RECOGNITION OF DESTROYED IMAGES BY SINGULAR VALUE
DECOMPOSITION
MOKRIŠ Igor1 – SEMANČÍK Ľubomír2
1
Institute of Informatics, Slovak Academy of Sciences, 846 07 Bratislava 45, Dúbravská
cesta 9, Slovak Republic, mokris@savbb.sk,
2 Dept. of Computers and Informatics, Military Academy, 031 01 Liptovský Mikuláš, Slovak
Republic, semancik@valm.sk
Presentation deals with extraction and recognition of images from real image scene.
For extraction of images the method of subtraction of background and real image scene was
used. Recognition of extracted images was performed by classifier with learning, which is
based on singular values of Singular Value Decomposition. Used approach was chosen due to
decreasing the computational complexity of recognition process.
Key words: image extraction, extracted image recognition, Singular Value Decomposition
1. Introduction
Images recognition of automatic investigation systems [23] needs solving the task of
destroyed extracted image recognition from real image scene. The images were obtained by
camera sensing in projection 2D from 3D and then they were extracted from real image scene.
The method of subtraction of background and real image scene for extraction of images was
used and extracted images by with way were obtained [22]. Because in the sensing process of
the real image scene undesirable influences can be shown, the extracted image is destroyed
and consists the fragments of background and because of the extracted image it is necessary to
reconstruct. From aspect of decrease the computational complexity of reconstruction problem
the threshold method for reconstruction of extracted images was used.
For recognition of extracted images the classifier with learning was used. The learning
set consists of the image etalons, which were obtained by manual extraction of images from
set of real images, which were prepared for this purpose. For learning set were used the
etalons in one projection in base position of chosen images. For recognition of real extracted
images the translation, dilation and rotation invariance furthermore was needed to solve in
relation to the recognized image. From this aspect the singular values of Singular Value
Decomposition of images as features for extracted image recognition were used [3,4,11,14,1726].
2. Extraction of images
The extraction of images from real image scene is needed for recognition of images.
For automatic investigation systems it is possible to solve the extraction of images by
subtraction of background of real image scene (background image) and a real image scene
(examples of that are in Fig. 1 and Fig. 2) [22]. The subtracted image, which is obtained this
way, consists of nonzero pixels in the position of actual image. Substituting of obtained image
pixel positions by the origin values of image pixels we can get extracted image. Next relation
can describe this approach
f (i, j), for f ris (i, j) - f back (i, j)  0
f extr (i, j)   ris
 0, for f ris (i, j) - f back (i, j)  0
(1)
where fris is brightness value of image pixels in real image scene, fback is brightness value of
image pixels in background image and fextr is brightness value of image pixels in extracted
image.
Fig.1 Real image scene
Fig.2 Background image
With respect to the fact, that in the real image scene and in the background image
there are various incorrect fragments, the result of the extraction process is not correct
extracted image, but destroyed extracted image, which consists of the image pixels of image
and also background image. I.e., it is not possible to obtain correct extracted image but only
destroyed extracted image.
a) Threshold of extracted image =30
b) Threshold of extracted image =40
Fig. 3 Destroyed extracted image with background fragments
The quality of extraction can be influenced by threshold of extracted image by relation
f (i, j), for f ris (i, j) - f back (i, j)  
f extr (i, j)   ris
 0, for f ris (i, j) - f back (i, j)  
(2)
where is threshold of extracted image, which is used for influence decrease of background
fragments (Fig. 3). The various results in dependency on threshold value were obtained for
extracted images. But, it is very important, that in extracted image is presented the destroyed
image and background fragments. In the next step, this way extracted images were recognized
by recognition system.
3. Extracted image feature generation
Let an image is represented by matrix F=[f(i,j)]; i = 1,2,...,n. Then for the
decomposition of an image matrix by singular values holds true [1,2,5,6,9,1012,13,15,16]
F  U S VT
(3)
where U=[u(i,j)] is a matrix of orthonormal row-oriented eigenvectors of matrix F.FT. For
matrix U holds true UT.U=I, where I is an unit matrix. V=[v(i,j)] is a matrix of orthonormal
column-oriented eigenvectors of matrix FT.F. For matrix V holds true VT.V=I and S = [s(i,j)]
is a diagonal matrix of singular values. For singular values si,j hold
s i, j  i  s i
(4)
where λi are eigenvalues of equation [7,8]
det(I - F T .F)  0
(5)
If the matrix U is expressed by U=[u1, u2, ... , uN] and matrix V by V=[v1, v2, ... , vN],
where ui and vi are column oriented eigenvectors and matrix S is expressed by sub-matrices
s1 0 ...
 0 0 ...
S
... ... ...

 0 0 ...
0
0
0

0
 ...  
...
...


0
0
0
0 
... ... ... 

