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Invariant object description with inverse pyramid based on the truncated
modified discrete Mellin-Fourier transform
Roumen Kountchev
Radiocommunications Department
Technical University of Sofia, Bulgaria
rkountch@tu-sofia.bg
Stuart Rubin
Space and Naval Warfare Systems Center
San Diego (SSC-PAC), CA, USA
stuart.rubin@navy.mil
Mariofanna Milanova
Department of ComputerScience
UALR, USA
mgmilanova@ualr.edu
Roumiana Kountcheva
T&K Engineering
Sofia, Bulgaria
kountcheva_r@yahoo.com
Abstract
In this paper is offered one new algorithm for global
description of pre-segmented objects in a halftone image.
The new algorithm has relatively low computational
complexity than the famous Mellin-Fourier transform and
permits multi-layer access which enhances the search by
content in indexed image databases. The new algorithm is
based on the previously developed methods of the inverse
pyramid decomposition and of the truncated modified
discrete Mellin-Fourier transform, which are combined
here.
Keywords: Invariant object representation, Inverse
difference pyramid decomposition, Modified Mellin-
Fourier transform.
1. Introduction
One of the most important tasks, which the creators of
computer vision systems have to solve, is related to
objects description, which permits their exact and reliable
classification. Objects description should satisfy
contradictory requirements [1,2]: to be invariable to
object rotation, translation and scaling; lighting
(respectively – contrast) changes; noises in the image; low
intra-class and high inter-class dispersion; maximum
compactness of the description and low computational
complexity of the operations performed.
The objects descriptions could be divided into two
basic groups: local and global. The first group represents
the local features of the objects, related to the structure of
the visible part of their surface (salient points, contours,
edges, texture, etc.) [3,4]. These descriptions are not
resistant enough against noises and have high
dimensionality and computational complexity. The
second group comprises characteristics, which represent
the general qualities of the visible surface, related for
example, to object’s geometrical parameters (invariable
moments), correlograms, coefficients of the contour lines’
Fourier decomposition, global brightness histograms,
coefficients of the generalized Hough transform,
coefficients of the Mellin-Fourier transform, etc. [5-8].
The advantages of these descriptions are their insensibility
to noises and the big compactness, but their computational
complexity is relatively high.
The basic methods for invariant object representation
with respect to 2D rigid transforms (combinations of
rotation, scaling, and translation, RST) are given in
significant number of scientific publications. Accordingly,
2D objects in the still grayscale image are depicted by
descriptors of two basic kinds: “shape boundary” and
“region”. To the first kind (shape boundary) are assigned
the chain codes, Fourier descriptors; Generalized Hough
Transform and Active Shape Model [7]. The skeleton of a
shape can be derived by the Medial Axis Transform. To
the second kind (region) are assigned some geometric
characteristics, such as for example: area, perimeter,
compactness, dispersion, eccentricity, etc., zero- and first-
order statistical moments, centre of gravity, normalized
central moments, seven rotation-invariant moments,
Zernike polynomial rotation- and scale-invariant, affine
transform invariant in respect to position, rotation and
different scales along the coordinate axes, co-occurrence
texture descriptor, etc.
The histogram descriptor is proved to be robust to
changes of object’s rotation, scale, and partial changes in
the viewing direction. The structural information however
is lost in the histogram. To solve this problem, the
combination of Discrete Wavelet Transform (DWT) or
Discrete Fourier Transform (DFT) with the feature
360
IEEE IRI 2011, August 3-5, 2011, Las Vegas, Nevada, USA
978-1-4577-0966-1/11/$26.00 ©2011 IEEE
extraction method is proposed. For the extraction of the
rotation-scale-translation (RST) - invariant features are
developed descriptors, based on the log-polar transform
(LPT) used to convert rotation and scaling into translation
[8] and on the 2D Mellin-Fourier Transform (2D-MFT)
[9]. As it is known, the modules of the spectrum
coefficients, obtained using the 2D-MFT, are invariant
with respect to the RST-transforms of the 2D objects in
the image. The basic problem for the creation of the RST-
invariant descriptors, in this case is the large number of
the calculated spectrum coefficients [10, 11]. With regard
to the necessity to reduce their number, and respectively –
the time needed for their calculation without decreasing
the objects description accuracy, should be solved
significant number of problems, regarding the choice of
the most informative MFT coefficients and the way of
creating the corresponding vector descriptor.
In this paper is offered a new algorithm for global
description of pre-segmented objects in a halftone image.
