1. Introduction

In modern society, indoor space is the place where people spend more than 80% of their daily time. Good lighting design provides people with a comfortable and healthy living and working environment and improves work efficiency. Traditional lighting design mainly focuses on providing sufficient illuminance on horizontal working surfaces, because it is directly related to people’s visual task performance [1–3]. However, with the change of work and lifestyles in recent decades, the attention to indoor lighting has gradually expanded from work-plane illuminance to the quality of the overall lit environment [4–6]. Firstly, due to the advance in technology, visual tasks requiring a high work-plane illuminance are diminishing [7]. For example, paper reading has been largely replaced by reading from self-luminous screens, and difficult manual and visual tasks can usually be completed through automation technology [7]; and secondly, people pay increased attention to the overall lit appearance of indoor spaces (how brightly lit, or dimly lit, the spaces appear) [1–3,7], because that is not only associated with visual satisfaction and comfort, but also related to the quality of visual communication. Thus, some researchers have even proposed to make the switch of the lighting design priority from task illumination to ambient illumination [1–3] with a focus on spatial brightness [7–10]. Spatial brightness describes a visual sensation related to the magnitude of the ambient illumination level within a space, which encompasses the overall sensation based on the response of a large part of the visual field extending beyond the fovea [11,12]. The spectral and spatial distributions of lighting are the two major aspects that affect one’s overall impression of spatial brightness [13]: the former is mainly based on the spectral photo-sensitive nature of various types of retinal photoreceptors that distribute across the retina [7–10]; while the latter is related to the pattern and intensity of light distribution within the field of view that affects the quantity (and maybe also the distribution) of the light received at eye-level [1–3].

Even in the same indoor space, the assessment of spatial brightness for a visual field could be very different depending on the observation location and field of view. Therefore, some researchers were committed to finding a general metric to quantify the overall spatial brightness of illuminated interiors [1–3,14–17]. Cuttle [1] proposed to use the concept of mean room surface exitance (MRSE) as a metric to quantify spatial brightness. MRSE is defined as the area-weighted average of surface exitance across the interior room surfaces. Duff et al. [14,15] investigated the suitability of using MRSE as a predictor of spatial brightness, and claimed that there did exist a clear correlation between MRSE and subjective evaluation of spatial brightness, while on the other hand, such a correlation did not exist between horizontal illuminance and spatial brightness. Besides, Loe et al. [16,17] examined the correlation between spatial brightness perception and the average luminance in a horizontal band with a width of 40° within the field of view (denoted as L_av,B40), and concluded that the values of L_av,B40 had a good correlation with the results of subjective spatial brightness assessments. Although these prior studies [1–3,14–17] have verified the correlation between their proposed metrics and spatial brightness perception through subjective evaluation experiments, some limitations prevented these metrics from wide adoption: firstly, the measurement procedures for MRSE and L_av,B40 are cumbersome and therefore are not easy to be mastered by lighting designers [18,19]; secondly, some results [14,15] were obtained based on experimental conditions with a relatively uniform lighting distribution, but the real-world scenario can be much more complicated; and lastly, both metrics of MRSE and L_av,B40 merely reflect the photometric properties of the illuminated room surfaces, and therefore cannot directly reflect the amount of light received by people’s eyes [20], which may be highly relevant to the sensation of spatial brightness.

Compared with MRSE and L_av,B40, indirect corneal illuminance (denoted as E_cor,i) has the potential to have a better correlation with subjective evaluation of spatial brightness, because it is directly related to the amount of light incident on the eyes and also reflects the spatial distribution of light. Moreover, the E_cor,i metric could be especially useful for scenarios where the perceived spatial brightness at a specific field of view is very important. For example, in the classroom, the most important direction of vision is towards the blackboard, and the lighting design focusing on this direction would be beneficial to students’ visual health [21]; in an office space, people’s perception of spatial brightness at various locations and viewing directions may be different, and the lighting design focusing on office workers’ frequent viewing directions could enhance their visual comfort and work efficiency [22,23]. Thus, it is important to investigate whether there is a high correlation between indirect corneal illuminance and the subjective assessment of spatial brightness for a visual field. Note that the spatial brightness perceptions for various visual fields together can compose one’s overall impression of an illuminated space.