0 ... s N 
0 ...
0 ...
then decomposition (3) is expressed by relation
N
N
i 1
i 1
F   u i .s i .v iT   Fi
(6)
Relations (3) and (6) enable to express an image F by a vector of singular values
S and matrices of eigenvectors U and V where Fi is sub-image of an image F. The matrices
U and V with vector S contain the whole information about an image, but substantial
information about an image is extracted in a vector S. Sub-image Fi expresses projection of an
image F into sub-space of U in direction of eigenvector ui in N-dimensional orthogonal space
of eigenvectors UN . [4,24-26].
SVD enables express an image by small number of singular values. The singular
values represent the energy of sub-image in direction of eigenvector ui and because of that
was analyzed the possibility their utilization as the features for invariant image recognition
[9,14,15,19,21].
4. System for extracted image recognition
System for extracted image recognition is developed as a classifier with learning (Fig.
4). It consists of feature generation subsystem and a classification subsystem [14,17,19,22].
Feature generation subsystem generates features for image recognition based on the singular
values. Classification subsystem solves the recognition problem by Euclidean metric based on
singular values of recognized extracted image in relation to the singular values of an image
etalon.
Image
etalon
Extracted
image
Image extraction
and threshold
determination
Feature generation
subsystem
Etalon
features
Extracted
image
features
Classification
subsystem
Image recognition
Fig. 4 System for extracted image recognition
For learning of recognition system was used the training set consists of the singular
values of image etalons, which express recognized images only by one projection in the base
position (examples are situated in Fig. 5-7). By means of that and by extracted destroyed
images were computed the singular values, which were used for extracted image recognition.
For practical utilization of image recognition in automatic investigation systems [23]
is important to determine, how is necessary to solve the quality of image extraction for
successful image recognition. I.e., what quality of extraction of an image is needed or satisfies
only partial extraction in which are also background fragments and extracted image is
destroyed and incomplete (Fig. 3). For verification of this considerations the experiments
were performed in which threshold was varied in relation (2). This way modified extracted
image was used for extracted image recognition in the recognition system.
Fig. 5 Image etalon No. 1
Fig. 6 Image etalon No. 2
Fig. 7 Image etalon No. 3
For determination of threshold the histogram of extracted images experimentally was
analyzed (Fig. 8). By histograms can be obtained information about incorrect background
fragments and faults in extracted image. Based on its analysis it is possible to determine
suitable threshold of extracted image for equation (2). From analysis of histograms resulted
that values of brightness function for background image are in interval (0,100) and values of
brightness function near to left region of interval represent fault fragments and values of
brightness function near to right region of interval represent extracted image. Based on this
reason were performed the experiments with the assistance of which were obtained extracted
images of variable quality [22].
Fig. 8 Histogram of extracted image
The extracted images entered into recognition system during next step. In the feature
generation subsystem there were computed singular values, which were used for the extracted
image recognition. In next table (Tab. 1) there are first 5 singular values for image etalons,
which are utilized in examples by Fig. 5 - 7. Tab. 2 shows singular values of extracted
image (Fig. 3) with variable threshold.
Tab. 1 Singular values of image etalons
Etalon No. 1
Etalon No. 2
Etalon No. 3
37398,2695
12566,1465
23037,6289
10311,0059
4238,6460
8506,0654
5443,0488
2968,4658
3910,4231
4723,8838
2055,8174
2839,3621
4183,0225
1334,2428
2339,1370
Tab. 2 Singular values of extracted image No. 3 with variable threshold τ
Singular
values
1
2
3
4
5
τ=0
τ=10
τ=20
τ=30
τ=40
43460,2852 28871,0469 22962,6934 18478,8984 12372,4209
3292,4609 8434,8047 9775,1152 9307,2012 8139,6626
2630,5002 5926,3364 5485,8813 5686,2900 5015,0605
1822,9987 4998,2520 4923,9268 4471,6626 4314,8887
1330,3317 4487,9224 4093,4053 3948,9622 3510,3481
τ=50
8889,6055
4904,8862
4222,2832
2771,7358
2577,2764
In the classification subsystem by Euclidean metric was find nearest vector of singular
values of extracted image in relation to the vector of singular values of image etalon. At the
same time the threshold of extracted image was accepted. The threshold of extracted image
influences background and also damage of extracted image. As an example of extracted
image recognition with variable threshold Tab. 3 can be used.
Tab. 3 Recognition of extracted image with variable threshold
Threshold of extracted
image
Recognition of extracted
image as image No. 3
Evaluation of recognition
T – true; F – false
τ=0 τ=10 τ=20 τ=30 τ=40 τ=50
1
3
3
3
2
2
F
T
T
T
F
F
Experiments show, that images with “middle threshold” are suitable for correct
recognition, what influences, that in extracted image is adequate represents recognized image,
which hereby contains adequate background fragments and adequate destroyed image parts.
Extracted images with little or big threshold were classified incorrect. In this case the
recognition was correct for threshold τ=10, 20, 30 and incorrect for threshold τ=0, 40, 50,
what cause great influence of background or minimize influence of background and extracted
image parts at the same time.
5. Conclusion
Experiments acknowledge the utilization of singular values as features for extracted
image recognition. Correctness of recognition was termed by fact which way the background
fragments influenced the extracted image. It shows that the recognition of extracted images is
better in that cases when during extraction in recognized image remain of background
fragments. It follows from fact that little background fragments influences singular values less
then big regions of extracted image. It is evident that this approach is limited, but on the other
hand, in the case of suitable application, can be very prosperous from point of decreasing of
the computational complexity for recognition process.
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