Modeling curve dynamics and spatial geometry characteristics of rice leaves

Journal of Integrative Agriculture 2017, 16(10): 2177–2190 Available online at www.sciencedirect.com

ScienceDirect

RESEARCH ARTICLE

Modeling curve dynamics and spatial geometry characteristics of rice leaves ZHANG Yong-hui1, 2, TANG Liang2, LIU Xiao-jun2, LIU Lei-lei2, CAO Wei-xing2, ZHU Yan2 1 2

Computer Engineering School, Weifang University, Weifang 261061, P.R.China National Engineering and Technology Center for Information Agriculture/Key Laboratory for Crop System Analysis and Decision Making, Ministry of Agriculture/Jiangsu Key Laboratory for Information Agriculture/Jiangsu Collaborative Innovation Center for Modern Crop Production/Nanjing Agricultural University, Nanjing 210095, P.R.China

Abstract The objective of this work was to develop a dynamic model for describing leaf curves and a detailed spatial geometry model of the rice leaf (including sub-models for unexpanded leaf blades, expanded leaf blades, and leaf sheaths), and to realize threedimensional (3D) dynamic visualization of rice leaves by combining relevant models. Based on the experimental data of different cultivars and nitrogen (N) rates, the time-course spatial data of leaf curves on the main stem were collected during the rice development stage, then a dynamic model of the rice leaf curve was developed using quantitative modeling technology. Further, a detailed 3D geometric model of rice leaves was built based on the spatial geometry technique and the non-uniform rational B-spline (NURBS) method. Validating the rice leaf curve model with independent field experiment data showed that the average distances between observed and predicted curves were less than 0.89 and 1.20 cm at the tilling and jointing stages, respectively. The proposed leaf curve model and leaf spatial geometry model together with the relevant previous models were used to simulate the spatial morphology and the color dynamics of a single leaf and of leaves on the rice plant after different growing days by 3D visualization technology. The validation of the leaf curve model and the results of leaf 3D visualization indicated that our leaf curve model and leaf spatial geometry model could efficiently predict the dynamics of rice leaf spatial morphology during leaf development stages. These results provide a technical support for related research on virtual rice. Keyword: rice, morphological models, leaf, geometry characteristics, virtual plant

1. Introduction Virtual crop, which simulates crop growth and development

Received 28 September, 2016 Accepted 16 February, 2017 Correspondence ZHU Yan, Tel: +86-25-84396598, Fax: +86-2584396672, E-mail: [email protected] © 2017 CAAS. Publishing services by Elsevier B.V. All rights reserved doi: 10.1016/S2095-3119(16)61597-6

in three-dimensional (3D) space, may find wide applications in the fields of high yield plant design, crop production and management, relevant teaching, crop breeding, and virtual experimentation, and has become an important research theme in recent years (Birch et al. 2003; Cao et al. 2008). Virtual crop might eventually permit the prediction of the growth of transformed genotypes or of combinations of alleles of genes of interest under arbitrary climate conditions (Tardieu 2003). Also, virtual crop is an intuitive tool to enhance our understanding of complex crop phenotypes, which will ultimately lead to new breeding approaches and improved crop cultivars (Xu et al. 2011).

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Significant progress has been made in the study of virtual crops of rice (Watanabe et al. 2005; Zhang et al. 2014a, b), maize (Fournier and Andrieu 1998; Guo et al. 2006), wheat (Fournier et al. 2003; Evers et al. 2007; Lei et al. 2011), cotton (Hanan and Hearn 2003), and other crops (Kaitaniemi et al. 2000; Dornbusch et al. 2007). Recently, virtual crop has been widely applied. For example, a 3D plant crown architecture model was developed and used to assess light capture and whole-plant carbon gain consequences of leaf display in understory plants (Pearcy and Yang 1996). A 3D architectural model was employed to specify rice plant types (Zheng et al. 2008). A 3D crop root architectural model was constructed to simulate root changes under various nutrient conditions (Fang et al. 2009). A virtual maize model of GreenLab was used to study both the stand and the individual plants in a field of maize with the aim of generating numerical representations of the crop at both stand and individual plant levels (Feng et al. 2014). Leaves are the most important crop organ for canopy light distribution and photosynthesis production. They are affected by development status and spatial distribution (Wang et al. 2007). Therefore, leaf spatial morphology and distribution is a vital part in virtual crops research. Rice is one of the most important food crops in the world. Morphological modeling and 3D visualization of the rice leaf is vital for virtual rice. Lately, there have been many studies on morphological modeling and 3D visualization of rice leaf. Based on analysis of the rice leaf, a process model of the rice leaf curve was constructed to study its spatial development (Shi et al. 2006). Various curves, such as Hermite (Mi et al. 2003; Watanabe et al. 2005), Bezier and quadratic curves (Yang et al. 2006), cubic B-splines (Zheng et al. 2009), and parabolic curves (Liu et al. 2009a), were chosen to simulate the rice leaf curve or leaf edge curve for leaf visualization. Meanwhile, non-uniform rational B-spline (NURBS) surface method (Liu et al. 2004), approximation method (Meng et al. 2005), and Bezier curved surface (Ma et al. 2010) were used to simulate the geometric morphology of the rice leaf blade. Besides, the logistic function was utilized to simulate the leaf elongation process, and the value of the leaf’s width-to-length ratio, thus the morphological modeling and 3D visualization of rice leaf were implemented by combining several sub-models (Watanabe et al. 2005). Geometric parameter models of the rice leaf blade were established based on leaf blade biomass (Liu et al. 2009b). A rice leaf shape model under different nitrogen and water statuses was constructed by Zhu et al. (2009). Until recently, the morphological models of the rice leaf curve were not supported by sufficient experimental data, and these previous models are not dynamic. Under natural growth conditions, the leaf blade appear as a spiral at early leaf appearance time, and then leaf blade gradually expands

and appears spatial curved surface from the tip to the base of leaf blade along the leaf length direction. However, most of the previous works just focused on the modeling of the expanded rice leaf blade, and lacked detailed investigation on spatial morphology changes of the unexpanded leaf blade. These deficiencies would affect the simulation of leaf blade spatial morphology and bring errors in application of the models. Our work therefore concentrated on the following aims: (1) develop a dynamic simulation model of rice leaf curve on the main stem based on field experimental data of different rice cultivars under different nitrogen (N) rates; (2) construct a detailed 3D geometry morphology model (including sub-models of unexpanded leaf blade, expanded leaf blade, and leaf sheath) of the rice leaf; and (3) achieve 3D dynamic visualization of the rice leaf based on our relevant models. We expected that our results would provide the technology support for further development of virtual crop.

