The Reproduction and Evaluation of Star Fields with the Milky Way in a Planetarium

applied
 sciences
Article
The Reproduction and Evaluation of Star Fields with the Milky
Way in a Planetarium
Midori Tanaka 1, *, Ken’ichi Otani 2 , Saori Setoguchi 2 and Takahiko Horiuchi 3

 1 Graduate School of Global and Transdisciplinary Studies, Chiba University, 1-33 Yayoi-cho, Inage-ku,
 Chiba 263-8522, Japan
 2 Engineering Division, Konica Minolta Planetarium Co., Ltd., 3-1-3 Higashi-Ikebukuro, Toshima,
 Tokyo 170-8630, Japan; kenichi.otani@konicaminolta.com (K.O.); saori.setoguchi@konicaminolta.com (S.S.)
 3 Graduate School of Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan;
 horiuchi@faculty.chiba-u.jp
 * Correspondence: midori@chiba-u.jp

 Abstract: In this study, we investigated the physical factors required to accurately reproduce the
 Milky Way in star fields in a planetarium using three evaluation indices: faithfulness, preference,
 and depth feeling. Psychometric experiments were conducted by manipulating three different
 physical factors (transmittance, representation size and star density) of the stars projected on a
 dome screen as experimental stimuli. The three evaluation indices were rated by observers for 12
 different reproductions of the Milky Way. By analyzing the experimental results, we developed a
 common model to estimate the scores for each evaluation index by changing the coefficients of the
 three physical factors. Our proposed model has good accuracy, and each evaluation index can be
 represented by transmittance, representation size and star density. The weighting values indicate that
 density reproduction was the pivotal factor for the majority of observers. In contrast, the observers
 



 were not affected by the size of the stars in the projected Milky Way.
 


Citation: Tanaka, M.; Otani, K.;
 Keywords: visual perception; psychophysics; planetarium; star
Setoguchi, S.; Horiuchi, T. The
Reproduction and Evaluation of Star
Fields with the Milky Way in a
Planetarium. Appl. Sci. 2021, 11, 1413. 1. Introduction
https://doi.org/10.3390/app11041413 Ever since the development of lighting technology, aimed at achieving a comfortable
 living environment at night, a sky full of stars has become a precious and rare sigh. Bright
Academic Editor: Vassilis Charissis
 nights cause light pollution [1,2], limiting the visible stars in the night sky to bright stars
Received: 28 December 2020
 such as stars with small apparent magnitude. In this study, we focus on the Milky Way. The
Accepted: 2 February 2021
 Milky Way is a mass of innumerable stars. The light we receive from most stars is weaker
Published: 4 February 2021
 than that of the bright stars. However, this light can still be seen as a thin cloud in the
 night sky because the light of all the stars is combined. In particular, the stars in the area
Publisher’s Note: MDPI stays neutral
 around the Scorpius and Sagittarius constellations are more evident than those in other
with regard to jurisdictional claims in
 areas. Therefore, under suitable conditions (a very dark night without light pollution and
published maps and institutional affil-
 moonlight, no clouds, etc.), it is easy to see the Milky Way in the summer season. However,
iations.
 in urban areas, the opportunity to observe the Milky Way is rare because of the bright
 night sky. To address this issue, planetariums worldwide have been working to artificially
 reproduce a starry sky using image reproduction systems [3]. In fields such as astronomy
 education and entertainment, planetariums have played a crucial role in communicating
Copyright: © 2021 by the authors.
 the majesty of the universe to the general public.
Licensee MDPI, Basel, Switzerland.
 Some studies have reported methods for observing and acquiring starlit sky images [4].
This article is an open access article
 There have also been reports on the use of computer-graphics (CG) reproduction methods to
distributed under the terms and
 generate displays [5,6]. However, there has been insufficient discussion of the appropriate
conditions of the Creative Commons
Attribution (CC BY) license (https://
 methods for reproducing astral images and the use of such methods in a planetarium that
creativecommons.org/licenses/by/
 seeks to reproduce the starry sky. Among those that do address this subject, previous
4.0/).
 studies have investigated the relationship between the physical representation factors

Appl. Sci. 2021, 11, 1413. https://doi.org/10.3390/app11041413 https://www.mdpi.com/journal/applsci

Appl. Sci. 2021, 11, 1413 2 of 13

 in a planetarium as well as the perceptual assessment of faithfulness and preference
 for the represented stars by conducting psychometric experiments in a real planetarium
 dome [7,8]. However, the target stars in previous studies were restricted to bright stars
 of small magnitude (from zero to seven), without considering fainter stars and objects
 (such as the Milky Way and nebulas), to investigate the effects of color, brightness, and size
 reproduction.
 In psychometric experiments [7,8], bright target stars were projected in a completely
 planar dark sky, which is an unlikely condition in the actual star field. The star field we
 observe in reality has nonuniform luminance owing to the presence of fainter stars, clouds,
 zodiacal light, and light pollution. In the planetarium industry, it is well known that adding
 natural light is essential for faithfully reproducing a star field that conveys a sense of the
 depth of the universe; however, the best reproduction methods have yet to be determined.
 Accurate reproduction of the Milky Way is particularly important because it has a
 fixed position in the sky and attracts attention as a beautiful element of the starry sky. From
 a scientific perspective, even though the individual stars of the Milky Way are dim, they
 can be perceived as a “visible mass”, forming a “dense cluster”. Although several studies
 have reported contrast perception [9] and density perception in relation to texture [10], we
 found no relevant studies regarding the perception of a group of illuminated point lights in
 scotopic vision. In a planetarium, people observe projected point light sources on a dome
 screen that mimics a star in scotopic vision. Because the physically same reproduction of
 stars is impossible, it is important to perceptually reproduce a starfield properly and not
 physically. In addition, there is no research on the best method to faithfully reproduce the
 Milky Way. Therefore, how to mimic the star field and represent a starry sky more faithfully,
 including fainter stars in the Milky Way, is a prevailing problem in planetarium research.
 Moreover, techniques for rendering the Milky Way that consider the characteristics of
 human perception have not been reported in the field of imaging science and technology.
 In this study, we investigate the relationship between the reproduction method and
 assessments of the faithfulness, preference, and depth of the reproduced Milky Way by
 conducting a psychometric experiment in a planetarium with human observers.

 2. Experiment
 In this experiment, we considered the factors influencing the successful reproduction
 of a starry sky with the Milky Way and nebulas, by evaluating three indices (faithfulness,
 preference, and depth feeling) relating to the images reproduced in planetarium projections
 by individually altering three parameters (transmittance, representation size and star
 density) of individual stars.

 2.1. Reproduction Apparatus
 There are two types of projection systems that produce images in a planetarium:
 optical and digital. Owing to the resolution and color tone limitations of digital projectors,
 it is difficult to reproduce a detailed Milky Way with sufficient resolution to evaluate image
 faithfulness. Therefore, in our experiment, a planetarium with an optical system using
 star plates was used to reproduce fixed stars, nebulas, and the Milky Way, as shown in
 Figure 1. Stars were projected on the dome screen by passing light through the star plates
 installed in the projector. The direct luminance of the stars was adjusted by inserting
 transmission filters in front of the light sources. The representation size and star density
 were controlled by the diameter of the hole and the number of stars in the star plates,
 respectively. Therefore, we defined the three parameters (transmittance, representation
 size and star density) of individual stars.

