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### L-System and

1. 1. L-System Tree Hugger Project Bauhaus University Weimar Hasibullah Sahibzada 25-Nov-2013
2. 2. Agenda L-System algorithm and its variations Axial Tree Brief on Related works ( plastic trees) Brief on Plastic trees
3. 3. L-System Lindenmayer systems (L-systems) created Aristid Lyndenmeyer http://www.avatar.com.au/courses/Lsystems/References.html#ref004 Developmental systems were introduced (Lindenmayer, 1968, 1971) in order to model morphogenetic (pattern-generating) processes in growing, multicellular, filamentous organisms. The emphasis was on plant topology , on the spatial relations between cells or larger plant modules. To model multicellular plant growth. They did not include enough detail to allow for comprehensive modeling of higher plants. -- original work
4. 4. L System Theory L-Systems are mathematical models to describe the development of plants in a visually convincing way. This method is preferable to 3D geometry modelling as it requires much less human interaction and often produces more convincing results in modeling the small, next to unnoticed details that a human often would not bother to model. Central Concept of L-System is that of Re-writing. Rewriting is a technique for defining complex objects by successively replacing Snowflaks parts of a simple initial object using a set of rewriting rules or productions. Classical Example from Von Koch snowflake 1: One begins with two shapes, an initiator and a generator. 2: The latter is an oriented broken line made up of N equal sides of length r. 3: Thus each stage of the construction begins with a broken line and consists in replacing each straight interval with a copy of the generator, reduced and displaced so as to have the same end points as those of the interval being replaced. B. B. Mandelbrot. The fractal geometry of nature. W. H. Freeman, San Francisco, 1982.
5. 5. Re-writing Many Examples are done based on rewriting technique. 1: Rewriting systems on graphs 2: Rewriting systems on rectangular arrays and matrices ( Cellular automata) 3: Rewriting systems on character strings History Begin 19th century: Thue provided first formal definition of a string rewriting system (srw) 1960: Backus and Naur used rewriting notation in the definition of programming language ALGOL 1968 the biologist A. Lindenmayer introduced L-Systems to model multicellular plant growth S. Ginsburg and H. G. Rice. Two families of languages related to ALGOL. J. ACM, 9(3):350–371, 1962.
6. 6. L-System (DOL) Simplest class of L-systems, those which are deterministic and context-free, called DOL- systems. Development of a filament of the bacteria Anabaena catenula 1: The symbols a and b represent cytological states of the cells 2: The subscripts l and r indicate cell polarity, specifying the positions in which daughter cells of type a and b will be produced. L-system describes development – startsAxiom(distinguished string) ar – p1:ar→albr – p2:al→blar – p3:br→ar – p4:bl→al Under a microscope, the filaments appear as a sequence of cylinders of various lengths, with a-type cells longer than btype cells. Graphical modeling using L-systems
7. 7. Turtle interpretation 1: A state of the turtle is defined as a triplet (x, y, α) 2: the Cartesian coordinates (x, y) represent the turtle’s position, and the angle α, called the heading, is interpreted as the between lines) Gets more complicated ( space direction in which the turtle is facing. 3: Given the step size d and the angle increment δ, the turtle can respond commands represented by the following symbols Approximations of the quadratic Koch island taken from Mandelbrot’s book [95, page 51].
9. 9. Axial Trees A rooted tree has edges that are labeled and directed. An axial tree is a special type of rooted tree 1: At each of its nodes, it has most one outgoing straight segment. 2: All remaining edges are called lateral or side segments. 3: A sequence of segments is called an axis, if: Sub-tree a: the first segment in the sequence originates at the root of the tree or as a lateral segment at some node, b: each subsequent segment is a straight segment. c: the last segment is not followed by any straight segment in the tree. Sample tree generated using a method based on Horton– Strahler analysis of branching patterns
10. 10. L System Model (Ijiri_wiss2005)Sketching L-System: Interface for designing Flactal Stractures by drawing Axis
11. 11. L-System Summary L-Systems are based on string rewriting using production rules ( graph grammars) Turtle graphics give them a graphical interpretation L-Systems serve as models of development in biology, but also in other areas. Small changes of rules have often surprisingly large impact. Axial trees are important branching structures in nature (rivers, botanical trees,
12. 12. Related Work Early Plant Models growth Procedural approach Early models of plants were based on procedural approaches that replicated growth by repetitive application of a small set of rules to an initial structure to yield very complex results. Captured Internal properties of trees branching angle branch length
13. 13. Procedural Approach The forms of nature based on spirals and ramification are generated not through the use of object data calculated by measurement, but through the use of an algorithmic structure based on the laws of nature. Explained Processes of recreating some forms of nature, including shells, horns, tusks, claws, and spiral plants. KAWAGUCHI, Y. 1982. A morphological study of the form of nature. In SIGGRAPH ’82: Proceedings of the 9th annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 223–232. 3-D Shapes Spiral Shells Spiral Growth into Tendrils Horn Mouth of shell Young plant
14. 14. Procedural Approach The whole form of actual trees,therefore, was speculated to be affected also by their branching angle and relative ratio of their branch lengths HONDA, H. 1971. Journal of Theoretical Biology 31, 331–338. Problem views of trees or their crown 1: Pattern-recognition. How Define a specie from the various forms (same crown ) 2: Morphogenesis. ( Gens specify the form of tree) Described erect trees as repeated branching structures. The whole form of actual trees seems to be determined by a great many factors, (branching angles and relative ratios of branch lengths)
15. 15. Textures The present work seeks to model trees with sufficient realism that they may be the subject of animation, rather than simple elements of the landscape. The model should have a well-defined structure; beneath the bark the limb should be smooth: leaves should be properly attached to twigs. Several spline segments interpolate the data points (asterisks) with C2 continuity. A strobe captures a disk as it passes along the curve. Surfaces Order of creation of limbs (red, then orange, yellow, green, blue, and white) BLOOMENTHAL, J. 1985. Modeling the mighty maple. SIG- GRAPH Computer Graphics 19, 3, 305–311.
