3-D Topography and Image Analysis (John Lu, organizer)


Kevin Coakley (National Institute of Standards and Technology)
Neutron Transmission Tomography of Fuel Cells

Friday 2:00-2:20, San Marino

Abstract:

At the National Institute of Standards and Technology (NIST), efforts are underway to image water density in fuel cells using neutron transmission tomography. In an experiment at the NIST Center for Neutron Research, transmission projection data for a fuel cell were collected. I compare reconstructions of two dimensional neutron attenuation images provided by a filtered background projection method and a penalized log-likelihood method using a Huber penalty function with two adjustable parameters. In a simulation study where I rescale attenuation, I demonstrate that for a fixed neutron exposure, the quality of the reconstruction is not a monotonic function of the scaling parameter.


James Yen (National Institute of Standards and Technology)
Analysis of 3-D Forms and Surfaces

Friday 2:20-2:40, San Marino

Abstract:

3-D data comes in many shapes and sizes. LADAR (laser radar) devices can scan objects from hundreds of meters away with accuracies of around 2-3 cm. In contrast, instruments like coordinate measuring machines and atomic force microscopes can measure objects from up close with uncertainties in the nanometer range. In both scenarios, there exist the problem of estimating shapes and surfaces from 3-D point clouds; in some cases, the shapes may already be known (e.g. spherical, parabolic) leading to parametric estimation problems; in other cases, the shapes do not follow specific forms (e.g. trees). There is also the problem of stitching together somewhat overlapping data from different areas of the same object or scene. In addition, 3-D data can often be seen and treated as image data, where the value of one dimension can be depicted symbolically or by pixel coloration, analogous to spatial data where the 3rd coordinate may be a quantity other than a dimensional measurement.


Nadia Machkour (National Institute of Standards and Technology)
Non-Parametric Estimation of Curvature and Uncertainty Analysis of the Geometry Measuring Machine (GEMM)

Friday 2:40-3:00, San Marino

Abstract:

The National Institute of Standards and Technology (NIST) is developing the Geometry Measuring Machine (GEMM) for free-form and aspheric surface metrology. Optical interferometry is a general, simple, and accurate method for form metrology of flat and spherical precision surfaces. However, interferometers are not well suited to testing free form and aspheric surfaces. The reason is that, unlike flat and spherical optics, a reference wavefront is difficult to generate. The idea behind GEMM is to reconstruct the profile of a surface under test from its local curvature measurements. In Euclidean spaces, curvature is an intrinsic property of curves and surfaces; and it is invariant under Euclidean motions, which means curvature is independent of the part position and angular orientation. In principle, the profile of a test part is given by solving a second-order differential equation involving curvature, though there remain challenging engineering and signal processing hurdles to overcome in order to realize this differential geometry theory.

To measure the curvature of a test part, GEMM uses a small interferometer, which looks at a small area of the test part to measure the topography. The local topography measurements are fitted to a circle close to the center of the field of view; the curvature of this circle is then assigned to that point on the test surface. Other mathematical model, such as polynomial function, can also be used; in this case the curvature can be deduced from simple derivation of the mathematical function. However, the GEMM-based profile reconstruction is sensitive to the uncertainties in the curvature estimation, and in order to improve the existing results and to provide a theoretical foundation, a nonparametric statistical theory of curvature estimation is being developed using high-order local polynomial regression. The application of this theory to the GEMM method will be illustrated with the measurement of the form of a free-form mirror. In addition, a comparison will be made to parallel measurements which were made using the NIST Moore coordinate measuring machine.


John Lu (National Institute of Standards and Technology)
Comments on 3-D Topography and Image Analysis

Friday 3:00-3:20, San Marino

Abstract:

In this session, we have seen that these are quite interesting statistical problems arising from measurement science or metrology. Unlike in geosciences, topography in metrology demands more precise measurements at a much smaller scale, often in the micron or even nanometer range. Uncertainty analysis is crucial for establishing repeatability and comparability of similar measurements in metrology. Unlike standard statistical image analysis, the geometry in image measurements plays an important role in 3-d topography and object recognition. We discussed, for example, how modern nonparametric regression could be used to solve a practical differential geometry problem in aspheric surface measurement in optical metrology. Sometimes, the image measurements themselves are only a partial projection of underlying 3-d objects, so the ill-posed nature of the problem poses another challenge for uncertainty analysis. Furthermore, even though most man-made objects can be treated as smooth, real-world objects such as trees and landscape can have a fractal dimension, so analysis of real world 3-d objects should encourage statisticians to look beyond traditional statistical toolkit.