Interactive design of authentic looking mosaics using Voronoi structures Lars-Peter Fritzsche1 Heino Hellwig1 Stefan Hiller2 Oliver Deussen2∗ 1 Dresden University of Technology, Germany 2 University of Konstanz, Germany Abstract In this work we introduce new and efficient tech- niques to interactively create visually pleasing synthetic mosaics. We extend the known Lloyd’s method for CVT computation to sets of small mo- saic tiles. We use this improved placement algo- rithm in an interactive tool that enables the user to arrange tiles of various shapes and sizes, so that the set has the appearance of a traditionally hand- crafted mosaic. The user controls the distribution process by adding contour lines and directional in- formation. Tiles can be sized or shaped in order to better approximate the master image features. This interactive procedure allows for variations and visually more pleasing results. Additionally, it is less time expensive than using heuristic con- trolled automatic methods. Keywords: mosaic illustrations, interaktive design, non-photorealistic rendering, Lloyd’s method, CVT, NPR, computer graphic Figure 1. Traditional mosaic consisting of only six fundamental colors. 1. Introduction styles and materials. The aesthetic pleasure of The goal of computer graphics has been and a mosaic results from the reduction of the visual still is the rendering of realistic appearing im- data and the manual arrangement of important ages. However in the area of non-photorealistic features. Some of the fundamental features are rendering, common drawing styles or illustration contour lines, colors, shapes, and positions of the techniques, such as cross-hatching, are imitated. basic primitives. These primitives are physical Also, traditional mosaics generate images with distinct sets of colored tiles that consist of either non-photorealistic characteristics. With our new stone, ceramic or wood. method, we aim at simulating such traditional From the technical point of view, the reduction mosaic appearance. Realizing the importance of by fine sets of primitives is a bandwidth limited preserving features of classical art works, an in- signal approximation, e.g. some kind of sampling. teractive mosaic editor was implemented that en- There are mainly two kinds of classical mosaic ables the user to control mosaic placement in var- arrangement. Firstly, the abstract ornamental de- ious ways. Mosaics have a history of more than signs, and secondly the often complex portraits or 5000 years and were realized in many different illustrations of animals. In this paper we focus more on the latter and less on the mathematical ∗[email protected], [email protected], [email protected], motivated portraits. By studying the real world [email protected] example at Figure 1 we can summarize a number of characteristic features: • constant splices between the tiles; • fundamental colors with stochastic variation in tone and luminance values; • slight variation of tiles size and shape; • tiles are arranged along feature lines of the underlying master image; • smooth changes in tile orientation. These features are necessary to keep inevitable vi- sual artifacts at a low level, and are precisely the features we are aiming to effectively simulate with Figure 2. Synthetic mosaic generated by Hausner. our proposed interactive editor. Several artefacts can be seen such as wrong tiles Hausner [6] presented a technique to gener- sizes too close together (a), random misalignment ate synthetic mosaics in an automatic way. It of tiles (b), unwanted tile overlapping (c). is based on modification of the known Lloyd’s method to approximate centroidal Voronoi tessel- lation (CVT)[11] and distributes quadratic tiles in pixel under the cone peak at the underlying source a mosaic like arrangement. Compared to classical image. In an orthographic projection this resulted mosaics (see Figure 1) the resulting pictures offer in a colored tessellation of the canvas. The cor- some flaws: responding regions represented approximations of the Voronoi-regions around the cone peaks. Hoff • some tiles appear misaligned; et al. [9] used the same technique to generate Voronoi-diagrams using graphics hardware. This • the splices are uneven; technique is widely known as cell based Voronoi • small tiles and large tiles are arranged to- approximation [11]. Deussen et al. [1] introduced gether; the Lloyds method and by this the CVT(ref. also [3]) for non-photorealistic rendering to distribute • the influence of the contour lines is too strong points during the interactive generation of stip- (see Figure 2). ple drawings. Hausner [6] also applied a modified Lloyd’s method to generate decorative mosaics. Hausner’s approach does not avoid these flaws, Small square tiles were distributed while apply- because his arrangement algorithm run heuristic ing the method. However, this method does not controlled. Additionally, his methods reflect over- really approximate a CVT of the underlying tiles. simplification