16. 16. Textures Resolution Issues A tree tends to maximize its surface area to volume ratio. If the viewpoint is close to a limb, a large number of limbs will be off-screen. Thus, a method is desired for polygonizing limbs that varies the axial and circumferential(enclosing boundry) resolutions according to the projection of the limb onto the screen and that culls off-screen limb sections. Verisimilar Bark X-raying Surface plaster Mapping Leaves by Camera Movable Joints BLOOMENTHAL, J. 1985. Modeling the mighty maple. SIG- GRAPH Computer Graphics 19, 3, 305–311.
17. 17. Botanical Structure & Development DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158. Concern here is the faithfulness of the models to the botanical nature of trees and plants. In the model They Integrated Botanical Knowledge of tree architecture Growth Mechanism to Occupy space Location of Leaves, flowers and fruits Important Thing Time Integration Viewing the aging of a tree Different Pictures Simulation of death of leaves and branches Wind and insects
18. 18. Botanical Structure & Dev DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158. Meristems Found in zones of a plant where growth Take place. Functionality 1: Growth Time 2: Ramification Diversions from small to big 3: Mortality. Each Bud is given two probabilities Break/abort Ramification
19. 19. Botanical Structure & Dev DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158.
20. 20. Botanical Structure & Dev DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158. Growth Simulation Bellow Parameters should be given -the age, - the clocks or growth speeds of the axes, - the number of possible buds at each node, as a function of order, - the probabilities of death, pause, ramification and reiteration By experiment Mathematical Law
21. 21. Botanical Tree Model Dynamic Modeling Technique Dynamic modeling and representation technique for trees that aims at incorporating aspects of the trees genotype into our models to allow them to react to the environment. Dependency of model Endogenous Exogenous
22. 22. Botanical Tree Model 1: Exogenous ( Environment )
23. 23. Related Works B: user-assisted plant modeling 1: complex parametric model. Weber and Penn [1995] 2: introduce decomposition graphs [Boudon et al. 2003] 3: Xfrog modeling technique Lintermann and Deussen [1999] which combines rule-based and procedural modeling and also allows for creating animated models. Not possible for models to dynamically react to their environment.
24. 24. Related Works C: Image-based techniques 1:Register Input Images to reconstruct the 3D shape While Reche-Martinez et al. [2004] 2: loosely arranged images Neubert et al. [2007] The main branches are determined by the user and the static model is constructed using a particle flow system and some botanic heuristics. D: sketch-based technique 1: Combination of rule-based and image-based techniques on procedural made trees Ijiri et al. [2006] and Zakaria and Shukri [2007] 2: 2D-3D Chen et al. [2008] Bringing Set of biologically motivated branching rules to infer the 3D structure of the tree from a given 2D sketch.
25. 25. Related Works E: Environment Modification of the environment itself can be used as a way of controlling the procedural model. 2: Climbing of plants that grow on support structures and are influenced by the light density. Arvo and Kirk [1988], Greene [1989], and later Benes and Millan [2002] F: Computing light Method 1: Simplified, technique proposed by Rudnick et al. [2007]. 2: Radiant energy transfer [Soler et al. 2003].
