1. Introduction

The CIE MES2 system [1] was proposed in 2010 as an intermediate between the USP-system developed by Rea et al. [2] in 2004, and the Move-system developed by Goodman et al. [3] in 2007. In the MES2 system, the spectral luminance efficiency function in the mesopic range from $0.005 cd m^{-2}$ to $5.0 cd m^{-2}$ is denoted by $V_{mes}\left(\lambda \right)$, and defined as

If we let

Moreover, the mesopic luminance $L_{mes }$, and the parameter $m$ are related by

Thus, if we let

Recently, Gao et al. [4] have shown that, with the values for $a$ and $b$ given by (6), the equation $F\left(m\right)=0$ may have either no solution or more than one, and, in agreement with Shpak et al. [5], they have recommended that the values for the parameters $a$ and $b$ should be better replaced by

Gao et al. [4] have shown that, with the new values for $a$ and $b$ given by (8), the equation $F\left(m\right)=0$ has a unique solution between $0$ and $1$, when the following condition is satisfied:

Thus, in this paper we will use the values for $a$ and $b$ given by (8), together with the remaining equations of the CIE MES2 system.

Note first that $a$ and $b$ given by (8), also satisfy

Now from (4), we note that when $m=0,$ we have $L_{mes}=L_s,$ and when $m=1,$ we have $L_{mes}=L_p.$ Hence, for the continuity of the luminance scale, from scotopic via mesopic to photopic visions, we should have:

Henceforth, in this paper all luminance units ($cd m^{-2}$) will be missed for simplicity.

Let

Note that, if the sequence ${\left\{m_n\right\}}_{n\ge 0} ,$ generated by the iteration method (14), converges to $m^{*}$, then we have

Hence, $m^{*}$ is a fixed point of the function $g,$ and this is the reason this iteration method is also named in the literature [6] as fixed-point iteration.

It is clear that the function $g,$ or the fixed-point iteration method, is dependent on both, $L_p$ and the ratio $L_s/L_p.$ In this paper, the ratio $L_s/L_p.$ will be denoted in an abbreviated form as $S/P,$ i.e.,

Recently, Gao et al. [4] and Shpak et al. [5] have reported that the convergence of the fixed-point iteration method depends on the ratio $S/P,$ and for large values of $S/P$ the method does not converge. Shpak et al. [5] suspected that, to achieve convergence, the ratio $S/P$ cannot be larger than 17. Since currently the fixed-point iteration method is officially recommended by the CIE MES2 system [1], and it may be also implemented in automatic devices, it is appropriate to provide a full theoretical consideration on the convergence of such method. This is the main goal of this paper.

2. Convergence analysis for fixed-point iteration method

We start quoting a result about a sufficient condition for the convergence of the fixed-point iteration method.

Lemma 1: (Fixed-Point Theorem [6, page 62, Chapter 2])

Let $f$ be a continuous function defined on $\left[c, d\right]\subset R,$ such that $f\left(x\right)\in [c, d]$ for all $x\in [c, d]$. Suppose, in addition, that $f^{\prime },$ the derivative of $f,$ exists on $\left(c, d\right),$ and there is a constant $k\in (0, 1)$ such that $\left|f^{\prime }(x)\right|\le k,$ for all $x\in [c, d]$. Then, for any number $x_0$ in $\left[c, d\right],$ the sequence ${\left\{x_n\right\}}_{n\ge 0}$ defined by

Now, we provide another sufficient condition for the convergence of the fixed-point iteration method.

Theorem 1: Let $f$ be a monotonically increasing and continuous function defined on $\left[c, d\right]\subset R,$ such that $f\left(x\right)\in [c, d]$ for all $x\in [c, d]$. Then, for any $x_0$ in $\left[c, d\right],$ the sequence ${\left\{x_n\right\}}_{n\ge 0}$ defined by (18) converges to a fixed point $x^{*},$ in $\left[c, d\right],$ of the function $f$.

Proof: It is obvious that ${\left\{x_n\right\}}_{n\ge 0}\subset [c, d]$. If $x_0=x_1=f\left(x_0\right),$ then $x_n=x_0$ for all $n\ge 1,$ and $x_0$ is a fixed point of the function $f.$ Now, suppose $x_0<x_1.$ In this case, it can be shown that the sequence ${\left\{x_n\right\}}_{n\ge 0}$ is monotonically increasing, and therefore has a limit $x^{*}\in \left[c, d\right]$. Due to the continuity of $f,$ we have

Now, from Theorem 1, we have:

Theorem 2: If $S/P\le 1,$ then the fixed-point iteration method given by (14) is convergent.

