Plant Modeling-Computer Sciences Applications-Project Report, Study Guides, Projects, Research of Applications of Computer Sciences

Download Plant Modeling-Computer Sciences Applications-Project Report and more Study Guides, Projects, Research Applications of Computer Sciences in PDF only on Docsity! i Acknowledgements Many thanks to Allah almighty, the divine lord who has always showered his countless blessings upon we humans who are no more than a speck of the dust. My head bends low in deep gratitude for it was HE who gave me the will to embark on journeys not believed to be conquered every now and then. It is Allah who revealed upon me the wonders of hope and consistency. I stand all numb in thy Prophet’s (P.B.U.H) court for he (P.B.U.H) is an icon of deep knowledge and sheer wisdom ever known to exist in this worldly life, the ever glowing beacon has kept my heart bright and far from the darkness of despondency. I am forever in debt to my parents without whom I would merely exist. Their never ending support, help and encouragement has always kept me in spirits high enough to manage all chores of life. I extend my sincere and cordial thanks to my project supervisor Mr. Kamran Safdar for his continued guidance, never ending support and the pin-point criticism & curiosity which has ever helped me improve my work a thousand folds. Without his consistent help, valuable discussions and thorough keenness for the subject matter it would have been impossible to take this work to level this far. docsity.com ii Furthermore I would offer my humblest credits to Dr. Muhammad Arif for his kind concern, direction and constructive suggestions throughout this project. And last but not the least I am thankful to the Mother Nature so chaste and pure whose inspiration travels deep to the beds of my soul. Table of Contents Chapter 1: Introduction .............................................................................................. 1 1.1 Thesis Organization ................................................................................................. 2 1.2 Project Title ............................................................................................................. 2 1.3 Aims & Objectives.................................................................................................... 2 1.4 Modeling the Natural Phenomena ........................................................................... 3 1.5 Difficulties in Modeling Plant Life .......................................................................... 4 1.6 Applications of Plant Modeling ............................................................................... 5 References ...................................................................................................................... 5 Chapter 2: Plant Modeling Techniques ..................................................................... 6 2.1 Image-based plant modeling .................................................................................... 6 2.2 Plant modeling using automata ............................................................................... 7 2.3 Concept Sketches ..................................................................................................... 8 2.4 Modeling plants using L-systems ............................................................................. 9 2.5 Geometric Modeling/ Positional information .......................................................... 9 2.5 Plants faithful to botanical structures.................................................................... 11 References .................................................................................................................... 12 Chapter 3: Theoretical Backdrop ............................................................................ 13 3.1 Fractals .................................................................................................................. 13 3.1.1 Fractals in nature ......................................................................................................... 14 docsity.com v Chapter 8: Graphical Realization of the Models ..................................................... 79 8.1 Main tasks to be performed.................................................................................... 79 8.2 Domain and environment ....................................................................................... 79 8.3 Intended users ........................................................................................................ 79 Chapter 9: Study Conclusion & Future .................................................................. 81 9.1 Summary & Conclusion ......................................................................................... 81 9.2 Future Recommendations ...................................................................................... 82 Bibliography ................................................................................................................ 82 Appendices ................................................................................................................... 85 A- FloraL User Manual ........................................................................................................ 85 B- FloraL Design Models .................................................................................................... 85 docsity.com vi List of Figures Chapter 2: Figure 2. 1 Segmentation procedure ........................................................................................................ 7 Figure 2. 2 Modeling plants using automata ........................................................................................... 8 Figure 2. 3 Modeling of plants using concept sketches ............................................................................ 8 Figure 2. 4 Virtual plant generated using L-systems (using LParser Software) ...................................... 9 Figure 2. 5 Simulated Model with positional information of Inflorescence ........................................... 10 Figure 2. 6 Plant structures generated using botanical theories and processes .................................... 11 Chapter 3: Figure 3. 1 Fractals in Nature (a) Broccoli; (b) Cumulus clouds; (c) Snowflake .................................. 14 Figure 3. 2 Fractals in Plant-life (a) Fractal Tree; (b) Fractal Fern .................................................... 14 Figure 3. 3 Self-similarity in plants ........................................................................................................ 15 Figure 3. 4 Koch snowflake demonstration for rewriting ...................................................................... 18 Figure 3. 5 Turtle interpretation of L-systems strings ............................................................................ 19 Chapter 4: Figure 4. 1 Cell division ......................................................................................................................... 23 Figure 4. 2 Koch curve ........................................................................................................................... 24 Figure 4. 3 A figure created using Turtle Interpreter ............................................................................ 25 Figure 4. 4 Quadratic Koch with angle variation .................................................................................. 26 Figure 4. 5 Modified Koch ..................................................................................................................... 27 Figure 4. 6 Snowflake with 90 o angle ..................................................................................................... 27 Figure 4. 7 Islands and lakes fractal ...................................................................................................... 28 Figure 4. 8 Steplike Fractal ................................................................................................................... 28 Figure 4. 9 A four-circle formation fractal ............................................................................................ 29 Figure 4. 10 Checkered Box fractal ....................................................................................................... 29 Figure 4. 11 Mosaic Fractal .................................................................................................................. 30 Figure 4. 12 Fractals inside a demagnetized DVD ................................................................................ 30 Figure 4. 13 Dragon-like fractal ............................................................................................................ 31 Figure 4. 14 Random Fractal ................................................................................................................. 31 Figure 4. 