Bull. Hiroshima Inst. Tech. Research Vol. 46 (2012) 187 – 196

ArticleComparative Study on Required Bit Depth of Gamma Quantization for
Digital Cinema using Contrast and Color Difference Sensitivities

Isao FURUKAWA* and Junji SUZUKI**(Received Oct. 28, 2011)

Abstract
A specification for digital cinema systems which deal with movies digitally from production to delivery and projection on the screens is recommended by DCI (Digital Cinema Initiative), and systems
based on this specification have been developed and installed in theaters. The parameters of the systems that play an important role in determining image quality include image resolution, quantization
bit depth, color space, gamma characteristics, and data compression methods. This paper comparatively discusses a relation between required bit depth and gamma quantization using both of a humanvisual system for grayscale images and a color difference model for color images. The required bit
depth obtained from a contrast sensitivity function against grayscale images monotonically decreases
as the gamma value increases, while it has a minimum value when the gamma is 2.9 to 3.0 from CIE
1976 L*a*b* color difference model. It is also shown that the bit depth derived from the contrast sensitivity function is one bit greater than that derived from the L*a*b* color difference model at the
gamma value of 2.9.
Key Words: digital cinema, human visual system, contrast sensitivity function, quantization bit depth,
color difference, CIE 1976 L*a*b*
cess. This gamma value was selected through

1. INTRODUCTION

practical testing in mastering process.

DCI (Digital Cinema Initiative) has decided to employ(2) The minimum modulation function with gamma

a bit depth of 12 bits and a gamma value of 2.6 forquantization matches the slope of the reciprocal ofimages in DCDM (Digital Cinema Distribution Mas-

the Barten’s contrast sensitivity function with 1/γ =ter) [1]. These specified values are derived from the

1/2.6 over the luminance range found in motion

result obtained by using Barten’s contrast sensitivity

pictures.

function for grayscale images [3] as discussed in [2].

Although the reason (1) seems to be supported

However, it is not necessarily clear about the optimum

by a subjective evaluation experiment to decide γ = 2.6

gamma value because the required bit depth is exper-of the DLP projector, it doesn’t seem necessary in theimentally determined after the gamma value, γ, is pri-reason (2) to adjust the slopes derived from both themarily determined as 2.6. Kennel [3] gives the follow-

Barten’s contrast sensitivity function and gamma quan-ing two points as the reason why the gamma value has

tization. It is based on the fact that, in order to avoid

been decided as 2.6.

pseud-contours resulted from the luminance quantiza-(1) DLP (Digital Light Processing) cinema projectors

tion, the luminance differences created by one quanti-

had been using γ = 2.6 for several years with suc-zation step change should be within one JND (Just* Department of Computer Science, Faculty of Applied Information Science, Hiroshima Institute of Technology** School of Information Science and Technology, Aichi Prefectural University
─ 187 ─

Isao FURUKAWA and Junji SUZUKI

Noticeable Difference) over the entire dynamic range

Mopt(u) is the optical MTF (Modulation Transfer Func-

rather than the slope of gamma quantization matchestion):

to that derived from the HVS (Human Visual System).

2 2 2

M opt ( u ) = e −2π σ u ,On the other hand, we proposed how to decide
the gamma value to make sure that the color difference artifact will not occur in the CIE 1976 L*a*b*(2)

σ is the standard deviation of the line-spread function

σ = σ 02 + (Cab d ) ,2

uniform color space and a calculation method of the(3)required quantization bit depth [5]. According to thisσ0 is a constant, Cab is the spherical aberration of themethod, the optimum gamma value is derived from aeye, and d is the pupil diameter described as

numerical evaluation, and the required quantization bit

(

(

))

d = 5 − 3 tanh 0.4 log LX 02 40 2 ,depth can be expressed by an equation if the signal(4)

dynamic range and a perceptual threshold of the color

E is the retinal illuminance in Trolands. The equationdifference are supplied.