26. 26. Plastic Trees Trees reaction in Environment changes: Studied 1: Geometrical 2: Topological 1: Study the environmental conditions Self Shadowing 2: Natural morphological properties. 3: Construct a procedural model . Isolated space Behavior of branch Controllable by environmental parameters Morphological parameters Self Shadowing
27. 27. Branch Age Factor Growth rate of entire tree. Growth rate in a branch The growth rate of an individual branch is determined by how many internodes (segments without buds) a given branch produces in one season. But Nodes Depends on resources Branch Age =>> Internode length and growth rate Distance from a given segment to its furthest leaf node, Threshold dt = 0.2dr Distance from the root of the tree to its furthest leaf node. Branch age Growth rate Distance from the segment to the root of the tree Minimum growth rate(all) Absolute growth rate Length of internode Length of branch segment Parent age
28. 28. Temporal Light Conditions Simulate effect for each leaf Leaf Cluster cast shadow Shadow calculated by Shadow volume that is attached to shadow caster point Leaf translucency Amount of light Normalization constant [0,1] == in coming light Visibility of hemisphere From P
29. 29. Inverse Tropism A tropism is the tendency of the branches to grow towards or away from some entity. Need to know Environmental changes + different stages of tree development Influence of tropisms on tree growth and shape. unit direction of the tropism tropism Tropism strength Tropism 1: Phototropism 2: Gravitropism
30. 30. Phototropism The tendency of a given branch to grow towards the light direction. Calculated on All branches using Temporal Light Model
31. 31. Gravitropism Gravitropism controls bending of the branches either away from or towards gravity Set the angle of branches Tropism New Direction Linear combination of all tropisms Normalized direction weights weight of the original direction of branch segment Branch segment lenght
32. 32. Inverse tropism To Compute the effect of tropism on a input tree. Inverse tropism d0
33. 33. Pruning estimation Natural pruning influences the tree structure Apical meristematic cells in a bud produce wood or plant organs If no light then die Sum of Leaf node distances Resource by their child leave cluster < Pruning Factor Normalized amount of light Leaf Cluster radius
34. 34. Pruning estimation User controllable parameter = 0.8 Local pruning Factor 5th percentile of local pruning factor topiary
35. 35. Dynamic Interaction After growth behavior and pruning Learn 1: Amount of changes in environment. 2: Tree response according to the changes 1: Tree Graph transformation 2: Modeling of leaf-Clusters 3: Types of Interaction
36. 36. Tree Graph Transformation Changes in tree growth according Age, Light conditions 1: binding 2: Rotation of a branch === > transferred to parent 3: Leaf Cluster changes ( shadow ) 4: Pruning
37. 37. Modeling a leaf cluster 1: Response to light Amount of light a. Branch creation, and branchlets ( Cluster Density ) b. Branch orientation c. Number of leaves per branch Initial density Normalize density For leaf cluster Incoming light
38. 38. Modeling a leaf cluster 1: Response to Obstacles Intersection of leaf New tree overlap Collide and then pruned. Hull cluster 1 Black = input tree color Red=binding Blue = pruned Exception for small branches 2
39. 39. Types of Interaction As seen before interaction A: Tree-Obstacle B: Tree-Tree A: Global Light Computation of transformation
40. 40. Implementation and Results Tree Modeling’s main focus is on the branches rather then leaf cluster Thickness Threshold set Big Rendered LOD ( Level of Detail) technique is created The amount of produced geometery Cluster Size Light situation LOD stage Depends on Camera Location too. Small Leaf Cluster
41. 41. Implementation and Results Main branching structure Stored Tree self similarities Branchlets ( patches) CPU Mem Mapped GPU / VBO Texture buffer GPU
42. 42. Results Models from 1: Xfrog 2: Open L-System 3: LiDar Plastic Tree: 1: Representation Model 2: Interaction model 3: Influence in tropism Limitation: 1: Prodxuction of new branches 2: Soil change effect. 3: Influence in tropism
43. 43. References HONDA, H. 1971. Journal of Theoretical Biology 31, 331–338. Ijiri_wiss2005 Sketching L-System: Interface for designing Flactal Stractures by drawing Axis R. E. Horton. Hypsometric (area-altitude) analysis of Erosional topology. Bull. Geol. Soc. America, 63:1117–1142, 1952. OPPENHEIMER, P. E. 1986. Real time design and animation of fractal plants and trees. SIGGRAPH Comput. Graph. 20, 4, 55– 64. BOUDON, F., PRUSINKIEWICZ, P., FEDERL, P., GODIN, C., AND KARWOWSKI, R. 2003. Interactive design of bonsai tree mod- els. Computer Graphics Forum. Proceedings of Eurographics 22, 3, 591–599. DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158. KAWAGUCHI, Y. 1982. A morphological study of the form of nature. In SIGGRAPH ’82: Proceedings of the 9th annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 223–232. BLOOMENTHAL, J. 1985. Modeling the mighty maple. SIG- GRAPH Computer Graphics 19, 3, 305–311. http://www.avatar.com.au/courses/Lsystems/References.html#ref004