1

Wavelet-based Spectral-Spatial Transforms for
CFA-Sampled Raw Camera Image Compression
Taizo Suzuki, Senior Member, IEEE

Abstract—Spectral-spatial transforms (SSTs) change a raw
camera image captured using a color filter array (CFA-sampled
image) from an RGB color space composed of red, green,
and blue components into a decorrelated color space such as
YDgCbCr or YDgCoCg color space composed of luma, difference
green, and two chroma components. This paper describes three
types of wavelet-based SST (WSST) obtained by reorganizing allof the existing SSTs covered in this paper. First, we introduce
three types of macropixel SST (MSST) implemented within each2 × 2 macropixel. Next, we focus on 2-channel Haar wavelet
transforms, which are simple wavelet transforms, and 3-channel
Haar-like wavelet transforms in each MSST and replace theHaar and Haar-like wavelet transforms with Cohen-DaubechiesFeauveau (CDF) 5/3 and 9/7 wavelet transforms, which arecustomized on the basis of the original pixel positions in twodimensional (2D) space. Although the test data set is not big,in lossless CFA-sampled image compression based on JPEG
2000, the WSSTs improve the bitrates by about 1.67 to 3.17
% compared with not using a transform and the WSSTs thatuse 5/3 wavelet transforms improve the bitrates by about 0.31
to 0.71 % compared with the best existing SST. Moreover, inlossy CFA-sampled image compression based on JPEG 2000, the
WSSTs show about 2.25 to 4.40 dB and 26.04 to 49.35 % in
the Bjøntegaard metrics (BD-PSNRs and BD-rates) comparedwith not using a transform and the WSSTs that use 9/7 wavelet
transforms improve the metrics by about 0.13 to 0.40 dB and2.27 to 4.80 % compared with the best existing SST.
Index Terms—Color filter array, color transforms, raw camera image, spectral-spatial transforms, wavelet transforms,
YDgCbCr, YDgCoCg.

I. I NTRODUCTION
OLOR transforms, which change a full-color image froman RGB color space composed of red (R), green (G),and blue (B) components into a decorrelated color space
expressed by one luma and two chroma components, arecommonly used in image (video) preprocessing. Since there is
not much correlation between the resulting luma and chroma
components, the color transforms greatly contribute to the effectiveness of image-processing implementations. Traditional
compression standards, such as JPEG [1], JPEG 2000 [2],and MPEG-2 [3], employ irreversible and reversible colortransforms (ICT and RCT) from RGB color space into YCbCr
color space (YCbCr-ICT and YCbCr-RCT), where Y , Cb, andCr mean luma, blue-difference chroma, and red-difference
chroma components, respectively. Newer compression standards, such as JPEG extended range (XR) [4], advanced
video coding (AVC) [5], and high efficiency video coding(HEVC) [6] use an RCT from RGB color space into YCoCg

C

T. Suzuki is with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki, 305-8573 Japan (e-mail:taizo@cs.tsukuba.ac.jp).

Fig. 1: Bayer pattern of a particular area of Parrot in Kodak
images [12] (each 2 × 2 pixel square is a macropixel): (left)
RGB full-color image and (right) simulated CFA-sampled
image with corresponding diagram.

color space (YCoCg-RCT) [7], where Y , Co, and Cg meanluma, orange chroma, and green chroma components, respectively. The YCoCg-RCT is composed of two calculations of
the mean and difference between each color component. It
decorrelates signals to a greater extent than the YCbCr-RCT
does, but this inferior-to-superior relationship may be reversed
depending on the image. Moreover, some studies [8–11] havediscussed a family of RCTs different from the YCbCr-RCT
and YCoCg-RCT.
Raw camera images captured using a color filter array(CFA), such as a Bayer pattern (shown in Fig. 1) (CFAsampled images), are usually compressed after they havebeen processed with demosaicing, denoising, deblurring, tonemapping, and so on. On the other hand, many studies [13–20]
have chosen to compress CFA-sampled images before processing, because professional photographers and designers tend
to prefer to work with them directly. This paper focuses on
the transforms in [17–20], which are called spectral-spatialtransforms (SSTs) in this paper.1 In [17], Zhang and Wu
discussed a lossless CFA-sampled image compression using
Mallat wavelet packets-based SSTs, which are composedof one-dimensional (1D) Cohen-Daubechies-Feauveau (CDF)
wavelet transforms [21], such as the 5/3 and 9/7 wavelet
transforms used in JPEG 2000, implemented horizontally andvertically. In [18], Malvar and Sullivan proposed a macropixel
SST (YDgCoCg-MSST2 ) that is implemented within each2 × 2 macropixel and changes a CFA-sampled image from
RGB color space into YDgCoCg color space for compression;
here, Dg is the difference green component. The YDgCoCgMSST is obtained by adding calculations of the mean anddifference between the green components (G1 and G2 ) in each
1 Although SST refers to all transforms that change a CFA-sampled image
from an RGB color space into a decorrelated color space such as YDgCbCr
or YDgCoCg color space, the conventional SSTs in [17–20] are reorganized
into the proposed WSSTs in this paper.
2 Although it is simply called MSST in [18], in this paper it is called
YDgCoCg-MSST to distinguish it from other transforms.

