THE JOURNAL OF VISUALIZATION AND COMPUTER ANIMATION, VOL. 7: 79-93 (1996)
Visual Simulation of Leaf Arrangement and
Autumn Colours
NORISHIGE CHIBA AND KEN OHSHIDA
Department of Computer Science, Faculty of Engineering, Iwate University, Morioka 020,
Japan (email [email protected])
KAZUNOBU MURAOKA
Morioka Junior College, Morioka 020, Japan
AND
NOBUJI SAITO
Department of Communication, Tohoku Institute of Technology, Sendai 982, Japan
SUMMARY
The representation methods of botanical trees have been presented so far by several researchers
in recent years. A great deal of effort has been dedicated mainly to designing a procedural
modelling method, especially a growth model of a tredplant skeleton. However, few previous
works have treated the problems of appropriate leaf arrangement and autumn colour evolution
of a leaf which are essential to giving natural and seasonal impressions in a close-range view of
a tree. In this paper, we will present the simulation methods for these attractive problems based
mainly on the estimation of the amount of sunlight and the brightest direction of each part of
every leaf.
KEY WORDS:
visual simulation; seasonal colour; leaf arrangement
1. INTRODUCTION
For realistic image synthesis of botanical trees based on conventional 3D CG
technology, which plays a vital role in the visual simulation of natural sceneries, at
least the following techniques are required:
1. a method giving a tree skeleton
2. the definition of the thickness for trunks and branches, and a method of
producing their surfaces
3 . a method providing the texture of bark
4. a method representing leaves.
This paper relates to the technique (4),especially to the simulation of leaf arrangement
based on phototropism,' and to the simulation of autumn coloured leaves based on
an ageing process.2 Few researchers have treated these problems so far.
In recent years, a great deal of effort has been dedicated mainly to designing a
CCC 1049-8907/96/020079-15
0 1996 by John Wiley & Sons, Ltd.
Received April 1995
Revised October 1995
80
N. CHIBA ET AL.
procedural modelling method, especially a growth model of a tree/plant skeleton, for
visually simulating natural tree/plant ~ h a p e s . ~Among
- ~ ~ those methods, environment~lly-sensitive'~growth models succeeded in simulating attractive irregular branching
patterns having natural visual impression^.'^^.'^.'^^'^-^' In this paper, we will use the
growth model presented in the literature2' only to produce a tree skeleton for
demonstrating our representation methods for leaves. Note that this growth model
has no ability to simulate leaves.
Several researchers presented how to model the geometry and the texture of a
leaf. Bloomenthal modelled a maple leaf as three hinged polygons, and mapped a
photograph where the veins are simply enhanced by using a paint ~ r o g r a m .Demko
~
et al. generated leaves as fractal sets by using iterated function systems (IFS).23
Oppenheimer also took a fractal approach where the external boundary shape is the
limit of growth of the internal veins.1° Lienhardt defined the topology of leaves by
using ramification operation^.^^ Lecoustre et al. prepared a subprogram capable of
generating calculated Begonia leaves based on the parameter file of a leaf containing
the geometrical instructions, such as length of internodes, branching angle, and
curvature of nerves.17 Holton prepared a small library of leaf types and patterns.22
The leaf shape is a quadrilateral or combination of quadrilaterals. Prusinkiewicz et
al. succeeded in modelling leaf shapes which are represented by polygons and in
animating the leaf developments by using L-systems and dL-systems.11~'8Viennot et
al. introduced a rapid drawing method for leaves by assuming that a leaf is flat and
the underlying arborescence is a ternary tree.14 Hammel et al. presented a modelling
method for compound leaves by using L-system and implicit contours defined by a
planar scalar field which is obtained by employing a 2D 'blobby' technique.2s
Inakage and Inakage proposed modelling techniques of simple leaf blades based on
leaf morphology.26 They used displacement, bump, and texture mapping techniques
according to the required resolutions to represent the venation pattern of a leaf. In
our visual simulation, we will also use a polygonized leaf with a bump mapping
technique for rendering veins and curved surfaces between them. From the viewpoint
of the science of form, Kaino clarified that the venation type, the margin type and
the folding manner of a leaf in a bud are closely related.27 The result is interesting
for animating leaf unfolding and development.
