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Topologically Consistent Leafy Tree Morphing

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We present a novel morphing technique to generate pleasing visual effects between 2 topologically varying trees while preserving the topological consistency and botanical meanings of any in-between shapes as natural trees. Specifically, we first efficiently convert leafy trees into botanically inspired chain-lobe representations in an automatic way. With the aid of branching-pattern aware, one-to-many correspondences between branches and leaves, we hierarchically interpolate branches of in-between trees while maintaining their topological consistencies. Finally, we simultaneously interpolate foliage, specifically every single leaf, during the morphing process, avoiding the generation of unpleasant “floating” leaves. We demonstrate the effectiveness of our approach by creating visually compelling tree morphing animations, even between cross-species.
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Received: 12 March 2017 Accepted: 18 March 2017
DOI: 10.1002/cav.1761
SPECIAL ISSUE PAPER
Topologically consistent leafy tree morphing
Yutong Wang1Luyuan Wang1Zhigang Deng2Xiaogang Jin1
1State Key Lab of CAD&CG, Zhejiang
University, Hangzhou, China
2Computer Science Department, University
of Houston, Houston, TX, USA
Correspondence
Xiaogang Jin, State Key Lab of CAD&CG,
Zhejiang University, Hangzhou 310058,
China.
Email: jin@cad.zju.edu.cn
Funding information
National Natural Science Foundation of
China, Grant/Award Number: 61472351
Abstract
We present a novel morphing technique to generate pleasing visual effects between 2
topologically varying trees while preserving the topological consistency and botan-
ical meanings of any in-between shapes as natural trees. Specifically, we first
efficiently convert leafy trees into botanically inspired chain-lobe representations
in an automatic way. With the aid of branching-pattern aware, one-to-many cor-
respondences between branches and leaves, we hierarchically interpolate branches
of in-between trees while maintaining their topological consistencies. Finally, we
simultaneously interpolate foliage, specifically every single leaf, during the morph-
ing process, avoiding the generation of unpleasant “floating” leaves. We demonstrate
the effectiveness of our approach by creating visually compelling tree morphing
animations, even between cross-species.
KEYWORDS
shape morphing, special effects, topological consistency, tree modeling
1INTRODUCTION
During the past decades, morphing has consistently proven
its usefulness in generating breath-taking visual effects and
has become an indispensable tool in the special effects and
animation industry. Similarly, the task of morphing between
topologically varying trees opens up to efficiently creating
aesthetically special effects for many unconventional anima-
tion applications, such as nurturing the mysterious atmosphere
of the extraterrestrial environment in games and movies.
The majority of existing efforts have been put in gener-
ating morphing between man-made objects, animals, even
human portraits (please refer to Alexa1for a comprehensive
survey), and they have produced pleasing results. However,
none of them is capable of generating the morphological and
topological transformations between trees. The reasons are as
follows: (a) they do not especially establish topological-aware
correspondences between branches; (b) they do not explicitly
maintain the hierarchically consistent morphing trajectories,
resulting in disconnected “floating” branches during the mor-
phing process;(c) they do not handle the problem of gener-
ating smooth transitions between foliage. In addition to the
fluid topological and geometric transformations maintained
in the existing works, morphing between tree requires special
care on in-between trees’ botanical features, such as cohesive
branching patterns, reasonable foliage shapes, and biomass
distributions. Indeed, none of the existing methods is able to
ensure the preservation of the above high-level tree features
during the morphing process.
The main technical challenges to the tree morphing
problem are twofold: (a) establishing botanical-aware, topo-
logically consistent correspondences between different tree
species; (b) generating botanically plausible foliage morph-
ing trajectories that avoid producing floating leaves during the
morphing process.
In this paper, we propose a novel tree morphing tech-
nique, which takes two topologically varying leafy trees
as input and efficiently generates botanically sound mor-
phing results by preserving the topological consistency of
in-between trees; an example result is shown in Figure 1. To
the best of our knowledge, our work reports the first approach
that seamlessly integrates foliage morphing into the tree ani-
mation framework and demonstrates the ability of generating
smooth foliage transformations even between different tree
Comput Anim Virtual Worlds. 2017;28:e1761. wileyonlinelibrary.com/journal/cav Copyright © 2017 John Wiley & Sons, Ltd. 1of10
https://doi.org/10.1002/cav.1761
2of10 WAN G ET AL.
