Subset Warping: Rubber Sheeting with Cuts

Image warping, often referred to as "rubber sheeting" represents the deformation of a domain image space into a range image space. In this paper, a technique is described which extends the definition of a rubber-sheet transformation to allow a polygonal region to be warped into one or more subsets of itself, where the subsets may be multiply connected. To do this, it constructs a set of "slits" in the domain image, which correspond to discontinuities in the range image, using a technique based on generalized Voronoi diagrams. The concept of medial axis is extended to describe inner and outer medial contours of a polygon. Polygonal regions are decomposed into annular subregions, and path homotopies are introduced to describe the annular subregions. These constructions motivate the definition of a ladder, which guides the construction of grid point pairs necessary to effect the warp itself

Automated visual tracking for studying the ontogeny of zebrafish swimming

The zebrafish Danio rerio is a widely used model organism in studies of genetics, developmental biology, and recently, biomechanics. In order to quantify changes in swimming during all stages of development, we have developed a visual tracking system that estimates the posture of fish. Our current approach assumes planar motion of the fish, given image sequences taken from a top view. An accurate geometric fish model is automatically designed and fit to the images at each time frame. Our approach works across a range of fish shapes and sizes and is therefore well suited for studying the ontogeny of fish swimming, while also being robust to common environmental occlusions. Our current analysis focuses on measuring the influence of vertebra development on the swimming capabilities of zebrafish. We examine wild-type zebrafish and mutants with stiff vertebrae (stocksteif) and quantify their body kinematics as a function of their development from larvae to adult (mutants made available by the Hubrecht laboratory, The Netherlands). By tracking the fish, we are able to measure the curvature and net acceleration along the body that result from the fish's body wave. Here, we demonstrate the capabilities of the tracking system for the escape response of wild-type zebrafish and stocksteif mutant zebrafish. The response was filmed with a digital high-speed camera at 1500 frames s–1. Our approach enables biomechanists and ethologists to process much larger datasets than possible at present. Our automated tracking scheme can therefore accelerate insight in the swimming behavior of many species of (developing) fish

Principal component and Voronoi skeleton alternatives for curve reconstruction from noisy point sets

Surface reconstruction from noisy point samples must take into consideration the stochastic nature of the sample -- In other words, geometric algorithms reconstructing the surface or curve should not insist in following in a literal way each sampled point -- Instead, they must interpret the sample as a “point cloud” and try to build the surface as passing through the best possible (in the statistical sense) geometric locus that represents the sample -- This work presents two new methods to find a Piecewise Linear approximation from a Nyquist-compliant stochastic sampling of a quasi-planar C1 curve C(u) : R → R3, whose velocity vector never vanishes -- One of the methods articulates in an entirely new way Principal Component Analysis (statistical) and Voronoi-Delaunay (deterministic) approaches -- It uses these two methods to calculate the best possible tape-shaped polygon covering the planarised point set, and then approximates the manifold by the medial axis of such a polygon -- The other method applies Principal Component Analysis to find a direct Piecewise Linear approximation of C(u) -- A complexity comparison of these two methods is presented along with a qualitative comparison with previously developed ones -- It turns out that the method solely based on Principal Component Analysis is simpler and more robust for non self-intersecting curves -- For self-intersecting curves the Voronoi-Delaunay based Medial Axis approach is more robust, at the price of higher computational complexity -- An application is presented in Integration of meshes originated in range images of an art piece -- Such an application reaches the point of complete reconstruction of a unified mes

Stenomaps: Shorthand for shapes

We address some of the challenges in representing spatial data with a novel form of geometric abstraction-the stenomap. The stenomap comprises a series of smoothly curving linear glyphs that each represent both the boundary and the area of a polygon. We present an efficient algorithm to automatically generate these open, C1-continuous splines from a set of input polygons. Feature points of the input polygons are detected using the medial axis to maintain important shape properties. We use dynamic programming to compute a planar non-intersecting spline representing each polygon's base shape. The results are stylised glyphs whose appearance may be parameterised and that offer new possibilities in the 'cartographic design space'. We compare our glyphs with existing forms of geometric schematisation and discuss their relative merits and shortcomings. We describe several use cases including the depiction of uncertain model data in the form of hurricane track forecasting; minimal ink thematic mapping; and the depiction of continuous statistical data