for E isThis paper comparatively discusses the optimum

24
π d 2   d   d  E=
L 1 −  +  12.4   .
4
  9.7 gamma values and the required gamma quantization
bit depths obtained from both the Barten and the CIE
1976 L*a*b* color difference models, and moreover(5)

In Eq. (1), other parameters are defined as follows.

shows the relationship between them. First of all, thek : signal-to-noise ratio of the eye,

Barten’s contrast sensitivity function for grayscale

T : integration time of the eye,images and the color difference resulted from the
gamma quantization in the CIE 1976 L*a*b* color

X0 : angular size of the object image,
Xmax : maximum angular size of the integration area,

space for color images are reviewed in Section 2 forNmax : maximum number of cycles over which theeye can integrate the information,

preparing the following discussion. Then, in Section 3,
the visual modulation function for the gamma quanti-η : quantum efficiency of the eye,zation is derived to compare the same modulationp : photon conversion factor,

obtained from the Barten model, and the required bit

Φ0 : spectral density of neural noise,

depth for grayscale images is computed by a numeri-u0 : maximum frequency of lateral inhibition.

cal method in such a way that the modulation from the

The parameters that Barten recommended for the cal-gamma quantization is always smaller than that fromculation are listed below. It should be noted that Barten

the Barten model over the predefined luminancedid not recommend a particular value of X0 to use.

dynamic range. In Section 4, the required quantizationk = 3.0, T = 0.1 (sec), η = 0.03, σ0 = 0.0083 (arc

bit depth for color images is computed numericallywhen the L*a*b* color difference model is applied,deg), Xmax = 12 (deg), Φ0 = 3 × 10-8 (sec deg2), Cab

and this result is compared with that derived in Sec-

(cycles/deg).

= 0.0013 (arc deg/mm), Nmax = 15 (cycles), u0 = 7
In [2], it is agreed to use a value of 1.285 × 106 pho-tion 3. Conclusions are given in Section 5.

tons/sec/deg2/Td as the value of the parameter p.

2. BASIC EQUATIONSTherefore, the factors that can be varied in a calcula-2.1 Visual modulation from Barten modeltion are the frequencies, u, the luminance, L, and the

The Barten’s contrast sensitivity function [3] is given

field of view, X0. The product of u and X0 gives theby Eq. (1). The following explanation of Eq. (1) is

number of cycles in a pattern, and it is defined in thebased on [2].
CSF( u ) =experiment of [2] as 13 cycles. Therefore, X0 can bemade a variable dependent on u:

M opt ( u ) k

Φ0
2 1
1
u2   1+2   η pE +
2
2
(
/
)
u
u
−
0
T  X 02 X max

N max 
1−e

,

X0 =

13
.
u

(6)

Thus, Eq. (1) can be reduced to two independent vari-(1)

ables, the luminance, L, and the spatial frequency, u.

─ 188 ─Comparative Study on Required Bit Depth of Gamma Quantization for Digital Cinema using Contrast and Color Difference Sensitivities

By assuming that the reciprocal of Eq. (1) is a

2.2 CIE 1976 L*a*b* color difference modelvisual modulation threshold to avoid pseud-contour inIn the DCI specification [1], the CIE XYZ tri-stimulus

images, it can be defined that a luminance change cor-values are calculated with a normalizing constant that

responding to one quantization step should be smaller

sets the Y tri-stimulus value equal to the absolute lumi-

than this threshold, m = 1/CSF(u). This conditionnance in cd/m2. The encoding transfer function isoften refers to that a luminance step induced by quan-defined bytization is lower than one JND. Because m = 1/CSF(u)1 2.6  X kx = INT 4095  52.37 


1
2
.
6


 Y 
ky = INT 4095 
 52.37 


1
2
.
6


 Z 
kz = INT 4095 ,
 52.37 

is a function of u, the modulation threshold can beobtained by the numerical calculation method: firstly
we define a luminance value, L, and secondly we compute u that maximizes CSF(u) by using Eq. (1). As aresult, the modulation threshold, m, is obtained fromthe reciprocal of the maximum contrast sensitivity
function, CSFmax(u).
In the following evaluation, the aim luminance
2