In this work, a quantitative method for comparison of spatial brightness within a visual field was developed. The indirect corneal illuminance required for the test scene to match the spatial brightness of the reference scene with a fixed corneal illuminance was obtained according to subjective assessment. Three lighting scenes with different patterns of spatial lighting distribution were compared against each other in pairs. Based on which, the applicability of using the E_cor,i metric for spatial brightness evaluation was investigated. Furthermore, the performance of the E_cor,i metric was compared with that of the previously proposed metrics including MRSE and L_av,B40.

2. Methods

2.1 Experiment setup

The experiment was carried out in a windowless room 4.0 m long, 3.2 m wide and 2.7 m high. The wall and ceiling surfaces were painted white and diffusive for efficient generation and reflection of indirect light. Carefully designed luminaires, which were intensity-, angle- and position- tunable, were located at strip rails hanged 0.2 m under the ceiling. By adjusting the angles of operating luminaires, light goes through at least one reflection in the room before reaching the subjects’ eyes. In other words, only indirect light was delivered to the subjects sitting about 1.6 m in front of the long-side wall, as shown in Fig. 1(a) and (b). In this work, we intended to investigate the performance of spatial brightness evaluation metrics under different patterns of lighting distribution, therefore a fixed general-lighting spectrum with a typical CCT of about 4400 K and a CRI (R_a) of 81.0 was adopted. The measured spectrum at subjects’ eye-level was stable during the experiment, as shown in Fig. 1(c).

 

Fig. 1. Illustrations of (a) a windowless lighting lab with luminaire layout, (b) an intensity-, angle-, and position- tunable luminaire, and (c) measured spectrum at subjects’ eye level.

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The observation position was fixed for all subjects during the experiment, and illuminance was measured at the position of the subjects’ eyes, which was all contributed by indirect light. The luminance distributions across room surfaces were measured using a luminance meter (JETI Spectraval 1511, luminance mode with a measuring angle of 2°).

A total of 44 subjects were recruited (24 females and 20 males, mean ± SD age: 23.2 ± 2.6; Ethnicity: Chinese). The inclusion criteria were college students, aged between 18 and 30 years old, normal or corrected-to-normal eyesight, and success in passing a color-blind test (Ishihara color vision test). Subjects were naïve (not an expert in lighting technology) and were paid for participation.

Three types of lighting scenes with different patterns of light distribution, named Scenes 1, 2 and 3, were compared against each other in pairs. As shown in Fig. 2, the left, middle, and right columns are schematic illustrations of light output from luminaires from the top view, photos taken by a Canon 6D Mark II DSLR camera with an 8 mm fisheye lens, and photos calibrated to luminance distributions taken by an LMK 6 video photometer, respectively. The schematic diagrams and photos of the Scenes 1-3 are shown in Figs. 2 (a)-(c), respectively. It can be found that the lighting distribution of Scene 1 is relatively more uniform compared with those of Scenes 2 and 3. Note that each photo in Fig. 2 was taken at the condition when corneal illuminance at the observation position was set to 100 lx (the illuminance at the observation position was all indirect).

 

Fig. 2. Schematic illustrations and photos of (a) Scene 1, (b) Scene 2 and (c) Scene 3. From left to right, each column represents: the schematic illustrations of the light output from luminaires (top view), photos taken by a DSLR camera with a fisheye lens, and luminance photos.

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Spatial brightness comparisons for the fixed visual field (see Fig. 2) were performed between the test scenes at different corneal illuminance levels and the reference scene with a fixed corneal illuminance of 100 lx. Among the three lighting scenes, the pattern of lighting distribution is quite different. The lighting intensity of each test scene was tuned to 11 levels of corneal illuminance from 50 to 180 lx (tuned to approximately 50 lx, 60 lx, 70 lx, 80 lx, 90 lx, 100 lx, 110 lx, 120 lx, 140 lx, 160 lx, and 180 lx). It was aimed to find the corresponding corneal illuminance (denoted as E_match) that made the subjective spatial brightness assessment of each test scene match that of the reference scene. The E_match values from all the paired comparisons were then used to investigate the suitability of the E_cor,i metric for spatial brightness evaluation.