FIGURE 1 Morphing between a Eucalyptus and a Staphylea pinnata by our approach
species; see Section 7. Our work has three main contribu-
tions: (a) an efficient clustering algorithm to automatically
segment foliage into botanically plausible groups, that is,
lobes; (b) a branching pattern-aware correspondence estab-
lishment method to allow sound one-to-many matches among
branches; (c) a novel foliage morphing algorithm to not only
avoid producing unpleasant floating leaves but also produce
time-varying foliage details, including geometric and textural
transformations of every single leaf.
2RELATED WORK
Morphing. To date, a large number of morphing tech-
niques (both 2D and 3D) have been proposed to continuously
transform one shape into another. Two-dimensional morph-
ing techniques, including polygonal morphing2,3 and image
morphing,4suffer from the lack of the three-dimensional
information of shapes and may produce unpleasant ghost
artifacts when applied to 3D shapes. Therefore, many mor-
phing methods specialized for 3D shapes have been proposed
(please refer to Alexa1for a comprehensive survey). How-
ever, none of the existing methods has explicitly handled the
tree morphing problem. The main reason is that morphing
between trees requires the preservation of high-level botanical
information (such as branching patterns, topological consis-
tency, and botanical plausibility) of the in-between trees,
which has not been addressed in all the previous morphing
methods.
Tree animations. Despite its long history, the study of tree
animations remains to be active. Existing literature can be
roughly classified into two categories: tree growth animations
and interactions between trees and the environment. The early
X-frog system5allows users to define several keyframes that
describe the developmental stages of trees, and then inter-
polate them to generate tree growth animations. Later, Pirk
et al6automatically infer the developmental parameters from
one single static tree and then interpolates them to generate
the tree growth animations. Environmental factors, includ-
ing obstacles,7,8 lighting,8winds,9,10 and competition with
other trees11 have been studied to date to realistically ani-
mate interactions between trees and the environment. How-
ever, none of them is capable of generating fluid morphing
between topologically varying trees. The reason is three-
fold: the developmental parameters of the growth model are
species-dependent; the topologically consistent correspon-
dences between different tree species are hardly addressed;
and the smooth transformations between foliage are not con-
sidered during the interpolation.
Creative tree modeling. Several exploratory approaches,
such as previous studies,12–15 have been proposed to assist
users to efficiently create high-quality 3D shapes. The recent
approach by Wang et al.16 is the most relevant work to ours.
It utilizes blending techniques as a novel shape creation tool,
targeting on generating as many morphologically diverse trees
as possible. However, it only establishes either one-to-one
or one-to-none correspondences between branches because
of its bipartite matching algorithm. Without considering the
branching patterns, visually unpleasant branches with abnor-
mally large substructures violating the biomass distributions
might be generated during the blending process, see Figure 2.
Furthermore, they do not tackle the problem of producing a
fluid transformation between foliage.
FIGURE 2 Morphing results generated by Wang et al.16 (top row) and our method (bottom row)
WAN G ET AL.3of10
FIGURE 3 Approach overview
3APPROACH OVERVIEW
The goal of tree morphing is to generate a sequence of
in-between trees that not only fluidly transform from the
source to the target but also maintain their topological con-
sistency and botanical plausibility. Figure 3 illustrates the
main steps of our approach. Given two leafy trees (Figure 3a),
line skeletons are extracted using Oscar et al.,17 which are
further segmented into branching-pattern-aware chain groups
based on a hybrid ordering strategy. After constructing the
branching-pattern aware topology tree (the BPTT) to pre-
serve the branching patterns and hierarchical topology of
chain groups, foliage are automatically clustered into lobes,
associating leaves with chains (Figure 3b). Based on the
BPTTs, branching-pattern aware, one-to-many correspon-
dences are hierarchically established in two steps: set up
either one-to-one or one-to-none correspondences between
chain groups by solving a minimum weight bipartite matching
problem, and then build one-to-many matches among chains
within the matched chain groups using a greedy algorithm.
Meanwhile, one-to-many correspondences between leaves
within the matched chains are also established by the greedy
algorithm (Figure 3c). The in-between leafy trees are hier-
archically interpolated by maintaining their topological con-
sistency, resulting in visually pleasing morphing results; see
Figure 3(d).