(7)

where kx, ky, kz (0 ≤ kx, ky, kz, ≤ 4095) are quantization

range is set from 0.0041 to 41 cd/m (i.e., 10,000:1

codes of XYZ tri-stimulus values, and the function

dynamic range) by considering the range used for theINT[s] returns the nearest integer value of s. There-

projection of motion pictures in a theater. Figure 1 showsfore, the bit depth of 12 bits and the gamma value ofresults of (a) umax that maximizes the contrast sensitivity

2.6 are defined for the gamma quantization of color

function, CSF(u), and (b) m = 1/CSFmax(u) at u = umax.

images. Although the XYZ tri-stimulus values are normalized by 52.37 (cd/m2) which is greater than thevalue of maximum luminance of 41 (cd/m2) assumed
in Section 2.1, this includes extra headroom reservedto accommodate a range of various white points suchas D55, D61 and D65 light sources.
A smallest required bit depth can be obtained insuch a way that the maximum color difference of adjacent quantized color values always becomes smaller
than the predetermined threshold of allowable color
difference in the CIE 1976 L*a*b* uniform color difference space. Let the quantized values of X, Y and Z(a)normalized their maximum values be x(kx), y(ky) andz(kz), the corresponding values of L*, a*, and b* at thecoordinate (kx, ky, kz) in the CIE 1976 L*a*b* space aredefined by

(
)
( )


a ( k , k , k ) = 500 { f ( x( k )) − f ( y( k ))}  (8)b ( k , k , k ) = 200 { f ( y( k )) − f ( z( k ))} , 
L* kx , ky , kz = 116 f y( ky ) − 16
*

*

x

y

z

x

y

x

y

z

y

z

where the function f(w) is defined as follows.
w1 3 ,if ( 24 116 )3 < w ≤ 1
f (w ) = 23
(1 3 )(116 24 ) w + (16 116 ), if w ≤ ( 24 116 ) .

(9)

(b)
Fig. 1 Results from Barten model. (a) L and umax, (b) L and m.

If we assume that the quantization codes kx, ky and kz
─ 189 ─

Isao FURUKAWA and Junji SUZUKI

in the L*a*b* color space change by δx, δy, and δz(these values are either -1, 0 or +1), respectively, theL*(kx, ky, kz), a*(kx, ky, kz), and b*(kx, ky, kz) values canbe written as L*(kx + δx, ky + δy, kz + δz), a*(kx + δx, ky + δy,kz + δz), and b*(kx + δx, ky + δy, kz + δz). Then, the quan-

Lhigh + Llow = 2 Lave .

Substituting Eq. (13) and Eq. (14) into Eq. (12), we
obtainm=

tized step sizes of L*, a*, and b* are

(
(
(

)
)
)

(
(
(

) (
) (
) (

)
)
)

∆L* kx , ky , kz = L* kx + δ x , ky + δ y , kz + δ z − L* kx , ky , kz ∆a* kx , ky , kz = a* kx + δ x , ky + δ y , kz + δ z − a* kx , ky , kz ∆b* kx , ky , kz = b* kx + δ x , ky + δ y , kz + δ z − b* kx , ky , kz . 

(10)

(14)

∆L
.
2 Lave(15)

The visual modulation, m, is determined by the
representation of ΔL (i.e., gamma quantization). Firstly,
we consider the required bit depth of the gamma quantization for which the minimum quantized luminanceis 0 and the maximum quantized luminance is P con-Consequently, the color difference, ΔEab, is obtained

sidering the input luminance (L) range is defined by 0

as follows by computing the Euclidean distance
between two points that are separated by ΔL*, Δa*,

≤ L ≤ P. Thus, the quantized luminance is given byγ k 
L= P n, 2 − 1 and Δb* in L*, a*, and b* directions,{ ({()} { (

2∆Eab =  ∆L* kx , ky , kz
+ ∆a* kx , ky , kz
2=  A f y( ky + δ y ) − f y( ky )