The new algorithm differs from the known algorithms in
the same group: it requires lower number of computations
when compared to the famous MFT and permits multi-
layer access which enhances the search on content in
indexed image database. The new algorithm is based on
the previously developed methods [12, 13] of the inverse
pyramid decomposition and of the truncated modified
discrete MFT, which are combined here.
The paper is arranged as follows: in Section 2 is
described the method for invariant object representation
based on pyramid of coefficients, calculated using the
truncated Mellin-Furier transform; in section 3 is given
the new approach for search-by-content of closest object
in image database; in section 4 are shown some
experimental results and in section 5 are the Conclusions.
2. Algorithm for invariant object
representation with pyramid of coefficients,
calculated using the truncated llin-Fourier
transform
The method for 2D object representation is aimed at the
preparation of the vector description of the segmented
object, framed by a square window. The description
should be invariant to 2D rotation (R), scaling (S),
translation (T) and contrast (C) changes. As a basis for the
RSTC description is used the discrete 2D Modified MFT
(2D-MMFT). As it is known, the MFT comprises DFT,
Log-pol transform (LPT) and DFT again. In order to
provide multi-layer search-by-content in the image
database, the 2D-MMFT coefficients are arranged in a
pyramid, called Inverse Pyramid Decomposition (IPD).
The algorithm, presented below, is aimed at digital
halftone images, and comprises the following stages:
For the initial (lowest) IPD level:
Step 1. The pixels B(k,l) of the original halftone image of
size M×N are transformed into bi-polar:
2/)1B()l,k(B)l,k(L max (1)
for k = 0,1,...,M-1 and l = 0,1,.., N-1,
where Bmax=255 is the maximum value in the pixel
quantization scale.
Step 2. The image is processed with 2D Discrete Fourier
Transform (2D-DFT). The Fourier matrix is of size n×n
(n=2m). The value of n defines the size of the window,
used to select the object image. For the invariant object
representation are used the complex 2D-DFT coefficients,
calculated in accordance with the relation:
]})lb/n)(ka[(2exp{-jl)L(k,b),F(a
1n
0k
1n
0l
S ¦¦
(2)
for 1n,..,1,0a and .1n,..,1,0b
The transform comprises two consecutive operations:
one-dimensional transform of the pixels L(k,l), first - for
the rows and after that - for the columns of the object
image. Since:
](lb/n)[2jsin-](lb/n)[2cos]}(lb/n)[2jexp{- SS S
and
](ka/n)[2jsin-](ka/n)[2cos]}(ka/n)[2jexp{- SS S
the 2D-DFT is performed as two consecutive one-
dimensional DFTs:
xFor the fixed values of 1n,..,1,0k and using
the 1D-Fast Fourier Transform (1D-FFT) are calculated
the intermediate spectrum coefficients:
¦
¦
¦
S
S
S
1N
0l
1n
0l
1n
0l
(lb/n)][2l)sinL(k,j
(lb/n)][2l)cosL(k,
}](lb/n)[2jexp{-)l,k(L)b,k(F
(3)
xFor 1n,...,1,0b and using the 1D-FFT again,
are calculated the final Fourier coefficients:
b),(aBjb),(aA(ka/n)]2sin[)b,k(Fj
(ka/n)]2[cos)b,k(F
]}(ka/n)[2jexp{-)b,k(F)b,a(F
FF
1n
0k
1n
0k
1n
0k
S
S
S
¦
¦
¦
(4)
where AF(a,b) and BF(a,b) are the real and the imaginary
components of b),F(a .
Step 3. The Fourier coefficients are centred in accordance
with the relation:
)b,a(F)b,a(F 2
n
2
n
0 for a,b = 0,1,..,n-1. (5)
361
Step 4. For the next operations some of the Fourier
coefficients are retained in accordance with the rule:
¯
®
.casesotherallin 0
;regionretained)b,a(if),b,a(F
b),(aF 0
0R (6)
The retained coefficients’ area is a square with a side
Hdn, which envelops the centre (0,0) of the spectrum
plane (H-even number). For H < n and
1)2/H(,..,1,0,1,..,1)2/H(),2/H(b,a this square
contains low-frequency coefficients only.
Step 5. The modules and phases of the coefficients
)b,(aj
F0R 0R
F
0R )eb,(aDb),(aF M
are calculated:
2
F
2
FF )]b,a(B[)]b,a(A[b),(aD R0R00R (7)
)]b,a(A/)b,a(B[arctgb),(a R0R00R FFF M (8)
Step 6. The modules b),(aD 0R
Fof the Fourier coefficients
b),(aF
0R are normalized in accordance with the relation:
)b,a(Dlnpb),D(a R0
F
(9)
where p=64 is the normalization coefficient.
