1

Lightning-Fast Dual-Layer Lossless Coding for
Radiance Format High Dynamic Range Images
Taizo Suzuki, Senior Member, IEEE, Sara Yukikata, Kai Yang, and Taichi Yoshida, Member, IEEE

Abstract—This paper proposes a fast dual-layer lossless coding
for high dynamic range images (HDRIs) in the Radiance format.
The coding, which consists of a base layer and a losslessenhancement layer, provides a standard dynamic range image(SDRI) without requiring an additional algorithm at the decoder
and can losslessly decode the HDRI by adding the residual
signals (residuals) between the HDRI and SDRI to the SDRI,
if desired. To suppress the dynamic range of the residuals in theenhancement layer, the coding directly uses the mantissa and exponent information from the Radiance format. To further reduce
the residual energy, each mantissa is modeled (estimated) as alinear function, i.e., a simple linear regression, of the encodeddecoded SDRI in each region with the same exponent. This iscalled simple linear regressive mantissa estimator. Experimental
results show that, compared with existing methods, our coding
reduces the average bitrate by approximately 1.57–6.68 % andsignificantly reduces the average encoder implementation timeby approximately 87.13–98.96 %.
Index Terms—Dual-layer lossless coding, high dynamic range
image, low computational cost, mantissa estimator, Radianceformat, simple linear regression.

I. I NTRODUCTIONADIANCE and OpenEXR are common formats for highdynamic range image (HDRI) coding and are expressedin terms of floating points (floats) with mantissa and exponent
information [1], [2]. To make a high-precision standard dynamic range image (SDRI) from these formats, traditional displays, i.e., decoders, require a tone mapping operator (TMO).
Considering the development costs and the market penetration,
new codecs should not use the TMO at the decoder andshould be able to use existing coders, such as JPEG [3],directly. Logarithmic methods [4–6] logarithmically convert
floats to integers and then encode HDRIs in accordance withstandards such as JPEG 2000 [7] and AVC [8], but do notguarantee the reversibility between the encoding and decoding.
Similarly, standards such as HEVC [9] and VVC [10] andtheir improvements [11–15] support HDRIs, but again do notguarantee the reversibility.
Dual-layer coding for HDRIs [16–20] is another popularform of HDRI coding; for instance, it is used in the JPEG

R

This work was supported by JSPS Grant-in-Aid for Scientific Research (C)under Grant 22K04084.
T. Suzuki is with the Institute of Engineering, Information and
Systems, University of Tsukuba, Ibaraki 305-8573, Japan (e-mail:taizo@cs.tsukuba.ac.jp).
S. Yukikata is with Sky Co., Ltd., Tokyo 108-0075, Japan (e-mail:yukikata@wmp.cs.tsukuba.ac.jp).
K. Yang is with Da-Jiang Innovations Science and Technology Japan Co.,
Ltd., Tokyo 108-0075, Japan (e-mail: roarkyang@gmail.com).
T. Yoshida is with the Department of Computer and Network Engineering,
Graduate School of Informatics and Engineering, The University of ElectroCommunications, Tokyo 182-8585, Japan (e-mail: t-yoshida@uec.ac.jp).

XT HDRI coding standard [21]. The encoder consists of abase layer for encoding a tone-mapped SDRI and an enhancement layer for encoding the residual signals (residuals)
between the HDRI and SDRI. It has two good properties,
image selectivity and backward compatibility. Boschetti etal. achieved a scalable coding for HDRIs by splitting an
HDRI into a pseudo-exponent image like a grayscale image
and a pseudo-mantissa like an SDRI [16]. Banterle et al.
employed a segmentation map to avoid seams or halos atthe boundary of each segmented zone due to the TMO [17].
Mai et al. focused on optimizing the TMO at the cost ofvisual quality of the SDRIs [18]. Watanabe et al. used thehistogram packing technique [22] to enhance the sparseness ofthe histogram values in the enhancement layer of the JPEG XT
framework [19]. Wei et al. proposed a two-layer local inverse
TMO (ITMO) using a fast global edge-preserving smoothing
technique [20]. However, since Boschetti et al.’s and Banterle
et al.’s methods [16], [17] were originally irreversible and theothers [18–20] use the logarithmic method [4], they are nottheoretically reversible.
Although irreversible (lossy) coding is commonly believed adesirable trade-off for HDR transmission, we believe that duallayer “reversible (lossless)” coding has the potential to become
the de facto standard in the future, as it has a third good
property, i.e., reversibility [23], [24]. Thought this property, it
is possible to obtain various SDRIs even after encoding andto utilize many applications of HDRIs. Whilst any dual-layer
coding can offer image selectivity and backward compatibility, the dual-layer lossless coding must have forward/inverse
reversible integer converters and coders in the enhancement
layer. Here, Iwahashi et al.’s method (basic method) [23] witha reversible integer converter is unable to produce highly
compressed HDRIs due to high residual energy. Yoshida et al.
reduced the residual energy by employing range compression
and an adaptive ITMO while using the integer converter of
JPEG XR to guarantee reversibility [24]. The drawback of
Yoshida et al.’s method is its high computational cost in encoding and decoding. For practical applications, the computational
cost in the coder should be as low as possible.
This study presents a “lightning-fast” dual-layer losslesscoding for HDRIs in the Radiance format (Radiance HDRIs).
Unlike existing methods that handle high dynamic range
signals in the enhancement layer, our approach directly uses
mantissa and exponent information, which has low dynamic
range.1 However, it is clear that the residual energy between
1 Note that Boschetti et al. used newly estimated pseudo-mantissa/exponent
information (images) in [16] and could not guarantee reversibility.