Proof: First, we note that when $S/P=1,$ the fixed-point iteration method (14) is convergent. In fact, when $S/P=1,$ we have $L_{mes}=L_p ,$ for any $m,$ and therefore

Thus, for any $m_0\in \left(0, 1\right),$ we have $m_n=g\left(m_{n-1}\right)=a+b lo{g}_{10}\left(L_p\right),$ for all $n\ge 1.$ Therefore, the sequence ${\left\{m_n\right\}}_{n\ge 0}$ converges to $a+b lo{g}_{10}\left(L_p\right),$ which is the unique fixed point of $g$ in $\left[0, 1\right].$

Now, suppose that $S/P<1,$ and let

It is easy to check that $g^{\prime }(m)$ is given by

Since $S/P<1,$ and $U\left(m\right)$ and $W\left(m\right)$ are positive for $m\in \left(0, 1\right),$ we have $g^{\prime }\left(m\right)>0$ for all $m\in \left(0, 1\right),$ and therefore, $g$ is a monotonically increasing and continuous function on $\left[0, 1\right].$ Moreover, since $S/P<1,$ then $0.005\le L_s<L_p<5,$ and therefore $0\le g\left(0\right)<g\left(1\right)<1,$ i.e., $g\left(m\right)\in [0, 1]$ for all $m\in \left[0, 1\right].$ And by Theorem 1, the fixed-point iteration method given by (14) is convergent.

We note that Lemma 1 cannot be applied to prove Theorem 2, since $g^{\prime }\left(m\right)$ is greater than $1,$ when $S/P$ and $m$ are sufficiently small.

Now, in order to investigate the convergence of the fixed-point iteration method when $S/P>1,$ we need the second derivative of the function $g,$ given by

Thus, it is clear that both $A$ and $B$ are linear functions of $S/P,$ and therefore $m_w=-B/A$ is a function of the ratio $S/P,$ i.e., $m_w=m_w\left(S/P\right).$ Fig. 1 shows the variation of the function $m_w=m_w\left(S/P\right)$ (vertical axis) with ratio $S/P$ (horizontal axis). The dotted vertical line corresponds to $S/P=1/C$. It can be seen that when $S/P<1/C$, we have that $m_w\left(S/P\right)$ is negative, and approaches to $-\infty$ when $S/P$ approaches from bellow to $1/C$. However, when $S/P>1/C, m_w\left(S/P\right)$ is positive and monotonically decreasing with respect to $S/P;$ and $m_w\left(S/P\right)$ approaches to $\infty$ when $S/P$ approaches from above to $1/C$. Furthermore, it can be easily seen that

 

Fig. 1 The function $m_w\left(S/P\right).$ Vertical dotted line is $S/P=1/C$.

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With the expression of $g^{″}\left(m\right)$ given by (23), we have:

Theorem 3: If $1<S/P\le (2-C)/C$, then $g^{″}\left(m\right)\ge 0$ for $0\le m\le 1.$ If $S/P>(2-C)/C$, then

Proof: Suppose that $1<S/P<1/C$ . Then, from (24), we have $A<0$ and $B<0$. And from (23), it is obvious that $g^{″}\left(m\right)\ge 0$ if, and only if, $Am+B\le 0$, i.e., $m\ge -B/A$. Since $-B/A\le 0$, we have $g^{″}\left(m\right)\ge 0$ for $0\le m\le 1.$

For $S/P=1/C$, it is clear from (24) that $A=0$ and $B<0$. Therefore, from (23), $g^{″}\left(m\right)>0$.

Now, suppose that $1/C<S/P\le (2-C)/C$. Then, from (24), we have $A>0$ and $B<0$. And from (23), it is obvious that $g^{″}\left(m\right)\ge 0$ if and only if $Am+B\le 0$, i.e., $m\le -B/A$. Since $-B/A=m_w\left(S/P\right)$ is a decreasing function of $S/P$, as it can be easily shown, we have

and, again $g^{″}\left(m\right)\ge 0$ for $0\le m\le 1.$

Finally, suppose $S/P>(2-C)/C$. Then, from (24), we have $A>0$ and $B<0$, and now, since $-B/A=m_w\left(S/P\right)$ is a decreasing function of $S/P$, we get

Therefore, since $A>0,$ we have $Am+B<0$ for $0\le m<m_w\left(S/P\right),$ $Am+B=0$ for $m=m_w\left(S/P\right),$ and $Am+B>0$ for $m_w\left(S/P\right)<m\le 1;$ and consequently $g^{″}\left(m\right)$ is positive in $\left[0, m_w\left(S/P\right)\right)$, negative in $\left(m_w\left(S/P\right), 1\right]$, and equal zero when $m=m_w\left(S/P\right).$

By using Theorem 3 above, we can prove the following result about the derivative of the function $g$ given by (13).