)}

) (

)} + {∆b ( k , k , k )} 
2

*

2

x

y

where n is a quantization bit depth, k is a code value

12of the gamma quantization (0 ≤ k ≤ 2n-1), and γ is az

gamma value. Therefore, the luminance step, ΔL, corresponding to one code value difference is

(
) (
( ))}
{

+C { f ( y( k + δ )) − f ( z( k + δ )) − ( f ( y( k )) − f ( z( k )))}  ,+ B f ( x( kx + δ x )) − f y( ky + δ y ) − f ( x( kx )) − f y( ky )

2

2

y

y

z

z

y

12

∆L =

z

(11)
2

2

(16)

2

where A = 116 , B = 500 , and C = 200 . The values of{(k + 1) − k } ,( 2 − 1)
P

γ

γ

γ

n

(17)

and the luminance average, Lave, of the two luminancevalues that have one code value difference isx(kx), y(ky), and z(kz) are determined by the quantiza-

Lave =

tion function (i.e., gamma quantization characteristics), and the specific expression for the gamma quan-{(k + 1) + k }.
2 ( 2 − 1)
P

n

γ

γ

γ

(18)

tization is discussed in Section 4. We expect any

Substituting Eq. (17) and Eq. (18) into Eq. (15), we

pseud-contours will not occur in images if the color

obtainγ
k + 1) − kγ(m=.
( k + 1)γ + kγ

difference, ΔEab, which is produced by the gamma
quantization as Eq. (11), is smaller than one JND. In

The visual modulation threshold, m, is computed

this paper, we employed the threshold as ΔEab = 1.

3. REQUIRED BIT DEPTH FROM BARTEN
MODEL
3.1 Gamma quantization of entire luminance
range

using Eq. (19) for gamma values of 1.5, 2.0 2.6 and 3.0,
and the results are shown in Fig. 2 together with thatderived from the Barten model. In a relatively highluminance region, k is sufficiently greater than 1 forgenerally 2n>>1, so that the following approximation

A visual modulation for a sinusoidal luminance pattern,

can be derived for k>>1

m, is given as follows when Lhigh and Llow are highest( k + 1)γ = kγ + γ kγ −1 ,and lowest luminance of the pattern, respectively [2].
Lhigh − Llow
m=
.
(12)
Lhigh + Llow

tion threshold, m, as,m=and Lave be an average of Lhigh and Llow:
(13)

(20)

and this leads to a modification of the visual modula-

Moreover, let ΔL be a difference between Lhigh and Llow,∆L = Lhigh − Llow ,

(19)

γ
.
2k + γ(21)

Because we are considering m at the relatively highluminance region (k >> γ), Eq. (21) are approximately

─ 190 ─Comparative Study on Required Bit Depth of Gamma Quantization for Digital Cinema using Contrast and Color Difference Sensitivities

(a)

(b)

(c)
(d)
Fig. 2 Visual modulation threshold for gamma quantization with ρ = 0 (10, 11, 12 bits). (a) γ = 1.5, (b) γ = 2.0, (c) γ = 2.6, (d) γ = 3.0.

represented bysimulation results in Fig. 2 illustrate that this relation-

γ
.
2k

m≈

(22)

On the other hand, k is derived from Eq. (16) as

(

)

 L
k = 2n − 1  
P

(23)

γ

 L
 
2 2 −1  P 

(

n

)

−1 γ

.

ing a minimum bit depth with maintaining the graphof m for the gamma quantization always below that for
the Barten model for a given γ within the concerned

Substituting this k into Eq. (22), we obtainm=

Next, we consider the minimum bit depth for agiven gamma value. This problem corresponds to find-1γ.

ship is maintained.

