Adaptive scanning methods for wavelet difference reduction in lossy image compression

Abstract

This paper describes methods for adapting the scanning order
through wavelet transform values used in the wavelet difference
reduction (WDR) algorithm of Tian and Wells (1996). These new methods
are called adaptively scanned wavelet difference reduction (ASWDR).
ASWDR adapts the scanning procedure used by WDR in order to predict
locations of significant transform values at half thresholds. These
methods retain all of the important features of WDR: low-complexity,
region of interest, embeddedness, and progressive SNR. They improve the
rate-distortion performance of WDR so that it is essentially equal to
that of the SPIHT algorithm of Said and Pearlman (1996) when arithmetic
compression is not employed. When arithmetic compression is used, then
the rate-distortion performance of the ASWDR algorithms is only slightly
worse than SPIHT. The perceptual quality of ASWDR images is clearly
superior to SPIHT

Full text

Page 1

ADAPTIVE SCANNING METHODS FOR WAVELET DIFFERENCE REDUCTION IN
LOSSY IMAGE COMPRESSION
James S. Walker
Department of Mathematics
University of Wisconsin-Eau Claire
walkerjs@uwec.edu
Truong Q. Nguyen
Dept. of Electrical & Computer Engineering
Boston University
nguyent@bu.edu
ABSTRACT
This paper describes methods for adapting the scanning or-
der through wavelet transform values used in the Wavelet
Difference Reduction (WDR) algorithm of Tian and Wells.
These new methods are called Adaptively Scanned Wavelet
Difference Reduction (ASWDR). ASWDR adapts the scan-
ning procedure used by WDR in order to predict locations
of significant transform values at half thresholds. These
methods retain all of the important features of WDR: low-
complexity, region of interest, embeddedness, and progres-
sive SNR. They improve the rate-distortion performance of
WDR so that it is essentially equal to that of the SPIHT al-
gorithmofSaid and Pearlmanwhen arithmeticcompression
isnotemployed. Whenarithmeticcompressionis used,then
the rate-distortion performance of the ASWDR algorithms
is only slightlyworse thanSPIHT. The perceptualqualityof
ASWDR images is clearly superior to SPIHT.
1. INTRODUCTION
This paper describes methods for adapting the scanning or-
der used in the Wavelet Difference Reduction (WDR) al-
gorithm of Tian and Wells [1]. These new methods im-
prove both the rate-distortion performance and the percep-
tual quality of the compressed images produced by WDR,
while retaining its desirable features [such as low complex-
ity, region of interest capability, embeddedness, and pro-
gressive SNR]. We refer to these new algorithms as Adap-
tively Scanned Wavelet Difference Reduction (ASWDR).
Two ASWDR algorithms will be described. The first
algorithm, called ASWDR1, adapts the scan order through
wavelet transform values using a parent/child and sibling
prediction procedure (sig. parent implies sig. children and
sig. value implies sig. siblings at half thresholds). The sec-
ond algorithm, called ASWDR2, modifies the prediction
scheme of ASWDR1 by using a weighted linear prediction
function (based on a set of context values) to predict signif-
icant values at half thresholds.
Bothofthesealgorithmsimprovetherate-distortionper-
formance of WDR. They have a performance essentially
equal to the performance of Said and Pearlman’s SPIHT al-
gorithm [2] when arithmetic compression is not used. Al-
though SPIHT has better rate-distortion performance when
arithmetic compression is used, the perceptual quality of
SPIHT images at low bit rates is clearly worse than either
WDR orASWDRimages. ASWDRimagesareslightlybet-
ter perceptually than WDR images. An objective measure,
edge correlation, will be used to quantify these improved
perceptual qualities.
2. TEST RESULTS
We shall compare the performance of WDR, ASWDR1,
ASWDR2, and the SPIHT algorithm of Said and Pearlman
[2], in compressing the
bpp test images Airfield, Lena,
Goldhill, and Barbara. (Airfield was obtained from [3],
and the other three images were obtained from [4].) In Ta-
ble 1 we list average PSNR values for all these algorithms
at three different bit rates:
bpp. These bit rates represent a fairly wide range of com-
pression ratios from low to high compression. The data in
Table 1 show that when arithmetic compression is not em-
ployed,thenbothASWDR1 andASWDR2 are equalin per-
formance to SPIHT, while WDR is just slightly worse. But,
when arithmeticcompressionis employed,then SPIHT pro-
duces the highest PSNR values. This points to the need for
furtherresearchonthe best modelforperformingarithmetic
compression with ASWDR and WDR.
PSNR values, however, are not always a reliable criteria
for image fidelity. In Fig. 1, we show compressions of Bar-
bara at
the ASWDR1 and WDR images over the ones produced
by SPIHT. The SPIHT compression has severely distorted
Barbara’s left eye and erased large parts of the striping on
the tablecloth. The ASWDR1 and WDR compressions are
more difficult to distinguish. The ASWDR1 image does
preserve some fine details which WDR erases or distorts,
we show this in the bottom two figures. Preservation of de-
?
???? bpp,
?????? bpp, and
????????
?????? bpp. All observers of these images preferred
To appear in Proc. ICIP2000, September 10–13, 2000, Vancouver, Canada
c
?
IEEE 2000

