An iterative design algorithm for the joint design of complexity- and entropy-constrained subband quantizers and associated entropy coders is proposed. Unlike conventional subband design algorithms, the proposed algorithm does not require the use of various bit allocation algorithms. Multistage residual quantizers are employed here because they provide greater control of the complexity-performance tradeoffs, and also because they allow efficient and effective high-order statistical modeling. The resulting subband coder exploits statistical dependencies within subbands, across subbands, and across stages, mainly through complexity-constrained high-order entropy coding. Experimental results demonstrate that the complexity-rate-distortion performance of the new subband coder is exceptional

Chung, Wilson C.

Kossentini, Faouzi

Smith Mark J. T.

English

NASA Technical Reports Server

NASA-CR-Z04400  IV ">
SUBBAND IMAGE CODING WITH JOINTLY OPTIMIZED QUANTIZERS
Faouzi Kossentini, Wilson C. Chang, and Mark J. T. Smith
School of Electrical & Computer Engineering,
Georgia Institute of Technology, Atlanta GA 30332
ABSTRACT
An iterative design algorithm for the joint design of
complexity- and entropy-constrained subband quantiz-
ers and associated entropy coders is proposed. Un-
like conventional subband design algorithms, the pro-
posed algorithm does not require the use of various bit
allocation algorithms. Multistage residual quantizers
are employed here because they provide greater con-
trol of the complexity-performance tradeoffs, and also
because they allow efficient and effective high-order sta-
tistical modeling. The resulting subband coder exploits
statistical dependencies within subbands, across sub-
bands, and across stages, mainly through complexity-
constrained high-order entropy coding. Experimental
results demonstrate that the complexity-rate-distortion
performance of the new subband coder is exceptional.
1. Introduction
The conventional approach to subband image coding
has been to design separate optimal or near-optimal
quantizers and associated entropy coders for each of
the subband images. A bit allocation algorithm is then
used to distribute bits among the subbands [1].
The subband image coder proposed here is different
in that the subband quantizers and associated entropy
coders are optimized jointly within and across the sub-
bands in a complexity- and entropy-constrained frame-
work. The algorithm used to design the coder employs
multistage residual vector quantizers [2] which feature
very low complexity and memory, and provide much
greater control over both design and encoding com-
plexity. An important aspect of the algorithm is that
no explicit bit allocation is needed. The bits are in-
directly (but optimally) allocated among the subbands
during the design process. The new coder exploits both
statistical intra-band and inter-band dependencies si-
multaneously, mainly through complexity-constrained
high-order conditional entropy coding, as discussed in
the remainder of this paper.
This work was supported in part by the National Sci-
ence Foundation under contract MIP-9116113 and the Na-
tional Aeronautics and Space Administration.
S tl tl
Ji il Jpid
-..
A
N
A
X L
y
S
1
i
J
Figure 1: Basic block diagram of the subband eneoder
2. System Description
As shown in Figure 1, the input signal is first decom-
posed into M subbands using an analysis transfor-
mation. Each subband is then encoded using a se-
quence of Pm (1 _< m _< M) residual vector quanti-
zation (RVQ) fixed length encoders. Any vector quan-
tization (VQ) encoder can be used, although we find
RVQs to achieve excellent rate-distortion-complexity
performance [3]. The output symbol for each stage
vector quantizer is fed into an entropy coder driven by
a high-order stage statistical model that is controlled
by a finite state machine (FSM). The FSM allows the
statistical model to switch among several zero-order
conditional models (represented by first order proba-
bilities) based on the state transitions. In this work,
a nonlinear mapping F given by u = F(sl, s2,..., sn),
where sl, s2,..., sn are n previously coded symbols or
outputs of some stage quantizers, will be used to de-
termine the conditioning state u. How to construct the
best mapping F will be described in Section 4. Finally,
the output bits of the entropy coders are combined to-
gether and sent to the channel. Since only previously
coded symbols are used by the FSM, no side informa-
tion is necessary and the decoder can track the state of