(24)

luminance range in Fig. 2. In practice, the required bit
depth should be an integer number, but we will searchThis equation can be represented as follows in thefor it as a real number. To determine the required

logarithmic domain.

minimum bit depth, we perform the following steps:
1γ

1γPlog m = − log L + log
γ
2 2n − 1

(

)

(25)

Eq. (25) shows that m and L have a linearly changing

1. Determine γ.
2. Determine an initial bit depth of integer value (e.g.,

relationship in the logarithmic coordinate and the slop

8 bits), and set the bit precision Δ = 1.

of the line is -1/γ in the relatively high luminance3. Compute the visual modulation threshold, mq, forregion. When both the quantization bit depth, m, andthe gamma quantization using Eq. (19) for k from 0

the highest luminance value, P, are constant, the line

to 2n-1.

of m against L moves downward as γ increases because
the decrement by P

1/γ

4. Compute the average luminance value, Lave, using

is greater than the increment of

γ in the second term of the right side in Eq. (25). The

Eq. (18) for the k.
5. Compare mq and the visual modulation threshold,─ 191 ─

Isao FURUKAWA and Junji SUZUKI

mb, from the Barten model for Lave. If mq < mb for all

quantization is performed according to the dynamic

Lave, then go to Step 6. If not, then go back to Step

range of the grayscale image. The quantized lumi-

3 after setting n = n + Δ.

nance, L, is given byγ

6. If Δ = 0.001, then this procedure is ended (i.e., the k ,
L = ρ + ( P − ρ)  n
 2 − 1 

precision of bit depth we found have 3 digits below(26)

decimal point). If not, go back to Step 3 after setting

where n is a quantization bit depth, k is a code valuen = n - Δ, Δ = Δ/10, and n = n + Δ.

of the gamma-quantization (0 ≤ k ≤ 2n-1), ρ is a lowest
luminance value, P is a highest luminance value, and γ

The result obtained by the above procedure is shown

is a gamma value of the quantizer. Therefore, the lumi-in Fig. 3 (a). This result coincides to that from Fig. 2,

nance step, ΔL, corresponding to one code value dif-and it is obvious that the required bit depth decreases

ference is

as γ increases. Fig. 3 (b) shows a critical luminance∆L =(i.e., a luminance that determines the required bit
depth), and the critical luminance shifts to higher
luminance as γ increases.

{(k + 1) − k } ,( 2 − 1)
P −ρ

γ

γ

γ

n

(27)

and the luminance average, Lave, of the two luminancevalues that have one code value difference is

3.2 Gamma quantization of limited luminancerange

Lave =In this subsection, we consider a case that the gamma{(k + 1) + k } + ρ.
2 ( 2 − 1)
P −ρ

γ

γ

γ

n

(28)

Substituting Eq. (27) and Eq. (28) into Eq. (15), we
obtainm=( k + 1)γ − kγ

( k + 1)γ + kγ + P2−ρρ (2n − 1)

γ

.

(29)

If the dynamic range of the luminance is represented
as Dr = P/ρ, the visual modulation threshold can be
written bym=( k + 1)γ − kγ

( k + 1)γ + kγ + 2 (2n − 1) ( Dr − 1)
γ

.

(30)

Eq. (30) coincides to Eq. (19) when the lowest luminance value, ρ, is 0 (i.e., Dr = ∞), so that the visual(a)modulation threshold decreases with a slop of -1/γ inthe relatively high luminance range as seen in subsection 3.1 when the dynamic range of the luminance issufficiently high.
The behavior of the visual modulation threshold,m, can be investigated by regarding m as a function of
the quantization code, k, that is,h( k )m=.
g( k )(31)

From Eq. (30), h(k) and g(k) are the functions of k as
follows.
(b)
Fig. 3 Gamma value dependence with Barten model and gamma
quantization with ρ = 0. (a) required bit depth, (b) criticalluminance.
─ 192 ─h( k ) = ( k + 1 )γ − kγ ,γγg( k ) = ( k + 1 ) + k + a.

