Exact medial axis of quadratic NURBS curves

International audienceWe study the problem of the exact computation of the medial axis of planar shapes the boundary of which is defined by piecewise conic arcs. The algorithm used is a tracing algorithm, similar to existing numeric algorithms. We trace the medial axis edge by edge. Instead of keeping track of points on the medial axis, we are keeping track of the corresponding footpoints on the boundary curves, thus dealing with bisector curves in parametric space. We exploit some algebraic and geometric properties of the bisector curves that allow for efficient trimming and we represent bifurcation points via their associated footpoints on the boundary, as algebraic numbers. The algorithm computes the correct topology of the medial axis identifying bifurcation points of arbitrary degree

Doctor of Philosophy

dissertationThe medial axis of an object is a shape descriptor that intuitively presents the morphology or structure of the object as well as intrinsic geometric properties of the object’s shape. These properties have made the medial axis a vital ingredient for shape analysis applications, and therefore the computation of which is a fundamental problem in computational geometry. This dissertation presents new methods for accurately computing the 2D medial axis of planar objects bounded by B-spline curves, and the 3D medial axis of objects bounded by B-spline surfaces. The proposed methods for the 3D case are the first techniques that automatically compute the complete medial axis along with its topological structure directly from smooth boundary representations. Our approach is based on the eikonal (grassfire) flow where the boundary is offset along the inward normal direction. As the boundary deforms, different regions start intersecting with each other to create the medial axis. In the generic situation, the (self-) intersection set is born at certain creation-type transition points, then grows and undergoes intermediate transitions at special isolated points, and finally ends at annihilation-type transition points. The intersection set evolves smoothly in between transition points. Our approach first computes and classifies all types of transition points. The medial axis is then computed as a time trace of the evolving intersection set of the boundary using theoretically derived evolution vector fields. This dynamic approach enables accurate tracking of elements of the medial axis as they evolve and thus also enables computation of topological structure of the solution. Accurate computation of geometry and topology of 3D medial axes enables a new graph-theoretic method for shape analysis of objects represented with B-spline surfaces. Structural components are computed via the cycle basis of the graph representing the 1-complex of a 3D medial axis. This enables medial axis based surface segmentation, and structure based surface region selection and modification. We also present a new approach for structural analysis of 3D objects based on scalar functions defined on their surfaces. This approach is enabled by accurate computation of geometry and structure of 2D medial axes of level sets of the scalar functions. Edge curves of the 3D medial axis correspond to a subset of ridges on the bounding surfaces. Ridges are extremal curves of principal curvatures on a surface indicating salient intrinsic features of its shape, and hence are of particular interest as tools for shape analysis. This dissertation presents a new algorithm for accurately extracting all ridges directly from B-spline surfaces. The proposed technique is also extended to accurately extract ridges from isosurfaces of volumetric data using smooth implicit B-spline representations. Accurate ridge curves enable new higher-order methods for surface analysis. We present a new definition of salient regions in order to capture geometrically significant surface regions in the neighborhood of ridges as well as to identify salient segments of ridges

A new geometric-and-physics model of milling and an effective approach to medial axis transforms of free-form pockets for high performance machining

Mechanical part quality and productivity depend on many parameters in CNC milling processes, such as workpiece material, cutters, tool paths, feed rate, and spindle speed, etc. To pursue high performance machining, the cutting parameter optimization is in high demand in industry, though it is quite challenge. This innovative research successfully addresses some essential problems in optimizing the cutting parameters by developing a new geometric-and-physics integrated model of milling and proposing an effective approach to the medial axis transforms of free-form pockets. In this research, an original geometric model of 21/2- and 3-axis CNC milling is developed and integrated with a well-established mechanistic model. A main research contribution is that this integrated model can predict complex milling processes in higher fidelity with instantaneous material remove rates, cutting forces and spindle powers, compared to prior machining models. In the geometric model, an in-process workpiece model is introduced by using a group of discrete Z-layers and applying the B-Rep scheme to represent the workpiece shape on each layer, in order to accurately represent instantaneous cutter-and-workpiece engagement in 2Yz- and 3-axis milling. Hence, the un-deformed chip geometry can be found even for complex part milling, which is then fed to the mechanistic model to predict instantaneous cutting forces. By using this integrated model, cutting parameters can be optimized for profiling, pocketing, and surface milling to ensure steady cut and the maximum material removal rates. This model has been verified by experiments, and will be implemented into a software tool for Bombardier Aerospace. Another important research in this work is to propose aggressive roughing of free-form pockets for ultimately high cutting efficiency. For this purpose, an accurate, efficient approach to the medial axis transforms of free-form pockets and an optimal approach to multiple cutters selection and their path generation are proposed. The main contributions of this research include (1) a new mathematical model of medial axis point, (2) an innovative global optimization solver, the hybrid global optimization method, (3) an optimization model of selecting multiple cutters for the maximum material removal rate. This research can substantially promote aggressive roughing in the machining industry to increase cutting efficiency of free-form pockets. The technique has been validated using considerable number of cutting tests and can be directly implemented into commercial CAD/CAM softwar

Region-based Multimedia Indexing and Retrieval Framework

Many systems have been proposed for automatic description and indexing of digital data, for posterior retrieval. One of such content-based indexing-and-retrieval systems, and the one used as a framework in this thesis, is the MUVIS system, which was developed at Tampere University of Technology, in Finland. Moreover, Content-based Image Retrieval (CBIR) utilising frame-based and region-based features has been a dynamic research area in the past years. Several systems have been developed using their specific segmentation, feature extraction, and retrieval methods. In this thesis, a framework to model a regionalised CBIR framework is presented. The framework does not specify or fix the segmentation and local feature extraction methods, which are instead considered as “black-boxes” so as to allow the application of any segmentation method and visual descriptor. The proposed framework adopts a grouping approach in order to correct possible over- segmentation faults and a spatial feature called region proximity is introduced to describe regions topology in a visual scene by a block-based approach. Using the MUVIS system, a prototype system of the proposed framework is implemented as a region-based feature extraction module, which integrates simple colour segmentation and region-based feature description based on colour and texture. The spatial region proximity feature represents regions and describes their topology by a novel metric proposed in this thesis based on the block-based approach and average distance calculation. After the region-based feature extraction step, a feature vector is formed which holds information about all image regions with their local low-level and spatial properties. During the retrieval process, those feature vectors are used for computing the (dis-)similarity distances between two images, taking into account each of their individual components. In this case a many-to-one matching scheme between regions characterised by a similarity maximisation approach is integrated into a query-by-example scheme. Retrieval performance is evaluated between frame-based feature combination and the proposed framework with two different grouping approaches. Experiments are carried out on synthetic and natural image databases and the results indicate that a promising retrieval performance can be obtained as long as a reasonable segmentation quality is obtained. The integration of the region proximity feature further improves the retrieval performance especially for divisible, object-based image content. Finally, frame-based and region-based texture extraction schemes are compared to evaluate the effect of a region on the texture description and retrieval performance utilising the proposed framework. Results show that significant degradations over the retrieval performance occur on region-based texture descriptors compared with the frame-based approaches