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.
HT is a standard pattern recognition technique originally used to detect straight lines in images, @cite_2 @cite_53 . Since its original conception, HT has been extensively used and many generalizations have been proposed for identifying instances of arbitrary shapes over images @cite_5 , more commonly circles or ellipses. The first very popular extension concerning the detection of any parametric analytic curve is usually referred to as the Standard Hough Transform (SHT). In spite of the robustness of SHT to discontinuity or missing data on the curve, its use has been limited by a number of drawbacks, like the need of a parametric expression, or the dependence of the computation time and the memory requirements on the number of curve parameters, or even the need of finer parameters quantization for a higher accuracy of results.
@cite_5 @cite_53 @cite_2
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Abstract The Hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve. The initial work showed how to detect both analytic curves (1,2) and non-analytic curves, (3) but these methods were restricted to binary edge images. This work was generalized to the detection of some analytic curves in grey level images, specifically lines, (4) circles (5) and parabolas. (6) The line detection case is the best known of these and has been ingeniously exploited in several applications. (7,8,9) We show how the boundaries of an arbitrary non-analytic shape can be used to construct a mapping between image space and Hough transform space. Such a mapping can be exploited to detect instances of that particular shape in an image. Furthermore, variations in the shape such as rotations, scale changes or figure ground reversals correspond to straightforward transformations of this mapping. However, the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes. This makes the generalized Hough transform a kind of universal transform which can be used to find arbitrarily complex shapes. Hough has proposed an interesting and computationally efficient procedure for detecting lines in pictures. This paper points out that the use of angle-radius rather than slope-intercept parameters simplifies the computation further. It also shows how the method can be used for more general curve fitting, and gives alternative interpretations that explain the source of its efficiency.