15 Construction of Edge-curve on square grid ...................................................................... 32 Figure 4. 16 Dragon curve ..................................................................................................................... 33 Figure 4. 17 Sierpinski Gasket ............................................................................................................... 33 Figure 4. 18 Hexagonal Gosper Curve .................................................................................................. 34 Figure 4. 19 Space filling FASS curve ................................................................................................... 34 Figure 4. 20 Space filling curve ............................................................................................................. 35 Figure 4. 21Subfigure A ......................................................................................................................... 35 Figure 4. 22 Node Rewriting demonstrating Hilbert curve .................................................................... 36 Figure 4. 23 Half-T space filling curve .................................................................................................. 36 Figure 4. 24 Hilbert Curve ..................................................................................................................... 37 Figure 4. 25 Fractal formation .............................................................................................................. 37 Figure 4. 26 Tree-like shape .................................................................................................................. 38 docsity.com vii Figure 4. 27 An Axial Tree ..................................................................................................................... 39 Figure 4. 28 Tree OL-systems ................................................................................................................ 39 Figure 4. 29 Branching using Bracketed DOL-systems ......................................................................... 40 Figure 4. 30 Vegetative Structure .......................................................................................................... 41 Figure 4. 31 Plant Structure ................................................................................................................... 41 Figure 4. 32 Tree-like Fractal Structure ................................................................................................ 42 Figure 4. 33 Tree with leaves ................................................................................................................. 42 Figure 4. 34 Symmetric Tree ................................................................................................................. 43 Figure 4. 35 Asymmetric Plant ............................................................................................................... 43 Figure 4. 36 Four different structures with Stochastic L-systems .......................................................... 44 Chapter 5: Figure 5. 1 Natural Shoot....................................................................................................................... 47 Figure 5. 2 Monopodial Branching in Nature ........................................................................................ 50 Figure 5. 3 Sympodial Branching in Nature .......................................................................................... 50 Figure 5. 4 Polypodial Branching in Nature .......................................................................................... 50 Figure 5. 5 Inflorescences in Plants ....................................................................................................... 51 Figure 5. 6 Open Raceme ................................................................................................................ 52 Figure 5. 7 Closed Raceme .................................................................................................................... 52 Figure 5. 8 Dibotryoid........................................................................................................................... 53 Figure 5. 9 Spiral Cyme (Simple) ........................................................................................................... 55 Figure 5. 10 Double Cyme ..................................................................................................................... 56 Figure 5. 11 Closed Cyme ...................................................................................................................... 57 Figure 5. 12 A Panicle ........................................................................................................................... 58 Figure 5. 13 Umbel Raceme ................................................................................................................... 60 Figure 5. 14 Spike Racemes ................................................................................................................... 61 Figure 5. 15 Spadix Raceme ................................................................................................................... 61 Figure 5. 16 A Daisy Capitulum............................................................................................................. 62 Chapter 6: Figure 6. 1 Block Diagram of System .................................................................................................... 65 Chapter 7: Figure 7. 1 Three-dimensional Vectors .................................................................................................. 76 Figure 7. 2 An L-systems generated 3D Plant Structure ........................................................................ 77 Figure 7. 3 A three-dimensional plant structure .................................................................................... 77 Chapter 8: Figure 8. 1 FloraL GUI System Block Diagram .................................................................................... 80 docsity.com 1 Chapter 1 "I recognized my Lord with the breakage of my determination” ~Hazrat Ali (RTA) Introduction Over the past few years, computer graphics has grown into a novel and versatile field of research that has its belongings to numerous fields of science. Although it is a splendor of its own, it often charms the observer, when simple mathematical conventions result into complex visual images. The graphical demonstration of virtual flora is hence an interdisciplinary neighborhood owing to the many diverse conceptions from mathematics, botany and computer graphics. On one hand, biologists are interested in plant models for understanding the fundamental systems underlying plant development and structure. Research is being carried out so that the models could be used for computer-assisted decision making in horticulture, agriculture, and forestry. On the other hand, the computer graphics people are concerned with using plants as elements of scenery for computer animations and various other graphical environments. State-of-the-art models have been constructed, which when combined with certain rendering procedures deliver convincingly lifelike results. Synthetic images of plants and other foliage are increasingly difficult to distinguish from photographs. Art-inspired non-photorealistic techniques successfully mimic drawings and paintings of plants [1]. Artificial landscapes are being generated and virtual gardens are being simulated with a variety of graphical plants. Generating complicated scenes with plants for virtual environments like games etc now takes no more than a fraction of seconds. Out of these complicated scenes the most complex one comes from the natural phenomena. These include clouds, water, fire, smoke, terrains, mountains and most importantly plants. Scientists have been scratching their heads for some decades now to conquer this domain. The project work starting off with the study of natural phenomena gets around the plant modeling revolving around the fractal theory using the concept of rewriting of stings. docsity.com 2 1.1 Thesis Organization The thesis has been organized into eight chapters. An overview follows: Chapter 1: Introductory chapter with project background and overview. Chapter 2: A brief insight into the many existing plant modeling techniques. Chapter 3: This chapter institutes a compact synopsis of L-systems literature. Chapter 4: Aims to discuss the project work for fractals and plants in 2D. Chapter 5: Modeling and simulation work of herbaceous plants is presented here. Chapter 6: Presents the research and development work for L-system model construction. Chapter 7: Provides an insight into the 3D modeling and simulation of fractals and plants. Chapter 8: Summarizes the development work done in lieu of the graphical user interface followed by the interface functionalities and generated results. Chapter 9: The concluding chapter wrapping up the thesis formally. 