Page 2

Error
0.5 bpp
PSNR
PSNR, AC
Edge Corr
Edge Corr, AC
0.25 bpp
PSNR
PSNR, AC
Edge Corr
Edge Corr, AC
0.125 bpp
PSNR
PSNR, AC
Edge Corr
Edge Corr, AC
? Method
WDR ASWDR1 ASWDR2 SPIHT
????????????????????????????
?????????????????????????
?????????? ????????
?
?
??????????????????
???
?
?
?
?
????????????????????????
????? ???????????????
?
???????
?????????
?
????
?
???????
???
?
????
?
????
?
???????
???????????????????? ???????????
????????????? ??????????????????
????????????????????
?
????? ?????? ????????????
Table1. AveragePSNRandedgecorrelationvaluesforfour
images, at three bit rates (
originals. (AC denotes the use of arithmetic compression.)
???? ,
?????? , and
???? bpp) for
?
bpp
tails at low bit rates is a vital property for applications such
as remote medical diagnosis via rapid transmission of com-
pressed images.
As an objective measure of preservation of edge detail
(which seems to better correspond to subjective quality rat-
ings than PSNR), we use edge correlations. To produce an
image that retains only edges, we do the following. A high-
pass filtered image is created by subtracting away the all-
lowpass subbandfroma 3-levelDaub 9/7wavelet transform
ofan image(andthenthe remainderis inversetransformed).
This is done for both the original image and a compressed
version. Both of these highpass filtered images have mean
values that are approximately zero. We define the edge cor-
relation
? by
?????
?
?
?
?
?
where
filteredversionofthecompressedimage,and
variance of the values of the highpass filtered version of the
original image. Thus
image captures the variation of edge details in the original
image.
?
?
? denotes the variance of the values of the highpass
?
?
? denotesthe
? measures how well the compressed
We list these edge correlations in Table 1. These data
show that the ASWDR algorithms are the best, with WDR
a close second, and SPIHT last. These results correspond
well with subjective ratings of the compressed images.
3. DESCRIPTION OF ASWDR1
The ASWDR1 algorithm is a simple modification of the
WDR algorithm ([1], [5]). Here is a
performing ASWDR1 on a grey-scale image:
? -step procedure for
Step 1. Perform a wavelet transform of the image.
We used a
? -level Daub 9/7 transform.
Step 2. Choose a scanning order for the transformed
image, whereby the transform values are scanned via
a linear ordering, say
???
?????
???
??????????
??? ?
?
where
scanning order is a zig-zag through subbands from
lower to higher [6]. Row-based scanning is used in
the low-pass/high-pass subbands and column-based
scanning is used in the high-pass/low-pass subbands.
?
is the number of pixels. In [1] and [5], the
Step 3. Choose an initial threshold,
least one transform value has magnitude less than or
equal to and all transform values have magnitudes
less than
? , such that at
?
??? .
Step 4 (Significance pass). Record positions for new
significant values: new indices
greater than or equal to the present threshold. Encode
these new significant indices using difference reduc-
tion ([1], [5]) .
?
for which

???
?!?" is
Step 5 (Refinement pass). Record refinement bits for
significant transform values determined using larger
threshold values. This generation of refinement bits
is the standard bit-plane encoding used in embedded
codecs ([6], [2]).
Step 6 (New scan order). Run through the signifi-
cant values at level
significant value, called a parent value, induces a set
of child values—four child values for all levels ex-
cept the last, and three child values for the last—
as described in the quad-tree definition in [2]. The
first part of the scan order at level
insignificant values lying among these child values.
Run through the insignificant values at level
wavelet transform. The second part of the scan or-
der at level
least oneof whose siblings is significant,lyingamong
the child values induced by these insignificant par-
ent values. The third part of the scan order at level
# in the wavelet transform. Each
#%$? contains the
# in the
#&$? contains the insignificant values, at
#'$
siblings are significant, lying among the child values
induced by these insignificant parent values. (Note:
Although this description is phrased as a three-pass
? contains the insignificant values, none of whose
2

