BAYESIAN SEPARATION OF DOCUMENT IMAGES WITH HIDDEN

MARKOV MODEL

Feng Su

†∗

, Ali Mohammad-Djafari

†

†

Laboratoire des Signaux et Systemes, UMR 8506 (CNRS-Supelec-UPS)

Supelec, Plateau de Moulon, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France

∗

State Key Laboratory for Novel Software Technology, Nanjing University, 210093 Nanjing, P. R. China

Keywords:

Blind Source Separation, document image, Bayesian estimation, HMM, MCMC.

Abstract:

In this paper we consider the problem of separating noisy instantaneous linear mixtures of document images in

the Bayesian framework. The source image is modeled hierarchically by a latent labeling process representing

the common classiﬁcations of document objects among different color channels and the intensity process of

pixels given the class labels. A Potts Markov random ﬁeld is used to model regional regularity of the classi-

ﬁcation labels inside object regions. Local dependency between neighboring pixels can also be accounted by

smoothness constraint on their intensities. Within the Bayesian approach, all unknowns including the source,

the classiﬁcation, the mixing coefﬁcients and the distribution parameters of these variables are estimated from

their posterior laws. The corresponding Bayesian computations are done by MCMC sampling algorithm. Re-

sults from experiments on synthetic and real image mixtures are presented to illustrate the performance of the

proposed method.

1 INTRODUCTION

Blind source separation (BSS) is an active research

topic of signal and image processing in recent years.

It considers separating a set of unknown signals from

their observed mixtures, with reasonable assumptions

of the form of the mixing process: linear or nonlin-

ear, instantaneous or convoluting, under or over de-

termined, noisy or noiseless, and so on. However, in

all cases the mixing coefﬁcients remain unknown and

have to be estimated as well as original source signals.

Various methods and models have been proposed

for BSS task, among which Principal Component

Analysis (PCA) seeks orthogonal directions of max-

imum variance exhibited by the data as source axes,

while Independent Component Analysis (ICA) (Hy-

varinen et al., 2001), in its basic form, assumes

statistical independency of sources and linear mix-

ing process and consists of seeking an inverse linear

transformation matrix applying on the data to achieve

maximum mutual independencybetween output com-

ponents. Both methods exploits basic statistical char-

acteristics of source signals to achieve the separation,

which makes them well generalizable and robust in

cases that as few prerequisite assumptions as uncor-

relatedness or independency can be made about the

source. Some variant algorithms are also proposed to

adapt to certain relaxation of model assumptions like

nonlinearity or noises (Harmeling, 2003; Almeida,

2005). However, in many other cases, we may ﬁnd

the availability or the needs of various types of prior

information to regulate the essentially ill-posed BSS

problem. Compared with PCA and ICA, Bayesian

framework allows convenient introduction of these

prior constraints about the sources and the mixing

coefﬁcients, and more important, supports ﬂexible

structuring and integrating multiple hierarchical clues

for separation purpose.

In the ﬁeld of image processing, BSS approaches

are being widely employed to separate or seg-

ment mixed images observed from, for example,

satelite and hyper-spectral imaging (Snoussi and

Mohammad-Djafari, 2004; Parra et al., 2000; Macias-

Macias et al., 2003), medical imaging (Calhoun and

Adali, 2006; Snoussi and Calhoun, 2005), and other

superimpositions of natural images (Bronstein et al.,

2005; Castella and Pesquet, 2004).

This paper focuses on one speciﬁc type of im-

151

Su F. and Mohammad-Djafari A. (2007).

BAYESIAN SEPARATION OF DOCUMENT IMAGES WITH HIDDEN MARKOV MODEL.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications, pages 151-156

DOI: 10.5220/0002064601510156

Copyright

c

SciTePress

ages - document images, where superimposition of

two images usually appears as a major type of degra-

dation encountered in digitization (Sharma, 2001) or

ancient documents (Drira, 2006). The former degra-

dation usually occurs as artifact during scanning a

double-sided document when the text on the back-

side printing shows through the non-opaque medium

and are mixed with the foreside text. Fig.1a shows

one such example. The latter cause of text superim-

position, usually called bleed-through, can often be

observed in old documentations due to ink blurring

or penetrating as illustrated by Fig.1b. Other forms

of overlapped patterns, like underwriting and water-

marks, are also common. Though the actual under-

lying mixing process may be quite complicated and

diverse in various mixture forms, the linear mixing

model usually serves as a resonable approximation

and beneﬁts analytical and computational simplicity,

thus is adopted in most document separation cases.

