USING PHYLLOTAXIS FOR DATE PALM TREE

3D RECONSTRUCTION FROM A SINGLE IMAGE

Ran Dror, Ilan Shimshoni

University of Haifa, Haifa, Israel

Keywords:

Phyllotaxis, 3D reconstruction, Palm tree.

Abstract:

Phyllotaxis is the study of the morphological order of plants. Remarkably, in spite of the overwhelming

diversity of plant morphology, there are common patterns that link a wide variety of species. The date palm,

having a phyllotactic order, possesses a simple, repetitive model. Only a small number of parameters are

needed to represent the phyllotactic order of the date palm. This a priori knowledge we have on the date

palm can help in the 3D reconstruction of the tree and can even make it possible to reconstruct a 3D model

from only one image. The proposed algorithm receives as input a single image of the date palm. Upon image

acquisition, the algorithm proceeds to search for, and locate, the trunk followed by a few prominent leaves.

From the location of the prominent leaves the algorithm proceeds to calculate tree model parameters, which

can then be used to search for additional, neighboring, leaves. Complete 3D reconstruction is achieved by

utilizing the calculated tree model parameters and by the known location of the leaves on the 2D image.

1 INTRODUCTION

Phyllotaxis

- a central area of research in plant mor-

phogenesis (Steeves and Sussex, 1989; Jean, 1994),

is the study of the arrangement of repeating units in

plants. For example, leaves around a stem, scales on

a pine cone or on a pineapple, the seeds in a sun-

ﬂower head, etc. (see Figure 1). Remarkably, in spite

of the overwhelming diversity of plant morphology,

there are common patterns that link a wide variety of

species. Though the study of phyllotaxis is traced to

the ﬁrst primitive observations in ancient times, it is

still a very active ﬁeld today (Adler et al., 1997; Smith

et al., 2006; Reinhardt, 2005).

Within the three main types of phyllotactic pat-

terns found in nature, the most prevalent is the spiral

phyllotactic pattern. This pattern appears in the ma-

jority of the 250,000 or more species of higher plants

(Cummings and Strickland, 1998). The phyllotactic

patterns are formed at the microscopic level at the

growing tip of the plant - the apical meristem (see

Figure 2). Botanical units such as leaves and petals

are generated at the meristem as bulges of fast grow-

ing cells known as primordia. As more primordia de-

velop, they are pushed farther and farther from the

from Greek - phyllo (leaf) and taxis (order)

apex thus developing into the familiar features of the

plant, be it a leaf, a ﬂower, or parts of a fruit. In spiral

phyllotaxis the angle between consecutive born pri-

mordia, called the divergence angle, is constant and

close to the Fibonacci angle of 137.5

◦

(a) (b) (c)

Figure 1: Examples of phyllotactic patterns in plants: (a) -

Echevaria subsseilis, (b) - Pineapple, (c) - Marguerite. (im-

ages from (Atela and Gol

e, 2008))

Spiral phyllotaxis is characterized by conspicuous

spirals, or contact parastichies, formed by sequences

of adjacent organs composing the structure (see Fig-

ure 2). Curiously, the numbers of parastichies run-

ning in opposite directions are usually two consecu-

tive Fibonacci numbers. The difference between age

indexes of two neighboring organs in the parastichy

288

Dror R. and Shimshoni I.

USING PHYLLOTAXIS FOR DATE PALM TREE 3D RECONSTRUCTION FROM A SINGLE IMAGE.

DOI: 10.5220/0001773802880296

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications (VISIGRAPP 2009), page

ISBN: 978-989-8111-69-2

spiral indicates the number of parastichy spirals.

Figure 2: The apical meristem of A Norway spruce (Picea

abies) (Electron micrograph from (Rutishauser, 1988)). Pri-

mordia are numbered according to their age - the higher the

number, the older the primordium. Three of the eight right

parastichies spirals are illustrated in green, two of the thir-

teen left parastichies spirals are illustrated in purple. The

divergence angle between primordia six and seven which is

equal to the Fibonacci angle is marked in the center.