2macropixel to the YCoCg-RCT; i.e., it comprises only threecalculations of the mean and difference between each color
component. In [19], Hernández-Cabronero et al. presented alossless CFA-sampled image compression pipeline to JPEG
2000. Although the SST is not explicitly stated in [19], byanalyzing it carefully, we can see that the YDgCbCr-MSST
described later can potentially be used. In [20], Lee et al.
analyzed the relationship between the HL and LH subbands3
generated in [17] and decorrelated them more with simple
calculations.
This paper describes three types of wavelet-based SST(WSST) that change a CFA-sampled image from RGB color
space into YDgCbCr or YDgCoCg color space. First, we
extend the YCbCr-RCT to YDgCbCr-MSST, which changes
a CFA-sampled image from RGB color space into YDgCbCr
color space, by adding calculations of the mean and difference
between the G1 and G2 components in each macropixelin accordance with the YDgCoCg-MSST described in [18].
This YDgCbCr-MSST has potentially been used in [19].
We focus on 2-channel Haar wavelet transforms, which aresimple wavelet transforms, and 3-channel Haar-like wavelet
transforms in the YDgCbCr-MSST. Since other wavelet transforms such as the 5/3 and 9/7 wavelet transforms can moreaccurately predict pixels of interest in the predict steps anddecorrelate signals to a greater extent than the Haar wavelet
transforms can, we replace the Haar and Haar-like wavelet
transforms in the YDgCbCr-MSST with the 5/3 and 9/7
wavelet transforms, which are customized on the basis of
the original pixel positions in two-dimensional (2D) space(2D-customized wavelet transforms). Second, we extend theexisting YDgCoCg-MSST in [18] with 2D-customized wavelet
transforms in accordance with the extension of the YDgCbCrMSST. Third, we find that the MSST derived from [20] generates components similar to those generated by the YDgCoCgMSST. From this finding, we reconfigure the existing SST in[20] with 2D-customized wavelet transforms. We call the three
WSSTs YDgCbCr-, YDgCoCg-, and YDgCoCg2-WSSTs. As
a result, we can consider that all existing SSTs coveredin this paper are particular classes of the proposed WSSTs
because the transforms are obtained by constraining part of
the wavelet transforms in the WSSTs to index matrices andHaar wavelet transforms. In lossless and lossy CFA-sampled
image compression based on JPEG 2000, the WSSTs thatuse 5/3 and 9/7 wavelet transforms improve the bitrates andthe Bjøntegaard metrics (BD-PSNRs and BD-rates) comparedwith the existing methods.
A preliminary version of this paper was presented in [22],where we discussed only the YDgCoCg-WSSTs for lossless
CFA-sampled image compression. This paper further presents
the YDgCbCr-WSSTs and YDgCoCg2-WSSTs and reconfigures the derivations for lossy and lossless CFA-sampled image
compression.
The remainder of the paper is organized as follows.
Section II reviews the CDF wavelet transforms, the tra-TABLE I: Coefficients of 5/3 and 9/7 wavelet transforms.
p0
u0
p1
u1

5/3
−1/2
1/4
0
0

9/7
−1.58613434205992−0.05298011857295
0.882911075530940
0.443506852043967

ditional YCbCr-RCT and YCoCg-RCT, and the existing
SSTs. Section III derives three types of WSST with 2Dcustomized wavelet transforms, i.e., YDgCbCr-, YDgCoCg-,
and YDgCoCg2-WSSTs. Section IV compares the resulting
WSSTs with the existing methods in JPEG 2000 for CFAsampled image compression. Section V concludes this paper.
Notation: Boldface letters represent matrices. I, J, O,
and superscript ·⊤ denote a 2 × 2 identity matrix, 2 × 2
reversal matrix, zero matrix, and transpose of a matrix/vector,
respectively. In addition, the size and dynamic range of theimages in the figures have been adjusted for display.
II. R EVIEW AND D EFINITIONS
A. Cohen-Daubechies-Feauveau Wavelet Transforms
CDF wavelet transforms [21] are common transforms used
for image processing. In practice, the 2-channel 1D wavelet
transforms W 2 (z) often consist of lifting steps with predictand update steps as follows [23]:
]
[
][
0
∏
1
0
1 Uk (z),(1)
W 2 (z) =Pk (z) 101
k=N −1 |{z}|{z}
update step

predict step

where z is a delay element, N is the number of iterations of thetwo lifting steps depending on the type of wavelet transforms,
and Pk (z) and Uk (z) are polynomials with coefficients pk anduk :Pk (z) = (1 + z −1 )pk

and Uk (z) = (1 + z)uk ,(2)

respectively. Although the original wavelet transforms havescaling coefficients, we will omit them for simplicity in thisstudy. The coefficients pk and uk in the 5/3 wavelet transforms(N = 1) and 9/7 wavelet transforms (N = 2) are shown in
Table I. Because the 5/3 and 9/7 wavelet transforms can moreaccurately predict pixels of interest in the predict steps thanthe Haar wavelet transforms can, they decorrelate signals more
as well. The (2-channel) Haar wavelet transforms have simple
predict and update steps:
[
] [
]
][
1/2 1/2
1 0
1 1/2
H2 =
=
.
(3)
−1
1
−1 1
0 1
| {z } | {z }update step

predict step

The above means that the calculation simply consists ofcalculating the mean and difference between the two input
signals.

3 The “HL subband” means the high-frequency (H) subband in the horizontal

B. Reversible Color Transforms to YCbCr or YCoCg ColorSpace for RGB Full-color Images

direction and the low-frequency (L) subband in the vertical direction after
wavelet transforms have been performed on the image; the “LH subband”
means vice versa.

The YCbCr color transform changes a full-color image from
RGB color space into YCbCr color space [2]. Although the

3Fig. 2: Lifting structures of RCTs (white circles represent
rounding operators): (top) YCbCr-RCT and (bottom) YCoCgRCT.

transforms are classified into irreversible and reversible versions, this paper uses the reversible version with lifting-based
low-complexity calculations (YCbCr-RCT). The YCbCr-RCT
Tbr is expressed as (see the top of Fig. 2) [9]
[
]⊤
[
]⊤
Y, Cb, Cr = Tbr G, B, R ,
(4)
where


1/2
Tbr =  −1−1

 
1/4 1/4
1
1
0  = 0
0
1
0



1/4 1/4
1 0 0
10  −1 1 0 .
0
1
−1 0 1(5)

On the other hand, the YCoCg-RCT Tog , which changes
a full-color image from RGB color space into YCoCg color
space with low-complexity lifting-based calculations, is expressed as (see the bottom of Fig. 2) [7]
[
]⊤
[
]⊤
Y, Co, Cg = Tog G, B, R ,
(6)
where



