[ieee 2010 data compression conference - snowbird, ut, usa (2010.03.24-2010.03.26)] 2010 data...

Local average-based model of probabilities

for JPEG2000 bitplane coder

Francesc Aulı-Llinas

Department of Information and Communications Engineering

Universitat Autonoma de Barcelona, Spain

Abstract

Context-adaptive binary arithmetic coding (CABAC) is the most common strategy of current

lossy, or lossy-to-lossless, image coding systems to diminish the statistical redundancy of

symbols emitted by bitplane coding engines. Most coding schemes based on CABAC form

contexts through the significance state of the neighbors of the currently coded coefficient, and

adjust the probabilities of symbols as more data are coded. This work introduces a probabilities

model for bitplane image coding that does not use context-adaptive coding. Modeling principles

arise from the assumption that the magnitude of a transformed coefficient exhibits some

correlation with the magnitude of its neighbors. Experimental results within the framework

of JPEG2000 indicates 2% increment on coding efficiency.

I. INTRODUCTION

Modern lossy, or lossy-to-lossless, image and video coding systems exploit three types

of redundancy to attain compression [1]: interpixel redundancy, psycho-visual redun-

dancy, and statistical redundancy. Typically, interpixel redundancy is removed through

the application of one or several reversible transforms such as the wavelet transform,

or alternative approaches. Psycho-visual redundancy is commonly employed by rate-

distortion optimization methods to irreversibly quantize transformed coefficients, which

produces a calculable loss on the image quality. Statistical redundancy is diminished by

means of a lossless entropy-coding stage.

Since the advent of bitplane coding strategies to successively refine image distortion,

context-adaptive binary arithmetic coding (CABAC) has been widely adopted in the final

entropy-coding stage of image coding systems. Arithmetic coding is a data compression

technique that utilizes the probabilities of the input symbols to represent the original

message by an interval of real numbers in the range [0, 1). The key to the compression

effectiveness of this technique is the probabilities model employed to code the input sym-

bols. With accurate models, the amount of information required to code each symbol can

be finely adjusted, thus the output of the arithmetic coder is a compacted representation

of the original message that requires less bytes than raw coding.

Commonly in image coding systems, the probabilities model is set up by both the

bitplane coding strategy and the arithmetic coder. The former identifies a context for the

coefficient that is going to be coded, whereas the latter uses that context to carry out an

adaptive process. More precisely, let [tK−1, tK−2, ..., t1, t0] be the binary representation

for an integer T which, in the discussion below, denotes the magnitude of the index

2010 Data Compression Conference

1068-0314/10 $26.00 © 2010 IEEE

DOI 10.1109/DCC.2010.12

59

obtained by quantizing a transformed coefficient, with K denoting a sufficient number

of bits to represent all coefficients. Bitplane coding strategies generally define a bitplane

j as the same bit tj from all coefficients, and encode the image from the most significant

bitplane K − 1 to the lowest bitplane 0. The first non-null bit of a coefficient, i.e., that

ts = 1 such that ∄ s′ > s with ts′ = 1, is called the significant bit of the coefficient. The

remaining bits tr, r < s are called refinement bits. The significance state of T in bitplane

j is defined as σ(T , j) =

0 if j > s

1 otherwise.

Most bitplane coding strategies determine the context for T in bitplane j considering

the set {σ(T k, j)}, 1 ≤ k ≤ N , with T k denoting the neighborhood of T including N

neighbors. The context indicates (possibly) the number and the position of the neighbors

that are significant in the current or previous bitplanes, which potentially helps to deter-

mine the probability of the currently emitted binary symbol tj , expressed as P (tj = 0),

either for significance or refinement coding. To avoid a preliminary coding pass that

computes the probability mass function of symbols, the arithmetic coder starts with pre-

defined probabilities for contexts that are adaptively adjusted as more bits are encoded.

This adaptive process captures the probabilities of the symbols emitted for each context,

adjusting them to the incoming message or, in other words, to the image that is coded.

