Image Vectorization with Depth: convexified shape layers with depth ordering

TL;DR

New image vectorization with depth is proposed which considers depth ordering among shapes and use curvature-based inpainting for convexifying shapes in vectorization process, and makes editing shapes and images more natural and intuitive.

Abstract

Image vectorization is a process to convert a raster image into a scalable vector graphic format. Objective is to effectively remove the pixelization effect while representing boundaries of image by scaleable parameterized curves. We propose new image vectorization with depth which considers depth ordering among shapes and use curvature-based inpainting for convexifying shapes in vectorization process.From a given color quantized raster image, we first define each connected component of the same color as a shape layer, and construct depth ordering among them using a newly proposed depth ordering energy. Global depth ordering among all shapes is described by a directed graph, and we propose an energy to remove cycle within the graph. After constructing depth ordering of shapes, we convexify occluded regions by Euler's elastica curvature-based variational inpainting, and leverage on the stability of Modica-Mortola double-well potential energy to inpaint large regions. This is following human vision perception that boundaries of shapes extend smoothly, and we assume shapes are likely to be convex. Finally, we fit Bézier curves to the boundaries and save vectorization as a SVG file which allows superposition of curvature-based inpainted shapes following the depth ordering. This is a new way to vectorize images, by decomposing an image into scalable shape layers with computed depth ordering. This approach makes editing shapes and images more natural and intuitive. We also consider grouping shape layers for semantic vectorization. We present various numerical results and comparisons against recent layer-based vectorization methods to validate the proposed model.

Figures (20)

• Figure 1: (a) Given color quantized raster input image $f$. (b) Typical vectorization result by adobeillustrator which considers each connected component separately. (c) Desired vectorization of curves using our propose method.
• Figure 2: [Shape Layer $S_i$] (a) The given color quantized raster image $f$. (b) Seven shape layers $S_i$$i=1,2,\dots, 7$ are colored by their associated colors $c_l$, with $l=1,2, \dots,5$ (black, orange, yellow, white and green).
• Figure 3: [Depth ordering $D(i,j)$] Consider $f$ from Figure \ref{['fig:shapelayer']}(a), (a) shows the orange sun shape layer $S_3$, and (b) shows the light yellow sky $S_4$. The red closed contours in (a) and (b), and yellow regions in (c) and (d) show the convex hull of each shapes. In (c) and (d), green area represents the numerators of $A(3,4)$ and $A(4,3)$ respectively. (c) $A(3,4)$ is close to 1, while $A(4,3)$ in (d) is near zero, thus $D(3,4)>0$, and $S_3$ is determined to be in front of $S_4$.
• Figure 4: [The graph $G(M,E)$ with a cyclic.] The given image $f$ in (a) gives a directed graph in (b). Considering the convex hull symmetric differences $V(1,2)$ (yellow is $\chi_2^{conv}\chi_1$), $V(2,3)$ (cyan is $\chi_3^{conv}\chi_2$) and $V(3,1)$ (magenta is $\chi_1^{conv}\chi_2$), $V(1,2)$ is the maximum and we set $E_{1,2}=0$. (d) shows the linear directed graphs which gives the global depth ordering.
• Figure 5: [Image Vectorization with depth flowchart] From the given color quantized image $f$, shape layers $S_i$s are defined, and depth ordering is determined. Euler's Elastica curvature-based inpainting is used to convexify shape layers considering the occluded region $O_i$ given by the depth ordering. Finally, each convexified layers $C_i$s are vectorized and stacked in SVG file format.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3