Theorem 4: Let

If $1<S/P<C_2$, then $g^{\prime }(m),$ given by (22), is negative for $0\le m\le 1.$ Moreover, there exists a constant $k\in (0, 1)$ such that $\left|g^{\prime }(m)\right|\le k,$ for $0\le m\le 1.$

Proof: From (22), we have

Finally, from Theorem 3, if $S/P>(2-C)/C,$ then $g^{\prime }\left(m\right)$ is increasing in $\left[0, m_w\left(S/P\right)\right)$, and decreasing in $\left(m_w\left(S/P\right), 1\right].$ Hence, we have

Figure 2 shows $h\left(S/P\right)$ versus the ratio $S/P$ for $S/P>(2-C)/C\approx 3.9751$. It is obvious that $h\left(S/P\right)$ is an increasing function of the ratio $S/P.$ It can also be checked that $h\left(C_2\right)=1.$ Since $S/P<C_2,$ there exists $\epsilon \left(0<\epsilon <1\right),$ such that $S/P<C_2-\epsilon .$ Thus, from (31), we have, for $0\le m\le 1,$

 

Fig. 2 The function $h\left(S/P\right)$ for $S/P>(2-C)/C\approx 3.9751$.

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Let $g$ be given by (13). Since $L_p<5,$ we have

Note that with the values for $a$ and $b$ given by (8), we have $a+b lo{g}_{10}5=1,$ and therefore $g_0\le 1$ when $L_s\le 5,$ and $g_0>1$ when $L_s>5.$ From Theorem 4, if $1<S/P<C_2,$ then $g\left(m\right)$ is a decreasing function of $m$. Therefore, there exists $m_L\in \left(0, 1\right),$ depending on the ratio $S/P,$ such that $g\left(m_L\right)=1.$ For $m_L=m_L\left(S/P\right),$ it can be shown that

Figure 3 shows $m_L\left(S/P\right)$ (solid curve), for some given $L_p$ ($L_p=3.0$ in black, and $L_p=0.5$ in blue), versus $S/P,$ between $5/L_p$ and $C_2$. It is also shown $g_1$ (dotted line), and it can be seen that $m_L\left(S/P\right)$ is less than $g_1,$ for a given $L_p.$

 

Fig. 3 The functions $m_L\left(S/P\right)$ (solid curves) and $g_1$ (dotted lines) for $S/P$ between $5/L_p$ and $C_2$. Black curves correspond to $L_p=3.0,$ and blue curves correspond to $L_p=0.5$.

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Now, we can state the following result regarding the function $g$.

Theorem 5: Suppose a fixed $L_p<5,$ and $1<S/P<C_2$. Then, we have:

Proof: Suppose that $1<S/P<C_2$. Then, from Theorem 4, $g$ is a decreasing function on $\left[0, 1\right]$, and therefore $g_0>g_1>0.$

If $L_s\le 5,$ then $g_0\le 1,$ and $g\left(m\right)\in \left[g_1, g_0\right]\subseteq [0, 1],$ for $m\in \left[0, 1\right].$

If $L_s>5,$ then $g_0>1,$ and, since $g$ is a decreasing function on $\left[0, 1\right]$, there exists $m_L=m_L\left(S/P\right)\in (0, 1)$ such that $g\left(m_L\right)=1.$ By solving the equation $g\left(m\right)=1,$ it can be found that $m_L=m_L\left(S/P\right)$ is given by (35). Moreover, we have

Therefore, for any fixed $L_p<5,$ $m_L$ is an increasing function of $S/P$ as shown by Fig. 3, where the solid black curve corresponds to $L_p=3,$ and the solid blue curve corresponds to $L_p=\mathrm{0.5.}$ It can be seen that for $S/P<C_2,$ we have $m_L\left(S/P\right)<g_1.$ For proving this, it is enough to show that $m_L\left(C_2\right)\le g_1,$ since $m_L$ is an increasing function of $S/P.$ If we define the function

