1 Elliptic HarmonicsIn the complex notation z(u) is represented as a series of complex exponentials.z(s) =infXn=− infzne2πjnsL= z0+infXn=1(zne2πjnsL+ z−ne−2πjnsL)where the complex coefficient zncan be expressed in polar notation, i.e.zn= rnejψn,with rn∈ R, rn≥ 0, and ψ ∈ R. The terms e2πjnsLdescribe rotations as a funct. of arclength s.We have demonstrated that the terms (zne2πjnsL+ z−ne−2πjnsL) form ellipses that are traversedn times while traversing the figure from 0 to L. Remember the demonstration with pairs ofphasors of different length |zn| rotating clockwise and counterclockwise, and resulting vectorbeing the sum of the clockwise phasor zn= {a, b} and the counterclockwise phasor z−n= {c, d}.1.1 Normalization in object and parameter spaceThe normalization proposed by Kuhl and Giardina is based on the ellipse defined by the 1storderFourier descriptors and is carried out both in object and in parameter space. Normalization inobject space effects the curve’s position, orientation and size, while that in parameter spaceapplies to the curve parametrization behind it. After normalization in object space the centerof the 1storder ellipse of a normalized contour concurs with the coordinate origin, its main axisoverlaps with the x -axis of the coordinate system has the length of 1. In parameter space thestarting point of the parametrization is moved to a standard position defined by the crossing ofthe 1storder ellipse and its main axis.1.1.1 Dependence on starting pointTo make the descriptors independent on the starting point of the parametrization, this can beshifted to a standard position, e.g. to the tip of the ellipse defined by the 1storder Fourierdescriptors. This can be thought of as a rotation in parameter space U given by the unit circle.The transformation is defined byzn|V= znejnθ, (1)where the notation |Vmarks the coefficients resulting from shifting by angle θ.1.1.2 Dependence on rotational positionIn the complex notation, rotation in object space by angle ψ is simply a multiplication by e−jψ.Applying it to 1 immediately reveals the coefficients of the rotated object.zn|R= znejψ(2)To achieve a standardized position of the curve, it is rotated in a way that its first ellipse’s mainaxis matches the horizontal (real) coordinate axis.1.1.3 Scale dependenceScaling the objects by factor α leads to multiplying its coefficients by the same factor:zn|S= αzn(3)The scaling factor α is usually set to normalize the half major axis to unity, meaningα =1|z1| + |z−1|=1r1+ r−1(4)1.1.4 Invariant Fourier descriptorsIgnoring z0, that is setting z0|T= 0, achieves translation invariance. Summing up all standard-izations; the invariant coefficients are denoted ˜zn:zn|V,R,S,T= ˜zn= znej(nθ−ψ)r1+ r−1(5)˜z0= 0 (6)Figure 1: Normalization steps of Fourier coefficients; shifting of the starting point to the tip ofthe ellipse (a), moving the center of gravity to the coordinate origin (b), rotating the main axisof the ellipse to the real axis (c), and finally scaling the half major axis to unity (d).1.2 Relationship to real valued notationThere is a close relationship between the complex and real notation (see the document Kelemen-EllipticHarmonicsOnly.pdf), i.e. they can be converted into each other.In the complex notation of Fourier coefficients real and imaginary parts of zncorrespond to thex and y coordinatesxyn=anbncndnsin2πnsLcos2πnsL,where the real valued coefficients an, bn, cn, and dnare defined as follows.an= Rezn+ Rez−nbn= −Imzn+ Imz−ncn= Imzn+ Imz−ndn= Rezn−