A reconstruction of digital parabolas from their least squares fit representation

It is known that the set of digital parabola segments and the set of their least squares parabola fit are in one-to-one correspondence. This enables constant space representation of a digital parabola segment. One of them is a representation of the form (x1,n,α,β,γ), where x1 and n are the x-coordinate of the left endpoint and the number of digital points of the segment, respectively, while α,β and γ are the coefficients of the least squares parabola fit Y = α·X² + β· X + γ for the given parabola segment. This paper gives an O(n·log²n) algorithm for recovering a given digital parabola segment from its proposed code.