Cell-based 2-Step Scalar Deadzone

Quantization for JPEG2000

Joan Bartrina-Rapesta, Francesc Aulı-Llinas

Ian Blanes, and Joan Serra-Sagrista

Department of Information and Communications Engineering

Universitat Autonoma de Barcelona, Barcelona, Spain

Abstract

Wavelet-based coding systems commonly employ uniform scalar deadzone quantization (USDQ)

together with a bitplane coding strategy to progressively refine image quality. Our previous

work presents a quantization scheme that employs 2 step sizes depending on the magnitude

of the coefficients. This 2-step scalar deadzone quantization (2SDQ) scheme is introduced in

the framework of JPEG2000 by modifying all coefficients within a codeblock to enhance the

quality of the image while transmitting fewer bitplanes than those needed with a conventional

USDQ scheme. This paper extends our prior work by applying the 2SDQ in a subblock level,

i.e., in small sets of coefficients, called cells, selected within a codeblock. Combined with

rate-distortion optimization techniques, the proposed cell-based 2SDQ can help to code high

quality images employing even fewer bitplanes than those needed with our previous strategy.

This may be especially useful for high-dynamic range images or for devices with constrained

resources.

I. INTRODUCTION

Nowadays, sensors with high-dynamic range (HDR) capabilities are becoming popular.

HDR imaging allows to differentiate with major clarity between the lightest and the

darkest areas of an image. Some examples of HDR sensors are found in digital cameras,

medical devices, or satellite sensors. In general, the coding of HDR images requires more

computational resources, mostly due to the increase in the bit-depth of the image.

Most modern image coding systems rely on a uniform scalar deadzone quantization

(USDQ) scheme to progressively reduce the distortion of the image. USDQ quantizes

the coefficients of a transformed image employing uniform quantization intervals of size

Δ except for the deadzone, which is the interval that contains zero and has size 2Δ. The

operation carried out by USDQ in the encoder is expressed as

v =

⌊|ω|

Δ

⌋, (1)

where �·� denotes the floor operation, and ω is a wavelet coefficient, with |ω| ∈ [0,W ],and W denoting the largest magnitude of the coefficients to be quantized. Let [bM−1,bM−2, ..., b1, b0], bi ∈ {0, 1}, be the binary representation for the quantized coefficient v,

with M denoting a sufficient number of bits to represent the coefficients to be quantized.

The collection of bits bj from all coefficients is referred to as a bitplane. Bitplane coding

2014 Data Compression Conference

1068-0314/14 $31.00 © 2014 IEEE

DOI 10.1109/DCC.2014.51

143

strategies encode bits from the most significant bitplane M − 1 to the least significant

bitplane 0. The first non-zero bit of the binary representation of v is denoted as bs and is

referred to as significant bit. The sign of the coefficient is transmitted immediately after

bs. If bj′ denotes the last bit transmitted for v, the reconstruction procedure applied in

the decoder by the dequantizer is expressed as

ω =

{0 if j′ > s

sign(ω)(v + δ)Δ2j′

otherwise, (2)

where v = [bM−1, bM−2, ..., bj′ ], and δ ∈ [0, 1) adjusts the reconstructed coefficient within

its quantization interval. In general, δ = 1/2. The transmission of each bitplane can be

conceptually seen as the reduction of the step size of the quantizer until reaching Δ.

Although there exists a large variety of quantization schemes for the purposes of image

coding, such as vector quantization [1]–[7], adaptive schemes that adjust their intervals

as more data are transmitted [8]–[10], or trellis coded quantization [11]–[14], USDQ

combined with a bitplane coding strategy is the most common approach because it is

adequate for a variety of sources [15]–[17] and,due to the use of the binary represen-

tation of the quantization indices, is very convenient for current hardware architectures.

However, it is not specifically designed to achieve optimal coding performance for a

selected range of decoding rates for current coding systems [18], [19]. The generalized

embedded quantization (GEQ) scheme, introduced in [20], is an approach devised to

explore the performance that can be achieved with a more flexible type of quantizers

that can arbitrarily vary the step size at each quantization stage. The 2 step deadzone

quantization (2SDQ) scheme described in [21] embodies a practical implementation of

GEQ that can be used with bitplane coding schemes reducing the number of bitplanes

coded without sacrificing image quality.

This work is built upon our previous 2SDQ scheme. The objective of the proposed

method is to achieve high quality images even when few bitplanes are transmitted. The

main insight to achieve so is to apply the 2SDQ scheme in sets of wavelet coefficients,

called cells, within conventional JPEG2000 codeblocks. Experimental results suggest that

quality levels of 38 dB (in peak signal to noise ratio (PSNR)), or more can be achieved

even when only 4 bitplanes are transmitted for 8 bits per samples (bps) natural images.