the encoder.
https://ntrs.nasa.gov/search.jsp?R=19970019704 2020-06-16T02:00:34+00:00Z
3. The Design Algorithm
Givenfixedanalysis/synthesistransformations,thepro-
posedesignalgorithmminimizestheexpectedistor-
tion E [d(X,J()], where X is the input and X is the
k J
output, subject to a constraint on the overall entropy
of the product of the M subband VQs. This design
algorithm is an iterative descent algorithm based on a
Lagrangian minimization, and is a generalization of the
entropy-constrained algorithms described in [4, 2, 5].
Given a fixed Lagrangian parameter A, the algorithm
attempts to satisfy simultaneously optimality condi-
tions, requiring the subband encoders, d_coders, and
entropy coders be designed jointly. Details of the op-
timality conditions and convergence of the algorithm
can be found in [3].
The parameter A is chosen based on the overall
rate and distortion of the subbands, and is used in the
entropy-constrained design of all subband quantizers.
Therefore, explicit bit allocation is not needed in the
design process. In fact, it can be shown [3, 6] that main-
taining the same slope A for all subband operational
R(D) curves results in a locally optimal allocation of
bits.
4. Complexity Issues
The complexity and memory associated with the de-
sign algorithm grows exponentially as a function of the
quantization and entropy coding parameters. To re-
duce them substantially, constrained VQs must gen-
erally be employed. In this work, we choose to use
multistage residual vector quantizers, mainly because
they require relatively low encoding complexity and
memory, and because they simplify the design pro-
cess by providing greater control over the complexity-
performance tradeoffs.
Asshown in [3], optimal encoding requires that the
synthesis transformation be embedded in the design
loop, which can result in much larger computational
and memory requirements. However, experimental re-
sults show that the subbands can be encoded accu-
rately by minimizing the distortion between the input
and the output of the subband quantizers (indepen-
dently) instead of minimizing the overall distortion of
the analysis-quantizer-synthesis system.
Exhaustive searching, which is generally necessary
for optimal encoding, can be circumvented effectively
by exploiting the multistage residual structure and us-
ing tree-structured searching techniques such as the
(M,L) algorithm. Such algorithms can be used to
search the stage codebooks in all subbands with rea-
sonable computational requirements. In particular, we
choose to use the dynamic M-search algorithm [7], which
employs a thresholding technique to decide the best
number of paths that should be saved at each RVQ
stage. Dynamic M-search provides a flexible way of
subband stage 1 A L
AK AM
sub, and stage 2
Figure 2: Inter-stage, inter-band, and intra-band con-
ditioning scheme within an image.
trading complexity for performance, and it can achieve
rate-distortion performance that is very close to that
of exhaustive searching while requiring only 20%-50%
more computations than sequential searching.
Optimal decoding requires a 2-dimensional opti-
mization procedure (see Figure 1) which consists of
using the iterative Gauss-Seidel algorithm [5] to mini-
mize the average distortion between the input and the
synthesized reproduction of all stage codebooks in all
subbands. Although the joint decoding optimization is
potentially more demanding than that associated with
non-constrained quantizers, its complexity can be dras-
tically reduced by, for example, grouping neighboring
stage eodebooks in neighboring subbands and jointly
optimizing each group independently. This typically
results in less than a 0.10 dB loss in PSNR.
The most important advantage of the multistage
residual structure is that it can substantially reduce
the complexity and memory by making the output al-
phabet of the stage quantizers small (e.g., 2, 3, or 4).
For example, a 4-stage RVQ with 4 code vectors per
stage codebook using a second-order conditional en-
tropy coder for each stage, generally requires that 64
probabilities per stage be computed and stored. For
a single-stage conventional VQ with 256 code vectors,
as many as 2563 probabilities may need to be com-
puted and stored. Multistage RVQs provide another
dimension upon which to capitalize. To illustrate this
point, Figure 2 shows the inter-stage, inter-band, and
intra-band conditioning scheme used in the system.
Each image shown is a multistage approximation of
the input image, and statistical dependencies among
