Leaf Deformation Taking Into Account Fluid FlowPaulo Silva

Yonghao Yue

Bing-Yu Chen

Tomoyuki Nishita

The University of Tokyo

The University of Tokyo

National Taiwan University

The University of Tokyo

Abstract—This paper proposes a method for animating leaf
structural deformation considering the leaf water content andits change due to evaporation. To achieve this we embed a massspring system into a mesh representing the leaf, and couple
the mass-spring parameters with a simulation representing thefluid flow and evaporation occurring in the leaf. This chain of
events aims to simulate the natural process that connects theleaf structural deformation to the water amount in the cells
through the turgor pressure. To the best of our knowledge no
existing method considers this phenomenon in the simulation ofleaf wilting in computer graphics.
Index Terms—Leaf Deformation, Fluid Flow, Aging.

I. I NTRODUCTION
Leaves curling and crinkling are commonplace during thecold months of the year, as they are part of most naturalsceneries. These this type of natural phenomena are useful inmany applications from the movies industry to the interactive
and entertainment industry. Therefore modeling leaves is an
important topic in computer graphics. Apart from possible
static applications, usually it is important to have a timevarying representation of natural phenomena. In this paper we
focus on the deformation leaves suffer as they age, dry and die.
Our objective is to provide a mechanism that is as automaticas possible by taking as base the actual natural processes
occurring in the plants. We believe this to be fundamental
for modeling ecosystems realistically in both interactive andnon-interactive applications.
While modeling trees and plants has seen quite a lot of
attention from the scientific community [1], [2], [3], [4], leaf
deformation has been much less focused on. Most current
research focuses on the modeling part. By contrast we assume
the object is already modeled and we concentrate on animatingit.
As the plant ages, changes in its metabolism and in thesurrounding environment lead to structural changes in the plant
itself. These changes include curling and crinkling, which varydepending on the underlying support structure and on factors
such as the amount of fluid in the leaf. It is our opinion
that this process needs to be simulated to faithfully capture
the deformation leaves undergo. In the method proposed inthis paper, we model the changes in internal fluid pressure
while taking into account the underlying structural support
offered by the leaf venation. We build our method on thework by [5]. In that work the authors propose a model for the
fluid flow in leaves. They consider a venation, stomata andfluid maps and use these to simulate the plant transpiration
cycle. We take the ideas in that paper for the fluid flow

(a)

(b)

Fig. 1: Leaf deformation example, (a) input image, (b)simulation result.

simulation and add our own deformation model on top. Ourdeformation model is composed by a mass-spring systemembedded in a mesh representing the input leaf. This massspring system includes springs to deform the leaf, accounting
for both shrinking and curling effects due to water loss.
Then we couple this simulation with the fluid flow simulationfrom [5] using their output to control the parameters of oursystem. Those parameters include spring constants and spring
rest lengths. Then by updating this system in a Newtonian
physical simulation we obtain an indirect relation between
the leaf shape and the fluid pressure in the leaf. This fluidpressure is called turgor pressure and is an important factor in
keeping the rigidity of the leaf structure [6]. Additionally wealso consider the influence of the leaf venation in the structure
support, by connecting it to the spring stiffness constant. The
main contribution of this paper is a method of animating leaf
deformation based on implicit turgor pressure variation. This
paper has the following structure. First we present present
some related work in senescence simulation in computer
graphics. In section III we present the general idea of our
method, and detailed descriptions of the fluid flow simulation,
mass-spring model and how it is embedded in the mesh, andhow we couple the two simulations. Then, in section IV, we
present animations generated by our method and we include
some details about our experimental environment. Finally
we conclude this paper with some ideas on possible future
directions.
II. R ELATED W ORK
There is a considerable amount of work modeling plants,
but the amount of work related to leaf deformation is scarce.
Here we focus on the works related to leaf deformation that

