On the Use of Parallel Genetic Algorithms for Improving the Efficiency of a Monte Carlo-Digital Image Based Approximation of Eelgrass Leaf Area I: Comparing the Performances of Simple and Master-Slaves Structures

Abstract

Eelgrass is a relevant sea grass species that provides important ecological services in near shore environments. The overall contribution of this species to human welfare is so important that upon threats to its permanence that associate to deleterious anthropogenic influences, a vigorous conservation effort has been recently enforced worldwide. Among restoration strategies transplanting plays a key role and the monitoring of the development of related plots is crucial to assess the restoration of the ecological features observed in donor populations. Since traditional eelgrass assessment methods are destructive their use in transplants could lead to undesirable effects such as alterations of shoot density and recruitment. Allometric methods can provide accurate proxies that sustain nondestructive estimations of variables required in the pertinent assessments. These constructs rely on extensive data sets for precise estimations of the involved parameters and also depend on precise estimations of the incumbent leaf area. The use of electronic scanning technologies for eelgrass leaf area estimation can enhance the nondestructive nature of associated allometric methods, because the necessary leaf area assessments could be obtained from digital images. But when a costly automatic leaf area meter is not available, we must rely on direct image processing, usually achieved through computationally costly Monte Carlo procedures. Previous results show that the amendment of simple genetic algorithms could drastically reduce the time required by regular Monte Carlo methods to achieve the estimation of the areas of individual eelgrass leaves. But even though this amendment, the completion of the task of measuring the areas of the leaves of a data set with an extension, as required for precise parameter estimation, still leads to a burdensome computational time. In this paper, we have explored the benefits that the addition of a master-slave parallel genetic algorithm to a Monte Carlo based estimation routine conveys in the aforementioned estimation task. We conclude that unless a suitable number of processors are involved, and also the proper mutation and crossover rates are contemplated the efficiency of the overall procedure will not be noticeably improved.

Appendix

Allometric methods have provided convenient tools that describe biomass or other traits of seagrasses [15, 32, 33]. With the aim of making this contribution self-contained, in this appendix we illustrate how allometric models can substantiate indirect assessment tools commonly used in eelgrass monitoring and that relate to the associated leaf area. Particularly, we will review the substantiation of the leaf length times width proxy of Eq. (11), which provides handy approximations to the involved leaf area, and also the verification of the formulae in Eqs. (6) and (8) that yield allometric proxies based on leaf area for the average of the biomass of leaves in shoots and corresponding leaf growth rates, respectively. In order to achieve our goals we will keep here the notation presented in the introduction and will let $l(t)$ denote the length of a representative eelgrass leaf at time t, this defined as the extent between the base and the tip. Likewise, we will let $h(t)$ stand for a typical width measurement taken at a fixed point over the span of leaf length. We will also let $a(t)$ and $w(t),$ respectively, denote the linked area and dry weight of leaves.

By taking into account the architecture and growth form of eelgrass leaves, we can use allometric models to scale leaf dry weight in terms of the matching length or area [16, 17]. These allometric assumptions can also be represented through the differential equations

$$\frac{1}{w}\frac{{{\text{d}}w}}{{{\text{d}}t}} = \frac{\theta }{l}\frac{{{\text{d}}l}}{{{\text{d}}t}}$$

(18)

$$\frac{1}{w}\frac{{{\text{d}}w}}{{{\text{d}}t}} = \frac{k}{a}\frac{{{\text{d}}a}}{{{\text{d}}t}},$$

(19)

where k and θ are positive constants [34]. Now from Eqs. (18) to (19), we can write

$$\frac{{{\text{d}}a}}{{{\text{d}}t}} = \frac{ca}{l}\frac{{{\text{d}}l}}{{{\text{d}}t}},$$

(20)

where $c = \theta /k$.

Since it has been observed, that through elongation, leaf architecture and growth form in eelgrass lead to slight values for width increments, it is sound to assume that for a characteristic leaf $h(t)$ remains fairly constant, and thus

$$\frac{{{\text{d}}h}}{{{\text{d}}t}} = 0$$

(21)

Hence using this assumption and Eq. (20) we may write

$$\frac{{{\text{d}}a}}{{{\text{d}}t}} = \frac{ca}{l}\frac{{{\text{d}}l}}{{{\text{d}}t}} + \frac{ca}{l}\frac{{{\text{d}}h}}{{{\text{d}}t}}$$

(22)

Moreover, if we consider the function $U(t)$ defined through

$$U(t) = ch(t)l(t) + c\int {e(t)\frac{{{\text{d}}\Phi }}{{{\text{d}}t}}} ,$$

(23)

where

$$e(t) = a(t) - h(t)l(t),$$

(24)

and

$$\Phi (t) = ln(h(t)l(t)),$$

(25)

then it is straightforward to show that

$$\frac{{{\text{d}}a}}{{{\text{d}}t}} = \frac{{{\text{d}}U}}{{{\text{d}}t}}.$$

(26)