The ability of the context model to capture estimates of emitted symbols has a strong

influence on the degree of compression achieved by the arithmetic coder. The more

precise –and skewed– the probabilities, the greater the compression. Commonly, the

neighborhood T k is set to the 8 immediate neighbors of T , and context models are

formed through {σ(T k, j)}, 1 ≤ k ≤ 8. Context formation models reduce the number of

combinations given by {σ(T k, j)} to avoid the context dilution problem, which arises

when the arithmetic coder is not fed with enough data to adjust the contexts’ probabilities

reliably. Popular approaches for context quantization rely on heuristics [2], statistical

analysis [3], or entropy-based concepts such as mutual information [1], [4]. Regardless

of the context modeling approach, it is worth noting that all coding schemes based on

CABAC have the same root.

The purpose of this paper is two-fold. First, we draw attention to experimental evidence

suggesting that non-elaborate modeling approaches achieve high efficiency for CABAC.

This evidence seems to indicate that to use {σ(T k, j)} might not be more advantageous

than other approaches to model the symbols’ probabilities. The second purpose of this

work is to introduce a simplified model for P (tj = 0) that, neither using {σ(T k, j)}

nor adaptive coding, achieves competitive coding performance within the framework of

JPEG2000. Our model is based on the assumption that the magnitude of a transformed

coefficient is similar to the arithmetic mean of the coefficient’s immediate neighbors.

This paper is organized as follows. Section II describes the JPEG2000 coding engine

as a representative bitplane coding strategy employing CABAC. Section III presents an

experimental evaluation of non-elaborated models for CABAC, and introduces our model

of probabilities. Section IV assesses the performance of the proposed model through

experimental results, and the last section provides concluding remarks.

60

II. JPEG2000 BITPLANE CODING ENGINE

JPEG2000 [5] is a powerful standard structured in 13 parts for the coding, transmission,

and security of images and video. Part 1 of the standard defines the core coding system,

which is wavelet-based with a two tiered coding scheme. The application of the wavelet

transform produces a multi-resolution representation of the image that captures vertically,

horizontally, and diagonally oriented features in high-pass subbands, referred to as HL,

LH, and HH, respectively. After the wavelet transform and quantization, the image is

logically partitioned in small blocks of wavelet coefficients that are independently coded

by the tier-1 coding stage. The subsequent tier-2 coding stage codes auxiliary information

to identify layers of quality within the final codestream. The attainment of target bitrates,

or qualities, for layers is conducted through rate-distortion optimization techniques.

The fractional bitplane coding strategy defined in tier-1 codes the wavelet coefficients

from the highest bitplane of the codeblock to the lowest one. In each bitplane, tier-1

carries out three sub-bitplane coding passes called Significance Propagation Pass (SPP),

Magnitude Refinement Pass (MRP), and Cleanup Pass (CP). Coefficients are scanned

in each bitplane by only one of these coding passes. SPP and CP encode whether

insignificant coefficients become significant or not. The difference between SPP and

CP is that the former scans coefficients that have at least one significant neighbor. MRP

refines the magnitude of those coefficients that became significant in previous bitplanes.

Emitted bits are coded by the context-adaptive binary arithmetic coder MQ.

Tier-1 employs 19 different contexts to code symbols [5, Ch. 8.3.2]: 9 for significance

coding, 5 for sign coding, 3 for refinement coding, and 2 for the run mode (see below).

The contexts devoted to significance coding are used in SPP and CP. They are defined

considering the significance state of the sample’s vicinity {σ(T k, j)}, 1 ≤ k ≤ 8.

Context formation principles arise from the data coded in each wavelet subband. Contexts

in vertical high-pass subbands (LH), for instance, are formed considering that these

subbands contain horizontally oriented features, therefore, each subband type has its

own context selection table. Once the coefficient becomes significant, its sign is coded

by a primitive that employs five contexts formed under the assumption of horizontal an

vertical symmetry among the sign of the immediate neighbors of the coefficient. After

that, refinement bits of the coefficient are coded by MRP using three contexts that are

selected depending on {σ(T k, j)}, and that are devised to capture the probability density

function of wavelet subbands. The remaining two contexts are employed by CP when the

run mode is active, which is a coding primitive that helps skipping multiple insignificant

coefficients with a single binary symbol.