On the other hand, few previous works treated the leaf arrangement and the colour
evolution. Bloomenthal arranged leaves on twigs according to actual measurement^.^
In order to arrange leaves naturally, Lecoustre et al. implemented phototropism of
a leaf by minimizing the difference in the direction of the light and the leaf's
normal." We will model the phototropism more precisely so as to form a natural
leaf arrangement by taking into account the light environment of each leaf by
determining the amount of sunlight and the brightest direction.
Lienhardt generated an image of colour evolution for maple leave^.^^,^* Soutome
et al. simulated autumn coloration of leaves in two levels of resolution, i.e. in a
single leaf and in a single tree, based on the mechanism of autumn c 0 1 o r a t i o n . ~ ~ ~ ~ ~
We will present a clear method, i.e. easy to understand, for visually simulating
autumn coloured leaves having natural impressions based mainly on the estimation
of the light environment for each part of every leaf. This computational effort
improves the quality of synthesized images drastically.
The remainder of this paper is constructed as follows,
VISUAL SIMULATION OF LEAF ARRANGEMENT
81
In Section 2, we describe our geometrical model of a leaf and the techniques for
giving its texture including veins.
In Section 3, we propose algorithms for realizing natural leaf arrangement. The
algorithms consists of one for bending a petiole so as to face the surface of a leaf
toward the brightest direction, one for estimating the amount of sunlight reaching
each part of every leaf and finding the brightest direction for each leaf, and its
efficient version.
In Section 4, we propose an algorithm for simulating autumn coloured leaves. We
first abstract the mechanism of colour evolution, and then describe the details of
the algorithm.
In Section 5, we demonstrate the advantage of our methods by giving several
simulated examples.
In Section 6, we conclude the paper by touching slightly on future works.
2. GEOMETRY OF LEAF AND TEXTURE
A typical simple leaf consists of a blade, a petiole, and a pair of stipules as shown
in Figure 1 . We represent an ovate blade by triangulated planes and a needle type
blade by a triangulated cylinder. Our ovate blade is hinged along a linear primary
vein, and our petiole consists of several links represented by triangulated cylinders.
We use a bump mapping technique for representing a venation, i.e. a pattern of
veins, and curved soft tissues of an ovate blade. The bump map is constructed
as follows:
Step 0:
prepare a 2D array M for a map.
db
Pe ole
Stipule
..................................
Figure 1. Structure of simple leaf
82
N. CHIBA ET AL.
Step I :
Step 2:
Step 3:
write the blade margin and the venation consisting of a primary vein
and secondary veins with integer 0.
find the distance of each element M ( i j ) included in the interior of the
margin by using a distance transformation technique developed in the
image processing field.31
transform the map to the height field representing the geometry of the
leaf by setting
M ( i j ) := H ( M ( i j ) )
Step 4:
where H is a 1D function to map distance to height, e.g. in this paper
we use H(x) = 1 - exp[-ax], where a is a constant (see Plate 1).
find the partial differences Ax(ij) and A y ( i j ) at each element (ij),and
construct the bump maps by setting D,(ij) := hx(ij) and D J i j ) := A y ( i j ) .
We can define the normal N(ij) of the surface of the blade at position (ij)by
where C, is an arbitrary constant which controls the magnitude of bumpiness.
Moreover we use a linear interpolation to determine the normal at an arbitrary point
in a rendering step.
3. LEAF ARRANGEMENT
Leaves of most species of trees have an ability of phototropism. ',I7 This phototropism
helps leaves to form a leaf mosaic, providing natural visual impressions, appropriate
to efficient sunlight absorption. The picture of a potted plant in Plate 2 shows
evidence of the ability. In the Plate, the sunlight comes from the right-side window
existing outside of the view. One can recognize that most leaves orient themselves
towards the sunlight by bending their petioles. Since other leaves often block the
sunlight, some leaves face to the opposite side of the window which is, however,
the brightest direction at their positions.