2.2 Experiment method for spatial brightness comparison

In general, the types of procedures for spatial brightness comparison can be divided into four categories, including adjustment, category rating, discrimination, and matching [8,11]. The first two methods are absolute evaluations without a presented reference: in the adjustment procedure, subjects are directed to tune the light quantity of a space to a preferred level; and in the category rating procedure, subjects use rating scales to describe the spatial brightness appearance of a visual scene. The last two methods are relative comparisons with the presence of a reference: in the discrimination procedure, subjects are presented with two scenes and instructed to report which one is brighter; while in the matching procedure, subjects are required to adjust the amount of light of the test scene until it matches the reference scene in spatial brightness. In this study, a reference fixed lighting intensity was presented: subjects were instructed to perform comparisons between the reference and test scenes in temporal juxtaposition. However, simply applying one of the two reference-based comparison procedures, discrimination or matching, may not be sufficient to overcome the following limitations: (i) there existed a significant difference in patterns of lighting distribution between the reference and test scenes, which made it difficult for subjects to make a judgement on which one has a higher level of spatial brightness; (ii) it was not easy for subjects to remember the spatial brightness appearance of the first scene when observing the second one; (iii) there existed individual difference among subjects’ judgments, and the corneal illuminance of the test scene required for each individual to match the spatial brightness of the reference scene generally follows the Gaussian distribution [13,24–26].

To address these problems, we developed a method of spatial brightness comparison for a visual field that combined the advantages of both discrimination and matching approaches: between the reference scene (fixed at a corneal illuminance of 100 lx) and the test scene (set to various corneal illuminance levels from 50 to 180 lx), subjects were asked to make a forced choice [27] on which scene has a higher level of spatial brightness from their visual field. The percentage of subjects who reported the test scene being brighter (denoted as P) versus corneal illuminance of the test scene (denoted as E_cor,i,t) was obtained. One could imagine that the result should be nearly 0% when the E_cor,i,t value was set to 50 lx and close to 100% when the E_cor,i,t value was set to 180 lx. Assuming that the corneal illuminance values of the test scene required for various individuals to match the spatial brightness of the reference scene generally follow a Gaussian distribution, then the experimental data of P vs. E_cor,i,t could be fitted by a psychometric function that is derived from the above-mentioned Gaussian function (probability density function) by integral approximate calculation [25]. In this work, the form of the Logistic function was adopted as the psychometric function, and the E_match value could be quantitatively calculated by fitting the data with the Logistic function [25,26] and finding the point with P = 50%.

2.3 Procedure

As shown in Fig. 3(a), the overall procedure of the study can be divided into three phases: (i) a null-condition trial, (ii) two experimental trials with a fixed reference and (iii) an experimental trial for direct comparison. For the first two phases, each trial was conducted by using one of the three lighting scenes (see Fig. 2) as the test scene to compare with a fixed reference scene (Scene 1 with a fixed corneal illuminance of 100.3 lx). First, a null-condition trail was implemented to confirm the removal of systematic biases and verify the accuracy of the experiment method. Scene 1, which shared the same pattern of spatial lighting distribution as the reference scene, was adopted as the test scene. The corneal illuminance of the test scenes was set to one of the eleven levels from 50 to 180 lx, while that of the reference scene was fixed at 100 lx. In the second phase, two experimental trials with the same fixed reference, Scene 2 vs. the reference and Scene 3 vs. the reference, were carried out to investigate the applicability of using the E_cor,i metric to assess spatial brightness by comparing with the performance of previously proposed metrics, including MRSE and L_av,B40, under very different patterns of lighting distribution. The illuminance level of the test scenes was also set to one of the eleven levels and the reference scene was also set to Scene 1 at 100 lx. For the third phase, an experimental trial was conducted to directly compare Scene 2 with Scene 3, which have the biggest difference in the pattern of spatial lighting distribution, to further compare among the metrics of E_cor,i, MRSE and L_av,B40 for spatial brightness evaluation. In this experimental trial, Scene 2 at 100 lx was adopted as the new reference scene and Scene 3 at illuminance levels from 50 to 180 lx were adopted as the test scenes.

 

Fig. 3. Illustrations of (a) the overall procedure of the experiment, (b) Latin square 22 × 22 disorder matrix for each experiment trial and (c) the detailed comparison procedure.

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Each comparison trial includes 11 corneal illuminance levels for the test scenes and 2 presentation sequences between the reference and test scenes, which makes a total of 22 comparison settings for each of the 44 subjects, as shown in Fig. 3(b). The corresponding order bias needs to be balanced. Therefore, a 22 × 22 Latin square disorder matrix was constructed for 22 groups of subjects (2 subjects in each group) and 22 comparison settings in different presentation sequences, as shown in Fig. 4 (a) and (b). For each group, spatial brightness comparisons were performed following a unique sequence of comparison settings according to the Latin square matrix, as shown in Fig. 4(b). Note that the two subjects in each group participated in the comparison experiment separately so that the field of view was identical for all the subjects.