we find most relevant.
Hong et al. [7] proposed a method based on a venation
skeleton and a surface membrane to model the aging leaves.
This system can be used to interactively deform the leaf. This
contrasts with our method where no user interaction is required
apart from specifying the simulation input.
Lu et al. [8] presented a method to animate leaf wilting due
to gravity. This method closely relates to the method by [7] asit also uses a venation skeleton to deform the leaf. However,this method only considers the effect of gravity.
Kidder et al. [9] presented a method for simulating fruit
senescence, which also includes animation of the fruit skin.
Like ours, this method also considers the amount of watercontent in the deformation of the fruit skin. Nevertheless, eachmethod has a completely different target, as Kidder et al.’s
method focuses fruits or volumetric objects, and ours focusesleaves or planar objects.
Recently Xiao and Chen [10] presented a strain based
method to explain the deformation of leaves. The author thenuses FEM based simulation to produce a simulation of drying
leaves. On the other hand, our method considers the effectof the fluid pressure or turgor pressure, which is essential forplants to maintain rigidity [6]. To the best of our knowledge,
no other work takes this into consideration in the simulationof leaf structural deformation.
III. L EAF D EFORMATION
Our method aims to reproduce the structural deformation
visible in senescent leaves. Leaves come in a wide varietyof shapes and sizes. In this work we focus on angiospermicand sheath leaf types. These leaf types tend to have a moreor less flat surface, with an underlying venation that serves
as both support to the leaf structure and also has transports
channels for nutrients, water, and so on. The fluid flow in theleaf, together with the leaf venation are the two factors that will
most influence the leaf shape in our model. To achieve the leaf
deformation effect we propose two separate simulations. One
simulation that accounts for the fluid flow and evaporation, anda mass-spring based simulation that actually deforms the leaf
shape. Both these simulations take into account the amount of
fluid and the venation distribution in the leaf. Next we explain
how we couple these two simulations and we give detailed
explanations on each.
A. Method Overview
The main input to our method is an image of a leaf
(see Fig. 1 (a)). First we extract the leaf texture to use asa color map. Together with this map we extract the leaf
boundary using computer vision techniques based on [11].
From this boundary we create a mesh representation of the leaf
using [12]. This textured mesh is then used in the subsequent
simulations by applying extra textures maps and embedding a
mass-spring model as explained next.
We consider the leaf has a two dimensional surface on which
we define three maps based on [5]. These maps are a fluid map,a stomata map and a venation map. The fluid map represents

(a)

(b)

(c)

Fig. 2: (a) venation map (b) stomata map, (c) fluid map.

the amount of water at each location in the leaf (see Fig. 2(c)). Stomata are small openings located mostly on the lower
surface of the leaf, which serve the purpose of transpiration,
moving water molecules out of the leaf, and also respiration.
The stomata map represents the distribution of these openings,
and is used to remove water out of the leaf simulating thewater lost by transpiration (see Fig. 2 (b)). The water flows in
a network of channels called Xylem which is part of a larger
vascular network or venation. In order to model the venation
distribution we use a venation map. A two dimensional map
that represents the venation implicitly by its flow capacity.
That is, the amount of water that can flow at each location in
the structure (see Fig. 2 (a)). Together with these maps we alsoneed to consider a location where the fluid enters the system.
An explanation of this simulation is presented in Sec. III-B.
Additionally to this simulation we add a mass-spring modelto the the input mesh. As the fluid map is updated, the spring
constants and their rest lengths also change to reflect the
new water content in the leaf. More precisely, the as thewater content changes at a certain location the springs atthat location, if any, will use those values to calculate theirparameters. This will cause the springs to contract or extend
accordingly, which aims to simulate the changes in the turgorpressure. Then, when the water amount is reduced the plant
will, for example, loose rigidity and wilt under its own weight,
or contract in case the spring stiffness increase. We update theleaf shape by updating the mass-spring system in a Newtonian
physics simulation. This simulation is described in Sec. III-C.
B. Fluid Diffusion
This component of our method uses the work by [5].
For the sake of clarity we make a short explanation of thismethod here. The fluid flow simulation main data structures
are the venation, stomata and fluid maps (see Fig. 2). Thenthe fluid flow is simulated using a continuous model based ona diffusion process as described by Eq. (1).
∂y(u, v)
= ∇ · (β(u, v)∇y(u, v)) + I(u, v) − O(u, v), (1)∂twhere y(u, v) represents the fluid map, β(u, v) represents thevenation map, I(u, v) represents the sources map and O(u, v)
represents the stomata map. The continuous model in Eq. (1)is discretized as shown in Eq. (2).
m+1
m+1
= yim . (2)+ (βi−1 + βi + 1)yim+1 − βi yi+1
−βi−1 yi−1

Fig. 3: Example of a leaf mesh used as input in our
deformation model.

where i is the pixel index and m is the time step index.
The author then separates the simulation into three steps. A
transpiration step where fluid is extracted from the system, ansource step where fluid enters the system and a diffusion step
where the fluid is diffused in the system. Then Eq. (2) willlead to a tridiagonal system which can be efficiently solved.
Solving this system for y will consequently update the fluid
map. For more details we refer the reader to the original paper.
C. Deformation ModelAt this stage we assume to have an input mesh such as theone in Fig. 3. We want to deform this mesh by both contracting
parts of it or just letting it yield under its own weight. To dothis we make use of a mass-spring system which we embed in
the mesh. Mass-spring systems are widely used in computer
graphics. The interested reader might want to consult [13] foran overview. One of the phenomena that we expect to seefrom a drying leaf is loss of mass and size. To model this,we start by positioning a mass particle at each mesh vertex.
Then we add a spring at each triangle edge. A spring sij onedge eij , which connects vertices ~vi to ~vj (see Fig. 4), exerts
a force F~ij on particle pi , located at ~vi , and an opposing force−F~ij on particle pj located at ~vj . This force is defined simply
by Eq. (3).
(3)
F~ij = −kij (∆xij − lij0 )~uij ,where kij is the spring constant, ∆xij = k~xi − ~xj k, ~xi and ~xj
are the current position of the particles pi and pj respectively,
lij0 is the initial spring rest length which is calculated as∆xij at time t = 0 and ~uij = (~xi − ~xj )/k~xi − ~xj k.
Changing the distance lij0 we can contract the leaf producing
a localized shrinking effect. Changing the spring constant kij
we can also let the leaf mesh stretch under its weight or
under external forces. Apart from shrinking, the leaf also tends
to curl. To model this behavior, and to lock the orientation

Fig. 4: Schematic of a very simple mesh.

between different triangles, we add another type of spring, thistime on each shared edge eij . We call these angle springs,
as they use the angle between faces sharing edge eij incomputing the spring force. For this we require the mesh tobe homomorfic to a disk. Or in other words, the mesh can
only have edges referenced by one face, boundary edges, oredges referenced by two faces, inner edges. We also require thenormal orientation to be consistent, as the angle is calculatedusing the direction of the triangle normal . The force these
springs generate is defined by Eq. (4).
Fmn = kθmn (θmn − θmn0 ),(4)

where Fmn is a scalar which represents the force that theangular spring smn exerts on particle pm and pn . These
two particles are located on opposite triangles which share
a common edge (see Fig. 4). Additionally kθmn is the angularspring constant, θmn is the current angle between triangles
and θmn0 is the initial rest angle. This force is applied on eachparticle in the triangle normal direction. When θmn −θmn0 < 0
the the force will be in the direction opposite to the normal.
This will force the spring to go back to its rest configuration.
Analogously for when θmn − θmn0 > 0. Therefore, theequation has no minus sign. The angular springs are initializedaccording to Alg. 1. The function trianglesRef erencing
Algorithm 1 Angular Spring Placement
for each inner edge eij do(Tk , Tl ) ← trianglesReferencing(eij )~vm ← vertexOpposing(Tk , eij )~vn ← vertexOpposing(Tl , eij )θmn0 = angle(Tk , Tl )addSpring(eij , ~vm , ~vn , θmn0 )end forreturns the triangles that share edge eij , vertexOpposing

TABLE I: Mesh information, number of triangles, edge
springs and angular springs for each input mesh.
NameBrambleVertices
1707

Triangles
3232

Edge Springs
4938

Angular Springs
4758

TABLE II: Average time step in milliseconds per update step
given the fluid map size. This includes fluid flow simulationand mass-spring simulation.
Fig. 5: We project the vertices opposite to edge eij onto theline that contains eij . The we calculate the vectors between
these projected points and the original. Finally we computethe angle between these vectors. This will be the angle of theangular spring associated with edge eij .