While phyllotactic order is common in the ar-

rangement of seeds in ﬂowers and leaves around a

stem, the branching of the trunk and branches is al-

most unpredictable. In contrast to other trees, both

date and oil palms do not have branched trunks,

rather incorporating a single, non-branching, trunk

with leaves arranged in spiral phyllotactic order its

entire length.

This simple and consistent arrangement makes it

possible to describe these trees by means of a simple

model. The proposed algorithm uses the aforemen-

tioned model to reconstruct a 3D image of the tree.

It receives as input a single image of the date palm.

Upon image acquisition, the algorithm proceeds to

search for, and locate, the trunk followed by a few

prominent leaves. From the location of the promi-

nent leaves the algorithm calculates tree model pa-

rameters, which are then used to search for additional,

neighboring, leaves. The 3D reconstruction is done

using the tree model parameters and the location of

the leaves on the 2D image.

The palm tree was chosen because of its economic

importance and possible future practical implementa-

tions. There are two species of the palm family which

have economic importance, the oil palm and the date

palm. Palm oil is now the most widely produced veg-

etable oil in the world, demand for which is expected

to climb even further as one of the raw materials for

biodiesel. The date palm is one of the most impor-

tant economic tree crops that thrive in the desert re-

gions of the world. In order to reduce labour costs

and the risk of working at the top of the palm trees, a

number of robotic prototypes for maintenance of palm

trees were developed (Aracil et al., 1999; Ripin et al.,

2000; Shamsi, 1998). This study proposes a further

step in that it will include automatic vision algorithms

to guide these robotic systems.

2 RELATED WORK

The challenging problem of 3D reconstruction of

trees has been studied in (Shlyakhter et al., 2001;

Martinez et al., 2004). A number of calibrated im-

ages from different views of the tree are segmented

into the tree, and the background. The background

segments are discarded and the tree segments are then

compiled to reconstruct a 3D model resembling the

speciﬁc tree. The target of these works was render-

ing trees in virtual environments, whereas in our work

we try to measure the tree skeleton from only one

non calibrated image. Segmentation in these works is

done either manually (Shlyakhter et al., 2001), or for

cases where the tree background is distinct, such as

sky background (Martinez et al., 2004). More sophis-

ticated tree segmentation is done using 51 values mea-

sured at each pixel in (Haering and da Vitoria Lobo,

1999).

In a recent study (Teng et al., 2006) trunk structure

is extracted. The main challenge in this work is to deal

with the general trunk structure. In our case however,

the trunk is cylindrical.

Phyllotactic models are used in the ﬁeld of com-

puter graphics for simulating realistic plant images.

An overview of these models is presented in Section

3. In a recent study (Kaewapichai et al., 2007) the

phyllotactic model was used to ﬁt the arrangement of

the scales on a pineapple.

3 PALM TREE MODEL

Several mathematical phyllotactic models were devel-

oped by computer graphics researchers for the pur-

pose of simulating realistic plant images. Synthe-

sized images of plant structures with predominantly

ﬂat, elongated or spherical geometry were created in

(Prusinkiewicz and Lindenmayer, 1990; Fowler et al.,

1989; Lintermann and Deussen, 1999) using three

USING PHYLLOTAXIS FOR DATE PALM TREE 3D RECONSTRUCTION FROM A SINGLE IMAGE

289

models: the planar model, the cylindrical model, and

the spherical model. These models relate phyllotaxis

to the packing problem of equally-sized organs.

Many phyllotactic plants do not ﬁt into these mod-

els. Phyllotactic plants with a variety of organ sizes

and surface shapes were dealt with in (Fowler et al.,

1992). In this model plant organs are procedurally

placed on the plant’s surface.

The palm tree has a spiral phyllotactic pattern.

Characteristics of the date palm phyllotaxis have been

studied in (Elhoumaizia et al., 2002; Ferry, 1998). In

(Elhoumaizia et al., 2002) it was found that the di-

vergence angle of consecutive leaves is similar for the

same cultivar and is approximately the Fibonacci an-

gle of 137.5

◦

. The handedness of the date palm is

the direction in which the leaves are born, clockwise

or counter clockwise. The frequencies of right or left

handedness in this study were more or less equal. Oil

palm measurements, testing for the vertical distance

between consecutive leaves, was measured in (Rees,

1964), showing that the horizontal distance between

leaves changes only gradually along the length of the

trunk.

