Nonoptimizational approach to invariant object detection
Предложен новый подход к инвариантному обнаружению объектов. Рассмотрен общий случай инвариантности к необходимому набору преобразований. Модельные экспериментальные результаты, подтверждающие эффективность предложенного подхода, представлены задачей обнаружения объектов изображений. A new approach...
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Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України
2013
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| Cite this: | Nonoptimizational approach to invariant object detection / T. Mandziy // Управляющие системы и машины. — 2013. — № 4. — С. 20-25. — Бібліогр.: 13 назв. — англ. |
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| citation_txt | Nonoptimizational approach to invariant object detection / T. Mandziy // Управляющие системы и машины. — 2013. — № 4. — С. 20-25. — Бібліогр.: 13 назв. — англ. |
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| description | Предложен новый подход к инвариантному обнаружению объектов. Рассмотрен общий случай инвариантности к необходимому набору преобразований. Модельные экспериментальные результаты, подтверждающие эффективность предложенного подхода, представлены задачей обнаружения объектов изображений.
A new approach to invariant object detection is proposed. A general case solution for invariance to required set of transformation is considered. Basic experimental results that justify proposed approach are presented on the task of image object detection.
Запропоновано новий підхід до інваріантного виявлення об’єктів. Розглянуто загальний випадок інваріантності до необхідного набору перетворень. Модельні експериментальні результати, які підтверджують ефективність запропонованого підходу, представлені задачею виявлення об’єктів зображень.
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20 УСиМ, 2013, № 4
UDC 004.932.72
T. Mandziy
Nonoptimizational Approach to Invariant Object Detection
Предложен новый подход к инвариантному обнаружению объектов. Рассмотрен общий случай инвариантности к необходи-
мому набору преобразований. Модельные экспериментальные результаты, подтверждающие эффективность предложенного
подхода, представлены задачей обнаружения объектов изображений.
A new approach to invariant object detection is proposed. A general case solution for invariance to required set of transformation is
considered. Basic experimental results that justify proposed approach are presented on the task of image object detection.
Запропоновано новий підхід до інваріантного виявлення об’єктів. Розглянуто загальний випадок інваріантності до необхідно-
го набору перетворень. Модельні експериментальні результати, які підтверджують ефективність запропонованого підходу,
представлені задачею виявлення об’єктів зображень.
1. Introduction. Object detection and recognition
is one of the most difficult tasks in computer sci-
ence. Up to date there exist many approaches to
object detection and recognition. Majority of those
approaches aims to solve the problem for a certain
narrow subset of object classes (faces, cars, tex-
tures, etc). The problem of such variety of appro-
aches consists mainly in absence of general solu-
tions to the object detection and recognition tasks.
For instance, majority of methods and algorithms
used for face recognition are inapplicable to tex-
ture recognition tasks and vise versa.
Let us consider general object detection on the
example of image object detection task. There are
many approaches for image object detection [1–3,
9]. Basically those approaches can be divided on
two groups. First group encapsulates approaches
for template based object detection [1–3]. This
group of methods aims to detect objects similar to
the given template in an input image. Template
can be represented as a set of features [4], a con-
tour representation [2, 9], a learned pattern [5], an
actual image of the target object [3], etc. This
group of methods is able to locate the object on
entire input image. Problems arise when some sort
of transformation or distortion is introduced to the
target object in an input image. In a presence of
such transformations as scale, rotation, projection,
shape variations etc. the problem rises to a new
level of algorithmical and computational complex-
ity. Presence of such transformations makes con-
ventional approaches more than useless. Existing
adaptations of those methods to such transforma-
tions are usually restricted to some small subset of
them, and do not solve the problem in general.
To the second group of object detection meth-
ods we refer to as object segmentation methods.
Those are methods, in general, incapable to locate
an object position in entire input image but given
such initial position they can adjust model pa-
rameters to fit a model image to an input image.
This second group includes mostly generative
models that are able to efficiently model object
appearances under some transformations on input
image given the approximal initial position of that
object in input image [6–8].
Those two groups of approaches to object de-
tection and object modeling exist and develop
separately. They both solve only a part of a more
general object detection task.
Absence of some kind of “holistic” approach to
object detection is a concern of this paper. It pre-
sents an attempt to combine advantages of exist-
ing object detection and object modeling tech-
niques to produce a new and to a degree general
approach to invariant object detection.
2. Invariant image object detection
Under invariance in object detection it is often
understood insensitivity of object detection algo-
rithm to a certain set of transformations. Problem
of invariance in general case is very complicated
and unsolved task. Many papers were dedicated to
the solution of invariance problem [1–5, 9, 10, 13].
Most of them attempt to solve this problem for a
certain limited set of transformations and those
solutions usually do not allow for further exten-
sion of transformation set.