2. Material and methods 2.1. Experiment site and design Two experiments were conducted in 2010–2011 at the Experiment Station of Nanjing Agricultural University (118°50´E, 32°02´N), Jiangsu Province, China. In Experiment 1, soil organic matter, total N, available phosphorous (P), and available potassium (K) were 14.5 g kg−1, 1.15 g kg−1, 41 mg kg−1, and 79 mg kg–1, respectively. The corresponding soil properties were 15.1 g kg−1, 1.32 g kg−1, 38 mg kg−1, and 85 mg kg−1, respectively for Experiment 2. Experiment 1: This field experiment was conducted in 2010. Oryza sativa L. ssp. japonica Wuxiangjing 14 with upright leaves (W14) and Oryza sativa L. ssp. indica Yangdao 6 with drooping leaves (YD6) were planted on 26 May, and transplanted on 13 June. The plot size was 4 m×2.5 m with a plant spacing of 28 cm×20 cm for YD6 and 20 cm×15 cm for W14, with one seedling per hill for each cultivar. Three N rates of 0 (N1), 125 (N2), and 250 kg ha–1 (N3) were applied in four splits (50% at pre-transplanting, 10% at tillering, 20% at spikelet promotion, and 20% at spikelet protection stages) for all treatments. For both cultivars under different N rates, P2O5 and K2O were applied as a basal dose at 82.5 and 164.7 kg ha–1, respectively. Other field managements followed local practices for high yield rice. Experiment 2: The experiment was conducted in barrels. Rice was direct seeding on 1 June 2011. The culture barrels were 35 cm in diameter at the upper underside, 20 cm at the lower underside, and were 40 cm in height. Two seedlings were planted per barrel for both cultivars on 1 June. The four N rates of 0 (N1), 1.5 (N2), 3.1 (N3), and 4.6 g (N4) for each barrel were applied in four splits (50%

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before planting, 10% at tillering, 20% at spikelet promotion, and 20% at spikelet protection stages) for all treatments. For both cultivars under different N rates, phosphorus (Ca3(PO4)2) and potassium (KCl) were applied as a basal dose at 4.69 and 2.88 g for each barrel, respectively. There were 40 pots for each cultivar and N rate combination. Other field management practices followed local practices for high yield of rice.

2.2. Data acquisition Four rice plants with uniform growth status were tagged for each cultivar and N rate. Spatial coordinate data of leaf curves and sheaths on the main stem in these tagged plants were measured by a 3D digitalization device (FastScan, Polhemus, USA) every other day from the first leaf appearance. In addition, five rice plants of each cultivar and N rate were destructively sampled at regular intervals. During the whole rice growth period, air temperatures were recorded daily at 30-min intervals with a ZDR-11 (made by Zhejiang University of China). The recorded data were downloaded every month and used for calculating the thermal time (TT, °C d). In addition, the relevant morphology data of samples for leaf modeling and 3D visualization were, the N contents collected, and the N contents and dry weights of rice samples’ different organs were measured.

2.3. Date processing and analysis The data of Experiments 1 and 2 were used to build and validate the models, respectively. Data fitness was done using Microsoft Excel 2010, CurveExpert 1.4, and Visual Studio 2010. Variance analysis was done using MatlabR2009a. Required parameters of leaf curve model were calculated in the following steps: Firstly, 3D coordinate data of leaf blade curve and leaf sheath were equivalently transformed into plane data by rotation, symmetry, and translation transformation (Table 1). Secondly, these plane data were used to calculate the elevation angles of points on the leaf curve, and the relative position parameter Sm of each point on the leaf curve where the elevation angle changed the fastest. Then, another parameter of leaf curve model Sd was calculated using Sm through the nonlinear least square method. Finally, plane coordinate data of the leaf curve were used to calculate the angle ∠BFC between the leaf blade and sheath in rice (Fig. 1).

2.4. Modeling spatial morphology of rice leaf curve Describing leaf curve The leaf curves of rice cultivars with upright leaves approximately keep linear during rice growth