1/2 1/4
1/4
1−1 
Tog =  0
1
−1/2 −1/2




0 1 0
1 0 0
1 −1 0= 0 0 1 1/2 1 0 0 1 0
0 0 1
0 0 1
1 0 0



1 0
0
1 0 0
· 0 1 1/2 0 1 0 .
0 0
1
0 −1 1(7)

C. Existing Spectral-Spatial Transforms for CFA-Sampled Images
Malvar and Sullivan proposed YDgCoCg-MSST, which is
implemented within each 2×2 macropixel and changes a CFAsampled image from RGB color space into YDgCoCg color
space for CFA-sampled image compression (see the middle ofFig. 3) [18]:
[
]⊤
[
]⊤
Y, Dg, Co, Cg = T og G1 , G2 , B, R ,(8)Fig. 3: Lifting structures of MSSTs (white circles represent rounding operators): (top) YDgCbCr-MSST, (middle)
YDgCoCg-MSST, and (bottom) YDgCoCg2-MSST.

where




1/4 1/4 1/4
1/4
 −1
1
0
0 

T og = 
 0
0
−1
1 
1/2 1/2 −1/2 −1/2




0 0 1 01
0 0 0 1 0 −1 0
 0 1 0 0  0


1 0 0
 0 1 0 0=
0 0 0 1 1/2 0 1 0 0 0 1 0
1 0 0 00
0 0 1 0 0 0 1

1 1/2 00
1 0 0 0
0


1
0
0 
 −1 1 0 0 .
·
(9)
0
0
1 1/2  0 0 1 0000
1
0 0 −1 1

The sets of G1 , G2 , B, and R are in macropixels of a CFAsampled image, as shown in Fig. 1. Hence, the transform isobtained by adding calculations of the mean and difference
between the G1 and G2 components to the YCoCg-RCT. It is
implemented as shown at the top of Fig. 7.
Hernández-Cabronero et al. presented a lossless CFAsampled image compression pipeline to JPEG 2000 [19].
However, the Haar wavelet transforms were only used as
preprocessing, and no SST was explicitly mentioned.
On the other hand, Zhang and Wu found that Mallat wavelet
packets are suitable for decorrelating CFA-sampled images andused them for lossless CFA-sampled image compression; i.e.,
1D wavelet transforms were directly applied to CFA-images
like wavelet transforms are to RGB full-color images [17].
In addition, Lee et al. found a redundancy in the HL and4

LH subbands generated in [17] and decorrelated them moreby calculating the mean and difference between the components [20]. However, the transforms in [17] and [20] were notexpressed as matrices.

Haar wavelet transforms H2 in (3):
1/2 1/4
1
H3 =  −1
−1
0

 


1/4
1 1/4 1/4
1 0 0
0  = 0 1
0  −1 1 0 .
1
0 0
1
−1 0 1
|
{z
}|
{z
}

III. WAVELET- BASED S PECTRAL -S PATIAL T RANSFORMS
FOR CFA-S AMPLED I MAGES
A. Preparation
This paper derives three types of WSST: YDgCbCr-WSST,
YDgCoCg-WSST, and YDgCoCg2-WSST. To derive them, weprepare 3-channel 1D wavelet transforms, 2 and 3-channel 2Dcustomized wavelet transforms including 3-channel Haar-like
transforms, and six permutation matrices.
First, let W 3 (z) be 3-channel 1D wavelet transforms, i.e.,
extended versions of the 2-channel 1D wavelet transforms
W 2 (z) in (1):
0
1 Uk (z)/2 Uk (z)/2
1
0 0
∏
0
1
0  Pk (z) 1 0 .
W 3 (z) =0
1
Pk (z) 0 1
k=N −1 0
|
{z
}|
{z
}
update step

update step(15)
The 3-channel Haar-like wavelet transforms H3 in (15) can beconsidered to be simpler versions of the 3-channel 1D wavelet
transforms W 3 (z1 , z2 ) in (12).
Finally, let Pl (l = 0, 1, · · · , 5) be the 4 × 4 permutation
matrices,01
P0 = 
00
1
0P2 = 
00
00P4 = 
10

predict step

(10)
Next, let W 2 (z1 , z2 ) and W 3 (z1 , z2 ) be 2 and 3-channel
2D wavelet transforms obtained by customizing the 2 and 3channel 1D wavelet transforms W 2 (z) in (1) and W 3 (z) in(10) as follows:
W 2 (z1 , z2 ) =

[
][
]
0
∏
1 Uk (z1 , z2 )1
0
0
1
Pk (z1 , z2 ) 1
k=N −1 |{z}|{z}
update step


0
1
∏
0
W 3 (z1 , z2 ) =k=N −1 0|predict step

1
0
0
0
0
0
0
1
1
0
0
0

0
0
1
0
0
1
0
0
0
1
0
0



0
0
1
0
 , P1 = 
0
0
1
0


0
0
1
0
 , P3 = 
0
1
0
0


0
0
0
0
 , P5 = 
1
0
1
0

0
0
1
0
0
0
0
1
0
1
0
0

1
0
0
0
1
0
0
0
1
0
0
0


0
0

0
1

0
0

1
0

0
0
.
0
1

B. Wavelet-based Spectral-Spatial Transforms to YDgCbCr
Color Space

predict stepUk (z1 )/2 Uk (z2 )/2

1
0
0
1
{z
}

(11)

Theorem-1: The YDgCbCr-WSSTs Tbr are expressed as(see the top of Fig. 4)
[
]⊤
[
]⊤
Y, Dg, Cb, Cr = Tbr G1 , G2 , B, R ,(17)update step1
0 0
· Pk (z1 ) 1 0,Pk (z2 ) 0 1
|
{z
}

(12)

where
[
]
1
O
W 2 (z1−1 , z2 )P0
−1
O W 3 (z1 , z2 )O[
Tbr = P0

predict step

where Pk (z1 , z2 ) and Uk (z1 , z2 ) are polynomials,
1(1 + z1−1 + z2−1 + z1−1 z2−1 )pk
2
1
Uk (z1 , z2 ) = (1 + z1 + z2 + z1 z2 )uk ,2
Pk (z1 , z2 ) =