Probably the most natural requirement is the
invariance to scale and rotation. They appear natu-
rally as result of image acquisition procedure. Most
УСиМ, 2013, № 4 21
of image object detection tasks have to cope with
these two transformations. Methods for invariant,
with respect to scale and rotation, object detection
and representation are given significant amount of
attention [1–5, 9, 10]. Shape changes and varia-
tions are another transformations that some object
detection and recognition algorithms where de-
signed to be invariant to [6–9, 11, 12].
Many of existing approaches to invariant object
detection suffer from lack of fundamental general-
ity of their solutions. Mostly offered solutions de-
signed specifically for certain type of invariance
that cannot be applied to transformations of dif-
ferent nature. As the result, absence of some kind
of general approach leads to continuous growth of
proposed solutions applicable only for narrow set
of transformations and object classes.
2.1. Direct MAX computation
The goal of invariant image object detection is
to detect a target object on an input two-dimensi-
onal image I (x, y) invariant to certain set of possi-
ble transformations of target object on I (x, y). Let
M () be a mathematical model of target object
image with some parameter vector . Every com-
ponent of is responsible for certain type of trans-
formation. For instance, first component of pa-
rameter vector 1 can be rotation angle or scale
s of target object, second component 2 can corres-
pond to appearance changes of target object (for
instance shape, texture, illumination etc), etc. Let
yxMIC ,,, be some similarity measure that
measures similarity of an input image I (x, y) with
target object model M () at (x, y).
So the objective is to detect object of interest
M () with arbitrary allowed parameter vector on
arbitrary input image I (x, y).
Basically described general task of invariant
object detection can be represented as follows:
yxMICyxC inv ,,,max,
. (1)
Any existing object detection algorithm can be
represented in form of (1). The difference is in the way
a particular algorithm solves yxMIC ,,,max
task. In practice this task falls into optimization
theory where yxMIC ,,,max is formulated
in terms of some conventional optimization tech-
nique (lest squares, dynamic programming, gradi-
ent based methods etc) or it is solved inexplicitly
by some technique applicable only to a very nar-
row class of objects. In general C (I, M (), x, y) is a
complex function of many variables and local mini-
mums. Thus optimization of (1) is difficult and ge-
nerally unsolvable with conventional methods task.
The only known general solution for (1) is
brute force approach. For object detection task, it
requires explicit computation of all possible out-
comes of similarity measure C for all possible
values of . In discrete case it implies dividing the
domain D of allowed values for by . Even
though such approach guaranties the solution of
(1) it is too computationally expensive for major-
ity of real life object detection tasks.
Computational complexity of brute force solu-
tion for (1) consists of two components. The first
is computation of values of C (I, M (), x, y) for all
possible D and the second is computation of
max function for obtained values. In discrete case,
computational complexity of the second component
is neglectable in comparison to first component.
But there is a case when computational comple-
xity of those two components can be reversed. For
the sake of simplicity and without loss of general-
ity of final solution, let us consider one-dimensional
parameter vector . As was mentioned, to solve (1)
numerically in terms of brute-force paradigm,
C (I, M (), x, y) should be computed for all possible
values of D. In this case (1) takes the next form:
,,,,,...,,,,
,...,,,,max,
21
1
yxMICyxkMIC
yxMICyxC inv
(2)
where 21, .
For “straightforward” solution of (2) it is pro-
posed to use analytical representation of max func-
tion. In discrete case analytical representation of
max function for N variables is the following:
n n
m
nn
nm xxxxxx
2121 lim,,,max . (3)
Rewriting of (2) in terms of (3) gives the next
expression:
.,,,lim,
0
1n
N
k
n
n
inv yxkMICyxC
(4)
22 УСиМ, 2013, № 4
At this point expression (4) only overcompli-
cates the solution by increasing computational com-
plexity of max function. But the transition from
discrete values of to continuous case in (4) dra-
matically decreases computational complexity of
abovementioned firs component – computation of
values of yxMIC ,,, for all possible D.
By heading to zero, sum in (3) transforms in
the definite integral:
n
n
n
inv dyxMICyxC
2
1
,,,lim, . (5)
Derived expression (5) is the definition of so-
called maximum norm for analytical functions:
1
lim n n
n Dn
f f d
, (6)
the only difference is in the condition for
C (I, M (), x, y) in (5) to be nonnegative.
So as one can see, given the analytical solution
to definite integral in (5) brings the complexity of
computation of C (I, M (), x, y) for all possible va-
lues of D practically to zero. On the other
hand complexity of max component computation
heavily increased.
Image object detection. For the set of transfor-
mations target object shape changes, scale s and
rotation were chosen. Let M (b, s, ) be a mathe-
matical model of target object image with some
parameter vector b responsible for shape changes,
scale s and rotation . Let yxsbMIC ,,,,,
be some similarity measure that measures similar-
ity of an input image I (x, y) with target object im-
age model M (b, s, ) at (x, y).
