Bull. Hiroshima Inst. Tech. Research Vol. 56（2022）49-62

ArticleAnalysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region
Isao FURUKAWA* and Junji SUZUKI**（Received Nov. 5, 2021）

Abstract
The Digital Cinema Initiative (DCI) in the U.S. has standardized a system for digital cinema
in 2005, which digitally processes movies from production to distribution/screening. In the DCI
standard, the required number of quantization bits is determined by applying the Barten’s model
as a visual model for luminance signals only. However, since most of the actual movies are in color,it lacks sufficient theoretical basis. In this study, we formulate the problem of applying varioustransformations corresponding to luminance, density, and gamma-corrected values in the discretizeddomain to the CIE XYZ color space. By using the asymptotic expansion method in the continuousvalued domain to these respective values, we compare the required quantization accuracy with thatobtained by the corresponding analysis in discretized domain. The results show that the required
number of quantization bits obtained by the analysis in the discretized domain is consistent
with the results of computer simulations and the approximations obtained in the continuousvalued domain for the cases of luminance (14 bits), density (12 bits), and gamma-corrected value
quantization (11 bits), respectively. The required accuracy of conventional TV signals, which havea smaller signal range than that of digital cinema, is also investigated, and its validity is confirmedby the simulation. The proposed analysis method makes it possible to obtain the required number ofquantization bits without time-consuming computer simulations.
Key Words: digital cinema, digital TV, required number of bits, color difference, XYZ signal, CIE
1976 L*a*b*
system, is the required quantization accuracy of the

1. INTRODUCTION

image signal. Various studies have been targeted

In recent years, there has been rapid progress

for the problem of quantization accuracy, but there

in improving the quality of imaging systems, suchare many obstacles such as the need for a huge

as digital cinema [1],[2] that can achieve quality

amount of computer simulations to obtain usefulexceeding HDTV for moving images, and archiving

design values. Therefore, if an analytical method

systems for art and natural history images [3] forthat formulates the required quantization accuracy

still images. There are several evaluation measures

based on some visual model is available, then it

to determine the image quality in such high-quality

is expected to contribute to the simple design ofimaging systems, including resolution, frequency

imaging systems for various applications.

transfer characteristics, and SNR [4]. Among them,

The color space of the digital cinema standard

one of the most important measures, regardlessspecified by the Digital Cinema Initiative (DCI)of whether the system is for video or still image

in the U.S. is capable of supporting future display

＊＊＊Department of Computer Science, Faculty of Applied Information Science, Hiroshima Institute of Technology
School of Information Science and Technology, Aichi Prefectural University (retired)― 49 ―

Isao FURUKAWA* and Junji SUZUKI**

devices with a wide color gamut by inputting andstoring images by the form of CIE XYZ signals[5],[6]. Therefore, it is desirable to study the required

2. COLOR DIFFERENCE FORMULA AND
ITS VARIABLE TRANSFORMATIONIn the following discussions, x, y, and z arequantization accuracy for CIE XYZ signals. The

normalized signals of X, Y, and Z with theirrequired number of quantization bits for color

white points of the maximum values X 0 , Y 0 , andsignals is determined so that quantized step changes

Z 0 , respectively, and let x(m1), y(m2), and z(m3) be

do not make any artifacts invisible in the continuous

their sample points quantized with M bits. Here,signals before quantization [7]-[10]. This is similar

m1, m2 , and m3 are integers that exist in the range

to the method of obtaining the required number ofdetermined by the number of quantization bits:

quantization bits for monochrome images based on
luminance signal [11]-[17].
Although there is a controversy on the evaluation
formula for color difference, we will focus on the



(1)

2.1 Q
 UANTIZATION FOR VARIOUS TRANSFORM
The minimum values of the luminance x, y, andCIE 1976 L*a*b* color difference, which is widely

z are set to ρ. The quantized values x(m1), y(m2),used and recommended by the CIE. Our previous

and z(m3) for the luminance, density, and gamma-

study [8] have analyzed the required quantization

corrected transform are considered in turn below

accuracy by approximating the quantization error

when these values are all uniformly quantized

as a continuous function under the assumption thatwith M bits. Here, m is used as an integer value

the quantization step size is sufficiently small, but it

to represent any of m1 , m 2 , and m 3 in Eq. (1).

cannot be applied for gamma values smaller than 1

Assuming that the uniform quantization functions

when the gamma-corrected value is used, and also

for luminance, density, and gamma-corrected value

it is impossible to identify the sample points thatquantization are Va (m), V b (m), and Vc(m), and thatgive maximum color difference. In this paper, wethese quantization step sizes are Δa , Δb, and Δc,make use of the same color difference analysis as

respectively, they are represented as follows:

the previous one in the discretized region to derive
the exact solution, and compare it with the resultsobtained by the approximate solution.