Page 3

process through the level
formed in one pass by linking together three separate
chains at the end of the one pass.)
# subband, it can be per-
Step 7. Divide the present threshold by 2. Repeat
Steps 4–6 until either a bit budget is exhausted or a
distortion metric is satisfied.
When decoding, the steps above are recapitulated to pro-
duce a quantized output. Finally each quantized value is
rounded into the midpoint of the quantization bin that it lies
in, and an inverse wavelet transform (followed by round-
ing to 8-bit integer grey levels) produces the decompressed
image.
4. DESCRIPTION OF ASWDR2
The ASWDR2 algorithm utilizes a different version of Step
6 above. This alternate version of Step 6 consists of testing
whether an insignificant child value is likely to be signifi-
cant. The test is to use related values definedin [7]: For ver-
tical and horizontal subbands these values are (Up), (Left),
(Parent),(Diag.Cousin), andfordiagonalsubbandstheyare
(Up), (Left), (Parent), (Horiz. Cousin), and (Vert. Cousin).
A predicted magnitude
for the child value is computed
by:, where
lated values andare weights. For our (preliminary) ver-
sion of ASWDR2, we used the following weights:
?
??
?????
????????
are magnitudes of the re-
?
?
Horizontal and Vertical
???
????? ,
???
????? ,
???
???? ? ,
?????
?????
Diagonal
???
????? ,
???
????? ,
???
????? ,
?????
?
?????
??????? .
Although these weights were chosen blindly, they still pro-
duced essentially the same results as the ASWDR1 algo-
rithm. Further research is needed in order to produce a bet-
ter set of weights (most probably weights that depend on
level as well as subband), using minimization of an error
metric (such as least squares, as described in [7]) over a set
of test images.
5. LOSSLESS COMPRESSION
Using an integer-to-integer wavelet transform [8] in Step
1, these ASWDR algorithms can be converted to lossless
compression algorithms. In Table 2 we compare ASWDR1
(using the 2+2,2 transform [8] and arithmetic compression)
with thealgorithm [9]. As with the lossy case, the
ASWDR1 algorithm is better in preserving details (as mea-
sured by edge correlations) at the cost of poorer PSNR re-
sults. Unlike thealgorithm, however, the ASWDR1
algorithm enjoys the ROI capability.
?????
?????
Image
PSNR
Barbara,
Lena,
Edge Corr.
Barbara,
Lena,
Lossless bpp
Barbara
Lena
? Method
?????
ASWDR1
?????? bpp
????????????????
?????? bpp
???????????
?
?
?????? bpp
???
?
????
?
?
?????? bpp
???
???
?????
??????
?
?
4.21
?????
Table 2. Comparison of the progressive, lossless compres-
sion algorithms,
????? and ASWDR1.
6. REFERENCES
[1] J. Tian, R.O. Wells, Jr., “A lossy image codec based
on index coding”, IEEE Data Compression Confer-
ence, DCC ’96, p. 456, 1996.
[2] A. Said, W.A. Pearlman, “A new, fast, and efficient
image codec based on set partitioning in hierarchi-
cal trees”, IEEE Trans. on Circuits and Systems for
Video Technology, 6, 3, pp. 243–250, June 1996.
[3] http://www.image.cityu.edu.hk/imagedb/
[4] ftp://ipl.rpi.edu/pub/image/still/usc/gray/
[5] J. Tian, R.O. Wells, Jr., “Embedded image cod-
ing using wavelet-difference-reduction”, Wavelet
Image and Video Compression, P. Topiwala, ed.,
pp. 289–301. Kluwer Academic Publ., Norwell,
MA, 1998.
[6] J.M. Shapiro, “Embedded image coding using ze-
rotrees of wavelet coefficients”, IEEE Trans. Signal
Proc., Vol. 41, pp. 3445–3462,Dec. 1993.
[7] R.W. BuccigrossiandE.P.Simoncelli,“Imagecom-
pression via joint statistical characterization in the
wavelet domain”, IEEE Trans. on Image Proc.,
Vol. 8, pp. 1688–1701,1999.
[8] A.R. Calderbank, I. Daubechies, W. Sweldens, B.-
L. Yeo, “Wavelet transforms that map integers to
integers”, Applied and Computational Harmonic
Analysis, Vol. 5, pp. 332-369, 1998.
[9] A. Said,
resolution representation for lossless and lossy im-
age compression,”IEEE Trans. Image Proc., Vol. 5,
pp. 1303–1310, 1996.
W.A. Pearlman, “An image multi-
3

Page 4

(a) Original (8 bpp)(b) ASWDR1 (0.25 bpp)
(c) SPIHT (0.25 bpp)(d) WDR (0.25 bpp)
(e) ASWDR1 (0.25 bpp)(f) WDR (0.25 bpp)
Fig. 1. Compressions of Barbara image.
4