Perfect representation of a convex 2D contour with only one Fourier component

Abstract

Shape analysis of a closed 2D contour is an important topic within biological shape analysis, where Fourier methods to reproduce the shape with a limited number of parameters have been and still are of vital importance. An example is within marine management research on fish, where shape analysis of otolith (earstone) contours is performed for species identification as well as for stock discrimination purposes. In both cases, it is expected that the fewer parameters that are needed in a method to reproduce the contour sufficiently good, the better. This contribution outlines how a convex contour of any shape can be represented to any wanted accuracy by only one Fourier component. The key idea is to allow a flexible choice of a predetermined number of x-values along an x-axis that goes through the two most distant points of the contour. The y-variable along the perpendicular y-axis is then monotonically transformed to a z-variable so that the minium and maximum z-values on the contour have the same distance from the x-axis. The x-values of the contour points are now chosen so that the corresponding z-values on the contour follows a perfect sinusoid if the x-values were equidistant. The method is illustrated by application to lasso contours of Norwegian Coastal Cod (NCC) and North East Arctic Cod (NEAC) otolith images, where the average new x-positions for the individual otolith contours were applied to all otoliths. The results show that a considerably better fit to the original individual otolith contours were obtained by applying the invers FFT to the new y-values than by the frequently applied 2D EFDs (Elliptical Fourier Descriptors) approach, for the same number, m < 11, of frequency components. A promising classification result was also obtained by the linear Fisher discrimination method and cross validation applied to the individual x-values for the NCC and NEAC otoliths, with 82% score for NCC and 80% score for NEAC with sample sizes 367 and 240, respectively.