Figure 2 outlines our model of phototropism of a leaf. In the case of Figure 2(a),
where the brightest direction is located in the direction of the tip of a leaf, the leaf
weeps. On the other hand, if the brightest direction is in the direction of the petiole,
the leaf stands up as shown in Figure 2(b).
Although most leaf mosaics are formed during their growing processes, from the
viewpoint of efficiency of computation, we implement the ability of leaf arrangement
by employing the following two steps:
Step I : arrange leaves according to the rules of phylotaxy.
Step 2: rearrange the leaves according to the rules of phototropism described
below.
Assuming that each petiole consists of several links, e.g. five links in this paper,
we simulate the phototropism of leaves as follows (see Figure 3).
Step I :
apply the following procedures simultaneously to every leaf appropriate times.
VISUAL SIMULATION OF LEAF ARRANGEMENT
side view
;OI ,
-0-
\'
' I ,
J
(b)
Figure 2. Model of phototropism
Figure 3. Implementation of phototropism
\'
83
84
N. CHIBA ET AL.
Step 1-1:
Step 1-2:
Step 1-3:
using the algorithm SUNLIGHT described below, find the amount of
sunlight S reaching a leaf and the brightest direction B.
if the amount of sunlight S is less than predefined threshold ST
implying the strength of the phototropism, then continue Step 1-3.
Otherwise, return.
let N be the normal vector of the leaf; determine the angle 0 between
the vectors N and B; let AO= 0/C, where C is an appropriate constant;
rotate each link with A0 as shown in Figure 3(b).
The algorithm SUNLIGHT is as follows. This algorithm determines the amount of
sunlight S for each mesh point defined on the surface of a concerned leaf and the
brightest direction B. (For the simplicity of the algorithm, we currently define B by
the average direction of sunlight.) This algorithm is also used in the model for
simulating autumn coloured leaves described in Section 3.
Algorithm SUNLIGHT (ZeufJ (see Figure 4)
(In this algorithm, we use a flat leaf model having no hinge along a primary vein
for computational simplification.)
Step 0:
partition the surface of the Zeufinto mesh of an appropriate resolution.
Step 1:
for each mesh point i, determine the amount of sunlight Si by using
a raycasting technique as follows: (The raycasting can be executed
efficiently by employing a 3DDDA algorithm in voxel ~ p a c e . ' ~ , ~ * )
Step 1-1: cast a predefined number of rays toward random directions, and
determine Si by
Step 2:
where sj is a vector having the same direction as the cast ray j and
the magnitude according to the light intensity of the intersection of
the celestial sphere and ray j , Niis the normal vector of the leaf at
the mesh point i, the operator - is an inner product and the sum is
taken over all cast rays j .
determine the total amount of sunlight S reaching the leaf by
Figure 4. Algorithm SUNLIGHT
VISUAL SIMULATION OF LEAF ARRANGEMENT
85
s = xsi
where the sum is taken over all mesh points i, and define the brightest
direction B by the average direction
where the sum is taken over all rays j cast from all mesh points.
The time complexity of a nalve version of SUNLIGHT is O(mm2),where rn is the
number of mesh points on a leaf, r is the number of rays cast from one mesh point,
and n is the number of leaves. If we employ the approximation technique employed
by the growth model of trees proposed in Reference 20, we can construct an efficient
algorithm so as to reduce the complexity to O(mrZ*(n/Z))= O(rnrZn), where Z is the
number of leaves growing from a single active shoot. Since n %- 1 holds normally,
calculation speed can be improved. Moreover if we use a uniform spatial subdivision
acceleration technique with 3DDDA in voxel space for solving the ray-leaf intersection problem, we can improve the efficiency of these algorithms. In the remainder
of this section, we will present an efficient approximation algorithm employing
‘leaf-balls’.