Compared to our previous strategy, the proposed cell-based 2SDQ reduces the rate of the

codestream up to 2 bps or more attaining the same image quality.

The paper is organized as follows. Section II briefly reviews GEQ and 2SDQ. Sec-

tion III describes the proposed method and details its implementation in a conventional

JPEG2000 codec. Section IV shows the benefits of the proposed method through exper-

imental results. The last section provides conclusions.

II. REVIEW OF GEQ AND 2SDQ

GEQ is designed to provide more flexibility than a typical USDQ scheme. To do so,

GEQ permits the quantization of wavelet coefficients employing intervals of arbitrary size.

[20] shows that well-designed GEQ schemes can achieve the same coding performance

as that of USDQ but using fewer quantization stages or, equivalently, bitplanes. Even

though [20] proposes a practical GEQ, its implementation requires important modifica-

tions in the coding engine. The main problem is that GEQ schemes can not use the binary

144

� ���������� ����� ��

� ���������

����

�� � � � � � � � � � � � � � �

���

Fig. 1: Quantization intervals employed by 2SDQ. Only the magnitude of the coefficient

is depicted since symmetry about 0 is assumed.

representation of the quantized indices, which is key in the adoption of USDQ for image

coding. Motivated by this issue, 2SDQ [21] introduces a strategy that converts wavelet

coefficients to 2SDQ indices using two quantization step sizes. 2SDQ indices can be then

coded through conventional bitplane coding. Figure 1 illustrates the 2SDQ scheme. The

step sizes defined by 2SDQ are referred to as ΔL for coefficients |ω| < αW , and as ΔH

for coefficients |ω| ≥ αW . Step sizes are selected with ΔH > ΔL, so coefficients whose

magnitudes are greater than αW are quantized more roughly than coefficients smaller

than αW . α ∈ (0, 0.5), and is chosen as α = 0.3 in [21]. It is important to note that, due

to the use of bitplane coding, the intervals above and below αW must contain the same

number of subintervals. This constrains the size of ΔH and ΔL to

ΔH =(1− α)ΔL

α. (3)

The operation to quantize the coefficients via the 2SDQ scheme is expressed as

v′ =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

⌊|ω|

ΔL

⌋if |ω| < αW

,⌈αW

ΔL

⌉+

⌊|ω| − αW

ΔH

⌋otherwise

(4)

where �·� denotes the ceiling operation. The decoder reconstructs the coefficients accord-

ing to

ω′ =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 if j′ > s

sign(ω)(v′ + δ)ΔL2j′ if j′ ≤ s and v′2j

′

<

⌈αW

ΔL

⌉,

sign(ω)αW +

((v′ + δ)2j

′

−

⌈αW

ΔL

⌉)ΔH otherwise

(5)

where v′ denotes the binary representation of v′ up to the last transmitted bit.

145

In [21] the 2SDQ is implemented in a JPEG2000 codec. A particularity of JPEG2000 is

that it codes independently sets of wavelet coefficients, called codeblocks. [21] applies the

2SDQ to all coefficients within codeblock x by decreasing in Rx the number of magnitude

bits employed to code the coefficients. To facilitate the implementation in the JPEG2000

codec, the conventional quantization stage of JPEG2000 is not modified. 2SDQ is applied

by performing an operation on wavelet coefficients belonging to selected codeblocks that,

after applying the conventional quantization of JPEG2000, results in 2SDQ quantization

indices. The reader is referred to [21] for more details of this operation. When 2SDQ has

been applied in a codeblock, M ′x = Mx−Rx bitplanes are coded, with Mx denoting the

number of bits to represent the coefficient with largest magnitude within x that would

be necessary if 2SDQ were not applied. The step sizes ΔL and ΔH are then

Δ′

L = Δα2Mx

2M ′

x−1

= Δα2Rx+1 (6)

and

Δ′

H = Δ(1− α)2Rx+1, (7)

with Δ representing the step size of the conventional USDQ stage of JPEG2000. Evi-

dently, the codeblocks in which 2SDQ is applied, and the number of bitplanes reduced,

are transmitted to the decoder as side information. The resulting coding scheme is not

compliant with JPEG2000.