(32)
(33)

Comparative Study on Required Bit Depth of Gamma Quantization for Digital Cinema using Contrast and Color Difference Sensitivities

where a is a variable independent from k:a=maximum visual modulation threshold is obtained byγn2( 2 − 1 ).
Dr − 1(34)

substituting Eq. (40) into Eq. (26) and setting γ = 2.
2


( P − ρ )  4( 2n − 1)2− 1 .
Lmax = ρ +n2 Dr − 14( 2 − 1 ) The fluctuation of dm/dk is determined byq( k ) = h ′( k )g( k ) − h( k )g ′( k )= γ  −2 kγ −1 ( k + 1)


γ −1

{

+ a ( k + 1)

γ −1

(41)

Now, we can compute Lmax by using ρ = 0.0041 (cd/}− kγ −1  .
m2) and P = 41 (cd/m2) which are the same as those(35)

employed in the evaluation of the Barten model. For

It is difficult to obtain a general solution analytically(41) are 0.0078, 0.0080 and 0.0081 (cd/m2), respec-

for q(k) = 0 because γ is a real number. If k = 0 (i.e.,tively. These values are obviously near the lowest lumi-

lowest luminance value),nance, so that the visual modulation threshold, m, has

q(0) = γa > 0(36)

is always satisfied for γ > 1, so that the visual modula-

10, 11 and 12 bits quantization, Lmax derived from Eq.

a maximum value at nearly lowest luminance anddecreases linearly with a slop of -1/γ at relatively highluminance.

tion threshold, m, is an increasing function at the lowest luminance.

In the followings, Eq. (30) is evaluated by a computer simulation. The visual modulation threshold, m,

According to the above discussion, it becomesis plotted for γ = 1.5, 2.0, 2.6 and 3.0 in Fig. 4 together

obvious that the visual modulation threshold, m, iswith that obtained from the Barten model. It can beincreasing function at the lowest luminance and also isseen from the figure that m of the gamma quantization

decreasing function at relatively high luminance.

have a maximum value at relatively low luminance and

Therefore, m has at least one relative maximum withindecreases with a slop of -1/γ at relatively high lumi-

this luminance range. A simplest example to solve

nance as discussed above, so that the constraint for

q(k) = 0 analytically is the case of γ = 2. In the follow-

the required bit depth of the gamma quantization isings, we outline the behavior of m for such a case.

alleviated as the gamma value increases. Figure 5 (a)Substituting γ = 2 into Eq. (35), we obtainshows the minimum bit depth required against gamma()

h( k ) = γ −2 k2 − 2 k + a .

(37)value. The procedure to compute this bit depth is thesame as that described in subsection 3.1. This result

Thus, the k that satisfies q(k) = 0 is a root of the qua-shows that the required bit depth decreases as thedratic equation, so that the k is simply derived by solv-

gamma value increases, which coincide with the result

ing the equation as

obtained in Fig. 4 (a). Figure 5 (b) shows that the−1 + 1 + 2a(38),2
where k should be a positive value. From Eq. (34), a iscritical luminance shifts to higher luminance as mono-

k=

(
a=

2 2n − 1)2

tonically as shown in Fig. 4 (b) when the gamma value
increases.

(39)

4. REQUIRED BIT DEPTH FROM COLOR
DIFFERENCE MODEL

for γ = 2. Substituting Eq. (39) into Eq. (38), the k that

Although the required bit depth can be derived fromsatisfies q(k) = 0 is

the contrast sensitivity for grayscale images as

Dr − 1
1  4( 2n − 1 )2
k= 
− 1 .
2
Dr − 1

described in the previous section, it also can be(40)

obtained by color difference model such as CIE 1976
L*a*b* [6]. The evaluation of bit depth by using the

The function, q(k), have a maximum value at the k ofcolor difference model might be a straightfor ward

Eq. (40) because q(k) changes its sign from positive tomethod, for the DCI specification specifies the images

negative as the k increases around the value deter-

by CIE XYZ tri-stimulus values for the purpose ofmined by Eq. (40). The luminance, Lmax, that gives

DCDM.

─ 193 ─

Isao FURUKAWA and Junji SUZUKI

(a)

(b)

(c)
(d)
Fig. 4 Visual modulation threshold for gamma quantization with Dr = 10,000 (10, 11, 12 bits). (a) γ = 1.5, (b) γ = 2.0, (c) γ = 2.6, (d) γ = 3.0.