4PROCESSING TREE MODELS
4.1 BPTT
Hybrid branch orderings. Line skeletons of the input trees
are first extracted using the Laplacian contraction method.17
On the basis of the classic strand model,18 we describe their
branching hierarchies using a hybrid of the gravelius and
the weibull orderings. It labels branches sharing the same
branching patterns with the same orders, contributing to an
efficient identification of branching patterns.
Branch orders are hierarchically assigned from the root
branches (with the order 1) towards tip branches. Given a
branch pwith ordering D(p), the branching angles and radii of
its child branches {ci}are computed, and the ordering D(ci)
is assigned using Equation 1:
D(ci)=Dw(ci),if 𝜎𝛼𝜃and 𝜎rrt,
Dg(ci),otherwise,(1)
where 𝜎𝛼and 𝜎rdenote the standard variance of the branching
angles and radii, respectively. 𝜃and rtare the user-defined
thresholds, and Dg(·) and Dw(·) are the gravelius and weibull
ordering functions; see Equation 2.
Dg(ci)=D(p),if 𝛼ci=𝛼min,
D(p)+1,if 𝛼ci𝛼min,
Dw(ci)=D(p),
(2)
where 𝛼ciis the branching angle and 𝛼min is the minimum
branching angle. The algorithm repeats until it reaches tip
branches.
BPTT. In real world, the growth of branches generally
exhibits one of three patterns19 (see inset). With alternative
branching, exact one branch
grows at a branching point.
In contrast, with opposite
and whorled branching, mul-
tiple branches grow at the
same branching point and are
arranged into special forms. Therefore, they should be con-
sidered as a group rather than individuals.
We use BPTT, to describe the topological hierarchies
among branches. Similar to Pirk et al.6and Wang et al.,16
consecutive branches sharing the same ordering are defined
as chains, which are then described by seven parameters listed
in Table 1. Branching patterns are identified using Equation3,
which cluster chains into groups, denoted as a set of GCi:
alternative,if #=1,
opposite,if #=2and𝛼dvg 180<𝜃,
whorled,otherwise
(3)
where #denotes the number of chains sharing the same order-
ing while growing at the same location, 𝛼dvg is the average of
divergence angles 𝛼dvg between chains (Figure 4a), and 𝜃is a
user-defined threshold, which is experimentally set to 30in
this paper.
4of10 WA N G ET AL.
TABLE 1 Parameters of chains
Name Description
chn piecewise linear representation of its geometry, including
radii information
OL the shape of the outer lobe
OR outer lobe ratio, denoting the lowest location where the
substructures grow
IL the shape of the inner lobe
IR inner lobe ratio, denoting the location where the leaves grow
vbranching vector
pgrowth location
FIGURE 4 Lobes (a) and chain parameters (b)
In the spirit of the multiscale topology tree,16 whose nodes
represent chains and edges encode their hierarchies, we take
one step further to preserve the hierarchy of branching pat-
terns and define the nodes of the BPTT as a quintuple:
Ni∶=<GCi,GOLi,GRi,Gvi,Gpi>,
where GCiis the group of chains exhibiting a certain branch-
ing pattern, GOLidenotes the group outer lobe of the group,
GRiis group lobe ratio, Gviis the average branching vector of
the chains, and Gpiis the growth location of the group. Niand
Njare connected by a BPTT edge if all the chains encoded by
Njcould find their parents in Ni.
4.2 Lobe geometry
We observe that people subconsciously tend to perceive trees’
foliage by dividing it into smaller parts according to their
branches, namely, the lobes. The existing lobe extraction
method20 usually takes nontrivial efforts to obtain an ideal
set of lobes because the thresholds differentiating the foliage
from the branches are needed to be manually set and tuned
according to the tree species. Instead, we propose an auto-
matic outer-to-inner lobe extraction method, which hierarchi-
cally groups leaves into lobes using K-means clustering.
Because leaves grow on the branches, we assume that the
foliage of a chain generally follows a crownlike shape formed
by its substructures. Three types of lobes are defined in our
approach; see Figure 4(b). The outer lobe of a chain (OL)
is defined as the leaves covering all of its substructure. The
group outer lobe (GOL), describing the general appearance
of a chain group’s foliage, equals to the union of OLsofthe
chains within the group. The inner lobe (or simply the lobe),
denoted as IL, defines the leaves growing at the tip of the
chain, that is, the actual leaves belonging to a chain.
