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9TECHNICAL SCIENCES
THE USE OF MOMENT OF INERTIA IN RECOGNITION OF INVARIANT IMAGES TO AFFINE
TRANSFORMATIONS
Mutallimova A.
Azerbaijan State Oil and Industry University, Baki, The Rebuplic of Azerbaijan, Assistant of the Department of Instrument Engineering
Abstract
The use of the moment of inertia for the recognition of images subjected to affine transformation is presented. The advantages and restrictions of using the moment of inertia during recognition are shown. The dependences of the moment of inertia of the section on the scale and the rotation of the image are presented.
Keywords: affine transformations, moments of inertia, recognition.
Introduction. Apart from mechanics, moments of inertia have found their application in the recognition of objects according to images [1,2,3]. The description of an object, in its turn, is a two-dimensional projection of a three-dimensional object, and that is why the formulas of the moments of inertia of the section are true for the descriptions of objects. According to the images, in the recognition of an object with the help of technical vision systems, certain difficulties associated with the linear transformations of images (orthogonal displacement and rotation on the projection surface) arise. Currently, in the process of object recognition, the solution of this problem is carried out by directly comparing each situation of the obtained image of the object (recognized image) with the reference image. If in any case the recognized image is placed exactly on top of the reference image, then it is possible to advance an idea about the conformity of the recognized object to the reference. However, such an approach to recognition has a significant drawback - the analysis requires significant time. Problem statement
The use of moments of inertia as the direction of the image significantly increases the recognition of objects according to the images [1,2,3]. As it is known, the moment of inertia of the section is determined by equations (1) ^ (3) [4] during the rotation of the axes:
Ju = Jx 'cos2 9 + JY ' sin2 Jxy • sin29 ;(1) Jv = Jx • sin2 9 + JY • cos2 9+ J^ • sin29 ;(2)
J - J
Juv = Jxy ' cos 29 + x Y • sin 29 ,(3) where
J X,JY,JXY
are axial and centrifugal inertia moments of the section relative to the initial axes; JuJyJuy are axial and centrifugal inertia moments of the section relative to the rotating axes; 9 is the angle of rotation.
Taking JX,JY,JXYfor for the moments of inertia of the reference image of the object, and
J , J , J for the moments of inertia of the recognized image of the object, and solving the above mentioned equations, the values of 9 angle are obtained, that is, the possible conditions of the recognized image of the object on the reference image, i.e. after initial calculations, each selected situation is limited to several situations. The process of recognizing an object according to its description is carried out by performing the rotation of the image, recognized by direct comparison of the image calculated and obtained by reference, to 9 angle. This approach ensures sufficient reliability of the recognition.
Descriptions of the objects have affine transformations. As a result, there is an additional problem of image distortion, which significantly slows down the recognition process. There is no information in the literature about the dependence of moments of inertia of the section during affine transformation. Therefore, the method of moments of inertia for distorted images is not practically used.
It follows from the above mentioned that finding the dependencies of the moments of inertia during the affine transformations of images is a topical scientific and technical issue. Its solution will allow to apply the moment method of a wide coverage of the image and increase the efficiency of recognition.
Problem solving method
In the article, the result of the dependence formulas of the moment of inertia is carried out by the analysis of the free shaft cross-section defined by the straight line (Fig. 1).
The section forms the surface area. Two mutually perpendicular coordinate axes Ox and OY were drawn from the center 0.
dSj elementary area has xl and y coordinates
showing the distance from the geometric center of the elementary area to axes OY and Ox, respectively.
As it is known from the course "Resistance of materials", the axial and centrifugal moments of inertia of the section are determined by the formula [4]:
Figure], Free shaft cross section
Jxi = J YiX =jj y12dx1dy1; (4)
Si Di
JY1 = J x12dS1 =JJ x12dx1dy1 ; (5)
Si D1
JXY1 = J (x1 • y1)dS1 =JJ (x1 • y1)dx1dy1 ,(6)
where JX1 is the axial moment of inertia about the section relative to axis OX; JY1 is the axial moment of inertia about the section relative to axis OY; JXYi are the centrifugal moment of inertia of the section relative to axes OXY; dSj is elementary area; xi- dSj is the distance from the geometric center of the elementary area to the OX axis; y - dS is the distance from the geometric center of the elementary area to axis OY; is the area of the sceiton; Dj is the integration area; dx is the dimension of dS elementary area on axis OX; dy is the dimension of dS elementary area on axis OY.
