Digital Restoration of Watermark Images
Andrey Karnaukhov(*), Igor Aizenberg($), Alois Haidinger(-),Victor Karnaukhov(*)
Nikolai Merzlyakov(*), Olga Milukova(*), and Emanuel Wenger(#)
(*)
Russian Academy of Sciences, Institute for Information Transmission Problems, Moscow, Russia
E-mail: avk@iitp.ru, victor.karnaukhov@iitp.ru, nick@iitp.ru, milukova@iitp.ru
($)
Neural Network Technologies, Tel Aviv, Israel
E-mail: igora@netvision.net.il
(-)
Austr. Acad. of Sci., Comm.of Paleography a. Codicology of Medieval Manuscripts, Vienna, Austria
E-mail: alois.haidinger@oeaw.ac.at
(# )
Austrian Academy of Sciences, Commission of Scientific Visualization, Vienna, Austria
E-mail: emanuel.wenger@oeaw.ac.at
Abstract
Can traditional methods of image restoration be applied for restoration of watermark images?
This paper deals with beta-radiographic hardcopies of watermark images taken from the old
books and documents. The fast Fourier transform techniques used for restoration of
watermark images. Software tools were implemented in the frame of an integrated system for
digital processing, identification of watermark images and database management. The
watermark images restored by using the Tikhonov’s regularization filter are presented.
1. Introduction
Watermarks are widely used for chronological identification and classification of historical
documents and corresponding historical events since more than 100 years. To identify a given
watermark, incunabulists, medievalists, historians, and other researchers use catalogues of
watermarks published in tens of volumes all over the world [1,2]. The images in these
catalogues represent graphic copies of watermarks were received by hand-draw sketching
from pages of historical documents, medieval handwritten books, and incunabula. The identity
of a watermark with one in the standard catalogues is a good indicator for the age of the
watermark, the paper, and the document in question. So, chronological identification and
classification of historical documents and corresponding historical events at such approach is
carried out by simple search of an identical or a similar watermark in catalogues and
performance of some measurements with the help of a ruler. A low efficiency of such
identification by using these methods is obvious.
Until the 19th century all paper was made by using a special paper-producing mold: a
rectangular sieve consisting of a frame and a bottom made of a wire mesh. From the thirteenth
century on nearly all European papermakers characterized their production by means of
watermarks. These watermarks were caused by wire figures, which were fixed on the wire
mesh. The molds were dipped into vats of liquid pulp. When lifting them out, the liquid
drained out through the wire mesh and the fibers became interwoven to a sheet of paper. The
pattern of the mesh with the watermark wire-figure attached on it was imprinted in the paper.
Due to some natural technological restrictions the longevity of paper producing molds was
short. Molds, and more particularly, meshes and wire-figures were in use only for one or two
years or even a few months. So, undated manuscripts can be accurate dated when identical
watermarks are found in dated ones. As a matter of fact, watermarks performed the function
of legal trademarks of the paper producers. A particular paper producer after breakage of the
paper-producing mold to save his/her trademark had to repair or usually recreate mold with
the same watermark. Such repaired or recreated meshes and/or watermarks slightly or
noticeably differed from their predecessor. For correct dating of such slightly deviated
watermarks the full mesh structure of the paper-producing mold has to be analyzed too.
2. Image Restoration Techniques
Watermarks and wire meshes of paper-producing molds are imprinted in the paper as small
deviations in the thickness of paper. These thickness deviations can be revealed, displayed,
and hardcopied by several methods. The electron radiography or betaradiography seem to be
the best in present time among the methods for a unique recording of watermarks because of
their good quality. Small deviations in the density of the paper are recorded sufficiently well
with these procedures. A typical watermark image scanned from a beta-radiographic hardcopy
is presented in Figure 1. The imprinted watermark (c) and mesh structure (laid lines (b, and
chain lines (c)) are recorded as inhomogeneous structure of low contrast smoothed lines
corrupted by a noise: contrast and smoothed black and white spots of different forms and
sizes.
Figure 1 A typical watermark image scanned from a beta-radiographic hardcopy.
Generally speaking, we can consider these smoothed images of watermarks, laid and chain
lines as blurred images of the original “ideal” images of these objects. From this point of
view, we can re-formulate our task to the task of digital restoration of watermarks, laid and
chain line images from their electron radiography or betaradiography hardcopies.
Image restoration is often defined as a process of recovering an original image from its blurred
degraded version [3]. The image restoration problem is usually formulated in the following
way: given a degraded image u~x, y  , it is required to estimate an original non-degraded
image zx, y  from the following equation:
u~ ( x, y )  u ( x, y )  n( x, y ),
(1)
x, y   , 
or in the matrix form:
u~  Az  n,
(2)
where A is a linear imaging operator, ux, y  is the blurred image, nx, y  is the noise,
u~x, y  is the output blurred image degraded by noise.