Step 7. The coefficients b),D(a are processed with Log-
Polar Transform (LPT). The centre (0,0) of the polar
coordinate system ),( TU coincides with the centre of the
image of the Fourier coefficients’ modules b),D(a (in
the rectangular coordinate system). The transformation of
coefficients b),D(a from the rectangular b),(a into the
polar ),( TU coordinate system is performed changing the
variables in accordance with the relations:
arctg(b/a) ,balog 22 (10)
The coordinate change from rectangular into polar is quite
clear in the continuous domain, but in the discrete domain
the values of and should be discrete as well. Since a
and b can only have discrete values in the range:
,1)2/H(,...,1,0,1),...,2/H(b,a some of the
coefficients ),D( TU will be missing. At the end of the
transform, the missing coefficients ),D( ii TU are
interpolated using the closest neighbours b),D(a in the
rectangular coordinate system b),(a in horizontal or
vertical direction (zero-order interpolation).
The number of discrete circles in the polar system with
radius Ui is equal to the number of the discrete angles Ti
for i = 1,2,..,H. The number of discrete circles in the polar
system with radius Ui is equal to the number of the
discrete angles Ti for i = 1,2,..,H. The radius of the
circumscribed circle is calculated in correspondence to the
relation:
H)2/2(r . (11)
The smallest step 'U between two concentric circles (the
most inside) is calculated:
)H/1(
r U' . (12)
As a result, for the discrete radius Ui and angle Ti for each
circle are obtained the relations:
H/ii
ir)( U' U for i = 1,2,..,H, (13)
i)H/2(
iS T for 1)2/H(,.,0),.,2/H(i (14)
Thus, instead of the logarithmic relation used in the
famous LP transform to set the values of the magnitude
bins (radiuses) in the LPT, here is used the operation
rising on a power. The so modified LP transform we
called Exponential Polar Transform (EPT). After the EPT
and the interpolation of the b),D(a coefficients is
obtained one new, second matrix, which contains
coefficients D(x,y), for x, y = 0,1,2,..,H-1.
Step 8. The 2D-DFT is performed once more for the
matrix with coefficients D(x,y), in accordance with the
relation:
]})ybax)(H/[(2j-exp{)y,x(D
H
1
b),S(a
1H
0x
1H
0y
2S ¦¦
(15)
for 1H,..,1,0a and .1H,..,1,0b
The second 2D-DFT coefficients are calculated in
correspondence with Eqs. 3 - 4, applying consecutively
the 1D-DFT on the rows of the matrix [D] first, and then -
on the columns of the intermediate matrix obtained.
Step 9. The modules of the complex coefficients b),S(a
are then calculated:
2
S
2
SS )]b,a(B[)]b,a(A[b),(aD (16)
where AS(a,b) and BS(a,b) are correspondingly the
real and the imaginary component of b),S(a .
With this operation the Modified MFT is finished. The
processing then continues in the next step with one more
operation, aimed at achieving the invariance against
contrast changes. In result is obtained the RSTC invariant
object representation.
Step 10. The modules b),(aDSof the Fourier
coefficients b),S(a are normalized:
)]b,a(D/)b,a(D[Bb),(aD maxSSmaxS0 , (17)
where )b,a(D maxS is the maximum coefficient in the
matrix )b,a(DS.
362
Step 11. The vector for the RSTC-invariant object
representation is based on the use of coefficients
b),(aD 0
Sof highest energy in the amplitude spectrum 2D-
MFT of size H×H. For the extraction of the retained
coefficients b),(aD 0R
Sis used the mask, shown on Fig. 1
(the part, coloured in yellow, corresponds to the area of
the complex-conjugated coefficients, b),(aD 0R
S
). The
shape of the mask approximates the area, where the
energy of the mean 2D-MFT spectrum is concentrated.
The parameter of the mask defines the number of
retained coefficients in correspondence with the relation:
H)1(22H2R 2DD DD . (18)
The vm components for m = 1,2,..,R of the corresponding
RSTC-invariant vector: T
R002010 ]v,..,v,v[V
&
are defined
by coefficients b),(aD 0R
S, arranged as one-dimensional
massif after lexicographic tracking of the 2D-MFT
spectrum in the mask area, colored in blue.