(16)

(13)
(14)

and z1 and z2 are horizontal and vertical delay elements. In
the proposed transforms described below, the suitable delay
elements are determined by taking into account the original
pixel positions in 2D space. Also, let H3 be 3-channel Haarlike wavelet transforms, which are completely equivalent to the
YCbCr-RCT Tbr in (5), obtained by extending the 2-channel

]
O
.
I
(18)

Note that the delay patterns in (18) are for a Bayer pattern, as
shown in Fig. 1, and they depend on the original CFA pattern.
The YDgCbCr-WSSTs Tbr are implemented as shown at thebottom of Fig. 5.
Proof-1: As in the case of YDgCoCg-MSST in [18], we
will extend the YCbCr-RCT to CFA-sampled images (see the
top of Fig. 3),[

Y, Dg, Cb, Cr

]⊤

[
]⊤
= T br G1 , G2 , B, R ,

(19)

5

Fig. 5: Implementations of YDgCbCr-WSSTs: (top) Haar case and (bottom) 5/3 and 9/7 cases.
TABLE II: Number of lifting steps in WSSTs.
YDgCbCr-WSSTs
YDgCoCg-WSSTs
YDgCoCg2-WSSTs

Haar
6
6
10

5/3
16
20
44

9/7
32
40
48

G1 and G2 components to the YCbCr-RCT. The YDgCbCrMSST T br is implemented as shown at the top of Fig. 5.
By using 2-channel Haar wavelet transforms H2 in (3) and3-channel Haar-like wavelet transforms H3 in (15), we canredefine the YDgCbCr-MSST T br in (20) as (see the top of
Fig. 6)
[
]
[
]
1 O
H2 O
T br = P0
P0
.
(21)
O H3
O I

Fig. 4: Lifting structures of WSSTs: (top) YDgCbCr-WSSTs,(middle) YDgCoCg-WSSTs, and (bottom) YDgCoCg2WSSTs.

where




1/4
1/4 1/4 1/4
 −1
1
0
0 

T br = −1/2 −1/2 1
0 −1/2 −1/2
0
1



1 0 1/4 1/4
1 0 0 0
0 1 00 
  0 1 0 0=
0 0 10  −1 0 1 0
0 0 0
1
−1 0 0 1
1 1/2 0 0
1 0 0 0
0 1


0 0
 −1 1 0 0 ,
·
0 0


1 0
0 0 1 00 0
0 1
0 0 0 1(20)by adding calculations of the mean and difference between the

To improve the transform’s performance, we can also use otherwavelet transforms, such as 5/3 and 9/7 wavelet transforms,
which can more accurately predict pixels of interest in thepredict steps, instead of the Haar and Haar-like wavelet
transforms in the YDgCbCr-MSST:
[
]
[
]
1
O
W 2 (z) OdT br = P0
P.
(22)
O W 3 (z) 0O
IMoreover, we can obtain the YDgCbCr-WSSTs Tbr in (18)by replacing the 1D wavelet transforms W 2 (z) and W 3 (z)with 2D wavelet transforms W 2 (z1−1 , z2 ) and W 3 (z1−1 , z2 ),to take into account the original pixel positions in 2D space.
Remark-1: Because the predict and update steps of the
YDgCbCr-WSSTs that use 5/3 and 9/7 wavelet transforms are
implemented with more pixels around the pixels of interest
than those that use Haar and Haar-like wavelet transforms,
which are predicted and updated with only an adjacent pixel,
the compression performance of the YDgCbCr-WSSTs thatuse 5/3 and 9/7 wavelet transforms is expected to be better thanthat of the transform that uses Haar wavelet transforms. We cansee that the pipeline system in [19], that uses the Haar wavelet
transforms between G1 and G2 components before YCbCrRCT in JPEG 2000, potentially uses the YDgCbCr-MSST(YDgCbCr-WSST that uses Haar wavelet transforms) in thesystem. Moreover, compared with other WSSTs introduced
later, the YDgCbCr-WSSTs are composed of fewer lifting
steps, as shown in Table II.

6Moreover, we can obtain the YDgCoCg-WSSTs Tog in(24) by replacing the 1D wavelet transforms W 2 (z) with
2D wavelet transforms W 2 (z1−1 , z2 ), W 2 (z1−1 , z2−1 ), andW 2 (z2−1 ), to take into account the original pixel positionsin 2D space.
Remark-2: For the same reason as in the case of
the YDgCbCr-WSSTs, the compression performance of the
YDgCoCg-WSSTs that use 5/3 and 9/7 wavelet transforms is
expected to be better than that of the transform that uses Haar
wavelet transforms. Because the YDgCoCg-WSSTs, in which
all wavelet transforms in (24) are Haar wavelet transforms,
are clearly equivalent to the existing YDgCoCg-MSST in [18],we consider that the existing YDgCoCg-MSST is a particularclass of YDgCoCg-WSST. In addition, the YDgCoCg-WSSTs
have more lifting steps than in the YDgCbCr-WSSTs andfewer lifting steps than in the YDgCoCg2-WSSTs describedin the next subsection, as shown in Table II.

