Artificial Intelligence in Light-Source Design

Snjezana Soltic and Andrew N. Chalmers

#### Abstract

The advent of light-emitting diode (LED) light sources has led to a new freedom in the design of light-source spectra, and it is now possible to optimise for different source performance parameters, which is the principal aim of the authors' work. LEDs and lasers are real or potential light sources, and are inherently monochromatic, that is, narrow-band sources, with typical optical bandwidths in the range 20–40 nm (nanometres) for LEDs and 1–5 nm for diode lasers. Mixtures of three or more can be used to produce nominally white light of the type acceptable for general purpose lighting. It is a characteristic of all types of sources that there is a trade-off between good colour properties and high efficiencies, and the methods described here are directed towards an optimum combination of such parameters. This chapter will explain the use of differential evolution (DE) as a highly effective heuristic approach to optimisation, and proceeds to explain the structure and operation of a DE algorithm designed as an optimisation tool for such purposes.

Keywords: differential evolution, optimisation, LED lighting, luminous efficacy, correlated colour temperature, colour rendering and fidelity

#### 1. Introduction

It has been stated by some authors that white LEDs are the eco-friendly light sources for the twenty-first century [1]. White light in LEDs can be produced either (i) by using blue (or violet) light to irradiate a phosphor that emits yellow light [2, 3] or (ii) by combining the outputs of a group of monochromatic LEDs [1, 4, 5]. In approach (i) the blue or violet is combined with the yellow to produce white light. We have adopted approach (ii) in which three (or, preferably, four) primary colours are mixed in appropriate proportions to produce high-quality white light with higher efficiency than approach (i) since there is no loss of energy as in the down-conversion in the phosphor [5, 6]. Both approaches aim to design LED-based white-light sources characterised by high colour rendering and fidelity properties combined with high luminous efficacy. It is an established fact in light-source design that these two qualities are generally contravariant, and that both depend on the emitted spectrum of the source. We have found that fine-tuning of the intensities of the individual sources, as well as the selection of their individual peak wavelengths, enables us to define the best balance of the significant spectral properties of the mixture. Our adopted optimisation technique implements a trial-and-error search algorithm within an n-dimensional space {I1, …, Ii, …, In} until an acceptable spectral power distribution (SPD) is found for the white-light mixture [5, 7, 8].

In this chapter, we will describe an approach to the design of the optimum LED mixture using a simple global optimisation algorithm based on differential evolution (DE) as proposed by Storn and Price [9]. We have optimised the SPDs of mixtures of real and simulated LEDs and diode lasers. For this purpose, a MATLAB program was developed that optimises the SPD of an arbitrary mixture of individual SPDs. Specific versions of this program were developed to implement several different techniques for defining (and calculating) the colour properties of sources.

This chapter proceeds as follows. In Section 2, the principles of the DE algorithm are described in detail. In Section 3, the theory of colour rendering and luminous efficacy is given, along with other relevant lighting terms and their usage. Section 4 presents a selection of key experimental results and comparisons and, finally, we state our conclusions in Section 5.

## 2. Differential evolution

Differential evolution is a powerful population-based evolutionary algorithm suitable for the optimisation of real-value multi-modal non-linear and nondifferentiable objective functions fo(x1, x2, ..., xn) [9, 10]. DE is simple and has proven to be powerful in solving a number of benchmark problems [11, 12].

The search for an optimal solution starts with a population of P randomly created solution vectors {v1, v2, ..., vP}. This population is maintained constant during the optimisation process, during which the solution vectors (i.e., candidate SPDs) undergo mutation, crossover, evaluation and selection over a number of generations G. The best choices for both the population size P and the number of generations G depend on the problem to be optimised. Storn and Price suggest P ∈ [5n, 10n], where n is the dimension of the objective function, but P must be ≥4 to provide sufficient mutually different solution vectors for the algorithm to function properly [10].

The operation of the algorithm is controlled by the use of a fitness function ffit designed to discriminate between solutions (SPDs here) with 'good' or 'poor' properties.

