36IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 1, JANUARY 2011
Adaptive Sequential Prediction of Multidimensional
Signals With Applications to Lossless Image Coding
Xiaolin Wu, Fellow, IEEE, Guangtao Zhai, Member, IEEE, Xiaokang Yang, and
Wenjun Zhang, Senior Member, IEEE
Abstract—We investigate the problem of designing adaptive se-
quential linear predictors for the class of piecewise autoregressive
multidimensional signals, and adopt an approach of minimum de-
the support on which the predictor operates. The design objective
is to strike a balance between the bias and variance of the predic-
tion errors in the MDL criterion. The predictor design problem is
particularly interesting and challenging for multidimensional sig-
nals (e.g., images and videos) because of the increased degree of
freedom in choosing the predictor support. Our main result is a
new technique of sequentializing a multidimensional signal into
a sequence of nested contexts of increasing order to facilitate the
MDL search for the order and the support shape of the predictor,
and the sequentialization is made adaptive on a sample by sample
basis. The proposed MDL-based adaptive predictor is applied to
lossless image coding, and its performance is empirically estab-
lished to be the best among all the results that have been published
Index Terms—Autoregressive process, context modeling, lossless
compression, minimum description length (MDL), prediction.
as information theory, signal processing, machine learning,
economics/finance, statistics, etc. A series of seminal papers
of Rissanen established the intimate connection between pre-
diction and universal source coding –. In parallel to his
coding, Rissanen also proposed a constructive algorithm called
Context  for universal sequential prediction and coding.
This algorithm was originally developed and analyzed for
sequences, or 1-D signals. Specifically, a sequence of symbols
REDICTING the future of a sequence from its past is a
common and important problem in many disciplines such
Manuscript received September 29, 2009; revised March 14, 2010 and June
01, 2010; accepted June 02, 2010. Date of publication July 29, 2010; date of
and by NSFC (60932006, 61025005, 61001145) and the 111 Project (B07022).
The associate editor coordinating the review of this manuscript and approving
it for publication was Dr. Antonio Ortega.
X. Wu is with the Department of Electrical and Computer Engineering, Mc-
Master University, Hamilton, ON, L8G 4K1, Canada (e-mail: firstname.lastname@example.org-
G. Zhai is with the Department of Electrical and Computer Engineering, Mc-
Master University, Hamilton, ON, L8G4K1, Canada, and also with the Institute
of Image Communication and Information Processing Shanghai Jiao Tong Uni-
versity, Shanghai, 200240 China (e-mail: email@example.com).
formation Processing Shanghai Jiao Tong University, Shanghai 200240, China
Color versions of one or more of the figures in this paper are available online
Digital Object Identifier 10.1109/TIP.2010.2061860
drawn from a finite alphabet is communi-
cated to a receiver. Upon observing
algorithm Context predicts the probability distribution of the
,. The performance criterion of the
predictor is one of minimum description length (MDL) that
prevents data overfitting (context dilution).
Generalizing universal prediction, modeling and coding to
gineering value, because it offers a more principled approach
sion, denoising, restoration, classification, etc. But more than
two decades after the theory of universal prediction, modeling
and coding, its potential in previously mentioned applications
is yet to be fully realized. This is due to, we believe, an opera-
tional challenge, which remains largely unanswered. A multidi-
mensional signal has to be sequentialized to fit into Rissanen’s
framework of universal sequential prediction. For example, a
2-D image signal
, before algorithm Context, which was
devised for sequential prediction of a 1-D random process, can
be applied to
. The algorithm will make different predic-
predetermined, signal-independent sequentialization of
can obscure statistical dependencies among the pixels and de-
grade the prediction performance. Adding to the complexity of
the problem is that the sequentialization should be allowed to
change in pixel position
because the image signal
is often nonstationary.
