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/n², so 5 harmonics should get us within 4% or so of the actual curve.

References

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