1.2 Project Title The title “The FloraL Genre” being a little abstract demands a brief limelight. The word “FloraL” is the combination of “Flora” and “L”. In botany „flora’ refers to all plant life and „L’ stands for the modeling methodology under discussion, that is, the L-systems. “Genre” is a French word meaning categories or types. So in a nutshell, the project title could be interpreted as “Modeling of different types of plant life using L-systems”. 1.3 Aims & Objectives The overall objective of this project is to study the modeling and design theory of the virtual flora using L-systems and to carryout a research as to how the different models have been developed. The base line is built upon the theory of fractals so the project work witnesses frequent discussions on this theory. The implementation and simulation of the otherwise mathematical models is also a major concern. In view of docsity.com 3 that an interpreter/simulator program is ought to be developed. The project work shall conclude with a graphical user interface which will compile all developments done throughout the project providing an interactive platform for generation and simulation of plant models using formalisms of L-systems. The honest goals underlying this project work could be summarized as:  Modeling and simulation of basic fractal structures  Modeling and simulation of branching structures  Basic Modeling and simulation of plant organs  Basic modeling and simulation of inflorescences  Basic modeling and simulation of phyllotaxis  Generation of models based on details of a plant structure  Development of simulator program for 2D plants and fractals  Development of an interactive interface for the simulator program 1.4 Modeling the Natural Phenomena The major task of computer graphics is to render images onto a two dimensional screen as per the projections of the higher dimensional underlying geometric and physical model of the objects. According to this point of view, computer graphics is basically mathematics [2]. However, Euclidean geometry has difficulty in modeling natural landscapes because a cloud, a shoreline, a terrain or a plant is not a cone, a cube, or a sphere that can be produced by some unfussy parameters. Although it is probable to model a plant by finding out the exact location and orientation of each stem, leaf, branch and flower, this is a squander of time and space (in the description of the plant) because these organisms exhibit a large degree of self- similarity. For instance, the relationship between the trunk of a plant and its branches is similar to the relationship between one of those branches and the smaller branches docsity.com 6 Chapter 2 "Everything is approximate, less than approximate, for when more closely and sharply examined, the most perfect picture is a warty, threadbare approximation, a dry porridge, a dismal mooncrater landscape. What arrogance is concealed in perfection. Why struggle for precision, purity, when they can never be attained. The decay that begins immediately on completion of the work was now welcome to me." ~Jean Arp, On My Way Plant Modeling Techniques For the basic flora elements like buds, branches, barks, leaves, flowers and fruits, some conventional methodologies for realistic static visual illustrations have been developed. Each technique in turn gives vent to many complicating questions and despite of their purposefulness many obstacles remain. People have been working on the modeling of plants since a long time now and came up with numerous many techniques. The most convincing ones were those which incorporated certain geometry and image synthesis. Geometry-based techniques use polygons to mathematically model plants. Image-based techniques use two-dimensional (2D) textures and create impressions of a 3D object. The prominent modeling methodologies with persuasive results are briefly discussed like so: 2.1 Image-based plant modeling There exist many different approaches for modeling of plants using images. There is one technique in which segmentation is performed in both image and 3D spaces, the user can easily visualize its effect. Using the segmented image and 3D data, the geometry of each plant is then automatically recovered from the multiple views by fitting a deformable plant model [1]. There is another method proposed by a group led by Kanade [2] which developed the approach on the lines of feature tracking and factorization. Both feature tracking and factorization have inspired and motivated many important algorithms in structure from motion, 3D reconstruction and modeling docsity.com 7 [3]. Another methodology known as the quasi-dense approach delivers the structure from motion. This technique is robust for practical modeling purposes and provides a cloud of sufficiently dense 3D points that allow the objects to be explicitly modeled [4]. There is another approach where joint segmentation is performed using a probabilistic framework. Figure 2.1 shows the modeling of plants using image joint- segmentation. Figure 2. 1 Segmentation procedure 2.2 Plant modeling using automata There are different approaches in this domain as well. There are those methods in which the automata are completely random [5] while in others it is entirely deterministic [5]. Another technique refers to the automata with elaborative behavior in which the automata are allowed to sense and adapt to the environment. The form of the plant is derived from the paths of one or more automata which begins as seeds and progressively explore the environment. [7] The generation process could be viewed in terms of particle systems as the basic structure of plants is simply a record of particle docsity.com 8 trajectories. The particles along with their governing rules are referred to as automata and hence the technique environment-sensitive automata. Figure 2.2 shows some results coming from this approach. Figure 2. 2 Modeling plants using automata 2.3 Concept Sketches The approach is based on the traditional illustration technique of concept sketching. The user sketches the key construction lines for the main plant body and lateral organs. The system then automatically constructs the 3D plant arrangement in phyllotactic patterns rendered as pen-and-ink line drawings. [8] Figure 2.3 shows some outputs from a system implementing this technique. Figure 2. 3 Modeling of plants using concept sketches docsity.com 11 2.5 Plants faithful to botanical structures The model integrates the botanical knowledge of the architecture of plants: how they grow, how they occupy space, where and how leaves, flowers or fruits are located, etc. The known botanical laws are included in the models which renders the structures coming from the models faithful to the botany [14]. The broad idea here is to model the activity of apices at dicretized times: an apex, at the release of a signal can:  Maybe become a terminal flower, or  Become inactive until next signal (sleep), or  Become a so called internode at the extremity of which one or several leaves appear with new so called lateral buds at their axel and a new so called apical bud at the end of the internode [14] , or  Become dead. These events occur according to specific stochastic laws characteristic for each variety and each species. The geometric parameters, such as the length and diameter of an internode or the branching angles are also calculated according to specific stochastic characteristic laws. The structures resulting from this modeling technique are fairly realistic and have their applications in horticulture and botanical simulation studies. Figure 2.6 demonstrates the technique through a rendered model. Figure 2. 