The arrangement of the stubs of pruned leaves on

the trunk of the palm tree can be modelled accurately

by the cylindrical model. Arrangement of the leaves

on the tree crown, on the other hand, deviates from the

cylindrical model because the radius of the trunk on

the crown decreases. In this paper we used the cylin-

drical model with adaptations allowing for changes in

the trunk’s radius. This model is relatively accurate,

while maintaining simplicity and offering an analyti-

cal solution.

3.1 Mathematical Model Description

The mathematical model of the palm tree describes

both leaf growth locations and leaf growth angles,

on the trunk. Leaf growth locations are described

in cylindrical coordinates (θ, r, H) with respect to the

trunk, where θ is the angle, r is the radius, and H is

the height. Leaves are numbered according to their

age. The youngest leaf, situated at the top of the tree,

is marked as number 1 with consequent leaves being

marked according to the order of descent along the

tree trunk. The earlier the leaf budded - the higher the

number assigned to that speciﬁc leaf.

Being a leaﬁng plant, the phyllotactic order of the

palm tree is not perfect, but still the location of the leaf

can be predicted quite accurately from the location

of its neighbour. Therefore, the phyllotactic model is

deﬁned recursively. The location of leaf number n + i

can be calculated from the location of leaf n by the

Figure 3: Model of the “skeleton” of the palm tree created

using the mathematical model.

following equations:

n+i

= θ

+ i · h · ψ (1)

n+i

= R

table

(n + i) · R (2)

n+i

= H

− i · d (3)

n+i

= α

table

(n + i) (4)

where:

• h - The handedness of the date palm, 1 for clock-

wise, -1 for counter clockwise.

• ψ - The Fibonacci angle of 137.5

◦

• R

table

( j) - A table with the ratio of trunk radius at

the growing point of leaf number j, to the radius

of the trunk at its widest point, R. The table was

measured from a reference tree assuming that this

value is representative.

• d - The vertical distance between two consecutive

leaves.

• α

n+i

- Leaf growing angle for leaf number n + i.

• α

table

( j) - A table with leaf growth angles for leaf

number j . The data was acquired from a reference

tree by measuring images taken from a perpendic-

ular angle to the leaf.

Only two parameters are unique for every tree, the

handedness of the tree h and the vertical distance be-

tween two consecutive leaves d. Knowing these pa-

rameters, the location of one leaf, and its age index,

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

290

we can predict the locations of its neighbors. All val-

ues are computed proportional to the radius of the tree

whose value can not be recovered from the image.

Figure 3 shows a model of the palm tree created using

the aforementioned mathematical model.

The mathematical model describes a spiral phyl-

lotactic pattern, which can have different parastichy

numbers according to r and d. But still for most date

palm cultivars, including the one considered here, the

parastichy numbers are 5 and 8 (Figure 4).

Another assumption of our model is that the leaves

grow outward from the axis of the trunk. In other

words the leaf midribs are located on a plane con-

taining the trunk axis and the leaf growing point (see

Figure 5).

(a) (b)

Figure 4: Contact parastichy pattern for (a) right handed

tree and (b) a left handed tree.

Figure 5: Leaves grow outward from the axis of the trunk.

4 ALGORITHM DESCRIPTION

An overview of our proposed algorithm is summa-

rized above. The different stages are summarized in

the following subsections and the output of each stage

is illustrated in Figure 6. The input of the algorithm

is an image of a palm tree, Figure 6(a) is an example

of such an image.

• Calculating a per pixel probability image of the

leaves and trunk (Figure 6(b), Subsection 4.1).

• Searching for the location of the trunk and “tree

center” using the aforementioned images (Figure

6(c), Subsection 4.2).

• Creating, using the leaf probability image and the

“tree center”, a leaf clues image (Figure 6(d), Sub-

section 4.3).

• Search for prominent leaves using the leaf clues

image (Figure 6(e), Subsection 4.4).

• Estimation of the model parameters by using the

location of the prominent leaves and their growth

angles (Figure 6(f), Subsection 4.5).

• Search for more leaves using prediction based on

the model parameters (Figure 6(g), Subsection

4.6).

