Image Representation on Curved Optimal Triangulation

Abstract

Image triangulation aims to generate an optimal partition with triangular elements to represent the given image. One bottleneck in ensuring approximation quality between the original image and a piecewise approximation over the triangulation is the inaccurate alignment of straight edges to the curved features. In this paper, we propose a novel variational method called curved optimal triangulation, where not all edges are straight segments, but may also be quadratic Bézier curves. The energy function is defined as the total approximation error determined by vertex locations, connectivity and bending of edges. The gradient formulas of this function are derived explicitly in closed form to optimize the energy function efficiently. We test our method on several models to demonstrate its efficacy and ability in preserving features. We also explore its applications in the automatic generation of stylization and Lowpoly images. With the same number of vertices, our curved optimal triangulation method generates more accurate and visually pleasing results compared with previous methods that only use straight segments.

Error‐bounded Image Triangulation

“…Color error Similar to [LG19, XCC22], the ℓ 2 ‐norm is used to define the color error ℰ ( t , ℐ ) between one triangle t ∈ T and the image ℐ : …”

“…However, the edges generated by the Delaunay triangulation may not match the image characteristics, usually leading to a large initial color error and hindering subsequent optimization. The second group detects feature edges of images using Canny edge detector [TA13, MA15] or edge drawing method [GW16,XCC22], and then treats these edges as constrained edges for the constrained Delaunay triangulation. These methods are also sensitive to the extracted features and initial triangulations.…”

“…These methods are also sensitive to the extracted features and initial triangulations. We also use [MCB17, XCC22] for initialization, but our experiments demonstrate that our method generates similar numbers of triangles using different initial triangulations for the same color error (Fig. 17).…”

A Survey of Smooth Vector Graphics: Recent Advances in Repr esentation, Creation, Rasterization, and Image Vectorization

“…The boundary Bézier curves are aligned with color discontinuities for a given image, which enables larger color variation at the boundaries and smooth colors inside the patches. More recently, Xiao et al [135] used quadratic Bézier curves as the patch edges. The control points, which are placed on the edges, are computed through an optimization process.…”

“…The control points, which are placed on the edges, are computed through an optimization process. However, neither thin plate spline fitting nor energy minimization guarantees C 1 continuity at non-edge patch boundaries [84], [135]. Alternatively, subdivision-based methods can be used to tackle the continuity problem, as illustrated in Fig.…”