Fig. 6: Lifting structures of the redefined MSSTs (whitecircles represent rounding operators): (top) YDgCbCr-MSST,(middle) YDgCoCg-MSST, and (bottom) YDgCoCg2-MSST.
C. Wavelet-based Spectral-Spatial Transforms to YDgCoCg
Color Space
Theorem-2: The YDgCoCg-WSSTs Tog are expressed as(see the middle of Fig. 4)
[
]⊤
[
]⊤
Y, Dg, Co, Cg = Tog G1 , G2 , B, R ,
(23)
where

[

W 2 (z2−1 )O

Tog = P2[W 2 (z1−1 , z2 )
·
O

]
O
P1
I

Theorem-3: The YDgCoCg2-WSSTs Tog2 are expressed as(see the bottom of Fig. 4)
[
]⊤
[
]⊤
Y, Dg, Co, Cg = Tog2 G1 , G2 , B, R ,

(27)

where
[

]

O
.
W 2 (z1−1 , z2−1 )D. Wavelet-based Spectral-Spatial Transforms to YDgCoCgLike Color Space(24)

Note that the delay patterns in (24) are for a Bayer pattern, as
shown in Fig. 1, and they depend on the original CFA pattern.
The YDgCoCg-WSSTs Tog are implemented as shown at thebottom of Fig. 7.
Proof-2: The YDgCoCg-WSSTs are derived from the
YDgCoCg-MSST in [18]. By using 2-channel Haar wavelet
transforms H2 in (3), we can redefine the YDgCoCg-MSST
T og in (9) as (see the middle of Fig. 6)
[
]
[
]
H2 O
H2 O
T og = P2
P1
.
(25)
O I
O H2
To improve the transform’s performance, we can use otherwavelet transforms, such as 5/3 and 9/7 wavelet transforms,
instead of the Haar wavelet transforms in the YDgCoCgMSST:[][]
W 2 (z) O
W 2 (z)Od
T og = P2
P1
.
(26)
O
I
O
W 2 (z)][]W 2 (z1 )O
W 2 (z1−1 , z2 ) O
Tog2 = P5P4POW 2 (z1 ) 3OI[]W 2 (z2 )
O
·
P .
(28)
O
W 2 (z2 ) 2Note that the delay patterns in (28) are for a Bayer pattern, as
shown in Fig. 1, and they depend on the original CFA pattern.
The YDgCoCg2-WSSTs Tog2 are implemented as shown at
the bottom of Fig. 8.
Proof-3: The YDgCoCg2-WSSTs are derived from the SSTs
in [20]. First, let us consider the MSST derived from [20],in which 2-channel Haar wavelet transforms are used forsimplicity.4 The MSST T og2 is represented as (see the bottom

4 In [20], the SSTs are composed of 5/3 and 9/7 wavelet transforms, notHaar wavelet transforms, which adds calculations of the mean and difference
between the HL and LH subbands.

7Fig. 7: Implementations of YDgCoCg-WSSTs: (top) Haar case and (bottom) 5/3 and 9/7 cases.

Fig. 8: Implementations of YDgCoCg2-WSSTs: (top) Haar case and (bottom) 5/3 and 9/7 cases.
YDgCoCg2-MSST T og2 in (29) as (see the bottom of Fig. 6)

of Fig. 3)

1/4 1/4 1/4 1/4
 −1
1
0
0 

T og2 = 
 0
0 1/2 −1/2
−1 −1
1
1


1 1/2 0 0
1 0
0 1
 −1 100

= P5 
0 0
1 0  0 0
0 0
0 1
0 0


1 0 0
0
1 0
 0 1 1/2 0 0 1
·
 0 0 10 0 −1
1/2 0
0 1
0 0


1 0
1/2 0
1 0
0 1
 0 1
0
0

·
0 0
1 0 −1 0
0 1/2
0 1
0 0

[
T og2 = P5

0 0
0 0

1 0
0 1
0
0
1
0
0
0
1
0


−1
0

0
1

0
−1
.
0
1

]
[
O
H2
P4
I
O

]
[
O
H2
P3
H2
O

]
O
P2 .
H2
(31)

To improve the transform’s performance, we can use otherwavelet transforms, such as 5/3 and 9/7 wavelet transforms,
instead of the Haar wavelet transforms in the YDgCoCg2MSST:[][]
W 2 (z) O
W 2 (z)
O
[
T
=
P
P
P
og2
5
4
O
I
O
W 2 (z) 3[]
W 2 (z)
O
·
P .
(32)
O
W 2 (z) 2(29)

It is implemented as shown at the top of Fig. 8. We can see thatthey generate components similar to those generated by the
YDgCoCg-MSST T og in (9) because the relationship between
T og and T og2 in the case of MSSTs is expressed by}{1
T og2 = diag 1, 1, − , −2 T og .
2

H2
O

(30)

We characterize the color space generated by T og2 as a
YDgCoCg-like (YDgCoCg2) color space. By using 2-channel
Haar wavelet transforms H2 in (3), we can redefine theMoreover, we can obtain the YDgCoCg2-WSSTs Tog2 in(28) by replacing the 1D wavelet transforms W 2 (z) with 2D
wavelet transforms W 2 (z2 ), W 2 (z1 ), and W 2 (z1−1 , z2 ), totake into account the original pixel positions in 2D space.
Remark-3: For the same reason as with the other WSSTs,
the compression performance of the YDgCoCg2-WSSTs thatuse 5/3 and 9/7 wavelet transforms is expected to be better
than that of the transform that uses Haar wavelet transforms.
Because the YDgCoCg2-WSSTs, in which the last wavelet
transforms W 2 (z1−1 , z2 ) in (28) are index matrices or Haar
wavelet transforms, are clearly equivalent to the existing SSTs
in [17] or [20], we consider that the existing SSTs are
particular classes of YDgCoCg-WSST. In addition, comparedwith the other WSSTs, the YDgCoCg2-WSSTs are composedof more lifting steps, as shown in Table II.