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and development. But those of rice cultivars with drooping leaves appear linear from leaf appearance time to leaf fixedlength time. With the dry matter accumulating in rice leaf blades, the weight of the leaf blade increases and the leaf gradually bends. When the weight of leaf reaches its maximum, the leaf stops bending, and the spatial morphology of the leaf basically remains unchanged until leaf senescence. In addition, the angle ∠FBE between sheath and stem also increases during the leaf blade bending process (Fig. 1). In Fig. 1, point B (coordinate origin) is the growth position of the sheath on the main stem (y axis). Point F is the base point of the leaf blade, which is on the leaf axis. Point E is the position of F on the main stem when the angle between sheath and stem (∠FBE) is zero. Line FC is the tangent of F. Point D is the projection of F on the main stem. ∠BFC is the angle between sheath and leaf blade. Line O1O2 parallels to the y axis, line O3O4 parallels to the x axis, and point O is the intersection. ∠HOO3 is the elevation angle of the distal end of leaf curve FGH. ∠HOO2 is the angle between the distal end of the leaf curve and the positive direction of the y axis, which is the complementary angle of ∠HOO3. In this study, all the relevant angles were calculated on the premise that the angle between sheath and stem∠FBE became zero by anti-clockwise rotating the whole leaf BFGH a ∠FBE degree around point B. And the relative position between the leaf and the sheath remained unchanged within this rotating process. The original spatial morphology of leaf is easy to obtain as long as the whole leaf BFGH is rotated clockwise an angle-∠FBE around point B. Describing leaf curve model The rice leaf curve is an approximately smooth curve or straight line in 3D space. In this study, a dynamic model of leaf curve was constructed based on the method of Dornbusch et al. (2007). Here, rice leaf curve was simulated by connecting a series of points (Fig. 2). As long as a set P composed of simulated points were obtained, the leaf curve could be simulated by eq. (1): xi (1) Pi+1 = Pi +| PP i i +1 |× ei , Pi = yi , P = { P1 , P2 ,... PN } (1≤ i ≤ N )

zi Where, |PiPi+1| denotes the distance between points Pi and Pi+1. N denotes the total number of points on simulated leaf curve. i is the direction vector from Pi to Pi+1 calculated by eq. (2):

cos(θi ) ×cos(Ai ) ei = sin(θi ) ×cos(Ai )

(2)

sin( Ai) Where, θi and Ai denote the azimuth angle and elevation angle of vector i, respectively. Each point Pi+1 of set P (starting point is the base point of leaf blade) can be calculated by eq. (1) with the spatial coordinate of Pi, the distance |PiPi+1|,

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Table 1 List of symbols used in this study Item

Symbol

Ai

The elevation angle of vector → ei

A1

The basal elevation angle of the leaf axis

AN

The distal elevation angle of the leaf axis

AH

The angle between the distal of leaf curve and positive directions of y axis

AHn

The value of AH corresponding to the nth leaf curve

ANCSH

The actual nitrogen content of the rice plant

C(l)

The length of cross-sectional spiral round the position l on the leaf axis

FAng

The final angle of ∂2

FLn

The final length of the nth leaf on the main stem

FTTn

The TT value when the nth leaf length is fixed

g(S)

The Gaussian function

I∂

The initial value of angle ∂2

ITTn

The TT value of the nth leaf appearance

l

The length from spire center to the base of leaf blade

MAHn

The maximum value of AH of the nth leaf on the main stem

MinSmn

The minimum value of the nth leaf’s Sm

MMAH7

The MAH7 corresponding to the nth leaf on the main stem in rice under appropriate N condition

r

The polar radius of the inner end of spiral

R

The polar radius of the outer end of spiral

Sd

The smooth degree of the leaf curve

Sm

The relative position on the leaf axis where the change rate of elevation angle (dA/dS) is maximum

Smn(TT)

The Sm value of the nth leaf at TT time

S

The relative position parameter of a point on the leaf curve

TLN

The total number of leaves on the main stem

TT

Thermal time

W(l)

The leaf width of position l on the leaf axis

ΔA

The differences between A1 and AN

ΔBTTn

The accumulated TT during the nth leaf bending

ΔETTn

The accumulated TT during the nth leaf elongation

∂1

The polar angle of the inner end of spiral

∂2

The polar angle of the outer end of spiral

β

The polar angle of arbitrary point on the cross-sectional spiral

θi

The azimuth angle of vector → ei

→

ei

The direction vector

di, j

The controlling vertexes of the NURBS surface

ωi, j

The weight factors associated with controlling vertexes di, j

dA/dS

The change rate of the elevation angle along the leaf curve

∠BFC

The angle between the leaf blade and sheath

∠FBE

The angle between the leaf sheath and stem

∠HOO2

The angle between the distal end of leaf curve and positive direction of y axis

∠HOO3

The elevation angle of the distal end of the leaf curve

and direction vector i. (xi, yi, zi) denotes the coordinate of the ith point (Pi) in set P. Pi+1 is the projection of Pi+1 on plane (xoz) (Fig. 3). Thus, the leaf curve can be simulated as long as the distances between two adjacent points, the azimuth angle, and the elevation angles of each point on the leaf curve are known. To simplify the calculation, the leaf axis was divided into equal N–1 segments by the N points on the leaf axis. The elevation angle A(S) of any point on the leaf curve could be

quantitatively modeled by the following eq.: S ∆ A× ∫ g ( S) dS A(S)=A1+ 1 0 (3) ∫0 g (S) dS Where, S denotes the relative position parameter of a point on the leaf curve (that is, the ratio of the length from the base of the leaf blade to this point and the leaf axis length). If the distance between two adjacent points was equal, S could be calculated by eq. (4). Let A1 denote the basal elevation angle of the leaf axis (∠IFG in Fig. 1). ΔA is

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the differences between A1 and AN (the distal elevation angle of the leaf axis), calculated by eq. (5). g(S) is the Gaussian function, and used to fit the change of the elevation angle along the leaf axis, calculated by eq. (6). i −1 (4) S= , i =1, 2, ..., N N −1 ΔA=A1–AN, AN=90°–∠HOO2 (5) S−Sm 2 (6) ) ) g(S)=exp( −( 2×Sd In eqs. (4)–(6), i denotes the sequence number of the labeled point on the leaf axis (the sequence number of the base point of the leaf blade is 1). ∠HOO2 (AH) is the angle between the distal of the leaf curve and the positive direction of the y axis (Fig. 1), calculated by eq. (7). Sm denotes the relative position on the leaf axis where the change rate of elevation angle (dA/dS) is the maximum (Fig. 4). Sd denotes the smooth degree of the leaf curve (Fig. 4). Modeling the leaf curve for rice cultivar with upright leaf blade Through observation and analysis of experimental data, we found that the leaf curve of the rice cultivar with upright leaf blade appeared linear during its development stages. Therefore, based on the previous simulation of the angle between stem and sheath ∠BFC in Fig. 1 (Zhang et al. 2012), the angle between stem and leaf blade ∠FCE in Fig. 1 (Chang 2007), and the leaf length (Zhu et al. 2009), the rice leaf curve of straight-line type could be modeled by linear functions.

y O

O4 E D

F

I

O1

C

x B

Fig. 1 Sketch map of the leaf curve in rice showing plane coordinate data and leaf and sheath angles.

