Estimating the shape and appearance of a three dimensional object from flat images is a challenging research topic that is still actively pursued. Among the various techniques available, Photometric Stereo (PS) is known to provide very accurate local shape recovery in terms of surface normals. In this work we propose to minimise non-convex variational models for PS that recover the depth information directly.Photometric Stereo consists in finding a depth map z that best explains all image irradiance equations (IIEs) I i = R(z; s i , ρ), for several images I i , considered under different lightings s i , with i ∈ {1, . . . , m}. The function R describes our reflectance model in terms of the depth z, the lighting s i , and the albedo ρ. We assume Lambertian reflectance, neglect shadows, and require m 3.Our approach uses a variational framework with a least-squares penalisation on the IIEs augmented with a zero-th order Tikhonov regularisation. The obtained energy (1) is non-convex and we make use of matrix differential theory and recent developments in non-convex and nonsmooth optimisation to determine good minimisers. min z,ρ
We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.
Estimating shape and appearance of a three dimensional object from a given set of images is a classic research topic that is still actively pursued. Among the various techniques available, photometric stereo is distinguished by the assumption that the underlying input images are taken from the same point of view but under different lighting conditions. The most common techniques provide the shape information in terms of surface normals. In this work, we instead propose to minimise a much more natural objective function, namely the reprojection error in terms of depth. Minimising the resulting non-trivial variational model for photometric stereo allows to recover the depth of the photographed scene directly. As a solving strategy, we follow an approach based on a recently published optimisation scheme for non-convex and non-smooth cost functions.The main contributions of our paper are of theoretical nature. A technical novelty in our framework is the usage of matrix differential calculus. We supplement our approach by a detailed convergence analysis of the resulting optimisation algorithm and discuss possibilities to ease the computational complexity. At hand of an experimental evaluation we discuss important properties of the method. Overall, our strategy achieves more accurate results than competing approaches. The experiments also highlights some practical aspects of the underlying optimisation algorithm that may be of interest in a more general context. G. Radow · L. Hoeltgen · M. Breuß Chair for Applied Mathematics
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