This work is an attempt on algorithmic solutions for sequen-
universal source coding algorithm Context to multidimensional
signals by coupling it with an adaptive sequentialization mech-
anism, and design an adaptive sequential predictor whose order
imum description length. To make the algorithm computation-
ally tractable, we confine ourselves to linear predictors.
Linear prediction is effective to decorrelate stationary
Gaussian random processes, and is widely used in predictive
coding of multidimensional signals. The best performing linear
predictor for image coding in the literature is the one of least
squares that adapts itself on a pixel by pixel basis , . The
model parameters are estimated for each next sample
the past samples
. This parametric modeling technique assumes that the image
signal is piecewise autoregressive (PAR).
The existing PAR predictors ,  deal with the sequential-
ization of past samples for sequential prediction in some ad hoc
way, ignoring the critical design issues of selecting the model
pixels has to be mapped into a
that lie in a local window of
1057-7149/$26.00 © 2010 IEEE
WU et al.: ADAPTIVE SEQUENTIAL PREDICTION OF MULTIDIMENSIONAL SIGNALS 37
order and the spatial configuration of the 2-D model support.
of the PAR model is fixed throughout the sequen-
tial prediction process and chosen empirically. The 2-D model
support is simply a rectangular causal region of size
is centered at the next pixel
. The underlying assumption of
this design is that the closer a sample is to
more related to
it is. While being valid for many 1-D sources
(random sequences) such as audio, this assumption no longer
holds for multidimensional sources for sample dependencies in
natural signals are often anisotropic. This can be easily seen in
the case of images when examiningthe correlation matrix in the
area of edges and textures. For such instances a signal-indepen-
dent predictor support is clearly suboptimal because it brings ir-
relevant past samples and miss relevant ones in the PAR model
estimation, generating biases. The drawbacks of predetermined
and Boulanger  proposed a model selector for choosing the
size of square support according to local image statistics. In ,
Memon et al. studied the optimal order of scanning pixels in the
analyzed the influence of the pixel scan order on the context
support but did not discuss the design of optimal sequential pre-
dictors. In their analysis, the image was modeled as an isotropic
Gaussian randomfield, and accordinglythecontextsupportwas
made of the causal pixels within a fixed radius of the current
pixel, which are too restrictive as we argued previously. In ,
Akimov et al. proposed several pixel ordering schemes for con-
text tree modeling in arithmetic coding of color map images.
In this paper, we propose to use correlation instead of Eu-
clidean distance between
samples to form spatially nested predictor supports for different
orders of the PAR model. Since the samples are sequentialized
with correlation coefficients rather than Euclidean distance, the
support of the model will have an arbitrary spatial configuration
and may not even be connected, whatever the signal structures
stipulate. For each increment of the PAR model order, the
predictor support template grows by one pixel to include the
next highest correlated past sample. To estimate the PAR model
, we also propose a technique to choose a
causal training set of past samples and the associated support
templates. For a fixed order of the PAR model, not only is
the spatial configuration of the model support uniquely deter-
mined, also given is a training set for parameter estimation.
These training sets are chosen according to context classifi-
cation (template matching) rather than by picking some local
causal window of the next sample
order increases, the training set decreases in size, similarly
to the tree node split mechanism in algorithm Context. This
allows us to apply the MDL principle to regulate the model
order with respect to the size of the training set, so as to prevent
model overfitting. Although the universality of the proposed
technique is not proven in this paper, it significantly improves
the prediction performance of its predecessors using a fixed
in space/time the
andto sequentialize past
as in , . As the model
For ease and clarity of our discussions, the technical de-
velopments are presented mostly around 2-D image sources.