III. CELL-BASED 2SDQ

The proposed cell-based 2SDQ (CB-2SDQ) scheme is also introduced in a conventional

JPEG2000 implementation, though any wavelet-based image codec might also incorporate

it. The main difference between the 2SDQ scheme of [21] and CB-2SDQ is that CB-

2SDQ quantizes sets of wavelet coefficients that may not correspond with the size of the

codeblock. The sets of coefficients employed by CB-2SDQ are the so-called cells. The

relation between a JPEG2000 codeblock and a cell is depicted in Figure 2. This finer

level of discrimination permits the proposed scheme to achieve higher quality levels per

same number of bitplanes coded.

The first step of the CB-2SDQ is to partition the codeblock in cells. Then, the number

of magnitude bits needed to represent all coefficients in each cell is computed. This

operation is computationally simple since it is similarly done for codeblocks. The number

of magnitude bits required to represent all coefficients within a cell if USDQ were solely

applied is used to select the cells in which CB-2SDQ is employed. As seen below,

the selection procedure may employ different strategies, such as the limitation of the

maximum number of bitplanes employed. The application of CB-2SDQ in the selected

cells is carried out in the same way as it is done in [21] for codeblocks. This is, before

the quantization stage of JPEG2000, the coefficients within selected cells are converted

so that they result in CB-2SDQ indices after JPEG2000 applies the conventional USDQ

stage. This simplifies the implementation without requiring structural modifications on

the JPEG2000 coding pipeline.

Key to achieve competitive coding performance when CB-2SDQ is applied is that

the rate-distortion optimization procedure employed to construct the final codestream

146

Fig. 2: Cell-based 2SDQ partitioning. In this example, cells are sixteen times smaller

than the codeblock.

takes into account that selected coefficients have been modified. This aspect is important

because coefficients within cells may have different distortion contributions depending

on whether CB-2SDQ has been applied on them or not. In general, rate-distortion op-

timization procedures determine the decrease in distortion and the increase in length

that is produced after a coding pass of a codeblock is transmitted. JPEG2000 employs

three coding passes per bitplane. The distortion decrements and the length increments

are then employed with Lagrange optimization [22] to construct the final codestream.

The introduction of CB-2SDQ in a JPEG2000 codec must modify the rate-distortion

optimization procedure to reckon the different distortion decreases produced in selected

cells. Typically, the distortion achieved at the end of coding pass l is computed as

the squared difference between the original and the reconstructed coefficients of the

codeblock, i.e.,

Dlx = G2

∑X∈x

(ω[X ]− ω[X ])2, (8)

with G representing the energy gain factor of the wavelet subband to which the code-

block belongs, ω[X ] and ω[X ] denote the original and the reconstructed coefficient in

a codeblock x, respectively. Then, the distortion decrease that is produced when coding

pass l is transmitted is determined as Dlx = Dl−1

x −Dlx.

In practice, the squared difference between the original and the reconstructed coeffi-

cients is approximated through the quantized indices as

Dlx ≈ G2Δ2

∑X∈x

((v[X ] + δ)−

{0 if j′ > s(v[X ] + δ)2j

′

otherwise

)2

, (9)

where v[X ] represents a coefficient in the codeblock, v[X ] + δ is the reconstructed coef-

ficient when all bits are transmitted to the decoder, and (v[X ]+ δ)2j′

is the reconstructed

coefficient when bits up to j′ are transmitted. The rate-distortion optimization method

must take into account that CB-2SDQ produces a deviation on the distortion decreases.

This deviation is related to the scale factor between USDQ and 2SDQ indices, which is

expressed as β = v/v′.

147

β relates the deviation produced on the squared error estimated in (9) when coding

2SDQ indices. In [21], this deviation is estimated as βL = α2Rx+1, for indices v′ < 2M′

x−1.

For indices v′ ≥ 2M′

x−1, this deviation is estimated as βH ≈ (α2ln2−α+1− ln2)2Rx+1.

Two different β are required due to the use of two different step sizes in (5).

Through βL and βH , the distortion decrease produced in codeblocks that contain cells

with CB-2SDQ indices can be finely adjusted. Let v[C] denote the wavelet coefficients

in cell c. The application of CB-2SDQ introduces these variations in (9) according to

Dlx ≈ G2Δ2

∑c∈x

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

β2

L,c

∑C∈c

((v[C] + δ)−

{0 if j′ > s

(v[C] + δ)2j′

otherwise

)2

if v[C] < 2M′

c−1

,

β2

H ,c

∑C∈c

((v[C] + δ)−

{0 if j′ > s

(v[C] + δ)2j′

otherwise

)2

otherwise

(10)

where βL,c, βH ,c, denote the deviations for cell c computed with M ′c instead of M ′

x. M ′c

is the number of bitplanes coded for cell c.To summarize, the implementation of CB-2SDQ in a JPEG2000 codec requires the

following operations. In the encoder, selected wavelet coefficients are transformed before

the conventional quantization stage of JPEG2000 so that they result in CB-2SDQ indices.

