Modeling and visualization of leaf venation patterns

1 Modeling and visualization of leaf venationpatternsAbstractWe introduce a class of biologically motivated algorithms for generating leaf venationpatterns. These algorithms simulate the interplay between three processes: (1) development ofveins towards hormone (auxin) sources embedded in the leaf blade; (2) modification of thehormone source distribution by the proximity of veins; and (3) modification of both the veinpattern and source distribution by leaf growth. These processes are formulated in terms ofiterative geometric operations on sets of points that represent vein nodes and auxin sources. Inaddition, a vein connection graph is maintained to determine vein widths. The effectiveimplementation of the algorithms relies on the use of space subdivision (Voronoi diagrams)and time coherence between iteration steps. Depending on the specification details andparameters used, the algorithms can simulate many types of venation patterns , both open(tree like) and closed (with loops).

2 Applications of the presented algorithms include textureand detailed structure generation for image synthesis purposes, and Modeling ofmorphogenetic processes in support of biological Runions, Martin Fuhrer, Brendan Lane, Pavol Federl, Anne Ga lle Rolland Lagan, and PrzemyslawPrusinkiewicz. Modeling and visualization of leaf venation patterns . ACM Transactions on Graphics 24(3),pp. 702 and visualization of leaf venation patternsAdam RunionsMartin FuhrerBrendan LanePavol FederlAnne-Ga elle Rolland-LaganPrzemyslaw PrusinkiewiczDepartment of Computer Science, University of CalgaryAbstractWe introduce a class of biologically-motivated algorithms for gen-erating leaf venation patterns . These algorithms simulate the in-terplay between three processes: (1) development of veins towardshormone (auxin) sources embedded in the leaf blade; (2) modifica-tion of the hormone source distribution by the proximity of veins;and (3) modification of both the vein pattern and source distribu-tion by leaf growth.

3 These processes are formulated in terms ofiterative geometric operations on sets of points that represent ,aveinconnectiongraphismaintained to determine vein widths. The effective implementationof the algorithms relies on the use of space subdivision (Voronoidiagrams) and time coherence between iteration steps. Dependingon the specification details and parameters used, the algorithms cansimulate many types of venation patterns , both open (tree-like) andclosed (with loops). Applications of the presented algorithms in-clude texture and detailed structure generation for image synthesispurposes, and Modeling of morphogenetic processes in support ofbiological [Computer Graphics]: Computational Ge-ometry and Object Modeling Geometric algorithms, languages,and systems; [Computer Graphics]:Three-DimensionalGraphics and Realism Color, shading, shadowing, and texture; [Simulation and Modeling ]: Types of simulation Visual.

4 [Life and Medical Sciences]: BiologyKeywords:realistic image synthesis, Modeling of natural phe-nomena, morphogenesis, vein development, leaf growth, auxin,Voronoi diagram, relative neighborhood1 IntroductionSimulation-based visual Modeling of patterns found in living or-ganisms has a long history, bridging biology, theoretical studies ofmorphogenesis, and computer graphics [Prusinkiewicz 1994]. Pre-vious models include reaction-diffusion models of animal coat pat-terns [Turk 1991] and sea shell pigmentation [Fowler et al. 1992],clonal mosaic models of animal coat patterns [Walter et al. 1998],diffusion-limited aggregation models of lichens [Desbenoit et ], and physically-based models of bark textures [Lefebvre andNeyret 2002; Federl and Prusinkiewicz 2004]. In this paper, we fo-cus on the Modeling of venation patterns in leaves. Together withspiral phyllotaxis and the branching structures of tree architecture, venation patterns are among the most admirable aspects of the nat-ural beauty of plants.

5 Yet, in comparison, venation patterns andtheir development are poorly understood [Dengler and Kang 2001].This makes the visual Modeling of venation patterns a particularlychallenging problem. As a step towards its solution, we propose apinnatepetioleleafletsimplecompoundenti retootheddissecteddigitatelobeblademargi nFigure 1:Terms pertinent to the description of leaf :Asampleleaf(a)andtheresultsofits:(b)mar ginalgrowth, (c) uniform isotropic (isogonic) growth, (d) uniformanisotropic growth, and (e) non-uniform anisotropic inspired by the current theories of hormonal control of veinmorphogenesis. The model generates visually realistic venationpatterns, reproduces in part their natural diversity, and captures theclose relation between venation and shapes of leaves. In image syn-thesis applications, this model offers a useful alternative to scannedtextures when leaf specimens are not readily available, leaves arenot flat (and therefore are difficult to scan), a large number of leafmodels with different yet related venations is needed, leaf devel-opment is animated, or when the topology of the leaf venation isneeded.

