# Layered Analysis of Irregular Facades via Symmetry Maximization

Layered Analysis of Irregular Facades via Symmetry Maximization Hao Zhang, Kai Xu, Wei Jiang, Jinjie Lin, Daniel Cohen-Or, Baoquan Chen Simon Fraser University Shenzhen Institutes of Advanced Technology National University of Defense Technology

Tel Aviv University Faade Rich interesting structure to analyze SFU, SIAT, NUDT, TAU Our goal A high-level understanding of the structure of irregular facades A generative model An explanation of how the input facade was seemingly generated

1. Decomposition 2. Instantiation SFU, SIAT, NUDT, TAU Layering Split Generative model A hierarchy of decompositions

SFU, SIAT, NUDT, TAU Generative model Two decomposition operations: split + layering + Input Split SFU, SIAT, NUDT, TAU

Two substructures Generative model Two decomposition operations: split + layering + Layering Input SFU, SIAT, NUDT, TAU

Two substructures (layers) Structure completion More compact generative model Two decomposition operations: split + layering input SFU, SIAT, NUDT, TAU split only

8 ops. split + layering 4 ops. A good generative model Two objectives Structural: Occams Razor (Simplest explanation) min. # ops.

SFU, SIAT, NUDT, TAU A good generative model Two objectives Structural: Occams Razor (Simplest explanation) Perceptual: Law of Gestalt (Symmetry maxi mization) Two substructures

are Symmetry maximization The two objectives can be optimized decomposition terminates the fastest most symmetric! simultaneously. SFU, SIAT, NUDT, TAU Symmetry-driven analysis

Symmetry maximization at each decomposition How good is the decomposition? How symmetric are the two substructures? SFU, SIAT, NUDT, TAU Symmetry-driven analysis Objective: sum of symmetry measure of all internal nodes 1 2

SFU, SIAT, NUDT, TAU 3 4 max . =1 + 2 + 3 + 4 Related works Structuring symmetry

Folding mesh [Simari et al. 06] SFU, SIAT, NUDT, TAU Structuring 3D Geometry [Martinet 2007] Symmetry hierarchy [Wang et al. 2007] Related works

Faade analysis Instant architecture [Wonka et al. 2003] Symmetry-summarization [Wu et al. 2011] Shape grammar parsing [Teboul et al. 2011]

Layering has so far not been considered. SFU, SIAT, NUDT, TAU Adaptive Partitioning [Shen et al. 2011] Related works Inverse procedural modeling Inverse L-system [Stava et al. 2010]

Partial symmetry and inverse procedural modeling [Bokeloh et al. 2010] We do not produce a shape grammar. SFU, SIAT, NUDT, TAU Overview Key components Element groups

Hierarchy optimization Box abstraction SFU, SIAT, NUDT, TAU Candidate selection

Interactive box abstraction SFU, SIAT, NUDT, TAU Element groups A set of well-aligned boxes whose content repeats Must be maximal: not contained by any other one

SFU, SIAT, NUDT, TAU Candidate decomposition selection Given a box pattern, find all candidates of split and layering Split cand. 1 Split cand. 2

Input Layering cand. 1 Layering cand. 2 SFU, SIAT, NUDT, TAU Candidate decomposition selection Principle 1: An element group can not divided into t wo components by a valid decomposition Some examples

SFU, SIAT, NUDT, TAU Candidate decomposition selection Examples Invalid split SFU, SIAT, NUDT, TAU Becomes valid after layering Candidate decomposition selection

Examples Invalid split SFU, SIAT, NUDT, TAU Becomes valid after split Candidate decomposition selection Examples Invalid layering

SFU, SIAT, NUDT, TAU Becomes valid after layering Candidate decomposition selection Principle 2: Candidate selection is carried out recursively Split cand. 1

Split cand. 1 Split cand. 2 Input SFU, SIAT, NUDT, TAU Combinatorial search!

Layering cand. 1 Layering cand. 2 Substructure Layering cand. 1 Finding the optimal hierarchy Genetic algorithm with tree representation Sample and evolve population of hierarchies

Genetic operators: Crossover Mutation Split Layering Altering decomposition SFU, SIAT, NUDT, TAU

Swapping sub-trees Finding the optimal hierarchy: Genetic algorithm Fitness function: 1 2 3 4

Symmetry measure of a node (a substructure) max . =1 + 2 + 3 + 4 SFU, SIAT, NUDT, TAU Symmetry measure at a node Requirements: Continuous measure for a discrete box pattern Behaves well at both ends of the symmetry spectrum

Integral Symmetry (IS): Integral of symmetry profiles along two directions X Perfectly symmetric asymmetric Y SFU, SIAT, NUDT, TAU

Symmetry measure of a discrete pattern Symmetry profile Intra-box profile () SFU, SIAT, NUDT, TAU

Symmetry measure of a discrete pattern Symmetry profile Intra-box profile ()

SFU, SIAT, NUDT, TAU Symmetry measure of a discrete pattern Symmetry profile Intra-box profile

SFU, SIAT, NUDT, TAU () Symmetry measure of a discrete pattern Symmetry profile Inter-box profile

1 2 , () 1 SFU, SIAT, NUDT, TAU

2 Symmetry measure of a discrete pattern Combining inter-box and intra-box profiles: Integral Symmetry SFU, SIAT, NUDT, TAU An application Faade retargeting

Top-down propagation SFU, SIAT, NUDT, TAU An application Faade retargeting Top-down propagation SFU, SIAT, NUDT, TAU An application Faade retargeting Top-down propagation

SFU, SIAT, NUDT, TAU An application Faade retargeting Top-down propagation input SFU, SIAT, NUDT, TAU retargeted

Structure-aware faade retargeting SFU, SIAT, NUDT, TAU Input User interactive structural Seam carving analysis [LinStructure-aware et al. 2011] SFU, SIAT, NUDT, TAU

Evaluation I: Integral symmetry Symmetry ranking tests A SFU, SIAT, NUDT, TAU B Evaluation I: Integral symmetry The accuracy score obtained is 88%

Consistent with human perception SFU, SIAT, NUDT, TAU Evaluation II: Symmetry-driven decomposition Our method max. SymScore( 1) SFU, SIAT, NUDT, TAU SymScore( 2)

Evaluation II: Symmetry-driven decomposition Compare to two alternatives Alternative 1: Global reflectional symmetry SFU, SIAT, NUDT, TAU Alternative 2:

Graph-cut segmentation Evaluation II: Symmetry-driven decomposition Please select which one, A or B, appears to offer the best high-level explanation of the facade structure: SFU, SIAT, NUDT, TAU Evaluation II: Symmetry-driven decomposition On 600 questions Obtain a winning percentage

of 73% against Alternative 1 of 79% against Alternative 2 SFU, SIAT, NUDT, TAU Limitations Box abstraction and element grouping carried out with user assistance Limited to axis-aligned structures Limited to binary decompositions Human perception related

learning/crowdsourcing? SFU, SIAT, NUDT, TAU Conclusion Hierarchical and layered analysis of irregular facades Generative model: hierarchy of decompositions A clearly defined objective: symmetry maximization Applications: Structure-aware faade retargeting/editing/exploration SFU, SIAT, NUDT, TAU

Acknowledgement Anonymous reviewers Yangyan Li, Niloy Mitra, and Ariel Shamir Participants of our user studies Research grants NSERC Canada, NSFC China, Guangdong Sci. and Tech. Program, Shenzhen Sci . and Inno. Program, CPSF, and the Israel Science Foundation. SFU, SIAT, NUDT, TAU Thank you! Code and data are available: kevinkaixu.net

SFU, SIAT, NUDT, TAU

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