 

Fig. 4. Illustrations of (a) the 22 × 22 Latin square disorder matrix processing and (b) the detailed disorder sequence for the 22 groups in each experimental trial.

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The detailed procedure in one round of comparison can be described as follows (also shown in Fig. 3(c)):

3. Results

3.1 Null-condition trial

First, a null-condition trial was performed, in which both the reference and test scenes shared the same pattern of lighting distribution. For the reference scene, the illuminance at the subject’s eye level was fixed at 100 lx, while that of the test scenes was set to one of the eleven levels (from 50 to 180 lx). For each corneal illuminance level of the test scenes (E_cor,i,t), the percentage of subjects who reported the test scene being brighter than the reference scene (P) was obtained, as shown in Fig. 5. Note that each data point is an average of 88 comparisons (44 subjects × 2 sequences) so that the result is statistically representative and the bias caused by comparison sequences and the individual difference can be well balanced.

 

Fig. 5. Null-condition trial: the percentage of subjects who reported the test scene being brighter than the reference scene (P) vs. corneal illuminance of the test scene (E_cor,i,t), and the fit curve.

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When the test scene was set to 100 lx, the reference and test scenes were identical, therefore theoretically, the percentage of subjects reporting the test scene being brighter should be 50%. From our null-condition trial, of the 88 comparisons, 52 (59.1%) reported the test scene being brighter (95% confidence interval (CI): 48.6% – 69.6%). It can be found that the expected value of 50% fell into the 95% CI. Moreover, when the test scene was set to 90 lx, 19.3% reported the test scene being brighter (95% CI: 10.9% – 27.7%), and when the corneal illuminance was set to 110 lx, 76.1% reported the test scene being brighter (95% CI: 67.1% – 85.2%). According to our 50% criteria, the test scenes at 110 lx and 90 lx could be clearly considered as brighter and less bright than the reference scene, respectively. Such results from the null-condition trial proved that our method could effectively balance the potential bias and make the difficult spatial brightness comparison task practical and accurate. Besides, the t-test statistical analysis also showed a significant difference between the experimental data of the test scene at 100 lx and that at all the remaining 10 illuminance levels (from 50 to 180 lx). All the p values were less than 0.05 (p = 0.015 for corneal illuminance at 110 lx, and p < 0.001 for the other corneal illuminance levels), which indicated that in the null-condition trial, the spatial brightness appearance of the reference scene at 100 lx could be distinguished from that of the test scenes with illuminance levels other than 100 lx. Therefore, the 10 lx interval setting could be considered effective and suitable for this study.

According to the analysis in Section 2.2, the experimental data in Fig. 5 can be fit by a 2-parameter logistic equation [7], as follows:

3.2 Experimental trials with a fixed reference

Next, two experimental trials with a fixed reference (Scene 1 at 100 lx) were carried out, in which Scene 2 and Scene 3 were adopted as test scenes. For each type of test scene, there were 11 corneal illuminance levels ranging from 50 to 180 lx, and the corresponding data of P vs. E_cor,i,t was obtained, as shown in Fig. 6. Fit curves based on Eq. (1) were applied to the experimental data: for Scene 2 vs. the reference, fitting results of k₁= 95.6 lx (95% CI: 93.7–97.5 lx) and k₂ = 6.7 were obtained, with the corresponding fit curve shown in Fig. 6 (a); and for Scene 3 vs. the reference, fitting results of k₁ = 97.1 lx (95% CI: 95.2–99.1 lx) and k₂ = 6.2 were obtained, with the fit curve shown in Fig. 6 (b).

 

Fig. 6. Experimental trials with a fixed reference (Scene 1 at 100 lx): (a) Scene 2 vs. the reference and (b) Scene 3 vs. the reference: the percentage of subjects who reported the test scene being brighter than the reference scene vs. corneal illuminance of the test scene, and the fit curves.

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The results of the two experimental trials showed that to match the spatial brightness of the reference scene (Scene 1 with a fixed corneal illuminance of 100.3 lx), corneal illuminances of 95.6 lx and 97.1 lx were needed for Scene 2 and Scene 3, respectively. It indicated that to achieve a certain level of spatial brightness, there existed only a minor difference in required indirect corneal illuminance among scenes with very different patterns of lighting distribution. In other words, indirect corneal illuminance could serve as a fair candidate for spatial brightness evaluation of lit spaces.