In the growth model of a tree having abilities of self-pruning due to lack of
sunlight, i.e. the death of a branch, and of phototropism of active shoots, the
following approximation technique is used.
To simulate the self-pruning and the phototropism of active shoots, we must
estimate the amount of sunlight and the brightest direction in the surrounding space
corresponding to each leaf of a tree. For the efficiency of this estimation, a ‘leafball’ with an appropriate radius defined as an approximation of a cluster of leaves
which are ‘descendants’ of each branch of a tree is introduced (see Figure 5). One
can assume that the leaf-ball is a translucent object, and that refraction does not
occur, if necessary.
Roughly speaking, the amount of sunlight reaching a particular leaf-ball can be
estimated by determining the area of the shadows projected from the other leaf-balls
on the celestial sphere at whose centre the leaf-ball concerned is located. We can
execute the estimation by applying hidden surface elimination algorithms having the
viewpoint on the leaf-ball concerned, such as a Z-buffer algorithm in which the Zbuffer is assigned to the celestial sphere, or an efficient raycasting algorithm
employing a 3DDDA algorithm in voxel space.
The brightest direction vector is defined as the sum of the vectors each of which
has the same direction as each ray which reaches the celestial sphere and has the
magnitude according to the light intensity of the intersection of the celestial sphere
and the ray.
Since, in this paper, we also use the growth model of a tree, as a result of
simulation we can obtain information about the light environment for each leaf-ball
from a hidden surface elimination algorithm, i.e. the celestial sphere storing the
information of shadows projected from the other leaf-balls on itself. We use this
information about the light environment for estimating the amount of sunlight
reaching each leaf located in the leaf-ball concerned as follows:
86
N. CHIBA ET AL.
Figure 5. Shadow of a ‘leaf-ball’ cast upon the celestial sphere2’
Step I :
Step 2:
Step 3:
prepare the celestial sphere corresponding to the leaf-ball concerned.
locate the leaves which were abstracted by the leaf-ball at appropriate
positions in the celestial sphere.
estimate the amount of sunlight for each mesh point of the leaves by
using a hidden surface elimination algorithm in the same manner
as the algorithm mentioned above for the simulation of self-pruning
and phototropism.
In the case of needle-leaf trees, such as a pine tree, almost straight thin leaves grow
thick on a tip of shoot in an orderly pattern, and most shoots bend their tips toward
the sky influenced mainly by their geotropism. Therefore, in the simulation of pine
trees, we determine the direction of cluster of leaves as follows by taking into
account the phototropism and geotropism of the shoot (see Figure 6):
Step 0:
Step 1.-
let A be the direction of a shoot which is determined in the growth
model of a tree skeleton influenced mainly by its phototropism, and let
4 be the angle between A and the zenith.
arrange a cluster of leaves at the tip of the shoot by inclining the
cluster with the angle A 4 = c4, 0 5 c 5 1 toward the sky from A.
4. AUTUMN COLOURED LEAVES
Most broad-leaf trees change their colour of leaves in autumn. There are two types
of leaves: one changes their colour into yellow, and the other into red. The abstract
VISUAL SIMULATION OF LEAF ARRANGEMENT
87
the zenith
a
V
Figure 6. Leaf arrangement of a pine tree
mechanisms of these phenomena are as follows2 (Figure7 shows our model of
the phenomena).
In the case of ‘yellow’: until the beginning of ageing, a leaf contains large
quantities of chlorophyll, i.e. a pigment of green, and small quantities of carotenoid,
i.e. a pigment of yellow. As ageing proceeds in autumn, accordingly chlorophyll is
dissolved and carotenoid remains as it is. Thus, the leaf turns yellow. Most broadleaf trees belong to this type.
In the case of ‘red’: to add to the above process, in some types of broad-leaf
trees, sugars contained in the cells of a leaf change themselves into anthocyan, i.e.
a pigment of red.
Now we show our model of autumn colour.
We assume that every leaf maintains constant amount Caro of carotenoid until it
falls. Moreover, on the basis of our field study and the literature,2 we make the
following assumptions.