Our goal is to extract ILs for chains. In order to main-
tain the botanical consistency, ILs are extracted hierarchically
downwards the BPTT in the following three steps.
Step1: initialization. Starting from the root node Ni
of the BPTT, the algorithm initializes its GOLlas all
the leaves in the foliage, where l
denotes the level of the node in the
BPTT, and l=0 for the root node.
Then, the algorithm collects its chil-
dren and computes the center umand
radius r
mof the crownlike shape formed by the substructures
of chains for the mth child chain group, which are the candi-
date centers and radii of their potential foliage. Besides, for
any chain chnnGCi, the algorithm assigns its potential
foliage center vnas the tip of the chain, and the radius r′′
nas
10%of the chain’s length (refer to inset).
Step 2: clustering. Leaves within the GOLlare itera-
tively divided into (m+n)groups by K-means clustering
method, where mis the number of child chain groups and n
is the number of chains contained in GCi. Initially, cluster
centers are set to be UV,whereU={u0,u1,,um}and
V={v0,v1,,vn}. Unlike the standard K-means clustering
algorithm, we associate each cluster with a distance threshold
to reduce the influence of outliers, which is experimentally
initialized as 35%of the radius r
ir′′
iand increased by 10.5%
during every iteration. For each unassigned leaf, the algorithm
measures its Euclidean distances to all the clusters and assigns
it to the nearest cluster when the distance is smaller than
the distance threshold. Then, the algorithm updates the clus-
ter centers and relaxes the distance threshold by 10.5%.The
clustering repeats until all of the leaves within the GOLlare
assigned.
Step 3: iteration. As a result, leaves assigned to clusters
that are initialized with uiUdefine the global outer lobe
GOLl+1
jof the child group and serve as the input of the next
iteration, whereas the others are the inner lobes of the chains
contained in the group GCi. Then, the algorithm moves down-
wards the BPTT and repeats the procedure for the child groups
at level l+1. The algorithm terminates when all of the leaves
are grouped into ILs.
4.3 Leaf geometry
The foliage usually consists of thousands of leaves. Due
to the large amount, we adopt the notion of billboard
clouds21 and generalize leaves into
textured quads. Specifically, a leaf is
represented as a septuple as li∶=<
chnidi,qi,wi,hi,r
i,s
i,t
i,Ti>,where
chnidirepresents the chain it grows on,
qiis the leaf position relative to its
corresponding chain, wiand hiare the
WAN G ET AL.5of10
width and height of the leaf, respectively, r
i,s
i,t
itogether
denotes the leaf’s 3D orientation, and Tidenotes its texture.
5CORRESPONDENCES
Considering the variety of branching patterns in trees, we
establish one-to-many (including one-to-one matching) and
one-to-none matchings between chains, see Figure 3(c),
where corresponding chains are rendered in the same color.
Other than constraining the matching candidates to be
nodes at the same level (Rule 1) and have matched parents
(Rule 2),16 we also require that only those chains belonging
to the matched groups are supposed to be matched in order to
maintain their topological consistency (Rule 3).
The cost of matching two BPTT nodes is measured by their
similarities (see Equation (4)):
CNi,Nj=kwk·nk
ink
j,(4)
where · denotes a distance function, nk
iand nk
jare the
kth element in node Niand Nj, respectively, and wkis the
weight term. Specifically, we use the Hausdorff distance, the
classic shape similarity measurement, to evaluate the simi-
larities between two GOLs. Similarities between branching
vectors, GRs, number of chains within the groups, and average
divergence angles are computed by the Euclidean distances.
Without loss of generality, matching nodes that violate any
of the rules is defined to be a positive infinity. Likewise,
the similarity between chains is defined by the weighted
sum of similarities between their feature parameters in
Table 1.
Similar to Wang et al.,16 we employ a top-down matching
strategy, which hierarchically establishes correspondences
based on the BPTTs. However, the correspondences at each
level are established via a two-phase protocol in order to pre-
serve their branching patterns: (1) constrained by the Rules
1 and 2, establish either one-to-one or one-to-none corre-
spondences between the BPTT nodes (i.e., the chain groups)
by solving a minimum weight bipartite matching problem as
Wan g et al .16; (2) according to the Rule 3, greedily match
chains with the most similar ones for the matched groups,
resulting in one-to-many correspondences.