P2

PN

P1 Pi+1=Pi+|PiPi+1|×ei

Fig. 2 The space curve formed by vectors.

Pi+1

Pi

x

Ai θi p´i+1

z

Fig. 3 Vector in space coordinate in the leaf curve model.

Sm=0.3, Sd=0.2

Sm=0.5, Sd=0.2

Sm=0.7, Sd=0.2

Sm=0.5, Sd=0.05

Sm=0.5, Sd=0.1

Sm=0.5, Sd=0.2

O3

H

PN−1

 ei

Fig. 4 Effects of parameter Sm (the relative position on the leaf axis where the change rate of elevation angle is the maximum) and Sd (the smooth degree of the leaf curve) on leaf curve, where A1=80°, ΔA=110°, azimuth angle θ=0°, leaf length l=40 cm.

O2 G

y

Modeling the leaf curve for the rice cultivar with drooping leaf blade The leaf curve of rice cultivars with drooping leaves present a linear shape from leaf appearance time to leaf fixed-length time; then, the leaf blade gradually bends until the weight of leaf reaches its maximum. Therefore the dynamic simulation of the bending leaf curve is complicated. In this study, the leaf curve of bending type was simulated based on the Gaussian function. (1) Modeling elevation angle of leaf curve. Our experimental data showed that the change of angle AH between the distal of the leaf curve and the positive direction of the y axis (complementary angle of the distal elevation angle of leaf curve) with the thermal time (TT) followed as slowfast-slow pattern (Fig. 5). After the angles AHs of leaves on the main stem in rice under different nitrogen rates during different growth stages were normalized, variance analysis revealed no significant difference among groups of AHs data (P>0.05) (Fig. 5). Therefore, the variation of the normalized AHs with normalized TT could be quantitatively described

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by the following formula: , TT ≤FTTn

0 AHn (TT ) MAHn

=

1 −Ab×(TT − FTTn )/∆BTTn

1+Aa×e

1 −Ab 1+Aa×e

, FTTn ≤TT ≤ FTTn +∆BTTn(7) , TT >FTTn+ ∆BTTn

Where, TT is the thermal time, calculated above a basal temperature (10°C for japonica and 12°C for indica cultivars) for the development from the appearance of the first leaf above soil surface (Gao et al. 1987). AHn (TT) denotes the AH value of the nth leaf on the main stem at time TT. The left side of eq. (7) represents normalized angle AH of the nth leaf. MAHn denotes the maximum value of AHn. Aa and Ab are constant coefficients which are 29.81 and 8.06 (R2=0.9976), respectively, according to the experimental data. FTTn denotes the TT value when the nth leaf length is fixed, given by eq. (8). ΔBTTn denotes the accumulated TT during the nth leaf bending, calculated by eq. (9):

FTTn=

n

1 b

(8)

a

ΔBTTn =

n+5

1 b

−

n

1 b

(9) a a Where, a and b are constants, which through analyzing the experimental data, are 0.0585 and 0.7936 (R2=0.9847), respectively. Our experimental data revealed the following phenomenon: under the same nitrogen rate, MAHn increased linearly with increasing leaf rank from the 1st to the 7th on the main stem and MAH7 reached its maximum. Then

MAHn decreased linearly with the increase of leaf rank after the 7th leaf on the main stem (Fig. 6). When MAHns on the main stem under the same nitrogen rate were divided by MAH7 to be normalized, variance analysis showed no significant differences between four data groups of the four nitrogen rates (P>0.05, Fig. 6). Therefore, the MAHn could be calculated by eq. (10): MAHn MAH7

=

Ma×n +Mb

, 1≤ n≤ 7

(10)

Mc×exp(Md × n), 8 ≤ n≤TLN

Where, Ma, Mb, and Mc, Md are equation coefficients, evaluated by 0.1338, 0.0428 (R2=0.9921) and 17.11, –0.358 (R2=0.9753), respectively. n is the leaf rank on the main stem. TLN is the total number of leaves on the main stem, set as a variety parameter, 17 for YD6. MAH7 is the maximum value of AH corresponding to the 7th leaf on the main stem, calculated by eq. (11): MAH7=MMAH7×FN (11) Where MMAH7, a variety parameter, denotes the MAH7 corresponding to the 7th leaf on main stem in rice under appropriate nitrogen condition, MMAH7=143° for YD6. FN, a nitrogen influencing factor, denotes the influence of nitrogen on leaf drooping degree, calculated by eq. (12):

ANCSH , ANCSH≤MNCSH FN= MNCSH 1 , ANCSH≥MNCSH

(12)

ANCSH is the actual N content of the rice plant, obtained by experiment measurement. MNCSH is the N concentra-

N1YD6

N2YD6

N3YD6

N4YD6

1.2 Normalized maximum value of AH

1.2

Normalized AH

1 0.8 0.6 0.4

0.8

0

0.2

0.4 0.6 0.8 Norrmalized TT

1

1.2

Fig. 5 Changes of normalized AH (the angle between the distal of the leaf curve and the positive direction of the y axis) with normalized TT (thermal time). Bars are SD.

y=17.11exp(–0.358x) R2=0.9753

0.6 0.4 y=0.1338x+0.0428 R2=0.9921

0.2 0

0.2 0

1

0

4

8 12 Leaf ranks on main stem

16

20

Fig. 6 Changes of normalized MAHn (the maximum value of AHn) with leaf rank, where MAHn denotes the maximum value of AH (the angle between the distal of the leaf curve and the positive direction of the y axis) of the nth leaf on main stem. YD6, Yangdao 6; N1, N2, N3, and N4 denote the nitrogen rates of 0, 1.5, 3.1, and 4.6 g respectively for each barrel in experiment 2.