8Fig. 9: Test images: (top and middle) images in [24] and (bottom) images in [25].
TABLE IV: LBRs [bpp] in lossless CFA-sampled image compression using JPEG 2000 lossless mode.
Direct
Test Images
In [24],
#0250
#0500
#0750
#1000
#1250
#1500
#1750
#2000
#2250
#2500
#2750
#3000
#3250
#3500
#3750
#4000
#4250
#4500
#4750
#5000
In [25],
Akademie
Arri
ChurchColor Test Chart
Face
Lake Locked
Lake Pan
Night at Odeonplatz
Swimming PoolNight at Siegestor
Average LBR

ReRGB

9.48
9.06
10.35
9.45
9.51
10.02
10.03
9.79
10.34
10.17
9.44
11.25
10.39
9.13
9.42
11.78
10.02
10.35
8.77
11.11
11.20
10.49
9.81
9.58
9.15
10.15
11.31
10.22
10.62
10.05
10.08

9.52
8.83
10.64
9.18
9.21
9.87
9.89
9.83
10.76
10.11
9.63
10.95
10.76
8.69
9.25
11.67
10.27
10.88
9.08
11.37
11.10
10.50
10.10
8.65
9.00
10.05
11.63
9.68
10.66
9.66
10.05

YDgCbCr-WSSTs
Haar [19]
5/3
9.41
9.31
8.94
8.81
10.39
10.17
9.21
9.09
9.10
9.00
9.77
9.69
9.86
9.75
9.62
9.48
10.36
10.16
10.03
9.95
9.41
9.28
11.10
11.05
10.44
10.20
8.49
8.37
9.17
9.00
11.71
11.60
10.02
9.89
10.60
10.43
8.88
8.77
11.13
10.95
10.91
10.68
10.23
9.97
9.77
9.54
8.94
8.83
8.91
8.79
9.88
9.71
11.29
11.03
9.69
9.60
10.40
10.23
9.70
9.59
9.91
9.76

TABLE III: MSE of Dg components of all images.
Exist. SSTs [18]Exist. SSTs [19]Exist. SSTs [20]YDgCbCr-WSSTs
YDgCoCg-WSSTs
YDgCoCg2-WSSTs

Haar
292554.87
292554.87
292560.78
same as [19]same as [18]same as [20]5/3——149287.56
89934.13
89934.1380473.37

9/7——142492.36
66916.36
66916.36
70006.71

IV. CFA-S AMPLED I MAGE C OMPRESSION

We compared the YDgCbCr-, YDgCoCg-, and YDgCoCg2WSSTs that use Haar, 5/3, and 9/7 wavelet transforms with
Direct, which directly compressed a CFA-sampled image,rearranged RGB (ReRGB), which merely rearranges components from a CFA-sampled image as shown at the top

YDgCoCg-WSSTs
Haar [18]
5/3
9.38
9.30
8.96
8.85
10.39
10.25
9.22
9.11
9.09
9.00
9.77
9.71
9.86
9.80
9.61
9.52
10.34
10.21
10.02
9.96
9.43
9.32
11.12
11.00
10.41
10.27
8.48
8.36
9.20
9.09
11.74
11.74
10.01
9.97
10.55
10.50
8.85
8.79
11.13
11.06
10.94
10.74
10.23
10.02
9.76
9.54
8.99
8.89
8.90
8.80
9.89
9.72
11.30
11.06
9.69
9.60
10.41
10.27
9.72
9.59
9.91
9.80

Haar
9.38
8.97
10.39
9.22
9.09
9.77
9.86
9.61
10.34
10.02
9.43
11.13
10.41
8.48
9.20
11.74
10.01
10.55
8.85
11.13
10.94
10.23
9.76
9.00
8.90
9.89
11.30
9.69
10.41
9.72
9.91

YDgCoCg2-WSSTs
5/3&Haar [20]
5/3
9.28
9.26
8.98
8.90
10.16
10.14
9.25
9.17
9.06
9.01
9.81
9.74
9.90
9.84
9.52
9.43
10.17
10.05
9.99
9.91
9.38
9.33
11.15
11.12
10.13
10.05
8.36
8.34
9.11
9.03
11.77
11.72
9.84
9.82
10.33
10.28
8.72
8.72
10.91
10.90
10.73
10.65
10.01
9.95
9.58
9.50
9.05
8.97
8.94
8.84
9.81
9.74
11.04
10.99
9.72
9.62
10.29
10.24
9.79
9.68
9.83
9.76

left of Fig. 10, and the existing SST in [18]5 , [19]6 , and[20]7 in terms of the lossless bitrate (LBR) [bpp] with theexisting codecs designed for non CFA-sampled images, JPEG
2000, by using imwrite.m in MATLAB. We used 5/3 and9/7 wavelet transforms for lossless and lossy compression,respectively, and a commonly used symmetric extension [26]at the image boundaries. The “wavelet packets” as in [20] wereapproximated by combining the SSTs with JPEG 2000, whichuses 5/3 and 9/7 wavelet transforms. As with the test images,
we used RGB full-color images of various sizes (about 3K
5 The YDgCoCg-MSST in [18] is equivalent to the YDgCoCg-WSST thatuses Haar wavelet transforms (Section III-C).
6 The pipeline system in [19] potentially used the YDgCbCr-WSST thatuses Haar wavelet transforms (YDgCbCr-MSST) (Section III-B).
7 The SSTs in [20] are equivalent to the YDgCoCg2-WSSTs, in which thelast wavelet transforms W 2 (z1−1 , z2 ) in (28) are Haar wavelet transforms(Section III-D).

9Fig. 10: Subband images of Color Test Chart (clockwise from the top left: R/Y , G1 /Dg, B/Cr/Cg, and G2 /Cb/Co components):(1st row) ReRGB and YDgCoCg2-WSSTs that used 5/3&Haar and 9/7&Haar wavelet transforms [20], (2nd row) YDgCbCrWSSTs that used Haar [19], 5/3, and 9/7 wavelet transforms, (3rd row) YDgCoCg-WSSTs that used Haar [18], 5/3, and 9/7
wavelet transforms, and (4th row) YDgCoCg2-WSSTs that used Haar, 5/3, and 9/7 wavelet transforms.

