New image compression artifact measure

using wavelets

Yung-Kai Lai and C.-C. Jay Kuo

Integrated Media Systems Center

Department of Electrical Engineering-Systems

University of Southern California

Los Angeles, CA 90089-256 4

Jin Li

Sharp Labs. of America

5750 NW Paci c Rim Blvd., Camas, WA 98607

ABSTRACT

Traditional ob jective metrics for the quality measure of co ded images such as the mean squared error (MSE)

and the p eak signal-to-noise ratio (PSNR) do not correlate with the sub jectivehuman visual exp erience well, since

they do not takehuman p erception into account. Quanti cation of artifacts resulted from lossy image compression

techniques is studied based on a human visual system (HVS) mo del and the time-space lo calization prop ertyofthe

wavelet transform is exploited to simulate HVS in this research. As a result of our research, a new image quality

measure by using the wavelet basis function is prop osed. This new metric works for a wide variety of compression

artifacts. Exp erimental results are given to demonstrate that it is more consistent with human sub jective ranking.

KEYWORDS: image quality assessment, compression artifact measure, visual delity criterion, human visual

system (HVS), wavelets.

1 INTRODUCTION

The ob jective of lossy image compression is to store image data eciently by reducing the redundancy of image

content and discarding unimp ortant information while keeping the quality of the image acceptable. Thus, the tradeo

of lossy image compression is the numb er of bits required to represent an image and the quality of the compressed

image. This is usually known as the rate-distortion tradeo . The numberofbitsusedtorecordthecompressed

image can b e measured easily.However, the \closeness" b etween the compressed and the original images is not

a pure ob jective measure, since human p erception also plays an imp ortant role in determining the qualityofthe

compressed image. Atpresent, the most widely used ob jective distortion measure is the mean squared error (MSE)

or the related p eak signal-noise ratio (PSNR). It is well known that MSE do es not correlate very well with the visual

quality p erceived byhuman b eing. This can b e easily explained by the fact that MSE is computed by adding the

squared di erence of individual pixels without considering the visual interaction b etween adjacent pixels.

In comparison with the amount of research on new compression techniques, work on visual quality measure of

compressed images is amazingly little. Most visual quality assessments use rating scales ranging from \excellent"

to \unsatisfactory" in measuring the p erceived image quality. Human assistance is needed in the establishmentof

quality scales and the comparison b etween compressed images and the set of training images. It is dicult to obtain

ob jective and quantitative measures by using this approach [15]. Results achieved are sub jective and qualitative.

Some work tries to mo dify existing quantitative measures to accommo date the factor of human visual p erception.

One is to improveMSEby putting di erentweights to neighb oring regions with di erent distances to the fo cal pixel

[17]. Most of them can b e viewed as a curve- tting metho d to comply with the rating scale metho d. Since late

1970's, researchers have started to pay attention to the imp ortance of the human visual system (HVS) and tried

to include the HVS mo del in the image quality metric [10], [11]. The development of the HVS mo del at that time

was not mature enough and the prop osed mo del could not interpret human visual p erceptual phenomena very well.

Recently, Karunasekera prop osed an ob jective distortion measure to evaluate the blo cking artifact of the blo ck-based

compression technique [12]. Watson [23] and van den Branden Lambrecht [1], [22] prop osed more complete mo dels

and extended their use to compressed videos.

It has b een shown by psychophysic exp eriments that the human visual system is comprised bymany units, each

of which is sensitive to the contrast at some space-frequency lo calized region and functions indep endently. The

overall visual p erception of the luminance or contrast of an ob ject is the aggregate p erformance of the bandpass

frequency resp onse of all units [4]. Based up on this space-frequency lo calized mo del, the visual p erception of co ding

artifacts can b e interpreted as the p erception of lo cal frequency inconsistency. When an image is compressed and

decompressed, the deco ded image has di erent lo cal frequency resp onses than the original image, thus resulting in

compression artifacts.

Since each cell can sense only a nite region with resp ect to the fo cal pixel in an image, Fourier analysis is

not suitable for the mo deling of HVS due to its global analytical prop erty. This issue has b een addressed and

inconsistency b etween theory and observed results by using the pure frequency analysis has b een found. Another

observation is that the common bandwidth of cortical cells is 1 o ctave[4].Itisknown from exp eriments that the

frequency resolution of the neighb orho o d is reduced by an o ctave when the distance from the fo cal p oint doubles [6].