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1 = 1+Sa×e−Sb×( TT− FTTn) /∆BTTn , FTTn ≤TT ≤ 1−MinSmn FTTn +∆BTTn (16) 1 , TT > FTTn+ −Sb 1+Sa×e ∆BTTn

Where, Smn (TT) denotes the Sm value of the nth leaf at time TT. The left side of eq. (16) denotes the normalized movement rate of Smn. MinSmn denotes the minimum value of Smn. Sa and Sb are constants, which are 43.28 and 7.59 (R2=0.9801), respectively. The other symbols have the same meanings as above. Our experimental results showed that MinSmn decreased linearly with the increase of leaf rank from the 1st to the 7th, and then increased linearly with increasing leaf rank after the 7th (Fig. 9). Variance analysis revealed no significant differences between MinSmns of the main stem in rice under different nitrogen rates (P>0.05). Therefore, the changes of MinSmn with leaf rank n on the main stem could be calculated by eq. (17):

MSa×n+MSb, 1≤n<7 MSc×n+MSd, 7≤n
A rice leaf is composed of the leaf blade and the leaf sheath. the leaf blade appears as a spatially spiral shape at the early time of leaf appearance, then gradually expands from the tip

25 20

N1YD6

(17)

Where, MSa, MSb, and MSc, MSd are constants, which

N2YD6

N3YD6

N4YD6

y=0.1288x+2.39 R2=0.9386

15 10 5 0 0

50

, TT ≤ FTTn

1−Smn (TT )

MinSmn=

2.5. Modeling spatial geometry of the rice leaf

100 AH

150

200

Fig. 7 The relationship between AH (angle between the distal of leaf curve and positive direction of y axis) and ∠BFC. YD6, Yangdao 6; N1, N2, N3, and N4 denote the nitrogen rates of 0, 1.5, 3.1, and 4.6 g respectively for each barrel in experiment 2.

1.2 1.0 Movement rate of Sm

0

are –0.0726, 0.0911 (R 2=0.9145) and 0.0436, 0.1193 (R2=0.9530), respectively. Parameter Sd denoted the smooth degree of the leaf curve (Fig. 4). Analyzing the data showed that Sds of all leaf curves of the main stem under different nitrogen rates were distributed between 0.2 and 0.3 (Fig. 10). Thus Sd was assigned a constant value 0.25 in our leaf curve model.

Angle between sheath and leaf (∠BFC)(°)

tion calculated by (13): MNCSH=c×AGB–d (13) c and d are constants, 5.18 and 0.52, respectively. AGB is above-ground biomass of rice plant (Gao et al. 1987), calculated by Meng (2002). In Fig. 1, A1 (∠GFI), elevation angle of point F (the base of the leaf axis), is the complementary angle (∠GFI+∠BFC= 90°) of ∠BFC (the angle between leaf blade and sheath). The correlation analysis of ∠BFC and AH under different nitrogen rates during rice growth and development stages, showed that there was a linear relationship between them (R2=0.9386) (Fig. 7). This relationship could be quantitatively described by eq. (14) or (15): ∠BFC=0.1288×AH+2.39° (14) 90°–A1=0.1288×(90°–AN)+2.39° (15) (2) Modeling parameter changes of the leaf curve model. Our experiment data showed that the model parameter Sm moved from the distal to the basal of the leaf axis during the leaf bending, and the movement rates of these Sm with TT presented a slow-fast-slow trend (Fig. 8). The movement rates of Sms from the distal to the basal of each leaf axis were normalized. These normalizations were assembled into one group. The variance analysis indicated that there were no significant differences between these normalizations of different leaves on the main stem in rice under different nitrogen rates (P>0.05) (Fig. 8). Therefore, the movement rate of Sm could be modeled by a uniform logistic function of eq. (16):

0.8 0.6 0.4 0.2 0

0

0.2

0.4 0.6 0.8 Normalized TT

1

1.2

Fig. 8 Movement rate of parameter the relative position on Sm (the leaf axis where the change rate of elevation angle is the maximum) with normalized TT (thermal time). Bars are SD.

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to the base of the leaf blade. The sheath basically appears as a spatial cylinder. In the following sections, the spatial geometry modeling of the unexpanded leaf blade, the expanded leaf blade, and the sheath are described separately.

the controlling points and the weight factor. Therefore, the NURBS curved surface was used to construct the spatial geometry model of the rice leaf in this study. A NURBS curved surface with k×l order is defined by the following eq. (18) (Watt 2005):

2.6. The NURBS curved surface

n

The NURBS curved surface has both non-uniform and rational spline properties, which not only describe the free-form curve and curved surface, but also can flexibly change the shape of a free-form curve and curved surface by adjusting

N1YD6 1.0

N2YD6

N4YD6

N3YD6

y=–0.0726x+0.9107 R2=0.9145

MinSmn

0.6 y=0.0436x+0.1193 R2=0.9530

0.2 0 0

4

8 12 Leaf ranks on main stem

16

20

Fig. 9 Changes of MinSmn (the minimum value of the nth leaf’s Sm) with leaf rank. YD6, Yangdao 6; N1, N2, N3, and N4 denote the nitrogen rates of 0, 1.5, 3.1, and 4.6 g respectively for each barrel in experiment 2. 0.4

Parameter Sd

0.3

0.2

0.1

0

0

0.2

0.4

0.6

i =0 j =0 m n

∑ ∑ wi , j Ni , k (u) N j , l (v)

0≤u, v≤1

(18)

i =0 j =0

Where, u and v denote the u and v axes on a curved surface, respectively. di, j and wi, j (i=0, 1, …, n; j=0, 1, …, m) denote the controlling vertices and the weight factors associated with these vertices, respectively. Ni, k (u) and Nj, l (v) denote the basis functions of the B-spline (Watt 2005).