• 3D reconstruction of the palm tree is achieved by

use of the tree model parameters and the location

of the leaves on the 2D image (Figure 6(h), Sub-

section 4.7).

4.1 Calculating Probability Image of

the Leaves and Trunk

The interesting part of the leaf of the date palm is its

midrib - the central “spine” of the leaf. These midribs

and the trunk are the tree’s “skeleton”. The leaﬂets

can be added easily to this “skeleton”, creating a com-

plete model of the tree. The prominent feature of the

leaves is their green color, while the distinctive fea-

ture of the midrib is its smoothness - low level of gra-

dient. Generating the leaf midrib probability image -

, denoted as the “leaf probability image”, is based

on simple histogram learning techniques described in

(Jones and Rehg, 2002). Midrib and non-midrib his-

tograms were collected using a training set of manu-

ally labelled images. Given these histograms a four

dimensional table with the probability that a given

color and gradient deﬁnes the leafs’ midrib is calcu-

lated (see Figure 7). The table is applied to the origi-

nal image to get the leaf probability image. The size

of the histogram is 30 bins/channel for the HSV chan-

nels, and 10 bins/channel for the gradient. In practice,

USING PHYLLOTAXIS FOR DATE PALM TREE 3D RECONSTRUCTION FROM A SINGLE IMAGE

291

(a) Original image. (b) leaf and trunk proba-

bility images.

“tree center”.

(d) Leaf clues image.

(e) Prominent leaves

search.

(f) Model prediction. (g) Search for more

leaves.

(h) VRML 3D model.

Figure 6: Output of algorithm stages. (e)-(g) show a zoomed in section of the original image.

the combination of color and smoothness has proved

to be both informative and robust.

The dominant feature of the trunk is its color. The

trunk probability image - P

is calculated in the same

way as the leaf probability image only without con-

sidering the gradient. The leaf and trunk probability

images are illustrated in Figure 6(b) in green and blue

respectively.

4.2 Search for Trunk Location and

“Tree Center”

The location of the trunk is represented by three pa-

rameters: the horizontal location on the image in pix-

els - x, trunk radius in pixels - r, and the leaning angle

of the trunk toward the camera - γ (see Figure 8(a)).

Assuming that the image was taken nearly parallel to

the horizon, and that the palm trunk is perpendicular

to the ground, knowing these three parameters, en-

ables the calculation of the 3D location of the trunk,

and the transformation between location on the 3D

trunk to the image and vice versa. We deﬁne the tree

center as the point on the line of symmetry of the

trunk, with the vertical location on the image in pix-

els - y, immediately beneath the oldest leaf (see Figure

Figure 7: 3D “slice” of the 4D leaf probability table, deﬁnes

the probability of a given color with gradient size: 6-8. The

probability is illustrated by a ball whose size is proportional

to the probability.

8(a)). We call it the tree center because it is almost the

intersection point of all the leaves midribs line. The

tree center will be used later for the removal of leaf

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

292

clues not in the direction “intersecting” with the tree

center.

Estimation of the four trunk location and tree cen-

ter parameters (x, r, y, γ) is done by searching for the

best location of a schematic model of the tree on both

the trunk and the leaf probability images. The model

is illustrated in Figure 8(b). Assuming that γ is not

too large, the trunk can be represented by a vertical

rectangle on the image plane. The “trunk rectangle”

is separated into two rectangles, D - descending and

U - ascending from the common “tree center”. Rect-

angles R and L represent the outer border of the trunk,

an area that should have pixels with low probability

in the trunk probability image. A leafy area is repre-

sented by rectangle M which should have a high prob-

ability for pixels in the leaf probability image. The

location and size of these rectangles are functions of

x, r and y.

For performance reasons the search is done in

phases. The ﬁrst phase searches horizontally for x

and r, using the “Integral Image” algorithm (Viola and

Jones, 2001) it maximizes the energy function:

(x, r) = ( f (U ∪ D, P

) − f (L , P

) − f (R, P

)) · r

(5)

where f (A, P) =

∑

(i, j)∈A

i, j

P(i, j)

represents the average

probability of the pixels in the rectangle A on the

probability image P. The term r

was added to pre-

vent very thin trunks from being accepted. For every

local maximum a search for y is done using pattern

search (Lewis and Torczon, 2000) by maximizing:

(x, r, y) = ( f (M, P

) + f (D, P

) − f (U, P

)). (6)

The best (x, r, y) combination is chosen by maximiz-

ing:

E(x, r, y) = E

(x, r) · E

(x, r, y). (7)

Finally the best γ is chosen. Both left and right trunk

line-boundary angles are searched for in order to max-

imize the derivative over the trunk probability image.

