Currents and finite elements as tools for shape space

The nonlinear spaces of shapes (unparameterized immersed curves or submanifolds) are of interest for many applications in image analysis, such as the identification of shapes that are similar modulo the action of some group. In this paper we study a general representation of shapes that is based on linear spaces and is suitable for numerical discretization, being robust to noise. We develop the theory of currents for shape spaces by considering both the analytic and numerical aspects of the problem. In particular, we study the analytical properties of the current map and the $H^{-s}$ norm that it induces on shapes. We determine the conditions under which the current determines the shape. We then provide a finite element discretization of the currents that is a practical computational tool for shapes. Finally, we demonstrate this approach on a variety of examples

Categorical Colormap Optimization with Visualization Case Studies

—Mapping a set of categorical values to different colors is an elementary technique in data visualization. Users of visualization software routinely rely on the default colormaps provided by a system, or colormaps suggested by software such as ColorBrewer. In practice, users often have to select a set of colors in a semantically meaningful way (e.g., based on conventions, color metaphors, and logological associations), and consequently would like to ensure their perceptual differentiation is optimized. In this paper, we present an algorithmic approach for maximizing the perceptual distances among a set of given colors. We address two technical problems in optimization, i.e., (i) the phenomena of local maxima that halt the optimization too soon, and (ii) the arbitrary reassignment of colors that leads to the loss of the original semantic association. We paid particular attention to different types of constraints that users may wish to impose during the optimization process. To demonstrate the effectiveness of this work, we tested this technique in two case studies. To reach out to a wider range of users, we also developed a web application called Colourmap Hospital