In section 2, we show the color difference

(2)
(3)



formula and its variable transformations, and obtain(4)

the quantization step size from the analytical modelof color difference. Section 3 shows the behavior ofThen the quantized values of x(m1), y(m2), andthe transform function for a system with a relativelyz(m 3) are obtained by applying above transformwide signal range (i.e., the minimum luminancefunctions to m1, m2 , and m3 [8].

value is extremely small), such as digital cinema. In
Section 4, the behavior of the transform function is

2.2 C
 OLOR DIFFERENCE BY QUANTIZATION

shown for a system such as digital TV, which has

To evaluate the color difference caused bya narrower dynamic range than the digital cinema.

quantization in the CIE 1976 L*a*b* color space,In section 5, we derive points where the transform

we make use of the following function f( ) of afunctions simultaneously reach their maximum color

luminance value

difference. In section 6, numerical examples of thetheoretical values are presented and their validity



[8]:
(5)

is verified by computer simulations. Section 7 gives
conclusions and future work.
― 50 ―(6)

Analysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region

The color difference caused by quantization is

Table 1 Δ m1, Δm2, and Δm3 values for each sample point.

evaluated between adjacent quantization samples.
Therefore, if we consider a sample point at m1, m2 ,and m3 as the reference for quantization values, thenthe adjacent quantization values are those that havechanged by +1 or -1 from at least one coordinate
value. Among those 26 adjacent samples, the color
difference due to quantization has a maximum value
when the change is, from the results of analysis
in the continuous domain, either m1 → m1-1, m2 →m2+1, and m → m3 -1, or m1 → m1+1, m2 → m2 -1, andm3 → m3+1. We prove that the results obtained fromthe analysis in the continuous domain also hold in
the discrete domain.

The color difference in CIE 1976 L*a*b* color
space between the reference and the adjacent sample

If we denote the sample points where at least oneof m1, m2 , and m3 changes by Δm1, Δm2 , and Δm3

points is denoted by ΔE(m1, m2 , m3). The function V
represents one of Va, Vb, or Vc.

(where Δm1, Δm2 , and Δm3 are either -1, 0, or +1,
except when they are all zero at the same time),
the 26 adjacent sample points can be represented by
Figure 1 in 3-D m1m2m3 space. Here, the neighboring
sample points are categorized to three sets in terms(7)of the number of differences in the coordinate values

where groups 1 to 3 correspond to all-axis-value

where A = 1162 , B = 5002 , C = 2002 . If we definechange (8 samples), 2-axis-value change (12 samples),	

1-axis-value change (6 samples), respectively, and thethensamples are indexed within each group. For example,(m) as(8)can be shown as

the sample point represented by (2-7) is the seventh
point in group 2. Table 1 summarizes Δm1 , Δm2 ,and Δm3 values for each sample point.

(9)In addition, m1, m2 , and m3 are transformed intom1ʼ, m2ʼ, and m3ʼ according to Δm1, Δm2 , and Δm3
values.
(10)Then, miʼ varies from 1 to 2M -1. The function(m1ʼ) can be represented as



(11)

where U(m) is defined by(12)
Similarly,(m2ʼ) and(m3ʼ) can be represented as(13)― 51 ―

Isao FURUKAWA* and Junji SUZUKI**

Therefore, by putting e{U(m1ʼ), U(m2 ʼ), U(m 3 ʼ)}
for {ΔE(m1 , m2 , m 3)} 2 , which is the square of the

3.2 B EHAVIOR FOR LUMINANCE QUANTIZATATION

If we let mθ(L) be mθ for luminance quantization as

color difference of the reference sample against its
26 adjacent samples, we can analyze their functional



(16)

then, U(m) is positive in the region of 1 ≤ m ≤ mθ(L) :behavior. The analysis is applied to 26 sample points



(17)

denoted (1-1) to (3-6) shown in Figure 1. It can beseen that the function h is isometrically largest if
the following condition holds.


(14)



(18)

and it becomes larger than 0 by using the relation

The function e{U(m1ʼ), U(m2ʼ), U(m3ʼ)} also



(19)

monotonically increases, and it has a maximum value
at the upper bound.