The ageing of a leaf depends on both the species of tree, i.e. the particular
average life span of its leaves, and the amount of sunlight, i.e. ‘inactive leaves’,
defined as those receiving little sunlight, age more rapidly than ‘active leaves’.
Moreover, in a single leaf, ageing begins first at the parts ‘far’ from both major
A
chlorophyll
i
carotenoid
i
anthocyan
T :the beginning of aging
Figure 7. Model of autumn coloration
veins and the petiole, and then spreads over the leaf. In an active leaf, the production
of anthocyan is more efficient than in an inactive leaf.
In our model, we define the starting time T of ageing for all leaves coming out
from the same shoot i as follows:
T = To + aD,
where To is the earliest starting time of the ageing, a is a non-negative coefficient,
and D j is the imaginary plant hormone density at the live shoot i defined in the
literaturez1 which implies the activity of i. We model the decrease of chlorophyll as
follows (see Figure 8):
Chlo = pc[1 - L(t; yo S,)]
Chlo
elapsed time from T
Figure 8. Model of decreasing chlorophyll
89
VISUAL SIMULATION OF LEAF ARRANGEMENT
C14P)
6, = C,/d(P)
Yc
=
L(x, a, b) = 1/{1 + exp[-a(x - b ) ] }
where Chlo is the amount of chlorophyll, p,, C , , and C2 are coefficients, t is an
elapsed time from T, and d(P) is the distance of the position P on a leaf from both
major veins and the petiole as defined later. Thus, the larger the distance d(P) is,
the more quickly the amount of chlorophyll at P decreases.
We model the increase of the amount of anthocyan Antho as follows (see Figure 9):
Antho = PaL(t; ya, 8,)
Y a = Cd’(P)
6, = C,/S(P)
where pa, C3, and C, are coefficients, t is an elapsed time from T, and S(P) is the
amount of sunlight received at position P on a leaf. Thus, the larger S(P) is, the
more quickly the amount of anthocyan at P increases.
We currently use the following definition of the distance d of a position P:
d(P) = h(P)Z(P)
h(P) = [M(P)/max M(P)] + 1.0
l(P) = 0. 9[Z’(P)/maxZ’(P)] + 0.1
where M(P) is the value of the height field at the position
used for rendering a leaf described in Section 2, and Z’(P) is
between the position P and the petiole. From the definition
we can consider that M(P) means distance from major veins
and secondary veins.
P, i.e. the bump map
the Euclidean distance
of M , as shown later,
such as primary veins
Antho
f
*t
T
Figure 9. Model of increasing anthocyan
90
N. CHIBA ET AL.
Using the variables Caro, Chlo, and Antho, we assign the colour 0 5 R, G, B
as follows:
R = (R,,Caro
G = (G,,Caro
B = (B,,Caro
I1
+ R,,Chlo + R,Jntho)/(Caro + Chlo + Antho)
+ G,,Chlo + G,Jntho)/(Caro + Chlo + Antho)
+ B,,,ChlO + B,Jntho)/(Caro + Chlo + Antho)
where (R,,, G,,, B,,), (Rch,Gch,Bch), and (R,,, G,,, Ban) are the predefined colours of
carotenoid, chlorophyll, and anthocyan, respectively. In this paper, we set
Car0 := 0.20.
5. EXAMPLES OF SIMULATION
Figure 10 shows the developments of simulated leaf arrangements for different initial
arrangements of leaves. In each image, the top left one is a top view, the bottom
left one is a front view, and the bottom right one is a side view. The light source
is located in the direction above a viewer with the angle of depression of 45”. Short
line segments on the leaves show the normals of the leaves. The dots implying leaf
shapes are the sampling points for determining the amount of sunlight. We used
16 x 16 sampling points. Since, for the simulation of the leaf colouring, we use the
amount of sunlight reaching each sample point, we should choose the appropriate
number of sampling points by taking into account the resolution of the visual simulation.