The cells in which CB-2SDQ is applied and the number of discarded bitplanes is coded

in the headers of the codestream as auxiliary information. Finally, the rate-distortion

optimization procedure is modified so that it computes distortion decreases as expressed

in (10). The decoder employs the auxiliary information to identify the cells in which

CB-2SDQ has been applied. After the conventional JPEG2000 dequantization stage,

the wavelet coefficients within the selected cells are transformed applying the reverse

operation as that of the encoder.

IV. EXPERIMENTAL RESULTS

The proposed CB-2SDQ scheme is compared to the 2SDQ scheme proposed in [21],

and to a conventional JPEG2000 codec. The images employed in the experiments are the

eight natural images with 2560x2048 samples that belong to the ISO 12640-1 corpus. All

images are 8 bit, gray scale. Coding parameters are: 5 levels of irreversible 9/7 wavelet

transform, single quality layer codestreams, and no precincts. The codestreams generated

with CB-2SDQ and 2SDQ are not compliant with JPEG2000. These two methods quantize

all wavelet subbands as described above except the lowest frequencies subband, in which

uniform quantization is applied.

The objective of the first test is to appraise the quality of the image recovered when

the number of transmitted bitplanes is limited to Mmax. Such a constrain may appear

in devices with limited memory resources, for example. CB-2SDQ is applied in cells

that have Mc > Mmax, removing Rc = Mc − Mmax magnitude bits. The JPEG2000

implementation employs a JPEG2000-compliant strategy that codes the highest Mmax

bitplanes of each codeblock only. Table I reports the image quality (in terms of PSNR),

in average for all the images of the corpus, achieved by JPEG2000, 2SDQ and CB-2SDQ

when using different configurations of codeblock and cell size, and different values of

148

Mmax

cell size codeblock size7 6 5 4

PSNR bps PSNR bps PSNR bps PSNR bps

JPEG2000

- 64x64 46.4 3.68 40.6 3.20 34.9 2.37 29.4 1.54

- 32x32 48.3 3.97 42.8 3.56 36.9 2.90 31.3 2.14

- 16x16 50.1 4.21 45.0 3.97 39.1 3.50 33.2 2.79

- 8x8 51.8 4.75 47.3 4.61 41.4 4.34 35.2 3.76

- 4x4 52.8 6.03 49.2 5.97 43.7 5.83 37.4 5.54

2SDQ [21]

- 64x64 49.2 3.85 43.9 3.56 38.1 2.96 31.7 2.08

- 32x32 50.7 4.10 45.6 3.86 39.8 3.44 34.0 2.68

- 16x16 51.9 4.32 47.5 4.16 41.8 3.88 35.7 3.29

- 8x8 52.6 4.77 48.9 4.68 43.4 4.48 37.3 4.07

- 4x4 53.1 6.01 49.9 5.96 44.8 5.83 38.8 5.60

CB-2SDQ

64x64 64x64 49.2 3.85 43.9 3.56 38.1 2.96 31.7 2.08

32x32 64x64 50.7 3.97 45.6 3.73 39.8 3.32 34.0 2.61

16x16 64x64 51.9 4.03 47.5 3.91 41.8 3.62 35.7 3.09

8x8 64x64 52.6 4.10 48.9 4.04 43.4 3.81 37.3 3.41

4x4 64x64 53.1 4.15 49.9 4.06 44.8 3.96 38.8 3.68

TABLE I: Evaluation of the coding performance achieved by a JPEG2000 implementa-

tion, 2SDQ and CB-2SDQ.

Mmax. For each value of Mmax, the PSNR and the length of the codestream (in bps)

is reported. JPEG2000 results are obtained encoding the images using different sizes

of codeblock. The CB-2SDQ scheme utilizes the maximum codeblock size allowed in

JPEG2000 (i.e., 64x64) since the codeblock size does not affect the quality achieved

by CB-2SDQ when the entire codestream is transmitted. Results indicate that CB-2SDQ

achieves equal or better image quality than that of JPEG2000 for the same number of

bitplanes transmitted. The smaller the Mmax, the larger the difference. Table I also shows

that the image quality obtained with 2DSQ and CB-2SDQ is the same when the codeblock

size of 2SDQ is equal to the cell size of CB-2SDQ. This is caused because both schemes

modify wavelet coefficients equivalently. Nonetheless, CB-2SDQ achieves lower rates

than 2SDQ. When the cell size is small, the rate difference between the codestream

generated by 2SDQ and CB-2SDQ is significant. Note, for example, that for cells of

8x8 and codeblocks of 8x8 and Mmax = 4, CB-2SDQ generates a codestream 0.66 bps

shorter than that of the 2SDQ. This is caused because the coding of small codeblocks

penalizes the coding efficiency of JPEG2000. The proposed CB-2SDQ scheme can be

combined with large codeblocks and so the coding efficiency is less penalized.

