Recognition and representation of curve and surface primitives in digital models via the Hough transform

Curve and surface primitives have an important role in conveying an object shape and their recognition finds significant applications in manufacturing, art, design and medical applications. When 3D models are acquired by scanning real objects, the resulting geometry does not explicitly encode these curves and surfaces, especially in the presence of noise or missing data. Then, the knowledge of the parts that compose a 3D model allows the reconstruction of the model itself. The problem of recognising curves and surfaces and providing a mathematical representation of them can be addressed using the Hough transform technique (HT), which in literature is mainly used to recognise curves in the plane and planes in space. Only in the last few years, it has been explored for the fitting of space curves and extended to different families of surfaces. Such a technique is robust to noise, does not suffer from missing parts and benefits from the flexibility of the template curve or surface. For these reasons, our approach is inspired by a generalisation of the Hough transform defined for algebraic curves. In this thesis, we present the methods we implemented and the results we obtained about the recognition, extraction, and representation of feature parts that compose a 3D model (both meshes and point clouds). Specifically, we first study the recognition of plane curves, simple and compound, expressed both in implicit and parametric form, with a focus on the application of cultural heritage and geometric motifs. Then, we analyse the extension of the method to space curves, concentrating on the improvement of the model through the insertion of the recognised curves directly on its surface. To overcome the limitation of knowing in advance the family of curves to be used with the HT, we introduce a piece-wise curve approximation using specific parametric, low-degree polynomial curves. Finally, we analyse how to recognise simple and complex geometric surface primitives on both pre-segmented and entire point clouds, and we show a comparison with state-of-the-art approaches on two benchmarks specifically created to evaluate existing and our methods.