Review of the mathematical foundations of data fusion techniques in surface metrology

The recent proliferation of engineered surfaces, including freeform and structured surfaces, is challenging current metrology techniques. Measurement using multiple sensors has been proposed to achieve enhanced benefits, mainly in terms of spatial frequency bandwidth, which a single sensor cannot provide. When using data from different sensors, a process of data fusion is required and there is much active research in this area. In this paper, current data fusion methods and applications are reviewed, with a focus on the mathematical foundations of the subject. Common research questions in the fusion of surface metrology data are raised and potential fusion algorithms are discussed

Nonaxisymmetric mathematical model of the cardiac left ventricle anatomy

We describe a mathematical model of the shape and fibre direction field of the cardiac left ventricle. The ventricle is composed of surfaces which model myocardial sheets. On each surface, we construct a set of curves corresponding to myocardial fibres. Tangents to these curves form the myofibres direction field. The fibres are made as images of semicircle chords parallel to its diameter. To specify the left ventricle shape, we use a special coordinate system where the left ventricle boundaries are coordinate surfaces. We propose an analytic mapping from the semicircle to the special coordinate system. The model is correlated with Torrent-Guasp’s concept of the unique muscular band and with Pettigrew’s idea of nested surfaces. Subsequently, two models of concrete normal canine and human left ventricles are constructed based on experimental Diffusion Tensor Magnetic Resonance Imaging data. The input data for the models is only the left ventricle shape. In a local coordinate system connected with the left ventricle meridional section, we calculate two fibre inclination angles, i.e. true fibre angle and helix angle. We obtained the angles found from the Diffusion Tensor Magnetic Resonance Imaging data and compared them with the model angles. We give the angle plots and show that the model adequately reproduces the fibre architecture in the majority of the left ventricle wall. Based on the mathematical model proposed, one can construct a numerical mesh that makes it possible to solve electrophysiological and mechanical left ventricle activity problems in norm and pathology. In the special coordinate system mentioned, the numerical scheme is written in a rectangular area and the boundary conditions can simply be written. By changing the model parameters, one can set a general or regional ventricular wall thickening or the left ventricle shape change, which is typical for certain cardiac pathologies

Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Structured light techniques for 3D surface reconstruction in robotic tasks

Robotic tasks such as navigation and path planning can be greatly enhanced by a vision system capable of providing depth perception from fast and accurate 3D surface reconstruction. Focused on robotic welding tasks we present a comparative analysis of a novel mathematical formulation for 3D surface reconstruction and discuss image processing requirements for reliable detection of patterns in the image. Models are presented for a parallel and angled configurations of light source and image sensor. It is shown that the parallel arrangement requires 35\% fewer arithmetic operations to compute a point cloud in 3D being thus more appropriate for real-time applications. Experiments show that the technique is appropriate to scan a variety of surfaces and, in particular, the intended metallic parts for robotic welding tasks

Elastic shape matching of parameterized surfaces using square root normal fields.

In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple \ensuremathL2\ensuremathL2 metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods