# Venation Skeleton-Based Modeling Plant Leaf Wilting

Authors: Shenglian Lu; Chunjiang Zhao; Xinyu Guo
Journal: International Journal of Computer Games Technology
 (2009)

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## Abstract

A venation skeleton-driven method for modeling and animating plant leaf wilting is presented. The proposed method includes five principal processes. Firstly, a three-dimensional leaf skeleton is constructed from a leaf image, and the leaf skeleton is further used to generate a detailed mesh for the leaf surface. Then a venation skeleton is generated interactively from the leaf skeleton. Each vein in the venation skeleton consists of a segmented vertices string. Thirdly, each vertex in the leaf mesh is banded to the nearest vertex in the venation skeleton. We then deform the venation skeleton by controlling the movement of each vertex in the venation skeleton by rotating it around a fixed vector. Finally, the leaf mesh is mapped to the deformed venation skeleton, as such the deformation of the mesh follows the deformation of the venation skeleton. The proposed techniques have been applied to simulate plant leaf surface deformation resulted from biological responses of plant wilting.


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## Body

## 1. Introduction

Realistic modeling of plant leaves has a long history in computer
graphics. This is partly due to their either beautiful or colorful images, and partly
they have a strong visual effect on the audience. Many techniques have been
proposed  for modeling the geometry or shape of leaves. Most of  these methods,
however, just describe the shape of leaves formed in normal natural conditions
and do not account for the shapes formed under stress,
for example, curled or withered leaves.Additionally, there has been a great deal of previous work on simulating
motions of plant, including plant growth, motion in the wind, and so on. But less
work has focused on modeling leaf surface deformation and simulating subtle
behaviors of plant, such as wilting of leaves suffering from insufficient water
supply. This paper presents a venation skeleton-based deformation method for
plant leaves and aims to develop
an approximately kinematic model of leaf for simulating motions of plant leaves, especially wilting. Firstly, an
initial leaf skeleton is exacted from scanned image of leaf. The leaf skeleton
plays two roles: it is used to generate a venation skeleton for later
deformation; a geometric mesh for the leaf surface is also constructed from the
leaf skeleton. Furthermore, a subdivision scheme is applied to generate a
detailed triangular mesh from the initial mesh, and each vertex in the mesh is
mapped to its nearest vertex in the venation skeleton. Then the venation
skeleton is deformed interactively to desired shape. Lastly, the detailed mesh
is deformed according to the deformed venation skeleton. Applications of our
approach to simulate wilting plant leaves with realistic results illustrate the
flexibility and effectiveness of our model.


## 2. Background and Related Work

### 2.1. Venation Patterns

Both the outline and venation system
of a leaf are essential in the recognition of plant species. Various venation structures can be found in plant
kingdom. It is believed that venation patterns correlate closely with the
taxonomic groups of plants and the shapes of leaves. Hickey [1] has given a classification
for the leaf venation patterns, in which pinnate venation and actinodromous
venation are two common found categories (see Figure 1). Pinnate venation characterized
by a single primary vein (the midvein) being attached by several secondary veins; the primary
vein originates at the base and extends toward the leaf tip. Whereas in actinodromous
venation, three or more primary veins diverge radially from a single point.
Primary veins support sequences of secondary (lateral) veins, which may branch
further into higher-order veins. The secondary veins and their descendants may
be free ending, which produces an open, tree-like venation pattern, or they may
connect, forming loops characteristic of a closed pattern.Some examples
of venation patterns: pinnate venation (a) and actinodromous venation (b).
(a)(b)Although the interrelationships between topological or
geometric properties of the various leaf venation patterns and functional
aspects are far from being well understood, it is believed that the leaf
venation system brings various functional properties. More information about
this can be found in [2].


#### 2.2.  Model Deformation and Motions of Plant Leaf in Computer Graphics

Some researchers have endeavored to generate curled shapes
of plant leaves. Prusinkiewicz et al. [3] provided a detailed representation of
combining interaction and parameterized algorithms for realistic plant modeling
and scene creation involving plants, including curled leaves. Mündermann et al.
[4] proposed a method for modeling lobed leaves; curled surface of leaves could
be generated by using free deformation in their framework. Recently, Hong et al.
[5] proposed an interactive method for modeling curled leaf surface. But it
could involve excessive manual
interactions for generating a desired curled shape of leaf by using their method.
Studies on curvature of plant leaves
from biophysical perspective have raised the question of what role, if any,
genes play in controlling the curvature of leaves [6]. Yet there are some
researchers study waved or wrinkled pattern in leaves with physical analysis [7].
But these may go
beyond our focus in this paper.In aspects of modeling motions of plant, most work has
been done on modeling dynamic motions of tree in the wind, such as the work
demonstrated in [8]. Based on the fact that plant growth is generally
influenced by gravity and tropisms, Jirasek and Prusinkiewicz [9] proposed a
biomechanical model for creating curved plant branches by using physically based
modeling. Hart et al. [10] extended their idea in modeling plant growth
considering physical properties of the plant. Note that none of the above
models simulated leaves behaviors. Wang et al. [11] had simulated physically
the growth of a plant leaf; the physical model used in their simulation is the
governing equations of fluid mechanics—the Navier-Stokes
equations. But they just tested their model in 2D.Much work has been done on surface deformation. And many
techniques, such as multiresolution mesh representations [12], skeleton-driven
global free-form shape deformations [13], and differential deformation [14, 15],
have been developed to help artists deform object shapes. But none of these techniques
has been tested by modeling leaf deformation.


### 3. Overview of the Proposed Method

Figure2 gives an overview how our proposed method works. The modeling
processes include constructing the venation skeleton from a leaf image,
deforming the leaf surface, and so forth. These schematics label the transition
processes in uppercase letters, A, B, and so forth. The interactive simulating system
that we propose tries to strike a pragmatic balance between processes that can
be automated and those that seem to require interaction to achieve the desired
level of realism. It has primarily been designed to support our experimentation
with interactive animation of leaf motions.Figure 2
Modeling processes.Each process will be detailed in the following sections.


### 4. Generating Venation Skeleton

The venation structure of a leaf plays
a major biological role in determining the leaf surface shape and controlling
its deformation; therefore, we use it to control the deformation of a leaf
blade. To generate the venation skeleton, currently we consider an interactive
method. In Figure2, process
A and C illustrate the steps for generating venation skeleton of a leaf.


### 4.1. Extracting a Leaf Boundary

We use a representation of a leaf skeleton
consisting of two boundary curves and a midvein curve which consists of feature
points, as shown in the left of process A. These boundary curves can be reconstructed
from feature points in the boundary of a leaf, while these feature points can
be extracted automatically from a scanned digital image by using a standard
edge detection algorithm, or obtained by using a 3D digitizer.To meet the need of interactive design,
the leaf skeleton can also be constructed automatically with a parametric
method, in which the length of midvein, the width of leaf blade, and the number
of feature points are all initialized with parameters.


### 4.2. Generating Venation Skeleton

The skeleton was originally
introduced by Blum [16] for 2D shapes in order to provide a symmetry-based
shape representation for shape perception and recognition. Recently the
skeleton in 3D has been studied in connection with a research on shape
organization [17] and shape manipulation. Practical extraction of the
skeleton of a 3D shape is usually based on 3D Voronoi diagram techniques [18].For our needs, we develop an
interface for generating interactively venation skeleton from a leaf skeleton.
As Figure2 shows (from process A to process C), the leaf skeleton can be obtained
from a scanned image; a venation skeleton is then generated from the leaf
skeleton, and each vein is segmented. The result of process C demonstrates  a
generated venation skeleton consisting of one midvein and four secondary veins;
the black vertices string segmented each vein into several line segments. The
process of generating  venation skeleton involves several manual interactions including
defining the start point and end point for each vein, and specifying parameters
for segmenting each vein. Note that the venation skeleton is not unique. It can
be created according to actual needs. Users can decide how the midvein crosses
the leaf skeleton, and how many secondary veins are attached to the midvein.


### 5. Leaf Surface Meshing and Banding

#### 5.1. Constructing Leaf Surface

We have constructed a leaf skeleton
with two boundary curves and a midvein curve as shown in Figure2  (process A). To
mesh the void area within these boundary curves, we employ Delaunay
triangulation scheme, because it can deal with the problem of existing concave area
in the leaf blade, which is difficult to render directly by simple polygon. For
example, lobed leaves often have irregular silhouette characterizing by a
number of concave outline. When using Delaunay triangulation, we can use
directly the feature points in the midvein and silhouette, or extract a series
of points from the midvein curve and silhouette curves with a fixed interval.
The mesh resulted from process B of Figure 2 is generated from the result of process
A by using Delaunay triangulation scheme.The initial mesh of a leaf surface
generated by using Delaunay triangulation is generally irregular and rough. It
is necessary to refine the mesh for later deformation. Currently, we use a
simple method to subdivide the initial mesh. Generally each triangle can be
divided into four triangles. To meet users’ requirements of interaction, we
provide two parameters for the subdivision: one is the times of iteration steps
of subdivision, the other is the minimal distance of an edge in the mesh. The iterations timeis used to control the number of iterations in the subdividing,
while  using the constraint of the minimal distance for an edge is to avoid generating
too short edge. The lower figure of process D in Figure2 illustrates a mesh
subdividing the initial mesh shown in the upper figure at the same process with
iteration steps and minimal distance being specified 2 and 0.5, respectively.


#### 5.2. Banding

The banding is to attach all the vertices
in the subdivision mesh to the initial venation skeleton of the leaf. The
banding is based on the distance of each vertex to the venation skeleton. In
other words, each vertex in the mesh is banded to the nearest vertex in the
venation skeleton. Figure3 gives an example of banding a leaf mesh to its
venation skeleton which consists of a midvein and two secondary veins.Figure 3
Example of banding a leaf mesh
to its venation skeleton.


#### 6. Venation Skeleton-Driven Leaf Surface Deformation

##### 6.1. Mechanism for Controlling Movement of the Venation Skeleton

Skeleton-based methods have been extensively
used for mesh deformation in computer animation and computer modeling. In this
method, two or more “bones” meet at each joint, to control shape deformation.
This allows intuitive control, naturally describing the way in which many
objects, for example, animals, deform the muscles and other tissues follow
motions of the underlying bones.Traditional skeleton-based methods commonly
require a tedious
process of weight selection to obtain satisfactory results. Note that the
natural deformation of plant leaves is different from the deformation of humans’
or animals’ organs, and spontaneous motions of a leaf blade are relatively simple.
The most regular motions of plant leaves which can be seen by our naked eyes
are curling and wilting. So we can constraint the movement of vertices in the
venation skeleton for simulating the movements of leaves. The major goal of our
approach is to develop an approximately kinematic model for simulating these
motions of plant leaves.For our needs, we restrict the
movement of each vertex in the venation skeleton by rotating it around a fixed
vector. Figure4(a) illustrates how a venation skeleton works. For convenience,
the venation skeleton includes a vein only, which consists of four segments.
The light black vertex serves as the root node. Note that the movement of a
leaf blade is always downward during its wilting, as such each joint in the skeleton
segment does spherical movement. Take vertex Vi, for example, vector Vi−1Vi   will align gradually vector Vi−1VN  during wilting,
in which Vi−1VN  is a downward vector reverse to Z-axis, while vector Vi−1VM  is perpendicular to the plane which contains vector Vi−1Vi and Vi−1VN. To the
movement of vertex Vi, it
can be looked as Vi is
rotated round vector Vi−1VM.How the venation skeleton
works. (a) The movement trajectory of a
skeletal joint; (b) child joints follow the parent’s movement.
(a)(b)To obtain a motion sequence of a
vertex in a skeleton, the simplest method is to rotate the vertex with a fixed
angle, such asθ in Figure 4(a), and
the angle is given commonly by the users. But a conventional method is using inverse
kinematics [19]. We have mentioned before that a vertex in the skeleton will do
spherical rotation during the wilting of its corresponding leaf surface. As
Figure 4(a) illustrates, we can calculate the new position of vertex Vi by using the following parametric
equation: (1)V(t)=k(t)*(Vi+t*(VN−Vi)), where k(t)=|Vi|/|Vi+t*(VN−Vi)|(0≤t≤1). Further, the
motion sequence of Vi can
be obtained by increasing parameter t.
This may simplify the rotation operation. Note that vector Vi−1VN needs to be recalculated when vertex Vi−1  is repositioned, but it
is always downward.And also, all child segment of a
vertex in the venation skeleton will follow the movement of the vertex; this
can be done by passing a displacement and rotation angle to its child vertices
when the vertex is being rotated. Figure4(b) gives results of rotating
four vertices in a skeleton successively, in which the blue is the rotated
vertices while the red is the base point.


##### 6.2. Leaf Surface Deformation

We have detailed the mechanism for controlling
movement of the venation skeleton and methods for constructing the leaf mesh in
our approach. The last step is to deform the leaf surface according to the
deformed venation skeleton. This process can be illustrated as process F in Figure2.Firstly all the vertices in the
subdivision mesh of the leaf are banded to the initial venation skeleton. Then
the initial venation skeleton is deformed by using the method described in
Section6.1. For example, at process E, we can generate the shape of the
venation skeleton shown bottom in Figure
2E  from the initial skeleton shown top in
Figure
2E (with different number of joints in each vein). Lastly the position of
each vertex in the mesh is recalculated according to the new coordinate of its
banded vertex in the venation skeleton. The left-bottom figure in Figure
2 shows the
resulted mesh.The right-bottom figure in Figure2 illustrates
the rendering result. The texture mapping is calculated before the deformation,
and no need to remap after the deformation.It needs to be noted that the number
of joints in each vein in the venation skeleton will influence the effects of
deformation. It is easy to imagine that the larger the number of joints is, the
smoother the deformed surface will be. And large deformation needs large number
of joints. But larger number of joints in the venation skeleton also means more
computation and more difficult controlling over the deformation. In our
experiments, we find that pleasing visual effects of the simulated wilted
leaves shape can be achieved when the number of joints in each vein is not
smaller than 4. Users can also obtain a satisfactory result by interactive experiments.


##### 6.3. Constraints and Collision Detection

Constraints
and collision detection are usually the common issues in surface deformation. For constraint, we have
stated that each vertex in the venation skeleton can rotate around a fixed
vector. Additionally, the rotation needs to meet some extra constraints. For
example, the leaf surface will be always drooped during its wilting. In
simulating the effect of wilting leaf surface, a vertex in the leaf mesh could
not be rotated after it had reached the maximal drooped distance. When
simulating curling of a leaf, it needs to avoid overlap of the leaf surface. This can be done by
keeping the included angle of two adjacent line sections on each vein being larger
than a predefined angle.Collision detection and response is
usually the most time consuming process for the overall simulation. Currently
we just consider a collision detection to avoid self-intersect in the
deformation. During deforming a leaf mesh, each handling currently vertex needs
to be checked if its movement will pierce some triangle in the mesh. Piercing
means that there is intersection in the triangle mesh. If no piercing occurs,
no response is made. If there is an intersection, then we calculate a maximal
displacement from the precalculated displacement for the vertex to move to avoid
intersection, and correct the displacement of corresponding vertex in the
venation skeleton.


#### 7. Applications and Discussion

We
implemented our algorithm for venation skeleton-driven leaf surface deformation
in C++ on a PC with a 2.8 GHz Pentium D processor and a NVIDIA GeFore 7900 GS
graphics card, and used
OpenGL to render the results. In this section we report the modeling results.Firstly we
simulate wilting effect of a plant leaf. The plant used in this experiment is
cucumber. Figure5  shows a photo
of a real wilting cucumber in greenhouse. In our experiment, we first
simulate the wilted effect of a cucumber leaf. As shown in Figure  6, Figure 6(a) is the initial shape of the simulated leaf surface while Figures 6(b)
and 6(c) are two deformed results corresponding to two different levels of wilted leaves. The initial leaf surface is created
interactively from a scanned image within 3 minutes, with 187 vertices and 286
triangles. The iterations time for subdividing the
leaf surface is specified to 0. In this experiment, the venation skeleton
consists of a midvein and two secondary veins parameters for segmenting these
veins are 20, 4, respectively, and the total time for simulating the leaf
wilting is 8 seconds
with 22 frames.Figure 5
Photo of a real wilting cucumber in
greenhouse.Simulated shapes of wilting cucumber leaf: (a) initial shape, (b) medium
wilting, and (c) acute wilting.
(a)(b)(c)We then
use the venation skeleton model to simulate the process of a cucumber wilting.
Figure7 demonstrates the simulated results, in which Figure 7(a)  is the
initial shape, Figure 7(b) simulates a slight wilting leaves, and Figure 7(c)
simulates an acute
wilting one. This initial plant is created within 5 minutes by using an
interactive interface that we have developed for designing crop structure. The number
of vertices and triangles of this model are 4082 and 6150, respectively. We use three instances
of leaf surface in the cucumber model, and the venation skeleton in each instance
is different from each other, but with the same number of veins (three). The
venation skeleton is deformed automatically by rotating the vertices in the venation
skeleton downward from the boundary to root of the leaf, by using (1), and the above
leaves start wilting later than the lower leaves do, whereas the speed of
wilting can be adjusted by modifying parameter t. This simulation consists of 300 frames and runs for approximately
2 minutes.Three
stages of a wilting cucumber plant: (a) none wilting, (b) slight
wilting, and (c) acute wilting.
(a)(b)(c)The second application example is simulating
a watermelon leaf wilting. Watermelon leaf is a typical lobed leaf. We use a
venation skeleton shown in Figure8(a)  to control its deformation of the leaf
blade. The venation skeleton consists of one midvein and four secondary veins,
and the parameter for segmenting the midvein is specified to 30; parameter for
segmenting secondary vein is 10. The initial shape of the leaf is shown in Figure  8(b), which contains 239 vertices and 316 triangles, while Figure  8(c) demonstrates three wilting effects, respectively. We do not apply
subdivision to the mesh of the leaf surface (the iterations time is 0), but the
results are plausible.A
skeleton structure of watermelon leaf and modeling effects: (a) venation
skeleton and mesh, (b) initial leaf
shape and (d) three simulated wilting effects.
(a)(b)(c)To test the influence of the iterations
time for subdividing leaf surface to the simulation of leaf wilting, we make an
experimental comparison, and the simulated results can be seen in Figure9. It
can be concluded that the larger the iterations time for subdividing, the
smoother the simulated wilted leaf surface.Comparison
of modeling watermelon leaf with different parameters: (a) the parameters
for segmenting the midveins is
10, iterations time is 0; (b) the parameters for segmenting the midveins is  20, iterations time
is 1; (c) the parameters for segmenting the midveins is 30, iterations time
is 3.
(a)(b)(c)The above application
examples demonstrate that the proposed venation skeleton-driven approach for simulating
wilting of leaf surface is effective and flexible. All of these simulations are
running in real time. It can generate realistic effects of wilted leaves
similar to natural shape. Currently generating the venation skeleton is manual
and interactive in our framework, and controlling the motions of leaves in the scale
of a plant is still simple. In fact, leaves wilting may be a natural response
for plant adapting to the environment basing on their inner state. An
attractive area for future work might involve combining our dynamic modeling
technique with physiological model of the leaf. In addition, we just consider a
single plant or a leaf in our framework. It is desirable to simulate the
motions of plant leaves in an ecosystem scale.Compared to previous related work on
modeling shape of plant leaves, in our knowledge, our method is a first attempt
to model plant leaves wilting in computer graphics. The proposed approach cannot
only generate wilted leaves shape, but also simulate the wilting process of a
plant. This provides an intuitive mechanism for animating subtle motions of
plant.


#### 8. Conclusion

We have
presented a model for modeling wilted leaf surface and simulating motions of
plant leaves. This model deforms a leaf surface by driving a venation skeleton
which is embedded into
the geometric mesh of a leaf. The venation skeleton can be created from any
polygonal mesh of leaf surface, whereas the polygonal mesh can be captured from
real leaves, which makes it easy to create highly realistic leaf appearance
models. Currently generating the venation skeleton is manual and interactive in
our framework, and with a parametric method.We have
demonstrated our model by simulating wilting of a cucumber plant and a
watermelon leaf. But it needs to be noted that motions of plant leaves would
result from a series of complex reasons, which are difficult to be revealed and
simulated, as such mechanism of motions of plant leaves is not easy to model.
The leaf deformation model presented in this paper is an example of a model
that provides intuitive control for simulating of some motions of plant leaves.
An exciting area for future work is the development of a framework for virtual
agronomic experiment for broader classes of plants.


## 2.1. Venation Patterns

Both the outline and venation system
of a leaf are essential in the recognition of plant species. Various venation structures can be found in plant
kingdom. It is believed that venation patterns correlate closely with the
taxonomic groups of plants and the shapes of leaves. Hickey [1] has given a classification
for the leaf venation patterns, in which pinnate venation and actinodromous
venation are two common found categories (see Figure 1). Pinnate venation characterized
by a single primary vein (the midvein) being attached by several secondary veins; the primary
vein originates at the base and extends toward the leaf tip. Whereas in actinodromous
venation, three or more primary veins diverge radially from a single point.
Primary veins support sequences of secondary (lateral) veins, which may branch
further into higher-order veins. The secondary veins and their descendants may
be free ending, which produces an open, tree-like venation pattern, or they may
connect, forming loops characteristic of a closed pattern.Some examples
of venation patterns: pinnate venation (a) and actinodromous venation (b).
(a)(b)Although the interrelationships between topological or
geometric properties of the various leaf venation patterns and functional
aspects are far from being well understood, it is believed that the leaf
venation system brings various functional properties. More information about
this can be found in [2].


### 2.2.  Model Deformation and Motions of Plant Leaf in Computer Graphics

Some researchers have endeavored to generate curled shapes
of plant leaves. Prusinkiewicz et al. [3] provided a detailed representation of
combining interaction and parameterized algorithms for realistic plant modeling
and scene creation involving plants, including curled leaves. Mündermann et al.
[4] proposed a method for modeling lobed leaves; curled surface of leaves could
be generated by using free deformation in their framework. Recently, Hong et al.
[5] proposed an interactive method for modeling curled leaf surface. But it
could involve excessive manual
interactions for generating a desired curled shape of leaf by using their method.
Studies on curvature of plant leaves
from biophysical perspective have raised the question of what role, if any,
genes play in controlling the curvature of leaves [6]. Yet there are some
researchers study waved or wrinkled pattern in leaves with physical analysis [7].
But these may go
beyond our focus in this paper.In aspects of modeling motions of plant, most work has
been done on modeling dynamic motions of tree in the wind, such as the work
demonstrated in [8]. Based on the fact that plant growth is generally
influenced by gravity and tropisms, Jirasek and Prusinkiewicz [9] proposed a
biomechanical model for creating curved plant branches by using physically based
modeling. Hart et al. [10] extended their idea in modeling plant growth
considering physical properties of the plant. Note that none of the above
models simulated leaves behaviors. Wang et al. [11] had simulated physically
the growth of a plant leaf; the physical model used in their simulation is the
governing equations of fluid mechanics—the Navier-Stokes
equations. But they just tested their model in 2D.Much work has been done on surface deformation. And many
techniques, such as multiresolution mesh representations [12], skeleton-driven
global free-form shape deformations [13], and differential deformation [14, 15],
have been developed to help artists deform object shapes. But none of these techniques
has been tested by modeling leaf deformation.


## 2.2.  Model Deformation and Motions of Plant Leaf in Computer Graphics

Some researchers have endeavored to generate curled shapes
of plant leaves. Prusinkiewicz et al. [3] provided a detailed representation of
combining interaction and parameterized algorithms for realistic plant modeling
and scene creation involving plants, including curled leaves. Mündermann et al.
[4] proposed a method for modeling lobed leaves; curled surface of leaves could
be generated by using free deformation in their framework. Recently, Hong et al.
[5] proposed an interactive method for modeling curled leaf surface. But it
could involve excessive manual
interactions for generating a desired curled shape of leaf by using their method.
Studies on curvature of plant leaves
from biophysical perspective have raised the question of what role, if any,
genes play in controlling the curvature of leaves [6]. Yet there are some
researchers study waved or wrinkled pattern in leaves with physical analysis [7].
But these may go
beyond our focus in this paper.In aspects of modeling motions of plant, most work has
been done on modeling dynamic motions of tree in the wind, such as the work
demonstrated in [8]. Based on the fact that plant growth is generally
influenced by gravity and tropisms, Jirasek and Prusinkiewicz [9] proposed a
biomechanical model for creating curved plant branches by using physically based
modeling. Hart et al. [10] extended their idea in modeling plant growth
considering physical properties of the plant. Note that none of the above
models simulated leaves behaviors. Wang et al. [11] had simulated physically
the growth of a plant leaf; the physical model used in their simulation is the
governing equations of fluid mechanics—the Navier-Stokes
equations. But they just tested their model in 2D.Much work has been done on surface deformation. And many
techniques, such as multiresolution mesh representations [12], skeleton-driven
global free-form shape deformations [13], and differential deformation [14, 15],
have been developed to help artists deform object shapes. But none of these techniques
has been tested by modeling leaf deformation.


## 3. Overview of the Proposed Method

Figure2 gives an overview how our proposed method works. The modeling
processes include constructing the venation skeleton from a leaf image,
deforming the leaf surface, and so forth. These schematics label the transition
processes in uppercase letters, A, B, and so forth. The interactive simulating system
that we propose tries to strike a pragmatic balance between processes that can
be automated and those that seem to require interaction to achieve the desired
level of realism. It has primarily been designed to support our experimentation
with interactive animation of leaf motions.Figure 2
Modeling processes.Each process will be detailed in the following sections.


## 4. Generating Venation Skeleton

The venation structure of a leaf plays
a major biological role in determining the leaf surface shape and controlling
its deformation; therefore, we use it to control the deformation of a leaf
blade. To generate the venation skeleton, currently we consider an interactive
method. In Figure2, process
A and C illustrate the steps for generating venation skeleton of a leaf.


## 4.1. Extracting a Leaf Boundary

We use a representation of a leaf skeleton
consisting of two boundary curves and a midvein curve which consists of feature
points, as shown in the left of process A. These boundary curves can be reconstructed
from feature points in the boundary of a leaf, while these feature points can
be extracted automatically from a scanned digital image by using a standard
edge detection algorithm, or obtained by using a 3D digitizer.To meet the need of interactive design,
the leaf skeleton can also be constructed automatically with a parametric
method, in which the length of midvein, the width of leaf blade, and the number
of feature points are all initialized with parameters.


## 4.2. Generating Venation Skeleton

The skeleton was originally
introduced by Blum [16] for 2D shapes in order to provide a symmetry-based
shape representation for shape perception and recognition. Recently the
skeleton in 3D has been studied in connection with a research on shape
organization [17] and shape manipulation. Practical extraction of the
skeleton of a 3D shape is usually based on 3D Voronoi diagram techniques [18].For our needs, we develop an
interface for generating interactively venation skeleton from a leaf skeleton.
As Figure2 shows (from process A to process C), the leaf skeleton can be obtained
from a scanned image; a venation skeleton is then generated from the leaf
skeleton, and each vein is segmented. The result of process C demonstrates  a
generated venation skeleton consisting of one midvein and four secondary veins;
the black vertices string segmented each vein into several line segments. The
process of generating  venation skeleton involves several manual interactions including
defining the start point and end point for each vein, and specifying parameters
for segmenting each vein. Note that the venation skeleton is not unique. It can
be created according to actual needs. Users can decide how the midvein crosses
the leaf skeleton, and how many secondary veins are attached to the midvein.


## 5. Leaf Surface Meshing and Banding

### 5.1. Constructing Leaf Surface

We have constructed a leaf skeleton
with two boundary curves and a midvein curve as shown in Figure2  (process A). To
mesh the void area within these boundary curves, we employ Delaunay
triangulation scheme, because it can deal with the problem of existing concave area
in the leaf blade, which is difficult to render directly by simple polygon. For
example, lobed leaves often have irregular silhouette characterizing by a
number of concave outline. When using Delaunay triangulation, we can use
directly the feature points in the midvein and silhouette, or extract a series
of points from the midvein curve and silhouette curves with a fixed interval.
The mesh resulted from process B of Figure 2 is generated from the result of process
A by using Delaunay triangulation scheme.The initial mesh of a leaf surface
generated by using Delaunay triangulation is generally irregular and rough. It
is necessary to refine the mesh for later deformation. Currently, we use a
simple method to subdivide the initial mesh. Generally each triangle can be
divided into four triangles. To meet users’ requirements of interaction, we
provide two parameters for the subdivision: one is the times of iteration steps
of subdivision, the other is the minimal distance of an edge in the mesh. The iterations timeis used to control the number of iterations in the subdividing,
while  using the constraint of the minimal distance for an edge is to avoid generating
too short edge. The lower figure of process D in Figure2 illustrates a mesh
subdividing the initial mesh shown in the upper figure at the same process with
iteration steps and minimal distance being specified 2 and 0.5, respectively.


### 5.2. Banding

The banding is to attach all the vertices
in the subdivision mesh to the initial venation skeleton of the leaf. The
banding is based on the distance of each vertex to the venation skeleton. In
other words, each vertex in the mesh is banded to the nearest vertex in the
venation skeleton. Figure3 gives an example of banding a leaf mesh to its
venation skeleton which consists of a midvein and two secondary veins.Figure 3
Example of banding a leaf mesh
to its venation skeleton.


### 6. Venation Skeleton-Driven Leaf Surface Deformation

#### 6.1. Mechanism for Controlling Movement of the Venation Skeleton

Skeleton-based methods have been extensively
used for mesh deformation in computer animation and computer modeling. In this
method, two or more “bones” meet at each joint, to control shape deformation.
This allows intuitive control, naturally describing the way in which many
objects, for example, animals, deform the muscles and other tissues follow
motions of the underlying bones.Traditional skeleton-based methods commonly
require a tedious
process of weight selection to obtain satisfactory results. Note that the
natural deformation of plant leaves is different from the deformation of humans’
or animals’ organs, and spontaneous motions of a leaf blade are relatively simple.
The most regular motions of plant leaves which can be seen by our naked eyes
are curling and wilting. So we can constraint the movement of vertices in the
venation skeleton for simulating the movements of leaves. The major goal of our
approach is to develop an approximately kinematic model for simulating these
motions of plant leaves.For our needs, we restrict the
movement of each vertex in the venation skeleton by rotating it around a fixed
vector. Figure4(a) illustrates how a venation skeleton works. For convenience,
the venation skeleton includes a vein only, which consists of four segments.
The light black vertex serves as the root node. Note that the movement of a
leaf blade is always downward during its wilting, as such each joint in the skeleton
segment does spherical movement. Take vertex Vi, for example, vector Vi−1Vi   will align gradually vector Vi−1VN  during wilting,
in which Vi−1VN  is a downward vector reverse to Z-axis, while vector Vi−1VM  is perpendicular to the plane which contains vector Vi−1Vi and Vi−1VN. To the
movement of vertex Vi, it
can be looked as Vi is
rotated round vector Vi−1VM.How the venation skeleton
works. (a) The movement trajectory of a
skeletal joint; (b) child joints follow the parent’s movement.
(a)(b)To obtain a motion sequence of a
vertex in a skeleton, the simplest method is to rotate the vertex with a fixed
angle, such asθ in Figure 4(a), and
the angle is given commonly by the users. But a conventional method is using inverse
kinematics [19]. We have mentioned before that a vertex in the skeleton will do
spherical rotation during the wilting of its corresponding leaf surface. As
Figure 4(a) illustrates, we can calculate the new position of vertex Vi by using the following parametric
equation: (1)V(t)=k(t)*(Vi+t*(VN−Vi)), where k(t)=|Vi|/|Vi+t*(VN−Vi)|(0≤t≤1). Further, the
motion sequence of Vi can
be obtained by increasing parameter t.
This may simplify the rotation operation. Note that vector Vi−1VN needs to be recalculated when vertex Vi−1  is repositioned, but it
is always downward.And also, all child segment of a
vertex in the venation skeleton will follow the movement of the vertex; this
can be done by passing a displacement and rotation angle to its child vertices
when the vertex is being rotated. Figure4(b) gives results of rotating
four vertices in a skeleton successively, in which the blue is the rotated
vertices while the red is the base point.


#### 6.2. Leaf Surface Deformation

We have detailed the mechanism for controlling
movement of the venation skeleton and methods for constructing the leaf mesh in
our approach. The last step is to deform the leaf surface according to the
deformed venation skeleton. This process can be illustrated as process F in Figure2.Firstly all the vertices in the
subdivision mesh of the leaf are banded to the initial venation skeleton. Then
the initial venation skeleton is deformed by using the method described in
Section6.1. For example, at process E, we can generate the shape of the
venation skeleton shown bottom in Figure
2E  from the initial skeleton shown top in
Figure
2E (with different number of joints in each vein). Lastly the position of
each vertex in the mesh is recalculated according to the new coordinate of its
banded vertex in the venation skeleton. The left-bottom figure in Figure
2 shows the
resulted mesh.The right-bottom figure in Figure2 illustrates
the rendering result. The texture mapping is calculated before the deformation,
and no need to remap after the deformation.It needs to be noted that the number
of joints in each vein in the venation skeleton will influence the effects of
deformation. It is easy to imagine that the larger the number of joints is, the
smoother the deformed surface will be. And large deformation needs large number
of joints. But larger number of joints in the venation skeleton also means more
computation and more difficult controlling over the deformation. In our
experiments, we find that pleasing visual effects of the simulated wilted
leaves shape can be achieved when the number of joints in each vein is not
smaller than 4. Users can also obtain a satisfactory result by interactive experiments.


#### 6.3. Constraints and Collision Detection

Constraints
and collision detection are usually the common issues in surface deformation. For constraint, we have
stated that each vertex in the venation skeleton can rotate around a fixed
vector. Additionally, the rotation needs to meet some extra constraints. For
example, the leaf surface will be always drooped during its wilting. In
simulating the effect of wilting leaf surface, a vertex in the leaf mesh could
not be rotated after it had reached the maximal drooped distance. When
simulating curling of a leaf, it needs to avoid overlap of the leaf surface. This can be done by
keeping the included angle of two adjacent line sections on each vein being larger
than a predefined angle.Collision detection and response is
usually the most time consuming process for the overall simulation. Currently
we just consider a collision detection to avoid self-intersect in the
deformation. During deforming a leaf mesh, each handling currently vertex needs
to be checked if its movement will pierce some triangle in the mesh. Piercing
means that there is intersection in the triangle mesh. If no piercing occurs,
no response is made. If there is an intersection, then we calculate a maximal
displacement from the precalculated displacement for the vertex to move to avoid
intersection, and correct the displacement of corresponding vertex in the
venation skeleton.


### 7. Applications and Discussion

We
implemented our algorithm for venation skeleton-driven leaf surface deformation
in C++ on a PC with a 2.8 GHz Pentium D processor and a NVIDIA GeFore 7900 GS
graphics card, and used
OpenGL to render the results. In this section we report the modeling results.Firstly we
simulate wilting effect of a plant leaf. The plant used in this experiment is
cucumber. Figure5  shows a photo
of a real wilting cucumber in greenhouse. In our experiment, we first
simulate the wilted effect of a cucumber leaf. As shown in Figure  6, Figure 6(a) is the initial shape of the simulated leaf surface while Figures 6(b)
and 6(c) are two deformed results corresponding to two different levels of wilted leaves. The initial leaf surface is created
interactively from a scanned image within 3 minutes, with 187 vertices and 286
triangles. The iterations time for subdividing the
leaf surface is specified to 0. In this experiment, the venation skeleton
consists of a midvein and two secondary veins parameters for segmenting these
veins are 20, 4, respectively, and the total time for simulating the leaf
wilting is 8 seconds
with 22 frames.Figure 5
Photo of a real wilting cucumber in
greenhouse.Simulated shapes of wilting cucumber leaf: (a) initial shape, (b) medium
wilting, and (c) acute wilting.
(a)(b)(c)We then
use the venation skeleton model to simulate the process of a cucumber wilting.
Figure7 demonstrates the simulated results, in which Figure 7(a)  is the
initial shape, Figure 7(b) simulates a slight wilting leaves, and Figure 7(c)
simulates an acute
wilting one. This initial plant is created within 5 minutes by using an
interactive interface that we have developed for designing crop structure. The number
of vertices and triangles of this model are 4082 and 6150, respectively. We use three instances
of leaf surface in the cucumber model, and the venation skeleton in each instance
is different from each other, but with the same number of veins (three). The
venation skeleton is deformed automatically by rotating the vertices in the venation
skeleton downward from the boundary to root of the leaf, by using (1), and the above
leaves start wilting later than the lower leaves do, whereas the speed of
wilting can be adjusted by modifying parameter t. This simulation consists of 300 frames and runs for approximately
2 minutes.Three
stages of a wilting cucumber plant: (a) none wilting, (b) slight
wilting, and (c) acute wilting.
(a)(b)(c)The second application example is simulating
a watermelon leaf wilting. Watermelon leaf is a typical lobed leaf. We use a
venation skeleton shown in Figure8(a)  to control its deformation of the leaf
blade. The venation skeleton consists of one midvein and four secondary veins,
and the parameter for segmenting the midvein is specified to 30; parameter for
segmenting secondary vein is 10. The initial shape of the leaf is shown in Figure  8(b), which contains 239 vertices and 316 triangles, while Figure  8(c) demonstrates three wilting effects, respectively. We do not apply
subdivision to the mesh of the leaf surface (the iterations time is 0), but the
results are plausible.A
skeleton structure of watermelon leaf and modeling effects: (a) venation
skeleton and mesh, (b) initial leaf
shape and (d) three simulated wilting effects.
(a)(b)(c)To test the influence of the iterations
time for subdividing leaf surface to the simulation of leaf wilting, we make an
experimental comparison, and the simulated results can be seen in Figure9. It
can be concluded that the larger the iterations time for subdividing, the
smoother the simulated wilted leaf surface.Comparison
of modeling watermelon leaf with different parameters: (a) the parameters
for segmenting the midveins is
10, iterations time is 0; (b) the parameters for segmenting the midveins is  20, iterations time
is 1; (c) the parameters for segmenting the midveins is 30, iterations time
is 3.
(a)(b)(c)The above application
examples demonstrate that the proposed venation skeleton-driven approach for simulating
wilting of leaf surface is effective and flexible. All of these simulations are
running in real time. It can generate realistic effects of wilted leaves
similar to natural shape. Currently generating the venation skeleton is manual
and interactive in our framework, and controlling the motions of leaves in the scale
of a plant is still simple. In fact, leaves wilting may be a natural response
for plant adapting to the environment basing on their inner state. An
attractive area for future work might involve combining our dynamic modeling
technique with physiological model of the leaf. In addition, we just consider a
single plant or a leaf in our framework. It is desirable to simulate the
motions of plant leaves in an ecosystem scale.Compared to previous related work on
modeling shape of plant leaves, in our knowledge, our method is a first attempt
to model plant leaves wilting in computer graphics. The proposed approach cannot
only generate wilted leaves shape, but also simulate the wilting process of a
plant. This provides an intuitive mechanism for animating subtle motions of
plant.


### 8. Conclusion

We have
presented a model for modeling wilted leaf surface and simulating motions of
plant leaves. This model deforms a leaf surface by driving a venation skeleton
which is embedded into
the geometric mesh of a leaf. The venation skeleton can be created from any
polygonal mesh of leaf surface, whereas the polygonal mesh can be captured from
real leaves, which makes it easy to create highly realistic leaf appearance
models. Currently generating the venation skeleton is manual and interactive in
our framework, and with a parametric method.We have
demonstrated our model by simulating wilting of a cucumber plant and a
watermelon leaf. But it needs to be noted that motions of plant leaves would
result from a series of complex reasons, which are difficult to be revealed and
simulated, as such mechanism of motions of plant leaves is not easy to model.
The leaf deformation model presented in this paper is an example of a model
that provides intuitive control for simulating of some motions of plant leaves.
An exciting area for future work is the development of a framework for virtual
agronomic experiment for broader classes of plants.


## 5.1. Constructing Leaf Surface

We have constructed a leaf skeleton
with two boundary curves and a midvein curve as shown in Figure2  (process A). To
mesh the void area within these boundary curves, we employ Delaunay
triangulation scheme, because it can deal with the problem of existing concave area
in the leaf blade, which is difficult to render directly by simple polygon. For
example, lobed leaves often have irregular silhouette characterizing by a
number of concave outline. When using Delaunay triangulation, we can use
directly the feature points in the midvein and silhouette, or extract a series
of points from the midvein curve and silhouette curves with a fixed interval.
The mesh resulted from process B of Figure 2 is generated from the result of process
A by using Delaunay triangulation scheme.The initial mesh of a leaf surface
generated by using Delaunay triangulation is generally irregular and rough. It
is necessary to refine the mesh for later deformation. Currently, we use a
simple method to subdivide the initial mesh. Generally each triangle can be
divided into four triangles. To meet users’ requirements of interaction, we
provide two parameters for the subdivision: one is the times of iteration steps
of subdivision, the other is the minimal distance of an edge in the mesh. The iterations timeis used to control the number of iterations in the subdividing,
while  using the constraint of the minimal distance for an edge is to avoid generating
too short edge. The lower figure of process D in Figure2 illustrates a mesh
subdividing the initial mesh shown in the upper figure at the same process with
iteration steps and minimal distance being specified 2 and 0.5, respectively.


## 5.2. Banding

The banding is to attach all the vertices
in the subdivision mesh to the initial venation skeleton of the leaf. The
banding is based on the distance of each vertex to the venation skeleton. In
other words, each vertex in the mesh is banded to the nearest vertex in the
venation skeleton. Figure3 gives an example of banding a leaf mesh to its
venation skeleton which consists of a midvein and two secondary veins.Figure 3
Example of banding a leaf mesh
to its venation skeleton.


## 6. Venation Skeleton-Driven Leaf Surface Deformation

### 6.1. Mechanism for Controlling Movement of the Venation Skeleton

Skeleton-based methods have been extensively
used for mesh deformation in computer animation and computer modeling. In this
method, two or more “bones” meet at each joint, to control shape deformation.
This allows intuitive control, naturally describing the way in which many
objects, for example, animals, deform the muscles and other tissues follow
motions of the underlying bones.Traditional skeleton-based methods commonly
require a tedious
process of weight selection to obtain satisfactory results. Note that the
natural deformation of plant leaves is different from the deformation of humans’
or animals’ organs, and spontaneous motions of a leaf blade are relatively simple.
The most regular motions of plant leaves which can be seen by our naked eyes
are curling and wilting. So we can constraint the movement of vertices in the
venation skeleton for simulating the movements of leaves. The major goal of our
approach is to develop an approximately kinematic model for simulating these
motions of plant leaves.For our needs, we restrict the
movement of each vertex in the venation skeleton by rotating it around a fixed
vector. Figure4(a) illustrates how a venation skeleton works. For convenience,
the venation skeleton includes a vein only, which consists of four segments.
The light black vertex serves as the root node. Note that the movement of a
leaf blade is always downward during its wilting, as such each joint in the skeleton
segment does spherical movement. Take vertex Vi, for example, vector Vi−1Vi   will align gradually vector Vi−1VN  during wilting,
in which Vi−1VN  is a downward vector reverse to Z-axis, while vector Vi−1VM  is perpendicular to the plane which contains vector Vi−1Vi and Vi−1VN. To the
movement of vertex Vi, it
can be looked as Vi is
rotated round vector Vi−1VM.How the venation skeleton
works. (a) The movement trajectory of a
skeletal joint; (b) child joints follow the parent’s movement.
(a)(b)To obtain a motion sequence of a
vertex in a skeleton, the simplest method is to rotate the vertex with a fixed
angle, such asθ in Figure 4(a), and
the angle is given commonly by the users. But a conventional method is using inverse
kinematics [19]. We have mentioned before that a vertex in the skeleton will do
spherical rotation during the wilting of its corresponding leaf surface. As
Figure 4(a) illustrates, we can calculate the new position of vertex Vi by using the following parametric
equation: (1)V(t)=k(t)*(Vi+t*(VN−Vi)), where k(t)=|Vi|/|Vi+t*(VN−Vi)|(0≤t≤1). Further, the
motion sequence of Vi can
be obtained by increasing parameter t.
This may simplify the rotation operation. Note that vector Vi−1VN needs to be recalculated when vertex Vi−1  is repositioned, but it
is always downward.And also, all child segment of a
vertex in the venation skeleton will follow the movement of the vertex; this
can be done by passing a displacement and rotation angle to its child vertices
when the vertex is being rotated. Figure4(b) gives results of rotating
four vertices in a skeleton successively, in which the blue is the rotated
vertices while the red is the base point.


### 6.2. Leaf Surface Deformation

We have detailed the mechanism for controlling
movement of the venation skeleton and methods for constructing the leaf mesh in
our approach. The last step is to deform the leaf surface according to the
deformed venation skeleton. This process can be illustrated as process F in Figure2.Firstly all the vertices in the
subdivision mesh of the leaf are banded to the initial venation skeleton. Then
the initial venation skeleton is deformed by using the method described in
Section6.1. For example, at process E, we can generate the shape of the
venation skeleton shown bottom in Figure
2E  from the initial skeleton shown top in
Figure
2E (with different number of joints in each vein). Lastly the position of
each vertex in the mesh is recalculated according to the new coordinate of its
banded vertex in the venation skeleton. The left-bottom figure in Figure
2 shows the
resulted mesh.The right-bottom figure in Figure2 illustrates
the rendering result. The texture mapping is calculated before the deformation,
and no need to remap after the deformation.It needs to be noted that the number
of joints in each vein in the venation skeleton will influence the effects of
deformation. It is easy to imagine that the larger the number of joints is, the
smoother the deformed surface will be. And large deformation needs large number
of joints. But larger number of joints in the venation skeleton also means more
computation and more difficult controlling over the deformation. In our
experiments, we find that pleasing visual effects of the simulated wilted
leaves shape can be achieved when the number of joints in each vein is not
smaller than 4. Users can also obtain a satisfactory result by interactive experiments.


### 6.3. Constraints and Collision Detection

Constraints
and collision detection are usually the common issues in surface deformation. For constraint, we have
stated that each vertex in the venation skeleton can rotate around a fixed
vector. Additionally, the rotation needs to meet some extra constraints. For
example, the leaf surface will be always drooped during its wilting. In
simulating the effect of wilting leaf surface, a vertex in the leaf mesh could
not be rotated after it had reached the maximal drooped distance. When
simulating curling of a leaf, it needs to avoid overlap of the leaf surface. This can be done by
keeping the included angle of two adjacent line sections on each vein being larger
than a predefined angle.Collision detection and response is
usually the most time consuming process for the overall simulation. Currently
we just consider a collision detection to avoid self-intersect in the
deformation. During deforming a leaf mesh, each handling currently vertex needs
to be checked if its movement will pierce some triangle in the mesh. Piercing
means that there is intersection in the triangle mesh. If no piercing occurs,
no response is made. If there is an intersection, then we calculate a maximal
displacement from the precalculated displacement for the vertex to move to avoid
intersection, and correct the displacement of corresponding vertex in the
venation skeleton.


## 6.1. Mechanism for Controlling Movement of the Venation Skeleton

Skeleton-based methods have been extensively
used for mesh deformation in computer animation and computer modeling. In this
method, two or more “bones” meet at each joint, to control shape deformation.
This allows intuitive control, naturally describing the way in which many
objects, for example, animals, deform the muscles and other tissues follow
motions of the underlying bones.Traditional skeleton-based methods commonly
require a tedious
process of weight selection to obtain satisfactory results. Note that the
natural deformation of plant leaves is different from the deformation of humans’
or animals’ organs, and spontaneous motions of a leaf blade are relatively simple.
The most regular motions of plant leaves which can be seen by our naked eyes
are curling and wilting. So we can constraint the movement of vertices in the
venation skeleton for simulating the movements of leaves. The major goal of our
approach is to develop an approximately kinematic model for simulating these
motions of plant leaves.For our needs, we restrict the
movement of each vertex in the venation skeleton by rotating it around a fixed
vector. Figure4(a) illustrates how a venation skeleton works. For convenience,
the venation skeleton includes a vein only, which consists of four segments.
The light black vertex serves as the root node. Note that the movement of a
leaf blade is always downward during its wilting, as such each joint in the skeleton
segment does spherical movement. Take vertex Vi, for example, vector Vi−1Vi   will align gradually vector Vi−1VN  during wilting,
in which Vi−1VN  is a downward vector reverse to Z-axis, while vector Vi−1VM  is perpendicular to the plane which contains vector Vi−1Vi and Vi−1VN. To the
movement of vertex Vi, it
can be looked as Vi is
rotated round vector Vi−1VM.How the venation skeleton
works. (a) The movement trajectory of a
skeletal joint; (b) child joints follow the parent’s movement.
(a)(b)To obtain a motion sequence of a
vertex in a skeleton, the simplest method is to rotate the vertex with a fixed
angle, such asθ in Figure 4(a), and
the angle is given commonly by the users. But a conventional method is using inverse
kinematics [19]. We have mentioned before that a vertex in the skeleton will do
spherical rotation during the wilting of its corresponding leaf surface. As
Figure 4(a) illustrates, we can calculate the new position of vertex Vi by using the following parametric
equation: (1)V(t)=k(t)*(Vi+t*(VN−Vi)), where k(t)=|Vi|/|Vi+t*(VN−Vi)|(0≤t≤1). Further, the
motion sequence of Vi can
be obtained by increasing parameter t.
This may simplify the rotation operation. Note that vector Vi−1VN needs to be recalculated when vertex Vi−1  is repositioned, but it
is always downward.And also, all child segment of a
vertex in the venation skeleton will follow the movement of the vertex; this
can be done by passing a displacement and rotation angle to its child vertices
when the vertex is being rotated. Figure4(b) gives results of rotating
four vertices in a skeleton successively, in which the blue is the rotated
vertices while the red is the base point.