Correspondences between chains also indicate matches
between their ILs. Leaves within the matched ILs are greed-
ily matched by finding the most similar ones. The similarity
between two leaves is equivalent to the weighted sum of
similarities between their leaf positions, widths, heights, and
orientations.
6MORPHING
6.1 Morphing trunks
Correspondences induce morphological transformations.
Five transformation operations are defined in this work; see
Figure 5. Inspired by Wang et al.,16 we apply morph oper-
ations to the one-to-one matched chains and grow or wilt
operations to chains, which have no corresponding sources or
targets. The transformation of one-to-many matched chains is
achieved by split or merge operations. The in-between trunks
are interpolated and then reconstructed using the method of
Alsweis et al.11
Specifically, a psplit of a chain is performed in four
steps: (a) make pcopies of the chain, denoted as SP =
{chn0,chn1,,chnp}; (b) at time t∈[0,1],segment
chniSP at the arc-length parametric location (1t)
into two parts, the chn(1t)
iand the chnt
i; (c) compute the
interpolation weights for every points n
jon each part as
w
j=b·t,if n
jchn(1t)
i,
t,if n
jchnt
i,
where b∈(0,1]is the slow-down coefficient, which is pro-
portional to the arc-length parametric value of the point n
j;
(d) perform morph operation to each copy using the point
weight w
js. Being the inverse operation of the split task, the
merge of chains is performed similarly except that we pro-
gressively speed up the morphing of the nodes nearby the
chains’ roots by calculating the interpolation weight wjusing
min(1.0,(2.0bt)when n
jchn(1t)
i.
6.2 Morphing foliage
The in-between leaves are computed by interpolating each
term of their feature vectors. The leaf position is equivalent
to the linearly interpolated parametric location on the newly
morphed chain. The width and height of the leaf are also lin-
early interpolated, and the orientations are computed through
quaternion interpolation. In order to avoid ghosting artifacts,
we separately interpolate contours and textons of the corre-
sponding leaves, and then superimpose the morphed texton on
the morphed contour and create the in-between leaf. Specif-
ically, leaf contours are extracted by the Canny boundary
FIGURE 5 Transformation operations
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FIGURE 6 Morphing between two leaves
FIGURE 7 Morphing between two Salix
FIGURE 8 Morphing between a Whitethorn acacia and an Acer japonicum
detection method.22 With the piecewise linear polygon repre-
sentations, the in-between leaf contours are interpolated using
2D polygon morphing method.3The in-between textons are
simply computed by pixel-wise linear interpolation. Although
more advanced leaf modeling method23 couldalsobeusedfor
our purpose, we experimentally found that our simple inter-
polation is sufficient to generate visually pleasing results; see
Figure 6.
7RESULTS AND EVALUATIONS
We collected 60 real-world trees, including arbors*and
shrubs, from the internet (i.e., http://www.evermotion.org/).
In addition, 10 designers’ hand-modeled virtual trees are
also incorporated into the dataset to evaluate the effec-
tiveness and flexibility of our approach. With randomly
paired inputs, our algorithm is capable of generating visually
pleasing and botanically plausible animations for both inner
species (Figure 7) and cross-species morphings (Figure 1
and Figure 8). Specifically, branching patterns are maintained
*An arbor is a woody plant that have an elongated, dominate stem, that is,
the trunk.
A shrub is a small woody plant that usually has multiple stems forming a
decurrent architecture.
during the morphing process, contributing to natural transi-
tions between matched branches; see Figure 1 and Figure 2.
Furthermore, the interpolation of leaves, including shapes
and textures, is performed in parallel to the trunk morph-
ing process based on the leaf–branch mappings, generating
smoothly transformed foliage sequences. Although the mor-
phing of flowers, fruits, and other factors are not explicitly
tackled currently, they can be interpolated in a similar way
as that of leaves, see Figure 9, where the source leafy tree
perched with butterflies smoothly transforms into the target
tree blooming purple flowers. One potential application of the
cross-species tree morphing is to enhance the supernatural
atmosphere when modeling the extraterrestrial environment
in games and movies. Please refer to our animation results in
the supplementary video.