to 5K) with a 16-bit dynamic range in each color component
from [24] and [25], as shown in Fig. 9. To simulate the CFAsampled images, we subsampled the test images in accordancewith the Bayer pattern in Fig. 1 and reduced their dynamic
range from 16 to 14 bits because the actual sensor data often
had only about 10 to 14-bit resolution at most. Since the
YDgCbCr and YDgCoCg color spaces have both positive andnegative values and the JPEG 2000 codec does not allow input
signals with negative values, we transmitted the transformed
coefficients, which have 15-bit positive values, by adding 214
to them on the encoder side and reconstructed the images after
subtracting 214 on the decoder side.
Figure 10 shows the subband images of the Color Test Chart
transformed by the WSSTs. The WSSTs that used the 5/3
and 9/7 wavelet transforms decorrelated each color component
more thoroughly than the transforms that used Haar wavelet
transforms (see especially the Dg components in Fig. 10).
In addition, Table III shows the mean square error (MSE)of the Dg components of all the images. Note that the Dg
components in the YDgCbCr-WSSTs and YDgCoCg-WSSTs
that used the same wavelet transforms were completely equiva-

lent because the components could be obtained from the same
calculations. In addition, we observed relationships between
the YDgCoCg-WSSTs and YDgCoCg2-WSSTs similar to
those described in Section III-D; they were not completely
equivalent because of the rounding error. Table IV shows the
LBRs [bpp] in lossless CFA-sampled image compression using
the lossless mode of JPEG 2000. Fig. 11 and Tables V andVI show the rate-distortion (R-D) curves with the bitrates[bpp] and the peak signal-to-noise ratios (PSNRs) [dB] andthe Bjøntegaard metrics (BD-PSNRs [dB] and BD-rates [%])
between about 0.0625-2 bpp in comparison with Direct inlossy CFA-sampled image compression using the lossy modeof JPEG 2000. Furthermore, Fig. 12 shows the particularareas of reconstructed G1 components with about 0.5 bpp.
In lossless CFA-sampled image compression based on JPEG
2000, the WSSTs improved the bitrates by about 0.17 to 0.32
bpp (1.67 to 3.17 %) compared with not using a transformand the WSSTs that used 5/3 wavelet transforms improved thebitrates by about 0.03 to 0.07 bpp (0.31 to 0.71 %) comparedwith the best existing SST in [20]. Moreover, in lossy CFAsampled image compression based on JPEG 2000, the WSSTs

10

TABLE V: BD-PSNRs [dB] in lossy CFA-sampled image compression using JPEG 2000 lossy mode.
ReRGB
Test Images
In [24],
#0250
#0500
#0750
#1000
#1250
#1500
#1750
#2000
#2250
#2500
#2750
#3000
#3250
#3500
#3750
#4000
#4250
#4500
#4750
#5000
In [25],
Akademie
Arri
ChurchColor Test Chart
Face
Lake Locked
Lake Pan
Night at Odeonplatz
Swimming PoolNight at Siegestor
Average BD-PSNR

1.88
3.43
-0.30
3.61
4.78
3.86
1.00
0.91
-0.10
4.55
-0.21
4.90
-0.74
5.00
3.94
1.14
0.76
-0.63
0.32
0.92
3.55
1.25
0.73
6.43
1.34
3.34
0.27
5.51
1.12
4.47
2.23

YDgCbCr-WSSTs
Haar [19]
9/7
2.50
3.78
2.86
4.00
0.71
2.58
2.73
3.43
5.19
6.27
4.03
4.96
0.97
2.19
1.82
3.20
1.68
3.03
4.15
4.81
1.26
2.91
4.08
4.90
0.68
2.72
6.11
7.76
3.87
5.19
0.39
1.00
1.55
2.93
0.20
2.20
0.90
2.42
1.49
3.42
4.14
6.05
2.59
5.05
2.61
4.82
5.64
6.50
1.65
2.95
4.06
5.72
1.94
4.10
4.98
6.08
2.47
4.25
3.65
4.61
2.70
4.13

YDgCoCg-WSSTs
Haar [18]
9/7
2.51
4.09
2.79
4.18
0.51
2.72
2.65
3.56
5.17
6.62
3.96
5.25
0.92
2.40
1.73
3.53
1.52
3.44
4.11
5.06
1.11
3.23
3.96
5.06
0.64
3.06
6.03
8.07
3.71
5.19
0.24
0.63
1.50
3.07
0.24
2.32
0.96
2.85
1.37
3.39
3.80
5.94
2.46
5.25
2.46
5.23
5.48
6.61
1.63
3.21
3.90
5.97
1.78
4.43
4.95
6.34
2.25
4.38
3.50
4.72
2.59
4.33

Haar
2.14
2.27
0.16
2.10
4.70
3.43
0.39
1.53
1.40
3.63
0.82
3.46
0.57
5.69
3.33
-0.24
1.28
0.18
0.70
1.13
3.51
2.35
2.35
4.90
1.11
3.61
1.75
4.41
2.02
2.94
2.25

YDgCoCg2-WSSTs
9/7&Haar [20]
9/7
3.98
4.24
3.55
4.01
2.81
3.08
2.86
3.28
6.19
6.51
4.68
5.09
1.91
2.38
3.18
3.66
3.14
3.64
4.52
5.05
2.76
3.25
4.54
4.90
3.13
3.57
7.66
8.01
4.82
5.26
0.28
0.62
3.07
3.49
2.47
2.97
2.75
3.08
3.49
3.85
5.60
5.94
5.05
5.41
4.75
5.07
5.88
6.34
2.76
3.20
5.61
5.99
4.50
4.77
5.75
6.33
4.19
4.53
4.00
4.54
4.00
4.40

Fig. 11: R-D curves in lossy CFA-sampled image compression using JPEG 2000 lossy mode: (left) #0250, (middle) #5000,
and (right) Night at Siegestor.