This observation suggests the use of dyadic wavelets for the analysis and mo deling of HVS.

In this work, we consider a wavelet approach for the mo deling of HVS and then build an ob jective quality metric

based on this mo del. To b e more precise, we rst prop ose a new de nition of contrast with resp ect to complex

images by taking the wavelet transform of the image, and using wavelet co ecients to estimate the contrast at

each resolution in the image. By using the multiresolution and space-frequency lo calization prop erties, manyknown

inconsistencies in the psychophysical literature can b e understo o d naturally.Known human visual system functions

such as the contrast sensitivity threshold [20] and the frequency masking e ect [21] can also b e incorp orated in this

mo del. We then prop ose an ob jective metric to measure the extent of the p erceived contrast in every resolution,

and derive an error measure by examining the weighted di erences of wavelet comp onents b etween the original and

compressed images at each resolution.

This pap er is organized as follows. In Section 2, we brie y review existing HVS theory and mo dels. Then, the

wavelet approachisintro duced to simulate the lo calization phenomenon of HVS in Section 3. A new HVS mo del

based on wavelets is prop osed and an image quality assessment system is describ ed in Section 4. Exp erimental results

are given in Section 5 to demonstrate that this new metric works for a wide range of compression artifacts and that

the resulting measure is consistent with human sub jective ranking.

2 MODEL FOR HUMAN VISUAL SYSTEM (HVS)

2.1 Contrast thresholds

Human visual systems (HVS) resp onse to light stimuli of surroundings. One imp ortant observation is that visual

p erception is sensitive to luminance di erence rather than the absolute luminance. Let L and L b e the

max min

maximum and minimum luminances of the waveform around the p ointofinterest. Michelson's contrast, de ned as

L L

max min

; (1) C =

L + L

max min

is found to b e nearly constant when used to represent the just noticeable luminance di erence.

Psychophysic exp eriments showed that HVS is comprised bymany units, eachofwhich is fo cused on a certain

p oint in the vision eld and only sensitive to the contrast in a certain frequency band. The overall visual p erception

of ob ject luminance or contrast is the aggregate p erformance of each cell's frequency resp onse [4]. Since HVS cannot

provide in nite luminance resolution, a contrast threshold exists in each frequency band. The contrast threshold

value is a function of the spatial frequency which can b e determined exp erimentally.Atypical contrast sensitivity

curve, de ned as the recipro cal of the contrast curve, is shown in Fig. 1 [4]. As shown in the gure, HVS has the

highest luminance resolution around 3-10 cycles p er degree, and the sensitivityattenuates at b oth high and low

frequency ends. An image artifact can b e sensed if its contrast is ab ove the threshold at sp eci c spatial frequencies.

3 10

2 10

Sensitivity Threshold 1 10

0 10 −1 0 1 2 10 10 10 10

Spatial Frequency (cycles/deg)

Figure 1: A typical contrast sensitivitycurveofhuman b eings.

2.2 Suprathreshold contrast

The subthreshold stimuli, i.e. stimuli with a contrast lower than the threshold, cannot b e sensed byhuman.

The suprathreshold contrast is of concern in this subsection. Visual resp onse to the suprathreshold contrast involves

human sub jective rating. Even though it is dicult to nd an precise formula for its mo deling, it is generally agreed

that the estimated resp onse R is a function of the spatial frequency and follows a p ower law [14]:

p

R = k (C C ) ; (2)

T

where C is the suprathreshold contrast, C is the contrast threshold, and the exp onent p varies b etween 0.38 and some

T

value more than 1.0 [7]. Moreover, the contrast constancy phenomenon [9] states that the suprathreshold resp onse

is almost constant for a given contrast at a wide range of spatial frequencies. These two seemingly contradicting

observations were mediated rst by Cannon [26] and then by Georgeson's 3-stage mo del [8].

2.3 Channel interactions

Although cells narrowly tune to di erent frequency bands, they are not strictly band-limited, and interactions

among adjacent frequency channels were well observed in the literature. Two primary e ects were often discussed in

the literature. They are the summation e ect and the masking e ect.