To separate document image mixtures, the com-

mon PCA and ICA algorithms can be used and have

shown their effectiveness in detecting independent

document features like watermarks, as inspected in

(Tonazzini et al., 2004) where each source was con-

sidered as random signal sequence in a whole without

further internal structuring. The Bayesian framework

has also been used before for document separation as

in (Tonazzini et al., 2006), where the source is mod-

eled by a Markov Random Field on the pixel values

to account for local smoothness inside one object, as

well as an extra line process enforcing the discontinu-

ity at object edges.

In this contribution, we propose a solution to

jointly separate and segment linearly mixed document

images. Besides considering the mixture in single

grayscale channel, we address the joint separation of

multi-channel mixture of multiple sources. In section

2, we give the probability formulation of the problem.

In section 3, the algorithm of Bayesian estimation for

model parameters is described. In section 4, simula-

tion results of the proposed algorithm are shown on

both synthetic and real images.

(a) (b)

Figure 1: Examples of mixed document images: a) show-

through mixture; b) bleed-through mixture.

2 MODEL ASSUMPTION AND

FORMULATION

Document images are created by various digitization

methods from vast types of documentation. Com-

monly, a color scanner can be used to produce three

different views of one document in the red, green,

and blue channels. With detectors working in non-

visible wavelengths such as infrared and ultraviolet,

even more information channels of data can be ob-

tained, depending on the object of interest in docu-

ments.

Given observations of M different mixtures, either

in grayscale or multiple channels, our work is thus

to obtain N corresponding source images (normally

M > N) in the same pixel format as the observations.

2.1 Data Model

In this work, the observations are M registered im-

ages (X

i

)

i=1...M

, which are deﬁned on the same set

of pixels R : X

i

= {x

i

(r)}

r∈R

. The observations are

noisy linear instantaneous mixture of N source im-

ages (S

j

)

j=1...N

also deﬁned on R , following the data

generation model given by:

x(r) = As(r) + n(r) r ∈ R (1)

where A = (a

ij

)

M×N

is the unknown mixing ma-

trix, n(r) is a set of independent zero-mean white

Gaussian noise for each observation with variance

σ

2

ε

= (σ

2

ε1

. . . σ

2

εM

), x(r) and s(r) are the observa-

tion and source vector at pixel r respectively. Let

S = {s(r), r ∈ R }, X = {x(r), r ∈ R }, and denote

the noise covariance matrix by R

ε

= diag[σ

2

ε1

. . . σ

2

εM

],

we have the Gaussian distribution for the observations

given the sources and the mixing parameters:

p(X|S, A, R

ε

) =

∏

r

N (As(r), R

ε

) (2)

2.2 Source Model

We model the distribution of pixel intensity for each

source images (and for each color channel) by a Mix-

ture of Gaussians (MoG), whose components cor-

respond to each object type (or class) that appears

roughly equal pixel values. For example, the simplest

model may consist of two components, one for fore-

ground text and the other for background blank. Fur-

thermore, to allow imposing constraints on distribu-

tion of class labels, for every source S

j

we represent

the class labels by a set of discrete hidden variables

Z

j

= {z

j

(r), r ∈ R } with z

j

(r) taking values from

{1, . . . , K

j

}, where K

j

is the total number of classes

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

152

in image S

j

. In the following, we assume all {K

j

}

equal to the same value K.

Given pixel labels, pixels of different classes can

be reasonably assumed independent, while concern-

ing the pixels inside a given class, there are usually

two choices:

1. We may assume pixel intensities are conditionally

independent given their labels;

2. Alternatively, we may explicitly take into account

the local dependency between neighboring pixels

of same class.

In the ﬁrst choice, the distribution of pixel r in the

j

th

source is modeled by:

p(s

j

(r)|z

j

(r) = k) = N (µ

jk

, σ

2

jk

) (3)

where µ

jk

and σ

2

jk

are the mean and variance of the

k

th

Gaussian component of the j

th

source. Assuming

independency between different sources and denoting

the set of labels corresponding to every source by Z =

{Z

j

, j = 1. . . N}, we have:

p(S|Z) =

∏

j

∏

k

∏

{r:z

j

(r)=k}

p(s

j

(r)|z

j

(r) = k)

which is by (3) also a Gaussian and spatially separable

on r.