and approximation, which prevents In [7], Hiller extents the 2D Lloyd’s method for the user’s input and restricts high quality output. CVT approximation to generator sets of heteroge- This prevents the user from influencing the tile neous simple polygons. By including the mass mo- placement which is very important in order to cre- ments of polygons and the related Voronoi-regions ate decorative mosaics of high quality. as additional parameters into the Lloyd’s iteration process, his method also optimizes the orientation 2. Related Work of the polygon set. Several commercial tools offer so called mosaic- Haeberli [5] was one of the first authors that filters that allow the user to generate mosaics from created non-photorealistic drawings by simulating images. The resulting pictures cannot be com- stroke brushes. The image was segmented and pared to traditional mosaics, because important every segment was represented by a stroke. For image features are missing. Dobashi et al. [2] one of his styles, he used cones to simulate brush improve this kind of automatic mosaic generation strokes. Each cone was assigned the color of the by partitioning the image using Voronoi-diagrams. 2 They minimize the color differences between pix- of their Voronoi-regions in such a way that the els in the reference and the Voronoi-regions. Here, iteration converges. the primitives are solely Voronoi-regions, and the As mentioned above, for a point set, the resulting pictures appear more like stained-glass- Voronoi-diagram can be computed using graphics style images than actual traditional mosaics, see hardware by placing cones at the position of the Figure 3. points in z-direction and analysing the resulting image from above (see [10, 9]). For our application the resulting approximation error is quasi negligi- ble. Hausner uses a slightly modified version of this algorithm to distribute small square tiles. He extends the Voronoi-diagram calculation process by using a modified metric, which is in fact not really a metric, and definitively not a manhattan metric. Hausner uses a predefined directional field (a) (b) for tile alignment. The main problem of Hausner’s Figure 3. Synthetic generated mosaics (a) by Dobashi et al. [2], (b) by Elber and Wolberg [4]. Observing that in classical mosaics the tiles are placed exactly along important feature curves, El- ber and Wolberg [4] presented a technique for cal- culating offset curves to these feature curves. Ad- jacent to the offset curves, the rectangular tiles are placed tangentially to the curve in a very tight packing. A disadvantage of this method is that the resulting regular tile arrangement appears quite artificial, especially since the influence of the fea- tures and offset curves is not restricted, see Fig- ure 3. In our approach we are able to model the influence of feature lines using our interactive tool. 3. Mosaic generation For an optimal placement of the mosaic tiles, we (a) (b) first discuss the existing methods and then give some suggestions for an improved version. Lloyds Figure 4. a) Approximative Voronoi-diagram ob- relaxation scheme for evenly distributing points tained by Hausner’s method and (b) the exact so- can be noted as follows: lution using Hiller’s method [7]. Input: set of (randomly) distributed points S Output: set of points that generates a CVT method is that the tiles cannot be automatically repeat placed in a artistic mosaic-like manner. 1. compute Voronoi-Diagramm V (S) of the point set The proper arrangement of individual tiles is a highly artistic process. This is why we decided to 2. move all points Si to the center of model our solution as an interactive tool rather gravity of their Voronoi region V (Si) until point movement is below a threshold or than simulating ”‘art”’ with a set of heuristic certain number of iterations steps reached. methods. We found that heuristic driven meth- ods very often produce unwanted artifacts such In the above algorithm, a CVT is formed once all as miss-aligned tiles or highlighting of unwanted of the objects are placed in the center of gravity image parts. 3 3.1. Generalized algorithm gons works as follows: Hausner’s method performs only a rough approx- Input: Set M of random distributed and random imation, which in practice leads to overlaping of orientated polygons P the tiles, if a tight packing is created (see Fig- Ouput: Set of polygons P 0, evenly distributed ure 4). If instead a frustum of pyramid with round and orientated in context to each other corners is used, the shape of the Voronoi-regions repeat can be much better approximated. For lines and 1. for each P approximate it’s Voronoi-region polygons the correct depth buffer encodings for by using geometric modelled distance obtaining valid Voronoi tessellations are given in 2. perform principle components analysis on Figure 5. An improved version of Lloyd’s method polygons and their Voronoi-regions 3. compute center of gravity of polygons and their Voronoi-regions 4.