Besides, the k₂ values obtained from both experimental trials were clearly smaller compared with that of the null-condition trial (6.7 & 6.2 for Scenes 2 & 3, respectively vs. 11.2 for Scene 1), which means that it was much more difficult to make a judgement on spatial brightness comparison when the patterns of lighting distribution between the reference and test scenes were different. According to the statistical results of the null-condition trial, it was suggested that the minimum interval of 10 lx was sufficient to distinguish the reference scene at 100 lx from the test scenes under different corneal illuminances other than 100 lx. In the two experimental trials with a fixed reference, the statistical t-test procedure was also applied to compare the data of the test scenes at 100 lx and that at the remaining 10 illuminance levels (from 50 to 180 lx), for Scenes 2 and 3. It was found that for both types of test scenes, there was no significant difference between the data at 100 lx and that at 110 lx (p > 0.05, the p values for Scenes 2 and 3 were 0.068 and 0.118, respectively), while between the data at 100 lx and that at the rest 9 illuminances levels, there were still significant differences, as the corresponding p values were all less than 0.05. It indicated that when the lighting distribution patterns of the reference and test scenes were different, the judgement on the spatial brightness comparison became a little more difficult, but the influence was limited so that the k₁ values (95.6 lx and 97.1 lx for Scene 2 and -Scene 3, respectively) could still be clearly determined by the same method, and they were found to be close to the ideal value of 100 lx.

3.3 Experimental trial for direct comparison

Furthermore, an experimental trial for direct comparison (named as the direct-comparison trial) between Scenes 2 and 3 was conducted by using the Scene 2 at 100 lx as the reference scene and the Scene 3 from 50 to 180 lx as the test scenes. The same experimental procedure in Fig. 3(c) was performed and the corresponding data of P vs. E_cor,i,t was obtained. As shown in Fig. 7, fit curve based on Eq. (1) was applied to the experimental data: the fitting results of k₁ = 103.0 lx (95% CI: 100.9–105.1 lx) and k₂ = 7.6 were obtained.

 

Fig. 7. Experimental trial of Scene 3 vs. the reference (Scene 2 at 100 lx): the percentage of subjects who reported the test scene being brighter than the reference scene (P) vs. corneal illuminance of the test scene (E_cor,i,t), and the fit curve.

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The result showed that to match the spatial brightness of the reference scene (Scene 2 with a fixed corneal illuminance of 100.1 lx), a corneal illuminance of 103.0 lx was needed for Scene 3, which was also very close to the expected theoretical E_match value of 100.1 lx. It indicated that the proposed metric is still capable of assessing spatial brightness even when the patterns of lighting distribution for the two scenes are quite different. Besides, the k₂ value obtained was clearly smaller compared with that of the null-condition trial, which means that it was more difficult to make a judgement on spatial brightness comparison when the patterns of lighting distribution between the reference and test scenes were very different.

4. Discussion

Based on the experimental results, it can be found that indirect corneal illuminance E_cor,i highly correlates with subjective assessment of spatial brightness, even under very different patterns of lighting distribution. In prior studies, other metrics, including MRSE [1] and L_av,B40 [16,17], were also proposed to quantify the spatial brightness of illuminated interiors. To investigate the relative effectiveness among the three metrics, the applicability of the MRSE and L_av,B40 metrics were also analyzed by directly comparing with that of the E_cor,i metric based on the same experimental data. Note that the MRSE values discussed in this section referred to the area-weighted average of room-surface exitance within the subject's field of view, as the distribution of light outside the visual field does not contribute to the subject’s perception.

Under a fixed pattern of lighting distribution, the values of E_cor,i, MRSE and L_av,B40 all increase linearly with the output flux of luminaires. In other words, there exist linear relationships between each of the two prior metrics (MRSE or L_av,B40) and our proposed metric of E_cor,i. As shown in Figs. 8 and 9, the three dashed lines represent the theoretical relationships between MRSE (or L_av,B40) and E_cor,i corresponding to the three lighting scenes, Scenes 1-3. The slope of each line was determined by the specific pattern of lighting distribution. According to the linear relationship between MRSE (or L_av,B40) and E_cor,i, once the slope has been determined, the values of MRSE and L_av,B40 required for test scenes to match the spatial brightness of the reference scene, denoted as MRSE_match and L_av,B40-match, can be obtained from the value of E_match. The solid dots in Figs. 8(a) and 9(a), (E_match, MRSE_match) and (E_match, L_av,B40-match), are the “matched points” for equal spatial brightness between a fixed reference and test scenes obtained from the first two phases of trials (a null-condition trial and two experimental trials with a fixed reference). The corresponding values of the reference scene are denoted as E_cor,i-ref, MRSE_ref, and L_av,B40-ref, and are marked as the grey dotted lines in Figs. 8 and 9.

