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

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.

A novel hypothesis for the original functionality of the Visual Word Form Area: Processing shape sequences

There is ongoing debate about what characteristics of left ventral occipitotemporal cortex drive development of the Visual Word Form Area (VWFA). We offer a new hypothesis. A summary of occipitotemporal organization indicates that the VWFA falls in a cortical region supporting action analysis, rather than object recognition. We discuss evidence that letters are serially processed in a top-down manner during the initial years of reading acquisition, and propose that this sequential activation of letter representations causes the VWFA to develop in motion-sensitive cortex specialized for processing of non-biological shape sequences. Supporting this hypothesis, a new fMRI analysis identifies a left-lateralized region that responds more strongly to dynamic motion of objects than humans; this region's location (-48, -55, -8) falls almost exactly at the canonical VWFA coordinates (-45, -57, -12).

Hydrodynamic performance of suction feeding is virtually unaffected by variation in the shape of the posterior region of the pharynx in fish

To capture prey by suction, fish generate a flow of water that enters the mouth and exits at the back of the head. It was previously hypothesized that prey-capture performance is improved by a streamlined shape of the posterior region of the pharynx, which enables an unobstructed outflow with minimal hydrodynamic resistance. However, this hypothesis remained untested for several decades. Using computational fluid dynamics simulations, we now managed to quantify the effects of different shapes of the posterior pharynx on the dynamics of suction feeding, based on a feeding act of a sunfish ( Lepomis gibbosus ). In contrast to what was hypothesized, the effects of the imposed variation in shape were negligible: flow velocity patterns remained essentially identical, and the effects on feeding dynamics were negligibly small. This remarkable hydrodynamic insensitivity implies that, in the course of evolution, the observed wedge-like protrusions of the pectoral surfaces of the pharynx probably resulted from spatial constraints and/or mechanical demands on the musculoskeletal linkages, rather than constraints imposed by hydrodynamics. Our study, therefore, exceptionally shows that a streamlined biological shape subjected to fluid flows is not always the result of selection for hydrodynamic improvement.

Landmark-free geometric methods in biological shape analysis

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map f between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy E sd ( f ) which we introduce. This measures the distance of f from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.