3. BEHAVIOR OF U(m) FOR ρ < θ
3.1 CLASSIFICATION OF U(m)ʼs FORM

In the case of quantization of signals withvarious transforms, if we let mθ be the value of mcorresponding to θ that determines the classification

So that U(m) is positive in the region II. Finally,of the function f( ), then mθ is generally a real

for mθ(L) + 2 ≤ m ≤ 2M -1, U(m) and its min/max

number. The form of U(m) differs at three regions

values can be shown as

on the line of m shown in Figure 2 as follows.



(20)



(21)

(15)



The minimum and maximum values of U(m)in each region are denoted by min[U(m)] andmax[U(m)], respectively, in the following discussion.

― 52 ―As a result, U(m) is positive in the region III.

(22)

Analysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region

3.3 B EHAVIOR FOR DENSITY QUANTIZATATION

Next, U(m) in the region of m = mθ(γ) +1 is

Let mθ(D) be mθ for density quantization as(23)Then, U(m) and its min/max values in the regionof 1 ≤ m ≤ mθ(D) can be shown as follows.



It can be seen that U(m) is positive by applying

(24)





the relation

(25)


(38)



(26)

(39)

Finally, U(m) in the region of mθ(γ) + 2 ≤ m ≤ 2M -1 isAs a result, U(m) is positive in the region I. Next,
U(m) in the region of m = mθ(D) +1 is(40)

As it is difficult to clarify the behavior of U(m)
 (27)

analytically, we consider the next functions h(m) andg(m) instead of U(m).

We can see that U(m) in the region II is positive
by applying

(42)

(28)



(41)


It is not difficult to see that the function g(m)Finally, U(m) and its min/max values in theregion of mθ

(D)+ 2 ≤ m ≤ 2 -1 are given as follows.

determines the sign of d{h(m)}/dm. If mc is zero for

g(m), then the mc gives a maximum value of h(m),

M

(29)



(30)

and mc values depend on γ as shown in Figure 3.
The behavior of U(m) for 0 < γ < 1 and 3 ≤ γ isshown in the same figure. Table 2 summarizes U(m)



(31)

As a result, U(m) is positive in the region III.

≤ m ≤ 2M -1.

3.4 B
 EHAVIOR FOR GAMMA-CORRECTEDVALUE QUANTIZATATION

Let mθ(γ) be mθ for gamma-corrected value quantization as

and its min/max values in the region of mθ(γ) + 2

The values of U(m) in Table 2 at m = mθ(γ) +2, 2M -1, mc , mc +1, and mc + 2 are respectively
given by

(32)

Then, U(m) in the region of 1 ≤ m ≤ mθ

(γ)



is


(33)

U(m) is constant for γ = 1 (corresponding luminance

(43)



(44)



(45)

quantization), and is monotonically increasing functionfor γ > 1, while monotonically decreasing and positive
function for 0 < γ < 1. The constant values of U(m) for
γ = 1 are given by


(34)


(35)

and min/max values of U(m) for 0 < γ < 1 are given by



(36)
(37)

(46)




(47)

As a result, it can be seen that U(m) in theregion III is positive because both U( mθ(γ) + 2) andU(2M -1), which give a minimum value of U(m), arepositive.

― 53 ―

Isao FURUKAWA* and Junji SUZUKI**

4.2 B
 EHAVIOR FOR LUMINANCE QUANTIZATATION

Table 2 Behavior of U(m) for mθ(γ) + 2 ≤ m ≤ 2 M -1.

When the function V in Eq. (48) is replaced by Va

in Eq. (2), the form of U(m) is(49)Then, U(m) is monotonically decreasing function
with respect to m, and it has a minimum value at m
= 2M -1, and has a maximum value at m = 1, so thatthey are given as follows.



(50)
(51)

As a result, U(m) is positive in this case.
(*) mmax and mmin give maximum and minimum
values of U(m), respectively. γθ = 3(θ/ρ ){2+(θ/ρ )},γ1 = 3(1/ρ ){2+(1/ρ )} [8].

4.3 B EHAVIOR FOR DENSITY QUANTIZATATION
When the function V in Eq. (48) is replaced by Vb

in Eq. (3), the form of U(m) is

4. BEHAVIOR OF U(m) FOR θ ≤ ρ(52)Then, U(m) is monotonically increasing function

In the previous section, we investigated thewith respect to m, and it has a minimum value at m

behavior of the function U(m) for relatively wide rage

= 1, and has a maximum value at m = 2M -1, so thatsignal (i.e., ρ < θ). This section studies the behavior

they are given as follows.

of U(m) for relatively narrow range signal (i.e., θ ≤ ρ).