 

Fig. 8. The first two phases of trials with a fixed reference. (a) MRSE vs. E_cor,i, and (b) the deviations of the E_match and MRSE_match values from the expected values of E_cor,i-ref and MRSE_ref, respectively. The solid dots represent the matched points (E_match, MRSE_match) obtained from the experiment.

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Fig. 9. The first two phases of trials with a fixed reference. (a) L_av,B40 vs. E_cor,i, and (b) the deviations of the E_match and L_av,B40-match values from the expected values of E_cor,i-ref and L_av,B40-ref, respectively. The solid dots represent the matched points (E_match, L_av,B40-match) obtained from the experiment.

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As shown in Figs. 8 (b) and 9 (b), when the spatial brightness of the test scenes match that of the reference scene (for the reference scene, E_cor,i-ref = 100.3 lx, MRSE_ref = 97.7 lm/m² and L_av,B40-ref = 35.3 cd/m²), the corresponding values of E_match are 98.7, 95.6 and 97.1 lx, the corresponding values of MRSE_match are 96.2, 109.8 and 84.3 lm/m², and the values of L_{av,B40, match} are 34.7, 42.1 and 29.8 cd/m² for the Scenes 1, 2 and 3, respectively. For the null-condition trial, as the lighting distribution pattern of the Scene 1 and that of the reference scene were identical, the matched point of (E_match, MRSE_match) lies in the straight line of MRSE versus E_cor,i that passes through the origin point and the point of (E_cor,i-ref, MRSE_ref) (see the grey dashed line in Fig. 8(a)), and the same conclusion holds for the scenario of L_av,B40 versus E_cor,i, (see the grey dashed line in Fig. 9(a)). Therefore, for the null-condition trial, the metrics of MRSE and L_av,B40 have the same deviation (−1.6%) as the metric of E_cor,i from the theoretically ideal values (MRSE_ref, L_av,B40-ref, and E_cor,i-ref). For the two experimental trials with a fixed reference, the deviations of E_match from the expected value E_cor,i-ref are still minor: −4.7% and −3.2% based on the comparisons of Scene 2 vs. the reference and Scene 3 vs. the reference, respectively. However, for the metric of MRSE, much greater deviations of 12.4% and −13.7% could be found from the comparisons of Scene 2 vs. the reference and Scene 3 vs. the reference, respectively, as shown in Fig. 8(b). Moreover, even more significant deviations, 19.3% and −15.6%, were found for the metric of L_av,B40 based on the comparisons of Scene 2 vs. the reference and Scene 3 vs. the reference, respectively, as shown in Fig. 9 (b). In addition, the deviations of the three metrics can also be reflected from the geometric representation: the deviations of the matched points from the grey dotted line of E_cor,i = E_cor,i-ref represent the degree of accuracy of the metric E_cor,i (the same conclusion holds for MRSE and L_av,B40). As shown in Figs. 8(a) and 9(a), the matched points of the three trials in the first two phases are much closer to the line of E_cor,i = E_cor,i-ref compared with the lines of MRSE = MRSE_ref and L_av,B40 = L_av,B40-ref, indicating that the metric of E_cor,i has the highest accuracy among the three metrics in the evaluation of spatial brightness for a visual field when comparing with the subjective assessment. Although a deviation of 10-15% by using the metric of MRSE may still be considered as acceptable in some lighting design practices, our recommended metric E_cor,i offers not only improved accuracy, but also much improved convenience in measurement, making it a better metric to quantify spatial brightness for a visual field.

The same analysis procedure is applied to the direct-comparison trial and the result is shown in the Fig. 10. The solid dots in Figs. 10 (a) and (b), (E_match, MRSE_match) and (E_match, L_av,B40-match), are the “matched points” for equal spatial brightness between the reference and test scenes from the experiment, while the hollow dots marked as #1 – #4 are the theoretically predicted points based on the predictions of the three metrics: the points #1 and #3 are based on the E_cor,i metric, the point #2 is based the MRSE metric, and the point #4 is based on the L_av,B40 metric. It can be found that when comparing two scenes with very different patterns of spatial lighting distribution, the performance of the three metrics can be significantly different.