Appl. Sci. 2021, 11, 1413 3 of 13
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 13

 Figure Mechanismof
 Figure1.1.Mechanism ofoptical
 opticalprojection.
 projection.

 2.2.Experimental
 2.2. ExperimentalStimulus
 Stimulus
 We selected
 We selected stars
 stars around
 around Scorpius
 Scorpius (constituting
 (constituting 1/32
 1/32 ofofthe
 thenight
 nightsky)
 sky)as asexperimental
 experimental
 stimuli because the area around Scorpius has more stars and thus, presents a brighter
 stimuli because the area around Scorpius has more stars and thus, presents a brighter
 segment of the Milky Way than other areas. Therefore, the area around Scorpius is a
 segment of the Milky Way than other areas. Therefore, the area around Scorpius is a suit-
 suitable star field to assess the reproduction of the Milky Way, as it ensures that observers
 able star field to assess the reproduction of the Milky Way, as it ensures that observers can
 can detect a difference when the reproduction pattern changes. For the faithful reproduction
 detect a difference when the reproduction pattern changes. For the faithful reproduction
 of a starry sky, including the Milky Way, we considered the astrophysical factors in the
 of a starry sky, including the Milky Way, we considered the astrophysical factors in the
 real environment, such as the brightness of the surrounding environment, effect of light
 real environment, such as the brightness of the surrounding environment, effect of light
 pollution, and gradation by zenith angle. The twinkling of stars is caused by atmospheric
 pollution, and gradation by zenith angle. The twinkling of stars is caused by atmospheric
 extinction [11]. However, we excluded external factors such as atmospheric extinction
 extinction [11]. However, we excluded external factors such as atmospheric extinction in
 in this experiment after determining that it was better to display a temporally stable star
 this experiment after determining that it was better to display a temporally stable star
 image. Therefore, the twinkling of stars was not included to prevent observers from paying
 image. Therefore, the twinkling of stars was not included to prevent observers from pay-
 attention to the twinkling star, instead of the Milky Way. In addition, it was necessary to
 ing attention
 eliminate thetoinfluence
 the twinkling star, instead
 of sunlight of the
 reflected Milkythe
 within Way. In system
 solar addition, it was necessary
 (zodiacal light) on
 to eliminate
 the perception the of
 influence
 stars andof sunlight
 that of airreflected
 and airwithin the solar
 lights from the system
 Earth’s (zodiacal
 atmosphere light) on
 (light
 the perception of stars and that of air and air lights from the Earth’s
 pollution), to avoid distracting the observers. The experiments were conducted in complete atmosphere (light
 pollution),
 darkness to toreproduce
 avoid distracting the observers.
 an environment The experiments
 of scotopic vision, which were conducted
 is typical of a in com-
 general
 plete darkness to reproduce
 planetarium environment. an environment of scotopic vision, which is typical of a gen-
 eral planetarium environment.
 We prepared 12 types of experimental patterns by changing the transmittance, rep-
 We prepared
 resentation size and12 types of experimental
 star density of the starpatterns
 image. by changing
 Table 1 lists the
 the transmittance,
 projection patternsrep-
 resentation size and star density of the star image. Table 1 lists the
 used as experimental stimuli. As explained previously, we assumed that the transmittance, projection patterns
 used as experimental
 representation size andstimuli. As explained
 star density in the Milkypreviously, we assumed
 Way influenced that the of
 the evaluation transmit-
 the star
 tance, representation size and star density in the Milky Way influenced
 image reproduced in the planetarium. Therefore, we first determined a standard pattern the evaluation of
 the star image reproduced in the planetarium. Therefore, we first determined
 (Std), following the procedure described in Section 2.2.1. This pattern ensures perceptual a standard
 pattern (Std),to
 faithfulness following
 the real the procedure
 starry described
 sky. Second, in Sectionadditional
 we prepared 2.2.1. Thispatterns
 pattern ensures
 in which per-
 an
 ceptual
 individualfaithfulness
 parameter to or
 thea combination
 real starry sky. Second, wehad
 of parameters prepared additional
 been changed. We patterns
 provideina
 which
 detailedanexplanation
 individual parameter
 of each pattern or a in
 combination
 the following of subsections.
 parameters had been changed. We
 provide a detailed explanation of each pattern in the following subsections.
 Table 1. List of experimental projection patterns.
 Table 1. List of experimental projection patterns.
 Changed Parameters, with
 Pattern Changed Remarks
 StdParameters,
 as the Benchmark
 Pattern Remarks
 Std with Std as the Benchmark
 N/A
 StdStd-L1 N/ATransmittance Pattern Std but half transmittance
 Std-L2
 Std-L1 Transmittance
 Transmittance Pattern
 Pattern Stdbut
 Std buthalf
 double transmittance
 transmittance
 Std-L2 S1 TransmittanceSize Pattern Std but half diameter
 Pattern Std but double transmittance
 S1-L1 Size and Transmittance Pattern S but half transmittance
 S1 S1-L2 Size
 Size and Transmittance Pattern
 Pattern Std but
 S but half diameter
 double transmittance
 S1-L1 D1 Size and Transmittance
 Density Pattern S but
 Pattern Std half transmittance
 but 2/3 times density
 S1-L2 Size and Transmittance Pattern S but double transmittance
 D1 Density Pattern Std but 2/3 times density
 D1-L1 Density and Transmittance Pattern D1 but half transmittance

Appl. Sci. 2021, 11, 1413 4 of 13

 Table 1. Cont.

 Changed Parameters, with
 Pattern Remarks
 Std as the Benchmark
 D1-L1 Density and Transmittance Pattern D1 but half transmittance
 D1-L2 Density and Transmittance Pattern D1 but double transmittance
 D2 Density Pattern Std but 3/2 times density
 D2-L1 Density and Transmittance Pattern D2 but half transmittance
 D2-L2 Density and Transmittance Pattern D2 but double transmittance