However, the proposed sequential prediction and modeling
techniques are general and can be readily applied to signals of
To validate the efficacy of the proposed MDL-based adap-
tive predictor, we apply it to lossless image coding and eval-
uate both the self-entropy of the prediction residuals and the
lossless code length with context modeling. An interesting and
important property of the proposed technique is that it tightly
couples predictor design and context modeling of the prediction
residuals. The 2-D support
of the MDL predictor for pixel
can be used as the modeling context for the prediction residual
, which drives adaptive arithmetic coding, can be
estimated from the same causal training set used to determine
. It should be stressed that the proposed predictor is causal and
the predictor coefficients are estimated using only past samples
that have similar casual neighborhoods as the current pixel, so
the encoding and decoding can be performed synchronously in
one-pass coding without any side information. In contrast, the
techniques of Takamura et al.  and Matsuda et al.  are
two-pass predictive coding schemes that transmit side informa-
tion about the predictor. Our new lossless technique achieves,
on average, 1.6 percent shorter code length than what was re-
technique has a high computational complexity. But it tightens
upper bounds for lossless code lengths of natural images, of-
fering new performance benchmarks for evaluating more prac-
In the remainder of the paper, the previously outlined new
technique of MDL-based sequential prediction of multidimen-
sional signals is developed in Section II. Algorithm details
for the construction of the proposed predictor are presented in
Section III. Experimental results are reported in Section IV,
including both self-entropy of prediction residuals and loss-
less code lengths of natural images when the proposed MDL
predictor is coupled with context modeling. This section also
presents thorough comparisons of the new technique with
almost all published lossless image compression methods,
demonstrating the progress made by this work. Section V
concludes this paper.
II. MDL-BASED ADAPTIVE PREDICTION
Linear predictors are widely used in signal compression, par-
ticularly in lossless compression –. The linear predictor
of the current sample
that is activated on a support
For 1-D signals such as audio the
-tap filter support
. For predictive image
is simply the
samples immediately preceding
coding the predictor support is 2-D. In the current practice
38IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 1, JANUARY 2011
Fig. 1. 2-D linear predictor of order 10 and the corresponding training sample
of predictive image coding, the 2-D support
sists of the
closest pixels to the current pixel
, being the th closest. Fig. 1 shows the 10-tap 2-D
for predictive image coding.
Given the spatial configuration of the 2-D predictor support
, thepredictor parameters
can be computed by solving the fol-
lowing minimization problem:
same spatial configuration as
expectation is taken over a set of sample vectors
where ’s are causal to
a key issue of our interest and will be cast in the light of MDL
The -norm of the prediction residuals is only one possible,
albeit commonly used, cost function for the prediction quality.
For data compression applications, the ideal objective is to min-
imize the entropy of the prediction residuals. But the design
of minimum-entropy predictor is computationally intractable in
general. Matsuda et al. proposed to search for the minimum-en-
tropy predictor by randomly perturbing the coefficients
. Fortunately, for most natural images the
residuals of linear predictors obey a Laplacian distribution ,
, hence, choosing
in (2) is equivalent to minimizing
the entropy of prediction residuals. The corresponding 1-norm
minimization problem can be solved by linear programming
. In practice, however, linear least-squares predictors are
most common because (2) can be solved far more efficiently by
One can make the predictor adaptive to local image statis-
tics by choosing space-varying training sets
the linear predictor can be designed on a pixel by pixel basis,
as proposed in , . In these papers, the causal training set
in (2) is a local window centered at the current pixel
marked in Fig. 1 in relation to the 2-D support
two critical issues were ignored in , , and they will be ad-
dressed in this paper. The first issue is how to determine the
order of the predictor
and the geometry of the 2-D support
the training set
to estimate the predictor coefficients. These
stands for the 2-D support ofpixels, having the
but with respect to . The
. The selection of the sample set is
. To the extreme,
questions are concerned with context modeling of the image
signal and the tradeoff between the estimation bias and variance
or the prevention of data overfitting. We view the predictor as
a 2-D piecewise autoregressive (PAR) image model and fit the
PAR model to the local image structure in the MDL principle
. Specifically, the objective function of our MDL problem
Given a 2-D support of the PAR model
and the training set
which the PAR model parameters
learnt, the first term
(measured by empirical entropy) of the prediction residual of
the PAR model, while the second term is the model cost per
symbol. The problem of MDL-based predictor design is then to
minimize (3) over all possible ,
Since an image is a 2-D signal, one needs to sequen-
tialize the pixels in the 2-D support
of the model order
. A heuristic sequentialization, which
was adopted by previous papers , , , is to traverse
the neighbors of the current pixel
and order them as such in the 2-D support
. For one-pass predictive coding,
the sequence of pixels
previously mentioned traversal fill outwards in a semicircle that
is causal to
. However, as we have discussed in Section I,
this scheme is suboptimal for anisotropic or/and periodic image
structures such as edges and textures. Given the predictor
, the resulting
, that are independent of
are not included in
To overcome the drawback of an image-independent prede-
termined shape of support
, we order the pixels in the 2-D support by their cor-
relations to the
, producing an ordered set
is the expected code length
, andcausal to.