 

Fig. 10. The direct-comparison trial. (a) MRSE vs. E_cor,i, (b) L_av,B40 vs. E_cor,i, and (c) the deviations of the E_match, MRSE_match and L_av,B40-match values from the expected values of E_cor,i-ref, MRSE_ref and L_av,B40-ref, respectively. The solid dots represent the matched points, (E_match, MRSE_match) and (E_match, L_av,B40-match), obtained from the experiment and the hollow dots #1 – #4 represent the theoretical predictions based on the three metrics.

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The results of the experiment show that the matched points are very close to the theoretically predicted points #1 and #3, which are based on the E_cor,i metric. On the other hand, the theoretically predicted points based on the metrics of MRSE and L_av,B40, the points #2 and #4, respectively, are far away from the matched points. The geometric representation in Figs. 10 (a) and (b) clearly shows that the metric of E_cor,i is much more accurate in quantifying spatial brightness compared with the prior metrics of MRSE and L_av,B40. Besides, a direct data overview of the deviations of the three metrics from the expected values is presented in Fig. 10 (c). It can be found that when the spatial brightness of the test scene matches that of the reference scene (for the reference scene, E_cor,i-ref = 100.1 lx, MRSE_ref = 115.0 lm/m² and L_av,B40-ref = 44.1 cd/m²), the corresponding values of E_match, MRSE_match and L_av,B40-match are 103.0 lx, 89.4 lm/m² and 31.6 cd/m², respectively. While the deviation of E_match from the expected value E_cor,i-ref is still minor (2.9%), the values of MRSE_match and L_av,B40-match show significant deviations of −22.3% and −28.3%, respectively, from the corresponding expected values. The result of the direct-comparison trial reconfirms that the metric of E_cor,i can serve as a better metric with much higher accuracy compared with the prior metrics of MRSE and L_av,B40 in assessing the spatial brightness for a visual field.

Therefore, in all possible comparison pairs among the Scenes 1-3, the results all suggest that our proposed metric of E_cor,i has much improved accuracy compared with the prior metrics of MRSE and L_av,B40. Especially, the metric E_cor,i can ensure a deviation within ± 5% in assessing spatial brightness for a visual field. With such a high accuracy and the additional benefit of convenience in measurement, the metric of E_cor,i has much superior performance compared with the prior metrics of MRSE and L_av,B40.

5. Limitations

Note that in this work, spatial brightness comparisons were performed with the reference scene fixed at a corneal illuminance of 100 lx. Future work is suggested to investigate the behavior with the reference scene set to other illuminance levels. Also, our proposed method relies on the acquisition of the indirect part of the corneal illuminance value. For those cases where the proportion of direct light is significant for the visual field, we recommend using the method of Duff et al. [18] to obtain the indirect portion of corneal illuminance via a method based on HDR imaging. The involvement of HDR imaging makes the measurement procedure less convenient, but it is still much simpler than the MRSE measurement method [18,19] adopting the same HDR imaging process.

6. Conclusion

In this work, a quantitative method was developed for comparisons of spatial brightness within a visual field among three lighting scenes with various patterns of lighting distribution. The indirect corneal illuminance required for each test scene to match the spatial brightness perception of the reference scene with a fixed corneal illuminance was obtained. It was demonstrated that the metric of indirect corneal illuminance, E_cor,i, had a very good correlation with the subjective evaluation of spatial brightness. Furthermore, the performance of the E_cor,i metric was demonstrated to be much better compared with that of the previously proposed metrics of MRSE and L_{av, B40} in terms of the correlation with the subjective evaluation of spatial brightness. Besides, the E_cor,i metric has a great advantage over the other two alternative metrics in the convenience of measurement, which is also critical for the acceptance of the metric by lighting practitioners. Therefore, we suggest that indirect corneal illuminance could serve as a preferred metric to quantify spatial brightness for a visual field, and the values of this metric at various visual fields together can be used to quantify people’s overall impression of an illuminated space. The authors believe that the outcome of this work could provide valuable insights toward the establishment of a widely accepted quantitative evaluation method for the spatial brightness of interior spaces.