III. PROBABILITIES MODEL BASED ON THE LOCAL AVERAGE

A. Evaluation of context-adaptive coding

The first purpose of this research is to appraise the effectiveness of approaches based on

CABAC. Rather than to deploy a theoretical framework such as [1], [4], our assessment

is based on an experimental evaluation that mixes the JPEG2000 context tables of

significance coding among subbands. Figure 1 depicts the difference between the mixed

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(a) Results for the “Portrait” image

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(b) Results for the “Cafeteria” image

Fig. 1. Evaluation of the JPEG2000 context selection for significance and refinement coding primitives. Columns

labeled “mixed context tables” use the context table belonging to HL for all wavelet subbands except for HL, in which

the LH table is used. Columns labeled “single context” use a single context to encode all bits emitted by MRP. Labels

on the top indicate the PSNR of the image at that bitplane, and the bitrate attained by the original JPEG2000 context

selection.

context selection and JPEG20001. In this figure, each triplet of columns depicts the

results achieved when all codeblocks are encoded from the highest bitplane of the image

to the one indicated in the horizontal axes. We note that bitplane boundaries are suitable

points to evaluate the performance of different context configurations since the coding

performance achieved at those bitrates/qualities is equivalent to that one achieved by more

sophisticated techniques of rate-distortion optimization at the same bitrates/qualities [6].

Results in Figure 1 suggest that non-elaborated context models achieve almost same

coding efficiency as elaborated models. Variations on the length of compressed files

between methods are less than 0.5 KB when all bitplanes are coded, which generates a

compressed file of more than 2 MB. In percentage, this variation is less than 0.025%.

Figure 1 also depicts the results achieved when the refinement coding primitive employs

only one context (i.e., {σ(T k, j)} is not used). Results indicate that to employ only one

context achieves a slight increment on compression efficiency, occasionally. Nonetheless,

we stress that at very low bitrates, commonly less than 0.1 bits per sample (bps), the two

observations (for significance and refinement coding) are not as consistent as at medium

and high bitrates.

The decrease on the efficiency of the JPEG2000 context selection at medium and high

bitrates was already known at the moment of standardization, hence the BYPASS coding

variation was introduced [5, Ch. 12.4.1]. This optional coding variation bypasses the MQ

1Images belong to the ISO 12640-1 corpus (8-bit gray-scale, size of 2560×2048). Coding parameters in this and

following experiments are: 5 levels of discrete wavelet transform (DWT), codeblock size of 64×64, restart coding

variation. The base quantization step sizes corresponding to bitplane 0 are chosen accordingly to the L2-norm of the

DWT synthesis basis vectors of the subband [5, Ch. 10.5.1]. This orthonormalizes wavelet coefficients, which is a

common practice in JPEG2000.

62

coder in SPP and MRP coding passes at the lowest bitplanes of the codeblock since this

decreases the computational load of the coder without penalizing coding performance.

Also related to this issue, the analysis carried out in [1] suggests that context selections

with a smaller number of contexts (not related to image features) also reach coding

performance comparable to that of JPEG2000.

B. Proposed model of probabilities

The probabilities model introduced in this work is based on the assumption that

the magnitude of a wavelet coefficient has some correlation with the magnitude of its

neighbors. Similar assumptions are found in the literature [7], [8], however, to the best of

our knowledge they have never been applied in the framework of bitplane image coding.

Let us define the local average of coefficient T as the arithmetic mean of the magnitude

of its 8 immediate neighbors according to

T =1

N

N∑

k=1

T k , (1)

where N is generally 8 except when T is on the boundaries of the subband. Figure 2(a)

evaluates the local average of non-null coefficients (i.e., all P that σ(P , 0) = 1) in

three wavelet subbands, for the “Portrait” image. Results are presented as the difference

between the magnitude of the coefficient and its local average, with a graph of frequencies.

Experimental evidence manifests that most coefficients have a magnitude similar to its

local average. Results hold for the other subbands and other images.

Through the local average, P (tj = 0) is generally modeled analyzing the probabilities

of significance and refinement bits emitted at each bitplane. Figure 2(b) depicts the

probability of significance bits, denoted as P ′(tj = 0) with j ≥ s, with relation to

T . Note the strong correlation between T and P ′(tj = 0), which corresponds to our

previous assumption. As an example, note that through that assumption one coefficient

is expected to have higher probability to become significant at bitplane v when its local

average is in the range [2v, 2v+1) than when T is lower than 2v. Generally, the lower the

local average, the lower the probability to become significant. Results hold for the other

subbands and other images.

Figure 2(c) analyzes the probabilities of the first refinement bit of coefficients at each

bitplane, referred to as P ′′(tj = 0) with j = s − 1, and relates this probability to T . As

above, there exists a strong correlation between T and P ′′(tj = 0). Results hold for the

remaining refinement bits, the other subbands, and other images.