2

B. Dual-Layer Lossless Coding
MR

Fig. 1.

MG

MB

E

The format has two layers, a base layer and an enhancement
layer, as illustrated at the top of Fig. 2. This dual-layer coding
has two good properties [16–20]:

Radiance format (a square means a bit).

the original mantissa, not the HDRI as it is, and the encodeddecoded SDRI, which is not to be modified any more, is rather
high, which leads to poor coding performance. To deal withthis problem, we generate an estimated mantissa, which ismodeled as a linear function, i.e., a simple linear regression, of
the SDRI. Note that the original mantissa has obvious discontinuities between regions with different exponents and the nonmodified SDRI causes more discontinuities in the residuals.
In order to prevent a coding performance degradation due tothe discontinuities, we estimate the mantissa in each region
with the same exponent. The proposed simple linear regressive
mantissa estimator (SLRME) does not require any ITMO orcomplex estimation such as iterative optimization and learning,
so it is very fast. The parameters calculated in the encoder aretransmitted to the decoder and are not recalculated afterwards.
The experimental results demonstrate that it outperformed existing methods in terms of bitrate, while significantly reducing
the encoder implementation time.
A preliminary version of this study was presented in [25],where we described an incomplete global approach for the
mantissa estimator. In this paper, we developed a lightningfast local approach that achieves higher coding performanceand significantly reduces the encoder implementation time.
Notation: In what follows, ·(p) denotes the p-th value and·E denotes the values according to the exponent E.

In the base layer, the original HDRI is first converted into an
SDRI by applying an arbitrary TMO at the encoder and theSDRI can be directly reconstructed without any TMO at the
decoder. In the enhancement layer, the residuals between theoriginal HDRI and another HDRI, generated by applying an
arbitrary ITMO to the encoded-decoded SDRI, are encodedat the encoder and the HDRI is reconstructed at the decoder
by applying the inverse operations. When the forward/inverse
integer converters and coders in the enhancement layer are
reversible, the dual-layer coding has the third good property:3) Reversibility: It can reconstruct the HDRIs losslessly.
This is called dual-layer lossless coding [23–25].

C. Simple Linear Regression for Image Processing
Simple linear regression is a standard low-computation
method for data analyses. It linearly estimates an objective
variable Y from an explanatory variable X using two parameters a and b, as follows:
Y (p) = a · X(p) + b.

A. Radiance Format

F (a, b) =The Radiance format (‘.hdr’) [1] is a widely used fileformat for representing HDRIs. Each pixel is represented
by assigning an 8-bit integer to each of three mantissasM ∈ {MR , MG , MB }, where M× is the mantissa for eachcolor, and an exponent E, resulting in a total of 8 × 4 = 32
bits/pixel (bpp) as illustrated in Fig. 1. The format eliminates
the sign bit and combines the exponent part into one valuethat does not depend on the mantissa, unlike the OpenEXR
format [2]. The RGBE converter, which maps 16-bit floatsf ∈ {fR , fG , fB }, where f× is the float for each color, to 8-bit
integers M and E, is defined as
E(p) = ⌈log2 (max(f (p))) + 128⌉ ,256 · f (p).
M (p) =
2E(p)−128(1)(2)

The inverse conversion, which maps M and E to f , is definedasM (p) + 0.5 E(p)−128
·2
.
256

(4)

Since decreasing the estimation error means increasing theestimation accuracy, one calculates

II. R EVIEW AND D EFINITIONS

f (p) =

1) Image selectivity: it provides both HDRIs and SDRIs,
2) Backward compatibility: existing coders can be employed without having to modify them.