these images generally exist. For each stage (re, p)
in each subband m, a 5-dimensional initial region of
support 7¢m,p containing a sufficiently large number
Rrn,p of conditioning symbols, or previous outputs of
fixed-lengthstageencoders,is first chosen.Sinceusing
the regionof support7¢,_,p in the conditioning pro-
cess generally results in unbearable complexity, we lo-
cate the nm,p, n,_,p << R,_,p, conditioning symbols
sl,...,s n'_',p such that the nm,pth order conditional
entropy H(Jm,p Is1,..., s ...... ) is minimized. To do this
we build a tree where the levels represent the orders (or
number of conditioning symbols) and the branches rep-
resent the possible combinations of conditioning sym-
bols at each level. As is described in [8], this tree is
symmetric, which can simplify the search process. Our
experimental results show that dynamic M-search pro-
vides an excellent balance between conditional entropy
and search complexity.
Since the objective is to minimize the average en-
tropy of all stage statistical models in all subbands
given a fixed level of complexity and memory of the
joint entropy coder, our approach is to first build a tree
M
with _,_=1 P-_ branches, where P,_ is the number of
stage codebooks in the rnth subband. Each branch is a
unary tree of length Lm,p, where Lm,p is the number of
complexity-entropy pairs. The dynamic M-search al-
gorithm is used to find the best n,_,p (1 < nm,p <_ Lm,p)
conditioning symbols given Rp,,_ conditioning symbols.
For each complexity-entropy pair, complexity is given
by A/m,p = Sin,pEru,p, where Sm,p is the number of all
combinations of realizations of the conditioning sym-
bols and Nm,p is the output alphabet size of stage p
in band m. Once all complexity-entropy pairs are ob-
tained, we then use the generalized BFOS algorithm
[9] to minimize the overall output entropy subject to
a constraint .Af max Oil the total number of conditional
probabilities, which is used here as a measure of com-
plexity and memory.
The FSM statistical model for each stage (re, p)
employs a mapping F to determine the state given
nm,p available symbols. The mapping F is one-to-one
and is actually given by a table that contains the num-
bers 0, 1,..., Sm,p - 1, representing each of the possible
combinations. As discussed in [10], a large number of
the tables representing the Sm,p states are usually ei-
ther not populated or scarcely populated. In addition
to this inefficiency, some of the empty tables may be
visited during actual encoding even though they were
never visited during the design process. This is the
so-called empty state problem that often arises in FSM
design. To address this, we use the PNN algorithm [11],
as described in [12], to drastically reduce the number
of states while still bounding the loss in entropy per-
formance to 1%. The PNN algorithm used here first
merges all of the empty states with the least probable
state into one conditioning state, thereby completely
removing empty states. Then, the two conditioning
states resulting in the lowest increase in entropy (when
merged) are combined into one conditioning state, and
so on until only one state, which represents one table
of first-order probabilities, is obtained. As described
M
above, another tree with _rn=l Pm branches is built,
where the nodes now represent complexity-entropy pairs
obtained by the PNN algorithm. The BFOS algorithm
is again used to minimize the overall output entropy
subject to the use of a much smaller number of condi-
tional probabilities.
Quantizing the conditioning states has the addi-
tional advantage that the stage statistical model orders
can be allowed to grow to relatively large numbers,
which tends to lower the overall entropy with only a
small increase in encoding complexity. Moreover, the
merging process improves the robustness of the sub-
band coder because only global statistics are carried
through, and the possibility of a strong mismatch be-
tween the test sequence and the coder is less likely.
5. Experimental Results
Several images of size 512 x 512 (excluding the test im-
ages) were used to design jointly optimized complexity-
and entropy-constrained subband residual scalar quan-
tization (RSQ) codebooks (i.e. the vector size is 1 x 1).
In the proposed framework, the use of larger vector
sizes resulted in a large increase in complexity but no
increase in rate-distortion performance.
For analysis/synthesis, we employ a 3-level bal-
anced tree-structured IIR allpass polyphase filter bank
as described in [13], resulting in 64 uniform subbands.