The objective is to detect object of interest
M (b, s, ) with arbitrary allowable parameter vec-
tor b on arbitrary input image I (x, y) regardless to
affine transformations (in this particular case scale
s and rotation ) of target object on input image.
So described task of invariant image object de-
tection according to (1) can be represented as the
following:
yxsbMICyxC
sb
inv ,,,,,max,
,,
, (7)
where yxCinv , is some invariant similarity measu-
re, yxsbMIC ,,,,, is a similarity measure sen-
sitive to affine transformations ,s and appe-
arance changes b .
Reformulating of (7) in terms of (5) gives the
following:
n
D
n
n
inv dbdsdyxsbMICC
1
,,,,,lim
, (8)
So basically such problem formulation brings
image object detection task down to “simple” in-
tegration of n th-power of similarity measure
yxsbMIC ,,,,, over a set of model parame-
ters b and affine transform parameters s and .
2.2. Practical difficulties
Analytical representation. Even though theo-
retically (3) can be used for general object detec-
tion and recognition tasks, it is crucial for practi-
cal reasons to build proper analytical model M ()
and similarity measure C (I, M (), x, y). The main
purpose of that is practical integrability of (5) and
simplicity of final result. To fully exploit all ad-
vantages of proposed approach it is required for
nyxMIC ,,, to be analytically integrable func-
tion with respect to parameter vector . In practice
it can be very difficult to build such function de-
pending on object representation and a set of trans-
formations. So in some practical cases integration
over part of components of may have to be done
numerically.
Computational complexity. Computational com-
plexity of max part in (1) given values of C (I,
M (), x, y) for all possible values of D is ne-
glectable in comparison to computational comple-
xity of similarity measure C (I, M (), x, y) values
for all possible values of D. In contrast to clas-
sical brute force approach, proposed solution in-
verts the computational load for those two stages.
All computational complexity of similarity meas-
ure C (I, M (), x, y) for all possible values of D
collapses practicly to zero once analytical solution
of integral in (5) is found. So now all computa-
tional complexity switched to computation of
max component and depends on the value of pa-
rameter n.
Computational complexity is one of the biggest
drawbacks of that approach. The main reason for
that is representation of input image and model of
УСиМ, 2013, № 4 23
target object. Importance of that is explained on
example where for similarity measure cross-corre-
lation is chosen. Generally speaking practical rep-
resentation of input image I and target object model
image M () would always be in a form of super-
position of their parts:
N
i
iII and
K
j
jMM
respectively. Given above, the n th-power of simi-
larity measure can be represented as follows:
.
n
N
i
K
j
ji
n
K
j
j
N
i
i
nn
MI
MIMIC
(9)
Thus computation complexity grows polyno-
mially with the growth of parameter n .
In practice b does not go to infinity but is cho-
sen depending on type of object of interest and
input image I (x, y) content, to be sufficiently large
enough to separate useful correlational peaks from
noise ones.
3. Basic experimental results
In this section basic experimental results ob-
tained for described above approach for image
object detection are presented. To make compu-
tation as simple as possible triangle was chosen as
target object. Triangle was represented as a super-
position of three line segments. For similarity mea-
sure cross-correlation measure was chosen.
Computational results sown on fig. 2 (in co-
lumns 2 and 3) depict shape invariant detection
of triangle on the input images (first column of
fig. 2). Contour image of triangle was modeled by
means of ASM [2, 5] with one-dimensional pa-
rameter vector b . Fig. 1 shows the possible shape
changes of target object depending on value of
shape parameter b. The difference between second
(for n = 1) and third (for n = 3) columns of fig. 2
shows that proposed approach allows to signifi-
cantly amplify useful correlational signal by sim-
ply increasing values of parameter n.
Fig. 3 shows results of affine invariant detec-
tion of target object. Each input image (first col-
umn on fig. 3) contains two triangles. The right-
most triangle corresponds to model image of tar-
get object and the leftmost triangle is a target ob-
ject triangle under some scale s and rotation
transformations. Second column depicts values of
invC computed for n = 2. As fig. 3 shows, picks of
invC correctly indicate the location of the target
object, subjected to affine transformations.
Experimental results shown on fig. 4 demon-
strate affine invariant detection of target object in
noisy input image. The noise takes a form of a
randomly placed line segments on input image.
The strongest pick of invC corresponds to a true
location of target object on input image. Due to
presence of noise invC contains numbers of addi-
tional picks with smaller amplitudes.
a b c
Fig. 1. ASM generated triangle shape samples: a) 1b ;
b) 0b ; c) 1b
Fig. 2. Shape invariant triangle detection results: first column
shows input images (top corresponds to object with 0b ,
bottom-b = 0,5 ); second and third columns show shape in-
variant Cinv for 1n and 3n respectively.