Mutation is the process of creating a new offspring vector ui,<sup>G</sup> + 1 by adding the weighted difference between two randomly chosen solution vectors, vr2,<sup>G</sup> and vr3,G, to a randomly chosen vr1,<sup>G</sup> [10]:

$$
\mu\_{i,G+1} = \nu\_{r1,G} + F \times (\nu\_{r2,G} - \nu\_{r3,G}) \tag{1}
$$

the vi,<sup>G</sup> vector is retained. Hence, only the fitter offspring become members of the

Proposals have been made for several variants of the original DE algorithm [10]. The most significant differences are in the creation of new solution vectors where:

1. the best solution vector from the current generation vbest,<sup>G</sup> is mutated rather

Our work is based on the original DE, and is described in pseudo-code given

2. more than two difference vectors are used in mutation; and

3. different crossover schemes are employed.

Require: G, P, F ∈[0,2], CR ∈[0,1] 1: Create vectors {v1,0,...,vP,0} 2: Evaluate {v1,0,...,vP,0}->ffit,<sup>0</sup>

Illustration of a typical crossover process for P = 6.

Illustration of the mutation process for P = 9.

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

5: Crossover {u1,G+1,...,uP,G+1} and {v1,G,...,vP,G}->{w1,G+1,...,wP,G+1}

6: Evaluate {w1,G+1,...,wP,G+1} 7: if ffit(wi,G+1) > ffit(vi,G) 8: vi,G+1 = wi,G+1

G + 1 generation.

Figure 2.

Figure 1.

than vr1,G;

3: for all G do

9: else

43

10: vi,G+1 = vi,G 11: end if 12: end for

4: Mutate vi,G -> ui,G+1

below:

where F is a mutation weight ∈ [0,2] and random indices r1, r2, r<sup>3</sup> ∈ [0, P�1] are chosen to be different from the index i. The mutation process is illustrated in Figure 1.

The offspring solution vectors ui,<sup>G</sup> + 1 undergo crossover which ensures that the offspring vectors wji,<sup>G</sup> + 1 differ from their parents [10]:

$$w\_{j\sharp, G+1} = \begin{cases} u\_{j\sharp, G+1} & \text{if } (j \le \text{CR}) \mid (j = i\ )\\ v\_{j\sharp, G} & \text{if } (j > \text{CR}) \mid (j \ne i\ ) \end{cases} \tag{2}$$

where the terms and parameters are defined as follows: i ∈ {1,2,..., P} is a randomly chosen integer, j ∈ [0, 1) is a randomly chosen real value, CR ∈ [0, 1] is a crossover constant influencing the number of elements to be exchanged. An example of the crossover process is shown in Figure 2.

The process compares the offspring vectors {w1,<sup>G</sup> + 1,..., wP,<sup>G</sup> + 1} against their parent vectors {v1,G,..., vP,G}. If the offspring wi,<sup>G</sup> + 1 has a better fitness function ffit than its parent vi,G, then it becomes a member of the next generation, G + 1. If not,

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

#### Figure 1.

In this chapter, we will describe an approach to the design of the optimum LED mixture using a simple global optimisation algorithm based on differential evolution (DE) as proposed by Storn and Price [9]. We have optimised the SPDs of mixtures of real and simulated LEDs and diode lasers. For this purpose, a MATLAB program was developed that optimises the SPD of an arbitrary mixture of individual SPDs. Specific versions of this program were developed to implement several different techniques for defining (and calculating) the colour properties of sources. This chapter proceeds as follows. In Section 2, the principles of the DE algorithm are described in detail. In Section 3, the theory of colour rendering and luminous efficacy is given, along with other relevant lighting terms and their usage. Section 4 presents a selection of key experimental results and comparisons and, finally, we

Computer Architecture in Industrial, Biomechanical and Biomedical Engineering

Differential evolution is a powerful population-based evolutionary algorithm

suitable for the optimisation of real-value multi-modal non-linear and nondifferentiable objective functions fo(x1, x2, ..., xn) [9, 10]. DE is simple and has proven to be powerful in solving a number of benchmark problems [11, 12]. The search for an optimal solution starts with a population of P randomly created solution vectors {v1, v2, ..., vP}. This population is maintained constant during the optimisation process, during which the solution vectors (i.e., candidate SPDs) undergo mutation, crossover, evaluation and selection over a number of generations G. The best choices for both the population size P and the number of generations G depend on the problem to be optimised. Storn and Price suggest P ∈ [5n, 10n], where n is the dimension of the objective function, but P must be ≥4 to

provide sufficient mutually different solution vectors for the algorithm to

The operation of the algorithm is controlled by the use of a fitness function ffit designed to discriminate between solutions (SPDs here) with 'good' or 'poor'