Retained
Spectral
coefficients
Retained
Spectral
coefficients
Complex
Conjugated
coefficients
Complex
Conjugated
coefficients
ǩ
ǩ -1
1
H-1
H/2
a
b
(0,0) H-1
H/2
Fig. 1. The mask of the retained coefficients in the amplitude
2D-MFT spectrum
The vector 0
V
&
is the RST-invariant description of the
processed image for the initial (zero) IPD level. For the
calculation of the vector 1
V
&
for the next IPD level (one)
of the processed image is performed inverse modified
discrete Mellin-Fourier transform for the coefficients
b)(a,DS. In result is obtained the approximation )l,k(L
ˆ
of the processed image. For this is performed the
following:
1. The denormalized coefficients are calculated:
maxmaxSSS B/])b,a(D)b,a(D[)b,a(D 0
c. (19)
2. The complex coefficients are then calculated:
b)(a,-j
SS
eb)(a,Db)(a,S M
c
c (20)
3. The so calculated coefficients are processed with
inverse 2D-DFT:
yb)]}(xa
H
2
exp{j[.)b,a(Sy),(xD
1H
0a
1H
0b
c
c¦¦
(21)
for x, y = 0,1,2,..,H-1.
4. Then the coefficients y),(xDc are processed with
inverse EPT, after replacing the variables (x,y) by (a,b)
correspondingly:
.)],y([sin)],x([),(b
,]),y([cos]),x([),(a
iiiiiii
iiiiiii
TUTU TU
TUTU TU (22)
In result are obtained the values of the coefficients
)b,(aD ii
c, from which after interpolation are restored the
coefficients, which are missing in the rectangular
coordinate system (a,b). For this is used zero interpolation
(each of the missing coefficients b),(aDcis replaced by
the closest one from the group of the existing coefficients
in horizontal or vertical direction in the system (ai,bi)).
5. The values of the interpolated coefficients b),(aDc
are denormalized in correspondence with the relation:
]b)/p,(aDexp[)b,a(D
ˆ
R0
Fc
. (23)
6. The retained complex coefficients are calculated:
)b,(a-j
F0R 0R
F
0R b)e,(aD
ˆ
b),(aF
ˆM
(24)
7. The coefficients b),(aF
ˆ0R are supplemented with
zeros in accordance with the rule below:
¯
®
.casesotherallin 0
;regionretained)b,a(if ,)b,a(F
ˆ
b)(a,F
ˆ0R
0 (25)
8. Second, inverse 2D-DFT on the coefficients
b),a(F
ˆ0is performed:
]})blak(n/[2jexp{)b,a(F
ˆ
l),(kL
ˆ
1
-a
1
-b
0
2
n
2
n
2
n
2
n
¦¦
. (26)
for 1n,..,1,0k and .1n,..,1,0l
As a result is obtained the approximated image with
pixels l),(kL
ˆ in the frame of the window of size n×n,
which contains the object.
For the First IPD level is performed the following:
¾The difference image E0(k,l) is calculated:
)l,k(L
ˆ
)l,k(L)l,k(E0 (27)
for ).1
2
n
,.0.,
2
n
(lk,
¾The so obtained difference image is divided into 4
equal sub-images and each is after that processed with the
already described direst Mellin-Fourier transform,
following steps 2-8. The only difference is that in this
case each sub-image is a square of size n/2.
¾In a way, similar with Step 11, are calculated the
corresponding RSTC vectors for the next (second) IPD
363
level. The length of the vectors is 4 times smaller. The so
calculated vectors are used to compose the general vector
for the processed pyramid level - correspondingly 1
V
&
,
whose length is equal with that of the vector 0
V
&
,
calculated for the first zero pyramid level.
With this, the building of the two-level pyramid
decomposition is finished. As a result are obtained the
vectors 0
V
&
and 1
V
&
, which are after that used for the search
of the closest object in the image database (DB). Each
image in the DB is represented by vectors, calculated in
similar way for each of the pyramid levels, following the
already algorithm, presented above. For each level of IDP
is calculated a corresponding vector, which carries the
information about the fine details, which represent the form
of the object in the image.
3. Search by content of closest images
represented through IPD/MMFT
The search of closest objects to the image request in
image DB is based on the detection of the minimum
squared Euclidean distance between their RSTC-invariant
vector representations.
The decision for the image request classification,
represented by the R-dimensional RSTC-vector V
&
is
taken on the basis of the image classes in the DB and their
RSTC-invariant vectors:
E
E
D
ECV
&
for DE=1,2,…,P and E=1,2,….,Q. (29)
P - the number of vectors in the DB, Q - the
number of image classes. After that, a classification rule is
applied, based on the -nearest neighbours (k-NN) and
“majority vote” algorithms [5]:
)V,V(d....)V,V(d)V,V(dif,CV K
K
2
2
1
1
0K21
E
D
E
D
E
D
EEEE ddd
&
&
&
&
&
&
&
(30)
where is an odd number; )V,V(d k
k
k
E
DE
&&
is the squared
Euclidian distance between vectors V
&
and k
k
VE
DE
&
for k =
1, 2, …, K; k
k
VE
DE
&
is the kth vector with index k
E
Dfrom the
class k
CE, which is at minimum distance from the query
V
&
(the indices ,
k
E
D Ek and E0 are in the ranges: [1, k
PE]
- for k
E
D, and [1, Q] - for Ek and E0).
