Baseline zone estimation in two dimensions

Abstract: We consider the problem of estimating the region on which a non-parametric regression function is at its baseline level in two dimensions. The baseline level typically corresponds to the minimum/maximum of the function and estimating such regions or their complements is pertinent to several problems arising in edge estimation, environmental statistics, fMRI and related fields. We assume the baseline region to be convex and estimate it via fitting a "stump" function to approximate p-values obtained from tests for deviation of the regression function from its baseline level. The estimates, obtained using an algorithm originally developed for constructing convex contours of a density, are studied in two different sampling settings, one where several responses can be obtained at a number of different covariate-levels (dose-response) and the other involving limited number of response values per covariate (standard regression). The shape of the baseline region and the smoothness of the regression function at its boundary play a critical role in determining the rate of convergence of our estimate: for a regression function which is "p-regular" at the boundary of the convex baseline region, our estimate converges at a rate N −2/(4p+3) in the dose-response setting, N being the total budget, and its analogue in the standard regression setting converges at a rate of N −1/(2p+2) . Extensions to non-convex baseline regions are explored as well

The role of convexity in the corner enhancement effect, in visual short- term memory, in perception of symmetry, and in shape interference.

Contour curvature information has been shown to have an impact on the visual perception of shape. We have conducted studies on perception of convexity and concavity in relation to memory and attention. Previous studies (Badcock & Westheirner, 1985; Krose & Julesz, 1989; Nakayama & Mackeben, 1989) have proposed that visual space is influenced by corners. Recent studies by Cole, Burton and Gellatly (2001) found that reaction times were faster for a stimulus located in the region of a corner of a figure. Cole et al (2001) believe that the role of corners is greater than that of straight edges, due to corners receiving a higher distribution of attentional resources relative to straight edges. The first part of this thesis considers the role figure-ground plays in the corner enhancement effect. Results demonstrate that the corner enhancement effect is only found when the probe is presented on the surface that owns the corner. Thus the corner enhancement effect is present for both concave and convex vertices. However, the effect disappears when the probe lay on the surface that does not own the corner. The second series of experiments made use of a shape with multiple concave or convex features as part of a change detection task, in which only a single feature could change. The results provided no evidence to suggest that convexities are special in visual short-term memory. Though coding of convexities as well as concavities provided a small advantage over an isolated contour. This finding is in accordance with the well documented effect of closure on shape processing (Elder & Zucker, 1993). It has been reported that deviations from symmetry were easier to detect when carried by convexities compared to deviations carried by concavities (Hulleman & Olivers, 2007). We extended this investigation to shapes that were repeated instead of reflected, to test whether there is a specific convexity advantage for bilateral symmetry. The results supported a convexity advantage for repetitions but not for reflections. Possible explanations for this are discussed. The final series of experiments involved a shape interference task; observers responded to circles or square in the context of irrelevant circles and squares. The findings suggest that interference between the shapes is much stronger when the contours that define the shapes belong to the same surface. In summary, we conclude that convexity and concavity are important aspects of shape analysis and representation, but there is no basic difference in how convexities and concavities are attended to, both in the corner enhancement effect, and in visual-short term memory. However, convexity plays a role in some perceptual tasks for example, when analyzing complex shapes observers may adopt strategies that focus on the convexities