The summation e ect is an inter-channel e ect saying that the neighb oring frequency channels contribute to the

total contrast. Therefore, a subthreshold contrast may still pro duce a small resp onse if there exist other excitory

stimuli in nearby frequency channels. This e ect can b e mo deled as a contrast threshold C decrease [13]:

T

1

!

s

N

i

X

s

i

; C = jC j

T Ti

i=0

th

where C is the contrast sensitivity threshold of the i closest channel, N is the total neighb oring channels a ecting Ti

the p erception of the target channel, C represents the original contrast without summation and s 's are the

T 0 i

exp onential comp onents to b e determined.

The masking e ect is another inter-channel e ect which states that the visibilityofa stimuli at some frequency

could b e impaired by the presence of other stimuli in nearby frequency channels. One well-known example in

compressed image artifact analysis is that the blo cking artifact of blo ck-based compression schemes, which consists

of high-frequency edge comp onents, is less visible in the texture regions. This e ect can b e viewed as a mo di cation

of contrast threshold and mo deled as an exp onential function of b oth their frequency and contrast di erences [1],

[21]:

!

   

N

m m

1;i 2;i

X

f C

i i

; (3) C = C

T T 0

C f

T 0 0

i=0

where C , C , N have the same de nition as in the subthreshold summation case, C and f are the actual

Ti T 0 i i

th

contrast and the spatial frequency of the i closest channel, resp ectively, and m 's and m 's are the co ecients

1;i 2;i

to b e determined. The validity of Eqn. (3) will b e examined in Section 5.2.

2.4 Spatial inhomogeneity

Most research results whichledtothecontrast threshold curveasshown in Fig. 1 were obtained with foveal vision,

i.e. measured very close the xation p oint. However, p eripheral vision researches showed that the p erceived image

gradually blurs with increasing distance from the xation p oint. In other words, the spatial frequency sensitivity

of the vision system decreases as the eccentricity from the fo cal p oint increases [6], [13], and it was shown that

the equal-sensitivity neighb orho o d radius is inversely prop ortional to the spatial frequency, i.e. the pro duct of the

spatial frequency and the resolvable radius from the fo cal p oint is almost constant [6]. Direct consequences of this

inhomogeneity phenomenon are that the resolvable region in the vision eld varies with the spatial frequency and

that contrasts at di erent resolutions play di erent roles in the visual system regarding their resp ective ranges.

3 SIGNAL LOCALIZATION AND WAVELET REPRESENTATION

3.1 Space-frequency lo calization

Limited by their spatial lo cation on the retina, the receptors can only fo cus on their own certain regions of the

visual eld. The frequency resp onse of the xation p ointischaracterized by the typical contrast threshold function

as shown in Fig. 1, but the high frequency resp onse will further attenuate as the eccentricity from the fo cal p oint

increases. Therefore, the frequency resp onse of visual stimuli is not only band-limited in the frequency domain but

also space-lo calized in the spatial domain.

The commonlyusedFourier frequency analysis is, however, a global pro cess whichgives all spatial comp onents

the same weighting.Itiswell known as Heisenb erg's uncertainty principle that the lo calization in b oth spatial and

frequency domains cannot b e achieved simultaneously. Gab or transform, which is a Gaussian-windowed Fourier

transform, was proved to achieve the limit of the Heisenb erg inequality. Gab or gratings were thus widely used in

mo dern psychophysical exp eriments.

The parameters of the Gaussian windowwere chosen at researchers' preferences and various degrees of lo calization

were achieved [19]. That is, byvarying the Gaussian envelop e parameters, the passbands of Gab or gratings were

overlapp ed to a di erent extent. There are some limitations in the Gab or representation. First, it is dicult to

analyze the stimuli whose frequency resp onses fall in the overlapp ed band. Second, since the Gab or lter is an I IR

(in nite impulse resp onse) lter, truncation is still needed for practical implementation. Thus, the lo calization is not

fully ensured after truncation. Toovercome these diculties, we adopt a wavelet approach for signal analysis in the next section.

3.2 Wavelet approach

The wavelet transform provides a go o d space-frequency lo calization prop erty [2] and can b e implemented by using

the multichannel lter banks. Compactly supp orted wavelets such as the Daub echeis lters [2] can b e implemented

with FIR ( nite impulse resp onse) lters. The space-frequency lo calization is optimized among all p ossible FIR lters

with the given length for the Daub echies lters. Another useful prop ertyofthewavelet lter is that the frequency

resp onse of the contrast threshold can b e easily obtained by prop erly cho osing the wavelet basis. Vision researches

with Gab or ltering usually used the resp onse amplitude, represented by dBs, to approximate the contrast threshold

function de ned by (1) [1]. However, the e ect of this approximation has not yet b een thoroughly investigated.

