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Aug 8, 2002 - Image Compression under Lossless/Lossy Coding Criterion. Somchart ... code modulation (DPCM) [2] is also a famous standard for lossless image ...... ample, if pdf. of image is similar to that of 2D AR(1) model, the best LLW ...
IEICE TRANS. FUNDAMENTALS, VOL.E85–A, NO.8 AUGUST 2002

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Special Section on Digital Signal Processing

Performance Evaluation of Lossless/Lossy Wavelets for Image Compression under Lossless/Lossy Coding Criterion Somchart CHOKCHAITAM†a) , Nonmember and Masahiro IWAHASHI† , Regular Member

SUMMARY In this paper, we propose lossless/lossy coding criterion as a new objective criterion to theoretically evaluate coding performance of the lossless/lossy wavelet (LLW). The proposed lossless/lossy coding criterion consists of three parameters: “lossless coding criterion ,” “quantization-lossy coding gain ” and “rounding errors .” The first parameter is a criterion to evaluate lossless coding performance of the LLW, whereas the second and the third parameters are criteria to evaluate lossy coding performance of the LLW at low bit rate and high bit rate, respectively. Relation among those three parameters is clearly illustrated in this paper. Performances of 15 kinds of the LLW are measured with two-dimensional (2D) octave-decomposition by applying some standard images and 2D AR(1) model as input signals. key words: lossless, lossy, wavelet, coding gain

1.

Introduction

The Joint Photographic Experts Group (JPEG) introduced many image compression standards, which are widely used to compress a huge amount of digital image data. For example, the JPEG algorithm [1] based on discrete cosine transform (DCT) [2] is a well-known standard for lossy image compression. The lossless JPEG (L-JPEG) algorithm based on differential pulse code modulation (DPCM) [2] is also a famous standard for lossless image compaction. Recently, the JPEG2000 [3] has been discussed as a new international standard that contains two operation modes—lossless mode and lossy mode. So far, some LLWs [4]–[9] are proposed as the JPEG-2000 candidates. The LLW can be used not only as a lossless coding but also as lossy coding because of including the “rounding” operations. Some researchers [9], [10] have been investigating the LLW in term of lossless compression performance and lossy compression performance separately. However a new objective measure to evaluate lossless and lossy compression performance based on the same hypothesis should be a great benefit. The conventional coding gain [2] has been proposed as an objective criterion to evaluate lossy coding performance of a measured coding technique comparing Manuscript received November 30, 2001. Manuscript revised March 1, 2002. Final manuscript received April 19, 2002. † The authors are with the Department of Electrical Engineering, Nagaoka University of Technology, Nagaoka-shi, 940-2188 Japan. a) E-mail: [email protected]

to PCM [2]. The coding gain for subband coding of nonorthogonal filter bank has been proposed by J. Katto and Y. Yasuda [15]. However, the existing coding gain is limited to evaluate only lossy coding performance. Moreover, the errors generated from rounding operation are newly occurred in the LWT-based coding system. Recently, we proposed the “lossless coding criterion” [11]–[13] as a new objective criterion to theoretically evaluate performance of the lossless/lossy wavelet transforms (LLWs). The relationship between the “lossless coding criterion” and the existing “coding gain (lossy coding gain)” is explicitly described. It becomes possible to theoretically evaluate lossless/lossy unified coding methods based on the LLW under the unified criterion—“lossless coding criterion” and “lossy coding gain.” However, the evaluation results in previous reports [11]–[13] neglect errors generated by rounding operations that mainly effect on coding performance of the LLW at high bit rate. In this paper, we propose a new lossless/lossy coding criterion as a new objective criterion to theoretically evaluate lossless and lossy coding performance of the LLW under the unified criterion. The proposed measure consists of three parameters: “lossless coding criterion,” “quantization-lossy coding gain” and “rounding errors.” The lossless coding criterion is a criterion to evaluate lossless coding performance of the LLW included in our previous reports [11]–[13]. The quantization-lossy coding gain and rounding errors are newly introduced in this paper as criteria to evaluate lossy coding performance of the LLW at low bit rate and high bit rate, respectively. Relation among those three parameters is explicitly illustrated. We measure compression performance of practical 15 kinds of recently proposed LLWs under the lossless/lossy coding criterion using 2D input signals. The analysis part of coding system in this report is based on 2D octavedecomposition, which is effective to image compression. Experimental results in this paper include applications of the two-dimensional (2D) LLW to 2D AR(1) model for theoretical evaluation and some standard images for practical evaluation. Effectiveness of the “optimum bit allocation” in the quantization procedure is also described. This report is organized as follows. The lossless/lossy wavelet (LLW) including rounding operations

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1883 Table 1

Parameters of JPEG-2000 candidates.

is reviewed in Sect. 2. The lossless/lossy coding criterion is introduced in Sect. 3. Evaluation results and their analysis are illustrated in Sect. 4. Finally, conclusions are described in Sect. 5. 2.

