A Primer on Inverse Procedural Modeling

Transcription

1 A Primer on Inverse Proceural Moeling CS535 Fall 2016 Daniel G. Aliaga Department of Computer Science Purue University

2 Recall: Proceural Moeling Apply algorithms for proucing objects an scenes The rules may either be embee into the algorithm, configurable by parameters, or externally provie 2

3 Proceural Moeling Fractals Terrains Image-synthesis Perlin Noise Clous Plants Cities An proceures in general 3

4 L-system Variables: a Constants: +, - (rotations of + or 90 egrees) Initial string (axiom): s=a Rules: a a+a-a-a+a

5 (Context-Free) L-system for Plants

6 L-system for Plants (stochastic)

7 L-system for Plants (3D)

8 Inverse Proceural Moeling (by Automatic Generation of L-systems) O. Šťava, B. Beneš, R. Měch*, D. Aliaga, P. Krištof Purue University, *Aobe Inc

9 Overview 2D Vector Image Analysis L-system Moifications R(m) A[ () f() (s) (β)r(m 1)] [+( ) f ( ) (s )+(β )R(m 1)]

10 Introuction Proceural Moeling Rules Scene Extensively stuie The most important: L-systems Applie to plants, builings, rivers, etc. Inverse Proceural Moeling Scene Rules Open problem

11 Motivation What is the L-system for these?

12 Motivation Writing proceural moels, be it L-systems or others is har Setting parameters is not intuitive Difficult for a non-expert to create a moel for a esire shape, especially when recursive branching

13 Approach Input: 2D vector image Output: L-system Inspire by symmetry etection Mitra et al., Partial an approximate symmetry etection for 3D geometry Pauly et al., Discovering structural regularity in 3D geometry

14 Approach Upate the affecte clusters Similarity Distance Scaling P(3) Cluster size Seq. length P(m) A f(1) -(30) P(m-1) 1. Calculate terminals by inverse instancing 2. Calculate transformations, fill transformation spaces, an perform clustering 3. Analyze clusters an calculate cluster significance 4. Create nonterminal symbols an L-system rules

15 Approach B4 B3 B2 B1 A4 A3 A1 A2 L-system 1 P1(m) :m>0 [A] T 1 P1(m-1) m=0 [A] P2(m) :m>0 [B] T 2 P2(m-1) m=0 [B] P3(m) [P1(3)] T 3 [P2(3)] S T s [P3]

16 Approach Different rules can generate the same result A4 L-system 2 P1(m) [A] T 1 [B] B4 B3 B2 B1 A3 A1 A2 P2(m) :m>0 [P1] T 2 P2(m-1) m=0 [P1] S T s [P2(3)]

17 Inverse Instancing inverse instancing transformation cluster Lsystem generatio n Terminal symbols Similar vector elements A4 L-system 2 P1(m) [A] T 1 [B] B4 B3 B2 B1 A3 A1 A2 P2(m) :m>0 [P1] T 2 P2(m-1) m=0 [P1] S T s [P2(3)]

18 Inverse Instancing inverse instancing transformation cluster Lsystem generatio n Compute similarity between all input elements Similar elements are represente by a terminal A4 L-system 2 P1(m) [A] T 1 [B] A : Terminals B4 B3 B2 A3 A2 P2(m) :m>0 [P1] T 2 P2(m-1) m=0 [P1] S T s [P2(3)] B : B1 A1

19 Transformation Analysis inverse instancing transformation cluster Lsystem generatio n Proceural rule Transformation between two symbols A4 L-system 2 P1(m) [A] T 1 [B] A : Terminals B4 B3 B2 A3 A2 P2(m) :m>0 [P1] T 2 P2(m-1) m=0 [P1] S T s [P2(3)] B : B1 A1 () f() (s) (β) Transformation between two coorinate systems

20 Transformation Analysis transformation Many possible transformations inverse instancing Use significant transformations for rules cluster Lsystem generatio n A4 A : Terminals B4 B3 B2 A3 A2 B : B1 A1

21 Transformation Analysis transformation inverse instancing cluster Lsystem generatio n Detect significant transformations Put all transformations into Transformation space Transformation = 4D Vector (2D transl., rotation, scale) A4 4D Transformation Space A : Terminals B4 B3 B2 A3 A2 B : B1 A1 0