 2.2.1. Standard Pattern
 For a reproduction faithful to astronomical observations in real life, it is ideal to re-
 produce actual physical factors such as star color, tone, size, and depth. For example, it
 might be possible to make star fields physically identical to the appearance of the night sky
 if we can prepare light sources of infinitesimal size with several color temperatures and
 high dynamic range (HDR) light-emitting performances and project them onto an infinitely
 large dome. However, such a reproduction of star images is impossible. This is because
 the planetarium has limited resources to manipulate the light source, construction design,
 dome size, and optical performance of the projector. Therefore, we first determined which
 projection conditions could reproduce star images that were perceptually the same as the
 real starry sky in a planetarium. The color and luminance of individual stars in the repro-
 duced standard pattern were designed to provide a perception of major stars equivalent to
 that obtained from the actual starry sky, as determined by five experienced observers. All
 observers were male, with an average age of 50 years and abundant experience in astronom-
 ical observation. The observers memorized the transmittance, representation size and star
 density of actual stars through real astronomical observations from a Japanese mountain
 under the conditions of a non-light-polluted, clear sky. The standard experimental pattern
 was then determined by memory matching with various star images whose transmittance,
 representation size and star density had been changed. Memory matching was selected
 because of the difficulty of a side-by-side comparison of the reproduced images with the
 actual starry sky. The luminance of the projections of the stars onto the dome screen was
 measured using a spectroradiometer (CS-2000, Konica Minolta). Since the size of one star
 in the Milky Way was too small to measure, we set up a larger star of 40 viewing angle with
 the same luminance. The size of the star was still smaller than the measurement angle of
 the spectroradiometer, we set it closer to the reflected light (star) on the dome screen, and
 the luminance of the size of one star was properly measured. At this time, the projector was
 set off the center of the dome, so we could measure perpendicular to the surface without
 shading of the screen by the instrument itself.
 Figure 2 presents the projected experimental stimulus prepared as the standard pattern.
 The projection size of stars in the stable stimulus was determined by the magnitude class,
 and the stars were projected in a viewing angle range of 0.20 to 40 from the viewing
 position of the observers, as shown in Figure 2b. The hole diameter sizes of the star plates
 were designed considering all projection conditions, including the intensity of the light
 source, design filter, and optical conditions. The size of each star in the variable stimulus
 was the same (an approximately 2-µm hole) because of the processing limit for metal
 plates. Furthermore, the size of projected stars in the variable stimulus was less than 0.20
 of the viewing angle. The standard parameters of physical factors such as luminance,
 representation size and star density were set as approximately 1.91 cd/m2 at the dome
 screen surface (luminance) with a viewing angle of 0.20 in diameter (size) of each individual
 star which was projected from a projector with a star plate.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 13

 as variable stimuli to change the physical factors in the reproduction of the Milky Way.
Appl. Sci. 2021, 11, 1413 The color and size of stars in the stable stimulus were reproduced by treating each 5star’s of 13
 magnitude information following Pogson’s rule [13] and the color temperature of individ-
 ual stars [14], and by controlling the hole sizes and filters of the star plate in the projector.

 (a) (b) (c)

 Figure2.2.Examples
 Figure Examplesof
 ofprojected
 projectedexperimental
 experimental stimuli.
 stimuli. (a)
 (a) Entire
 Entire stimulus;
 stimulus; (b)
 (b) Stable
 Stable stimulus
 stimulus (bright
 (bright stars);
 stars); (c)
 (c) Variable
 Variable
 stimulus
 stimulus(the
 (theMilky
 MilkyWay,
 Way,faint
 faintstars).
 stars).

 2.2.2.The field of the
 Additional Milky Way used as the experimental stimulus included 60,000 stars
 Pattern
 (density), as shown in Figure 2c. The colors of the stars in the Milky Way were set to
 • theTransmittance-based
 be same in all projection Pattern
 patterns; white. People can recognize stars of magnitude
 6.0 with the naked eye [12]. However,
 The brightness of the night sky has people can perceive
 a significant the brightness
 influence of fainter
 on the perception ofstars
 stars
 with a magnitude greater than 6.0, although their eyes cannot resolve
 during actual observation. When the darkness of the night sky is altered by light pollution the dim star as a
 point.
 and zodiacal light, the visible area of the Milky Way is limited. Changes in contrast 6.0
 Based on this knowledge, we set bright stars that had a magnitude less than be-
 as a stable
 tween stimulus inofour
 the brightness theexperiment.
 surroundingThe fainter
 night sky andstarsthe
 with a magnitude
 faint stars in thehigher
 Milky than
 Way
 6.0
 may were set asorvariable
 increase decrease stimuli to change
 visibility and thethe physical of
 perception factors in the reproduction
 the brilliance of the
 of the star. There-
 Milky Way. The color and size of stars in the stable stimulus were reproduced
 fore, we changed the contrast with the background of the night sky and visibility by pre- by treating
 each
 paringstar’s
 the magnitude information
 L1 and L2 patterns. For following
 these, the Pogson’s
 projectedrule [13] andofthe
 luminance thecolor temperature
 variable stimulus
 of individual stars [14], and by controlling the hole sizes and
 was relatively increased or decreased using neutral density (ND) filters on the lensfilters of the star plate in
 barrel
 the projector.
 of the projector. The transmittance in patterns L1 and L2 was approximately half and twice
 the transmittance
 2.2.2. of the standard pattern, respectively. Furthermore, these ND filters (L1
 Additional Pattern
 for half transmittance, L2 for double transmittance) were set on other patterns (S1, D1,
 • Transmittance-based Pattern
 and D2) to prepare combination patterns with different sizes, densities, and transmit-
 The brightness of the night sky has a significant influence on the perception of stars
 tances.
 during actual observation. When the darkness of the night sky is altered by light pollution
 • zodiacal
 and Size-based Pattern
 light, the visible area of the Milky Way is limited. Changes in contrast
 between the brightness of the surrounding
 For the variable stimulus, we controlled nighttheskysizeand the faint
 of the stars stars in theon
 projected Milky Way
 the dome
 may increase or decrease visibility and the perception of the brilliance
 screen by expanding or contracting the hole diameter of the star plate in the standard of the star. Therefore,
 we changed
 pattern. Thisthe contrast
 allowed uswith the background
 to investigate of the night
 the projection size sky and stars
 of faint visibility
 that by
 arepreparing
 too small
 the L1 and L2 patterns. For these, the projected luminance of the
 to observe as individual stars, but are appropriate for observing the Milky Way. Com- variable stimulus was
 relatively increased or decreased using neutral density
 pared to the standard pattern, pattern S1 had a half diameter. (ND) filters on the lens barrel of the
 projector. The transmittance in patterns L1 and L2 was approximately half and twice the
 • Density-based
 transmittance of thePattern
 standard pattern, respectively. Furthermore, these ND filters (L1 for
 half transmittance, L2
 A star is reproduced for double transmittance)
 by processing a holewerein a set
 staronplate.
 otherThe patterns
 number (S1,of
 D1, and D2)
 processed
 to prepare combination patterns with different sizes, densities, and
 faint stars in the Milky Way depends on the brightness, which affects visibility, becausetransmittances.
 •of the
 Size-based
 masking Pattern
 area removed in the star plate. In contrast, an increase in the number of
 processed
 For the variable also
 faint stars creates
 stimulus, we a lack of tint the
 controlled in the Milky
 size of theWay.
 starsTherefore,
 projected the sizedome
 on the of the
 star field
 screen by was stable for
 expanding or all projectedthe
 contracting patterns. To investigate
 hole diameter of the the
 starappropriate
 plate in thenumber
 standard of
 processed
 pattern. faint
 This stars us
 allowed in to
 theinvestigate
 Milky Way,the we changed
 projection theofdensity
 size to that
 faint stars make arethe
 toostar holes
 small to
 observe as individual stars, but are appropriate for observing the Milky Way. Compared to
 the standard pattern, pattern S1 had a half diameter.
 • Density-based Pattern

Appl. Sci. 2021, 11, 1413 6 of 13

 A star is reproduced by processing a hole in a star plate. The number of processed
 faint stars in the Milky Way depends on the brightness, which affects visibility, because
 of the masking area removed in the star plate. In contrast, an increase in the number of
 processed faint stars also creates a lack of tint in the Milky Way. Therefore, the size of
 the star field was stable for all projected patterns. To investigate the appropriate number
 of processed faint stars in the Milky Way, we changed the density to make the star holes
 relatively dense (pattern D1, 2/3 density) or sparse (pattern D2, 3/2 density) in the star
 plate of the standard pattern for the variable stimulus.

