A vector sketch is a popular and natural geometry representation depicting a 2D shape. When viewed from afar, the disconnected vector strokes of a sketch and the empty space around them visually merge into positive space and negative space , respectively. Positive and negative spaces are the key elements in the composition of a sketch and define what we perceive as the shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior or boundary of a 2D shape, the empty space may or may not belong to the shape's exterior. For standard discrete geometry representations, such as meshes or point clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs to define the positive space of a vector sketch, or the sketch shape. Even though extracting this 2D sketch shape is mathematically ambiguous, we propose a robust algorithm, Alpha Contours , constructing its conservative estimate: a 2D shape containing all the input strokes, which lie in its interior or on its boundary, and aligning tightly to a sketch. This allows us to define popular differential operators on vector sketches, such as Laplacian and Steklov operators. We demonstrate that our construction enables robust tools for vector sketches, such as As-Rigid-As-Possible sketch deformation and functional maps between sketches, as well as solving partial differential equations on a vector sketch.

Line drawing vectorization is a daily task in graphic design, computer animation, and engineering, necessary to convert raster images to a set of curves for editing and geometry processing. Despite recent progress in the area, automatic vectorization tools often produce spurious branches or incorrect connectivity around curve junctions; or smooth out sharp corners. These issues detract from the use of such vectorization tools, both from an aesthetic viewpoint and for feasibility of downstream applications (e.g., automatic coloring or inbetweening). We address these problems by introducing a novel line drawing vectorization algorithm that splits the task into three components: (1) finding keypoints, i.e., curve endpoints, junctions, and sharp corners; (2) extracting drawing topology, i.e., finding connections between keypoints; and (3) computing the geometry of those connections. We compute the optimal geometry of the connecting curves via a novel geometric flow --- PolyVector Flow --- that aligns the curves to the drawing, disambiguating directions around Y-, X-, and T-junctions. We show that our system robustly infers both the geometry and topology of detailed complex drawings. We validate our system both quantitatively and qualitatively, demonstrating that our method visually outperforms previous work.