Runtime statistics. We have implemented our morph-
ing algorithm in C++ on a desktop equipped with Intel©
Core i7 clocked at 3.50 GHz, 8 GB of RAM, and NVIDIA©
Geforce GTX 660 GPU. The runtime statistics are presented
in Table 2. In sum, our method demonstrates its efficiency
of generating smooth morphing results on an off-the-shelf
computer.
Comparison with topological-aware tree-blending
method. Figure 2 compares our approach with the most rele-
vant work by Wang et al.,16 which generates diverse inspiring
novel trees by blending among topologically varying trees.
WAN G ET AL.7of10
FIGURE 9 A morphing scene, where the source leafy tree perched with butterflies smoothly transforms into the target tree blooming purple flowers
TABLE 2 Runtime statistics
Time
Input statistics Preprocessing Set up Morphing
Examples # of chains # of leaves Ordering (s) Lobe extraction (s) matches (s) (per frame, s)
Figure 1 5,220/20,195 4,690/5,919 1.82 3.12 0.49 11.35
Figure 3 4,507/4,590 4,436/3,765 0.61 0.99 0.19 1.52
Figure 8 5,660/4,370 2,937/2,844 1.34 0.97 0.55 7.31
Figure 7 7,782/21,138 6,459/2,874 4.39 1.81 0.85 7.92
Figure 9 5,224/8,272 2,351/5,564 1.64 1.61 0.81 10.08
In contrast to massively growing branches from the main
trunk using the method in Wang et al.16 (top row), our
algorithm gradually splits the source main trunk into mul-
tiple stems, producing a more natural morphing sequences
(bottom row).
In addition to visual comparison, we quantify the “natu-
ralness” of the morphing sequences using the trunk support
weight in the well-known pipe model theory.24 Denoted as
T(z), it defines the weight sustained by a trunk as all the
weights of leaves and branches from the top of its substruc-
tures to a given location zalong the trunk. Because people
are sensitive to the dominant trunks of trees, we compute
the T(OR)s for the main trunks of every in-between tree and
take the heaviest weight as the maximum main trunk support
weight (MTSW). Without loss of generality, we normalize
the MTSW by the total weight of the tree. Figure 10 plots
the MTSWs of the in-between trees generated by both meth-
ods against the morphing process. Evidently, our method (red
FIGURE 10 Comparison of the MTSWs of the morphing sequences
generated by our method and11
plot) smoothly “shifts” the weight of leaves and branches sus-
tained by the source main trunk to multiple target stems under
the splitting operations. By contrast, the results by Wang
et al.16 (blue plot) shows steep weight drops at the interval of
[30%,40%]and a slow raise at the interval of [70%,100%].
8of10 WA N G ET AL.
FIGURE 11 Comparison between our method (top) and the baseline method (bottom)
The above can be explained by our branching pattern pre-
serving one-to-many correspondences (see the red circled
trees in Figure 2), where the multiple stems exhibiting the
whorled branching patterns are grouped and interpolated at
the same time during the whole morphing process, leading
to naturally sharing of the trees’ weights. In contrast, the
correspondence establishment scheme in Wang et al.16 only
allows either one-to-one or one-to-none matches between
branches; see the blue circled tree in Figure 2. Therefore,
two other unmatched main stems should be interpolated by
the grow operations during the morphing process. Because of
their young age and small radii, the gradually grown chains
are supported by the main trunk during early morphing time
(e.g., [0%,30%]).Inthemiddletime(e.g.,[30%,40%]), when
their radii reach a certain threshold, the ordering strategy
marks them as main stems. As a result, they share the weight
originally supported by the main trunk, leading to the sudden
drop of the MTSW. Because the tree weight are nonuni-
formly shifted to growing stems, the MTSWs might be lower
than the target ones, see the MTSW values at the later time
(e.g.,[70%,100%]). In a nutshell, our method not only pre-
serves branching patterns during the morphing process but
also maintain a smooth MTSW change, contributing to more
natural and pleasant animations compared to Wang et al.16
Comparison with baseline foliage morphing method.
Because few, if any, existing works have put explicit efforts
on the smooth morphing between foliage, we implemented
a naive foliage morphing method as the baseline by linearly
interpolating randomly matched leaves and compared it with
our method, see Figure 11. Apparently, the baseline method
produces floating leaves (bottom row, circled leaves), leading
to noticeable visual artifacts.