The Haar wavelet is the simplest basis function in the compactly supp orted wavelet family. It provides a capability

to compute the contrast directly from the resp onses of the low and high frequency subbands. For the Haar wavelet,

the lter co ecients for the low and high frequency lter banks are given by



1

p

n = 1; 0;

2

h [n] = (4)

0

0 otherwise

8

1

p

n =0;

<

2

1

p

n = 1; (5) h [n] =

1

2

:

0 otherwise,

resp ectively. Assume that a discrete-time input signal x[n] is the staircase contrast pattern



L n<0

max

x[n]= (6)

L n  0;

min

The resp onses y [n]andy [n] at the 0th resolution after ltering with h [n]and h [n], resp ectively,are

0;0 1;0 0 1

p

8

2L n<1

max <

1

p

(L + L ) n = 1

y [n] = (7)

max min

0;0

2

p

:

2L n>0;

min



1

p

( L L ) n = 1

max min

2

(8) y [n] =

1;0

0 otherwise.

Then, the contrast C at the interval (1; 0) and the 0th ( nest) resolution can b e computed by the ratio of y [n]

0 1;0

and y [n]:

0;0

y [1] L L

1;0 max min

: = C =

0

L + L y [1]

max min 0;0

At the 1st (second nest) resolution, the low frequency band resp onse y [n]isdownsampled bytwo, and fed into

0;0

the same lter bank. The resp onses are

8

2L n< 1

<

max

y [n] = L + L n = 1 (9)

0;1 max min

:

2L n> 1;

min



L L n = 1

max min

y [n] = (10)

1;1

0 otherwise.

Again, we can compute the contrast at this resolution as

L L y [1]

max min 1;1

C = : =

1

L + L y [1]

max min 0;1

Following this way,we can compute the contrast at any resolution by simply dividing the high band resp onse by

the low band resp onse at n = 1. Fig. 2 illustrates this constant-ratio relationship across resolutions. 200

150

100 −1 0 (a)

250 50 200

150 0 −1 0 −1 0 (b) (c)

400 100

300 50

200 0 −1 0 −1 0 (d) (e)

500 100

400 50

300 0 −1 0 −1 0

(f) (g)

Figure 2: Contrast computation using lter resp onses, where (a) is the original staircase signal, (b) and (c) are low

and high band resp onses at the 0th resolution, (d), (e), (f ) and (g) are resp onses at the 1st and 2nd resolutions.

There are several reasons to de ne multiple contrasts in di erent resolutions. First of all, since human contrast

sensitivity is highly dep endent on the spatial frequency,multiple contrasts can b e used to address di erentvariations

at di erent resolutions across the image [18]. Second, the spatial inhomogeneity phenomenon detailed in Section 2.4

requires the resp onse in di erent frequency bands to have di erent supp orted radii. Furthermore, it was shown that

each frequency channel in the HVS has the bandwidth of ab out 1 o ctave[18].Thedyadic wavelet transform satis es

these requirements naturally. Finally, p erfect reconstruction is p ossible with resp onses obtained from di erent scales,

and no visual information will b e lost during the pro cess. In contrast, to p erfectly reconstruct the visual information

using the Gab or analysis, all lters must have the same length and, as a result, the space-frequency lo calization

prop erty is less exible.

4 PROPOSED QUALITY MEASURE SYSTEM

The prop osed quality measure system is shown in Fig. 3. Both the original image and the distorted image

are passed through the system for dyadic wavelet decomp osition. Contrasts were computed at every resolution of

interest. The contrast threshold adjustments due to the masking e ect are then made according to Eqn. (3) at each

resolution. The summation e ect is not implemented in the system at present since its e ect is much less signi cant

than the masking e ect in the frequency range of natural images.