Though the correlation exhibited in the previous evaluation is clear, it can not be

directly employed to estimate the probabilities of symbols since T is computed using

the magnitude of the coefficient’s neighbors, which is not available at the decoder until

the lowest bitplane is transmitted. Let us define T ′ as the local average computed using

the quantized representation of reconstructed coefficients at the decoder. We note that to

use quantized coefficients –instead of {σ(T k, j)}– to determine probabilities of emitted

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Fig. 2. Tests to evaluate the local average of coefficients. All experiments are carried out for the HL1 subband of

the “Portrait” image except when indicated. The subindex of the subband label indicates the decomposition level. (a)

Evaluates the difference between the coefficient’s magnitude and its local average; (b) evaluates the probabilities of

symbols emitted in significance coding, related to the local average of coefficients; (c) evaluates the probabilities of

symbols emitted in the first refinement bit, related to the local average of coefficients; and (d) evaluates the probabilities

of symbols emitted by SPP, related to the local average of quantized coefficients.

symbols is not novel [9]. Again, to the best of our knowledge, it has never been employed

in the proposed framework. Figure 2(d) depicts the same analysis carried out previously

for significance coding, but using T ′ instead of T . In this case, only bits emitted by

the SPP coding pass of JPEG2000 are analyzed (see below). Although the correlation is

not as strong as above, it is still clear. This observation holds for refinement coding and

also when the analysis considers sets of wavelet coefficients (codeblocks as defined in

JPEG2000) rather than with the whole wavelet subband.

C. Application to JPEG2000

Our experience indicates that in the framework of JPEG2000 a rough approximation for

P (tj = 0) is sufficient to achieve high efficiency. The main principles behind the proposed

model is that probabilities for significance coding (refinement coding, respectively) drop

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to nearly 0.5 when T ′ is equal or higher than 2j (2j + δ · 2j , respectively, where δ is the

reconstruction factor2 used at bitplane j). When T ′ is lower, the probability of emitted

symbols can be approximated by considering only j. The higher the j, the higher the

P (tj = 0). Formally expressed, the proposed model determines the probabilities for bits

emitted by SPP coding passes of codeblock Bi according to

P (tj = 0) =

0.5 if T ′ ≥ 2j

j

Ki

· 0.15 + 0.65 otherwise, (2)

where Ki is a sufficient number of bitplanes to represent all coefficients within Bi. The

probabilities for bits emitted by MRP are determined according to

P (tj = 0) =

0.5 if T ′ ≥ 2j + δ · 2j

j

Ki

· 0.15 + 0.7 otherwise. (3)

Equations 2 and 3 set fixed probabilities for symbols emitted in SPP and MRP, thus

the adaptive process carried out by the MQ coder is avoided. Again, to avoid the context-

adaptive coding process to model the symbols’ probabilities of bitplane coding engines

is not new in the literature [11]–[13]. Parameters of equations 2 and 3 are determined

empirically, though small variations do not change results significantly.

To use a local average-based approach for bits emitted by CP is not appropriate since

CP mostly scans coefficients with none significant neighbor, so T ′ becomes highly biased.

Experience indicates that in this case context-adaptive coding achieves a slight increment

on coding efficiency compared to local average-based approaches. With the “Portrait”

image, for instance, lengths of CP coding passes are 0.7% smaller for context-adaptive

coding than for a local average-based approach, on average. However, we stress that

most significance bits (around 70%) are encoded by SPP. In the experiments of the next

section the significance coding primitive of CP uses context-adaptive coding. Rather than

to use the original JPEG2000 context selection, our approach employs only two contexts,

determined as

c =

0 if∑M

k′=1 σ(T k′

, j) = 0

1 otherwise, (4)

where T k′

denotes the horizontal and vertical immediate neighbors of T , thus M is

generally 4 except when the coefficient is on the codeblock boundaries. This context

selection is aimed to validate our hypothesis stating that simple context models achieve

high efficiency. The sign and run mode coding primitives are left as formulated in

JPEG2000.

2Commonly in bitplane image coders δ = 0.5. Slight variations on this parameter does not affect significantly the

efficiency of the probabilities model. In this work we use the δ parameter as suggested in [10].