6 The model can also be used as a stepping stone to studyand visualize leaf venation patterns for biological purposes. In thiscontext, realistic visualization plays a critical role as an element ofmodel evaluation and validation [Prusinkiewicz 1998], since cur-rent objective measures for comparing complex venation patternswith reality only capture a limited set of features [Bohn et al. 2002].2 Background and related leaf shape patterns are strongly correlated with leaf shapes [Denglerand Kang 2001] and thus must be considered in that context. A use-ful summary of the terminology for describing leaf shape is givenby Juddet al.[1999]. A typical leaf consists of aleaf blade(lam-ina), attached by apetiole(stalk) to the stem (Figure 1).Simpleleaves have a single, connected blade. A simple leaf is calledentireif itsmargin(edge) forms a smooth arc,toothedif the margin hassmallprotrusions,andlobedifthemargini ssignificantlyindented,dividing the blade into distinguishablelobes.

7 Lobed leaves are fur-ther categorized asdissected, with the indentation approximatelyperpendicular to the leaf axis, anddigitate, with the lobes orga-nized radially (like fingers on the hand). In contrast to the simpleleaves,compoundleaves have blades partitioned into separate sub-units calledleaflets. In this paper we do not consider compoundleaves, assuming that their venation can be modeled at the level ofindividual Mathematical description of leaf development of venation patterns is correlated with the growthof leaf blades. Growth can be characterized by thegrowth ten-sorfield [Hejnowicz and Romberger 1984], which specifies themagnitude of the expansion of infinitesimal surface regions in var-ious directions, and may include a possible rotation of each re-gion. The growth tensor is a generalization of therelative ele-mentary rate of growth(RERG), which is defined as the rate atwhich an infinitesimal distanceDs, measured in the direction ofa linelat a pointpof a growing object, increases over rate is normalized with respect to the distanceDs, yieldingRERGl(p)=(1/Ds)(dDs/dt)[Hejnowic zandRomberger1984].

8 Growth ismarginalif it is concentrated on the border anddiffuseif it is spread throughout the surface [Roth-Nebelsick et al. 2001].Diffuse growth is calledisotropicif expansion is equal in all direc-tions, andanisotropicotherwise. Furthermore, growth isuniformif the growth tensor is the same at all points of the surface, andnon-uniformif it is not. A uniform isotropic growth is calledisogo-nic[Coen et al. 2004]. These variations are illustrated in Figure are two approaches for Modeling leaf growth. The firstapproach is to specify the progression of leaf shape over growth was first simulated in this way by Scholten andLindenmayer [1981]; diffuse growth can be described in terms ofwarping and morphing of graphical objects [Prusinkiewicz et ; Gomes et al. 1999]. The second approach is to specify thegrowth tensor field, either explicitly or as a result of a physically-based expansion model [Rolland et al.]

9 2003; Wang et al. 2004].Neither approach provides a generally convenient method for spec-ifying arbitrary growth. Consequently, in our implementation wehave focused on simple special venation patternsWe describe leaf venation pat-PercurrentTertiaryveinsSecondaryvein sPrimary veinReticulateTertiaryveinsFigure 3:Some terms perti-nent to the description of vena-tion using the terminology ofHickey [1979] and its simpli-fication by Juddet al.[1999].A fundamental notion is thatof veinorder. Generally, thefirst-order veins are the widestveins originating at the leafbase(the point of attachmentto the petiole), and finer veinsand veinlets have progressivelyhigher orders (Figure 3). Vena-tion patterns are correlated withthe taxonomic groups of plantsand with the shapes of of monocotyledons ( , grasses) usually have approxi-mately parallelprimary(first-order) veins, which is consistent withthe highly elongated leaf shape and wide leaf base.

10 Dicotyledonswith simple entire leaves often havepinnatevenation, characterizedby a single primary vein (themidvein) that originates at the base andextends towards the leaf tip. Dicotyledons with digitate leaves typi-cally haveactinodromousvenation, in which three or more primaryveins diverge radially from a single point. Primary veins support se-quences ofsecondary(lateral) veins, which may branch further intohigher-order veins. The secondary veins and their descendants maybefree-ending, which produces anopen, tree-like venation pattern,or they may connect (anastomose), forming loops characteristic of higher-order veins usually link the sec-ondaries, forming a ladder-like (percurrent) or netlike (reticulate)pattern (Figure 3). Mechanism of vein pattern developmentThe most widely accepted theory of vein pattern formation is thecanalization hypothesis[Sachs 1981]. According to this hypothe-sis, vein development is controlled by a signal that propagates in theleaf blade and causes vein differentiation.