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This paper proposes a computational approach to seasonal changes of living leaves by combining the geometric deformations and textural color changes. The geometric model of a leaf is generated by triangulating the scanned image of a leaf using an optimized mesh. The triangular mesh of the leaf is deformed by the improved mass-spring model, while the deformation is controlled by setting different mass values for the vertices on the leaf model. In order to adaptively control the deformation of different regions in the leaf, the mass values of vertices are set to be in proportion to the pixels' intensities of the corresponding user-specified grayscale mask map. The geometric deformations as well as the textural color changes of a leaf are used to simulate the seasonal changing process of leaves based on Markov chain model with different environmental parameters including temperature, humidness, and time. Experimental results show that the method successfully simulates the seasonal changes of leaves.

The seasonal changes of trees vary the appearances of trees through seasons, which include shapes and textures of the leaves, flowers, and fruits. Among these, the change of leaves constitutes the most important part of the seasonal changes of trees. In this paper, we focus on how to compute the leaf changing during different seasons.

As we observe the changes of leaves from spring to winter, most leaves become withered and curled up due to the influences of environmental factors [

In order to simulate textural colors of leaves, the Phong lighting model with a diffuse component derived from leaf pigments is adopted to directly compute the reflections on the surfaces of leaves [

In order to efficiently simulate the seasonal changes of leaves, we combine the changes of geometric shape and textural color of the above methods in our algorithm to produce the results. The Markov chain model is used to show the state transfer of leaves in the dynamic growing process of trees. The following sections are arranged as follows. In Section

The work related to the simulation of seasonal changes of leaves includes leaf modeling, leaf deformation, and leaf appearances rendering. For leaf modeling, there are L-system-based and image-based methods. The L-system-based methods model leaves with self-similarity [

The leaves gradually become withered and curled up during the transitions of different seasons. The deformation of geometric shapes of leaves is very important to simulate the seasonal changes. The 3D deformation algorithms are mainly classified into two categories, which are free-form-based deformation methods [

In order to simulate color changes of leaf surfaces in various environmental conditions, Phong lighting model considering leaf’s pigments [

In this paper, we apply the image-based approach to model the geometric shapes of three-dimensional leaves [

Instead of adopting the automatic edge detection methods to extract the leaf contour, we provide the interface to make the user interactively select the edge points of the leaf. After the selection of edge points, the smooth B-spline curve running through these points is automatically generated to approximate the leaf edges [

(a) The B-spline curve with key points selected by the user; (b) the Delaunay triangulated mesh of the leaf.

The Delaunay triangulation method is usually used to generate a triangulated irregular network (TIN) [

The high-resolution triangular mesh produces more natural and smooth deformations. However, more triangles in the mesh would lead to more time to compute the deformation. According to the triangulation algorithm, the subdivision level of triangular mesh is related to the number of iterations. Usually, we set the number of iterations to be 160 in our implementation, which is enough to produce the subdivided triangular mesh capable of natural deformation within acceptable time. In Figure

Triangular meshes of the maple leaf produced by a different number of iterations.

Leaves become slowly curled up as the season changes. This phenomenon is mainly caused by the different structures of the upper and bottom surfaces of a leaf, which have different amounts of contraction during the dehydration process. To take into account the differences between the upper and bottom internal structures of a leaf, we introduce the improved mass-spring model to make leaf deformation more realistic.

The mass-spring model is widely used in the simulation of the deformation of soft fabrics [

There are internal and external forces acting on the springs, and we denote the joined forces as

In the above equations, the mass of a particle is denoted as

Actually, the deformation curve of a leaf under forces is not ideally linear. If we directly compute the deformation with the above equations, the problem of “over elasticity” would occur, that is, the deformation of the springs would exceed 100%. To overcome this problem, we adopt the method of constraining velocities to constrain the deformation of the springs [

The key of shape deformation is to compute the changes of the position of each particle. If each particle has the same mass value, the relative displacements in directions

According to Newton’s law of motion

(a) The texture of a maple leaf; (b) mask map of the maple model.

According to the texture coordinates of the particles of triangular mesh, we find in the mask map the pixels which correspond to particles in the leaf mesh model. The gray values of pixels in the mask map are mapped to the value of particle masses

In (

The detailed steps to implement deformation process are shown as follows.

Generate the

Initialize parameter values in our improved mass-spring model. Set the initial velocity and acceleration of particles to be zero. Initialize masses of the particles according to the

Establish constraints among particles. The connection between particles (i.e., the mesh topology) determines what other particles directly exert forces on the current particle for the computation of displacements. The constraints are built by three steps as follows.

Exert the force, and compute the change of position of each particle by numerical calculation in one time step.

Repeat the numerical calculation in each time step to obtain the new velocities and accelerations, and update particle positions accordingly to produce deformation effects at different time steps.

For example, the deformation results at different time steps of the maple leaf under the

Several deformations using the mask map in Figure

The deformation results in Figure

Another mask map of the maple leaf model.

Different deformation results of the maple leaf for mask map shown in Figure

To simulate the seasonal changes of leaves, we need to take the transitions of textural colors of leaves into account besides geometric deformations. The whole seasonal changing process of leaves can be regarded as the sequences of a series of discrete states. The leaves transform from one state to the other with certain probabilities conditioned by environmental factors. This transformation can be approximated by Markov chain model [

Markov chain model has two properties.

The leaf’s state is denoted as

Transition relationship for Markov chain model.

The arc

Function

The probability that the leaf transfers to other states is denoted as

Function

The parameters of time, temperature, and humidness are set by users. Taking the maple leaves in Figure

Seven texture states of a maple model.

Several states which combine changes of textures and shapes in different seasons are showed in Figure

The basic triangular mesh model of the maple leaf, and seven states combining textures and geometric deformations.

To summarize, the seasonal changing process of leaves under certain environmental parameters is showed in Figure

Seasonal changing process of leaves based on Markov-chain model.

To produce the results of seasonal changes of trees, we grow the leaves on the trees and simulate their distributions for different seasons. In order to get the 3D model of the tree, we adopt the L-system method to produce the trunks and branches of the tree. The trunks and branches of the tree are drawn with quadratic surface, and the leaves grown on branches are modeled as triangular meshes. In Figure

Tree growing process based on L-system.

Seasonal changes of a maple tree based on Markov chain model.

In this paper, we propose a computational approach to simulate the seasonal changes of living leaves by combining the changes in geometric shapes and textural colors. First, the key points are selected on the leaf image by user interaction. Then, the triangular mesh of the leaf is constructed and optimized by improved Delaunay triangulation. After the models of leaves have been obtained, the deformations of leaves are computed by improved mass-spring models. The seasonal changes of trees under different environmental parameters are computed based on Markov chain. The improved mass-spring model is based on the user-specified

In the future, we are interested in the following work.

Work on how to generate the mask map more naturally according to the characteristics of the deformations of leaves.

Intend to simulate the dynamic procedure of the leaves falling onto ground out of gravity.

Develop a more precise model to compute the colors of leaves which takes into account of the semitransparency of leaves.

This work is supported by National Natural Science Foundation of China (61173097, 61003265), Zhejiang Natural Science Foundation (Z1090459), Zhejiang Science and Technology Planning Project (2010C33046), Zhejiang Key Science and Technology Innovation Team (2009R50009), and Tsinghua-Tencent Joint Laboratory for Internet Innovation Technology.