6 Plant structures generated using botanical theories and processes docsity.com 12 References [1] L. Quan, P. Tan, G. Zeng, L. Yuan, J. Wang & S. B. Kang. Image-based plant modeling. IEEE, 2006 [2] Tomasi, C. and Kanade, T. Detection and tracking of point features. Technical report CMU-CS-91-132, Carnegie Mellon University, 1991 [3] Tomasi, C. and Kanade, T. Shape and motion from image streams under orthography: A factorization method. International Journal of Computer Vision, 1992, pp: 137–154. [4] Lhuillier, M. and Quan, L. 2005. A quasi-dense approach to surface reconstruction from uncalibrated images. IEEE, 2005, pp: 418–433. [5] W. T. Reeves. Particle-Systems: a technique for modeling a class of fuzzy objects. Computer Graphics Conference, 1983, pp: 359-376. [6] A. R. Smith. Plants, fractals and formal languages. Computer Graphics Conference, 1984, pp: 1-10. [7] J. Arvo and D. Kirk. Modeling plants with environment-sensitive automata. Australia, 1996. [8] F. Anastacio, M. C. Sousa, F. Samavati, J. A. Jorge. Plant modeling using Concept sketches. France, 2006. [9] T. E. Burk, N. D. Nelson, and J. G. Isebrands. Crown Architecture of Short- rotation, Intensively Cultured Populus. III. A Model of Firstorder Branch Architecture. Canadian Journal of Forestry Research, 1983, pp: 1107–1116. [10] W. R. Remphrey and G. R. Powell. Crown Architecture of Larix laricina Saplings: Quantitative Analysis and Modeling of (nonsylleptic) Order 1 Branching in Relation to Development of the Main Stem. Canadian Journal of Botany, 1984, pp: 1904–1915. [11] B. Lintermann and O. Deussen. XFROG 2.0. December 1998. [12] B. Lintermann and O. Deussen. Interactive Modeling of Plants. IEEE Computer Graphics and Applications, 1999, pp: 56–65. [13] P. Prusinkiewicz, L. Mundermann, R. Karwowski and B. Lane. The use of positional information in the modeling of plants. Los Angeles, 2001. [14] P. de Reffye, C. Edelin, J. Francos, M. Jaeger & C. Puech. Plant Models faithful to botanical development. France, 1988. docsity.com 13 Chapter 3 “On every stem, on every leaf ... and at the root of everything that grew, was a professional specialist …” ~ Oliver Wendell Holmes Theoretical Backdrop For many centuries attention of scientists and mathematicians along with artists and philosophers got attracted by the beauty of plants. The striking geometric features such as the bilateral symmetry of leaves, the rotational symmetry of flowers, and the helical arrangements of scales in pine cones have been studied most widely [1]. This focus is reflected in a quotation from Weyl [2]: “Beauty is bound up with symmetry” Two factors systematize the plant structures and hence play a part in their beauty. The first is the elegance and relative simplicity of developmental algorithms, that is, the rules which describe plant development in time [1]. The second is self-similarity, characterized by Mandelbrot [3] as follows: “When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar.” L-systems or Lindenmayer systems are formal grammars (a set of rules and symbols) which could both be used to model the growth processes of plant development, that is, using the developmental algorithms and to generate self-similar fractals. 3.1 Fractals In 1975 Benoit Mandelbrot coined this term “Fractal” derived from the Latin word fractus meaning “fractured” or broken. In colloquial usage, a fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole" [3]. As a geometric object, a fractal has following features [4]: docsity.com 16 but at ever smaller scales. Figure 3.3 demonstrates this phenomenon. The concept has been discussed by Rozenberg, Herman and Lindenmayer as: In many growth processes of living organisms, especially of plants, regularly repeated appearances of certain multicellular structures are readily noticeable... In the case of a compound leaf, for instance, some of the lobes (or leaflets), which are parts of a leaf at an advanced stage, have the same shape as the whole leaf has at an earlier stage [6]. The shape of a living plant is formed in the framework of space and time, as a process of developing based on the structure it already has. By including the development in fractal models, Aristid Lindenmayer invented a formalism that describes the growing of plants, known as L-systems. As an extension to the static fractal models, the most significant features of this formalism are its simplicity that is, the plainness of the rules which describe plant development in time and self- similarity. 3.2 L-systems 3.2.1 Origin L-systems were introduced in 1968 by Lindenmayer as a theoretical framework of plant modeling for the study of simple multicellular organisms‟ development, which was then applied to investigate higher plants and plant organs. An L-system is only mathematical symbol formalism open to many interpretations. For its element of self- similarity, it can model the “form-less” patterns of many natural growing processes. In the simulation and rendering of growing plant, different symbols are interpreted as geometric elements of the plants, such as internodes and leaves. After the inclusion of geometric features, plant models expressed using L-systems is detailed enough to allow the use of computer graphics for realistic visualization of plant structures and developmental processes. The L-systems notion is impressed from the formal language theory, where there are rules corresponding to each symbol and a symbol upon its encounter in a string gets replaced with its rule forming a new longer sequence of strings. In subsequent recursions whenever that symbol is encountered it gets replaced or rewritten. This rewriting of rules happens for each symbol in the string and the replacements for the docsity.com 17 symbols are parallel rather than sequential as in the case of Chomsky‟s formal grammar. This rewriting of strings forms the base line of the L-systems theory. 3.2.2 Rewriting and Grammars In mathematics, computer science and logic, rewriting covers a wide range of methods for replacing sub terms of a formula with other terms. What is considered are rewriting systems which in its most basic form, consist of a set of terms, plus relations on how to transform these terms. The central concept of L-systems is that of rewriting. In general, rewriting is a technique for defining complex objects by successively replacing parts of a simple initial object using a set of rewriting rules or productions [2]. The rules and productions together are known as generative grammars by mathematicians and programmers. The basic strength of rewriting lies in the ability to define complex object by successively replacing parts of a simple initial object using a set of rewriting rules or productions. Productions are applied in parallel, with all letters being rewritten simultaneously in a given word, since development of plants takes place simultaneously in all parts of an organism. 3.2.2.1 Generative Grammars Assume the alphabet consists of 'a' and 'b', the start symbol is 'S' and we have the following rules: 1. S aSb 2. S ba Then we start with "S", and can choose a rule to apply to it. If we choose rule 1, we replace 'S' with 'aSb' and obtain "aSb". If we choose rule 1 again, we replace 'S' with 'aSb' and obtain "aaSbb". This process is repeated until we only have symbols from the alphabet (i.e., 'a' and 'b'). Finishing off our example, if we now choose rule 2, we replace 'S' with 'ba' and obtain "aababb", and are done. We can write this series of choices more briefly, using symbols: . The language of the grammar is the set of all the strings that can be generated using this process: . docsity.com 18 3.2.2.2 Difference between Chomsky grammars & L-systems  Difference lies in the method of applying productions.  In Chomsky productions are applied in series.  In L-systems productions are applied in parallel.  This difference reflects the biological motivation of L-systems.  Productions are intended to capture cell divisions in multicellular organisms, where many divisions may occur at the same time. 3.2.2.3 Rewriting Classical Example The "limit curve" defined by repeating this process an infinite number of times, adding more and more, smaller and smaller triangles at each stage, is called the Koch's SNOWFLAKE CURVE, named after Niels Fabian Helge von Koch (Sweden, 1870-1924). Figure 3. 4 Koch snowflake demonstration for rewriting Starting with an equilateral triangle whose sides have length 1. On the middle third of each of the three sides, an equilateral triangle is constructed with sides of length 1/3. Next if the base of each of the three new triangles is erased the structure looks like as in figure 3.4 (b). On the middle third of each of the twelve sides, an equilateral triangle with sides of length 1/9 is built. Erasing the base of each of the twelve new triangles yields the snowflake. The triangle in figure is the initiator or the axiom while the small triangle drawn at each side of the triangle of figure is the generator or the production rule. Under a magnifying glass, a little piece of the snowflake looks identical to a larger, unmagnified chunk. Objects that exhibit this kind of self-similarity are called FRACTALS and are of great importance in modeling of plants. docsity.com 21 Chapter 4 “The sun and moon cling to heaven, and grain, grass and trees cling to the earth” ~ Friedlaender, B. Fractals & Plant Structures in 2D The specification of L-systems varies from one class to another. The addition of turtle symbols makes the specification more flexible. There are a total of five classes which are able enough to model every plant form having a simple or complex structure. The model for one structure is quite different from another. So the exact amount of detail required for modeling a particular structure should be at hand that is the developmental process. By far the main issue was how to construct such a model so that it‟s easy to convert it into a simulation routine to come up with a graphical structure. L-systems models carry with them all the equipment necessary for the simulation routine. Each symbol in a string has a graphical interpretation and thus the simulation made quite easy. The first task would then be to take the most basic specification of the L-system and to come up with a graphical interpreter and simulator. This basic class with the simplest specification is termed as DOL-systems. 