2.7. Spatial geometry modeling of the unexpanded leaf blade

0.8

0.4

p(u, v)=

m

∑ ∑ wi , j di , j Ni , p (u) N j , l (v)

0.8

1

1.2

Normalized TT

Fig. 10 Changes of parameter Sd (the smooth degree of the leaf curve) with normalized TT (thermal time). Bars are SD.

The leaf blade appears as a spiral shape at the early time of leaf appearance, then gradually expands to a curved surface from the tip to the base of the leaf blade during leaf elongation. The unexpanded leaf blade appears as a spiral surface, whose cross-section perpendicular to the leaf axis is a plane spire (Fig. 11-A and B). Analysis showed that there was a linear relationship between the radial and angular coordinates. Therefore, the rectangular coordinate (xβ, yβ, zβ) of a point with polar angle β(∂1≤β≤∂2) on the spiral could be calculated by eq. (19):

xβ = xq + (r + ( β −∂1 )×( R − r )/(∂ 2−∂1 ))×cos(β) yβ = yq

(19)

zβ = zq + (r + ( β −∂1 )×( R − r )/(∂ 2−∂1 ))×sin(β) Where, ∂1 and r are the angular coordinate and radial coordinate of the inner end of the spire, respectively. ∂2 and R are the coordinates of the outer end of the spire, respectively. To simplify the calculation, ∂1 was assigned a constant value of zero, i.e., the starting point was set on the polar axis as shown in Fig. 11-B, and there was no influence on the geometric morphology of the leaf blade. Q (xq, yq, zq) is the center of the spire. Obviously, the length C(l) of a cross-section of the spiral (Fig. 11-B), was equal to the leaf width of position l. ∂ That was, C(l)=∫ ∂12(r+(x−∂ 1)(R−r)/(∂ 2−∂ 1))dx=W(l), then ( R +r )(∂ 2−∂1) =W(l). Therefore, we have: 2 2W (l ) ∂ 2– ∂1= (20) R+r Where, C(l) denotes the length of of the cross section of the spiral with the center point on the leaf axis, and the length from this center to the base of the leaf blade is l. W(l) denotes the leaf width of position l on the leaf axis, estimated by the leaf shape model (Zhu et al. 2009). Therefore, inserting eq. (20) into eq. (19), we obtain:

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xβ =xq +(r + 2( β −∂1 )×( R 2−r 2 )/W (l))×cos( β ) yβ = yq

(21)

zβ =zq +(r + 2( β −∂1 )×( R 2−r 2 )/W (l))×sin( β ) If a cross-section of an unexpanded leaf blade (Fig. 11-B) is equally divided into n–1 segments according to its angular coordinate, then β=∂1+i×(∂1–∂1)/n. This formula could be used to equivalently transform eq. (19) or (21). The rice leaf blade is an axial symmetric image with the leaf axis as the symmetric axis when the leaf is tiled. Therefore, the midpoint of an arbitrary cross-section of a spiral is on the leaf axis. Supposing that ∂c is the polar angle of this mid-point, eq. (22) holds according to eq. (20): (22) ∂c=∂1+W(l)/(R+r) Besides, the following eq. (23) also holds: (23) ∂c=∂2–W(l)/(R+r) Therefore, the spatial coordinates of points on the leaf axis may be calculated by eq. (19) or (21) with parameter ∂c, then, the leaf axis may be simulated by these points. ∂2 is the angular coordinate the outer end of the spire (Fig. 11-B), which varies during leaf blade expansion. In this study, the dynamic simulation of ∂2 was implemented by eq. (24). l×(ΔETTn –(TT – ITTn )) × I ∂ ∂ 2= +FAng (24) FLn×ΔETTn Where, FLn denotes the final length of the nth leaf on the main stem, estimated by rice leaf shape model (Zhu et al. 2009). ITTn denotes the TT value of the nth leaf appearance, calculated by:

ITTn=

n −1

1 b

ΔETTn denotes the accumulated TT during the nth leaf elongation, calculated by

ΔETTn=

n a

−

2.8. Spatial geometry modeling of the expanded leaf blade In our study, the geometric model of the expanded rice leaf blade was constructed using the NURBS method with controlling points. This mesh curved surface of the leaf blade was composed of the leaf length (u) and the leaf width (v) directions (Fig. 13). To simplify the modeling, the leaf axis was located on the xoy plane as in Fig. 1. Thus the leaf width direction (v) was paralleled to the z axis. A geometric model of the leaf blade was obtained by the following steps: Firstly, m points on the leaf axis were obtained by equally dividing the leaf axis into m–1 segments, and the leaf (axis) length was estimated by the leaf shape model (Zhu et al. 2009). Secondly, the leaf width of each point position of m points on the leaf axis was output by the leaf shape model (Zhu et al. 2009), each leaf width was equally divided into t–1 segments by t points with the mid-point on the leaf axis. Then, mt controlling points on the mesh curved surface of the

(25)

a

1 b

on each cross-sectional spiral of the unexpanded leaf blade could be calculated at different leaf elongation times. Then, we could obtain m and t controlling points along the leaf length direction and the leaf width direction, respectively, i.e., there were m cross-section spirals, and t controlling points on each spiral for each unexpanded leaf blade during leaf elongation. Consequently, the dynamic simulation of the spatial geometry of the leaf blade was constructed by the NURBS curved surface method with these controlling points during leaf elongation (Fig. 12).