The intersection point of these lines is used in order

to calculate γ. Results of the search for trunk loca-

tion and tree center are illustrated in Figure 6(c). The

tree center is marked by a white cross, and the trunk

borders are drawn in magenta.

4.3 Creating the Leaf Clues Image

The leaf clues image is created from the leaf proba-

bility image - P

by thresholding. After thresholding,

groups of connected pixels that we call leaf clues re-

main. The dominant clue direction is calculated using

PCA. Clues whose extensions do not intersect with

a circle around the tree center (whose radius is

· r)

are removed. Figure 6(d) shows an example of a leaf

clues image, with the tree center marked.

(a) (b)

Figure 8: (a) Trunk location and “Tree Center” parameters

description. (b) Rectangles U, D, R, L, and M, used in the

search for the trunk location and “Tree Center”.

4.4 Search for Prominent Leaves

The purpose of prominent leaf search is to ﬁnd

enough leaves to be able to calculate the model pa-

rameters. Figure 6(e) illustrates the result of this

search. The leaf search is done on the leaf clues im-

age from bottom up because the lower leaves (older)

always hide the younger leaves above them. Only the

frontal leaves are considered because they are more

prominent and distinct. The process is iterative and

includes three parts: locating the leaf’s start point in

the leaf clues image, tracking this leaf from its start

point, and removing the leaf found from the leaf clues

image and the leaf probability image. These parts are

explained below.

Locating the Leaf Start Point: The leaf start point

is chosen as the lower point of a clue in the leaf clues

image. The clue is chosen from a region in the center

of the trunk, above the tree center.

Tracking the Leaf: The leaf is tracked from its

growing point using the leaf probability image and

the original image in HSV color space. Tracking is

done using segments whose length is one third of the

trunk’s radius. The end of each segment is the be-

ginning of the following segment but in a different

direction (Figure 9). The width and length of every

segment is calculated according to the model and the

leaf starting point. The tracking is done using a par-

ticle ﬁlter (Arulampalam et al., 2002), with a weight

function which is the product of the following terms:

• Sum of leaf pixels on the leaf probability image

plus the number of pixels that belong to this spe-

ciﬁc clue in the proposed leaf. This encourages

the leaf model to be on the leaf pixels and on the

starting clue (illustrated in green in Figure 9).

• 1/sum of the standard deviations of the pixels

USING PHYLLOTAXIS FOR DATE PALM TREE 3D RECONSTRUCTION FROM A SINGLE IMAGE

293

along the lines of the leaf segments (illustrated by

the blue lines in Figure 9). This discourages cross-

ing over to other leaves because lines along a leaf

tend to be smooth.

• Number of pixels of the proposed leaf that are not

sky pixels. A simple deﬁnition for a sky pixel in

HSV color space was determined empirically.

• 1/sum of the ascending angles. Consider the an-

gles between successive leaf parts, illustrated in

Figure 9 as β

−β

. Ascending angles are deﬁned

as negative angles. They are counted to discour-

age the leaf from ascending against gravity.

Leaf samples are propagated by adding segments

with the mean direction of the previous segment per-

tubrated by Gaussian noise with σ =

leaf radius

relative

to the end of the segment.

Figure 9: Particle ﬁlter tracking model.

Removing the Leaf: Once the geometric shape of

the leaf has been found, it is removed from leaf clues

image and the leaf probability image.

4.5 Estimating Model Parameters

The exact location of neighboring leaves depends

on the leaf parameters and the tree model equa-

tions. Nevertheless, parameter similarities between

trees give similar parastichy patterns, making it possi-

ble to estimate the location of a neighboring leaf while

accounting for the handedness (possible locations are

calculated by using a range of possible parameters).