4.1 Determinism in Rewriting The concept of rewriting as demonstrated in the previous chapter came from the Chomsky‟s formal grammar. From the generative grammars it follows that a symbol has a rule associated with it. Now if a symbol has only one rule and none other then it would be possible to back track a derivation to yield the starting symbols. While if the symbol has more than one rules it would not be possible to find out as to which rule has been applied out of the possible ones in a complex derivation. This idea of a single rule for each symbol leads to the concept of determinism. Furthermore, if in a string a rule is applied without any condition on it with respect to its neighbor symbols then the grammar is said to be context-free as opposite to context-sensitive grammar which is the vice versa. These two concepts i.e. Determinism and Context- free grammar construct the class of L-systems known as DOL-systems. docsity.com 22 4.2 DOL-systems  DOL-systems present the most basic and simple class of L-systems.  The rules for each symbol are Deterministic & Context-free.  Here all plants are considered identical on contrary to nature.  The movements of the turtle are randomized during interpretation.  The topological changes are not catered under DOL-systems. 4.2.1 Anabaena Catenula The following example provides an illustration of the operation of DOL-systems. The formalism is used to simulate the development of a fragment of a multicellular filament such as that found in the blue-green bacteria Anabaena Catenula and various algae [1]. The symbols x and y represent cytological states of the cells (their size and readiness to divide). The subscripts a and b indicate cell polarity, specifying the positions in which daughter cells of type x and y will be produced. The development is described by the following L-systems: n=4 ω: xa p1: xa xb ya p2: xb yb xa p3: ya xa p4: yb xb Starting from a single cell xa (the axiom), the following sequence of words is generated: xa xb ya yb xa xa xb xb ya xb ya yb xa yb xa xa yb xa xa … Under a microscope, the filaments appear as a sequence of cylinders of various lengths, with x-type cells longer than y-type cells. docsity.com 23 Figure 4. 1 Cell division The corresponding schematic image of filament development is shown in Figure 4.1. L-systems are discrete so the continuous growth of cells between subdivisions is not captured by this model. 4.2.2 Algorithm Step 1: Set the Initial Angle equal to „Angle‟. Step 2: Set the Angle Increment equal to „Delta‟. Step 3: Set the Derivation Length equal to „Level‟. Step 4: Set the Initiator Segment Length equal to „SegLength‟. Step 5: Set the Generator Segment Length equal to „DivSegments‟; Step 6: DO Set Delta equal to „Angle‟. Rewrite the Initiator segment using initial values of Level, SegLength and Delta. Increment Level and Delta at each iteration. WHILE END OF AXIOM; xa ya xb yb docsity.com 26 Table 4.1 demonstrates the formation of quadratic Koch structure at subsequent derivations. It follows the snowflake curve idea where the initiator-generator combination results into a complex structure. The structure here is different from the snowflake because of the angle which in this case is 90 o . Hence, even a slight change in angle can render an entirely different structure. Quadratic Koch Curve Figure 4. 4 Quadratic Koch with angle variation Other modifications in strings, angles and derivation lengths yield a variety of structures. The results section demonstrates the corresponding change in structures. 4.2.4.1 Results The development of the turtle interpreter was extended to the development of the DOL-systems interpreter. The specification of the DOL-system results in perfect self- similar structures. This self-similarity makes it possible to generate fractals using this class. The varieties of models and equivalent graphical interpretations have been studied using this simulator. The models generated are some modifications of the Koch curve studied earlier using context-free list representation. The slightest change in angle results into an entirely different sequence. The results are affected through the changes made in each part of the model like axiom strings, production strings or derivation level etc. The generated structures reveal the fact that how L-systems have simplified the rendering of such complex shapes which were really tedious to generate using the previous approaches. The figures 4.5 through 4.14 demonstrate the DOL-systems based models and their corresponding simulations. docsity.com 27 L-systems Simulation Axiom F-F-F-F F F-F+F+FF-F-F+F Delta = 90 n = 3 Figure 4. 5 Modified Koch L-systems Simulation Axiom F-F-F-F F F+FF-FF-F-F+F+FF-F- F+F+FF+FF-F Delta = 90 n = 3 Figure 4. 6 Snowflake with 90 o angle docsity.com 28 L-systems Simulation Axiom F+F+F+F F F+f-FF+F+FF+Ff+FF- f+FF-F-FF-Ff-FFF f ffffff Delta = 90 n = 2 Figure 4. 7 Islands and lakes fractal L-systems Simulation Axiom -F F F+F-F-F+F Delta = 90 n = 3 Figure 4. 8 Steplike Fractal docsity.com 31 L-systems Simulation Axiom F-F-F-F F F-FF- -F-F Delta = 90 n = 2 Figure 4. 13 Dragon-like fractal L-systems Simulation Axiom F-F-F-F F F-F+F-F-F Delta = 90 n = 3 Figure 4. 14 Random Fractal docsity.com 32 4.2.4.2 Conclusion From the results it could be deduced that random modification of productions gives little insight into the relationship between L-systems and the figures they generate. However, we often wish to construct an L-systems which captures a given structure or sequence of structures representing a developmental process. This is called the inference problem in the theory of L-systems. 4.3 Synthesis of DOL-systems Although some algorithms for solving the inference problem were reported in the literature, they are still too limited to be of practical value in the modeling of higher plants. Consequently, some methods were introduced that appeared more intuitive in nature. They exploit two modes of operation for L-systems with turtle interpretation, called edge rewriting and node rewriting using terminology borrowed from graph grammars [1]. 4.3.1 Edge Rewriting Edge rewriting can be viewed as an extension of Koch constructions. For example, Figure 4.15 shows the curve on the grid and the L-system that generated it. Both the Fl and Fr symbols represent edges created by the turtle executing the “move forward” command. The productions substitute Fl or Fr edges by pairs of lines forming left or right turns. Many interesting curves can be obtained assuming two types of edges, “left” and “right.” Examples are included in the implementation. Figure 4. 15 Construction of Edge-curve on square grid docsity.com 33 4.3.1.1 Results L-systems Simulation Axiom L R L+R+ L -L -R Delta = 90 n = 3 Figure 4. 16 Dragon curve L-systems Simulation Axiom R L R+L+R R L -R -L Delta = 90 n = 3 Figure 4. 17 Sierpinski Gasket docsity.com 36 Figure 4. 22 Node Rewriting demonstrating Hilbert curve Assuming that the contact points and directions of Recursive subfigures Ln and Rn are as in Figure 4.22, the figures Ln+1 and Rn+1 formula are captured by the following formulas: Ln+1 = +RnF − LnFLn − FRn+ Rn+1 = −LnF + RnFRn + FLn− 4.3.2.1 Results L-systems Simulation Axiom -L R -LFLF+RFRFR+F+RF -LFL-FR L LF+RFR+FL-F-LFLFL -FRFR+ F F Delta = 90 n = 3 Figure 4. 23 Half-T space filling curve docsity.com 37 L-systems Simulation Axiom -L R -LFLFLF+RFR+FL-F- LF+RFR+FLF+RFRF- LFL-FRFR L LFLF+RFR+FLFL-FRF- LFL-FR+F+RF-LFL-FRFRFR+ F F Delta = 90 n = 3 Figure 4. 24 Hilbert Curve L-systems Simulation Axiom L R RFLFR+F+LFRFL-F- RFLFR L LFRFL-F-RFLFR+F+ LFRFL F F Delta = 90 n = 3 Figure 4. 25 Fractal formation docsity.com 38 4.3.3 Conclusion The curves are generated by either edge or node rewriting method. As in the case of edge rewriting, the relationship between node rewriting and tiling of the plane extends to branching structures. It offers a method for synthesizing L-systems that generate objects with a given recursive structure, and links methods for plant generation based on L-systems. 4.4 Plant Structures So far, we can see that the turtle understands a string as a series of line segments. Based on the length of the segments and the angles between them, the resulting line is either self-intersecting or not, can be more or less complicated, and may have some segments drawn many times and others made invisible, but it always remains just a single line. Figure 4. 26 Tree-like shape Conversely, the plant empire is subjugated by branching structures; thus a mathematical description of tree-like shapes and methods for generating them are needed for modeling purposes. Figure 4.26 demonstrates a tree-like shape. docsity.com 41 4.4.3.1 Implementation & Results L-systems Simulation Axiom F F F[+F]F[-F]F Delta = 25.7 n = 5 Figure 4. 