Mutation is the process of creating a new offspring vector ui,<sup>G</sup> + 1 by adding the weighted difference between two randomly chosen solution vectors, vr2,<sup>G</sup> and vr3,G,

where F is a mutation weight ∈ [0,2] and random indices r1, r2, r<sup>3</sup> ∈ [0, P�1] are chosen to be different from the index i. The mutation process is illustrated in Figure 1. The offspring solution vectors ui,<sup>G</sup> + 1 undergo crossover which ensures that the

wji,Gþ<sup>1</sup> <sup>¼</sup> uji,Gþ<sup>1</sup> if ð Þ <sup>j</sup>≤CR <sup>∣</sup> ð Þ <sup>j</sup> <sup>¼</sup> <sup>i</sup>

The process compares the offspring vectors {w1,<sup>G</sup> + 1,..., wP,<sup>G</sup> + 1} against their parent vectors {v1,G,..., vP,G}. If the offspring wi,<sup>G</sup> + 1 has a better fitness function ffit than its parent vi,G, then it becomes a member of the next generation, G + 1. If not,

where the terms and parameters are defined as follows: i ∈ {1,2,..., P} is a randomly chosen integer, j ∈ [0, 1) is a randomly chosen real value, CR ∈ [0, 1] is a crossover constant influencing the number of elements to be exchanged. An

vji,G if ð Þ j>CR ∣ ð Þ j 6¼ i

ui,Gþ<sup>1</sup> ¼ vr1,G þ F � ð Þ vr2,G � vr3,G (1)

(2)

state our conclusions in Section 5.

2. Differential evolution

function properly [10].

to a randomly chosen vr1,<sup>G</sup> [10]:

offspring vectors wji,<sup>G</sup> + 1 differ from their parents [10]:

example of the crossover process is shown in Figure 2.

properties.

42

Illustration of the mutation process for P = 9.

#### Figure 2.

Illustration of a typical crossover process for P = 6.

the vi,<sup>G</sup> vector is retained. Hence, only the fitter offspring become members of the G + 1 generation.

Proposals have been made for several variants of the original DE algorithm [10]. The most significant differences are in the creation of new solution vectors where:


Our work is based on the original DE, and is described in pseudo-code given below:

```
Require: G, P, F ∈[0,2], CR ∈[0,1]
```

```
1: Create vectors {v1,0,...,vP,0}
```

```
10: vi,G+1 = vi,G
```

```
11: end if
```
12: end for

### 3. Light-source properties

Our purpose in this chapter is to present an approach to intelligent spectral design for any white-light source, aiming to achieve an optimum combination of luminous efficacy and colour rendering which, as previously noted, are contravariant characteristics of the SPD. We next introduce a brief outline of these, and other important, source properties.

symbolised as Rf min, from the full set of 99 individual indices Rf i, as well as the

This was a precursor to TM-30-15, first proposed by Davis and Ohno of NIST (USA) [21]. The following serves as a brief introduction for the purpose of the present discussion. Again using the previously mentioned colour-shift concept, the CQS metric employs 15 saturated test colour samples, on the premise that certain light sources may render saturated colours more poorly than the de-saturated colours of the CIE's CRI method. The chromatic differences are calculated using the

The full calculation procedure has also been explained in [23]. It employs multiple steps, several of which are non-linear, resulting in the general CQS index, Q <sup>a</sup>. As with the previous two cases, the 'special CQS' (Q <sup>i</sup>) for each test colour sample may be calculated for a more thorough investigation of a test source. We have used the minimum value of Q <sup>i</sup> (designated Q min) from the set of 15 Q <sup>i</sup> values,

The luminous efficacy of the radiation (LER) of a light source assesses the 'lighting content' of the spectrum by comparing the visible light output (in lumens)