The ﬁrst step in model parameter estimation is

done twice; once for the possibility of right hand-

edness and once for left handedness. Leaves found

in the last step are matched into a list of neighboring

couples according to their relative location. d is cal-

culated for every couple from the location of the two

leaves according to Equation 3. From the list of the

d’s outliers are removed and the handedness of the

tree is chosen to be the one with the most inliers and

d is computed as the mean value of the inliers.

The relative age index between leaf couples is

known according to their relative location and hand-

edness. All leaves are numbered according to their

age index in relation to the lowest leaf. The leaf index

of the lowest leaf is chosen as the one with the mini-

mal sum of differences between the leaf angle as seen

in the image in relation to the prediction of the leafs’

angle according to the model using the projection of

α from Equation 4 on to the image.

Figure 6(f) is the tree image. Prominent leaves are

marked in green with their age indexes, while those

marked in magenta predict the location and angle of

other leaves according to the model.

4.6 Search for More Leaves

Model parameters and prominent leaf age indexes en-

able the prediction of neighboring leaf locations and

growth angles using the tree model equations. The

search for more leaves is done iteratively by search-

ing for neighboring leaves from the bottom up. Every

tree location and growth angle of neighboring leaves

is calculated. If a leaf clue exists in the leaf clue im-

age in that area, then leaf tracking is performed in the

same way as in Section 4.4. After the search a leaf

is accepted if the percentage of pixels in the geomet-

ric shape of the leaf, found in the leaf clue image, is

beyond a given threshold. Figure 6(g) illustrates the

result of this search. All the leaves found, including

the ones found in the search for the prominent leaves

are marked in magenta.

4.7 3D Reconstruction

As mentioned in Section 3.1, one of the assumptions

of the tree model is that the leaves grow outward from

the axis of the trunk. Using this assumption, knowing

the location of the trunk and the location of the leaf

midrib on the 2D image and including the leaf grow-

ing point, we are able to calculate the location of the

3D midrib. Figure 6(h) is a 3D VRML model of the

date palm tree based on these calculations.

5 EXPERIMENTS

The algorithm described in this paper has been imple-

mented in MATLAB and tested on nearly 50 images

of date palm trees. In the ﬁrst stage the leaf, trunk and

sky probability image were generated from a small

number of manually labeled images. The radius table

table

( j) and the leaf angle table α

table

( j) were mea-

sured from a small set of images of a tree.

Several results of the execution of the algorithm

are presented in Figure 10. Additional results as well

as the VRML models can be viewed at:

http://mis.hevra.haifa.ac.il/∼ishimshoni/Phyllotaxis/.

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

294

6 CONCLUSIONS AND FUTURE

WORK

In this paper, we proposed a new 3D reconstruction

algorithm which reconstructs the 3D model of the

date palm tree from a single uncalibrated input image

by using the spiral phyllotactic pattern. To demon-

strate the effectiveness of the proposed method, we

show several output VRML model images.

3D reconstruction from a single input image is,

naturally, only the ﬁrst step in the development of au-

tomatic vision algorithms to guide autonomous palm

tree maintenance machines. We envision the next step

to be the further development of the algorithm by in-

corporating structure from motion techniques in or-

der to improve the accuracy of the algorithm and to

integrate the algorithm into a robotic prototype. The

proposed algorithm can also be applied to additional

phyllotactic plants.

REFERENCES

Adler, I., Barabe, D., and Jean, R. (1997). A history of the

study of phyllotaxis. Ann. of Botany, 80(3):231–244.

Aracil, R., Saltar

en, R., and Sabater, J. (1999). Trepa: Par-

allel climbing robot for maintenance of palm trees and

large structures. In Proc. of Int. Workshop and Conf.

on Climbing and Walking Robots, pages 453–461.

Arulampalam, M. S., Maskell, S., Gordon, N., and Clapp,

T. (2002). A tutorial on particle ﬁlters for on-

line nonlinear/non-gaussian bayesian tracking. IEEE

Transactions on Signal Processing, 50:174–188.

Atela, P. and Gol

e, C. (2008). Phyllotaxis - an interactive

site for the mathematical study of plant pattern forma-

tion. www.math.smith.edu/∼phyllo/.