To mo del spatial inhomogeneity, one has to nd the weights to each frequency band as a function of distance

to the xation p oint. Since a continuous weighting function is to o involved mathematically, discrete stack structure

approximations were used in practical implementations when applied to complex images [6]. The staircase mo del

approximates the weighting function as a step function of the distance to the fovea. The resolvable suprathreshold

contrast radius obtained from exp eriments [5] and its staircase approximation are shown in Fig. 4. We see that the

staircase function with a constant radius-frequency pro duct ts the exp erimental data prettywell. The weights of

eachchannel is set to unity, and the visual information out of the radius at a sp eci c spatial frequency,asshown in the dash-dotted staircase line in Fig. 4, is discarded. Original Compressed Image Image

Wavelet Wavelet Decomposition Decomposition

Masking Masking Effect Effect

Inhomogeneity Inhomogeneity Truncation Truncation

Suprathreshold Suprathreshold Computation Computation

R o Rc

D

Figure 3: Contrast sensitivity threshold of the Gab or and Haar lters.

Although Georgeson's 3-stage mo del [8] is more complete, it requires additional work on parameter estimation.

Instead, the p ower law as describ ed in (2) is used with Cannon's result [26] at the last stage. The exp onent p varies

from 0.45 to 0.5 with resp ect to the distance from the actual contrast to the threshold. Here a constantvalue 0.45

is used for p.

Let the subscripts c and o represent the compressed image and the original image, resp ectively. Then, the quality

error measure is

1 0

H V N

X X X

1

2

A @

; (11) (R (i; j ) R (i; j )) D =

c;k o;k

HV

i=1 j =1

k =1

where V and H are the vertical and horizontal sizes of the image, resp ectively,andN is the numb er of ltering

channels. The resulting error measure D is dimensionless since the contrast itself is dimensionless.

It is known that HVS is also orientation-dep endent. In other words, the contrast sensitivity threshold as well

as p erceived contrast vary with resp ect to the grating orientation [27]. This feature is not implemented in our

system since exp erimental results at di erentorientations are not available at present, but it can b e incorp orated

conveniently into the system by using the summation of more channels with di erentorientations.

5 EXPERIMENTAL RESULTS

5.1 Validation of the Haar wavelet

Since the cortical cells have a Gaussian-shap ed reception pro le [3], it is often used as an evidence that the

Gab or lter is preferable in the vision exp eriment. Since the Haar lter do es not p ossess the same Gaussian-shap ed

passband as the Gab or lter do es, one may susp ect the validity of using the Haar lter in vision analysis. Tovalidate 50

Data from García−Pérez, 1988 Bilinear approximation 40 Staircase approximation

30

Radius (deg) 20

10

0 2 4 8 16 32

Spatial frequency (cycle/deg)

Figure 4: Exp erimental data from Garca-Perez (1988) and the corresp onding staircase mo del approximation.

the use of the Haar wavelet, psychophysical exp eriments were conducted on a 17" Silicon Graphics color graphic

2

display GDM-17E11. The luminance range of the displaywas adjusted from 0 to 80 cd=m (candela/squared meter)

by using a Photoresearch sp ectroradiometer. There were 256 discrete gray scales present in the exp eriments. The

relationship of the luminance versus the gray scale is shown in Fig. 5. This curvewas used to compute the actual

contrast in the following exp eriments.

80

70

60

50

40

30 Luminance (cd/m^2)

20

10

0 0 50 100 150 200 250 300

Gray level

Figure 5: The plot of the luminance versus the gray level for the color graphic display used in exp eriments.

Both Gab or and Haar ltered patches were used in the exp eriments. The spatial frequency range of the test

patches was from 0.069 to 19.2 cycles p er degree, whichcovers virtually the whole frequency band wewould sense

from digital images. The result is shown in Fig. 6, where the sensitivity threshold, de ned as the recipro cal of the

contrast threshold, is plotted as a function of spatial frequency. The closeness of these two curves shows that the

Haar lter has a comparable p erformance in comparison with the Gab or lter. This exp erimental result clearly

demonstrates that the Haar lter can also b e used in the quality assessment system. 2 10

1 Sensitivity Threshold 10

Haar Filtering Gabor Filtering

−1 0 1 10 10 10

Spatial Frequency (cycles/deg)

Figure 6: Contrast sensitivity threshold by using the Gab or and Haar lters.

5.2 Suprathreshold masking

The masking e ect can b e viewed as a function of the ratio of target signal and masking signal frequencies

[24] or as a function of the ratio of target signal and masking signal contrasts [25]. We constructed (3) based on

the assumption that these two e ects are separable. Toverify this assumption, psychophysical exp eriments were

conducted to nd the parameters of this mo del.