4.1 FUNCTIONAL FORM OF U(m)Figure 2.


(48)

(54)

As a result, U(m) is positive in this case.

The functional form of U(m) for θ ≤ ρ is given

as follows similar to the region III of θ ≤ ρ in

(53)

4.4 B
 EHAVIOR FOR GAMMA-CORRECTEDVALUE QUANTIZATATION

For the gamma-corrected value quantization, the― 54 ―

Analysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region

form of U(m) is

Table 3 Behavior of U(m) for θ ≤ ρ(55)

as the function V in Eq. (48) is replaced by Vc in Eq.
(4). The behavior of U(m) is as described in 3. except
the domain is 1 ≤ m ≤ 2M -1 instead of mθ(γ) + 2 ≤m ≤ 2M -1, so that the min/max values are different.
The γ-dependency of mc, which gives a maximum
value of h(m) in eq. (41) is shown in Figure 4. U(m)and its min/max values for θ ≤ ρ are summarizedin Table 3.
It is obvious that U(m) has its maximum value ofU( mc +1) or U( mc + 2) for h( mc +1) > h( mc ),and U( mc ) or U( mc +1) for h( mc +1) < h( mc ).
The minimum value of U(m) at m = 1 and 2 M -1
is obtained as follows, and these values are both
positive.
(*) mmax and mmin give maximum and minimum
values of U(m), respectively. γ1 = 3(1/ρ){2+(1/ ρ)} [7].
maximum value of e{U(m1ʼ), U(m2ʼ), U(m3ʼ)} in each(56)

region gives the maximum color difference of the(57)

is determined from the condition that the maximum

whole, and the required number of quantization bits
color difference does not exceed 1.

As a result, U(m) is positive in this case.

5.2 D
 OMAIN OF m1, m2, m3 AND ITS CLASSIFICATIONLet each quantization step size be given by

5. MAXIMUM COLOR DIFFERENCE



(58)

5.1 R
 EQUIRED NUMBER OF QUANTIZATION
BITS

Then, the relation between miʼ and mi (i = 1, 2, 3) is

The previous discussion proves that U(m) is

The ranges of m1, m2 , and m3 have been defined

positive for all luminance, density, and gamma-

in Eq. (1). Since we only need to find the color

corrected value quantization against any values

difference by limiting the change of quantization

within the range specified by Eq. (1) for both

points in the m1m2m3 space to either 0 or ±1, the set

conditions ρ < θ and θ ≤ ρ . Therefore, it canof quantized samples to be considered contains 3 ×

be seen that e{U(m1ʼ), U(m2ʼ), U(m3ʼ)}, the square of

3 × 3 = 27 points. Figure 5 shows the domain of m1,

the color difference, gives the maximum value form2 , and m3 in the 3-D space for θ ≤ ρ , where thecoordinate changes of m1 → m1-1, m2 → m2+1, andquantization points are indicated by the numbers

m 3 → m 3 -1, together with m1 → m1+1, m 2 → m 2 -enclosed in square symbols. Table 4 shows the1, and m3 → m 3+1. In addition, the square of thisdomain of m1, m2 , and m3 in each region.

color difference takes its maximum value at the
point where U(m 1ʼ), U(m 2 ʼ), and U(m 3 ʼ) havesimultaneously their maximum values because
the partial differentiations with respect to U(m1ʼ),U(m2ʼ), and U(m3ʼ) are all positive.
In this section, we derive the point whereU(m1), U(m2), and U(m3) simultaneously reach theirmaximum in each region of m1m2m3 space in thecases of m1 → m1-1, m2 → m2+1, and m3 → m3-1, andcalculate e{U(m1ʼ), U(m2ʼ), U(m3ʼ)} at that point. The



(59)

5.3 Q UA NTI ZED POI NTS FOR COLOR
DIFFERENCE COMPUTATIONThe candidate points that possibly give themaximum color difference can be determined in
3-D m1m2m3 space for ρ < θ. Figure 6 and 7 showsthese points for luminance and density quantization,
respectively. Note that m1 and m3 are identical to m

itself, while m2 is subtracted by 1 from m according

to Eq. (59). In the case of gamma-corrected value― 55 ―

Isao FURUKAWA* and Junji SUZUKI**

Table 4 Domain of m1, m2 , and m3 in each region for ρ < θ.

quantization, these candidate points are shown inθ ≤ ρ as discussed in 4., respectively.

