Abstract Feature curves are largely adopted to highlight shape features, such as sharp lines, or to divide surfaces into meaningful segments, like convex or concave regions. Extracting these curves is not sufficient to convey prominent and meaningful information about a shape. We have first to separate the curves belonging to features from those caused by noise and then to select the lines, which describe non-trivial portions of a surface. The automatic detection of such features is crucial for the identification and or annotation of relevant parts of a given shape. To do this, the Hough transform (HT) is a feature extraction technique widely used in image analysis, computer vision and digital image processing, while, for 3D shapes, the extraction of salient feature curves is still an open problem. Thanks to algebraic geometry concepts, the HT technique has been recently extended to include a vast class of algebraic curves, thus proving to be a competitive tool for yielding an explicit representation of the diverse feature lines equations. In the paper, for the first time we apply this novel extension of the HT technique to the realm of 3D shapes in order to identify and localize semantic features like patterns, decorations or anatomical details on 3D objects (both complete and fragments), even in the case of features partially damaged or incomplete. The method recognizes various features, possibly compound, and it selects the most suitable feature profiles among families of algebraic curves.
Sharp edges, ridges, valleys, and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in a feature-sensitive metric, features are recognized on multiple scales via integral invariants of local neighborhoods. Morphological and smoothing operations are then used for feature region extraction and classification into basic types such as ridges, valleys, and prongs. The resulting representation of feature regions is further used for feature-specific editing operations We propose a simple and effective method for detecting view-and scale-independent ridge-valley lines defined via first- and second-order curvature derivatives on shapes approximated by dense triangle meshes. A high-quality estimation of high-order surface derivatives is achieved by combining multi-level implicit surface fitting and finite difference approximations. We demonstrate that the ridges and valleys are geometrically and perceptually salient surface features, and, therefore, can be potentially used for shape recognition, coding, and quality evaluation purposes. Curves on objects can convey the inherent features of the shape. This paper defines a new class of view-independent curves, denoted demarcating curves. In a nutshell, demarcating curves are the loci of the "strongest" inflections on the surface. Due to their appealing capabilities to extract and emphasize 3D textures, they are applied to artifact illustration in archaeology, where they can serve as a worthy alternative to the expensive, time-consuming, and biased manual depiction currently used. The crest lines, salient subsets of the extrema of the principal curvatures over their corresponding curvature lines, are powerful shape descriptors which are widely used for shape matching, interrogation, and visualization purposes. In this paper, we develop fast, accurate, and reliable methods for detecting the crest lines on surfaces approximated by dense triangle meshes. The methods exploit intrinsic geometric properties of the curvature extrema and provide with an inherent level-of-detail control of the detected crest lines. As an immediate application, we use of the crest lines for adaptive mesh simplification purposes. The patch layout of 3D surfaces reveals the high-level geometric and topological structures. In this paper, we study the patch layout computation by detecting and enclosing feature loops on surfaces. We present a hybrid framework which combines several key ingredients, including feature detection, feature filtering, feature curve extension, patch subdivision and boundary smoothing. Our framework is able to compute patch layouts through concave features as previous approaches, but also able to generate nice layouts through smoothing regions. We demonstrate the effectiveness of our framework by comparing with the state-of-the-art methods. Graphical abstractDisplay Omitted HighlightsA hybrid framework that can extract patch layouts that are better than the state-of-the-art.A novel criterion to evaluate the quality of patch layouts.Improved algorithms for smoothing curves on surfaces. A new theorem is discussed that relates the apparent curvature of the occluding contour of a visual shape to the intrinsic curvature of the surface and the radial curvature. This theorem allows the formulation of general laws for the apparent curvature, independent of viewing distance and regardless of the fact that the rim (the boundary between the visible and invisible parts of the object) is a general, thus twisted, space curve. Consequently convexities, concavities, or inflextions of contours in the retinal image allow the observer to draw inferences about local surface geometry with certainty. These results appear to be counterintuitive, witness to the treatment of the problem by recent authors. It is demonstrated how well-known examples, used to show how concavities and convexities of the contour have no obvious relation to solid shape, are actually good illustrations of the fact that convexities are due to local ovoid shapes, concavities to local saddle shapes.  Introduction Plane Curves Main Definitions and Facts Some General Classes of Plane Curves Atlas of Plane Curves Structure of the Atlas Atlas of Curves Geometric Indices for Atlas of Curves Space Curves Main Defintions and Facts Some Classes of Space Curves Applications of Curves Dynamic Properties of Plane Curves Plane Curves and Mechanisms Geometric Splines Appendices Smooth Functions of One and Two Variables Vector Functions of Scalar Argument Coordinate Systems. Coordinate Transformations Search of Some Geometric Characteristics Greek Roots of Names for Some Curves English-German-French Indices Bibliography Index of Terms