 2.3. Experimental Index
 We conducted a psychometric experiment to assess the perception of three indices:
 the faithfulness, preference, and depth feeling of star image reproduction, in a planetarium
 for 12 types of projection patterns, as summarized in Table 1. These three indices were
 selected to assess the appearance of the Milky Way by considering two image reproduction
 methods, faithful reproduction and preferred reproduction, and a design concept for a
 dome screen that includes the greatest features of the sky. The definitions of these indices
 in this experiment are as follows:
 • Faithfulness: whether the observation target is faithful to one’s impression of the
 actual Milky Way.
 • Preference: whether the observation target can meet one’s impression of an expected
 Milky Way in a planetarium.
 • Depth feeling: whether the observation target conveys the depth of the universe.
 In our previous study [8] investigating faithfulness and preferred star reproduction
 without the Milky Way, we found that the independence between these two indices de-
 pended on the observers; male observers evaluated the faithfulness as a preference, but
 female observers did not. We hypothesize that depth feeling might be related to the other
 indices of faithfulness and preference. To clarify star reproduction with the Milky Way, we
 used these three indices in this study.

 2.4. Experimental Procedure
 The observers evaluated the results compared with those of the recalled actual Milky
 Way using opposite word pairs (“faithful”/“non-faithful”) and five integer levels from
 −2 to +2, and wrote their evaluation values down on answer cards based on a 5-point
 Likert scale. The meanings of each evaluation level were −2 (not faithful), −1 (slightly not
 faithful), 0 (neither), +1 (slightly faithful), and +2 (faithful). In the preference and depth
 feeling evaluations, the observers performed evaluations using another opposite word
 pair (“preference”/“non-preference,” “deep”/“shallow”) with the same five integer levels
 from −2 to +2 without a comparison target. The answer task was conducted in darkness
 with only the projected star images in order to maintain dark adaptation. However, there
 was no other bias to discriminate against particular answers. In the evaluation, there was
 no designated fixation point, and the observers were able to observe the star image freely.
 Therefore, they could judge the total appearance of all projection stimuli based on foveal
 vision with the cones and peripheral vision with the rods [15]. Snapshot images of the
 experimental environment are shown in Figure 3. Each star pattern was projected onto the
 position of the oval mark in the figure. The diameter of the dome screen was 23 m, and the
 zenith of the dome screen was slanted 15◦ frontward. There was no other illumination in
 the space where the experiment was conducted than the projected starry sky image. The
 room appeared completely dark. It was not possible to verify the low light level using the
 spectroradiometer, as it was too dark to measure (

with the Milky Way, we collected experienced/inexperienced observers, each group half
 the total observers; 19 were experienced and 18 were inexperienced. Furthermore, in this
 study, the condition for selecting the observers considered general customers who visited
 a planetarium. Therefore, the observers were randomly gathered without bias to their
Appl. Sci. 2021, 11, 1413 characteristics such as age, gender, or experience of astronomical observation. This 7means
 of 13
 that if observers do not have enough experience observing the Milky Way in the actual
 starry sky, they answered the faithfulness evaluation compared to the imagination of ac-
 tual Milky Way. This psychometric experiment was conducted using the dome of the
 a planetarium.
 relaxed posture. The
 After thedome was maintained
 observers at a suitable
 had taken their seats in temperature,
 the dome, theand the conditions
 illumination in the
 remained comfortable
 dome was turned off. and uniform throughout the experiment.

 Figure 3. Snapshot of experimental environment.
 Figure 3. Snapshot of experimental environment.

 Table
 A 2. Number
 total of 37 of observers(22
 observers for males
 each age.
 and 15 females aged 20–50 and older) participated
 in thisAgeexperiment,
 Bracket as shown in Table 2. All experiments
 Male Female were conducted according
 Total
 to the principles outlined in the Helsinki Declaration. Written informed consent was
 20s 4 5 9
 obtained from all participants. As a result of previous experiments [8], evaluation bias was
 30s 5 5 10
 independent of astronomical observation experience. To confirm this tendency for star
 40s 6 5
 fields with the Milky Way, we collected experienced/inexperienced observers, each 11 group
 Over 50 7 0
 half the total observers; 19 were experienced and 18 were inexperienced. Furthermore, 7
 Total
 in this study, 22
 the condition for selecting 15
 the observers considered 37
 general customers who
 visited a planetarium. Therefore, the observers were randomly gathered without bias to
 At the beginning
 their characteristics suchofasthis
 age,psychometric experiment,
 gender, or experience the observers
 of astronomical received instruc-
 observation. This
 tions for
 means thatthe evaluationdo
 if observers experiment and performed
 not have enough exercises
 experience observingusingthe
 allMilky
 12 patterns.
 Way inInthethe
 experimental
 actual starry sky, instructions, the observers
 they answered heard the
 the faithfulness entire process
 evaluation of thetoexperiment,
 compared rating
 the imagination
 ofmethods including
 actual Milky Way. definitions of the three
 This psychometric indices was
 experiment and conducted
 time managementusing thefor eachofevalu-
 dome the
 ation, and a After
 planetarium. questionnaire for their
 the observers hadpersonal
 taken theirinformation
 seats in the such as age,
 dome, gender, and in
 the illumination experi-
 the
 dome was
 ence of turned off. observation in a planetarium and an actual sky. Both steps of exercise
 astronomical
 and experiment included evaluations for all 12 patterns using three indices. After hearing
 Table 2. Number of observers
 the instructions, for each
 the exercise stepage.
 was performed before the experiment. Therefore, the
 observer understood
 Age Bracket the rangeMale fluctuations in the
 of star images to be evaluated
 Female Total in the
 experiment. It was assumed that the observers had completed dark adaptation by this
 20s 4 5 9
 time so that
 30s
 they were able to observe
 5
 the detailed differences
 5
 among patterns. 10
 The40sexperimental step started 6 after the dome shifted 5 to dark for 35 min11for the dark
 adaptation
 Over 50of rods. There was no 7 illumination except from 0 the projected star-field
 7 image,
 and theTotal
 experiment was preceded 22 by oral instructions
 15 using a microphone 37 in the dark
 dome. In both the exercise and experiment steps, the observers did not know which star
 pattern was projected. In this experiment, the 12 patterns were randomly projected, and
 the At the beginning
 observers of thisevaluated
 sequentially psychometric experiment,
 one evaluation the observers
 index received
 (faithfulness, instruc-or
 preference,
 tions for the evaluation experiment and performed exercises using all 12 patterns. In
 depth feeling) of each pattern within 15 s after observing the star image for 30 s. Between
 the experimental instructions, the observers heard the entire process of the experiment,
 pattern projections, observers had a short break of several minutes while the projection
 rating methods including definitions of the three indices and time management for each
 evaluation, and a questionnaire for their personal information such as age, gender, and
 experience of astronomical observation in a planetarium and an actual sky. Both steps of
 exercise and experiment included evaluations for all 12 patterns using three indices. After
 hearing the instructions, the exercise step was performed before the experiment. Therefore,
 the observer understood the range of fluctuations in the star images to be evaluated in the
 experiment. It was assumed that the observers had completed dark adaptation by this time
 so that they were able to observe the detailed differences among patterns.
 The experimental step started after the dome shifted to dark for 35 min for the dark
 adaptation of rods. There was no illumination except from the projected star-field image,

pattern was reset. The illumination of the dome was turned on after all evaluation tasks
 related to this experiment were completed. Thereafter, the observers answered a ques-
 tionnaire in a lit place and left the room. The duration for evaluating one pattern by the
 three evaluation indices was 75 s. The total time needed from the first introduction to the
 last evaluation of the overall psychometric experiment, including exercises, was approxi-
Appl. Sci. 2021, 11, 1413 8 of 13
 mately 1 h.

 2.5. Experimental Results
 and theTheexperiment
 significancewas of the evaluations
 preceded given
 by oral for each pattern
 instructions using awas verified using
 microphone in thea t-test
 dark
 dome. In both the
 after excluding exercise
 outlier dataandusing experiment steps, the observers
 the Smirnoff–Grubbs did not know
 test and verifying which star
 the distribution
 pattern
 equalitywas projected. Inusing
 of evaluations this experiment, the 12
 an F-test. Here, we patterns
 assumed were
 therandomly
 normalityprojected, and the
 of the evaluated
 observers
 data. These sequentially evaluated one
 tests were conducted evaluation
 separately for allindex (faithfulness,
 66 pattern preference,
 combinations (12 ×or depth
 11/2) for
 feeling) of each pattern within 15 s after observing the star image for
 each evaluation index (faithfulness, preference, and depth feeling). Using parametric sta- 30 s. Between pattern
 projections, observers
 tistical techniques had agenerated
 on data short break fromof several
 the Likert minutes
 scale while
 is stillthe projection pattern
 controversial [16,17].
 was reset. The
 However, illumination
 we used the ratingof the dome
 score was turned
 directly obtained on from
 after all
 theevaluation
 Likert scale tasks
 for related
 naïve ob-to
 this experiment
 servers who did were completed.
 not have experienceThereafter, the observers
 of psychophysical answered aFurthermore,
 experiments. questionnaire inin
 ad-a
 lit place and left the room. The duration for evaluating one pattern
 dition to the above-mentioned analysis, we carried out an analysis of variance (ANOVA) by the three evaluation
 indices
 to confirmwasthe75 influence
 s. The total oftime needed from
 the interaction bythe
 thefirst introduction
 combination to the last
 of physical evaluation of
 factors.
 the overall psychometric experiment, including exercises, was
 First, we checked whether there was a tendency for rating faithfulness based approximately 1 h. on as-
 tronomical observations. However, we could not find any relationship between rating
 2.5. Experimental Results
 scores and observers’ characteristics. Therefore, we treated all answers of 37 observers as
 The significance
 experimental of the evaluations
 results without categorizing given
 thefor each pattern
 observer’s priorwas such aast-test
 verified using
 information age,
 after excluding outlier data using the Smirnoff–Grubbs test
 gender, and experience of astronomical observation. The average rating value and verifying the distribution
 for each
 equality
 pattern isofshown
 evaluations using
 in Figure an F-test.
 4. The Here,
 projection we assumed
 pattern with the the normality
 highest ratingofwas
 the evaluated
 Std-L1 for
 data. These tests were conducted separately for all 66 pattern
 faithfulness and Std for preference and depth feeling, respectively (p < 0.01). In combinations × 11/2)
 (12addition,
 for each evaluation index (faithfulness, preference, and depth feeling).
 D2-L2 for all indices was rated as the lowest score (p < 0.01). For all three indices, we found Using parametric
 statistical
 a tendencytechniques
 of ratings on fordata
 eachgenerated
 pattern. We from the Likert
 focused on thescale is still controversial
 luminance represented[16,17].
 by the
 However, we usedand
 projected patterns the calculated
 rating score thedirectly obtained from the Likert scale for naïve ob-
 total luminance.
 servers who did not have experience of psychophysical experiments. Furthermore, in
 Figure 5 shows the average rating value with the standard error among all 37 observ-
 addition to the above-mentioned analysis, we carried out an analysis of variance (ANOVA)
 ers for each pattern in luminance, ordered from low (left) to high (right). We confirmed
 to confirm the influence of the interaction by the combination of physical factors.
 the tendency from the peak (high rating) to the valley (low rating) along the luminance
 First, we checked whether there was a tendency for rating faithfulness based on
 order for each evaluation index. In the faithfulness evaluation, there was a broad peak
 astronomical observations. However, we could not find any relationship between rating
 from Std-L1 to S1-L2 and a valley for the brighter pattern D2. This valley result was almost
 scores and observers’ characteristics. Therefore, we treated all answers of 37 observers as
 identical for the other two indices of preference and depth feeling. However, the tendency
 experimental results without categorizing the observer’s prior information such as age,
 of the peak range differed, as shown in Figure 5b,c. There were two peaks for preference
 gender, and experience of astronomical observation. The average rating value for each
 evaluation, and the peak range for the evaluation of depth feeling was rather narrow
 pattern is shown in Figure 4. The projection pattern with the highest rating was Std-L1 for
 (from D2-L1 to D1-L2). This result indicates that both luminance and other factors affect
 faithfulness and Std for preference and depth feeling, respectively (p < 0.01). In addition,
 the judgment of preference and depth feeling evaluations.
 D2-L2 for all indices was rated as the lowest score (p < 0.01). For all three indices, we found
 a tendency of ratings for each pattern. We focused on the luminance represented by the
 projected patterns and calculated the total luminance.

 (a) (b) (c)

 Figure 4. Evaluation
 Figure 4. Evaluation results
 results for
 for all
 all observers
 observers for
 for each
 each projection
 projection pattern.
 pattern. (a)
 (a) Faithfulness;
 Faithfulness; (b)
 (b) Preference;
 Preference; (c)
 (c) Depth
 Depth feeling.
 feeling.

 Figure 5 shows the average rating value with the standard error among all 37 observers
 for each pattern in luminance, ordered from low (left) to high (right). We confirmed the
 tendency from the peak (high rating) to the valley (low rating) along the luminance order
 for each evaluation index. In the faithfulness evaluation, there was a broad peak from
 Std-L1 to S1-L2 and a valley for the brighter pattern D2. This valley result was almost
 identical for the other two indices of preference and depth feeling. However, the tendency
 of the peak range differed, as shown in Figure 5b,c. There were two peaks for preference
 evaluation, and the peak range for the evaluation of depth feeling was rather narrow (from

Appl. Sci. 2021, 11, 1413 9 of 13
Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 13

 D2-L1 to D1-L2). This result indicates that both luminance and other factors affect the
 judgment of preference and depth feeling evaluations.

 (a) (b) (c)

 Figure 5.
 Figure Evaluation results
 5. Evaluation results for
 for all
 all observers
 observers for
 for each
 eachprojection
 projection pattern
 pattern in
 inluminance
 luminance order.
 order. (a)
 (a)Faithfulness;
 Faithfulness; (b)
 (b)Preference;
 Preference;
 (c) Depth feeling.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 13
 3. Modeling
 Modeling
 We constructed a common model for evaluation
 evaluation indices
 indices such
 such as
 as faithfulness,
 faithfulness, prefer-
 prefer-
 ence,
 ence, and depth feeling by using physical factors (transmittance, representation size and
 star
 star density)
 density) in
 in a planetarium
 planetarium system
 system in order to describe the evaluation for the Milky
 2, Φ ∈ faithfulness, preference, depth feeling , (1)
 Way.
 Way.√ To
 To derive
 derive the
 the model
 model presented
 presented in
 in this
 this work,
 work, we
 we considered
 considered which factor was more
 , important
 , for representation
 of a starry
 sky:
 faithfulness,
 , preference
 or depth
 1 feeling. (2)
 3.1.
 where
 3.1. and Model
 Estimation
 Estimation 
 Model for Physical
 represent
 for Physical
 the meanFactors
 and standard deviation of the natural logarithm of
 Factors
 the variable,
 The respectively.
 The physical
 physical factors of
 factors of 1,2,3 represents
 transmittance,
 transmittance, representation
 representation the weights
 size and
 size of star
 and ( ,density
 , of
 star density the,
 of the
 where
 standard w represents
 pattern the
 were scaling
 normalized factor.
 standard pattern were normalized as ( , , as By(Cleast
 tra , C mean-square
 size , C density ) fitting,
 = ( 1, as
 1, 1 shown
 ) to in Figure
 integrate
 1,1,1 to integrate the the7,
 changes
 they werefor each factor.
 calculated as 4.02
 According andto their0.38,
 ratioswhere to another
 the pattern,
 fittest
 changes for each factor. According to their ratios to another pattern, the transmittance weights the ( 
 transmittance
 , , ) and
 component
 w were (0.25,
 component was
 was calculated
 0.25, 0.50) andby
 calculated by using
 134.15,
 using thetotal
 totalarea
 respectively.
 the area ofofholes
 Comparisonholes onon ofthe
 the the star
 star plate
 differences
 plate because
 becausein weights
 one oneof
 of
 the the
 across standard patterns
 the evaluation
 standard patterns had had
 indices a value
 indicates
 a value of
 of 1.0.that 1.0.
 Thethe The representation
 change in density
 representation size and size
 wasstar and star
 thedensity
 strongest density
 factor
 compo-
 components
 in evaluating
 nents were derived
 the Milky
 were derived thefrom
 from Way. thecorrelation
 The
 changes changes incoefficient
 in the physical the physical
 values. values.
 forFigure
 all evaluationFigureindices
 6 shows a6summary
 shows was a
 summary
 0.92,
 of eachand of each
 the
 component component
 correlation
 for each for each
 coefficients
 projection projection
 forpattern. For pattern.
 each evaluation example, index For example,
 in were
 the as high
 case inasthe
 of pattern case
 0.97, of
 0.97,
 S1-L1,
 pattern
 and S1-L1,
 0.86. This we obtained
 result indicates three
 thatcomponents
 our hypotheses (Ctraare , Csize , Cdensity ) for
 appropriate = (0.5, 0.25, 1) the
 estimating because
 eval-
 we obtained three components , , = (0.5, 0.25, 1) because the pattern S1-
 the pattern
 uation S1-L1
 of the Milky hadWay halfusing
 transmittance,
 the common a 1/4 area ratio, and in theEquations
 same density as the
 L1 had half transmittance, a 1/4 area ratio, andestimation
 the same density model as the standard (1) pattern.
 and (2).
 standard
 This model pattern. By
 indicates using
 that these physical
 we can reproduce components
 the Milky to represent the
 Way with anofappearance characteristics
 well-bal- of
 By using these physical components to represent the characteristics each projection pat-
 each
 anced projection
 inconstructedpattern, we
 faithfulness,a preference,constructed
 and depth a common
 feelingthe model to
 if apsychometric estimate
 planetarium evaluation the psychometric
 can be made with forthe
 tern, we common model to estimate values
 evaluation
 estimated values for faithfulness,
 appropriate parameters. preference, and depth feeling.
 faithfulness, preference, and depth feeling.
 In our modeling process, we had three hypotheses regarding the evaluation results
 as follows:
 1. Psychometric evaluations can be described using a logarithmic scale that conforms
 to the Weber–Fechner law.
 2. Each psychometric evaluation follows a normal distribution in the logarithmic do-
 main because the average rating value for each evaluation is unimodal, as depicted
 in Figure 5.
 3. Humans evaluate faithfulness, preference, and depth feeling using all physical fac-
 tors comprehensively, and this process can be described as addition in a numerical
 formula.
 Most psychophysical experiments using direct scaling followed a lognormal distri-
 Figure
 bution
 Figure 6.6. Component
 [18], of each
 each projection
 projection
 and a multi-layered
 Component of pattern. model is often expressed by a linear model
 perceptual
 pattern.
 [19]. Considering these hypotheses, we proposed a common estimation model to rep-
 resentInthe
 ourevaluation
 modelingfor
 process, we had three
 the evaluation indexhypotheses regarding
 Φ by a log-normal the evaluation
 distribution for theresults
 Milky
 as follows:
 Way as follows:

Appl. Sci. 2021, 11, 1413 10 of 13
 Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 13

 1. Psychometric evaluations can be described using a logarithmic scale that conforms to
 the Weber–Fechner law. 2, Φ ∈ faithfulness, preference, depth feeling , (1)
 2. √ Each psychometric evaluation follows a normal distribution in the logarithmic domain
 , , because the average
 rating
 value for each , evaluation
 is unimodal,
 1as depicted in (2)
 Figure 5.
 where
 3. and evaluate
 Humans represent the meanpreference,
 faithfulness, and standard anddeviation of theusing
 depth feeling natural logarithm
 all physical of
 fac-
 the variable, respectively. and this
 tors comprehensively, 1,2,3
 processrepresents the weights
 can be described of ( in
 as addition , a numerical
 , ,
 where w represents
 formula. the scaling factor. By least mean-square fitting, as shown in Figure 7,
 they were calculated as 4.02 and 0.38, where the fittest weights ( , , ) and
 Most psychophysical experiments using direct scaling followed a lognormal distribu-
 w were (0.25, 0.25, 0.50) and 134.15, respectively. Comparison of the differences in weights
 tion [18], and a multi-layered perceptual model is often expressed by a linear model [19].
 across the evaluation
 Considering indices indicates
 these hypotheses, that the
 we proposed change inestimation
 a common density was the strongest
 model factor
 f to represent
 in evaluating the Milky Way. The correlation coefficient for all evaluation
 the evaluation for the evaluation index Φ by a log-normal distribution for the Milky Way indices was
 0.92, and
 as follows: the correlation coefficients for each evaluation index were as high as 0.97, 0.97,
 and 0.86. This
 (ln xresult
 − µ)2indicates that our hypotheses are appropriate for estimating the eval-
 !
 (Φ) 1
 f ( x ) = w √ uation exp of−the Milky − 2, Φ ∈ {faithfulness, preference, depth feeling}, (1)
 2πσx 2σ2 Way using the common estimation model in Equations (1) and (2).
 This model  indicates that we can reproduce the Milky Way with an appearance well-bal-
 (Φ) (Φ) (Φ) (Φ) planetarium
 (Φ) (Φ)
 
 (Φ)
 x Ctra , Csize , Cdensity = p1 Ctra preference,
 anced in faithfulness, + p2 Csize and
 + p3depth feeling
 Cdensity , p1if a + p2 + p3 can = 1be made with (2) the
 estimated appropriate parameters.
 where µ and σ represent the mean and standard deviation of the natural logarithm of
 (Φ)
 the variable, respectively. pi (i = 1, 2, 3) represents the weights of (Ctra , Csize , Cdensity ),
 where w represents the scaling factor. By least mean-square fitting, as shown in Figure 7,
 they were calculated as µ = 4.02 and σ = 0.38, where the fittest weights (p1 , p2 , p3 ) and w
 were (0.25, 0.25, 0.50) and 134.15, respectively. Comparison of the differences in weights
 across the evaluation indices indicates that the change in density was the strongest factor
 in evaluating the Milky Way. The correlation coefficient for all evaluation indices was 0.92,
 and the correlation coefficients for each evaluation index were as high as 0.97, 0.97, and
 0.86. This result indicates that our hypotheses are appropriate for estimating the evaluation
 of the Milky Way using the common estimation model in Equations (1) and (2). This
 model indicates that we can reproduce the Milky Way with an appearance well-balanced in
 faithfulness, preference, and depth feeling if a planetarium can be made with the estimated
 appropriate parameters.
 Figure 6. Component of each projection pattern.

 Figure7.7.Fitting
 Figure Fittingresult
 resultby
 byproposed
 proposedcommon
 commonestimation
 estimationmodel.
 model.

 When estimating a specific evaluation index, it is possible to enhance the estimation
 accuracy of the model by setting the parametric mean and standard deviation in Equation
 (1) as follows:

Appl. Sci. 2021, 11, 1413 11 of 13

Appl. Sci. 2021, 11, x FOR PEER REVIEW When estimating a specific evaluation index, it is possible to enhance the estimation
 11 of ac-
 13
 curacy of the model by setting the parametric mean and standard deviation in Equation (1)
 as follows: !
 
 (Φ)
  1 (ln x − µΦ )2
 f x = w(Φ) √ exp − − 2,
 Φ 2πσΦ x 2σΦ 2 2, Φ ∈ (3)
 √ (3)
 Φ ∈ {faithfulness,
 faithfulness,preference,
 preference,depth
 depthfeeling , },
 feeling
 where µ
 where Φ and
 and σ Φ represent
 representthe themean
 meanand andstandard
 standarddeviation
 deviation ofof
 thethe
 natural
 natural logarithm
 logarithm of
 thethe
 of variable
 variableforfor
 evaluation
 evaluation index
 index Φ.Φ.w( Φ)Φrepresents
 representsthethe
 scaling factor
 scaling forfor
 factor thethe
 evaluation
 evalua-
 index
 tion Φ. By
 index Φ.least-mean-square
 By least-mean-square fitting, as shown
 fitting, in Figure
 as shown 8, they8,were
 in Figure theycalculated as shown
 were calculated as
 in Table 3. Here, the fittest weights
 shown in Table 3. Here, the fittest weights (p , p
 1 2 ( , p 3 , , ) and w were (0.26, 0.27, 0.47) and
 ) and w were (0.26, 0.27, 0.47) and 129.93 for
 faithfulness,
 129.93 (0.26, 0.23,(0.26,
 for faithfulness, 0.52)0.23,
 and 129.31
 0.52) andfor 129.31
 preference, and (0.23, and
 for preference, 0.25,(0.23,
 0.52) 0.25,
 and 130.08
 0.52) andfor
 depth feeling, respectively. The correlation coefficients for each evaluation
 130.08 for depth feeling, respectively. The correlation coefficients for each evaluation in- index improved
 to 0.98,
 dex 0.97, and
 improved 0.94.0.97,
 to 0.98, Thisand
 fitting model
 0.94. This for each
 fitting evaluation
 model for eachindex fulfills aindex
 evaluation condition
 fulfillsfor
 a
 reproducing
 condition foran outstandinganappearance
 reproducing outstanding with faithfulness,
 appearance withpreference, or depth
 faithfulness, feeling or
 preference, as
 desired
 depth by observers
 feeling as desiredin abyplanetarium.
 observers in a planetarium.

 (a) (b) (c)
 Figure
 Figure 8.8.Fitting results
 Fitting for for
 results eacheach
 evaluation indexindex
 evaluation by proposed estimation
 by proposed model. model.
 estimation (a) Faithfulness; (b) Preference;
 (a) Faithfulness; (c) Depth
 (b) Preference;
 feeling.
 (c) Depth feeling.

 Mean and
 Table 3. Mean
 Table and standard
 standard deviation values of estimation model.

 Index Φ
 Index Mean
 Meanµ Deviation σ 
 Standard Deviation
 Standard
 Faithfulness
 Faithfulness 4.004.00 0.37
 0.37
 Preference 4.03 0.35
 Preference
 Depth feeling 3.944.03 0.35
 0.40
 Depth feeling 3.94 0.40
 3.2. Evaluation Trends of Observers
 3.2. Evaluation Trendsof
 In the analysis of the
 Observers
 rating scores provided by observers, a specific tendency was
 foundIn the analysis of the ratingWe
 for each evaluation index. classified
 scores the observers
 provided based aonspecific
 by observers, the rating scores of
 tendency all
 was
 non-outlier observers for each pattern using hierarchical clustering by Ward’s
 found for each evaluation index. We classified the observers based on the rating scores of method [20].
 Observers
 all wereobservers
 non-outlier sorted into fortwo
 eachclusters
 pattern (Cluster 1 and Clusterclustering
 using hierarchical 2). The numbers
 by Ward’sof observers
 method
 in Cluster
 [20]. 1 andwere
 Observers Cluster 2 were
 sorted into28two
 and clusters
 4 for faithfulness
 (Cluster 1(5and
 outliers),
 Cluster29 2).
 andThe3 fornumbers
 preference
 of
 (5 outliers), and 19 and 8 for depth feeling (10 outliers), respectively.
 observers in Cluster 1 and Cluster 2 were 28 and 4 for faithfulness (5 outliers), 29 and 3 Figure 9 shows
 thepreference
 for normalized weights obtained
 (5 outliers), and 19 and from
 8 forthe application
 depth ofoutliers),
 feeling (10 our proposed equations
 respectively. for
 Figure
 each cluster in the evaluation indices. For all evaluation indices,
 9 shows the normalized weights obtained from the application of our proposed equations Cluster 1 focused on
 density. The weight results for Cluster 1 were stable across all evaluations;
 for each cluster in the evaluation indices. For all evaluation indices, Cluster 1 focused on however,
 those forThe
 density. Cluster
 weight2 changed
 results for depending on the
 Cluster 1 were evaluation
 stable indices,
 across all as shown
 evaluations; in Figure
 however, 9d.
 those
 These
 for findings
 Cluster indicate
 2 changed that the on
 depending mosttheimportant
 evaluationphysical
 indices, factor
 as shownin the reproduction
 in Figure 9d. Theseof
 the Milky Way for the majority of observers is density control (Cluster
 findings indicate that the most important physical factor in the reproduction of the Milky 1). In Cluster 2,
 luminance reproduction by controlling the transmission filter of the preferred Milky Way
 Way for the majority of observers is density control (Cluster 1). In Cluster 2, luminance
 is a more important factor than density reproduction. Size control is not very important for
 reproduction by controlling the transmission filter of the preferred Milky Way is a more
 either cluster.
 important factor than density reproduction. Size control is not very important for either
 cluster.