in the optimization
by their Euclidian
generated by the
may include some pixels,
, whereas others
and adapt it to local features
that may not even be connected. As
optimal order of the PAR model in MDL sense,
order as more and more observations become available in the
MDL design of one-pass adaptive predictor.
of the PAR model of order
and . The
given by the cor-
increases in search for the
III. ALGORITHM DEVELOPMENT
In the design of the proposed MDL-based adaptive predictor,
we need to first estimate the correlations
and a neighboring pixel
images are nonstationary, we have to judiciously collect sam-
ples in the part of the image causal to
between the cur-
. Since most natural
when estimating .
WU et al.: ADAPTIVE SEQUENTIAL PREDICTION OF MULTIDIMENSIONAL SIGNALS 39
Fig. 2. Support and training set of the PAR model for a pixel being predicted.
(a) Example 1. (b) Example 2.
Define the context of pixel
causal to , with
to be the 2-D template of
an algorithm parameter
is the th closestpixel to . We form a training
of pixels whose contextssufficiently match
is a threshold. Or alternatively the training set
pixels whose contexts are the most nearest to
norm or norm. Then we estimate
using the sample vectors in the matched training set
these estimated correlation coefficients
adaptive nested 2-D supports
, can be obtained by (4).
Having developed a scheme of generating nested 2-D sup-
for increasing order
, we next propose an MDL approach of designing the
of the PAR model,
Fig. 3. Test images used in our experiments and comparison studies. From left
to right and top to bottom: “Lena,” “Barbara,” “Man,” “Boat,” “Harbor,” “Of-
fice,” “Cafe1,” “Cafe2,” “Sail,” “Fingerprint,” “Airplane,” “Coach,” “Fashion,”
“Bottle,” “Airfield,” and “Liver38.”
SELF ENTROPY (BITS PER PIXEL) COMPARISONS
PAR predictor. Given a
set of samples
we need to select an appropriate
to estimate the PAR model parameters
in (2). Note that the cardinality of
acts with the model order
in the MDL objective function (3).
In reminiscence to Rissanen’s algorithm Context , we refine
the sample set
by matching the higher order statistics of the
current context template
recursively define the training set
as increases. Specifically, we
40 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 20, NO. 1, JANUARY 2011
Fig. 4. Residual images for “Lena.” (a) MED. (b) GAP. (c) EDP. (d) MDL-PAR.
Fig. 5. Residual images for “Barbara.” (a) MED. (b) GAP. (c) EDP. (d) MDL-PAR.
Fig. 6. Residual images for “Bottle.” (a) MED. (b) GAP. (c) EDP. (d) MDL-PAR.