(a)(b)
Fig. 5 Gamma value dependence with Barten model and gamma quantization with Dr = 10,000. (a) required bit depth, (b) critical luminance.

If one of the tri-stimulus values, X, is normalized
by its maximum value, the gamma-corrected value ofX, respectively. This leads to the following equation byletting the signal dynamic range be Dr = P/ρ.

the signal, xγ, is given by
1γ

xγ = ( x − ρ P ) , γ > 1,x = xγγ + Dr −1 .
(42)

where ρ and P are minimum and maximum values of(43)

As the gamma-corrected quantization uniformly digitizes x (0 ≤ x ≤ (1-Dr -1)1/γ) with n bits, the quantization

─ 194 ─Comparative Study on Required Bit Depth of Gamma Quantization for Digital Cinema using Contrast and Color Difference Sensitivities

step size is given by−1 1 γ(1 − Dr ).
(44)
2n − 1Substituting the tri-stimulus X signal into the lumi∆=

nance, L, in Eq. (26), we obtainγ k 
X = ρ + ( P − ρ ) n x  , 2 − 1(45){

n = log 2 γ (1 − 1 Dr )1γ

( A + 4 B + 4C )1 2 µ (3∆Eab) + 1} .

(50)

Thus, the required bit depth for the gamma quantization using the CIE 1976 L*a*b* color difference model
can be determined from both the dynamic range oftri-stimulus XYZ signals and the gamma value, γ, if thewhere kx is a code value of the quantizer (0 ≤ kx ≤ 2 -1).

perceptual threshold, ΔEab, is given. Although the

When the both sides of Eq. (45) is divided by P anddynamic range of log10Dr = 3.2, which is a typical value

the normalized quantized value of X is replaced byof the movie films [7], is employed in [5], the followingx(kx), Eq. (45) is rewritten byevaluations for the optimum gamma value and the

n

γ

 k 
x( kx ) = Dr −1 + (1 − Dr −1 ) n x  .
 2 − 1(46)

required bit depth extend the dynamic rang to 10,000:1
as described in Section 2.
Figure 6 shows the result of calculating the

By using the relation of Eq. (44), this equation becomesx( kx ) = Dr −1 + ( ∆kx ) .
γ(47)

required quantization bit depth against γ based on Eq.
(50) when the dynamic range, log10Dr, of the color

Because y(ky) and z(kz) are similarly obtained for thesignals is set to 3.0, 3.2, 3.4, 3.6, 3.8 and 4.0. Here,tri-stimulus Y and Z values, the color difference, ΔEab,

ΔEab = 1 is assumed to be a permissible perceptual

can be calculated numerically by substituting x(kx),threshold, and the dynamic range of 10,000:1 used fory(ky), and z(kz) into Eq. (11). This numerical evaluation

the evaluation of the Barten model in Section 2 cor-

is applied to all the quantized points in the threedimensional color space, so that the number of operations in this calculation becomes enormous andimpractical when the number of quantization bits is
relatively large.
In order to solve the problem of this amount ofoperations, we proposed the method to compute therequired number of bits for color signals analyticallyassuming that the change of color difference inducedby the quantization in the CIE 1976 L*a*b* color space
expressed in Eq. (11) are small enough in the continuous values of L*, a* and b* [5]. As a result, the required

Fig. 6 Required bit depth from color difference model.

bit depth, n, can be calculated by the following equation.

