Computing Layouts with Deformable Templates ChiHan Peng 1
Computing Layouts with Deformable Templates Chi-Han Peng 1, Yong-Liang Yang 2, Peter Wonka 1, 2 1 Arizona 2 State University King Abdullah University of Science and Technology
Layout computation Forbid gaps Allow deformations Jigsaw image mosaics, Arcimboldo-like collages, [Kim 2002, Huang 2011] Forbid deformations Packing (furniture, mosaics, texture atlas, etc. ) [Yu 2011, Kim 2002, Levy 2002] Allow deformations Floor planning, urban layouts, “Paving”-based meshing, etc. [Merrell 2010, Yang 2013, Blacker 1991] Forbid deformations Tiling (e. g. , Wang tiles), escherization, etc. [Cohen 2003, Yeh 2013, Kaplan 2000]
Problem statement • How to cover a 2 D problem domain in a gap-less manner with templates that can deform in certain admissible ways?
Key observation • Many real-world layout applications actually resemble the process of “tiling” on a semi-quadrilateral domain. Urban layouts Mall floor plans Office floor plans
A discrete-continuous framework
A discrete-continuous framework Tessellating (quadrangulating) the domain and the templates Discrete tiling by integer programming Continuous geometric optimization (Optional) user-specified discrete constraints
Discrete tiling
Definitions • A tiling is an as complete as possible cover of the mesh faces into non-overlapping tiles. • A tiling is maximum if it has the fewest possible number of uncovered faces for the given templates. A non-maximum tiling found by a simple flooding. A maximum (complete) tiling
An integer programming (IP) approach 1. First, enumerate all possible tile placements. Denote their presences as Boolean variables Ti. 2. The maximum tiling problem can be formulated as:
Solving the maximum tiling problem • Can be solved very efficiently by modern, off-the-shelf solvers such as Gurobi. (Tiling a mesh with Tetris templates on a quad core laptop machine. ) Random floodings in 1000 seconds: still 12 uncovered faces. Gurobi IP solver: found a maximum tiling in <1 seconds.
Geometric optimization
Admissible transformations
Shape registration • Register step by step (each is an affine registration with a closedform solution) in the reverse order. Multiple passes. Template Tile (rigid transformation) (local translation)
Geometric optimization 1. Retrieve tilings with more suitable tile shapes. 2. Global shape optimization (multiple QP iterations). Error Tiling w/o shape scores (avg. error: 0. 851) Tiling w/ shape scores (avg. error: 0. 613) After shape optimization (avg. error: 0. 391)
Advantages of our tiling-based approach
Randomize tilings • Just add a small random factor to the weighting scheme. … No random factor. Add a 1% random factor to the weights.
Control distributions of template types 1. Impose preferences to the template types by adjusting weights. W: 4 W: 3 W: 2 W: 4. 4 W: 3 W: 2 (+10% of the Weight) W: 4 W: 3. 3 W: 2 W: 4. 4 W: 3 W: 2. 2
Control distributions of template types 2. Control the number of occurrences of each template type by linear constraints, e. g. , lower and upper bounds. =1
Higher-level connectivity constraints 1. Adjacency constraints that span over multiple tiles, e. g. , a multi-level branching tree. Root Lv 1 branch Lv 2 branch Leaf
High-level connectivity constraints 2. Control the “regularity” of tilings. 28 T-junctions 1 T-junction Both: x 7 (W: 2) x 39 (W: 4. 4) x 1 (W: 3) With imaginary “joint” tiles at regular (valence-4) junctions
Results and applications
Floor planning • Meet both accessibility (corridors) and aesthetic (room shapes) criteria of floor plans of large facilities.
Urban layouts • Offer more users controls than previous work. Default More regular Occurrence control
Art and design • A powerful solution-finding tool for tiling-based designs. Default More Ingrained characters by tile boundary constraints
Future work 1. Tiling on tessellations other than pure quad meshes. 2. Tiling on 3 D surfaces and volumes. [Floor Plan Manual Housing, Friederike Schneider and Oliver Heckmann, ISBN: 376436985 X]
Future work 3. Alternative initial quadrangulations of the domain. More regular Just a triangle template More irregular
Acknowledgements • We thank the anonymous reviewers for their insightful comments, Yoshihiro Kobayashi and Christopher Grasso for the renderings, and Virginia Unkefer for the proofreading. This research is supported by the NSF #0643822 and the Visual Computing Center at King Abdullah University of Science and Technology. More details about this project are available at: http: //www. public. asu. edu/~pchihan/tiling/ Thank you for your listening!
Timing table
- Slides: 28