## 6.2. Leaf Surface Deformation

We have detailed the mechanism for controlling
movement of the venation skeleton and methods for constructing the leaf mesh in
our approach. The last step is to deform the leaf surface according to the
deformed venation skeleton. This process can be illustrated as process F in Figure2.Firstly all the vertices in the
subdivision mesh of the leaf are banded to the initial venation skeleton. Then
the initial venation skeleton is deformed by using the method described in
Section6.1. For example, at process E, we can generate the shape of the
venation skeleton shown bottom in Figure
2E  from the initial skeleton shown top in
Figure
2E (with different number of joints in each vein). Lastly the position of
each vertex in the mesh is recalculated according to the new coordinate of its
banded vertex in the venation skeleton. The left-bottom figure in Figure
2 shows the
resulted mesh.The right-bottom figure in Figure2 illustrates
the rendering result. The texture mapping is calculated before the deformation,
and no need to remap after the deformation.It needs to be noted that the number
of joints in each vein in the venation skeleton will influence the effects of
deformation. It is easy to imagine that the larger the number of joints is, the
smoother the deformed surface will be. And large deformation needs large number
of joints. But larger number of joints in the venation skeleton also means more
computation and more difficult controlling over the deformation. In our
experiments, we find that pleasing visual effects of the simulated wilted
leaves shape can be achieved when the number of joints in each vein is not
smaller than 4. Users can also obtain a satisfactory result by interactive experiments.


## 6.3. Constraints and Collision Detection

Constraints
and collision detection are usually the common issues in surface deformation. For constraint, we have
stated that each vertex in the venation skeleton can rotate around a fixed
vector. Additionally, the rotation needs to meet some extra constraints. For
example, the leaf surface will be always drooped during its wilting. In
simulating the effect of wilting leaf surface, a vertex in the leaf mesh could
not be rotated after it had reached the maximal drooped distance. When
simulating curling of a leaf, it needs to avoid overlap of the leaf surface. This can be done by
keeping the included angle of two adjacent line sections on each vein being larger
than a predefined angle.Collision detection and response is
usually the most time consuming process for the overall simulation. Currently
we just consider a collision detection to avoid self-intersect in the
deformation. During deforming a leaf mesh, each handling currently vertex needs
to be checked if its movement will pierce some triangle in the mesh. Piercing
means that there is intersection in the triangle mesh. If no piercing occurs,
no response is made. If there is an intersection, then we calculate a maximal
displacement from the precalculated displacement for the vertex to move to avoid
intersection, and correct the displacement of corresponding vertex in the
venation skeleton.


## 7. Applications and Discussion

We
implemented our algorithm for venation skeleton-driven leaf surface deformation
in C++ on a PC with a 2.8 GHz Pentium D processor and a NVIDIA GeFore 7900 GS
graphics card, and used
OpenGL to render the results. In this section we report the modeling results.Firstly we
simulate wilting effect of a plant leaf. The plant used in this experiment is
cucumber. Figure5  shows a photo
of a real wilting cucumber in greenhouse. In our experiment, we first
simulate the wilted effect of a cucumber leaf. As shown in Figure  6, Figure 6(a) is the initial shape of the simulated leaf surface while Figures 6(b)
and 6(c) are two deformed results corresponding to two different levels of wilted leaves. The initial leaf surface is created
interactively from a scanned image within 3 minutes, with 187 vertices and 286
triangles. The iterations time for subdividing the
leaf surface is specified to 0. In this experiment, the venation skeleton
consists of a midvein and two secondary veins parameters for segmenting these
veins are 20, 4, respectively, and the total time for simulating the leaf
wilting is 8 seconds
with 22 frames.Figure 5
Photo of a real wilting cucumber in
greenhouse.Simulated shapes of wilting cucumber leaf: (a) initial shape, (b) medium
wilting, and (c) acute wilting.
(a)(b)(c)We then
use the venation skeleton model to simulate the process of a cucumber wilting.
Figure7 demonstrates the simulated results, in which Figure 7(a)  is the
initial shape, Figure 7(b) simulates a slight wilting leaves, and Figure 7(c)
simulates an acute
wilting one. This initial plant is created within 5 minutes by using an
interactive interface that we have developed for designing crop structure. The number
of vertices and triangles of this model are 4082 and 6150, respectively. We use three instances
of leaf surface in the cucumber model, and the venation skeleton in each instance
is different from each other, but with the same number of veins (three). The
venation skeleton is deformed automatically by rotating the vertices in the venation
skeleton downward from the boundary to root of the leaf, by using (1), and the above
leaves start wilting later than the lower leaves do, whereas the speed of
wilting can be adjusted by modifying parameter t. This simulation consists of 300 frames and runs for approximately
2 minutes.Three
stages of a wilting cucumber plant: (a) none wilting, (b) slight
wilting, and (c) acute wilting.
(a)(b)(c)The second application example is simulating
a watermelon leaf wilting. Watermelon leaf is a typical lobed leaf. We use a
venation skeleton shown in Figure8(a)  to control its deformation of the leaf
blade. The venation skeleton consists of one midvein and four secondary veins,
and the parameter for segmenting the midvein is specified to 30; parameter for
segmenting secondary vein is 10. The initial shape of the leaf is shown in Figure  8(b), which contains 239 vertices and 316 triangles, while Figure  8(c) demonstrates three wilting effects, respectively. We do not apply
subdivision to the mesh of the leaf surface (the iterations time is 0), but the
results are plausible.A
skeleton structure of watermelon leaf and modeling effects: (a) venation
skeleton and mesh, (b) initial leaf
shape and (d) three simulated wilting effects.
(a)(b)(c)To test the influence of the iterations
time for subdividing leaf surface to the simulation of leaf wilting, we make an
experimental comparison, and the simulated results can be seen in Figure9. It
can be concluded that the larger the iterations time for subdividing, the
smoother the simulated wilted leaf surface.Comparison
of modeling watermelon leaf with different parameters: (a) the parameters
for segmenting the midveins is
10, iterations time is 0; (b) the parameters for segmenting the midveins is  20, iterations time
is 1; (c) the parameters for segmenting the midveins is 30, iterations time
is 3.
(a)(b)(c)The above application
examples demonstrate that the proposed venation skeleton-driven approach for simulating
wilting of leaf surface is effective and flexible. All of these simulations are
running in real time. It can generate realistic effects of wilted leaves
similar to natural shape. Currently generating the venation skeleton is manual
and interactive in our framework, and controlling the motions of leaves in the scale
of a plant is still simple. In fact, leaves wilting may be a natural response
for plant adapting to the environment basing on their inner state. An
attractive area for future work might involve combining our dynamic modeling
technique with physiological model of the leaf. In addition, we just consider a
single plant or a leaf in our framework. It is desirable to simulate the
motions of plant leaves in an ecosystem scale.Compared to previous related work on
modeling shape of plant leaves, in our knowledge, our method is a first attempt
to model plant leaves wilting in computer graphics. The proposed approach cannot
only generate wilted leaves shape, but also simulate the wilting process of a
plant. This provides an intuitive mechanism for animating subtle motions of
plant.


## 8. Conclusion

We have
presented a model for modeling wilted leaf surface and simulating motions of
plant leaves. This model deforms a leaf surface by driving a venation skeleton
which is embedded into
the geometric mesh of a leaf. The venation skeleton can be created from any
polygonal mesh of leaf surface, whereas the polygonal mesh can be captured from
real leaves, which makes it easy to create highly realistic leaf appearance
models. Currently generating the venation skeleton is manual and interactive in
our framework, and with a parametric method.We have
demonstrated our model by simulating wilting of a cucumber plant and a
watermelon leaf. But it needs to be noted that motions of plant leaves would
result from a series of complex reasons, which are difficult to be revealed and
simulated, as such mechanism of motions of plant leaves is not easy to model.
The leaf deformation model presented in this paper is an example of a model
that provides intuitive control for simulating of some motions of plant leaves.
An exciting area for future work is the development of a framework for virtual
agronomic experiment for broader classes of plants.



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*Source: 890917-2009-04-12.xml*