Even though experiments were conducted on a
such simple object as triangle, it should be noted
that there is no algorithmical restrictions on the
complexity and topology of a target object shape.
It is only a computational complexity (or to be
more precise, the number of parts the target object
is represented with) that can put restrictions on
chosen target object.
24 УСиМ, 2013, № 4
a
b
c
d
e
f
Fig. 3. Affine invariant triangle detection results: the first column
shows input images that contains triangles with different
scales and rotations: a – s = 1, = 0; b – s = 1, = 10; c –
s = 1, = 45; d – s = 0,7, = 0; e – s = 1.3, = 0; f – s = 0,8,
= 70); the second column show values of Cinv for n = 2
УСиМ, 2013, № 4 25
Fig. 4. Affine invariant triangle detection results: left image: input
image with random noise; right: values of Cinv for input image
4. Conclusions
Proposed in this paper approach to invariant
object detection has two main advantages. The
first advantage of formulated in (5) approach is its
generality. Expression (5) neither put any con-
strictions on the type and nature of objects to be
detected, nor it puts any constrictions on set of
transformations we want to achieve invariance to.
The second advantage is absence of optimization
procedure. The detection process is straightfor-
ward computation of expression (5). It is a strong
statement of this approach because majority of
existing fundamental object detection techniques
strongly rely on the search of the optimum solu-
tion to a certain optimization task. And as a result
they suffer from all the existing problems in the
domain of multivariable nonlinear functions opti-
mization.
To be able fully exploit advantages of the
proposed approach one would have to cope with
few of its drawbacks. The major disadvantage is
the requirement of analytical integrability of
nyxMIC ,,, over . Another disadvantage is
high computational complexity of (5) for large
values of n .
Presented experimental results for image object
detection justify the validity of the proposed
approach. It allows one to build an object detec-
tion system invariant to certain set of transforma-
tion as long as they can be properly represented in
yxMIC ,,, .
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E-mail:teodor_mandziy@ipm.lviv.ua
© Т.С. Мандзий, 2013
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-83176 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0130-5395 |
| language | English |
| last_indexed | 2025-12-07T17:50:23Z |
| publishDate | 2013 |
| publisher | Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України |
| record_format | dspace |
| spelling | Mandziy, T. 2015-06-16T14:40:04Z 2015-06-16T14:40:04Z 2013 Nonoptimizational approach to invariant object detection / T. Mandziy // Управляющие системы и машины. — 2013. — № 4. — С. 20-25. — Бібліогр.: 13 назв. — англ. 0130-5395 https://nasplib.isofts.kiev.ua/handle/123456789/83176 004.932.72 Предложен новый подход к инвариантному обнаружению объектов. Рассмотрен общий случай инвариантности к необходимому набору преобразований. Модельные экспериментальные результаты, подтверждающие эффективность предложенного подхода, представлены задачей обнаружения объектов изображений. A new approach to invariant object detection is proposed. A general case solution for invariance to required set of transformation is considered. Basic experimental results that justify proposed approach are presented on the task of image object detection. Запропоновано новий підхід до інваріантного виявлення об’єктів. Розглянуто загальний випадок інваріантності до необхідного набору перетворень. Модельні експериментальні результати, які підтверджують ефективність запропонованого підходу, представлені задачею виявлення об’єктів зображень. en Міжнародний науково-навчальний центр інформаційних технологій і систем НАН та МОН України Управляющие системы и машины Автоматическая обработка и распознавание изображений Nonoptimizational approach to invariant object detection Неоптимизационный подход к инвариантному обнаружению объектов Неоптимізаційний підхід до інваріантного виявлення об’єктів Article published earlier |
| spellingShingle | Nonoptimizational approach to invariant object detection Mandziy, T. Автоматическая обработка и распознавание изображений |
| title | Nonoptimizational approach to invariant object detection |
| title_alt | Неоптимизационный подход к инвариантному обнаружению объектов Неоптимізаційний підхід до інваріантного виявлення об’єктів |
| title_full | Nonoptimizational approach to invariant object detection |
| title_fullStr | Nonoptimizational approach to invariant object detection |
| title_full_unstemmed | Nonoptimizational approach to invariant object detection |
| title_short | Nonoptimizational approach to invariant object detection |
| title_sort | nonoptimizational approach to invariant object detection |
| topic | Автоматическая обработка и распознавание изображений |
| topic_facet | Автоматическая обработка и распознавание изображений |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/83176 |
| work_keys_str_mv | AT mandziyt nonoptimizationalapproachtoinvariantobjectdetection AT mandziyt neoptimizacionnyipodhodkinvariantnomuobnaruženiûobʺektov AT mandziyt neoptimízacíiniipídhíddoínvaríantnogoviâvlennâobêktív |