implying that fewer samples are available to learn the PAR
model of higher order. In algorithm Context, the match of con-
texts is exact (i.e.,
and property (9) follows. In our case, we relax the matching
threshold and allow
because it is highly unlikely to have
exactly matched templates in natural multidimensional signals
Given the 2-D support
PAR model coefficients
can be computed by (2), and the re-
sulting model can be used to predict
It is well known that predictive errors of many natural sig-
nals, such as images, can be satisfactorily fit to a Laplacian
distribution. In such cases, the estimate of the residual entropy
can be computed by calculating the variance
the prediction errors
for the samples in
and the training set, the
is the maximum amplitude of pixel values, and
Out of the technical development mentioned previously, an
interesting property manifests itself that is useful when the
proposed MDL predictor is applied in lossless signal compres-
sion. If the prediction residuals are to be arithmetic coded, then
in (11) naturally becomes an estimate of the conditional
in context and can be directly used to drive adap-
tive arithmetic coding of . In the next section, we will discuss
the application of the new technique in lossless image coding
and demonstrate its superior performance over the current state
of the art.
IV. EXPERIMENTAL RESULTS
To illustrate the adaptation power of the proposed
MDL-based predictor, we plot the 2-D predictor support
and the training set in relation to the current pixel
being predicted in Fig. 2. The pixel to be predicted is marked
in red, the predictor support and training set in green, where
the green intensity indicates the correlation value in (4) and the
degree of template matching in (8). For the periodic texture
pattern in Fig. 2(a), the proposed MDL design technique selects
a predictor support that consists of spatially disjoint past sam-
ples. The selected samples exhibit the local signal structure.
Likewise, the training set is spatially disjoint and anisotropic,
containing samples of similar local texture pattern. For the
second example in Fig. 2(b), where the pixel being predicted
is on an edge, both the predictor support and the training set
WU et al.: ADAPTIVE SEQUENTIAL PREDICTION OF MULTIDIMENSIONAL SIGNALS41
BIT RATES (BITS PER PIXEL) OF LOSSLESS IMAGE CODECS
are chosen adaptively along the edge direction. The statistical
consistency between the pixel being predicted and those in the
training set can be clearly seen in the figure.
To evaluate the performance of the proposed sequential pre-
diction technique, we compare the self-entropy of its residuals
with that of Li and Orchard’s edge directed predictor (EDP) ,
context-based predictors such as the median predictor (MED)
, , and the gradient adaptive prediction (GAP) used in
which is the best case reported in . The predictors are applied
to sixteen test images (shown in Fig. 3). The test set is chosen
to cover a large variety of natural images and, thus, lends statis-
tical significance to our experimental results. The self-entropies
tropy on average as compared to GAP, MED, and EDP.
The performance of an adaptive image predictor can be eval-
uated by its ability to capture fine image details, or equivalently
by the absence of image structures in the residual image. Let us
is evident that the residual images of the MDL-based PAR pre-
dictor contain the least visible signal structures. This property
suggests that the proposed technique can be an effective tool for
ural images achieved by the new MDL-based prediction tech-
nique coupled with context modeling as described in the end of
the proceeding section. We take this opportunity to summarize
the proposed MDL-based technique with all other existing loss-
less image codecs for which we have publicly available exe-
cutables. Their lossless bit rates are also tabulated in Table II. A
fairly large set of JPEG and other commonly used test images
are used (see Fig. 3), and they span a wide range in source en-
1The parameters of the predictor are set to ? ? ??, ?
? ??? in this
The MDL-PAR technique obtains shorter lossless code lengths
thanallother methods,although themarginovertheMRDtech-
nique , the second best in the comparison group, is quite
images “Lena” and “Barbara” can be losslessly compressed to
below 3.8 and 4.0 bits per pixel, respectively.
Theoretically, the source entropy of natural images still re-
mains an open problem. Nevertheless, this work improves the
known upper bounds and, hence, provides new benchmarks for
the evaluation of future lossless image codecs. In a note of prac-
tical interest, the gap from our new upper bound is 10% for the
JPEG 2000 lossless mode and 17.66% for the JPEG-LS stan-
dard on average.
An MDL-based sequential predictor for multidimensional
sources is devised. The main contribution is a novel correla-
tion-based sequentialization technique that makes the universal
coding algorithm Context, which was originally developed
for 1-D random processes, applicable to multidimensional
sources, though at the cost of high computational complexity.