{n = log 2 γ (1 − ρd )1γ

( A + 4 B + 4C )1 2 µ (3∆Eab) + 1}

(48)

where ρd = 10-D (D is a dynamic range of the signal
when converting it into the density value: D = log10Dr),

μ = wc-2/3(wc - ρd)1-1/γ, wc = {(2/3)/(1/γ - 1/3)}ρd.
Therefore,

ρd =

1
ρ
= .
Dr P

(49)

Using this relationship, Eq. (48) can be rewritten by
Fig. 7 Gamma value that gives minimum bit depth from color difference model.
─ 195 ─

Isao FURUKAWA and Junji SUZUKI

responds to D = 4.0. This result shows that the bit

CIE 1994 color difference model and CIE2000 color

depth greater than 10 bits is enough to avoid pseud-difference formula [8], etc. The reason why we used
the CIE 1976 L*a*b* model in this paper is that it iscontours in color images.
The value of γ that gives the minimum quantiza-

easy to analyze the color difference compared to othertion bit depth is different for various dynamic ranges,

uniform color space models. Although a result obtained

and this value is obtained by differentiating Eq. (48)here shows that the required quantization bit depth forwith γ and searching γ that gives the minimum value

color images is not severe against that for the gray-of n. There is no essential meaning in the ordinate

scale images, it can be thought that this result is sig-

value in Fig. 7 though it shows the change in the dif-nificant to some degree because actual digital cinema

ferential coefficient of n with γ. It can be seen from theemploys color images. Moreover, it is verified that thefigure that the value of γ at which the differential coef-

requirement for DCDM bit depth of 12 bits, which isficient becomes 0 (i.e., n is minimized) lies in thedetermined based on the subjective evaluation experi-

range of 2.9 to 3.0. For example, if we assume the opti-

ment, is sufficient based on the theoretical consider-mum gamma value is γ = 2.9, the required bit depth

ations using contrast and color different sensitivities.

derived from the Barten model for grayscale images inFig. 5 (n is about 11 bits) is about one bit greater thanthat from the CIE 1976 L*a*b* color difference model

REFERENCES[1] Digital Cinema Initiatives, “Digital cinema system
specification,” Ver. 1.2, Mar. 2008.

(n is about 10 bits).

[2] M. Cowan, G. Kennel, T. Maier, and B. Walker,

5. CONCLUSION“Contrast sensitivity experiment to determine theIn this paper, we examined theoretically the relation-

bit depth for digital cinema,” SMPTE Motion

ship between the required quantization bit depth andImaging Journal, Vol. 113, pp. 281–292, Sept.

the gamma value for DCDM assuming both the Barten

2004.

model as a perceptual model for grayscale images andthe CIE 1976 L*a*b* color space as a perceptual model[3] P. Barten, Contrast Sensitivity of the Human Eye
and Its Effects on Image Quality, SPIE Press,

for color images. The criterion used for avoiding
pseud-contour is that the luminance change or color

1999.
[4] G. Kennel, Color and Mastering for Digital

difference induced by one gamma-quantization step

Cinema, Elsevier Inc., 2007.

change becomes smaller than one JND. As a result,[5] J. Suzuki and I. Furukawa, “Required number ofthe required bit depth obtained from the Barten model

quantization bits for CIE XYZ signals applied todecreases gradually as the gamma value increases,

various transforms in digital cinema systems,”

and the required bit depth obtained from the CIE 1976
L*a*b* color difference model is minimized when the

IEICE Trans. Fundamentals, Vol. E90-A, No. 5, pp.

gamma value is 2.9 to 3.0. Moreover, the required[6] The Color Science Association of Japan, ed.,quantization bit depth derived from the Barten model

Handbook of Color Science [Second Edition],based on the grayscale images is about one bit greaterthan that obtained from the CIE 1976 L*a*b* color

University of Tokyo Press, 1998.

1072–1084, May 2007.

[7] G. Kennel and D. Snider, “Gray-scale transforma-difference model based on color images at γ = 2.9.

tions of digital film data for display, conversion,and film recording,” SMPTE Journal, Vol. 102, pp.

The Barten model is widely accepted as a contrast
sensitivity function for grayscale images, while the CIE
1976 L*a*b* model does not supply a complete uni-1109–1119, Dec. 1993.
[8] H.-C. Lee, Introduction to Color Imaging Science,

form color difference. Actually the improvement has
been done for the color difference models such as the─ 196 ─Cambridge University Press, 2005.