article, in equation 6 on page 3, also expands a function of 2 variables in this way Parametric estimate of intensity inhomogeneities applied to MRI), where on page 10 expand a function of 2 variables in a sum of two-dimensional legendre polynomials:
My question is why in this case they do not use all the orthogonal combinations Pi (x) Pj (y) in the summation as is usual in other cases such as for example the discrete expansion of fourier:
