=Paper=
{{Paper
|id=Vol-2744/short35
|storemode=property
|title=Modeling the Luminance Spatial-Angular Distribution in Lighting Scenes (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2744/short35.pdf
|volume=Vol-2744
|authors=Vladimir Budak,Victor Chembaev,Tatiana Meshkova,Victor Zheltov
}}
==Modeling the Luminance Spatial-Angular Distribution in Lighting Scenes (short paper)==
https://ceur-ws.org/Vol-2744/short35.pdf
Modeling the Luminance Spatial-Angular Distribution in
Lighting Scenes*
Vladimir Budak [0000-0003-4750-0160], Victor Chembaev [0000-0002-2488-4184],
Tatiana Meshkova [0000-0003-1008-2161] and Victor Zheltov [0000-0001-6768-2518]
Department of lighting technology,
Moscow Power Engineering Institute, Moscow, Russia
budakvp@mpei.ru
Abstract. Now days regulatory documents for non-special lighting systems, the
illuminance and various parameters derived from it are normalized as a
quantitative characteristic. In most cases, all calculations are carried out for
illuminance on the floor of the room or on an imaginary working plane located
at the height of the table. However, illuminance is an integral characteristic of
incident light, while the human eye responds to light reflected from the surface.
That is, if we take a completely black surface with a reflection coefficient equal
to zero, then formally you can get the required illuminance on it, while visually
we will not see anything, since nothing will be reflected from the surface. In
terms of the human eye, luminance must be normalized instead of illuminance.
Recently, the calculation and measurement of luminance was an extremely
difficult task, so the it is understandable, that current regulatory documents
describes almost illuminance normalization, but not luminance. This paper aims
to modeling luminance spatial-angular distribution, which enables us to run the
assessment of the lighting quality.
Keywords: Luminance angle distribution, Quality of lighting, Global
illumination.
1 Introduction
One of the key areas of lighting engineering is the design of lighting systems (LS). The
number of characteristics should be determined during LS designing: the types of
lighting devices, their number, location, direction, etc. Now days, the designer is guided
by regulatory documents that determine the qualitative and quantitative characteristics
of lighting.
In current regulatory documents for non-special LS (office, industrial, commercial
and others), the illuminance and various parameters derived from it are normalized as
a quantitative characteristic (ratio of minimum to maximum illumination, etc.) In most
Copyright © 2020 for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0).
*
Publication financially supported by RFBR grant №19-01-00435
2 V. Budak, V. Chembaev, T. Meshkova, V. Zheltov
cases, all calculations are carried out for illuminance on the floor of the room or on an
imaginary working plane located at the height of the table. However, illuminance is an
integral characteristic of incident light, while the human eye responds to light reflected
from the surface. That is, if we take a completely black surface with a reflection
coefficient equal to zero, then formally you can get the required illuminance on it, while
visually we will not see anything, since nothing will be reflected from the surface. In
terms of the human eye, luminance must be normalized instead of illuminance.
Recently, the calculation and measurement of luminance was an extremely difficult
task, so the it is understandable, that current regulatory documents describes almost
illuminance normalization, but not luminance.
Introduction of Unified Glare Rating (UGR), as one of possible quality indicators for
non-specialized LS, can partially eliminate this problem:
0, 25 N L2i i
UGR = 8lg 2 , (1)
La i =1 pi
where Li –glare light source luminance, cd/m2, 𝜔i –glare light source angular size,
steradian, pi – light source position index relative to line of sight, Lа – adaptation
luminance, cd/m2.
Thus, UGR allowed to express the quality of lighting with just one number and was
accepted into regulatory documents. Today the designer is guided by illuminance as a
quantitative characteristic and UGR as a qualitative assessment of lighting, when
designing non-special OS, such as retail, office, public, industrial premises and many
others, where UGR can answers the question - how comfortable will it be for a person
within the lighting installation.
However, the formula is valid only for small-angle uniform gloss sources. UGR cannot
take into account extended uneven glare. The matter is complicated by the fact that the
DIALux and Relux simulation programs widely used in lighting design are based on
the finite element method and do not solve the global lighting equation with respect to
brightness, but the radiosity equation in the diffuse approximation. Which obviously
leads to the fact that secondary glare cannot be taken into account. A significant step
forward in lighting design is the introduction of DIALux Evo based on the method of
Photon maps. However, the methodology for calculating UGR has not changed.
Thus, LS are designed only approximately assessing how comfortable it will be for a
person in it and normalizing an invisible characteristic – illuminance. However, at the
beginning of the last century, it was suggested that the spatial-angular distribution of
luminance plays a key role in the comfort perception [1].
2 Luminance spatial-angular distribution
The modeling of lighting installations is based on the well-known in computer graphics
global lighting equation, first obtained by James T Kajiya in 1986 [2]
1
L(r, ˆl ) = L0 (r, ˆl ) + L(r, ˆl )(r; ˆl , ˆl ) (N
ˆ , ˆl ) dˆl , (2)
where L(r, ˆl ) – luminance at point r in direction l̂ , (r; ˆl, ˆl ) – bidirectional reflection
Modeling the Luminance Spatial-Angular Distribution in Lighting Scenes 3
function (reflection or transmission), L0 – direct luminance component, directly from
sources, N̂ – normal at point r to the scene surface element.
The equation is written relative to the point r located on the surface of the scene,
however, in the task of assessing the quality of lighting, the observer is in the volume
of the scene. After a series of transformations, one can obtain an equation written with
respect to a point in the volume of the scene [3]
1
LV (r, ˆl ) = L0 (r , ˆl ) + C01 L(r1 , ˆl )(r ; ˆl , ˆl )G (r1 , r )
r − r 3 d 3 r
( (r − r − r ˆl )) ˆl − d r1 , (3)
r − r (r − r ) 2
where G (r1 , r ) - equation kernel.
Equation (3) describes the luminance spatial-angular distribution (LSAD) at each point
in the scene space. It allows us to approach the question of assessing the quality of
lighting not only based on the assessment of individual glare light sources as in UGR,
but based on the analysis of a continuous spatial-angular distribution of luminance.
Figure 1. Figure a) shows a point on the surface of the scene, figure b) a point in the volume of
the scene at which we are determining the luminance in a given direction
In 1997, the work "Instant Radiosity" was published [4]. This publication laid the
foundation for a new approach to solving the global illumination equation, which
ultimately did not become widespread in computer graphics. The author formulated the
algorithm of the new method in a phenomenological approach, without giving its full
mathematical justification, using only disparate formulas to describe individual parts
of the modeling process.
In our work, we propose applying Local estimation method of the Monte Carlo method
to solving the global illumination equation.
The solution of the global illumination equation (2) can be expanded in the Neumann
series [5]
1
L(r, ˆl ) = L0 (r, ˆl ) + L0 (r1 , ˆl1 )(r; ˆl1 , ˆl )G (r1 , r ) d 3 r1 +
1 1
L0 (r1 , ˆl1 )(r2 ; ˆl1 , ˆl 2 )G (r1 , r2 ) d 3 r1(r; ˆl 2 , ˆl )G (r2 , r ) d 3 r1 +
+ (4)
4 V. Budak, V. Chembaev, T. Meshkova, V. Zheltov
After a series of transformations, expression (4) can be interpreted as a Markov chain
with a transition probability determined by the kernel of the equation
(r; ˆli , ˆl )G(ri , r)
k ( xi → x) = . (5)
p2 ( xi → x)
As a result of the construction of the Markov chain, we can evaluate the luminance at
a given point in a given direction on the surface of the scene. Such estimation can be
called a Local estimation of the Monte Carlo method, since it allows you to directly
calculate the luminance at a given point on the scene surface in a given direction.
We cannot construct a similar scheme for the global illumination equation for a point
located in the volume of the scene (3). However, we have to determine the quality of
lighting based on the spatial-angular distribution of luminance - so, it is necessary to be
able to obtain an angular distribution of luminance for an arbitrary point in the scene.
Two additional δ-functions and appear in the equation, ( (r − r − r ˆl )) и
r − r
ˆl − , which depend on the desired direction l̂ . This makes it impossible to
r − r
directly simulate the equation. When modeling the brightness at a given point and
direction of the scene space (r , ˆl ) , we cannot get into the direction l̂ we need from the
nodes of the trajectory of the Markov chain. To do this, we need one more additional
node fixing the intermediate point r of the equation (3). This approach is called
Double Local estimation [5]. A Double Local estimation allows modeling the global
illumination equation for a point in the scene in space and thereby obtain a spatial-
angular distribution of luminance.
3 Lighting quality
3.1 Lighting quality criterion based on LSAD
Today, when designing non-specialized LS, such as office premises, public places,
shops, shopping centers, etc., in fact, there is only one criterion that describes the
quality of the LS - the Unified Glare Rating (UGR). The discomfort in the luminance
spatial-angular distribution is affected not only by the absolute value of the observed
luminance, but also by the ratio of the source-background luminance difference to the
background luminance (adaptation brightness) known as contrast [6]. The ratio of
contrast to threshold contrast can serve as a criterion for lighting quality. In the case of
a continuous spatial-angular distribution over the lighting scene, a natural
generalization of contrast is the ratio of the gradient of the distribution of luminance
over the observation field to the average over the luminance field [7][8]. Increasing the
gradient value, the boundary between the glossy source and the background will
become more defined, and the quality of lighting, accordingly, will decrease. That is,
the larger the source and the higher the luminance gradient around the bright source,
the greater the contribution to the discomfort of this source. Note that in real life,
extended glare are both sources of discomfort and contribute to the luminance of the
Modeling the Luminance Spatial-Angular Distribution in Lighting Scenes 5
LSAD adaptation. The generalized contrast at the scene point can be determined [9]:
grad ( L( x, y ) p( x, y ) )
K ( x, y ) = , (6)
L
where
1
A (A)
L= L( x, y) p( x, y)dxdy, A = dxdy , (7)
( A)
x, y – coordinates of the point on the projection of the scene, L – luminance of a given
point in the direction of observation, L – field-average luminance, p(x,y) – some weight
function, taking into account the different contribution to the reaction of the eyes of
points located in the center of the field of view and on the periphery. In the criterion p
formula, it carries the same physical meaning as the position index in the UGR formula.
Thus, we can formulate the lighting quality criterion Q as the field-weighted contrast
K(x, y), assigned to a certain threshold:
1
Kthr
Q= K ( x, y)dxdy , (8)
where Kthr – contrast threshold value.
It is assumed that the threshold contrast will be determined by the light engineering
problem.
3.2 Determination of threshold luminance
It is obvious that in the proposed form, any subtle change of luminance will contribute
to the quality criterion, since there will be a change in the luminance gradient. It is also
obvious that changes in contrasts in luminances below a certain limit will not make a
real contribution. So, for example, if there is a direct source of light in the field of view
or a glare from it in the room, contrasts in the dark corner most likely will not play a
role in the perception of lighting quality. But if we formally carry out the calculation,
then they will give a contribution. Thus, contrasts below a certain threshold Lthr should
not actually be taken into account.
In 1946, Blackwell conducted an enormous amount of research to establish
threshold contrasts in solving the detection problem [10]. In the experiment, the
relationship between the threshold contrast and the luminance of adaptation for various
angular sizes of the target was established. Figure 2. graphs of this dependence are
presented.
6 V. Budak, V. Chembaev, T. Meshkova, V. Zheltov
Figure 2. Dependence of the logarithm of the threshold contrast on the logarithm of the
brightness of adaptation for 5 angular sizes: 121.0, 55.2, 18.2, 9.68, 3.60 angular minutes.
Based on this study, a number of thresholds can be considered as Lthr. By setting the
minimum size of the element that you want to detect - for example, it can be the size of
the character when reading text or signage from a certain distance. Then, knowing the
brightness of the adaptation, which can be taken as the average brightness over the
field, can be determined from the results of Blackwell’s research L - a threshold
change in brightness to solve the detection problem. Then, by setting a certain number
of threshold excesses, one can already determine the threshold brightness as
Lthr = N L , (9)
where N – is a certain number, depending on the difference between the really solved
problem and the threshold.
So, as the threshold brightness of the cutoff of contrasts that do not affect the quality
criterion, we take a certain number of thresholds, for example, 10ΔLthr.
Then the expression for contrast can be written as
L( x, y ) Lthr → 0
K ( x, y ) = . (10)
L( x, y ) Lthr → K ( x, y )
As part of our work, an experiment was conducted to determine the quality of lighting
at stations of the Moscow Metro. Figure 3 shows a scatter map of the average observer
rating from the lighting quality criterion. A linear correlation coefficient was also
calculated between the average rating of observers and the values of the quality
criterion, which amounted to 0.61. The correlation coefficient can reach 1, in this case
all points must lie strictly on the diagonal. To interpret the value of the correlation
coefficient, you can use the Cheddock scale.
Modeling the Luminance Spatial-Angular Distribution in Lighting Scenes 7
Average observer rating
Quality criterion
Figure 3. The scattering of the average observer rating and the lighting quality criterion with an
adaptation luminance equal to the average luminance at the station
As part of the work, we examined the effect of the number of thresholds N on the
correlation coefficient. Figure 4 shows a graph of the correlation coefficient of the
results on the number of thresholds.
Correlation coefficient
Number of thresholds
Figure 4 The dependence of the correlation coefficient on the number of thresholds
8 V. Budak, V. Chembaev, T. Meshkova, V. Zheltov
4 Conclusion
From studies back in 1915, it is known that the luminance spatial-angular distribution
significantly affects the quality of illumination perceived by the observer. Based on
LSAD, it is possible to assess the quality characteristics of lighting, including mainly
normalized today, Unified Glare Rating UGR. However, until recently, there were
neither mathematical methods, nor computational capabilities to model exactly LSAD.
Thus, the existing regulatory documents and tools of engineers by the lighting designer
are built around radiosity modeling.
Today we have a situation when, from the point of view of theory, mathematical
solution methods, algorithms, computational capabilities, physical measurement tools,
lighting engineering is ready to switch to the analysis of a characteristic that is truly
perceived by the human eye - luminance. And as a consequence of this process, it is
possible to change the design paradigm from design to specified quantitative
characteristics, to design taking into account the quality of lighting based on the LSAD.
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