30 Vegetative Structure L-systems Simulation Axiom F F F[+F]F[-F][F] Delta = 22.5 n = 4 Figure 4. 31 Plant Structure docsity.com 42 L-systems Simulation Axiom F F FF-[-F+F+F]+[+F-F-F] Delta = 22.5 n = 4 Figure 4. 32 Tree-like Fractal Structure L-systems Simulation Axiom X F FF X F[+X]F[-X]+X Delta = 20 n = 7 Figure 4. 33 Tree with leaves docsity.com 43 L-systems Simulation Axiom X F FF X F[+X][-X]FX Delta = 25.7 n = 7 Figure 4. 34 Symmetric Tree L-systems Simulation Axiom X F FF X F-[[X]+X]+F[+FX]-X Delta = 22.5 n = 5 Figure 4. 35 Asymmetric Plant docsity.com 46 Chapter 5 “A weed is no more than a flower in disguise” ~ James Russell Lowell Plant Growth Models So far the work was progressing under the conception of database amplification. The term refers to the generation of complex looking objects from very concise descriptions [1]. L-systems are nothing more than a set of concise rewriting rules. The small size of specifications makes L-systems a technique popular for modeling plants and other fractal patterns. Still the construction of L-systems for generating a particular structure is not an easy task. The initial L-systems conceptions like edge- rewriting & bracketed OL-systems etc. were suited for structures with a high degree of self-similarity. But, when the task of generating a variety of patterns is at hand, these simple L-systems classes are not assistance. In order to model a structure, we need to get hold of the underlying developmental process which yields the particular plant form. This way, the simulation of the development of real plants could be made possible. The approach can be seen favorable with the help of the following two unique characteristics: Importance of the space-time connection between different plant organs: most of the plants have organs in different stages of development at the same time. For instance, some flowers may still be in the bud stage, others maybe fully developed, and still others may have been transformed into fruits [1]. If the development of individual organs is modeled individually, we can imitate these phase effects. Innate potential of plant growth simulation: The mathematical model can be used to generate biologically correct images of plants at different ages and to create sequences of images illustrating plant development in time [1]. There is a certain type of controlling authority which supervises the growth process in different life forms. Hence, we can say that growth process in plants is also controlled by the plant itself. Herbaceous plants carry this phenomenon of internal control mechanism. Herbaceous plants or non-woody plants are different from woody plants docsity.com 47 in which the final form is based on the environmental factors like competition amongst branches and accidents etc. The L-systems for herbaceous plants have different levels of model specification depending on the objectives in mind. These models drive the development. These levels further bring the different types of branching patterns into lime light. The compound flowering structures in herbaceous plants i.e. inflorescence has a variety of different models made out of it. So, the different structures could be modeled disjointedly using different specifications of L- systems. Moreover, the development could be generalized too based on the apical growth so that the L-systems could be generated based on different characteristics of a certain plant structure. The details follow in the succeeding headings. 5.1 Levels of Model Specification L-systems vary depending upon the objectives in mind, for example, L-systems for image synthesis purposes are quite different from those which generate simple branching structures. Under the heading of herbaceous developmental models, L- systems are specified at thee levels of detail. First and most abstract level is known as partial L-systems. The level second to partial L-systems is known as L-system schemata which try to model the control mechanisms within plants. The third and last level with geometric aspects added to the L-system schemata is known as complete L- systems. 5.1.1 Partial L-systems Partial L-systems employ the notation of non-deterministic OL-systems to define the realm of possibilities within which structures of a given type may develop. Partial L-systems capture the main traits characterizing structural types and provide a formal basis for their classification [1]. Figure 5.1 shows a shoot with a terminal flower. The partial L-systems for this shoot could be given as in Table 5.1. Figure 5. 1 Natural Shoot docsity.com 48 L-systems Simulation Derivation ω : a p1 : a → I[L]a p2 : a → I[L]A p3 : A → K a I[L]a I[L]I[L]A I[L]I[L]I[L]A I[L]I[L]I[L]K I[L]I[L]I[L]K Table 5. 1 Partial L-systems and simulated shoot The L-systems in Table 5.1(a) show that a shoot [Table 5.1(b)], after a period of vegetative growth produces a single flower. The symbol „a‟ represents the vegetative apex and „A‟ is for flowering apex which is capable of forming reproductive organs. Symbol „L‟ stands for a leaf and „I‟ for an internode while „K‟ stands for a flower. A possible derivation using the above stated L-systems is given in Table 5.1(c). Each transformation is known as a developmental switch. 5.1.2 L-Systems Schemata As partial L-systems does not specify the moments in which developmental switches occur so the timing of these switches is specified at the level of L-system schemata. As mentioned previously, this level incorporates mechanisms that control plant development. By control mechanism it is meant that certain information is transferred amongst different plant organs. So, the problem of non-determinism is resolved. Simplest method for the implementation of a developmental switch is stochastic. Each production has a certain probability associated with it. Certain plants change their states depending on the environment. For instance, factors like temperature or number of daylight hours etc. Such effects could be modeled using sets of productions, in the form of tables. Based on environmental factors a table is replaced by another. Delay mechanism assumes that the apex undergoes a number of changes in state and when a certain state is reached, the switch is applied. Similar to delay mechanism is the accumulation of components which highlights the physiological nature of counting process. docsity.com 51 Sympodial (Figure 5.3) and monopodial (Figure 5.2) are terms borrowed from biology while terminal and polypodial (Figure 5.4) have been spawned to denote the remaining cases that exist in nature. 5.2.1 Models of Inflorescences A general elaboration of branching structures is known as inflorescences. Inflorescences are composite flowering structures. A complete flowering-shoot system could be considered as inflorescence and a structure in which only some of the branches bear flowers are also termed as inflorescences. Figure 5. 5 Inflorescences in Plants 5.2.1.1 Monopodial Inflorescences Simple Racemes (open) In Raceme, a shoot has lateral apices with terminal structures and a main apex that continues to development. Hence, naturally racemes are monopodial inflorescences. Simple racemes are widely used. Table 5.4 shows partial L-systems and corresponding racemes. L-systems Simulation ω : a p1 : a → I[L]a p2 : a → I[L]A p3 : A → I[K]A Table 5. 4 (a) Partial L-systems Raceme (Open) (b) Elongated Open Raceme (c) Planar Open docsity.com 52 This system differs from that modeling a shoot with a single flower in Table 5.1(a) only in production p3. Here it is designed to repeatedly produce lateral flowers shown in Table 5.4(b), while in the previous system „A’ produces a single flower. The flowering sequence is acropetal in open racemes that is, from base to the top. At each developmental stage the inflorescence contains a sequence of flowers of different ages. The flowers newly created by the apex are delayed in their development with respect to the older ones situated at the stem base. Figure 5.6 shows an open raceme in nature. Figure 5. 6 Open Raceme Simple racemes (closed) In closed racemes, the main apex eventually terminates its development and produces a terminal flower. A very good example of this type of branching structure is that of an Apple tree. Developmental switches are associated with two symbols, „a’ and „A’. Thus, in order to obtain an L-system scheme it is necessary to specify how both of these switches will be controlled [2]. Flowering sequences could be both acropetal (base to top) and basipetal (top to root). Table 5.5 shows this case. Figure 5. 7 Closed Raceme L-systems Simulation ω : a p1 : a → I[L]a p2 : a → I[L]A p3 : A → I[K]A p4 : A → K Table 5. 