The class 0
CE of the vector V
&
in Eqs. 30 is defined
by the most frequent value E0 of the indices Ek of the
vectors k
k
VE
DE
&
:
0 = max{h(Ek)} for Ek =1,2,..,Q and k=1,2,..,K. (31)
Here h(Ek) is the histogram of the indices Ek, for the
relations in Eqs. 31.
For the enhancement of the object search, the smallest
distance for the image request is calculated to a group of
images, using the vectors defined for the zero level of the
corresponding couple of pyramid decompositions. After
that the smallest distance is calculated in the so selected
group only, using the vectors for the next (first) pyramid
level of the corresponding pyramids, etc.
4. Experimental results
For the experiments was used the software
implementation of the method, developed in the Technical
University of Sofia. An example of the difference between
the LPT and the EPT is shown on Fig. 2: Fig. 2.a is the
experimental image “Lena”, on which are indicated the
points, which participate in the EPT (the black points are
not retained and the image is restored after corresponding
interpolation); on Fig. 2.b are shown the points, which
participate in the LPT (the used points are black). This
experiment confirms the efficiency of the new approach,
because in the well-known LPT the retained central part of
the processed image is much smaller. The new approach
permits the user to choose the size of the square and the
radius, and in result – to set the retained area size and the
density of the original points, used for the vector
calculation.
a. Points retained by EPT; b. Points retained by LPT
Fig.2. Test images showing the retained points for EPT and LPT
The main part of the experiments was aimed at the
content-based object retrieval. For this were used the
specially developed image database of the Technical
University of Sofia, which contains 200 faces of adult
people and children. All images were of size 256 x 256
pixels, grayscale (8bpp). In accordance with the method,
described above, the database comprises two large parts,
i.e.: the coded images and their vector representations.
The search was performed on the basis of the smallest
distance between the corresponding vectors. The photos
were taken in various lighting conditions with many
shadows, different views, etc. Most of the faces in the
databases were cropped from larger images.
On Fig. 4 is shown the search result obtained for the
Image request 1 (The test image “Lena”, shown on Fig.
3). The detected closest images are of adults only. The
closest image is the upper left. There are no photos of
children in the closest 15 images, i.e. no mistakes in the
364
selection. The first closest images detected in the database
are some variations of the test image “Lena” – obtained
after rotation on different angles, cropped and enlarged.
Fig. 3. Image request – enlarged part of the test image “Lena”
Fig. 4. Result of the object search for Image request 1
(the image "Lena" from the image database)
The results obtained show that the object
representation is really RSTC invariant, because the first
closest 4 images are same as the image request, but of
different size, rotation and contrast.
5. Conclusions
In this paper is presented a method for invariant object
representation with inverse difference pyramid
decomposition and Modified MFT, suitable for efficient
object search in image databases. The main differences
from the famous MFT are: the first DFT is performed for
limited number of coefficients only. In result is obtained
an approximating image, suitable for object
representation; instead of the Log-Pol Transform, here
was used the Exponential-Polar Transform (EPT). As a
result, the part of the retained points from the matrix of
the Fourier coefficients’ modules is larger (i.e. bigger
central part of the image participates in the EPT). The
number of coefficients, used for the object representation
is limited in accordance with the vector length selection.
In result, the MMFT described in this work has the
following advantages over the MFT:
The number of transform coefficients used for the
object representation is significantly reduced and from
this follows the lower computational complexity of the
method, which permits real-time applications.
The choice of coefficients, used for the vector
calculation offers various possibilities by setting large
number of parameters, which permits the method use in
content- and context-based object retrieval.
The user can select the size of the retained part of
the image, which to be used for the vector preparation.
Knowing that this is the central part of the investigated
image, this approach could be used to fix the Region Of
Interest (ROI) for some special kind of search (machine
parts, faces, etc).
The new method, presented in this work, offers high
flexibility and permits object detection in various
positions, lighting conditions and view points.
Acknowledgements
This research work was supported in part by the Joint Research Project
Bulgaria-Romania 02/09 and by the System Research and Application
(SRA) Contract No. 0619069, USA.
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