For this implementation, all stage codebooks in alI sub-
bands contain 3 scalars. The initial maximum allowed
number of conditional probabilities N m_x is set to 2048.
After using the BFOS algorithm, we employed the PNN
algorithm to populate another tree as described be-
fore. The BFOS algorithm is used again to locate the
best numbers of conditioning states for each stage in
each subband subject to using a maximum number
of 256 probabilities. The output of each of the stage
fixed-length RSQ encoders is encoded using an adap-
tive arithmetic coder driven by the corresponding FSM
statistical model just generated.
For each rate-distortion point, the total memory
required to store the subband RSQ codebooks and as-
sociated mapping tables as well as tables of conditional
probabilities is approximately 1.5 kilobytes. For en-
coding (including analysis) using dynamic M-search,
approximately 8 multiplies and 11 adds per pixel are
required. Only 3 multiplies and 9 adds are required for
decoding (including synthesis). In this example, the
design time is approximately 4 CPU hours on a Sun
Sparc 10 workstation. Not only are the complexity
and memory relatively small, but the performance is
exceptional. For instance, the PSNRs obtained for the
image LENA at 0.50 and 0.25 bpp are approximately
37 and 34.1 dB, respectively. Moreover, the subjective
quality is also very good. Figure 3 shows the coding
result of the image LENA at a bit rate of 0.25 bpp using
(a) OR.IGINAL
(b) IIR/RSQ SUBBAND CODER
Figure 3: The LENA image coded using the IIR/KSQ
SUBBAND CODER. The bit rate is 0.25 bpp and the
PSNR is 34.07 dB.
the subband coder. This is almost 4 dB better than
the JPEG standard at the same rate.
These experimental results show that the proposed
subband coder can achieve very good compression re-
sults while maintaining relatively low complexity and
memory. Moreover, it also compares favorably with
standard JPEG in terms of complexity. The proposed
coder does require more encoding/decoding computa-
tions, but multiplication-free implementations of it [14]
can achieve complexity very close to that of JPEG with
only a small loss (less than 0.50 dB) in quality.
6. References
[1] J. W. Woods, ed., Subband Image Coding. Nor-
well, MA: Kluwer Academic Publishers, 1991.
[2] F. Kossentini, M. Smith, and C. Barnes,
"Entropy-constrained residual vector quantiza-
tion," in Proc. IEEE ICASSP, vol. V, (Minneapo-
lis, MN, USA), pp. 598-601, Apr. 1993.
[3] F. Kossentini, W. Chung, and M. Smith, "A
jointly optimized subband coder," Submitted to
Trans. on Image Processing, July 1994.
[4] P. A. Chou, T. Lookabaugh, and R. M. Gray,
"Entropy-constrained vector quantization," IEEE
Trans. on ASSP, vol. ASSP-37(1), pp. 31-42, Jan.
1989.
[5] F. Kossentini, M. Smith, and C. Barnes, "Neces-
sary conditions for the optimality of variable rate
residual vector quantizers," Submitted to Trans.
on Information Theory in June 1993. Revised in
May 1994.
[6] K. Ramchandran and M. Vetterli, "Best wavelet
packet bases in a rate-distortion sense," IEEE
Trans. on Image Processing, vol. 2, pp. 160-174,
April 1993.
[7] F. Kossentini and M. Smith, "A fast searching
technique for residual vector quantizers," Signal
Processing Letters, vol. 1, pp. 114-116, July 1994.
[8] F. Kossentini, W. Chung, and M. Smith, "Con-
ditional entropy-constrained residual VQ with ap-
plication to image coding," Submitted to Trans. on
Image Processing, July 1994.
[9] E. A. Riskin, "Optimal bit allocation via the gen-
erated BFOS algorithm," IEEE Trans. on Infor-
mation Theory, vol. 37, pp. 400-402, Mar. 1991.
[10] S. M. Let, T. C. Chen, and K. H. Tzou, "Subband
HDTV coding using high-order conditional statis-
tics," IEEE Journal on Selected Areas in Commu-
nications, vol. 11, pp. 65-76, Jan. 1993.
[11] W. H. Equitz, "New vector quantization cluster-
ing algorithm," IEEE Trans. on ASSP, vol. 37,
pp. 1568-1575, Oct. 1989.
[12] F. Kossentini, W. Chung and M. Smith, "Subband
coding of color images with multiplierless encoders
and decoders," submitted to IEEE Int. Sym. on
Circuits and Systems, (Seattle, WA), May 1995.
[13] M. Smith and S. Eddins, "Analysis/synthesis tech-
niques for subband image coding," IEEE Trans.
on ASSP, vol. 38, pp. 1446-1456, Aug. 1991.
[14] F. Kossentini, W. Chung, and M. Smith, "Sub-
band coding of color images with multiplierless
encoders and decoders," in Abstract Submitted to
IEEE ISCAS, (Seattle, WA, USA), May 1995.


Subband Image Coding with Jointly Optimized Quantizers

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server