We introduce an extremely scalable and efficient yet simple palette-based image decomposition algorithm. Given an RGB image and set of palette colors, our algorithm decomposes the image into a set of additive mixing layers, each of which corresponds to a palette color applied with varying weight. Our approach is based on the geometry of images in RGBXY-space. This new geometric approach is orders of magnitude more efficient than previous work and requires no numerical optimization. We provide an implementation of the algorithm in 48 lines of Python code. We demonstrate a real-time layer decomposition tool in which users can interactively edit the palette to adjust the layers. After preprocessing, our algorithm can decompose 6 MP images into layers in 20 milliseconds.

In digital image editing software, layers organize images. However, layers are often not explicitly represented in the final image, and may never have existed for a scanned physical painting or a photograph. We propose a technique to decompose an image into layers. In our decomposition, each layer represents a single-color coat of paint applied with varying opacity. Our decomposition is based on the image’s RGB-space geometry. In RGB-space, the linear nature of the standard Porter-Duff [1984] “over” pixel compositing operation implies a geometric structure. The vertices of the convex hull of image pixels in RGB-space correspond to a palette of paint colors. These colors may be “hidden” and inaccessible to algorithms based on clustering visible colors. For our layer decomposition, users choose the palette size (degree of simplification to perform on the convex hull), as well as a layer order for the paint colors (vertices). We then solve a constrained optimization problem to find translucent, spatially coherent opacity for each layer, such that the composition of the layers reproduces the original image. We demonstrate the utility of the resulting decompositions for recoloring (global and local) and object insertion. Our layers can be interpreted as generalized barycentric coordinates; we compare to these and other recoloring approaches.

We present a system for 3D modeling of free-form surfaces from 2D sketches. Our system frees users to create 2D sketches from arbitrary angles using their preferred tool, which may include pencil and paper. A 3D model is created by placing primitives and annotations on the 2D image. Our primitives are based on commonly used sketching conventions and allow users to maintain a single view of the model. This eliminates the frequent view changes inherent to existing 3D modeling tools, both traditional and sketchbased, and enables users to match input to the 2D guide image. Our annotations-same-lengths and angles, alignment, mirror symmetry, and connection curves-allow the user to communicate higherlevel semantic information; through them our system builds a consistent model even in cases where the original image is inconsistent. We present the results of a user study comparing our approach to a conventional "sketch-rotate-sketch" workflow.

1 2 3 4 5 composite RGB space original recoloring Figure 1: Given a digital painting, we analyze the geometry of its pixels in RGB-space, resulting in a translucent layer decomposition, which makes difficult edits simple to perform. Artwork c Adelle Chudleigh. AbstractIn digital painting software, layers organize paintings. However, layers are not explicitly represented, transmitted, or published with the final digital painting. We propose a technique to decompose a digital painting into layers. In our decomposition, each layer represents a coat of paint of a single paint color applied with varying opacity throughout the image. Our decomposition is based on the painting's RGB-space geometry. In RGB-space, a geometric structure is revealed due to the linear nature of the standard Porter-Duff [1984] "over" pixel compositing operation. The vertices of the convex hull of pixels in RGB-space suggest paint colors. Users choose the degree of simplification to perform on the convex hull, as well as a layer order for the colors. We solve a constrained optimization problem to find maximally translucent, spatially coherent opacity for each layer, such that the composition of the layers reproduces the original image. We demonstrate the utility of the resulting decompositions for re-editing.

Discrete curvature and shape operators, which capture complete information about directional curvatures at a point, are essential in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In many of these applications, objects are represented by meshes. Currently, a spectrum of approaches for formulating curvature operators for meshes exists, ranging from highly accurate but computationally expensive methods used in engineering applications to efficient but less accurate techniques popular in simulation for computer graphics.We propose a simple and efficient formulation for the shape operator for variational problems on general meshes, using degrees of freedom associated with normals. On the one hand, it is similar in its simplicity to some of the discrete curvature operators commonly used in graphics; on the other hand, it passes a number of important convergence tests and produces consistent results for different types of meshes and mesh refinement. IntroductionDiscrete curvature is a key ingredient in a variety of applications: simulation of deformable two-dimensional objects, variational modeling and geometric data processing. In these applications it is often necessary to approximate a solution of a continuous problem involving curvature-based energy or forces. Such energy may either capture the physics of the problem (bending energy for thin deformable objects) or our intuition about desirable behavior (variational approaches to surface modeling or curvature flow smoothing of complex geometry).Curvature and related surface Laplacian discretizations range from highly accurate but complex and computationally expensive high-order finite elements used in engineering to efficient and simple approximations for meshes used in graphics and interactive geometric modeling. Unfortunately, the latter type of approximations, while essential for interactive applications, lacks the predictability and convergence of more mathematically complex and computationally expensive formulations. The lack of convergence leads to mesh-dependent behavior, visual artifacts and fundamental difficulties with adaptive refinement and remeshing of variationally defined shapes.In this paper we present a simple and efficient discrete curvature operator exhibiting good convergence properties for general meshes. This operator is closely related to widely used discrete geometric operators on one hand, and classical nonconforming finite elements on the other. The discretization we propose uses mesh normals as additional degrees of freedom. One of the essential observations is that the normal field can be defined by scalars, assigned to edges in a natural and geometrically invariant way, similar to the definition of discrete