The Lossless/Lossy Wavelet (LLW)

We analyze analysis part and synthesis part of the one-dimensional (1D) LLW in Sects. 2.1 and 2.2, respectively. Then, our analysis is extended to twodimensional (2D) octave-decomposition in Sect. 2.3. In this report, we use z-transform expression defined by X(z) =

K−1 

x(k)z −k

(1)

k=0

where x(k) and K denote signal’s intensity and its value for image signal is given in “integer” and range of image pixel, respectively. We analyze all signals in this paper under the following assumptions: (1) all filters are linear and time-invariant. (2) correlations between each of the errors and the signals are zero (statistical independence), and (3) their power spectrums are approximately flat. With the previous assumption, we can represent

non-linear rounding operation by an additive noise [16] and variance of output signal of rounding operation is 2 σS2 Ro = σS2 Ri + σN R

(2)

2 where σS2 Ri , σS2 Ro , σN denote variance of input signals, R output signals and additive noises of rounding operation, respectively. Since power spectrum of noise are assumed to be flat, we can approximate variance of additive noise as 1 2 (3) σN = R 12 We renamed the LLW according to the numbers of taps in low-pass and high-pass filters as indicated in the first row from left in Table 1. The parameters in ith lifting structure of each LLW as shown in Table 1 are

Pi (z) =

3 

aki z k

(4)

k=−3

2.1 Analysis Part and Its Equivalent Expression In this section, we investigate analysis (encoding) part of the 1D LLW. In the left side of Fig. 1, the original

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Fig. 2 Synthesis part of the 1D LLW and its equivalent expression.

Fig. 1 Analysis part of the 1D LLW and its equivalent r denotes rounding operation. expression. The 

signal X(z) is divided into two groups by      1 1 1 X(z 1/2 ) X0 (z) = X1 (z) X(−z 1/2 ) 2 z 1/2 −z 1/2



(5)

After this, Finite Impulse Response (FIR) filters (P1 , P2 , P3 , and P4 ) are applied to produce two band signals Y0 (z) and Y1 (z) by        X0 (z) P4 (z)Y1 (z) Y0 (z) = + Y1 (z) X1 (z) P3 (z)X0 (z)   NR4 (z) + (6) NR3 (z) and



X0 (z) X1 (z)



 =





X0 (z) P2 (z)X1 (z) + X1 (z) P1 (z)X0 (z)   NR2 (z) + NR1 (z)



(7)

where NRi (z) denotes additive noises generated from ith rounding operation of analysis part of the 1D LLW. Note If the LLW is performed by double or triple lifting structure, parameters in extended lifting structure are set to be zero as shown in Table 1. In lossless mode, the band signals Y0 (z) and Y1 (z) are entropy coded without any quantization. In lossy mode, before the entropy coding, the band signals are quantized with step size (αb SY ) by        Y0 (z)/(α0 SY ) NQ0 (z) Y0 (z) = + (8) Y1 (z) Y1 (z)/(α1 SY ) NQ1 (z) where SY is a quantization step size and two parameters αb , b = 0, 1, are determined in Sect. 3.2. NQi (z) denotes additive noises generated from ith quantization of analysis part of the 1D LLW. Therefore, analysis part of the 1D LLW in the left side of Fig. 1 can be expressed by an equivalent expression in the right side of Fig. 1. The analysis filters HL (z) and HH (z) in the right side of Fig. 1 are       1 P4 (z 2 ) 1 P2 (z 2 ) 1 0 HL (z) = HH (z) 0 1 P3 (z 2 ) 1 0 1    1 0 1 · (9) P1 (z 2 ) 1 z and rounding errors NRL,A (z) and NRH,A (z) in analysis part are