Solving equations (4) ^ (6) , the following equations are obtained
_ Axi • Ayi3 _ Axi3 • Ayi
JXi ~ ; JYi
JXY1
3
Ax^ • Ay12 4
3
(7)
where Axj is upper integration limit on axis OX; Ay is upper integration limit on axis OX.
As affine transformations, the article considers the change in the scale of an image with a curve around a horizontal or vertical coordinate axis. This corresponds to the change in the size of section M times on axis OX and N times on axis OY with respect to the point 0 given by a dotted line (Figure 1).
Similarly, in the formulas (4) ^ (6), the axial and centrifugal moments of inertia of the distorted sections are determined by the following formulas:
2jo ff 2
J v>
JY2
J (x
= (x
= J y22dS2 =JJ y2 2 dx 2 dy 2
S2 D2
J x22dS2 =JJx22dx2dy 2 ;
S2 D2
y2)dS2 =JJ(x 2 • y 2 )dx2dy 2
(8)
(9)
■(10)
where J„, is axial moment of inertia of the resized
JY2 is axial moment of
section relative to axis OX;
1 2
inertia of the resized section relative to axis OY; JXY2
is the centrifugal moment of inertia of the resized section relative to axis OXY.
For the case under consideration, the area of the section and the area of integration change M • N times. The upper limit of integration on axis OX and the size of the elementary areas on axis OX change M times. The upper integration limit on axis OY and the size of the elementary areas on axis OY change N times. Thus, if the geometric dimension of the elementary area retains its coordinates during the scale change, then the distance from the geometric centers of the elementary areas to axes OX and OY, as well as the number of elementary areas will remain unchanged.
Then the following equation will be obatined:
D2 = M ■ N - Dt;
dS2 = M -N ■ dSt;
dy2 = d(N ■ yt); x2 = x,
dx2 =d(M-x0l
y2 = yi J
(11)
S
D
J
2
S
D
2
2
where dS2 is the elementary area; x2 - dS2 is the distance from geometric center of the elementary area to axes OX; y - dS2 distance from the
geometric center of the elementary area to axis OY; S2 is the section area; D2 is the integration area; dx2 -dS2 is the size of the elementary area on axis OX; dy2 - dS2 is the size of the elementary area on axis OY; Ay2 is the upper limit of integration on axis OY. Formulas (8)^(10) are obtained respectively by
solving them for D2.
X v9 y 2
dS2, Ax2 , Ay2, dx2, dy2
JX2 = M2 • N4 • JX1 ;
(12)
Comparing the results of equation (12) with the results of equation (7), the following equation can be obtained:
JY2 = M4
• N2 • J
Y1
J = M3 • N3 • J
JXY2 M N JXY1
(13)
If two mutually perpendicular coordinate axes OU and OV rotated counterclockwise to angle 9 (Figure 1) relative to axes OX and OY are passed through the center of section 0, then according to formulas (1) ^ (3) moments of inertia are related to the following equations when resizing the section and rotating the axes.
• cos2 9+J72 • sin2 9- JXY2 •2 9+ JY2 • cos2 9+ J
JU2 JX2
JV2 JX2 ' sin
JU2 = M2 JV2 = M2
' Y2 r4
• sin 2<;(14) XY2 • sin2< ;(15)
N4 • J •N4 • J
X1
X1
•cos2 < + M4 • sin2 < + M4
N2 • J
N2 • J
JUV2 = J
XY2
T - T • cos2< + Jx2 Jy2 2
• sin 2<. (16)
Substituting equation (13) into equations (14) ^ (16), the equation of dependence of the moment of inertia according to the change in the size of the section M times on axis OX and N times on axis OX relative to the center and during the rotation of the axes of the section is obtained:
Y1
Y1
sin2 <-M3 • cos2 < + M3
N3 • J
N3
XY1 J XY1
sin 2< ; • sin 2< ;
Jw2 = M3 • N3 • JXY1 • cos2< +
M2 • N4 • JY1 - M4 • N2 • J,
2
■ • sin 2< .
(17)
(18)
(19)
(17) ^ (19) is a system of three equations with three variables. Solving these equations, the values of M, N and 9 are determined. If the recognized image is rotated to angle 9 , according to quantities M and N, resizing, directly comparing the obtained image with the reference, and the process of recognizing the object is carried out.
Computer modeling. Computer modeling was performed to verify the accuracy of the obtained theoretical research.
In formulas (17) ^ (19), the positive angle of rotation of the section axes is calculated counterclockwise. Rotation of the section axes counterclockwise is equal to rotation of the image clockwise relative to the coordinate origin.