The proposed MDL-based predictor is also applied to lossless
image coding and it obtains the best compression performance
up to now, improving the known upper bounds for achievable
lossless code lengths of natural images.
The authors would like to thank the associate editor for his
professionalism in handling the reviewing of this paper. The
authors would also like to thank an anonymous reviewer for
pointing out a subtle difference between algorithm Context and
their algorithm, and other three reviewers who insisted on the
inclusion of actual compression results.
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Xiaolin Wu (M’88–SM’96–F’11) received the B.Sc.
degree in computer science from Wuhan University,
Wuhan, China,in 1982, andthe Ph.D.degree in com-
puter science from the University of Calgary, Cal-
gary, AB, Canada, in 1988.
He started his academic career in 1988 and
has since been on the faculty of the University of
Western Ontario, New York Polytechnic University,
and currently McMaster University, Hamilton, ON,
Canada, where he is a Professor in the Depart-
ment of Electrical and Computer Engineering and
holds the NSERC-DALSA Industrial Research Chair in Digital Cinema. His
research interests include image processing, multimedia compression, joint
source-channel coding, multiple description coding, and network-aware visual
communication. He has published over 180 research papers and holds two
patents in these fields.
He is an Associate Editor of the IEEE TRANSACTIONS ON PROCESSING.
Guangtao Zhai (M’10) received the B.E. and M.E.
degrees from Shandong University, Shandong,
China, in 2001 and 2004, respectively, and the
Ph.D. degree from Shanghai Jiao Tong University,
Shanghai, China, in 2009.
From August 2006 to February 2007, he was an
Intern at the Institute for Infocomm Research, Sin-
gapore. From March 2007 to January 2008, he was
a Visiting Atudent at the School of Computer Engi-
neering, Nanyang Technological University, Singa-
pore. From October 2008 to April 2009, he was a
Visiting Scholar at the ECE Department, McMaster University, Hamilton, ON,
Canada, where he is currently a Post Doctoral Fellow. His research interests
include image and video processing, perceptual signal processing, and pattern
Xiaokang Yang (M’00–SM’04) received the B. S.
degree from Xiamen University, Xiamen, China, in
1994, the M.S. degree from the Chinese Academy of
gree from Shanghai Jiao Tong University, Shanghai,
He is currently a Full Professor and Deputy Di-
rector of the Institute of Image Communication and
Information Processing, Department of Electronic
Engineering, Shanghai Jiao Tong University. From
September 2000 to March 2002, he was a Research
Fellow in Centre for Signal Processing, Nanyang Technological University,
Singapore. From April 2002 to October 2004, he was a Research Scientist
with the Institute for Infocomm Research, Singapore. He has published over
80 refereed papers, and has filed six patents. His current research interests
include video processing and communication, media analysis and retrieval,
perceptual visual processing, and pattern recognition. He actively participates
in the International Standards such as MPEG-4, JVT, and MPEG-21.
Dr. Yang received the Microsoft Young Professorship Award 2006, the Best
Young Investigator Paper Award at IS&T/SPIE International Conference on
Video Communication and Image Processing (VCIP2003), and awards from
A-STAR and Tan Kah Kee foundations. He is a member of the Visual Signal
Systems Society. He was the Special Session Chair of Perceptual Visual Pro-
cessing of IEEE ICME2006. He is the local cochair of ChinaCom2007 and the
technical program co-chair of IEEE SiPS2007.
Wenjun Zhang (M’02–SM’10) received the B.S.,
M.S., and Ph.D. degrees in electronic engineering
from Shanghai Jiao Tong University, Shanghai,
China, in 1984, 1987, and 1989, respectively.
As a Group Leader, he was in charge of devel-
oping the first Chinese HDTV prototype system in
1998. He is a Changjiang Scholarship Professor in
the field of communications and electronic systems
with Shanghai Jiao Tong University. His research
interests include digital media processing and
transmission, video coding, wireless wideband, and