questionnaires. However, we found no relationships between the rating score and per-
 sonal data such as age, gender, or experience of astronomical observation. It is interesting
 that the faithfulness evaluation did not depend on the experiences of actual astronomical
 observations or planetarium observations. In other words, there is no difference in the
Appl. Sci. 2021, 11, 1413 evaluation of faithfulness between amateur and experienced observers. This result 12 sug-
 of 13
 gests that people can perceptually evaluate the Milky Way accurately, even if they have
 never seen the actual starry sky.

 (a) (b) (c) (d)

 Figure 9.
 Figure 9. Weights
 Weightsin
 intwo
 twoclusters
 clustersfor
 foreach
 eachevaluation.
 evaluation.(a)(a) Faithfulness;
 Faithfulness; (b)(b) Preference;
 Preference; (c) (c) Depth
 Depth feeling;
 feeling; (d) (d) Total
 Total compar-
 comparison.
 ison.
 We tried to find common characteristics within clusters using the answers to the
 4. Conclusions However, we found no relationships between the rating score and personal
 questionnaires.
 data A such as age,
 natural gender,
 starry or the
 sky with experience of astronomical
 Milky Way is created by aobservation. It issources,
 set of point light interesting
 and
 thatreproduction
 its the faithfulnesshasevaluation did not depend
 not been sufficiently on the experiences
 investigated. of actual
 In this study, astronomical
 to investigate the
 observations
 important or planetarium
 physical factors thatobservations.
 reproduce theInstar
 other words,
 field there
 with the is noWay
 Milky difference in the
 in a planetar-
 evaluation
 ium, of faithfulness
 we analyzed between amateur
 three evaluation and experienced
 indices—faithfulness, observers.and
 preference, Thisdepth
 resultfeeling—
 suggests
 with psychometric experiments, using the projected stars as experimental stimulinever
 that people can perceptually evaluate the Milky Way accurately, even if they have and
 seen the actual
 changing three starry sky.factors (transmittance, representation size and star density). A
 physical
 standard projection pattern was designed by a group of experienced observers with abun-
 4. Conclusions
 dant astronomical observation experience. The standard was faithful to the actual starry
 A natural starry
 sky, perceptually but sky
 not with the Milky
 physically. Way is created
 In evaluation by a set 37
 experiments, of point lightwere
 observers sources,
 en-
 and its reproduction has not been sufficiently investigated. In this study,
 gaged to evaluate 12 types of star patterns projected on a dome screen. Based on the eval-to investigate
 the important
 uation physical
 results, we factors
 proposed that reproduce
 a common estimation themodel
 star field with the Milky
 for describing Way in a
 the faithfulness,
 planetarium, we analyzed three evaluation indices—faithfulness, preference,
 preference, and depth feeling with a log-normal distribution. The resultant model exhib- and depth
 feeling—with psychometric experiments, using
 ited good accuracy with high correlation coefficients. the projected stars as experimental stimuli
 and changing threeabout
 Many studies physical
 humanfactors (transmittance,
 perception representation
 of the real sky at nightsize
 haveand
 beenstardiscussed
 density).
 since Galileo Galilei’s works. As future work, extensive and insightful discussions arewith
 A standard projection pattern was designed by a group of experienced observers fur-
 abundant astronomical observation experience. The standard was faithful to the actual
 ther required based on such studies. In a further analysis involving clustering observers,
 starry sky, perceptually but not physically. In evaluation experiments, 37 observers were
 we identified two clusters defined by evaluation tendencies. The major cluster focused on
 engaged to evaluate 12 types of star patterns projected on a dome screen. Based on the
 density reproduction for the appearance of the Milky Way in a planetarium.
 evaluation results, we proposed a common estimation model for describing the faithfulness,
 preference, and depth Conceptualization,
 Author Contributions: feeling with a log-normal distribution.
 K.O.; methodology, Thevalidation,
 M.T.; resultant K.O.
 model exhibited
 and S.S.; for-
 good accuracy with high correlation coefficients.
 mal analysis, M.T.; investigation, M.T. and Horiuchi; resources, K.O. and S.S.; writing—original
 draft Many studies
 preparation, about
 M.T.; human perception
 writing—review of the
 and editing, real
 M.T. sky
 and at night
 T.H.; have been
 supervision, discussed
 T.H. All authors
 sinceread
 have Galileo Galilei’s
 and agreed works.
 to the As future
 published versionwork,
 of the extensive
 manuscript.and insightful discussions are
 further required based on such studies. In a further analysis involving clustering observers,
 Funding: This research received no external funding.
 we identified two clusters defined by evaluation tendencies. The major cluster focused on
 Institutional Review Board
 density reproduction Statement:
 for the The study
 appearance was
 of the conducted
 Milky Way inaccording to the guidelines of the
 a planetarium.
 Declaration of Helsinki, and approved by the Institutional Review Board of Konica Minolta Plane-
 tarium
 AuthorCo., Ltd.
 Contributions: Conceptualization, K.O.; methodology, M.T.; validation, K.O. and S.S.; formal
 analysis, M.T.; investigation,
 Informed Consent Statement: M.T. and Horiuchi;
 Informed consentresources, K.O.from
 was obtained and S.S.; writing—original
 all subjects involved indraft
 the
 preparation,
 study. M.T.; writing—review and editing, M.T. and T.H.; supervision, T.H. All authors have
 read and agreed to the published version of the manuscript.
 Data Availability Statement: Data not available due to commercial restrictions.
 Funding: This research received no external funding.
 Conflicts of Interest: The authors declare no conflict of interest.
 Institutional Review Board Statement: The study was conducted according to the guidelines of
 the Declaration of Helsinki, and approved by the Institutional Review Board of Konica Minolta
 Planetarium Co., Ltd.
 Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.
 Data Availability Statement: Data not available due to commercial restrictions.
 Conflicts of Interest: The authors declare no conflict of interest.