In the second choice, the local dependency can be

accounted by extra smoothness constraints, like the

mean value, between neighboring pixels. We ﬁrst as-

sign a binary valued contour ﬂag q

j

(r) for every pixel

r of every source j, which is deterministicly computed

by:

q

j

(r) =

1 if z

j

(r

′

) = z

j

(r), ∀r

′

∈ V (r)

0 else

where V (r) denotes the neighbor sites of the site r.

Then, based on the value of the contour ﬂag and

possibly current values of the neighboring pixels, the

distribution of intensity of individual pixel is formu-

lated as:

p(s

j

(r)|z

j

(r) = k, s

j

(r

′

), r

′

∈ V (r)) = N ( ¯s

j

(r),

¯

σ

2

j

(r))

(4)

with,

¯s

j

(r) = q

j

(r)µ

jk

+ (1− q

j

(r))

1

|V

jk

(r)|

∑

r

′

∈V

jk

(r)

s

j

(r

′

)

¯

σ

2

j

(r) = q

j

(r)σ

2

jk

+ (1− q

j

(r))σ

2

j

where V

jk

(r) denotes the intersection of V (r) with

the site set R

jk

= {r : z

j

(r) = k}, σ

2

j

is the a prior

variance of pixel values inside a region. Eqn.(4) states

that at the contour pixel intensities follow the Gauss

distribution whose parameters are determined by the

class labels as (3), while inside a region the distri-

bution parameters are computed from the neighbor-

ing pixels. Note that under this assumption, p(S|Z)

is no longer separable on r, but with the paral-

lel Gibbs sampling scheme proposed in (Feron and

Mohammad-Djafari, 2005), it can still be simulated

efﬁciently.

As a commonly observed property of visual ob-

jects, pixels belonging to the same object usually con-

nect to each other in a neighborhood of space, form-

ing several connected regions of uniformly classiﬁed

pixels, for instance, the multiple components consti-

tuting a text. By class labels deﬁned earlier, this im-

plies regional smoothness of the spatial distribution of

class labels. This can be naturally modeled by a prior

Potts Markov Random Field for every label process

z

j

(r):

p(z

j

(r), r ∈ R ) ∝ exp

β

j

∑

r∈R

∑

r

′

∈V (r)

δ(z

j

(r) − z

j

(r

′

))

(5)

The parameter β reﬂects the degree of smoothing in-

teractions between pixels and controls the expected

size of the regions. In our work, all {β

j

} are as-

sumed equal and assigned an empirical value within

[1.5, 2.0].

2.3 Multiple Channels

When multi-channel image data are considered, there

are multiple options for the processing model. We can

perform separation of sources independently in each

channel and by some measures merge the results in

the end. Or, we may consider joint demixing for all

channels. In the latter case, the mixing model can still

have more alternatives:

a) all channels are equally mixed with the same mix-

ing matrix;

b) the mixing occurs separately in each channel with

different mixing matrices;

c) cross-channel mixing is assumed to be present.

In the case of a), samples from different channels of

the same observation can be concatenated for esti-

mation of the mixing coefﬁcients, which is similar

to the monochrome case. In the case of c), an ex-

panded mixing matrix A

ML×NL

(supposing L chan-

nels) is used for all channels of all sources.

In this work, we assume the model b), where

the mixing in different channels are mutually in-

dependent and with their own separate coefﬁ-

cients. Thus, in RGB color format, the sources

BAYESIAN SEPARATION OF DOCUMENT IMAGES WITH HIDDEN MARKOV MODEL

153

and observations are actually {S

r

j

, S

g

j

, S

b

j

}

j=1...N

and

{X

r

i

, X

g

i

, X

b

i

}

i=1...M

. Correspondingly, there are

{A

r

, R

r

ε

, µ

r

jk

, . . . A

b

, R

b

ε

, µ

b

jk

} and so on. But for each

source S

j

, only one classiﬁcation ﬁeld Z

j

is main-

tained and shared by all channels, as a natural way

to enforce the common segmentation among dif-

ferent channels. This two-level hierarchical source

model, which also facilitates introducing segmenta-

tion constraints like discontinuity and local regional

dependency, is the main difference with the work of

(Tonazzini et al., 2006), where a one-level MRF mod-

eling of sources is deﬁned on the single-channel pixel

intensities along with an explicit binary edge process.