5 (a) Partial L-systems - Racemes (Closed) (b) Elongated Closed Raceme (c) Planar docsity.com 53 Compound raceme (Open dibotryoid) Dibotryoid means two-level branching structures. Compound racemes can hence generate complex branching structures. The simplest combination is one with open raceme on both first-order branches and the main axis. Table 5.6 (a) demonstrates the developmental models and resulting structure Table 5.6 (b). Figure 5. 8 Dibotryoid L-systems Simulation ω : a p1 : a → I[L]a p2 : a → I[L]A p3 : A → I[L][b]A p4 : A → I[L][b]B p5 : b → I[L]b p6 : b → I[L]B p7 : B → I[K]B Table 5. 6 (a) Partial L-systems for Open dibotryoids (b) Branching pattern of Open dibotryoid Within each component raceme, the flowering sequence is always acropetal, but the timing of switches has a crucial impact on the overall flowering sequence and appearance of the plant. Compound racemes (Closed dibotryoids) Differs from close dibotryoids in only one way and that is, each branch including the main axis, bears a terminal flower. Example: mint plant. Compound racemes (tribotryoid) Compounding raceme into higher number of levels is possible. Here, a possibility is when closed raceme occurs on second-order branches plus terminal portions of first- order branches. Table 5.7-8 carry the models and branching of compound racemes. docsity.com 56 Figure 5. 10 Double Cyme Rose Campion‟s branching structure is a very good example. Table 5.10 shows the partial L-systems for double cymes and corresponding pattern. Figure 5.10 shows a double cyme pattern in nature. L-systems Simulation ω : a p1 : a→ I[L]a p2 : a→ I[L]A p3 : A→ I[A][A]K Table 5. 10 (a) Partial L-systems for Double Cymes (b) Double Cyme branching structure Cymes (closed) If the sympodial inflorescences produce terminal flower at some point, then the branching structure is known as closed cymes. Both single and double cymes could be closed too. They result from the addition of production p4: A → K to the L-systems specified in Table 5.9, which define open single and open double cymes. Figure 5.11 chows a closed cyme plant structure in nature. docsity.com 57 Figure 5. 11 Closed Cyme Thyrsus (closed) A thyrsus is an inflorescence with branches of cymes borne on a monopodially branching axis [1]. So, it‟s a mixture of monopodial and sympodial branching organizations. Difference is made depending on the orientation of flowers that either the Thyrsus with cymes is in a spiral form or in a zigzag form (Table 5.11-b). A thyrsus may have double cyme structures. L-systems Simulation ω : a p1 : a → I[L]a p2 : a → I[L]A p3 : A→ I[L][B]A p4 : A→ K p5 : B→ I[B]K p6 : B→ K Table 5. 11 (a) Partial L-systems-Thyrsus (b) Spiral form-Thyrsus (c) Zigzag Thyrsus (d) Double There are three developmental transformations in closed structures. First transformation characterizes change from vegetative to flower development on main axis. The second transformation is essential for the closing of main axis with a docsity.com 58 terminal flower. Finally, the third one generates flowers that terminate development of sympodial structures. 5.2.1.3 Polypodial Inflorescences Panicle Polypodial inflorescence is not a term or conception from the botanical literature. It was coined by the authors of L-systems in order to represent the rest of the branching structures i.e. that are not entirely monopodial or sympodial. These types of branching structures represent continuing development of both main axis and lateral apices of a branch. This kind of inflorescence is known as Panicle. Figure 5. 12 A Panicle The presence of two continuing apices at each new node is expressed by the following production [2]: A → I[L][A]A As nodes near the base of the plant don‟t have branches so we can include the initial rules so that the transition from purely vegetative to a flowering state is possible. The docsity.com 61 Figure 5. 14 Spike Racemes Figure 5. 15 Spadix Raceme docsity.com 62 Capitulum Capitulum or “head” is a fleshy, spherical or disk-shape raceme. The head of a sunflower is an inflorescence of this kind; the oldest flowers begin at margin at youngest at the center. Members of the family Compositae commonly have this type of structure [1]. Figure 5.16 shows capitulum of a daisy. Figure 5. 16 A Daisy Capitulum The spatial arrangement of certain components like flowers etc. is a characteristic feature. These components form early discernable spiral patterns. Table 5. 14 (a) Spike (b) Spadix (c) Capitulum docsity.com 63 5.3 Conclusion The branching in herbaceous plants depends upon the triggering environmental factors. This is achieved by having mode than one production rules for an apex. The resulting structures could thus be more plant-like and would avoid the symmetry caused by node and edge rewriting. References [1] P. Prusinkiewicz & A. Lindenmayer, “The algorithmic beauty of plants”, Springer-Verlag, New York, pp 66-97,1990. [2] P. Prusinkiewicz & Lila Kari, “Subapical bracketed L-systems”, Proc.\ 5th Int.\ Workshop on Graph Grammars and their Application to Computer Science, vol. 1073, pp 3-9 (1996). docsity.com 66 The data other than strings output from the rule base is fed to the geometry module where factors like angle, elongation and branching length are finalized. The production strings from the string collection module is fed to the subapical growth qualifier which checks as to whether the generated production is subapical or not. The botanically correct strings are then combined with the geometric data to generate the final DOL-system model. This model could be sent to the render program where the model is simulated to yield the corresponding plant structure. The generated models could be saved inside a repository for future use and could be simulated whenever required. The block diagram of complete system is shown in figure 6.1. 6.2 Working of the System 6.2.1 User Input Keeping in view the ease of the user the system gathers input by asking simple questions from the user and provides certain linguistic options to choose from. Certain botanical terminology has been explained to the user inside the help manual. Parameter Options Parameter Options Type Leaf Age Young Vegetation Grown Mature Overgrown Branching Bush Orientation Left Plant Right Tree Random Root Stemmed Branch Contour Gentle Curve No Stem Some Directed Directed Straight Regularity Random Length None One-sided Single Symmetric Double Asymmetric Maximum Thickness Fine Bold Thick Table 6. 1 Input Parameters The different parameters and their options available to the user are listed in Table 6.1. The input comes to the rule base module where the decisions regarding modeling parameters are made. docsity.com 67 6.2.2 System Repositories The system operates upon a knowledge base and a rule base both of which are developed using expert‟s knowledge. The knowledge base could be seen as a number of repositories with certain organized data ready for use in the rule base. These are string repository and geometric data repository. There‟s another repository with the name of models repository in the system‟s block diagram in figure 6.1. This repository stores the constructed models for use in the render module and for future simulations etc. 6.2.2.1 String Repository The basic theme behind a string repository is to make the system available with every possible string for the construction of a wide variety of branching productions. The repository consists of a number of classes where each class is named under the basic principles of subapical growth depicting the nature of strings in each class. A précis of the subapical terminology follows. String Classes Any structure with the mentioned growth patterns would be categorized as a structure possible in nature. Modeling the exact growth of each and every plant is almost impossible or is time consuming. This follows the idea that instead of modeling each structure by hand attempts should be made to model each growth possibility under different growth rules and branching patterns. Classes Simple Branched Complex A F X - + [+X] [X+] F[+X][-X] F[+X]F[-X] F[-X]F[+X] B + F + X F + X + [-X] [X-] F[+X]F F[-X]F F[+X][-X]F C - F - X F - X - F[+X] [+X]F [+X][-X]F [+X]FF [-X]FF [+X][-X]FF D [+X] [-X] F[-X] [-X]F [X-[+X]] [X+[-X]] Table 6. 2 String Classes inside the string repository docsity.com 68 The combination of these rules under certain conditions would then yield a wide range of structures with correct growth principles. Hence, almost every type of structure whether it is existent in nature or not could be generated. The string repository for the system is shown in Table 6.2. There are a total of twelve classes. Under each branching pattern there are four classes. The system therefore is available with almost every bit of string required to come up with almost any bracketed DOL- system possible. The strings in each class follow the basic principle of subapical growth. The symbols have been limited to „F‟, „f‟ and „X‟ only to this point. Addition of symbols to the strings would result in an increased number of classes hence providing us with more possible variations in structures. 