Plate 3 is a simulated example using a growth model of a tree. Plate 3(a) shows
an initial arrangement of leaves, i.e. the phyllotaxis is opposite, and (b) an effect of
rearrangement. The colours of the leaves imply the difference in the amounts of
sunlight, i.e. the amount increases from green to red. One can see that leaf
arrangement in (b) successfully improves the light environment for each leaf. This
simulation requires about two hours on HP9000/730 workstation. Plate 4 shows
shaded images of the simulations. The leaves in (a) give an impression of an artificial
‘spherical distribution’.
If we can prepare a growth model of leaves, then we can expect that applying
the second step to all leaves in each growth step, we can realize more natural
leaf mosaics.
Plate 5 is an example of a ‘pine tree’. In Plate 5(a), each cluster of needle leaves
is located so as to face itself toward the zenith to emphasize the imbalanced abilities
of geotropism and phototropism. Plate 5(b) is obtained by taking into account
the phototropism.
Plate 6 shows simulated autumn coloured leaves including cases of both ‘yellow’
and ‘red’ coloration. Plate 7(a) is a closed view of a broadleaf tree. Plates 8 and 9
are simulated bonsai-like trees. One can see in Plates 6, 7 and 9 that the ageing
proceeds from the parts far from both the major veins and the petiole, and the
colours of parts receiving a large amount of sunlight change fast to red. Moreover,
as we can see in nature, there appears in Plate 9 the difference in the progress of
coloration of leaves according to the activities of shoots. On the other hand, as we
encounter in various geometrical modelling situations, the ‘penetration problem’ on
leaves remains, i.e. one can find such a leaf in the Plate. One possible solution is
Plute I (Cliihu et ul.). Height field M
I’lutt 3 (Clzihu et ut.). Siniuluted leuj’urrungenient on u
gcnerutd tree: ( a ) top; (bj bottom
Plute 2 (Chiba et ul.). Phototropism of an actual leaf
PIure 4 (Chiha ef uE.). Shaded imugea ofthe tree in Plate 3
Plate 5 (Chiba et al.). A simulated pine tree: (a) lefi; ( b ) right
Plate 6 (Chiba et al.). Simulated autumn colouration
Plate 9 (Chiba et al.), Japanese-elm-like potted free
VISUAL SIMULATION OF LEAF ARRANGEMENT
91
Figure 10. ( a ) Simulated leaf arrangement: simulated in the order top-left, top-right, bottom-left, and
bottom-right
(h)Simulated leaf arrangement
92
N. CHIBA ET AL.
to introduce virtual repulsive forces acting mutually on every part of leaves during
the simulation of leaf arrangement. However, this solution increases the computational
time extremely.
6. CONCLUSION
In this paper, we have presented the algorithms for simulating appropriate leaf
arrangement and autumn colour evolution of a leaf based mainly on the estimation
of the amount of sunlight and the brightest direction of each part of every leaf.
Presenting several simulated examples, we have demonstrated that these computational
effort improves the quality of synthesized images drastically. The animation of the
simulated autumn coloration can be found in SVR.33
In further work, it will be interesting to develop the following techniques:
1. more precise autumn coloration models, e.g. taking into account geographical
and weather conditions
2. growth models of leaves, blossoms, and fruits
3. a method for simulating changes of leaf shapes according to their ageing processes
4. a model for realizing the obstacle-avoidance growth in thickness of trunks,
branches, and roots
5. a growth model for producing the texture of bark
6. a growth model of tree skeletons having the abilities of realizing the following
phenomena: (a) an activity of each shoot and root depends on the gradient of
itself, (b) there is a strong relation between the activities of branches and those
of roots, and (c)physical perturbation for a shoot (e.g. caused by the wind,
poured water, and bonsai enthusiast’s hands) suppress its growth in length.
ACKNOWLEDGEMENT
We would like to thank the referees for many helpful comments and suggestions.
This work was supported partly by a Grant-in-Aid for Scientific Research of the
Ministry of Education, Science, Culture, and Sport of Japan under Grant: General
Scientific Research (B)06452406 and of The Telecommunications Advancement Foundation.
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