 NRL,A (z 2 ) NRH,A (z 2 )      1 P4 (z 2 ) 1 0 0 1 P2 (z 2 ) = 0 1 P3 (z 2 ) 1 0 1 NR1 (z 2 )       1 P4 (z 2 ) NR2 (z 2 ) 1 0 + 2 0 1 P3 (z ) 1 0      2 1 P4 (z ) NR4 (z 2 ) 0 + + (10) 0 1 NR3 (z 2 ) 0

2.2 Synthesis Part and Its Equivalent Expression Similarly, synthesis part of the 1D LLW in the left side of Fig. 2 can be replaced by an equivalent expression in the right side of Fig. 2. The synthesis filters GL (z) and GH (z) in the right side of Fig. 2 are       GL (z) 1 0 1 −P2 (z 2 ) 1 0 = −P1 (z 2 ) 1 0 GH (z) 1 −P3 (z 2 ) 1    1 −P4 (z 2 ) 1 · (11) 0 1 z −1 and rounding errors NRL,S (z) and NRH,S (z) in synthesis part are   NRL,S (z 2 ) NRH,S (z 2 )     1 0 1 −P2 (z 2 ) 1 0 = −P1 (z 2 ) 1 0 1 −P3 (z 2 ) 1       NR4 (z 2 ) 1 0 1 −P2 (z 2 ) · + 0 −P1 (z 2 ) 1 0 1       0 1 0 NR2 (z 2 ) · +  NR3 (z 2 ) −P1 (z 2 ) 1 0   0 + (12)  NR1 (z 2 ) where NR i (z) denotes additive noises generated from ith rounding operation of synthesis part of the 1D LLW. 2.3 Two-Dimensional (2D) Octave Decomposition In this paper, the coding system for LLW evaluation is based on 2D octave decomposition, where the lowest band signal is further divided into a few bands as shown in Fig. 3. The one-dimensional signal processing described in previous sections is applied

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+ NR5,S (z12 , z22 )

Fig. 3

Two-dimensional octave decomposition of the LLW.

(17)

where z1 and z2 denote horizontal dimension and vertical dimension respectively. Equations (15)–(17) illustrate how rounding errors are propagated through lifting structure [16]. Up/Down sampling factor wp in Fig. 4 is   64 for p = 0, 1, 2, 3 16 for p = 4, 5, 6 wp = (18)  4 for p = 7, 8, 9 “NQi ” in Fig. 4 denotes errors generated from quantization in ith subband. Figure 4 confirms that the errors in LWT coding are generated from two sources: quantization and rounding operation. 3.

Fig. 4

The equivalent expression of coding system in Fig. 3.

twice to an image signal—in horizontal and in vertical. Namely, the input signal (XI ) is divided into 10 subbands: “YHL1 ,” “YLH1 ,” “YHH1 ” for 1st stage, “YHL2 ,” “YLH2 ,” “YHH2 ” for 2nd stage and “YLL3 ,” “YHL3 ,” “YLH3 ,” “YHH3 ” for 3rd stage. We denote {LL3 , LH3 , HL3 , HH3 , LH2 , HL2 , HH2 , LH1 , HL1 , HH1 } as {0, 1, 2, · · · , 9}, respectively. Notice that number of stages and number of subbands of evaluation coding system can be changed. Similarly, Fig. 3 can be replaced by its equivalent expression in left side of Fig. 4. The equivalent expression of the LLW can be written in term of filters in Eq. (9) and Eq. (11). For example, filter parameters in its equivalent expression of HL2 subband are H5 (z1 , z2 ) = HL (z1 )HL (z2 )HH (z12 )HL (z22 ) G5 (z1 , z2 ) = GL (z1 )GL (z2 )GH (z12 )GL (z22 )

(13) (14)

and rounding errors in its equivalent expression are

Lossless/Lossy Coding Criterion

In this section, we propose the lossless/lossy coding criterion consisting of three parameters: “lossless coding criterion,” “quantization-lossy coding gain” and “rounding errors.” Simulation results for confirmation of relation between our proposed lossless/lossy coding criterion and lossless/lossy coding performance of the LLW are shown in Sect. 3.4. 3.1 Lossless Coding Criterion [11]–[13] There are two kinds of the lossless coding criterion: “bit-rate-lossless coding criterion” and “variancelossless coding criterion.” Firstly, we define bit-ratelossless coding criterion (CLSL ) to evaluate system’s performance as a ratio between the total bit rate of PCM (BP CM ) and that of lossless coding (BLSL ) by CLSL = 20 log10