The next test evaluates the progressive lossy coding performance of CB-2SDQ and

the JPEG2000 codec. The test employs different cell sizes to appraise the performance

that can be achieved. The codeblock size for CB-2SDQ is again 64x64. Figure 3 depicts

the results achieved. The horizontal axis of these figures is the rate, whereas the vertical

axis is the PSNR. These results suggest that CB-2SDQ obtains significantly better coding

performance than JPEG2000, especially when the cell size is small. Note, for instance,

that for a rate above 1 bps, CB-2SDQ achieves approximately 2 dB more than JPEG2000

when the cell size and the codeblock are of 32x32. Only at low rates (from 0 to 0.5 bps),

149

22

24

26

28

30

32

34

36

0 0.5 1 1.5 2 2.5 3

PS

NR

(in

dB

)

rate (in bps)

CB-2SDQ (cell = 32x32, codeblock = 64x64)CB-2SDQ (cell = 64x64, codeblock = 64x64)

JPEG2000 (codeblock = 32x32)JPEG2000 (codeblock = 64x64)

22

24

26

28

30

32

34

36

38

40

42

0 0.5 1 1.5 2 2.5 3 3.5 4

PS

NR

(in

dB

)

rate (in bps)

CB-2SDQ (cell = 4x4, codeblock = 64x64)CB-2SDQ (cell = 16x16, codeblock = 64x64)

JPEG2000 (codeblock = 4x4)JPEG2000 (codeblock = 16x16)

(a) (b)

Fig. 3: Evaluation of the coding performance achieved by JPEG2000 and CB-2SDQ with

Mmax = 4 for the “Portrait” image. Different sizes of codeblock and cell are used in (a)

and (b) for JPEG2000 and CB-2SDQ, respectively.

the conventional JPEG2000 codec achieves slightly better performance. This is caused

due to the auxiliary information transmitted by CB-2SDQ. Similar results hold for the

other images of the corpus.

The final test provides a visual comparison among the images obtained with the

different coding strategies proposed. Figure 4 depicts the “Portrait” image when coded

with 2SDQ and JPEG2000 using a codeblock size of 64x64 and 16x16, and CB-2SDQ

using a cell size of 4x4 and codeblock size of 64x64 and 16x16. The number of magnitude

bitplanes to encode is limited to Mmax = 4, and the image is coded at a target rate of 1

bps. The best image quality is provided by CB-2SDQ.

V. CONCLUSIONS

Recently, the 2-step scalar deadzone quantization (2SDQ) was integrated in a JPEG2000

implementation to obtain higher quality factors than USDQ while processing the same

number of bitplanes. 2SDQ quantizes the wavelet coefficients using two different step

sizes at a codeblock level. This work presents a cell-based 2SDQ (CB-2SDQ) built upon

the insights provided by 2SDQ. The main idea is to apply 2SDQ at a cell level instead

of a codeblock level, with cells representing small sets of coefficients in a codeblock.

The cell structure permits to quantize the wavelet coefficients with higher accuracy. The

CB-2SDQ introduced in this work improves the coding rate achieved by 2SDQ. Future

research will explore the use of CB-2SDQ when coding high dynamic range images.

VI. ACKNOWLEDGMENTS

This work has been partially funded by the Spanish Government (MINECO), by

FEDER, by the Catalan Government, under Grants TIN2012-38102-C03-03, RYC-2010-

05671, and2009-SGR-1224.

150

(a) 30.23 dB (b) 31.15 dB (c) 35.35 dB

(d) 34.36 dB (e) 34.87 dB (f) 35.75 dB

Fig. 4: “Portrait” image reconstructed at 1 bps when employing (a) JPEG2000 with

codeblock size of 64x64, (b) 2SDQ with codeblock size of 64x64, (c) CB-2SDQ with

a cell size of 4x4 and codeblock size of 64x64, (d) JPEG2000 with codeblock size of

16x16, (e) 2SDQ with codeblock size of 16x16, and (e) CB-2SDQ with a cell size of

4x4 and codeblock size of 16x16.

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