## Fourier interpolation of a Plane Curve

**Introduction**

Ghosh and Jain (1) showed how to charactize a simple, closed, piece-wise linear
curve as a Fourier series in a single parametric variable. Such a formulation
allows:

- interpolation of curves by sampling the parameter more densely.
- morphing of one curve into another by locating matching parametric points
on the two curves and smoothly interpolating the Fourier components.
- simplification of curves by removing Fourier components. This maintains
the overall shape, but eliminates sharp corners.
- location of the curve in the plane as the DC components of the series.
- ability to "add" curves.
- ability to add band-limited noise to a curve.
- ability to connect serveral curves (possibly representing serial sections)
to make a 3D object.

We intend to use the Fourier fit scheme as a way to simplify and characterize
the outlines of moving animals. The animals will be videotaped. The videotape
will then be analysed to extract the outline of the animal, Fourier fit the
outline, then use the low bandwidth components to characterize the motion. For
example, the DC components give the position of the animal. The fundamental
frequency coefficients of the fit give the ellipse (2) which best fits the animal
outline.

**Code**

A matlab program was written to animate a test
object, extract its outline, then do the Fourier fit.

**Results**

The shape that was animated was chosen to be a very
simple approximation of a insect larvae turning. There are 4 reconstructions
linked below:

- When the reconstruction is limited to the fundamental (n=1), we get an ellipse.
The x in the center of the elipse is the program's estimate of the DC component
(position).
- Adding in n=2 shows some bending.
- Adding n=3 shows some of the rectangular shape, since
right angle information is carried in the 3rd harmonic.
- Adding n=4 and 5 shows fairly good reconstruction.
The Fourier series terms diminish in amplitude as 1/n
^{2}, so 5 harmonics
should get us within 4% or so of the actual curve.

**References**

- Ghosh, PK and Jain, PK (1993)
*An Algebra of Geometric Shapes*, IEEE
Computer Graphics and Applications, vol 13 pp 50-59, issue 5
- Rouben Rostamian,
*Equation of an Ellipse*, http://mathforum.org/epigone/geom.puzzles/27/ce2iei180me9@forum.mathforum.com