Cube ArtNaoki Kita∗
Kazunori Miyata
Japan Advanced Institute of Science and Technology(a)(b)(c)(d)(e)Figure 1: Cube Art representing a 3D model assembled from same cubes. (a) Given a model, the shaded model is captured from sixorthogonal directions. (b) A voxelization of the model is also computed and the visibility of each voxel is calculated. (c) Under thesevisibility constraints, the optimal cube poses (rotations) are computed as an energy (defined by the dissimilarity of a cube face patch anda corresponding shaded image patch) minimization problem. (d-e) The resulting cube models (Cube Art and Dice Art) have enhanced
perceptions of the original model due to the cube’s face textures compared to no rotation and random rotation on the upper rightof (e) and Figure 2. The cubes used are same cubes that only allow orientation-preserving rotations and translations.

Abstract
In this study, we propose Cube Art, a representation of a 3D model
assembled from same cubes. Here the term same is defined as twocubes that are regarded as the same, in that they can be rotated inspace such that the textures (or the Dice numbers) on the corresponding sides become equal. Since such a cubic model lacks mostof original geometric properties, the model has poorer expressiveness than the original one. To enhance the resulting shading to be
more expressive, each cube needs to be properly positioned suchthat the textures on each face of the cube approximate the shadingof the original models. The goal of this study is to determine thepositioning of each cube so that the textures on each face of thecubes best approximate the original shading model under face visibility constraints of the model’s orthogonal directions. We measure
the similarities between each face’s texture of the cubes and the textures resulting from the orthogonal projections onto a hexahedronof the original shading model for each cube position, and choose
the best cube position. Our results show that properly rotated cubes
enhance the perception of the shading and its aesthetics.
Keywords: computational design, human visual perception
Concepts: •Computing methodologies → Perception;

1

Motivation

Cubes are one of the most basic primitives and are used as a unit inwide various fields from architectures (e.g., Bricks) to recreational
puzzles (e.g., Rubik’s Cube and Dice). A cube’s basic structure isvery simple, and its mathematical properties have been explored indepth. In group theory, the symmetries of a cube have some intriguing properties (e.g., there are 24 orientation-preserving symmetries∗ e-mail:naoki-kt@jaist.ac.jp

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or rotations of a cube), and therefore recreational math attracted theinterest of mathematicians and puzzle lover’s since a long time. Although there are researches of generating cubic puzzles (e.g., [Songet al. 2012]), little attention has been paid to the use of textured
cubes to give a model composed of assembled same cubes a similar
expressiveness to the original model.

2Our Approach

We first generate a texture of each face of the bounding volume bymeans of an orthographic projection of a shaded model. We thenvoxelize a given polygonal model and compute the visibilities ofeach voxel by casting a ray from each face in its normal direction.
If there are no obstacles and the ray reaches the model’s bounding volume, the face is considered to be visible. Next, we computethe dissimilarity between a voxel and a cube. The score obtained
from this computation is the weighted sum of each face’s dissimilarity between a voxel and its corresponding cube. We employ
the sum of squared difference (SSD), or a log-polar histogram approach for better capturing the texture’s local alignments, and theweight is calculated by the cube’s visibility multiplied by its importance (e.g., viewing direction of the human eye). In addition, if acube texture is not specified by a user, we can extract a color palettefrom a shaded target model. In this case, we need to consider how
to assign colors to each cube face, which in total amounts to 30
patterns (= permutation of the six colors divided by the number oforientation-preserving symmetries/rotations; 6!/24). This coloring
problem is also known as MacMahon’s Cube in mathematical puzzles. We can compute such mapping in the same way we computetexture similarity. Figure 2 shows that our result has enhanced perceptions of the original model compared to initial cube positioning(no rotation) and random rotation.
Figure 2: Left toright: no rotation,
random rotation,and our results.

References
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