Ð

where Km is the maximum luminous efficacy of radiation (≈683 lumen per watt), S(λ) is the spectral distribution of the light source, and V(λ) is the CIE spectral sensitivity function for human photopic vision [24]. The LER is an important determinant of the overall economy of a light source since the overall luminous efficacy is given by the product of LER with the energy conversion efficiency of

From the perspective of the lighting system designer, the correlated colour temperature (CCT) is the key feature in the selection of a light source since the CCT serves as an indicator of, not only the colour of the source, but also, the 'atmo-

The CCT is defined in [25], and its significance is that it describes the chromaticity of the source (which must be close to the Planckian locus) in the CIE (u, v) chromaticity diagram.<sup>2</sup> Note that it is possible for many different spectral power distributions (SPDs) to have the same CCT; and that CCT is not essentially linked

; 2

<sup>3</sup> <sup>v</sup><sup>0</sup> � � diagram based on CIE 1976 (u<sup>0</sup>

, v<sup>0</sup> )

Ð

<sup>λ</sup>Vð Þλ Sð Þλ dλ

<sup>λ</sup>Sð Þ<sup>λ</sup> <sup>d</sup><sup>λ</sup> (3)

LER <sup>¼</sup> Km

referenced colour sample number imin (in the set i<sup>1</sup> to i99).

3.3 Colour quality scale

CIE 1976 (CIELAB) colour model [22].

Artificial Intelligence in Light-Source Design DOI: http://dx.doi.org/10.5772/intechopen.88094

as an optimisation parameter.

the particular light source.

sphere' it will create.

coordinates.

45

3.5 Correlated colour temperature

with colour rendering, quality or fidelity.

<sup>2</sup> In CIE documentation, this is now replaced by the u<sup>0</sup>

3.4 Luminous efficacy of radiation

to the total radiant output (in watts) as in Eq. 3.

#### 3.1 Colour rendering index

Colour rendering has been defined by the Commission Internationale de l'Éclairage, International Commission on Illumination (CIE) who have published recommendations for the method of calculation of their colour rendering index (CRI) [13] based on a knowledge of the light-source spectrum. It represents an evaluation of the average colour shift of eight defined moderate-chroma colour samples when compared under the test source and a reference source having the same correlated colour temperature (CCT).<sup>1</sup> The system includes 14 test colours in total, and the additional 6 comprise 4 highly saturated colours (red, yellow, green, and blue) plus samples representing skin and foliage colours, respectively. As of the time of writing, this is the internationally agreed method. Note that CRI and associated technology have also been covered in [14]. The two most widely quoted colour rendering terms are: Ra—the general colour rendering index, based on the colour shifts of the eight principal test colour samples; and R9—the 'special' (individual) index for the highly saturated red colour (sample 9).

In our optimisation work (see Section 4.1), we also made use of several derived indices symbolised as Rb, Rc and Rmin, based, respectively, on: the 6 additional test colours; the full set of 14 test colours and the minimum individual value from the Rc set.

#### 3.2 Colour fidelity index

Some dissatisfaction with the CIE method has arisen since the widespread adoption of LED lighting. As a consequence, the Illumination Engineering Society of North America (IES) has adopted a recommended method (TM-30-15) [15, 16] which recommends two new indices (Rf and Rg) for the classification of the colour properties of light sources. The underpinning research leading to the development of TM-30-15 [16–18] identified several weaknesses in the CIE's earlier CRI method [13], claiming that it does not adequately sample wavelength space and hence tends to over-estimate colour performance.

The method is also based on the colour-shift concept, but now using a set of 99 test colours considered to provide uniformity of both wavelength sampling and colour-space sampling. In addition, it uses a more modern colour-difference calculation technique, CAM02UCS [19] which is a development of the basic CIECAM02 colour appearance space [20].

The new index Rf gives an overall assessment of colour fidelity, while gamut index Rg indicates the relative magnitudes of colour shifts for sample colours in different regions of colour space. Also available is the skin colour index, Rf skin, which is an average of two specific sample-colour indices, selected as representative of human skin. In our optimisations, we also called up the minimum value of Rf,

<sup>1</sup> Defined in Section 3.5.

symbolised as Rf min, from the full set of 99 individual indices Rf i, as well as the referenced colour sample number imin (in the set i<sup>1</sup> to i99).