65

Though the evaluation of the computational complexity of the proposed model is

beyond the scope of this paper, we note that the most complex operation is to com-

pute T ′, which requires integer sums and divisions by eight. The former operation is

computationally not complex, whereas the latter can be implemented as a bit-shift.

IV. EXPERIMENTAL RESULTS

The performance of the proposed model is compared to three other strategies within

the framework of JPEG2000. The first one is to code all emitted symbols without using

arithmetic coding, labeled as “raw coding”. The second strategy uses the same context for

all bits emitted in each coding pass, i.e., 1 context for SPP, 1 context for MRP, 1 context

for CP, 1 context for sign coding, and the 2 original contexts for the run mode. This

strategy is labeled “single context”. The third strategy is the context selection formulated

by JPEG2000, which is considered near-optimal for context-adaptive approaches [1].

The strategy based on the local average is labeled “L-AVE”. Results are provided for

the “Portrait” and “Cafeteria” images of the ISO 12640-1 corpus. Similar performance

is achieved with the other images of the corpus.

For each image, Figure 3 depicts three graphs corresponding to the SPP, MRP, and CP

coding passes. In all figures, results are presented as the difference between the evaluated

strategy and raw coding. As well as in Figure 1, each triplet of columns depicts the result

achieved when all codeblocks of the image are encoded from the highest bitplane of the

image to the one indicated in the horizontal axes. Results suggest that the proposed model

achieves a performance slightly superior to that of JPEG2000 for SPP, and superior to

that of JPEG2000 for MRP. Note that the performance of “L-AVE” is not degraded at the

lowest bitplanes. This is intuitively explained since at the lowest bitplanes of the image

the magnitude of most coefficients is reconstructed with high accuracy, so the model

becomes very reliable. For CP coding passes, the performance achieved by the proposed

model and JPEG2000 is virtually the same.

Table I provides the bitrate of the codestream constructed by the aforementioned

strategies at the end of selected bitplanes. At low bitrates, JPEG2000 and our approach

achieve similar performance, whereas at medium and high bitrates our approach enhances

coding efficiency around 2%. At the highest bitrate depicted in Table I, the performance

gain between the “single context” strategy and JPEG2000 is about the same as the

performance gain between JPEG2000 and the local average-based approach.

V. CONCLUSIONS

The purpose of this paper is to draw attention to approaches different to those ones

based on context-adaptive coding to remove the statistical redundancy of symbols emitted

by bitplane coding strategies. This purpose is motivated by an experimental evidence

that seems to indicate that non-elaborate context modeling approaches achieve high

performance. Since the formation of contexts might not be especially advantageous to

estimate the probabilities of symbols, we introduce a conceptually simple model of

probabilities based on the assumption that the magnitude of a transformed coefficient

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Fig. 3. Evaluation of the coding passes lengths generated by the proposed model of probabilities, compared to three

other strategies. Labels on the top indicate the PSNR of the image at that bitplane, and the bitrate attained by the

original JPEG2000 context selection.

is correlated with the magnitude of its 8 immediate neighbors. The proposed model

is implemented in the JPEG2000 coding engine. Experimental evidence suggests that

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TABLE I

EVALUATION OF THE CODESTREAM BITRATE ACHIEVED BY THE PROPOSED MODEL OF PROBABILITIES,

COMPARED TO THREE OTHER STRATEGIES. RESULTS ARE GIVEN IN BITS PER SAMPLE.

“Portrait” “Cafeteria”

PSNR (in dB) 27.21 34.92 43.71 55.34 24.47 33.64 43.98 56.05

raw coding 0.27 1.06 2.65 5.04 0.66 1.94 3.89 6.20

single context 0.10 0.58 1.85 4.18 0.32 1.21 2.90 5.23

JPEG2000 0.01 0.56 1.80 4.11 0.30 1.14 2.80 5.12

L-AVE 0.01 0.56 1.78 4.04 0.30 1.15 2.78 5.03

virtually same efficiency as that of JPEG2000 is achieved at low bitrates, and enhanced

efficiency is achieved at medium and high bitrates. Conclusions can be generalized to

bitplane image coding.

ACKNOWLEDGMENT

This work has been partially supported by the Spanish Government (MICINN), by

FEDER, and by the Catalan Government, under Grants 2008-BPB-0010, TIN2009-14426-

C02-01, and 2009-SGR-1224.

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