showed about 2.25 to 4.40 dB and 26.04 to 49.35 % in the
Bjøntegaard metrics compared with not using a transform andthe WSSTs that used 9/7 wavelet transforms improved themetrics by about 0.13 to 0.40 dB and 2.27 to 4.80 % comparedwith the best existing SST in [20]. However, the inferior-tosuperior relationship between the SSTs reversed depending on
the image. It may be better to apply adaptive selection ofseveral SSTs to each divided local area, as was done in [9],to design more effective wavelet transforms in the WSSTs
for CFA-sampled images, and/or to design the transforms byconsidering the rounding error as in [27] and [28].
V. C ONCLUSION
This paper described three types of WSST that change a
CFA-sampled image from RGB color space into YDgCbCr

or YDgCoCg color space. First, we extended the YCbCrRCT to YDgCbCr-MSST by adding calculations of the mean
and difference between the G1 and G2 components to theexisting YDgCoCg-MSST. We focused on 2-channel Haarand 3-channel Haar-like wavelet transforms in the YDgCbCrMSST, and replaced the Haar and Haar-like wavelet transforms
in the YDgCbCr-MSST with 2D-customized wavelet transforms. Second, we extended the existing YDgCoCg-MSST
with 2D-customized wavelet transforms in the manner ofextending the YDgCbCr-MSST. Third, from the finding that
other existing SST implemented within a macropixel generatecomponents similar to those generated by the YDgCoCgMSST, we reconfigured the existing SST with 2D-customized
wavelet transforms. As a result, we can consider that all of
the existing SSTs covered in this paper are particular classes

11

TABLE VI: BD-rates [%] in lossy CFA-sampled image compression using JPEG 2000 lossy mode.
ReRGB
Test Images
In [24],
#0250
#0500
#0750
#1000
#1250
#1500
#1750
#2000
#2250
#2500
#2750
#3000
#3250
#3500
#3750
#4000
#4250
#4500
#4750
#5000
In [25],
Akademie
Arri
ChurchColor Test Chart
Face
Lake Locked
Lake Pan
Night at Odeonplatz
Swimming PoolNight at Siegestor
Average BD-rate-31.09-38.83
5.56-47.35-54.05-52.24-18.38-16.682.81-41.592.53-58.52
13.98-52.17-47.69-24.26-12.0912.20-3.22-12.23-37.40-14.93-8.91-62.89-23.55-37.91-2.26-61.59-15.15-52.22-26.34

YDgCbCr-WSSTs
Haar [19]9/7-39.96-51.05-29.16-40.79-11.72-36.81-34.29-44.31-54.19-63.94-51.42-59.08-18.05-36.12-27.05-44.82-25.62-41.29-32.35-40.86-19.81-39.29-50.36-54.69-9.53-34.45-60.55-70.51-43.27-55.39-6.69-22.36-23.38-39.68-1.42-24.35-11.28-29.74-19.43-40.61-39.86-57.79-27.67-49.89-32.63-53.23-53.26-59.27-27.20-44.16-40.53-55.38-24.43-47.60-57.06-65.01-32.36-50.34-41.34-51.76-31.53-46.82

YDgCoCg-WSSTs
Haar [18]9/7-39.65-53.52-28.66-41.96-8.59-38.15-33.46-45.91-54.20-66.31-50.80-61.62-17.05-39.04-26.20-48.15-23.20-45.41-34.07-47.00-17.66-42.52-50.36-57.49-9.28-37.94-60.03-71.99-42.12-54.93-4.23-20.30-23.02-40.88-2.74-25.38-12.29-34.67-18.13-39.82-37.16-56.78-26.97-51.30-31.50-56.54-52.04-59.59-27.31-46.81-39.14-56.82-23.47-50.21-56.89-66.45-30.31-51.14-39.53-52.83-30.67-48.72

Haar-33.28-21.09-3.08-28.22-48.73-44.05-7.96-22.68-22.05-22.93-14.00-46.06-8.13-57.46-37.502.71-20.06-1.93-7.58-15.03-34.85-25.60-31.04-45.97-17.86-35.57-21.81-50.62-27.74-31.00-26.04

YDgCoCg2-WSSTs
9/7&Haar [20]9/7
-52.63-55.14-31.47-39.10-39.47-42.37-37.12-42.76-62.34-65.37-55.03-59.55-30.77-38.41-43.57-49.54-42.06-47.28-35.39-47.40-37.85-42.86-50.19-54.15-38.70-42.99-70.11-71.78-50.35-55.31-8.89-17.43-40.99-45.89-26.30-32.96-32.97-36.67-41.14-45.62-54.12-56.85-49.72-52.65-52.47-55.24-52.28-56.90-40.89-46.41-53.50-57.10-51.23-53.51-62.24-66.21-49.61-52.69-43.06-50.33-44.55-49.35

Fig. 12: Particular areas of reconstructed G1 components in lossy CFA-sampled image compression using JPEG 2000 lossy
mode with about 0.5 bpp: (top) #0250, (middle) #5000, (bottom) Night at Siegestor, and (left-to-right) original image, Direct,
ReRGB, YDgCbCr-WSSTs that used Haar [19] and 9/7 wavelet transforms, YDgCoCg-WSSTs that used Haar [18] and 9/7
wavelet transforms, and YDgCoCg2-WSSTs that used Haar, 9/7&Haar [20], and 9/7 wavelet transforms.

of the WSSTs. In lossless and lossy CFA-sampled image
compression based on JPEG 2000, the WSSTs that used 5/3
and 9/7 wavelet transforms improved the bitrates and the
Bjøntegaard metrics compared with the existing methods.
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for providing many constructive suggestions that significantly improve the presentation of this paper. This work was
supported by Grant-in-Aids for Young Scientists (B), Grant

Number 16K18100, from the Japan Society for the Promotionof Science (JSPS).
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Taizo Suzuki (SM’17) received the B.E., M.E., andPh.D. degrees in electrical engineering from Keio
University, Japan, in 2004, 2006, and 2010, respectively. From 2006 to 2008, he was with Toppan
Printing Co., Ltd., Japan. From 2008 to 2011, he
was a Research Associate of the Global Center of
Excellence (G-COE) at Keio University, Japan. From2010 to 2011, he was a Research Fellow of the Japan
Society for the Promotion of Science (JSPS) anda Visiting Scholar at the Video Processing Group,
the University of California, San Diego, CA. From2011 to 2012, he was an Assistant Professor with Nihon University, Japan. In
2012, he joined University of Tsukuba, Japan as an Assistant Professor, andsince 2019, he has been an Associate Professor. His current research interests
include signal processing and filter banks/wavelets for image and video. Since
2017, he has been an Associate Editor of IEICE Trans. Fundamentals.