3 BAYESIAN ESTIMATION OF

MODEL PARAMETERS

The unknown variables we want to estimate in the

models givenaboveare {S, Z, A, θ}, θ representing all

hyperparameters. The Bayesian estimation approach

consists of deriving the posterior distribution of all

the unknowns given the observation and then based

on this distribution, employing appropriate estimators

such as Maximum A Posteriori (MAP) or the Poste-

rior Means (PM) for them. With our model assump-

tions, this posterior distribution can be expressed as:

p(S, Z, θ|X) ∝ p(X|S, A, R

ε

)p(S|Z, θ

s

)p(Z)p(θ)

(6)

where, θ

s

= {(µ

jk

, σ

2

jk

), j = 1. . . N, k = 1. . . K} and

θ = {A, R

ε

, θ

s

}.

3.1 Prior Assignments for Model

Parameters

According to the linear mixing model and all

Gaussian assumptions, we choose correspondingcon-

jugate priors for model hyperparameters.

• Gaussian for source means

µ

jk

∼ N (µ

k0

, σ

2

k0

)

• Inverse Gamma for source variances

σ

2

jk

∼ I G(α

k0

, β

k0

)

• Inverse Wishart for noise covariance

R

−1

ε

∼ W

i

(α

ε

0

, β

ε

0

)

In this work, we assign uniform prior to A for sim-

plicity and no preference of the mixing coefﬁcients,

while in other cases prior distributions like Gamma

may be used to enforce positivity.

3.2 Estimation by MCMC Sampling

Given the joint a posteriori distribution (6) of all un-

known variables, we use the Posterior Means as the

estimation for them. Since direct integration over z

is intractable, MCMC methods are employed in the

actual Bayesian computations. In our work, a Gibbs

sampling algorithm is used to generate a set of sam-

ples for every variable to be estimated, according to

its full-conditional a posteriori distribution given all

other variables ﬁxed to their current values. Then, af-

ter certain burn-in runs, sample means from further

iterations are used as the Posterior Means estimation

for the unknowns. The algorithm takes the form:

Repeat until converge,

1. simulate S

′

∼ p(S|Z, θ, X)

2. simulate Z

′

∼ p(Z|S

′

, θ, X)

3. simulate θ

′

∼ p(θ|Z

′

, S

′

, X)

Below we give the expressions of related conditional

probability distributions.

• Sampling Z ∼ p(Z|X, S, θ) ∝ p(X|Z, θ)p(Z):

p(X|Z, θ) =

∏

r

p(x(r)|z(r), θ)

=

∏

r

N (Am

z(r)

, AΣ

z(r)

A

t

+ R

ε

)

where, m

z(r)

= [µ

1z

1

(r)

, . . . , µ

Nz

N

(r)

]

t

and Σ

z(r)

=

diag[σ

2

1z

1

(r)

, . . . , σ

2

Nz

N

(r)

].

Notice p(Z) =

∏

N

j=1

p(z

j

), and as mentioned ear-

lier, p(z

j

) takes the form of Potts MRF as (5).

An inner Gibbs sampling is then used to simulate

z

j

with the likelihood p(x(r)|z(r), A, θ) marginal-

ized over all conﬁgurations of {z

j

′

(r), j

′

6= j}.

• Sampling S ∼ p(S|X, Z, θ):

p(S|X, Z, θ) ∝ p(X|S, A, R

ε

)p(S|Z, θ)

=

∏

r

N (m

apost

s

(r), R

apost

s

(r))

R

apost

s

(r) =

h

A

t

R

−1

ε

A+ Σ

−1

z(r)

i

−1

m

apost

s

(r) = R

apost

s

(r)

h

A

t

R

−1

ε

x(r) + Σ

−1

z(r)

m

z(r)

i

• Sampling R

ε

:

p(R

ε

|X, S, A) ∝ p(X|S, A, R

ε

)p(R

ε

)

Considering we assign an inverse Wishart distri-

bution to p(R

ε

), which is conjugate prior for the

likelihood (2), R

ε

is a posteriori sampled from:

R

−1

ε

∼ W

i

(α

ε

, β

ε

)

α

ε

=

1

2

(|R |− n), β

ε

=

1

2

|R |(R

xx

− R

xs

R

−1

ss

R

t

xs

)

where, the sample statistics R

xx

=

1

|R |

∑

r

x

r

x

t

r

,

R

xs

=

1

|R |

∑

r

x

r

s

t

r

, R

ss

=

1

|R |

∑

r

s

r

s

t

r

.

VISAPP 2007 - International Conference on Computer Vision Theory and Applications

154

• Sampling A ∼ p(A|X, S, R

ε

):

p(A|X, S, R

ε

) ∝ p(X|S, A, R

ε

)p(A)

Given uniform or Gaussian prior for A, the poste-

rior distribution of A is a Gaussian:

Vec(A) ∼ N (µ

A

, R

A

)

µ

A

= Vec(R

xs

R

−1

ss

), R

A

=

1

|R |

R

−1

ss

⊗ R

ε

where ⊗ is the Kronecker product and Vec(.) rep-

resents the column-stacking operation.

