Ph.D Thesis
Ph.D Student Harary Gur On 3D Spirals and Surface Completion Department of Electrical Engineering Professor Ayellet Tal

Abstract

In this thesis we discuss both 3D spirals and surface completion. We show how the former can be utilized by the latter.

Spirals have attracted the attention of mathematicians, as well as biologists, artists and architects. Most of the mathematical research on spirals has focused on planar curves. We explore three-dimensional spirals. In particular, we define 3D Euler spirals and 3D logarithmic spirals. We prove via a series of theorems that our curves satisfy properties that characterize eye-pleasing curves. In addition, we demonstrate the utility of our curves both in curve completion applications and in modeling a variety of natural structures in wildlife, such as seashells and horns.

Physical artifacts often contain holes of missing geometry. Such holes are not the result of faulty scanning, but rather come from the source objects themselves, which might be broken or incomplete. We introduce two novel algorithms to synthesize missing geometry for a given triangle mesh that has ``holes.'' Similarly to previous work, our algorithms are context-based in the sense that they fill the hole by synthesizing geometry that is similar to the remainder of the input mesh. Our first algorithm further imposes a coherence objective. A synthesis is coherent if every local neighborhood of the filled hole is similar to some local neighborhood of the input mesh. This requirement avoids undesired features that can occur in context-based completion.

When the holes are very large, an automatic algorithm may not suffice.

Our second algorithm is designed to take the user's input into account. We formulate the completion as a global energy minimization problem that considers the user constraints - four points on the mesh that lie on a broken feature curve crossing the hole. We show how our Euler spiral and our previous completion algorithm can be utilized together for this purpose. We demonstrate our algorithms' ability to fill holes that were difficult or impossible to fill in a compelling manner by earlier approaches.