NameBrambleName
Bramble128 × 12811
512 × 51240

256 × 256171024 × 1024
145

A. Performance
returns the vertex from face Tk that is not part of edge eij ,and angle calculates the angle between faces Tk and Tl . The
function angle is defined in Alg. 2. The function project
Algorithm 2 Angular Spring Angle′← project(~vm , eij )~vm~vn′ ← project(~vn , eij )′~vm′ m ← normalize(~vm − ~vm)~vn′ n ← normalize(~vn − ~vn′ )cos(θmn ) = ~vmm · ~vnn
θmn ← cos−1 (cos(θmn ))~nl = normal(Tl )if ~vm′ m · ~nl < 0 thenθmn ← θmn + π
end ifreturn θmn

Our test environment was equipped with an Intel Core 2
Duo CPU clocked at 2.13GHz, 2GB of main memory anda NVIDIA GeForce 9400M GPU with 256M B of dedicatedvideo memory.
We tested our method with several input pictures each atseveral resolutions. Triangulation parameters were similar in
all cases. In these tests the number of triangles, number of edge
springs and number of angular springs can be seen in Tbl. I.
The timings presented in Tbl. II include the average timeit took for a complete update step of both the fluid flow
simulation and the mass-spring simulation. The fluid flow
simulation has a very big weight at higher resolutions, while
for the smaller fluid map resolutions both simulations takeabout the same time. Note that the triangulation resolution isfixed. Even in our unoptimized implementation we still obtainreal-time performance.
B. Example Deformations

returns the projection of ~vm on the line defined by eij , thefunction normalize normalizes the vector passed as argument
and the function normal returns the vector normal to thetriangle Tl passed as argument (see Fig. 5). The rest anglesof the springs are updated directly by the amount of fluid
present at the center point of each spring. For example, for aspring on edge eij that would be the point halfway between
vertices vi and vj . Analogously for the angular springs. On
the odder hand, the venation map values weight the spring
constants. That is, using a normalized version of this map(values ∈ [0, 1]) we can make areas with thicker venation
have stiffer spring constants. Finally this mass-spring system
is integrated in time, and the change in the fluid map will
indirectly cause a change in the leaf structure. This aims tosimulate the change in turgor pressure.

Here we show the results of applying our method to a
few input meshes. The data for the fluid flow simulation was
mostly hand made. The mesh generation however, as with thespring placement, were both done automatically. In Fig. 6 weshow a few frames of the deformation the leaf suffers while
drying. However we believe our method is versatile enoughto be applied in a much larger variety of plants than the few
presented here.
C. Limitations

IV. R ESULTS

Since our method is based on the method by [5] it does
suffer from the same limitations. Our method speed is mostly
influenced by the resolution of the fluid map. However wefound small maps, such as 256 × 256, still provide compelling
results. The result depends on the triangulation. We found an
uniform triangulation to be suited for our purposes. But webelieve the triangulation can be improved using the venation
map as constraints.

This section provides information about our test environment, simulation timings, example deformations and a brief
discussion of our method limitations.

We presented a method for leaf deformation by implicitly
accounting for the turgor pressure. This pressure controls the

V. C ONCLUSIONS AND F UTURE W ORK

(a)

(b)

(c)

(d)

Fig. 6: Above, example of a fluid diffusion sequence for abramble leaf at time (a) t = T0 and (b) t = T1 . Below, thecorresponding deformation (c) and (f) respectively.

rigidity in plants [6], and we think it’s important to takeit into consideration. To model this we used leveraged thework by citeSilva12 which accounts for the plant internal
fluid flow. On top of this method we introduced a massspring model to deform the leaf mesh. The parameters of thismass-spring model are controlled by the fluid flow simulation.
In this way, the plant natural transpiration cycle indirectly
controls the leaf structure deformation. This is equivalent toindirectly accounting for the influence of the turgor pressure incontrolling the leaf structure. We shown this with some results
generated by our method.
We leave for future work the usage of the venation as aform of improving the triangulation result. We don’t consider
this essential, since the mesh might be artist created. But we
aim to achieve a fully automatic system, and this step canpossibly increase the quality of the generated mesh. Although
we the shape is a very important indicator of the stage ofsenescence in which a leaf is in, other factors also contributeto a more realistic result. Color change is one such factor [14].
We believe simultaneous simulation of leaf structural andcolor changes is important to achieve a realistic senescencesimulation, and therefore that is another important future
direction. We think another interesting direction would be toextend this method to generate an animation of an entire plant.
That is, to consider other plant structures such as the stalk,roots and so on.
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