n −1 a

1 b

A R r Q

B x

(26)

Where, a and b are equation coefficients, and their values are the same as those in eq. (8). I∂, an input parameter, denotes the initial value of angle ∂2, assigned the same value for each spiral of unexpanded leaf blade, between [6.5, 12.5]. FAng is an input parameter, and denotes the final angle of ∂2, assigned the same value for each spiral of unexpanded leaf blade, between [0.1, 0.2] radian. By eq. (20), we obtain R+r=2×W(l)/(∂2–∂1); since ∂1 is zero and R+r=2×W(l)/∂2, we can let R=d×W(l)/∂2 and r=(1–d)×W(l)/∂2, where d is between 1.5 and 2. During simulation of the leaf blade expansion process, the initial and terminal values of required parameters could be assigned with the corresponding observation values. Through the process described above, controlling points

Fig. 11 Cross-section spiral of an unexpanded leaf blade in rice.

Fig. 12 Mesh geometry model of an unexpanded leaf blade in rice.

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leaf blade were constructed (m for the leaf length direction, t for the leaf width direction), and the (x, y, z) coordinates of these mt controlling points were calculated by the spatial geometry method. Finally, the mesh curved surface of the expanded leaf blade was constructed by these controlling points by the NURBS method (Fig. 13).

2.9. Spatial geometry modeling of the rice leaf sheath The geometric shape of the leaf sheath is roughly cylindrical. Geometric modeling of the sheath was similar to that of the leaf blade in our work. Each sheath was simulated by a NURBS curved surface with m rows of controlling points, the distances between adjacent rows were equal, and t controlling points in each row had serial numbers from 0 to t–1 (Fig. 14-A). The rows from the first to the (m–1)th were composed of seven controlling points, which were uniformly distributed on a circle with the first and last controlling points coincident (Fig. 14-A). A non-closed curve was defined by the controlling points on the mth row in the sheath as in Fig. 14-B. Obviously, the coordinates of the t controlling points on each of these m rows of the sheath could be calculated by the sheath length and diameter estimated by the simulation model of the sheath morphology (Chang et al. 2008). Each sheath geometric model could be simulated using the NURBS curved surface with these controlling points (Fig. 15).

3. Results 3.1. Validation of the leaf curve model Experimental observation and data analysis showed that the degree of drooping of the rice leaf was great at both the middle tiller stage and the early jointing stage, then the leaf blade gradually became upright, and basically appeared linear after the jointing stage. The spatial coordinate data of the 4th to 9th and the 6th to 12th leaves on the main stem under three nitrogen rates at the middle tiller stage and

Fig. 13 Mesh geometry model of expanded leaf blade in rice.

the early jointing stage respectively in Experiment 1 were measured, then those data were used to validate our model. In order to validate the effectiveness of the leaf curve model, we input the measured plant height, coordinate of the leaf blade base point, and leaf length into our model. First, the coordinate of the blade leaf base was used to calculate the angle between the leaf blade and sheath (∠EBF in Fig. 1). Second, eqs. (8) and (9) were utilized to calculate TT of each leaf blade according to the leaf age, then these TTs were normalized. Third, the variety parameter MMAH7 and eq. (11) were used to calculate MAH7 of N1, N2, and N3 rate, which were 113°, 132°, and 142°, respectively. Then, the parameters of the leaf curve model (the distal elevation angle of leaf curve, the angle between leaf blade and sheath, the initial elevation angle of leaf curve and Sm, Sd) were calculated according to these results. Finally, the measured coordinate of the leaf blade base in Experiment 1, the required rotating angle (∠EBF in Fig. 1), all the model parameters, and the azimuth angle of each leaf curve were used to simulate the natural spatial morphology of each leaf curve on the main stem under different nitrogen rates. The distribution of measured coordinates of the leaf curve and the simulated leaf curve showed that the leaf curve of the same leaf rank on the main stem under the higher nitrogen rate bent more than that under the lower nitrogen rate (Fig. 16). It also illustrated that the drooping degree of the former was greater. At the tillering stage, the A

B 2 3

2

1 0(q–1)

1(0)

3

x

x

q–1 z

z

q–1(q)

Fig. 14 The distribution of controlling points on a cross-section of a leaf sheath in rice.

Fig. 15 Mesh geometry model of a leaf sheath in rice.

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3D dynamic visualization of a single rice leaf from leaf appearance to leaf senescence (Fig. 17) and of leaves on a rice plant on different growing days (Fig. 18) with our leaf curve model and the spatial geometry model of rice leaves. C#.NET programming and OpenGL library were used for computer analysis. Light and color rendering techniques were also involved in 3D visualization of the rice leaves and rice plants. The simulation results (Figs. 17 and 18) showed that the 3D visualization of a single leaf and leaves on the rice plant could be satisfactorily predicted during the growth and development of rice leaves.

average distances (Lu et al. 2011) between the measured points and the simulated leaf curves were 0.80, 0.85, and 0.89 cm for N1, N2, and N3, respectively, the rates between these average distances and the average length of leaves on the main stem were 3.65, 3.30, and 3.11% for N1, N2, and N3, respectively. At the early jointing stage, the average distances between the measured points and the simulated leaf curves were 1.07, 1.16, and 1.20 cm for N1, N2, and N3, respectively, the rate between these average distances and the average length of leaves on the main stem were 3.20, 3.02, and 2.71% for N1, N2, and N3, respectively. The above analysis showed that the simulated leaf curve was highly consistent with the measured points on the leaf axis (Fig. 16), which indicated that our leaf curve model satisfactorily predicted leaf curvature.