∗ = 10 log10 CLSL

σP2 CM 9  −1 (σy2b )wb

(20)

b=0

= NRL,A (z1 )HL (z2 )HH (z12 )HL (z22 ) σP2 CM

+ NRL,A (z2 )HH (z12 )HL (z22 ) (15)

NR5,S (z12 , z22 ) = NRL,S (z22 )GH (z12 )GL (z2 )GL (z1 ) + NRH,S (z12 )GL (z2 )GL (z1 ) + NRL,S (z2 )GL (z1 ) + NRL,S (z1 ) (16) Therefore, total rounding errors in HL2 subband NT R5 (z1 , z2 ) is calculated from NT R5 (z12 , z22 ) = NR5,A (z12 , z22 )G5 (z1 , z2 )

(19)

∗ and variance-lossless coding criterion (CLSL ) is defined from

NR5,A (z12 , z22 )

+ NRH,A (z12 )HL (z22 ) + NRL,A (z22 )

2BP CM 2BLSL

σy2b

and denote variance of PCM signal where and that of bth band signal in lossless technique. A relation between bit-rate-lossless coding criterion (CLSL ) ∗ ) [11]–[13] and variance-lossless coding criterion (CLSL is CLSL = 20 log10

2BP CM ∗ = CLSL + c1 2BLSL

(21)

where c1 is constant depending on signal’s probability density function. Evaluation results of both criteria are ∗ has an advantage similar meaning. However, the CLSL because a total bit rate of signal is not necessary to be computed.

IEICE TRANS. FUNDAMENTALS, VOL.E85–A, NO.8 AUGUST 2002

1886 Table 2 Quantization-lossy coding gain CLSY,Q and variance of rounding error for “Cartoon.”

Table 3

Lossless/lossy coding criterions based on 2D AR(1) model.

Next, we find optimum quantization steps [15] by minimizing the Ω and its solutions are given [11]–[13] by

3.2 Quantization-Lossy Coding Gain In this section, we consider relation between the “lossless coding criterion” in Eq. (21) and the conventional “lossy coding gain” [2] defined by CLSY = 10 log10

σP2 CM 2 σLSY

(22)

where σP2 CM denotes variance of total errors in PCM 2 denotes variance of total errors in coding and σLSY lossy coding calculated from 2 σLSY

=

2 σN Q

+

2 σN R

σP2 CM 2 σN Q

(24)

A relation between quantization-lossy coding gain ∗ [11]–[13] is CLSY,Q and lossless coding criterion CLSL ∗ CLSY,Q = CLSL + c1 − Ω

||G0 || , ||Gb ||

where ||Gb || =

b = 0, 1, · · · , 9

  k2

(25)

gb2 (k1 , k2 )

(26)

(27)

k1

In this case (optimum bit allocation), Ω becomes

(23)

2 where σN denote variances of errors generated from Q quantization. With this reason, the conventional coding gain of the LWT is not constant since it depends on a measured bit rate as illustrated in Table 3. For example, the conventional coding gain for “5/11-C” with the same step size at BLSL − 1 [bpp] is 3.24 but that at BLSL − 2 [bpp] is 6.59. Next, quantization-lossy coding gain is defined from conventional lossy coding gain with neglecting rounding error as

CLSY,Q = 10 log10

αb =

Ωopt = 10 log10

9 

||Gb ||2

wb−1

(28)

b=0

Note The minimum of Ωopt = 0 for orthogonal transform such as ST and lossless DCT [14]. 3.3 Rounding Errors Rounding errors cannot be neglected when quantization errors are relatively small comparing to rounding errors at high bit rate. Therefore, variance of rounding errors must be considered. Based on the assumptions mentioned in the previous section, variance of rounding errors of analysis part in Eq. (8) is calculated from 2 σNR L,A 2 σN R H,A

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  1 ||P4 (z 2 )||2 1 0 = 0 1 ||P3 (z 2 )||2 1     0 1 ||P2 (z 2 )||2 · 2 σN 0 1 R1     2  σNR 1 ||P4 (z 2 )||2 1 0 2 + 0 1 ||P3 (z 2 )||2 1 0    2   σNR 0 1 ||P4 (z 2 )||2 4 + + (29) 2 σN 0 1 0 R3 2 is one-twelfth where variance of rounding error σN Ri as shown in Eq. (3). Similar to Eq. (29), we can calculate variance of total rounding errors of the equivalent expressions from Eq. (17). Next, we can calculate variance of total rounding error in the coding system for LLW evaluation from 2 = σN R