The left-hand side of the above inequality is a constant, while the right-hand side depends on $L_p$. If we let

Hence, $p$ is a decreasing function of $L_p$ in the interval $\left(5/C_2, C_3\right)$, and increasing in the interval $\left(C_3, 5\right).$Furthermore, it can be verified that

Therefore, the inequality (40) is true for $L_p\in \left(5/C_2, 5\right),$ and consequently $q,$ given by (37), is an increasing function of $L_p$ in the interval $\left(5/C_2, 5\right)$. Then, $q\left(L_p\right)\le q\left(5\right)=0,$ which is equivalent to $m_L\left(C_2\right)\le g_1,$ and then, $m_L\left(S/P\right)<g_1$ for $S/P<C_2,$ which concludes the proof.

We are now ready to state another convergence theorem for the fixed-point iteration method when $S/P>1$.

Theorem 6: Suppose a fixed $L_p<5,$ and $1<S/P<C_2$. Then, the fixed-point iteration method given by (14), is convergent for any $m_0\in \left[0, 1\right]$ when $L_s\le 5,$ and for any $m_0\in \left[m_L\left(S/P\right), 1\right]$ when $L_s>5$.

Proof: From Theorem 4, there exists a constant $k\in (0, 1)$ such that $\left|g^{\prime }(m)\right|\le k,$ for $0\le m\le 1.$ Moreover, if $L_s\le 5,$ from Theorem 5 we have $g\left(m\right)\in \left[0, 1\right],$ for $m\in [0, 1]$. Then, by using Lemma 1, the fixed-point iteration method given by (14) is convergent for any $m_0\in \left[0, 1\right]$. In the case $L_s>5,$ again from Theorem 5 we have $g\left(m\right)\in \left[m_L\left(S/P\right), 1\right],$ for $m\in \left[m_L\left(S/P\right), 1\right],$ and according to Lemma 1, the fixed-point iteration method given by (14) is convergent for any $m_0\in \left[m_L\left(S/P\right), 1\right]$.

Theorems 2 and 6 provide a sufficient condition for the convergence of the fixed-point iteration method (14), namely $S/P<C_2$. However, the fixed-point iteration method may also be convergent when this condition fails, i.e., $S/P>C_2$.

Moreover, in both Theorems 2 and 6, the convergence is guaranteed when choosing properly the initial value $m_0$. CIE recommended the initial guess $m_0=0.5$ for the fixed-point iteration method, i.e., the middle point of the interval $\left[0, 1\right].$ Gao et al. [4] proved that the function $g$ has a unique fixed point in $\left[0, 1\right],$ which is equivalent to assert that the equation $F\left(m\right)=0,$ where $F$ is given by (7), has a unique solution in $\left[0, 1\right].$ However, when $L_s>5,$ Theorem 6 indicates that the initial guess $m_0$ should be in the interval $\left[m_L\left(S/P\right), 1\right]$ to ensure the convergence of the fixed-point iteration method. It is well known that the performance of the iteration method depends on the initial guess $m_0$. Therefore, if $m_0$ is chosen “close” to the fixed point $m^{*},$ then the number of iterations to get a good estimation of $m^{*}$ will be smaller. From the results presented in this paper, it follows that

The above information can help to choose a “better” initial guess $m_0$, and will be discussed in the next section.

3. Performance of fixed-point iteration method with new initial strategy

We have shown that the fixed-point iteration method (14) is convergent for $S/P<C_2,$ whenever a proper initial value $m_0$ is chosen. In order to test numerically this result, we have taken some values for $L_p$ from $0.1$ to $4.9,$ namely $0.1, 0.3, 0.5, 0.7, \dots , 4.9,$ that is a total number of $25$ values for $L_p.$ Similarly, we have taken some values for the ratio $S/P$ from $0.1$ to $1,$ namely $0.1, 0.15, 0.2, 0.25, \dots , 1,$ and from $1.1$ to $18.1$ we have taken the values $1.1, 2.1, 3.1, 4.1, \dots , 18.1,$ which make a total number of $37$ values for the ratio $S/P.$ Therefore, we have considered $925=25\times 37$ cases for testing the performance of the fixed-point iteration method with the original initial guess $m_0=0.5$ (the current CIE MES2 method [1]), and also with a new initial strategy. Regarding the selected range of values for the ratio $S/P$ from 0.1 to 18.1, it must be remarked that, for most current conventional light sources, these values are low, in a range around 0.0-3.0 [7]. However, higher values, up to a maximum of around 73.0, which are associated to blue monochromatic lights, are also possible [5]. For example, Nizamoglu et al. [8] have reported $S/P$ values of 5.15 for nanocrystal hybridized LEDs, and previous researchers [4] have considered $S/P$ values up to 50, for theoretical light sources based on Hung et al. method [9]. The initial value $m_0$ in the new strategy, that we propose, is given by

