ABSTRACT
We study a problem that we call Shape-from-Template, which is the problem of reconstructing the shape of a deformable surface from a single image and a 3D template. Current methods in the literature address the case of isometric deformations, and relax the isometry constraint to the convex inextensibility constraint, solved using the so-called maximum depth heuristic. We call these methods zeroth-order since they use image point locations (the zeroth-order differential structure) to solve the shape inference problem from a perspective image. We propose a novel class of methods that we call first-order. The key idea is to use both image point locations and their first-order differential structure. The latter can be easily extracted from a warp between the template and the input image. We give a unified problem formulation as a system of PDEs for isometric and conformal surfaces that we solve analytically. This has important consequences. First, it gives the first analytical algorithms to solve this type of reconstruction problems. Second, it gives the first algorithms to solve for the exact constraints. Third, it allows us to study the well-posedness of this type of reconstruction: we establish that isometric surfaces can be reconstructed unambiguously and that conformal surfaces can be reconstructed up to a few discrete ambiguities and a global scale. In the latter case, the candidate solution surfaces are obtained analytically. Experimental results on simulated and real data show that our isometric methods generally perform as well as or outperform state of the art approaches in terms of reconstruction accuracy, while our conformal methods largely outperform all isometric methods for extensible deformations.
ABSTRACT
BACKGROUND: Just as power, type I error of cluster detection tests (CDTs) should be spatially assessed. Indeed, CDTs' type I error and power have both a spatial component as CDTs both detect and locate clusters. In the case of type I error, the spatial distribution of wrongly detected clusters (WDCs) can be particularly affected by edge effect. This simulation study aims to describe the spatial distribution of WDCs and to confirm and quantify the presence of edge effect. METHODS: A simulation of 40 000 datasets has been performed under the null hypothesis of risk homogeneity. The simulation design used realistic parameters from survey data on birth defects, and in particular, two baseline risks. The simulated datasets were analyzed using the Kulldorff's spatial scan as a commonly used test whose behavior is otherwise well known. To describe the spatial distribution of type I error, we defined the participation rate for each spatial unit of the region. We used this indicator in a new statistical test proposed to confirm, as well as quantify, the edge effect. RESULTS: The predefined type I error of 5% was respected for both baseline risks. Results showed strong edge effect in participation rates, with a descending gradient from center to edge, and WDCs more often centrally situated. CONCLUSIONS: In routine analysis of real data, clusters on the edge of the region should be carefully considered as they rarely occur when there is no cluster. Further work is needed to combine results from power studies with this work in order to optimize CDTs performance.