• Sampling (µ

jk

, σ

2

jk

):

With (Z, S) sampled in earlier steps and conjugate

priors assigned, the means µ

jk

and the variances

σ

2

jk

can be sampled from respective posteriors as

follows:

µ

jk

|s

j

, z

j

, σ

2

jk

∼ N (m

jk

, v

2

jk

)

m

jk

= v

2

jk

µ

k0

σ

2

k0

+

1

σ

2

jk

∑

r∈R

( j)

k

s

j

(r)

v

2

jk

=

n

( j)

k

σ

2

jk

+

1

σ

2

k0

−1

and,

σ

2

jk

|s

j

, z

j

, µ

jk

∼ I G(α

jk

, β

jk

)

α

jk

= α

k0

+

n

( j)

k

2

β

jk

= β

k0

+

1

2

∑

r∈R

( j)

k

(s

j

(r) − µ

jk

)

2

where, label region R

( j)

k

= {r : z

j

(r) = k} and the

region size n

( j)

k

= |R

( j)

k

|.

4 SIMULATION RESULTS

For evaluating the performance of the proposed algo-

rithm, we use both synthetic and real images in the

test. The synthetic images were generated according

to the model setting that each source is composed of

pixels of two classes (text and background) and two

source images are linearly mixed in every color chan-

nel independently to produce two observation images.

This was done in three steps:

1. Two binary (K

j=1,2

= 2) text image were scanned

from real documents or created by graphic tools.

They were used as the class labels Z

j=1,2

for each

source;

2. With known means and variances for pixel value

of each class, the source images were generated

according to (3);

3. For each color channel, a random selected A

2×2

was used to mix the sources and ﬁnally white

Gaussian noises R

ε

were added (SNR=20dB).

(a) (b) (c) (d)

Figure 2: Separation of synthetic image mixtures: a) origi-

nal sources; b) image mixtures; c) demixed sources; d) clas-

siﬁcation labels.

Fig.2 shows the synthetic image mixtures, demixed

sources and the label ﬁelds.

The real image for test was scanned from a duplex

printed paper, where show-through causes the super-

imposition of text. The separation result is shown in

Fig.3.

(a) (b) (c)

Figure 3: Separation of real show-through image mixtures:

a) image mixtures; b) demixed sources; c) classiﬁcation la-

bels.

For comparison, we also employed the FastICA

algorithm (Hyvarinen, 1999) on the sample images

with typical parameter set. The results on the show-

through examples of Fig.3 are shown in Fig.4. All

three channels of the two observed mixtures were

used as inputs simultaneously to the ICA algorithm.

The two demixed sources can be found in two of

six independent components (IC) outputed, while

the other four output ICs usually contain unintended

noise-like signals, which, along with the permutabil-

ity property of the ICA algorithm, bring difﬁculties

to reconstructing color representation of the sources.

On the other hand, when less color channels are ex-

ploited in demixing, we noticed that the separation

result does not necessarily degrade or improve, owing

to the possible presence of cross-channel correlations.

The MCMC computation involvedin the proposed

Bayesian separation method is time-consuming. For

the example image of 300x240 pixels in Fig.3, which

is small relative to ordinary document sizes and res-

olutions, the typical computation time of the experi-

BAYESIAN SEPARATION OF DOCUMENT IMAGES WITH HIDDEN MARKOV MODEL

155

(a) two ICs corresponding to the demixed sources

(b) other ICs containing noise-like signals

Figure 4: Separation results by ICA.

mental implementation can come up to hours without

speciﬁc optimizations. However, various computing

alternatives such as Mean Field and variational ap-

proximation can be exploited to achieve higher efﬁ-

ciency.

5 CONCLUSION

We proposed a Bayesian approach for separating

noisy linear mixture of document images. For source

images, we considered a hierarchical model with

the hidden label variable z representing the common

classiﬁcation of objects among multiple color chan-

nels, and a Potts-Markov prior was employed for the

class labels imposing local regularity constraints. We

showed how Bayesian estimation of all unknowns of

interest can be computed by MCMC sampling from

their posterior distributions given the observation. We

then illustrated the feasibility of the proposed algo-

rithm on joint separation and segmentation by tests

on sample images.

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