The contrast ratio C =C , where C and C represent the contrast of masking signal and target signal,

mask mask

resp ectively, ranged from 0.5 to 2.5. The frequency ratio f =f ,meanwhile, ranged from 3to3octaves. To

mask

isolate individual e ects, we rst xed the frequency ratio, and varied the contrast of each signal to investigate the

e ect of the contrast ratio. The result is shown in Fig. 7 (a), and an exp onential tting function was determined

using the data. As shown in Fig. 7 (a), we see that when the contrast of masking signal b ecomes larger with resp ect

to the target signal, the contrast sensitivity decreases (or equivalently the least detectable contrast increases) [25].

We then varied b oth the contrast ratio and the frequency ratio of target and masking signals. During the

computation pro cess, we scaled exp erimental data with resp ect to their contrast ratios according to Fig. 7 (a). The

scaled data showed very little deviation, thus indicating that (3) is a very go o d approximation to the masking mo del.

The means of exp erimental data and tting functions are shown in Fig. 7 (b), which are in go o d agreement with

those in [24]. From the tting functions, the sensitivity threshold changes for frequency ratios of 2, 1, 1, 2, and

3 o ctaves are o.96, 0.84, 0.76, 0.87, and , resp ectively.

5.3 Application to compression image artifacts

The new image quality assessment system was constructed according to Fig. 3 with parameters determined by

ab ove exp eriments. Compressed Lena images of size 256  256 were used for image quality assessment. Twotyp es

of compression schemes were applied. One is the DCT-based compression with the JPEG default quantization table

and a quantization factor of 5, co ded at 0.34 bpp with PSNR=26.43. The other is wavelet-compressed Lena by using

the emb edded zerotree wavelet (EZW) algorithm prop osed by Shapiro, co ded at 0.32 bpp with PSNR=28.47. These

two images are shown in Fig. 8.

Since the HVS contrast sensitivity threshold is characterized by the spatial frequency, de ned as cycles p er degree,

one should exp ect the quality measure varies with the ratio of D , the distance b etween the observer and the image,

and W , the width of the image. Fig. 9 showed the relation b etween the ratio and the quality measure. As the distance 0.9 1

Experimental data Exponential fitting function 0.8 0.9

0.7

0.8

0.6 Sensitivity threshold change Sensitivity threshold change

0.7 Experimental data Fitting function 0.5 0.5 1 1.5 2 2.5 3 0.65 −3 −2 −1 0 1 2 3 Contrast ratio (Cmask/C)

Frequancy ratio f_mask/f (octaves)

(a) (b)

Figure 7: Illustration of the masking e ect: sensitivity threshold changes under di erent (a) contrast and (b)

frequency ratios.

between the observer and the image increases, the spatial frequencies of the details (high frequency comp onents)

b ecome even higher that the visual system attenuation will fail to p erceive the compression artifact. Therefore, the

error will approach zero as the viewing distance increases. On the other hand, once the viewing distance is decreased

to a certain extent, the whole details of the image to the pixel level are p erceivable. The quality measure will b e

thus almost constant when the viewing distance is smaller than this distance, until the inhomogeneity phenomenon

and the HVS attenuation at low frequencies takeover at extremely close distances, whichwilllower the p erceived

error by a quite small extent. The \b est" viewing distance shown in the gure is ab out 6{10 times the image width,

which is consistent with the rule of thumb in practical image viewing situations.

We can also see from Fig. 9 that the EZW-compressed image has a lower quality error measure than that of the

JPEG-compressed image, although the compression ratio is smaller. This is consistent with the sub jective ranking

of human observers.

6 CONCLUSION

In this research, we prop osed a new approach to mo del human visual system (HVS) bywavelet transform.

Haar wavelets were proved to provide exact contrast values at each resolution, and the masking e ect and spatial

inhomogeneitywere conveniently incorp orated into the HVS mo del bymultiresolution analysis. Exp eriments showed

that Haar lters have comparable p erformance with the Gab or lters. The resulting new metric was successful in

measuring compressed image artifacts.

7 ACKNOWLEDGMENT

This work was supp orted byIntel, the Integrated Media Systems Center, a National Science Foundation Engineer-

ing ResearchCenter, and the National Science Foundation Presidential FacultyFellow (PFF) Award ASC-9350309.

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Figure 8: Two test images: (a) JPEG-compressed Lena and (b) EZW-compressed Lena.

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0.2

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0.1

0.05 JPEG−compressed Lena image EZW−compressed Lena image

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