(3)

The forward and inverse RGBE converters are perfectly reversible.

NX| Y (p) − (a · X(p) + b) |2 ,
{z
}
|
p=1

(5)

estimation error

where N is the total number of p. When each of the two
equations obtained by partially differentiating F (a, b) in (5)with respect to a and b is set to 0, a and b are uniquelydetermined minimum solutions, as follows:PPP
N · p (X(p) · Y (p)) − p X(p) · p Y (p)
P
P
a=
, (6)
N · p (X(p))2 − ( p X(p))2PP
p Y (p) − a ·p X(p)b=.
(7)
N
Because of its simplicity, this sort of regression is often applied
to images under certain conditions, such as guided filter [26]and chroma from luma (CfL) prediction in AV1 [27].
III. O UR C ODING M ETHOD
The lightning-fast dual-layer lossless coding for Radiance
HDRIs is illustrated at the bottom of Fig. 2. We can see thatthe difference between the basic procedure at the top of Fig. 2
and the one at the bottom of Fig. 2 is only in the enhancement
layer.

3

HDRI

f

IntegerConverter

Encoder 2
IntegerConverter

TMO

Encoder 1

Inverse Integer
Converter

Decoder 2

Decoder 1

S

ITMO

ITMO

Decoder 1

S

Encoder Sidef

Encoder 2

Encoder 2

Encoder 3

Encoder 3

E
M⋆(aE, bE)⋆

TMO

M – M⋆

Encoder 1

Decoder 1

Inverse RGBE
Converter

f

HDRI

E
M⋆

(aE, bE)

⋆

SLRME

SLRME

S⋆

S⋆
⋆

GF

GF
Enhancement Layer
Base Layer
Decoder 1

Encoder Side

Fig. 2.

M

⋆

⋆

S

SDRI
Decoder SideM

RGBE
Converter

HDRI

IntegerConverter

Enhancement Layer
Base Layer

HDRI

f

S

SDRI
Decoder Side

Flows of dual-layer lossless coding: (top) basic procedure and (bottom) our procedure.

A. Incorporating RGBE Converter
Our coding incorporates the RGBE converter and the inversein (1)-(3) into the forward/inverse integer converters at theupper line of the basic procedure. Direct use of the mantissa
and exponent reduces the dynamic range of the information
handled in the enhancement layer and may increase compression efficiency. However, since the residual energy between
the original mantissa M and the encoded-decoded SDRI S
is rather high if S is not to be modified any more, thecompression efficiency will not be as high as expected. Here,existing codecs work well when the residuals are smooth andclose to the color gray.

B. Simple Linear Regressive Mantissa Estimator
We developed SLRME with the goal of making the residuals smooth and gray ones for good coding. Here, SLRME
estimates the mantissa from the encoded-decoded SDRI at alow computational cost. Since the mantissa has discontinuities
between regions with different Es and simple linear regression
works well in the narrowest-possible area, SLRME uses simple
linear regression to estimate M in each region with the same
E. Also, since irreversible degradation of S in the base layer
may adversely affect coding performance, the method slightly
restores S at the expense of a small computation; specifically,
it applies a Gaussian filter (GF) as a preprocessing to S.2
Consequently, it makes use of the estimated mantissa M ⋆ ,which is modeled as a simple linear regression of S, as
2 Any restoration technique can be employed if its cost is low.

follows:
M ⋆ (q) = round(aE · S ⋆ (q) + bE )[0,255] ,(8)

where q means the index determined for each E, S ⋆ is the S
restored with the GF, and round(·)[0,255] denotes a rounding
operation to [0, 255] ∈ N. In accordance with (6) and (7),aE and bE are uniquely determined, i.e., determined without
requiring iterative optimization and learning, in each region
with the same E, as follows:PPP
NE · q (S ⋆ (q) · M (q)) − q S ⋆ (q) · q M (q)
P
P
,
aE =
NE · q (S ⋆ (q))2 − ( q S ⋆ (q))2
(9)
P
P ⋆
q M (q) − aE ·q S (q)
bE =
,
(10)
NE
where NE is the total number of q. As a result, it generates
residuals, which are clearly smoother and closer to gray thanthose produced by Yang et al.’s method [25], as shown in
Fig. 3. In particular, Yang et al.’s method could not estimate
the mantissa in ICICS at all.
C. Influence on Coding Efficiency by Parameters
To provide both HDRIs and SDRIs, our dual-layer losslesscoding transmits the SDRI, the residuals between M and M ⋆ ,and the parameters (aE , bE ) to the decoder. Each of (aE , bE )has only a 32-bit value and in the experiments, there were at
most 23 such parameters, making them of negligible impact
on coding performance. It is also computationally inexpensive,as the decoder does not have to recompute the parameters.