9 σ2  NT R i=0

i

(30)

wi

2 where σN denote variance of total rounding error T Ri in ith subband. Finally, the conventional lossy coding gain is determined from Eqs. (22)–(24) as

CLSY = 10 log10

σP2 CM 2 10−CLSY,Q σP2 CM + σN R

(31)

From Eq. (31), it’s implied that lossy coding performance at high bit rate (when quantization errors are relatively small comparing to rounding errors) mainly depends on variance of rounding errors. Therefore, variance of rounding errors is applied as a criterion to evaluate lossy coding performance of the LLW at high bit rate. As a conclusion, we propose lossless/lossy coding criterion consisting of three parameters. The lossless coding criterion is applied as an objective criterion to evaluate lossless coding performance of the LLW. The quantization-lossy coding gain and variance of rounding errors are applied as objective criteria to evaluate lossy coding performance of the LLW at low bit rate and

high bit rate, respectively. The lossless coding criterion and quantization-lossy coding gain are varied depending on image inputs, whereas variance of rounding errors is constant depending on parameters of the LLW. We confirm accuracy of proposed criteria to evaluate lossless/lossy coding performance of the LLW in the next section. 3.4 Confirmation of Accuracy of Our Proposed Lossless/Lossy Coding Criterion In this section, we confirm accuracy of our proposed lossless/lossy coding criterion as a criterion to evaluate lossless/lossy coding performance of the LLW. First, we endorse similarity between evaluation results of bit-rate-lossless coding criterion (CLSL ) and those ∗ ). Figure 5 of variance-lossless coding criterion (CLSL illustrates bit-rate-lossless coding criterion CLSL and ∗ averaged over variance-lossless coding criterion CLSL all standard image listed in Table 4. We calculate correlation between them (Cor ) defined from

∗ ) (CLSL CLSL (32) Cor =

1/2 ∗ 2 ( (CLSL ) ) ( (CLSL )2 )1/2 Correlation between them is 0.9994, so it strongly confirms similarity of both evaluation results. Next, we apply “Cartoon” image as an example

Fig. 5 Bit-rate-lossless coding criterion CLSL and ∗ . variance-lossless coding criterion CLSL

∗ . The (o) and (e) denote “odd-tap” Table 4 The variance-lossless coding criterion CLSL LLW and “even-tap” LLW, respectively.

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Fig. 6

PSNR and a total bit rate for “Cartoon” at low bit rate.

of practical standard image to confirm accuracy of our proposed lossless/lossy coding criterion. Table 2 shows quantization-lossy coding gain CLSY,Q and variance of rounding errors for “Cartoon.” From lossless/lossy coding criterion listed in Table 2, we can interpret as follows: (1) Based on our proposed criteria, quantization-lossy coding gains of the LLW with optimum bit allocation is better than those of the LLW with same step size by 0.46 [dB] for “9/7-F” and 1.84 [dB] for “13/7-C,” so lossy coding results of the LLW with optimum bit allocation is expected to be better. Based on practical “Cartoon” image, Fig. 6 confirms an effectiveness of bit allocation that PSNR of decoded image with optimum bit allocation is higher than PSNR of decoded image with same step size by about 0.8 [dB] for “9/7-F” and about 3.2 [dB] for “13/7-C.” (2) Based on our proposed criteria, quantization-lossy coding gain of “9/7-F” with optimum bit allocation is higher than that of “13/7-C” with optimum bit allocation by 0.18 [dB], so lossy coding results of “9/7-F” is expected to be better than those of “13/7-C” at low bit rate. Based on practical results, Fig. 6 confirms that PSNR of decoded image of “9/7-F” is better than that of “13/7-C” at low bit rate by about 0.2 [dB]. (3) Variance of rounding operations of “9/7-F” is higher than that of “13/7-C” by 1.426, so lossy coding results of “13/7-C” is expected to be better than those of “9/7F” at high bit rate. Based on practical results, Fig. 7 confirms that PSNR of decoded image of “13/7-C” are better than that of “9/7-F” about 1 [dB] at 4 [bpp]. 4.

Fig. 7 rate.