Figure 4 shows the contours with the number of iterations needed for the convergence of the fixed-point iteration method, using the initial value $m_0=0.5,$ as recommended by CIE [1]. In $921$ cases, from the total of $925,$ the convergence is obtained when computing less than $100$ iterations. In two cases, namely $S/P=18.1$ and $L_p=4.3,$ and $S/P=18.1$ and $L_p=4.5,$ have been necessary $120$ and $157$ iterations, respectively, for the convergence. And for $S/P=18.1$ and $L_p=4.7,$ and $S/P=18.1$ and $L_p=4.9,$ the convergence is not achieved after $200$ iterations.

 

Fig. 4 Contours plots with the number of iterations for the convergence of the fixed-point iteration method with initial value $m_0=0.5,$ as a function of $S/P$ and $L_p$. Different colors represent different numbers of iterations needed, as shown on the vertical bar on the right.

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Figure 5 shows the contours with the number of iterations needed for the convergence of the fixed-point iteration method, using as initial value $m_0,$ in each case, the value provided by the new strategy proposed in (47). Now, in all $925$ cases under study, the convergence is obtained when computing less than $70$ iterations. This fact proves the validity of our convergence analysis. In addition, let $N_1$ and $N_2$ be the number of iterations needed for the convergence, when using $m_0=0.5$ and the initial value provided by (47), respectively. It has been found that $N_2=N_1+1$ only for $3$ cases, $N_2=N_1$ for $175$ cases, and $N_2<N_1$ for $747$ cases. Thus, with the proposed new initial strategy (47), the fixed-point iteration method converges faster than with the CIE recommended initial value [1] in the $80\%$ of the cases.

 

Fig. 5 Contours plots with the number of iterations for the convergence of the fixed-point iteration method with the new initial strategy (see (47)) as a function of $S/P$ and $L_p$. Different colors represent different numbers of iterations needed, as shown on the vertical bar on the right.

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4. Conclusions

The MES2 system was recommended by CIE [1] to compute the mesopic luminance, using a fixed-point iteration method (see (14)). In this process of computation of the mesopic luminance, a numerical solution of a nonlinear equation $F\left(m\right)=0$ is searched (see (7)). Shpak et al. [5] have proposed new values for the parameters $a$ and $b$ involved in that equation (see (8)). With these new values for $a$ and $b$, Gao et al. [4] have shown that the nonlinear equation $F\left(m\right)=0$ has a unique solution in $(0, 1)$, whenever a condition for $L_s$ and $L_p$, given by (3), is satisfied (see (9)). However, Gao et al. [4] and Shpak et al. [5] have found that the fixed-point iteration method may be not convergent for large values of $S/P=L_s/L_p$ . Shpak et al. [5] pointed out that this ratio should not be larger than 17 in order to have convergence. In this paper a theoretical consideration on the convergence has been given, and it has been found that the fixed-point iteration method converges when using appropriate initial values, and the ratio $S/P$ is smaller than $C_2\approx 18.1834$. For values of $S/P$ larger than $C_2$, there is no guarantee for the convergence of the fixed-point iteration method. Values of the ratio $S/P$ for current light sources are usually lower than 3.0 [7], but Nizamoglu et al. [8] have reported higher $S/P$ values of 5.15 for nanocrystal hybridized LEDs. The theoretical upper limit for the ratio $S/P$ is around 73.0 [5]. Therefore, our current analyses considering sources with high $S/P$ values make sense, because we are proposing a valid CIE method for both current and future light sources. Moreover, for values of $S/P$ smaller than $C_2$, a new strategy (47) for the choice of the initial value $m_0$ has been proposed. In many cases, this new strategy produces a remarkable reduction of the number of iterations needed to achieve convergence, compared when using $m_0=0.5$ as initial value, as currently recommended in CIE MES2 method [1].