6.2.2.2 Geometric Data Repository Apart from the axiom and production strings there is some other information necessary for the construction of the branching structure. This includes the branching angle, thickness, length and age of the structure. These factors both add to the appearance of the plant and contribute towards the execution of the L-system strings. The proposed system relieves the user from entering the exact value of each by building a repository of such data. This data store like string repository consists of classes as well with the same idea that almost each possibility is covered. Classes Branch Contour Age Thickness A Gentle Curve Young None B Somewhat Directed Grown Single C Directed Mature Double D Straight Overgrown Maximum Table 6. 3 Geometric Data Repository The repository for geometric data is shown in table 6.3. Each variable in this table corresponds to a class and has a multitude of values corresponding to it. Table 6.4 demonstrates the possible values or members of each class in this category. Classes Branch Contour Age Thickness A 5-22.5 1,2 0 B 23-30 3,4 1 C 31-40 5,6 2 D 40-90 7,8 3 Table 6. 4 Parameter Values for Geometric Data docsity.com 71 Definition 3. A bracketed DOL-system (BDOL-system) is a DOL-system S = <∑E; [w0]; P> where the axiom [w0] is a well nested word over the alphabet ∑E. Definition 5 from [10] states: Definition 5. A BDOL-system S = <∑E; [w0]; P> is subapical with respect to the main axis (of the generated branches) iff for any [w] ε L(S) with the standard decomposition [w] = [x1[α1] x2[α2] x3[α3]… xn[αn] xn+1] the internodes x1, x2, x3,… xn do not contain branching letters: x1, x2, x3,… xn. The rule states that a bracketed DOL-system would be subapical if the symbol inside the brackets is different to those outside the brackets. The system checks for this rule and if the string abides by this rule it is said to follow the subapical growth. The resulting plant structure would thus be fairly plant like avoiding any sort of artificial look despite of the random selection of strings. 6.2.5 Simulation & Renderings The models generated using the system could be rendered using the render program. The interpreter takes in the L-system models the way it was generated by the system and renders the corresponding structure. The simulation p to this level is two dimensional and the system constructs models using bracketed DOL-systems alone. The simulator could be used to construct all types of models with this DOL-system specification. 6.3 Results Simulation L-systems Axiom: X Elongation: F Branching: FF[+X][-F]F[+X][-X]F Angle: 26 Derivation level: 5 Table 6. 7 Generated Model and Structure docsity.com 72 Table 6.7 demonstrates the model and corresponding structure generated using the proposed system. The user here intuitively decides if the generated structure is as per requirement or not. The variation of plant structures generated using this system is demonstrated in Table 6.8. The first three structures represent the branching structures of leaves followed by different tree-like, bush-like and plant-like structures. These structures are of course not found in nature but can sure fit in any virtual environment giving the impression of virtual foliage. Table 6. 8 Simulated Results demonstrating variation docsity.com 73 6.4 Conclusion The generation of plants for image synthesis purposes and artificial landscape generation need not to be an exact representation of the botanical plant structures. The technique presented in the shape of the system discussed in this chapter provides with an idea as to how the L-systems could be generated at run-time if a certain amount of linguistic data is given. Also, how the models could still be made to follow the subapical growth. The users with no or little knowledge of L-systems and botany would now be able to model plants as per their requirements. References [1] P. Prusinkiewicz & Lila Kari, “Subapical bracketed L-systems”, Proc.\ 5th Int.\ Workshop on Graph Grammars and their Application to Computer Science, vol. 1073, pp 3-9 (1996). docsity.com 76 Figure 7. 1 Three-dimensional Vectors The symbols in Table 7.2 control turtle orientation in three-dimensional space. Symbols Meaning + Roll counter-clockwise to positive Z-axis by angle dz, using rotation matrix Rz(dz). - Roll clockwise to positive Z-axis by angle dz, using rotation matrix Rz(-dz). & Roll counter-clockwise to positive Y-axis by angle dy, using rotation matrix Ry(dy). ^ Roll clockwise to positive Y-axis by angle dy, using rotation matrix Ry(-dy). \ Roll counter-clockwise to positive X-axis by angle dx, using rotation matrix Rx(dx). / Roll clockwise to positive X-axis by angle dx, using rotation matrix Rx(-dx). | Turn around, using rotation matrix Ry(180). Table 7. 2 Turtle Interpretation for 3D 7.1.2 Plants in 3D The L-systems for a 3D plant structure could be given as below. The generated structure is given in Figure 7.2 n=7, δ=22.5◦ ω : A p1 : A → [&FL!A]/////‟[&FL!A]///////‟[&FL!A] p2 : F → S ///// F p3 : S → F L p4 : L → [‟‟‟∧∧{-f+f+f-|-f+f+f}] docsity.com 77 Figure 7. 2 An L-systems generated 3D Plant Structure Figure 7. 3 A three-dimensional plant structure docsity.com 78 7.2 Conclusion The plant structures to be demonstrated in 3D require a detailed specification on L- systems part. The development of models as well as the generation on screen are both a tedious task and require a lot of effort. References [1] http://en.wikipedia.org/wiki/3D_computer_graphics [2] P. Prusinkiewicz & A. Lindenmayer, “The algorithmic beauty of plants”, Springer-Verlag, New York, pp 66-97,1990. docsity.com 81 Chapter 9 “What is a weed? A plant whose virtues have not yet been discovered” ~ Ralph Waldo Emerson Study Conclusion & Future This chapter concludes the work done under the heading The FloraL Genre. 9.1 Summary & Conclusion The project started with the very basics of modeling. The first thing to do was to explore the different plant modeling techniques. A technique from amongst the various existing was chosen that is The Lindenmayer Systems. The detailed study resulted in a hypothesis that plants could be termed as fractals based on the self- similarity concept they carry. The modeling of fractals using L-systems was the first task which acted as a baseline for the detailed plant modeling. The branching structures are the most important and main feature of a plant from. The modeling and simulation of the branching structures was explored and some development work was carried out in this regard. The simulations from the developed program were verified with the ones generated using different plant simulators and modelers. The developmental process is a key concern when talking about modeling the natural plants. The subapical growth in herbaceous plants was studied and different branching patterns for the herbaceous plants were co-related with the L-systems formalisms. At this point a detailed research work was carried out and a system was proposed for the generation of different L-system models based on certain input parameters giving the details of the structure required by the user of the system. A research paper was written for formal hypothesis and result discussions. The next step was to study the modeling of plant organs like flowers and leaves. The models were developed and simulations were performed to generate the structures. The project now is in its conclusion phase and the deliverable that is the FloraL GUI is in its completion. docsity.com 82 9.2 Future Recommendations Most of the work done in the project has already been carried out by different people. A new technique for generation of models using L-system specification was introduced. The work done in this project could be extended to plant modeling using image synthesis and animation of the plant growth and movements due to environmental factors. docsity.com 83 Bibliography 1. P. Prusinkiewicz. Modeling plants and plant ecosystems. Washington DC, 1999. 2. M. F. Barnsley. Fractals everywhere. Academic Press, San Diego,1988. 3. H. S. M. Coxeter. Introduction to geometry. J. Wiley & Sons, New York, 1961. 4. http://home.wanadoo.nl/laurens.lapre/lparser.html 5. http://algorithmicbotany.org/virtual_laboratory 6. http://www.xfrog.com 7. http://www.kurtz-fernhout.com/PlantStudio/index.htm 8. L. Quan, P. Tan, G. Zeng, L. Yuan, J. Wang & S. B. Kang. 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Crown Architecture of Short- rotation, Intensively Cultured Populus. III. A Model of Firstorder Branch Architecture. Canadian Journal of Forestry Research, 1983, pp: 1107–1116. 17. W. R. Remphrey and G. R. Powell. Crown Architecture of Larix laricina Saplings: Quantitative Analysis and Modeling of (nonsylleptic) Order 1 Branching in Relation to Development of the Main Stem. Canadian Journal of Botany, 1984, pp: 1904–1915. docsity.com