It turns out that there exists a constant p such that we have

$$a(t) = ch(t)l(t) + p + z(t),$$

(27)

where

$$z(t) = c\int {e(t)\frac{{{\text{d}}\Phi }}{{{\text{d}}t}}} .$$

(28)

We can then assume that if $h(t)$ has been selected in a way that $e(t)$ in Equation (24) is negligible, then $z(t)$ will also be negligible, so we can consider the linear regression model criterion

$$a(t) = ca_{p}(t) + p,$$

(29)

where

$$a_{p} (t) = h(t)l(t).$$

(30)

This result can provide criteria to test the assumption that $h(t)l(t)$ provides a reliable proxy for $a(t).$ This can indeed be ascertained if after considering data on eelgrass leaf area $a(t),$ lengths $l(t)$ and suitably taken width $h(t)$ measurements, the linear regression model of Eq. (29) was reliably identified as having a slope near 1 and an intercept 0. As it is explained in Echavarría-Heras et al. [19] who tested the criterion of Eq. (29) for eelgrass leaf width and length data the proxy $h(t)l(t)$ will provide accurate assessments of leaf area $a(t)$ only in case width is measured half way between the base and tip of the leaf. This establishes the result

$$a(t) = a_{p} (t) + \varepsilon_{p} (t),$$

(31)

where $\varepsilon_{p} (t)$ is the involved approximation error.

By assuming that Eqs. (18) and (19) hold, Echavarría-Heras et al. [35] generalized the model of Hamburg and Homann [15] and established that $w(t)$ can be scaled by using an allometric model of the form

$$w(t) = \rho l(t)^{\alpha } h(t)^{\beta } ,$$

(32)

where $\rho > 0$, $\alpha$ and $\beta$ are constants. We will first show that we can similarly represent the above scaling for $w(t)$ only in terms of the considered proxy for leaf area, that is, $a_{p} (t) = h(t)l(t)$. To be precise, rearranging Eq. (32) we can obtain the alike form

$$w(t) = \rho a_{p} (t)^{\alpha } h(t)^{\theta } ,$$

(33)

where $\theta = \beta - \alpha$. Now, let’s consider a time increment $\Delta t \ge 0$. Then direct algebraic manipulation of Eq. (33) yields

$$\frac{{w(t +\Delta t)}}{w(t)} = \left( {\frac{{a_{p} \left( {t +\Delta t} \right)}}{{a_{p} (t)}}} \right)^{\alpha } \left( {1 + \frac{{\Delta h}}{h(t)}} \right)^{\theta }$$

(34)

where $\Delta h = h\left( {t +\Delta t} \right) - h(t).$ Since we observed that during a growing interval of negligible length $\Delta t$ eelgrass leaf architecture and growth form yield small values for $\Delta h$, we may expect the ratio involving $a_{p} (t)$ in Eq. (34) to be dominant. On the other hand, if we assume that $w(t)$ can be allometrically scaled in terms of $a_{p} (t)$, then as it is stated by Eq. (19), there will be a constant m [34] such that

$$\frac{1}{w}\frac{{{\text{d}}w}}{{{\text{d}}t}} = \frac{m}{{a_{p} }}\frac{{{\text{d}}a_{p} }}{{{\text{d}}t}}$$

(35)

$$\frac{{w(t +\Delta t)}}{w(t)} = \frac{{a_{p} \left( {t +\Delta t} \right)^{m} }}{{a_{p} (t)^{m} }}.$$

(36)

Therefore, the dominance of the term containing $a_{p} (t)$ in Eq. (34) is consistent with the assumption of an allometric scaling of $w(t)$ in terms of $a_{p} (t) .$ Besides, supposing that Eq. (36) is satisfied for all values of t, then both sides must be equal to a constant. Let c be such a constant. Then we will have

$$w(t) = ca_{p} (t)^{k} ,$$

(37)

The result for $w(t)$ in the form of Eq. (34) can be also obtained for real leaf area values $a(t)$. In fact, by recalling that $a_{p} (t)$ is an estimator for the real area $a(t)$ then in accordance with Eq. (31) there exist an approximation error $\varepsilon_{p} (t).$ Moreover $a_{p} (t)$ would provide a reliable approximation for $a(t)$ if for the ratio

$$s(t) = \frac{{\varepsilon_{p} (t)}}{a(t)},$$

(38)

we set the consistency condition

$$s(t) \ll 1,$$

(39)

Solving for $a_{p} (t)$ in Eq. (31) and replacing in Eq. (34), after few steps we can obtain

$$\frac{{w(t +\Delta t)}}{w(t)} = \left( {\frac{{a(t +\Delta t)}}{a(t)}} \right)^{m} \left( {1 - \frac{{s\left( {t +\Delta t} \right) - s(t)}}{1 - s(t)}} \right)^{m} .$$

(40)

By virtue of the consistency condition (38), the right-hand side of Eq. (40) will be dominated by the ratio $a(t +\Delta t)^{m} a(t)^{ - m}$. Therefore, for $\alpha$ and $\beta$ constants can also propose an allometric model, for eelgrass leaf dry weight $w(t)$ and corresponding leaf area values $a(t)$, e.g.