Figure 8 and 9 as examples for cases of 0 < γ < 1 and

The following values of mθ(L) , mθ(D) , and mθ(γ) are

γ1 < γ, respectively. The points for other conditions

used for luminance, density, and gamma-correctedof gamma values are readily found by examining thevalue quantization. Note that these constants are not

behavior of the color difference function shown in

defined when the minimum value of luminance is ρ

Table 2.

= 10 -2.0 , since θ ≤ ρ in this case.

The points that give the maximum color
difference with θ ≤ ρ for various quantization



(60)

are shown in 3-D m1m2m3 space in Figure 10.

In this case, color differences for luminance and(61)

density quantization are maximized at m = 1 andm = 2M -1, respectively. The color difference forgamma-corrected value quantization is determinedaccording to its behavior shown in Table 3.



(62)

γθ and γ1 are given as follows

6. NUMERICAL EXAMPLES



6.1 CONSTANTS

We use ρ = 10 -3.2 and 10 -2.0 , which are typicalfor cinema film screens [3] and television monitors

(63)
(64)

[4], respectively. As already shown in Eq. (6), θ= (24/116) 3≈ 0.00885645 and= (116/24) 2

23.36111111. Thus, the cases of ρ = 10
10

-2.0

-3.2

≈

and ρ =

corresponds to ρ < θ as discussed in 3. and6.2 REQUIRED NUMBER OF BITS

― 56 ―

In numerical evaluations, the required number ofAnalysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region

quantization bits is obtained by calculating the color
difference ΔE and applying the condition that it
does not exceed 1. Substituting M = 14 into Eq. (61)for luminance quantization, mθ(L) ≈ 134.843 and mθ(L)= 135 are obtained. Figure 11 (a) shows the resultof ΔE for ρ = 10 -3.2 . The point 5 in the figure gives
maximum color difference ΔEmax = 0.514562 at (m1,m2 , m3) = (134, 133, 134).
In the case of density quantization, by substituting
M = 12 into Eq. (62), mθ(D) ≈ 1468.13 andmθ(D) =

1468 are obtained. Figure 11 (b) shows the result of
ΔE for ρ = 10 -3.2 . The point 27 in the figure gives Δ
Emax = 0.649522 at (m1, m2, m3) = (4095, 4094, 4095).
I n t he c a s e o f g a m m a - c o r r e c t e d v a lue
quantization, mθ(γ) and mθ(γ) depend on both γ and
M. Figure 11 (c) shows the result of ΔE for ρ = 10 3.2

and γ = 2.6. The point 27 in the figure gives Δ

Emax = 0.558289 at (m1, m2 , m3) = (325, 324, 325). On
the other hand, Figure 11 (d) shows the result of ΔE
for ρ = 10 -3.2 and γ = 2.8. The point 46 in the figure
gives ΔEmax = 0.53105 at (m1, m2 , m3) = (479, 478,
479).
Table 5 shows maximum color differences between
the quantized points and their adjacent points
calculated from the computer simulation only forcritical situations that the maximum color differencechanges from greater than 1 to less than 1. It is
confirmed from the simulation that the maximum
color difference for luminance quantization is obtained
at m = 1 in Eq. (17), while that for density quantization
is obtained at m = 2M -1 in Eq. (31) In addition, themaximum color difference for gamma-corrected value
quantization is determined by the value of m in Table
2 which depends on γ as described in 3.4. These
results show that the theoretical values agree with thecomputer-simulated values, and therefore the validityof the theory has been verified.

― 57 ―

Isao FURUKAWA* and Junji SUZUKI**― 58 ―

Analysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region― 59 ―

Isao FURUKAWA* and Junji SUZUKI**

14

12

7. CONCLUTION
The proposed analysis method for calculatingcolor differences in the discretized region has thefollowing advantages over that in the continuous
approximation.
・	The analysis can be applied even when thequantization step size is not sufficiently small.
・	It is possible to analyze the case of 0 < γ < 1
in gamma-corrected value quantization, whichcannot be handled by the continuous approximation
analysis.
・	It is possible to identify the exact quantizedsample coordinates that give the maximum color
difference.
On

the

other

hand,

the

analysisoftheapproximations for continuous signals has theadvantage that the required number of quantization
bits can be obtained by calculating only one single
equation.
Future work includes confirming the effectivenessof this analysis method in more appropriate color
difference formulas and color spaces, and studying
contrast and gamut mapping algorithms that take
into account the three-dimensional distribution ofcolor differences due to quantization.
― 60 ―

Analysis of Required Quantization Precision for CIE XYZ Signals in Discrete Region

Table 5 Discrete sample coordinates and maximum

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