Zhisnyakov A.L., Fomin A.A.,
Privezencev D.G., Baranov A.A.
Murom Institute of the Vladimir
State Univercity
APPROACH TO
IMAGES FEATURES DETECTION
BASED ON THE
CONTINUOUS WAVELET TRANSFORM
The task of signs detection appears very often in
different applications of digital images processing. For example they are photo
documents restoration, analysis of various pictures (metal microstructures or
butt-welded joints), and so on. The heterogeneities searching is one of first
phases of same tasks solving. In this connection the problem of correct
detecting of local images heterogeneity is actual because the quality of this
subtask solving has an influence on the quality of entire task.
We can relate to local heterogeneity wide sphere of
elements that are presented on the images and have characteristics that are
different from characteristics in the ones areas. In example, these
heterogeneities are various spots or image objects.
The different disturbed facts can influence on the
local heterogeneity searching procedure. For example while spots detecting
these obstacles can appear: the spots can be translucent, can have different
form and measures, can possess same statistic characteristics with another
image areas, in addition the image can be noised. All of these facts can
influence to the result of detection procedure. Using the wavelet-transform for
image features detecting we can appreciably reduce disturbed facts influencing because
of some characteristics of one.
For the
solving of this task we can imagine analyzed image as the matrix of
instantaneous values:
, 
, 
,
where N, M are the measures of image.
For the getting of area-frequency image representing
notion we can use both discrete and continuous wavelet-transform. Traditionally
for analysis of transitional stochastic signals (images) and detecting of ones
features, continuous wavelet-transform is applied because it contains the
surplus information about analyzed signal (image).
The two-dimension wavelet-representation of image can
be got both separable one-dimension (by image rows and columns) and
two-dimension wavelet-transform.
One-dimension continuous wavelet-transform of the
function 
is the function of two arguments:
, (1)
where
,
where the wavelets 
are scaled by value a and shifted by value b copies of generative wavelet 
.
For each pair a
and b the function 
defines the amplitude of the corresponding wavelet. In other words
the function 
measures f(x) changing
around the point b size of which is
proportional to a.
Equivalence of wavelet-transform to the convolution
with the filter allows to detect local signal heterogeneities by amplitude
maximum of wavelets corresponded to the feature area with commensurable
measures of feature and filter. The chance of the filter size changing by changing
of scale coefficient a allows to
select wavelets-transform parameters adapted to the signal.
The using of separable one-dimensional
wavelet-transform for the two-dimensional wavelet-representations getting is
justified in the ways when features characteristics, in example measures, is
changed in connection with viewing direction, in other words they are
anisotropic. Such method allows more correctly to take into account the
changing of characteristics of local images features.
One of possible enlargements of (1) on two-dimensional
case is the selection of wavelet 
that has or has not spherical symmetry, then:
, (2)
Taking into account (1) and (2) for wavelet-representation
getting of two-dimensional signal (image) 
we can get:
, (3)
where
.
The most widely distributed example of basis function
applied for calculation of one-dimensional wavelet-transform is the wavelet
“Mexican hat”. For two-dimensional case it is noted as
, (4)
where h(x,y)
looks like
The examples of such wavelets are represented on the
figure 1.
Figure 1. Analogues of “Mexican hat” wavelets for the
two-dimensional case
Using of two-dimensional wavelet-transform is
rationally in cases of detecting of image features with circle or elliptic
form.
The offered method to the image features detecting is
based on the applying two-dimensional wavelet-transform for getting
wavelet-representation of anylized image and is as follows.
The instant values matrix of analyzed image Z is
exposed by two-dimensional wavelet-transform (3) with wavelet filter (4) on the
stated interval of scaling coefficients values 
, 
.
For each derived set of wavelet-coefficients 
the value of Shannon’s entropy is defined as the function like:
where P is appearance probability of appropriate
wavelet-coefficient.
The values of scaling coefficients at which the value
of entropy amount to the maximum is defined as
, 
, at 
.
The set of wavelet-coefficients, that corresponds to
some values aopt1, aopt2, is selected as optimal.
The elimination of the artifacts on the resulting
images (fig. 2), is realized by image binarization and carrying out the
morphologic operations or applying of filter algorithms.
Figure 2. The result
of features detecting.
The continuous wavelet-transform represents wide
possibilities for solving the task of image features detection. Possibility of
information reception about image frequency constituent allows doing the
detecting procedure more qualitative.
At first, while transiting to the greater scale of
wavelet-transform, high-level components of the signal (that contains the
noise) are lost. Secondly, the brightness of image elements influences not to the
detection procedure. And thirdly, possibility of free scale selecting of
wavelet-transform allows to detect the features of any measures and forms.
References.
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and its application to pattern recognition. – Singapore: Regal press (S) Pte.
Ltd., 2000. – 359 p.
2. Kolers, P Recognizing patterns. Studies in living and automatic
systems / P. Kolers, D. Murray. – M.I.T. Press, 1968. – 288 p.
4. Daubechies, I Ten lectures on wavelets /
I. Daubechies. – SIAM, 1992. – 460 p.
5. Mallat, S. A wavelet
tour of signal processing / S. Mallat. – Academic press, 1999. – 671 p.