There are a lot of different approaches in solving this problem. The most universal techniques
are given by the regularization and statistical estimation approaches [3-5]. In both cases, the
solution of the restoration problem are transformed to estimation of either the conditional, or
the unconditional extremum of a functional.
Unfortunately, a direct implementation of the minimization task is difficult in the general twodimensional case. A simplification of the minimization task is usually based on usage of
specific features of the integral operator A (see equation (2)). For example, if the imaging
system can be described by a homogeneous operator A and the degraded image u~x, y  is
defined on the whole of plane x, y   ,  the equation (1) comes to the convolution type.
In such case, the blurred image is defined by the following equation:
u~ ( x, y )   h( x   , y   ) z ( , )dd  n( x, y )
(3)
where u~ ( x, y ) is the blurred image, h( x   , y   ) is the blurring function, n( x, y ) is the
noise, and z ( , ) is the desired, original image. This equation can be solved by using the fast
Fourier transform technique [3]. We should specially emphasize that the degraded image
u~x, y  is usually defined on a bounded domain. It does not permit to apply Fourier transform
directly to the equation (3). To overcome this restriction an additional procedure is required to
extend the definition of the degraded image onto the whole plane x, y   ,  [6]. The
convolution nature of the above equation implies that its equivalent Fourier representation is
~
U ( f x , f y )  Z ( f x , f y ) H ( f x , f y )  N ( f x , f y ),
~
where U ( f x , f y ),
Z ( f x , f y ), and
N ( f x , f y ) are the Fourier spectra of the blurred image,
original image, and noise, respectively, and H ( f x , f y ) is the blurring transfer function. It is known
that the general linear solution of the equation (3) can be written as
zˆ ( , )   K (  s,  t )u~ (s, t )dsdt ,
where the kernel of the inversion has the following form:
K ( s, t ) 
1
4 2
 R( f
x
, f y )e
i ( sf x  tf y )
df x df y 
1
4 2

H * ( f x , f y )e
i ( sf x  tf y )
2
H ( f x , f y )  ( f x , f y )
df x df y
and R( f x , f y ) is the restoration filter, and  ( f x , f y ) is a given function.
In the presence of noise the optimal restoration filter (in the MSE criteria) is the least squares
filter or the Wiener filter [3]
H *( fx, fy )
R( f x , f y ) 
,
S nn ( f x , f y )
2
H( fx, fy ) 
S zz ( f x , f y )
(4)
where H ( f x , f y ) is like above, S nn ( f x , f y ) and S zz ( f x , f y ) are the noise and the object power
spectra, which are assumed to be known. It is also assumed that the noise added is white
noise, i.e., its spectral density is constant, and the picture and the noise are uncorrelated. This
method works well for images with a high signal to noise ratio (SNR), which is defined to be
the ratio between the variance of the picture and the variance of the noise, and performs
poorly for images with low SNRs.
3. Software implementation and experimental results
For the experimental implementation of this approach we selected the linear model of image
degradation (3) and used Fourier transform techniques for restoration of watermark images.
For restoration of watermark images we used the Wiener filter (4) and a particular realization
of the Tikhonov’s regularization filter. The second filter had the following transfer function:
R( f x , f y ) 
H *( fx, fy )
2
H ( f x , f y )   (1   ( f x2  f y2 ))
,
(5)
where H ( f x , f y ) is like above,  and  are non-negative regularization parameters.
Experimental results are given in Figure 2 and Figure 3, where Figure 2(a) the initial blurred
watermark image; Figure 2(b) the watermark image restored by using the Wiener
regularization filter (5), Figure 3(a) the initial blurred watermark image; Figure 3(b) the
watermark image restored by using the regularization filter (5).
All required software tools were implemented in the frame of an integrated system for digital
processing, identification of watermark images and database management. Its detailed
description is given in [7]. User interaction with this system is realized through a graphical
user interface containing a set of graphical forms, menus, buttons, and other controlling
objects. This graphical user interface is designed in the Windows-style.
The integrated system is implemented on a PC-based platform and runs under
Windows 9*/NT/2000 or higher operating system.
Acknowledgements
This work is supported partially by the Russian Foundation for Basic Research under the
project numbers 99-07-90017 and 01-07-90354.
References
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а
b
Figure 2 An experimental result:
(a) blurred image;
(b) restored image by using the Wiener filter (4)
а
b
Figure 3 An experimental result:
(a) blurred image;
(b) restored image by using a regularization filter (5)