PSNR and a total bit rate for “Cartoon” at high bit

4.1 Theoretical Evaluation Based on 2D AR(1) Model 4.1.1

2D AR(1) Model

In this paper, we theoretically evaluate the LLW based on 2D AR(1) model with the frequency characteristics given by |X(ej(ω1 +ω2 ) )| (1 − ρ1 )(1 − ρ2 ) = 2 (1 + ρ1 − 2ρ1 · cos ω1 )(1 + ρ22 − 2ρ2 · cos ω2 ) (33) where ρ1 and ρ2 are correlation coefficients in horizontal and in vertical, respectively. 4.1.2

Lossless/Lossy Coding Criterions for 2D AR(1) Model

To calculate the variance-lossless coding criterions ∗ for 2D AR(1) model in Eq. (39), we rewrite the CLSL Eq. (21) in frequency domain [11]–[13] as ∗ CLSL = 10 log10

(34)

b=0

where ||Fb ||2 =

2 σX 2 σY b

Evaluation Results

We apply the lossless/lossy coding criterion described in Sect. 3 to evaluate the LLW in Sect. 2. The 2D AR(1) models are applied as input signal of theoretical evaluation in Sect. 4.1 and standard images are applied as input signals of practical evaluation in Sect. 4.2. We analyze evaluation results in Sect. 4.3

9 

w−1 ||Fb ||2 b

 2π 



|X(ejω1 , ejω2 )|2 dω1 dω2 0

=  2π 

0



|Hb (e

jω1

0

, ejω2 )X(ejω1 , ejω2 )|2 dω1 dω2

0

(35) Table 3 indicates lossless/lossy coding criterions for 2D AR(1) model. From Table 3, we find the following facts:

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1889 Table 5 The bit-rate-lossless coding criterion CLSL . The (o) and (e) denote “odd-tap” LLW and “even-tap” LLW, respectively.

(1) In average, “5/11-C” has the highest variancelossless coding criterion followed by “5/11-A,” “9/7-M” and “13/7-T,” respectively. (2) In average, “9/7-F” has the highest variancequantization lossy coding gain for coding system with the same step size (non-optimum bit allocation). This result is similar to that of M.D. Adams’s practical evaluation [9]. (3) In average, “13/7-C” has the highest variancequantization lossy coding gain for coding system with optimum bit allocation followed by “5/11-A,” “13/7-T” and “13/3,” respectively. (4) “2/2” produces the lowest variance of rounding errors followed by “13/3,” “9/3-S,” “9/3-K” and “5/3,” respectively. (5) The convention coding gain of the LWT depends on a measured bit rate. 4.2 Practical Evaluation Based on Standard Images 4.2.1 Lossless Coding Performance of the LLW ∗ and CLSL in Eq. (21) The lossless coding criterion CLSL for each of standard images are summarized in Table 4 and Table 5, respectively. In this paper, total bit rate is calculated from the zero-th order entropy rate defined as  Ps log2 Ps (36) H=− s

where Ps indicates probability of a symbol “s.” In this report, we apply the entropy rate in lossless coding criterion instead of a practical bit rate for a fair comparison. This is because a practical bit rate depends on what kind of encoder is applied. The highest result in each category is highlighted. From Table 4 and Table 5, we find the following facts: (1) The best LLW in term of lossless coding criterion

depends on input image’s characteristics. (2) In average, “13/11” has the highest variancelossless coding criterion followed by “13/7-T,” “9/7-M” and “5/11-C.” The “13/7-T” has the highest bit-ratelossless coding criterion followed by “9/7-M,” “5/11C,” “13/7-C” and “13/11.” These results agree with results of M. D. Adams’s paper [9]. (3) The results in Table 4 and Table 5 also confirm that evaluation results of variance-lossless coding crite∗ is similar to those of bit-rate-lossless coding rion CLSL criterion CLSL . (4) “Odd-tap” LLW has an average of lossless coding criterion higher than that of “even-tap” LLW. (5) Among “odd-tap” LLWs, difference between the best and the worst in term of bit-rate-lossless coding criterion is 0.24 [dB]. It means the difference is approximate 0.045 [bpp] in term of total bit rate. (6) From the result of medical image “Chest (X-ray) in Table 5, all of LLWs get negative value of CLSL . It means coding performance of LLW for “Chest (X-ray)” is worse than performance of PCM. 4.2.2

