Differential operators on sketches via alpha contours
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ACM Transactions on graphics
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Association for Computing Machinery
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- Computing methodologies
- Parametric curve and surface models
- Shape analysis
- Vector graphics
- Sketch processing
- Differential operators
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Résumé
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.
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