$$w(t) = \beta a(t)^{\alpha } .$$

(41)

Readily, using Eq. (41) from Eq. (1) we can obtain an allometric proxy for $w_{s} (t)$, this is denoted through the symbol $w_{s} (\alpha ,\beta ,t)$, and it turns out to be,

$$w_{s } (\alpha ,\beta ,t) = \sum\limits_{nl(s)} {\beta a(t)^{\alpha } } .$$

(42)

Since the parameters $\alpha$ and $\beta$ in Eq. (41) are identified by means of regression methods, the uncertainty ranges for their estimates imply that $w_{s} (t)$ and $w_{s} (\alpha ,\beta ,t)$ are linked through the expression

$$w_{s} (t) = w_{s} (\alpha ,\beta ,t) + \in_{s} (\alpha ,\beta ,t),$$

(43)

that is, $w_{s} (\alpha ,\beta ,t)$ provides only an approximation to the true value of $w_{s} (t)$ being $\upvarepsilon_{s} (\alpha ,\beta ,t)$ the associated approximation error. Similarly from Eq. (42), we can obtain an allometric proxy for $w_{ms} (t),$ the average leaf biomass in shoots at a time t. This surrogate which we denote here through the symbol $w_{ms} (\alpha ,\beta ,t)$ is given by

$$w_{ms} (\alpha ,\beta ,t) = \frac{{\sum\nolimits_{ns(t)} w_{s} (\alpha ,\beta ,t)}}{ns(t)}.$$

(44)

Similarly, we have

$$w_{ms} (t) = w_{ms} \left( {\alpha ,\beta ,t} \right) + \upvarepsilon_{ms} \left( {\alpha ,\beta ,t} \right),$$

(45)

where $\upvarepsilon_{ms} \left( {\alpha ,\beta ,t} \right)$ is the resulting approximating error.

Moreover, as it has been explained by Echavarría-Heras et al. [17], we can use Eqs. (3) and (4) in order to obtain an allometric approximation for $L_{sg} (t,\Delta t)$, which here we symbolize by means of $L_{sg} \left( {\alpha ,\beta ,t,\Delta t} \right)$ and that we express through

$$L_{sg} \left( {\alpha ,\beta ,t,\Delta t} \right) = \frac{{\sum \nolimits_{nl(s)}\Delta w\left( {\alpha ,\beta ,t,\Delta t} \right)}}{{\Delta t}},$$

(46)

where $\Delta w\left( {\alpha ,\beta ,t,\Delta t} \right)$ is an allometric surrogate for $\Delta w(t,\Delta t).$

Therefore, Eq. (41) yields

$$\Delta w\left( {\alpha ,\beta ,t,\Delta t} \right) = \beta a\left( {t +\Delta t} \right)^{\alpha } - \beta a(t)^{\alpha } .$$

Factoring $a\left( {t +\Delta t} \right)^{\alpha }$ then considering that $a(t +\Delta t) = a(t) +\Delta a$ where $\Delta a$ stands for the increment in leaf area attained over the interval $\left[ {t,t +\Delta t} \right]$ we get,

$$\Delta w\left( {\alpha ,\beta ,t,\Delta t} \right) = \beta a(t +\Delta t)^{\alpha } \left( {1 - \left( {1 - \rho (t,\Delta t)} \right)^{\alpha } } \right)$$

where

$$\rho (t,\Delta t) = \frac{{\Delta l}}{{a(t +\Delta t)}},$$

and by letting

$$\delta (t,\Delta t) = \left( {1 - \left( {1 - \rho (t,\Delta t)} \right)^{\alpha } } \right)$$

we obtain

$$\Delta w\left( {\alpha ,\beta ,t,\Delta t} \right) = \beta a(t +\Delta t)^{\alpha }\Delta (t,\Delta t).$$

Therefore, from (46) we have

$$L_{sg} \left( {\alpha ,\beta ,t,\Delta t} \right) = \frac{{\sum\nolimits_{nl(s)} {\beta a(t +\Delta t)^{\alpha }\Delta (t,\Delta t)} }}{{\Delta t}}.$$

(47)

Similarly, if $L_{g} \left( {\alpha ,\beta ,t,\Delta t} \right)$ stands for the allometric proxy for $L_{g} (t,\Delta t)$, then according to Eq. (4), this is given by

$$L_{g} (\alpha ,\beta ,t,\Delta t) = \frac{{\sum\nolimits_{{ns(t,\Delta t)}} {L_{sg} (\alpha ,\beta ,t,\Delta t)} }}{{ns(t,\Delta t)}},$$

(48)

this explains Eq. (8) and in turn we have

$$L_{g} (t,\Delta t) = L_{g} (\alpha ,\beta ,t,\Delta t) + \upvarepsilon_{g} (\alpha ,\beta ,t,\Delta t)$$

(49)

being $\upvarepsilon_{g} (\alpha ,\beta ,t,\Delta t)$ the involved approximation error.