4. Discussion The leaf blade is an important nutrition and photosynthesis organ, and a critical component of crop morphology and canopy structure (Wang et al. 2007). The leaf curve is the vital support for spatial morphology of a crop leaf, and its change can affect the spatial morphology and distribution of the leaf, and affect the crop canopy structure and light distribution (Zhang et al. 2015). Therefore, the modeling

3.2. 3D visualization of the rice leaf Based on our previous studies on the leaf shape model (Zhu et al. 2009), the leaf color model (Zhang et al. 2014a), and the panicle model (Zhang et al. 2014b), we implemented Simulated data A

110

N1

110

N2

110

N3

88

88

66

66

44

44

44

22

22

22

88 66

–40

Measured data

–20

0

40 –40

20

–20

0

20

40

–40

–20

0

20

40

Middle tillering stage B

110

N1

–40

–20

110

N3

110

N2

88

88

88

66

66

66

44

44

44

22

22

22

0

0

20

40

–40

–20

0

20

40

–40

–20

0

20

40

Early jointing stage

Fig. 16 Comparisons between measured and simulated leaf curves for rice leaves at two growth stages. N1, N2, and N3 denote the nitrogen rates of 0, 125, and 250 kg ha–1 respectively in experiment 1.

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of the leaf curve is very critical for virtual crop. In previous, the crop leaf curve was mainly simulated with various mathematical functions and methods (Guo and Li 1999; Zhang et al. 2001; Deng et al. 2005; Watanabe et al. 2005; Shi et al. 2006; Dornbusch et al. 2007; Yang et al. 2008; Liu et al. 2009a; Zheng et al. 2009). However, quadratic functions are not suitable for simulating the leaf curve with small or large degrees of bending. B-spline curve and Hermite curve interpolation methods need coordinate data of the leaf curve to obtain the corresponding interpolation curve, and these two methods lack utility in multivariate crop growth conditions. Dornbusch et al. (2007) constructed a leaf curve model for barley using the Gauss function

2d

36 d

5d

8d

42 d

11 d

45 d

32 d

48 d

Fig. 17 3D dynamic visualization of the 5th leaf of main stem of YD6 at different days after leaf appearance under normal nitrogen rate.

30 d

110 d

60 d

90 d

120 d

140 d

Fig. 18 3D dynamic visualization of leaves on YD6 plant at different days after the first leaf appearance under normal nitrogen rate.

with several input parameters, but did not investigate the dynamics of leaf curves. In this study, the parameters of our curve model were of biological significance, and the 3D measurements by FastScan of leaf curve were frequently performed to accumulate sufficient information on the spatial characteristics of the leaf curve. The method described could build the relationship between model parameters and TT, and was useful for more accurately simulating the dynamic changes of the leaf curve (Fig. 11). Further, the cultivar parameters MMAH7 and the nitrogen impact factor FN were used to quantify the impacts of the cultivar and nitrogen on leaf drooping degree, respectively. Compared to the previous models (Guo and Li 1999; Zhang et al. 2001; Deng et al. 2005; Watanabe et al. 2005; Shi et al. 2006; Dornbusch et al. 2007; Yang et al. 2008; Liu et al. 2009a; Zheng et al. 2009), our leaf curve model exhibits better applicability on simulations of leaf curves with arbitrary bending degrees (including the straight line leaf blade), and thus provides a more accurate simulation of the leaf curve in rice under different growth conditions. The leaf blade morphology affects the crop canopy structure and photosynthesis production (Stewart et al. 2002). Therefore, simulating dynamics of leaf blades is vital for virtual crop. To construct a practical crop visualization model realistically, it is necessary for an organ morphology model to adjust the organ deformation to the constantly changing environment. With rice leaf development, the leaf blade shape gradually changes from a spiral surface, to a curved surface. Previous studies have ignored simulation of curling leaf and its expanding process (Mi et al. 2003; Meng et al. 2005; Watanabe et al. 2005; Yang et al. 2008; Liu et al. 2009a; Ma et al. 2010). In this study, the 3D morphology dynamics of rice leaves was finely simulated during the whole leaf growth and development process. The simulation covers all leaf processes-including leaf appearance, expansion, leaf deformation, and leaf senescence. Our work should improve the simulation precision of rice leaf morphology. We constructed a detailed 3D model of the rice leaf. Integrated with our previous studies of leaf shape (Zhu et al. 2009), leaf color (Zhang et al. 2014a), and leaf blade and sheath spatial geometry, the proposed models were used to implement 3D dynamic visualizations of a single rice leaf and leaves on the rice plant on different growth days by computer visualization technology. The simulated results (Figs. 17 and 18) satisfactorily showed changes in rice leaf spatial morphology in time and space and gave a better prediction than previous studies (Mi et al. 2003; Meng et al. 2005; Watanabe et al. 2005; Yang et al. 2008; Liu et al. 2009a; Ma et al. 2010). The leaf curve and the spatial morphology of the rice leaf were studied under normal rice plant density and

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water conditions with experimental data of two different years. Our results may fluctuate under other density and water conditions, and may be influenced by other factors. New experimental data will be used to revise and validate our model. We will design and conduct further rice field experiments to investigate the influence of other factors on the leaf curve and the spatial morphology of rice leaf in future studies.

5. Conclusion In the present study, a dynamic rice leaf curve was firstly developed using our experimental data. Then, a detailed geometric model of the rice leaf was constructed with spatial geometry and NURBS methods. Finally, integrated with our previous models, the relevant models in this work were employed to achieve 3D simulations of a single rice leaf and leaves on the rice plant on different growing days. The results should improve the research and application of virtual crop.

Acknowledgements The work was supported by the National High-Tech R&D Program of China (2013AA100404), the National Natural Science Foundation of China (31201130, 61471269, 31571566), the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), China, the Natural Science Foundation of Shandong Province, China (BS2015DX001), the Science and Technology Development Project of Weifang, China (2016GX019), and the Doctoral Foundation of Weifang University, China.

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