22 Transformation Analysis transformation inverse instancing Clustering in the transformation space Large clusters ~ significant transformations cluster Lsystem generatio n A4 4D Transformation Space A : Terminals B4 B3 B2 A3 A2 B : B1 A1 0

23 Transformation Analysis transformation inverse instancing cluster Lsystem generatio n One rule might not represent one cluster All transformations must be between same symbols P2(m) :m>0 [B] T 2 P2(m-1) P1(m) :m>0 [A] T 1 P1(m-1) m=0 [B] m=0 [A] A4 4D Transformation Space A : Terminals B4 B3 B2 A3 A2 B : B1 A1 0

24 Transformation Analysis transformation inverse instancing cluster Lsystem generatio n One transformation space for each pair of terminal symbols 4x 4D Transformation Spaces A4 A-A A-B A : Terminals B4 B3 B2 A3 A2 0 B-A 0 B-B B : B1 A1 0 0

25 Transformation Analysis transformation inverse instancing cluster Lsystem generatio n Cluster = Transformations between the same symbols 4x 4D Transformation Spaces A-A A-B A4 A : Terminals B4 B3 B2 A3 A2 0 B-A 0 B-B B : B1 A1 0 0

26 Cluster Analysis inverse instancing transformation cluster Lsystem generatio n One cluster One rule Each rule is unique 4x 4D Transformation Spaces A-A A-B A4 A : Terminals B4 B3 B2 A3 A2 0 0 P(m) :m>0 [A] T P(m-1) m=0 [A] B : B1 A1 P(m) [A] T [B]

27 Cluster Analysis inverse instancing transformation cluster Lsystem generatio n Rules are generate from clusters sequentially Orer of clusters is important 4x 4D Transformation Spaces A-A A-B A4 A : Terminals B4 B3 B2 A3 A2 0 0 P(m) :m>0 [A] T P(m-1) m=0 [A] B : B1 A1 P(m) [A] T [B]

28 Cluster Analysis inverse instancing transformation cluster Lsystem generatio n Compute importance of each cluster Weighte cluster importance function w=w n n + w h h + w Φ Φ + w l l n: number of points in the cluster h: proximity of two elements Φ: similarity between terminals l: average length of the sequences in a cluster Sort clusters accoring to their importance

29 Cluster Analysis inverse instancing transformation cluster Lsystem generatio n Weighte cluster importance function Weights etermine the final rules Prefer sequences Prefer proximity

30 L-system Generation transformation inverse instancing New rule = new non-terminal symbol cluster Lsystem generatio n Terminals A-A A-B A : A4 C4 B : B4 B3 B2 B1 A3 C3 A2 C2 A1C1 0 C : 0 Non-Terminals

31 L-system Generation transformation inverse instancing cluster Lsystem generatio n Clusters no longer vali Upate them using the new non-terminal symbol Compute importance of upate clusters Terminals A-A C-C A-B A : C4 B : C3 C2 0 0 Non-Terminals C1 C :

32 L-system Generation transformation inverse instancing cluster Lsystem generatio n Generate new rules until there are no clusters Axiom Last non-terminal Non-Terminals A : Terminals C4 C-C C : B : C3 D : C2 0 C1D1 Axiom: S T S D(3)

33 Final L-system L-system Generation transformation inverse instancing cluster Lsystem generatio n Terminals L-system C(m) [A] T 1 [B] A : B : D(m) :m>0 [C] T 2 D(m-1) m=0 [C] S T s [D(3)]

34 Summary Upate the affecte clusters Similarity Distance Scaling P(3) Cluster size Seq. length P(m) A f(1) -(30) P(m-1) 1. Calculate terminals by inverse instancing 2. Calculate transformations, fill transformation spaces, an perform clustering 3. Analyze clusters an calculate cluster significance 4. Create nonterminal symbols an L-system rules

35 Results

36 Results

37 Conclusion Important step towars the solution of the problem of inverse proceural moeling Key concepts Multiple transformation spaces Cluster

38 Future Work General rules More complex expression in the L-system rules Polynomials Context sensitivity 3D structures