Lossy Coding Performance of the LLW

Quantization-lossy coding gain CLSY,Q in Eq. (25) of standard images is illustrated in Table 6. Table 7 summarizes lossy coding performance of the LLW averaged over all of the images listed in Table 4. The best result of each category is highlighted. From Table 6 and Table 7, we find the following facts: (1) The best LLW in term of quantization-lossy coding gain also depends on input image characteristics. (2) In average, the “13/7-C” has the highest quantization-lossy coding gain with optimum bit allocation followed by 13/7-T, 9/7-F, 5/11-A, respectively. (3) “Odd-tap” LLW has an average of quantizationlossy coding gain higher than that of “even-tap” LLW, regardless of optimum bit allocation

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1890 Table 6 The quantization-lossy coding gain CLSY,Q of the LLW with optimum bit allocation. The (o) and (e) denote “odd-tap” LLW and “even-tap” LLW, respectively.

Table 7 Lossy coding performance of the LLW averaged based on all standard images indicated in Table 3.

(4) In Table 7, the optimum bit allocation can improve quantization-lossy coding gain by 0.57 [dB] in average. These results confirm effectiveness of optimum bit allocation. (5) If we apply the same quantization step size (nonoptimum bit allocation) to all standard images, we find that “9/7-F” has the best quantization-lossy coding performance at low bit rate. This result is similar to results in M.D. Adams’s report [9]. (6) “2/2” has the lowest variance of rounding errors followed by “9/3-S,” “13/3,” 9/3-K” and “5/3,” respectively. However, lossy coding performance of “2/2” is not the best at high bit rate because quantization-lossy coding gain of “2/2” is the worst. (7) In Table 7, the conventional coding gain with optimum bit allocation of “9/7-F” at BLSL − 1 [bpp] is less than zero. It means that coding performance of the “9/7-F” at BLSL − 1 [bpp] is worse than PCM. This is because variance of rounding errors of “9/7-F” is big.

4.3 Analysis on Evaluation Results From evaluation results in Sect. 4.1 and Sect. 4.2, we can conclude as follows: (1) Lossless coding performance of the LLW mainly depends on probability density function (pdf.) of each image and LLW-filter characteristics. The best LLW in term of lossless coding criterion of each standard image is different depending on pdf. of each image. For example, if pdf. of image is similar to that of 2D AR(1) model, the best LLW (for all correlations) in term of lossless coding criterion is “5/11-C.” We also find that lossless coding performance of “odd-tab” LLW is better than that of “even-tab” LLW averaged over all standard image and 2D AR(1) model. (2) The optimum bit allocation in Eq. (31) improves lossy coding performance of the LLW comparing to that of same step size. (3) Lossy coding performance of the LLW at low bit rate mainly depends on its lossless coding performance and Ωopt where the Ωopt mainly depends on ||Gb || as

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shown in Eq. (31). Therefore, the best LLW in term of lossy coding gain of each standard image is different depending on pdf. of each image and ||Gb ||. We suggest to consider parameter ||Gb || to minimize the Ωopt when a new LLW is designed in the future. The minimum of Ωopt = 0 for orthogonal transform. (4) Lossy coding performance of the LLW at high bit rate mainly depends on variance of rounding errors. Variance of rounding operations depends on LLW-filter characteristics and number of lifting structure. The more number of lifting structure, the higher in term of variance of rounding errors. Therefore, “9/7-F,” which has 4 times lifting structure, does not have a good lossy coding performance at high bit rate. (5) Under our proposed criteria, “13/7-C” and “13/7T” are two of the best evaluation results. 4.4 Other Factor Although our proposed lossless/lossy coding criterion is a good objective criterion to evaluate the LLW, some factors are missing to be considered in this paper such as hardware complexity, subjective lossy compression performance, and etc. 5.

Conclusion

We have proposed new objective criteria to theoretically evaluate performance of the LLW in both lossless compaction mode and in lossy compression mode. Our proposed criterion consists of three parameters: “lossless coding criterion,” “quantization-lossy coding gain” and “rounding errors.” The rounding errors are newly considered as a criterion to evaluate lossy coding performance of the LLW at high bit rate. We found that (1) Performances of “odd-tap” LLW are better than performances of “even-tap” LLW (2) the LLW named “13/7C” and “13-7-T” are two of the best results in this paper. (3) variance of rounding errors, which effects on lossy coding performance of the LLW at high bit rate, depends on LLW-filter characteristics and number of lifting structure. References [1] JPEG CD10918-1, “Digital compression coding continuous-tone still images,” JPEG-9-R6, Jan. 1991.

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