8CONCLUSION
In this paper, we present the first approach to efficiently gen-
erate smooth morphing sequences between two leafy trees.
Unlike the conventional tree modeling techniques,6,8,16 which
usually emphasize trunk modeling but treat foliage model-
ing as postprocessing operations, we seamlessly integrate
trunk morphing and foliage morphing into a unified animation
framework and generate a novel type of visual special effect.
We also develop an automatic foliagesegmentation method to
efficiently associate leaves with branches using the structure
lobe. Our method is advantageous in preserving both geomet-
ric and topological consistency of the in-between trees during
the morphing process. Therefore, it is highly flexible to create
visually compelling morphing effects between topologically
varying leafy trees.
Limitations and future work. Our current method still
leaves room for improvement. Because our goal is to gener-
ate visually pleasing morphing effects, issues such as smooth
ramifications25 and collision-free branches26 are not explic-
itly addressed. In terms of the lobe extraction, environmental
factors, such as lighting conditions, wind effects, and so forth,
should be considered in the future. In addition, we also plan to
incorporate user interaction to our morphing pipeline, where
users may control the general crown shapes of the in-between
shapes to meet their artistic needs.
ACKNOWLEDGEMENTS
We thank the anonymous reviewers for their constructive
comments that helped improve this paper. Xiaogang Jin was
supported by the National Natural Science Foundation of
China (Grant 61472351).
REFERENCES
1. Alexa M. Recent advances in mesh morphing. Comput Graphics
Forum. 2002;21(2):173–198.
2. Carmel E, Cohen-Or D. Warp-guided object-space morphing. The
Visual Comput. 1998;13(9):465–478.
3. Liu L, Wang G, Zhang B, Guo B, Shum Heung-Yeung. Perceptually
based approach for planar shape morphing. Proc Pacific Graphics.
2004:111–120.
WAN G ET AL.9of10
4. Wolberg G. Image morphing: a survey. The Visual Comput.
1998;14(8):360–372.
5. Lintermann B, Deussen O. Interactive modeling of plants. IEEE
Comput Graphics Appl. 1999;19(1):56–65.
6. Pirk Sören, Niese T, Deussen O, Neubert B. Capturing and ani-
mating the morphogenesis of polygonal tree models. ACM Trans
Graphics (TOG). 2012;31(6):169.
7. Palubicki W, Horel K, Longay S, et al. Self-organizing tree models
for image synthesis. ACM Trans Graphics (TOG). 2009;28(3):58.
8. Pirk S, Stava O, Kratt J, et al. Plastic trees: interactive self-adapting
botanical tree models. ACM Trans Graphics. 2012;31(4):50.
9. Zhao Y, Barbiˇ
c J. Interactive authoring of simulation-ready plants.
ACM Trans Graphics (TOG). 2013;32(4):84.
10. Pirk S, Niese T, Hädrich T, Benes B, Deussen O. Windy trees: com-
puting stress response for developmental tree models. ACM Trans
Graphics (TOG). 2014;33(6):204.
11. Alsweis M, Deussen O. Procedural techniques for simulating the
growth of plant leaves and adapting venation patterns. Proceed-
ings of the 21st ACM Symposium on Virtual Reality Software and
Technology; 2015. p. 95–101.
12. Talton JO, Gibson D, Yang L, Hanrahan P, Koltun V. Exploratory
modeling with collaborative design spaces. ACM Trans Graphics
(TOG). 2009;28(5):167.
13. Jain A, Thormählen T, Ritschel T, Seidel H-P. Exploring shape
variations by 3d-model decomposition and part-based recombina-
tion. Comput Graphics Forum. 2012;31(2pt3):631–640.
14. Guo X, Lin J, Xu K, Jin X. Creature grammar for creative modeling
of 3d monsters. Graphical Models. 2014;76(5):376–389.
15. Guo X, Lin J, Xu K, Chaudhuri S, Jin X. Customcut: On-demand
extraction of customized 3d parts with 2d sketches. Comput Graph-
ics Forum. 2016;35(5):89–100.
16. Wang Y, Xue X, Jin X, Deng Z. Creative virtual tree model-
ing through hierarchical topology-preserving blending. IEEE Trans
Visual Comput Graphics. 2016. https://doi.org/10.1109/TVCG.
2016.2636187.
