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Polynomials

A 1d-polynomial is defined as:

$\displaystyle f(x) = \sum_{i=0}^{d} \alpha_{i} x^{i}$ (D25)

A 2d-polynomial is defined as:

$\displaystyle f(x,y) = \sum_{i=0}^{d}\sum_{j=0}^{i} \alpha_{i-j, j} x^{i-j} y^{j}$ (D26)

Thus the order of the coefficients (line,pixel) is independent of degree d:
d=0: $ A_{00} $ (1)
d=1: $ A_{10} A_{01} $ (2, 3)
d=2: $ A_{20} A_{11} A_{02} $ (4, 5, 6)
d=3: $ A_{30} A_{21} A_{12} A_{03} $ (7, 8, 9, 10)

Thus the number of coefficients (unknowns in least squares estimation) equals for a 2d-polynomial of degree d:

$\displaystyle \ensuremath{{1\over2}}((d+1)^{2}+d+1)$ (D27)

And the degree of a polynomial with N coefficients is equal to:

$\displaystyle d = \ensuremath{{1\over2}}(int32(\sqrt{1 + 8N}) - 1) - 1$ (D28)



Subsections

Leijen 2009-04-14