We present a generalized method for reconstructing the shape of an object from measured gradient data. A certain class of optical sensors does not measure the shape of an object but rather its local slope. These sensors display several advantages, including high information efficiency, sensitivity, and robustness. For many applications, however, it is necessary to acquire the shape, which must be calculated from the slopes by numerical integration. Existing integration techniques show drawbacks that render them unusable in many cases. Our method is based on an approximation employing radial basis functions. It can be applied to irregularly sampled, noisy, and incomplete data, and it reconstructs surfaces both locally and globally with high accuracy.
In general, it is crucial to note that the reconstruction method depends on the slope-measuring sensor and the properties of the acquired data. For example, slope data acquired by Shape from Shading is rather noisy, exhibits curl, and is usually located on a full grid of millions of points. Here, a fast subspace approximation method like the one proposed by Frankot and Chellappa @cite_13 is appropriate. On the other hand, wavefront reconstruction deals with much smaller data sets, and the surface is known to be rather smooth and flat. In this case, a direct finite-difference solver can be applied @cite_11 . Deflectometric sensors deliver a third type of data: It consists of very large data sets with rather small noise and curl, but the data may not be complete, depending on the local reflectance of the measured surface. Furthermore, the measuring field may have an unknown, irregularly shaped boundary. These properties render most of the aforementioned methods unusable for deflectometric data. In the following sections, we will describe a surface reconstruction method which is especially able to deal with slope data acquired by sensors such as Phase-measuring Deflectometry.
@cite_13 @cite_11
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An approach for enforcing integrability, a particular implementation of the approach, an example of its application to extending an existing shape-from-shading algorithm, and experimental results showing the improvement that results from enforcing integrability are presented. A possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions, and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes. The integrability projection constraint was applied to extending an iterative shape-from-shading algorithm of M.J. Brooks and B.K.P. Horn (1985). Experimental results show that the extended algorithm converges faster and with less error than the original version. Good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces. > This thesis describes a general method for automatic reconstruction of accurate, concise, piecewise smooth surfaces from unorganized 3D points. Instances of surface reconstruction arise in numerous scientific and engineering applications, including reverse-engineering--the automatic generation of CAD models from physical objects. Previous surface reconstruction methods have typically required additional knowledge, such as structure in the data, known surface genus, or orientation information. In contrast, the method outlined in this thesis requires only the 3D coordinates of the data points. From the data, the method is able to automatically infer the topological type of the surface, its geometry, and the presence and location of features such as boundaries, creases, and corners. The reconstruction method has three major phases: (1) initial surface estimation, (2) mesh optimization, and (3) piecewise smooth surface optimization. A key ingredient in phase 3, and another principal contribution of this thesis, is the introduction of a new class of piecewise smooth representations based on subdivision. The effectiveness of the three-phase reconstruction method is demonstrated on a number of examples using both simulated and real data. Phases 2 and 3 of the surface reconstruction method can also be used to approximate existing surface models. By casting surface approximation as a global optimization problem with an energy function that directly measures deviation of the approximation from the original surface, models are obtained that exhibit excellent accuracy to conciseness trade-offs. Examples of piecewise linear and piecewise smooth approximations are generated for various surfaces, inc