17. Au OscarK-C, Tai C-L, Chu H-K, Cohen-Or D, Lee T-Y. Skele-
ton extraction by mesh contraction. ACM Trans Graphics (TOG).
2008;27(3):44.
18. Holton M. Strands, gravity and botanical tree imagery. Comput
Graphics Forum. 1994;13(1):57–67.
19. Dirr MA. Manual of woody landscape plants: their identification,
ornamental characteristics, culture, propagation and uses, stipes
publ. Co., Champaign, Illinois 554; 1990.
20. Livny Y, Pirk S, Cheng Z, et al. Texture-lobes for tree mod-
elling. ACM Trans Graphics (Proc SIGGRAPH 2011). 2011;30(4):
53.
21. Behrendt S, Colditz C, Franzke O, Kopf J, Deussen O. Realistic
real-time rendering of landscapes using billboard clouds. Comput
Graphics Forum. 2005;24(3):507–516.
22. Canny J. A computational approach to edge detection. IEEE Trans
Pattern Anal Mach Intell. 1986;8(6):679–698.
23. Mundermann L, MacMurchy P, Pivovarov J, Prusinkiewicz P.
Modeling lobed leaves. Proceedings of CGI, 2003. p. 60–65.
24. Shinozaki K, Yoda K, Hozumi K, Kira T. A quantitative analysis
of plant form-the pipe model theory: I. basic analyses. Jpn J Ecol.
1964;14(3):97–105.
25. Zhu X, Jin X, You L. High-quality tree structures modelling
using local convolution surface approximation. The Visual Comput.
2015;31(1):69–82.
26. Kim Y-J, Woo J-H, Kim M-S, Elber G. Interactive tree modeling
and deformation with collision detection and avoidance. Comput
Anim Virtual Worlds. 2015;26(3-4):423–432.
Yutong Wang received her BSc
degree in software engineer-
ing from Chongqing University,
China, in 2012. She is currently
a Ph.D candidate at the State Key
Lab of CAD&CG, Zhejiang Uni-
versity. Her main research inter-
ests include creative modeling,
computer animation, and sketch-based modeling.
Luyuan Wang received her
BSc degree in digital media
from Hunan University, China,
in 2016. She is currently a Ph.D
candidate at the State Key Lab
of CAD&CG, Zhejiang Univer-
sity. Her main research interests
include creative modeling and
virtual try-on.
Zhigang Deng is currently a Full
Professor of Computer Science
at the University of Houston
(UH) and the Founding Director
of the UH Computer Graphics
and Interactive Media (CGIM)
Lab. His research interests
include computer graphics,
computer animation, virtual human modeling and ani-
mation, and human computer interaction. He earned
his Ph.D. in Computer Science at the Department
of Computer Science at the University of Southern
California in 2006. Prior to that, he also completed
B.S. degree in Mathematics from Xiamen Univer-
sity (China) and M.S. in Computer Science from
Peking University (China). He is the recipient of a
number of awards including ACM ICMI Ten Year
Technical Impact Award, UH Teaching Excellence
Award, Google Faculty Research Award, UHCS
Faculty Academic Excellence Award, and NSFC
Overseas and Hong Kong/Macau Young Scholars
Collaborative Research Award. Besides being the
CASA 2014 Conference General Co-chair and SCA
2015 Conference General Co-chair, he currently
serves as an Associate Editor of several journals
including Computer Graphics Forum and Computer
Animation and Virtual Worlds Journal. He is a senior
member of ACM and a senior member of IEEE.
10 of 10 WAN G ET AL.
Xiaogang Jin is a professor of
the State Key Lab of CAD&CG,
Zhejiang University, China.
He received his BSc degree in
computer science in 1989, MSc
and PhD degrees in applied
mathematics in 1992 and 1995,
respectively, all from Zhejiang
University. His current research interests include traffic
simulation, insect swarm simulation, physically based
animation, cloth animation, special effects simula-
tion, implicit surface computing, computer-generated
marbling, nonphotorealistic rendering, and digital
geometry processing. He received an ACM Recogni-
tion of Service Award in 2015. He is a member of the
IEEE and ACM.
How to cite this article: Wang Y, Wang L, Deng
Z, Jin X. Topologically Consistent Leafy Tree Mor-
phing. Comput Anim Virtual Worlds. 2017;28:e1762.
https://doi.org/10.1002/cav.1761

Supplementary resource (1)

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