Cummings, F. W. and Strickland, J. C. (1998). A model of

phyllotaxis. J. of Theo. Biology, 192(4):531–544.

Elhoumaizia, M. A., Lecoustreb, R., and Oihabic, A.

(2002). Phyllotaxis and handedness in date palm

(phoenix dactylifera l.). Fruits, 57(5-6):297–303.

Ferry, M. (1998). The phyllotaxis of the date palm (phoenix

dactylifera l.). In Proc. Inter. Conf. on Date Palms, Al-

Ain, UAE, pages 559–571.

Fowler, D. R., Hanan, J., and Prusinkiewicz, P. (1989).

Modelling spiral phyllotaxis. Computers and Graph-

ics, 13(3):291–296.

Fowler, D. R., Prusinkiewicz, P., and Battjes, J. (1992). A

collision-based model of spiral phyllotaxis. In Proc.

ACM Conf. on Computer Graphics and Interactive

Techniques, pages 361–368.

Haering, N. C. and da Vitoria Lobo, N. (1999). Features

and classiﬁcation methods to locate deciduous trees in

images. Comp. Vis. and Im. Under., 75(1-2):133–149.

Jean, R. V. (1994). Phyllotaxis: a systemic study in plant

morphogenesis. Cambridge University Press.

Jones, M. J. and Rehg, J. M. (2002). Statistical color mod-

els with application to skin detection. International

Journal of Computer Vision, 46(1):81–96.

Kaewapichai, W., Kaewtrakulpong, P., Prateepasen, A., and

Khongkraphan, K. (2007). Fitting a pineapple model

for automatic maturity grading. In ICIP, pages I: 257–

260.

Lewis, R. M. and Torczon, V. (2000). Pattern search algo-

rithms for linearly constrained minimization. SIAM

Journal on Optimization, 10(3):917–941.

Lintermann, B. and Deussen, O. (1999). Interactive model-

ing of plants. IEEE Computer Graphics and Applica-

tions, 19(1):56–65.

Martinez, A. R., Mart

ın, I., and Drettakis, G. (2004).

Volumetric reconstruction and interactive rendering

of trees from photographs. ACM Trans. Graph,

23(3):720–727.

Prusinkiewicz, P. and Lindenmayer, A. (1990). The Algo-

rithmic Beauty of Plants. Springer-Verlag.

Rees, A. R. (1964). The apical organization and phyllotaxis

of the oil palm). Annals of Botany, 28:57–69.

Reinhardt, D. (2005). Regulation of phyllotaxis.

International Journal of Developmental Biology,

49(5/6):539–546.

Ripin, Z. M., Soon, T. B., Abdullah, A. B., and Samad, Z.

(2000). development of a “low cost” modular pole

climbing robot. TENCON, 1:196–200.

Rutishauser, R. (1988). Symmetry in Plants, chapter Plas-

tochron ratio and leaf arc as parameters of a quantita-

tive phyllotaxis analysis in vascular plants, pages 171–

212. World Scientiﬁc Publications.

Shamsi, M. (1998). Design and development of a tree

climbing date harvesting machine. PhD thesis, Silsoe

College, Cranﬁeld University, UK.

Shlyakhter, I., Rozenoer, M., Dorsey, J., and Teller, S.

(2001). Reconstructing 3D tree models from instru-

mented photographs. IEEE Computer Graphics and

Applications, 21(3):53–61.

Smith, R. S., Guyomarc’h, S., Mandel, T., Reinhardt, D.,

Kuhlemeier, C., and Prusinkiewicz, P. (2006). A plau-

sible model of phyllotaxis. Proceeding - National

Academy of Sciences USA, 103(5):1301–1306.

Steeves, T. A. and Sussex, I. M. (1989). Patterns in plant

development. Cambridge University Press.

Teng, C. H., Chen, Y. S., and Hsu, W. H. (2006). An

approach to extracting trunk from an image. IEICE

Transactions, E89-D(4):1596–1600.

Viola, P. and Jones, M. (2001). Rapid object detection using

a boosted cascade of simple features. In Proceedings

CVPR, pages I:511–518.

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Original images Segmented leaves VRML 3D model

Figure 10: Results of the algorithm’s operation.

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