Figure 2 presents the initial image. In Figure 3 a and 3b, the dimensions of the initial image were modified twice on axis OX and three times on axis OY, and three times on axis OX and twice on axis OY, respectively. In Figure 3c and 3d, these images rotated 35° clockwise respectively.
X
Figure 2. Initial image
The results of the calculations showed that the area of the initial image and the moment of inertia: S= 5410;
Jx1 = 7.636049454 •1010; Jy1 =
1.671401647 • 1010; Jxy1 = 92743630
Table 1 shows the area of the resized image and the results of the calculation of moment of inertia.
Table 2 shows the area of the image resized and rotated 350, the results of the calculation of moment of inertia.
I
Figure 3. Modified images
Table 1.
Resizing the image The area and moment of inertia of the image resized M times on axis OX and N times on axis OY
M=2, N=3 S = 32460 Jx2 = 2.473978449264 -1013 Jy2 = 2.4066483462 -1012 Jxy2 = 1.916964252 -1010
M=3, N=2 S = 32460 Jx2 = 1.099551409812-1013 Jy2 = 5.41500686844 -1012 Jxy2 = 1.911286998 -1010
Table 2.
Resizing and rotating the image The area, moment of inertia and rotation of the image resized M times on axis OX and N times on axis OY
M=2, N=3, 9=35 S = 32470 Ju2 = 1.737935884778-1013 Jv2 = 9.77543787682 -1012 Juv2 = 1.050338141061-1013
M=3, N=2, 9=35 S = 32465 Ju2 = 9.143978694545 -1012 Jv2 = 7.26666783102 -1012 Juv2 = 2.630119802185 -1012
Conclusion. As it is seen from Table 1, the ratios obtained in formula (13) are confirmed by the results of the calculations. It can be seen from Table 2 that the ratios obtained in (17) ^ (19) are also confirmed by the results of the calculations, i.e. reliable recognition of objects can be carried out according to the images subjected to fine transformation using the formulas of dependence of change of moment of inertia on change and rotation of the image.
REFERENCES:
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УДК 631.362:665.335.5
IMPROVING THE EFFICIENCY OF PRESSING THE MALE OF CASTOR SEEDS IN THE SCREW
PRESS
Nadikto V.
doctor of technical sciences, professor of Department of Machine Usage in Agriculture, Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Ukraine
Zhuravel D.
doctor of technical sciences, professor of Department of Technical Systems and Technology in Livestock, Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Ukraine
Chebanov A.
candidate of technical sciences, senior lecturer of Department of Electric Power Engineering and Automation, Dmytro Motornyi Tavria State Agrotechnological University, Melitopol, Ukraine
Verechaga O.
postgraduate of Department of Technical Systems and Technology in Livestock, Dmytro Motornyi Tavria
State Agrotechnological University, Melitopol, Ukraine
ЩДВИЩЕННЯ ЕФЕКТИВНОСТ1 ПРЕСУВАННЯ МЕЗГИ НАС1ННЯ РИЦИНИ У
ШНЕКОВОМУ ПРЕС1
Надикто В.Т.
Доктор технгчних наук, професор кафедри машиновикористання в землеробствI Тавртського державного агротехнологгчного унгверситету 1мет Дмитра Моторного
Журавель Д.П.
Доктор технгчних наук, професор кафедри технгчного сервгсу та систем в АПК Тавртського державного агротехнологгчного унгверситету 1мет Дмитра Моторного
Чебанов А.Б.
Кандидат технгчних наук, старший викладач кафедри електроенергетики I автоматизацИ Тавртського державного агротехнологгчного унгверситету 1мет Дмитра Моторного
Верещага О.Л.
Аспгрант кафедри технгчного сервгсу та систем в АПК Тавртського державного агротехнологгч-
ного унгверситету 1мет Дмитра Моторного
Abstract
In the article it is offered to carry out calculation of constructive parameters of screw presses of different productivity at pressing of pulp of castor seeds through experimentally established law of its compression. This law provides for the calculation of free volumes between specific numbers of pressing turns. The efficiency of pressing the castor seed pulp of each pressing turn was established, due to which the required number of pressing turns of the press was substantiated.
Анотащя
У статп запропоновано виконувати розрахунок конструктивних napaMeipiB шнекових npeciB pi3Hoi' продуктивное^ при пресуванш мезги насшня рицини через експериментально установлений закон и сти-снення. Такий закон забезпечуе розрахунок вшьних об'eмiв мiж конкретними номерами пресувальних витав. Встановлено ефектившсть пресування мезги насшня рицини кожного пресувального витка, за раху-нок чого обгрунтовано потpiбну шльшсть пресувальних витав пресу.