4

139

128

138
137

126
124

136
135
134

122
120

Fig. 3. Differences between mantissas (when Q = 85): (left) part of C23 in [28], (right) part of ICICS in [29], (1st rows) E, S, M ′ , and M ⋆ , and (2ndrows) M , M − S, M − M ′ , and M − M ⋆ , where E has been pseudo-colored and M ′ is the mantissa estimated by Yang et al.’s method [25].

TABLE IE XPERIMENTAL BITRATES [ BPP ], AVERAGE BITRATES [ BPP ], AND
AVERAGE ENCODER IMPLEMENTATION TIMES [ SEC ] ( WHEN Q = 85).
Images[28], [29]
C05
C14
C23
C32
Avg. of 42
Avg. Time
WalkingGirl
WalkingOnSnow
ICICS
UBC
Avg. of 4
Avg. Time

Yoshida+
[24]
18.56
11.49
15.86
13.30
14.81
104.28
12.92
11.06
17.60
12.54
13.53
99.49

Yang+
[25]
17.41
11.02
15.75
12.50
14.04
18.93
13.57
10.90
17.56
13.19
13.81
8.55

Prop.
w/o SLRME
with SLRME
18.47
17.36
11.02
10.72
16.51
15.14
12.73
12.22
14.55
13.82
0.90
1.08
13.65
12.62
11.11
10.60
17.59
16.40
13.29
12.33
13.91
12.99
0.91
1.10

bitrate [bpp]

17

16

15
5

15 25 35 45 55 65 75 85 95

Yoshida+ [24]Yang+ [25]Prop. w/o SLRME
Prop. with SLRME

bitrate [bpp]18

excluded from consideration methods that are irreversible. We
used 1920 × 1080 full-color Radiance HDRIs from the HDREye dataset [28] and DML-HDR dataset [29] and measured
the performance on MATLAB (2021b) running on an Intel
Core i9-11900K CPU. We used Reinhard et al.’s method [30]for the TMO, Huo et al.’s method [31] for the ITMO,3 JPEG
for the base layer coder (Encoder/Decoder 1 in Fig. 2), and
JPEG 2000 lossless mode for the enhancement layer coder(Encoder/Decoder 2 and 3 in Fig. 2).4 While the selectionof methods somewhat affects coding performance, the above
methods are representative. We adjusted the residuals to bepositive values, as JPEG 2000 does not allow negative input signals, and transmitted the adjusted parameter as extra
information. Finally, all parameters that were transmitted as
extra information were considered not to affect the bitratecalculation as the amount of extra information was negligible.
Table I shows examples of bitrates [bits per pixel (bpp)],average bitrates [bpp], and average encoder implementation
times [seconds (sec)] of lossless coding for HDRIs with a
JPEG quality factor of Q = 85. Figure 4 shows the relationships between Q and bitrate in C23 in [28] and ICICS in[29] as representative examples. Our coding with the SLRME
reduced the average bitrate by approximately 5.02 % relative
to the coding without the SLRME and did so at only asmall increase in computational cost. Moreover, it reduced
the average bitrate by approximately 1.57–6.68 %, while significantly reducing the average encoder implementation timeby approximately 87.13–98.96 % compared with the existing
methods. These results show that SLRME can achieved an
excellent level of performance.

17

16
5

15 25 35 45 55 65 75 85 95
Q

Yoshida+ [24]Yang+ [25]Prop. w/o SLRME
Prop. with SLRME

Fig. 4. Relationships between Q and bitrate: (top) C23 and (bottom) ICICS.

IV. E XPERIMENTS ON D UAL -L AYER L OSSLESS C ODING
In the experiment, we compared our coding (with/without
the SLRME) with two existing dual-layer lossless codingsfor HDRIs: Yoshida et al.’s method [24] and Yang et al.’s
method [25]. Since we were interested in lossless coding, we

V. C ONCLUSIONSWe proposed a lightning-fast dual-layer lossless coding
for Radiance HDRIs. To efficiently suppress the dynamic
range of residuals between the HDRI and SDRI in the enhancement layer, our coding directly uses the mantissa andexponent information from the format. To further reduce theresidual energy at a low computational cost, we devised the
SLRME for the encoded-decoded SDRI. Experimental results
demonstrated that our coding outperformed existing methods
in terms of bitrate, while significantly reducing the encoder
implementation time.
3 Only Yang et al.’s method used the ITMO.
4 JPEG and JPEG 2000 were implemented by ‘imwrite.m’ in MATLAB.

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