The study deals with the problem of estimating the accuracy and stability of 3D face models obtained by a stereo pair. The problem of the conditionality of the fundamental matrix, which is a mathematical stereo pair model, is considered. We prove that small changes of stereo camera parameters result in small changes in the solution of the problem of reconstructing three-dimensional coordinates. Several types of three-dimensional reconstruction optimization problems that are based on quality criteria are formulated. The paper also considers the issues of determining an object orientation in a three-dimensional space by position lines. A 3D image coding system utilizing invariant moments is proposed and the theoretical sensitivity of 3D invariants to geometric distortions is investigated. These results are used to obtain scaling invariants. Designing and studying such models is the important step to solve the analysis problem and determine the proximity of images, which is therefore necessary for their clustering and recognition.
Copyright © 2018 for the individual papers by the papers’ authors. Copying permitted for private and academic purposes. This volume is published and copyrighted by its editors. In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the 1st Workshop (Summer Session) in the framework of the Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”, Tampere, Finland, 20–23 August, 2018, published at http://ceur-ws.org
· = ,
There are now many diferent methods and algorithms for face recognition related
both to the identification of local features (lips, nose, facial contours or profile) and
methods aimed at analyzing the entire image as a whole [
models have limited application, because it is dificult to use
them for face recognition if there are diferent head angles, natural facial expressions,
grimaces and other disturbances. Thereby, more and more attention is paid to 3D-models
obtained with the use of high-resolution cameras, which allow increasing the accuracy
and completeness of recognition [
A 3D image is a piecewise continuous three-variable function (, ,
) defined on a compact support nonzero integral. An example of such a function is the brightness function also known as a halftone image. The digital image typically results from the discretization of the continuous brightness function (, ,
) and is stored as a three-dimensional array is typically a power of two (for example, 64, 256) and is called the image depth. element of this array is a pixel with an intensity ranging from 0 to (, , ), where = 0, 1, . . . , − 1, = 0, 1, . . . , − 1 and = 0, 1, . . . ,
− 1. Each − 1. The
value ⊂ × × and having a finite 2.
Let us consider the issue of designing a 3D model by reconstructing an image and the problem of the conditionality of the fundamental matrix, which is a mathematical stereopair model. The three-dimensional reconstruction reduces to solving the following problem where: ⎣ ⎡ ⎤ ⎦ = ⎢ ⎥ , = ⎢ ⎡ 111 − 114 1* ⎢ 112 − 114 1* ⎢⎣ 121 − 124 2* 211 − 214 1* 212 − 214 1* 221 − 224 2* 311 − 314 1*⎤ 312 − 314 1*⎥ 321 − 324 2*⎦⎥ and shows how close the square matrix is to degeneracy. If matrix is almost degenerate, then we can expect small changes in
and to cause significant changes in . ⃦
⃦ ⃦⃦ ⃦⃦ ⃦
⃦ ⃦ ⃦⃦ ⃦
Let us consider system ( + Δ ) = + Δ . With a relative change in the right-hand side ( ⃦⃦⃦Δ⃦⃦⃦ ), relative error ⃦⃦Δ ⃦⃦⃦ can be ⃦ ( ) ⃦⃦⃦Δ⃦⃦⃦ . If = 0, then ( ) = +∞, that is a rank-deficient (degenerate) matrix. The bigger ( ), the closer matrix
to degeneracy and vice versa – the closer the matrix to the identity matrix, the closer the ( ) value to 1 and the matrix is far from degeneracy.
The study of conditionality is an important link in determining the stability of the solution to the problem of reconstructing 3D information. From this point on, let us assume that the ⃦
⃦ ⃦ ⃦⃦ ⃦ matrix is well-conditioned.
Let fundamental matrix be well-conditioned.
equation (1) are changed for small quantities and absolute term column in the
problem solution obtained by using a least-squares method will change for small value Δ = ( ) = ⃦( 0 + 1) ( 0 + 1)⃦⃦ − ⃦⃦⃦ 0 0⃦⃦ .
⃦ ⃦ ⃦ ⃦ also consider the system
and absolute term column are changed for small quantities, then the solution obtained by using a least-squares method will change insignificantly.
To do this, let us introduce disturbing matrix 1, disturbing vector 1 and require singular numbers (i.e. square roots of the eigenvalues) of matrix 1 to be bounded from above by some constant. In addition to the system (1), let us ( 0 + 1) + 0 + 1, where is a small parameter. Its solution is given by
= (( 0 + 1 )( 0 + 1))−1( 0 + 1 )( 0 + 1).
Let us consider the expression
= ( 0 + 1 )( 0 + 1)Δ, where Δ = | − 0| is a deviation. Value can be written as = ( 0 + 1 )( 0 + 1) − ( 0 + 1 )( 0 + 1)( 0 0)−1 0 0 = = 0 0 + ( 0 1 + 1 0) + 2 1 1 − 0 0( 0 0)−1 0 0− − suggest diferent criteria for finding recovered point of the problem (1), as it can be seen in Table 1. = (, , ) being an approximation Minimizing root-meansquare deviation
from the solution
system of equations (1)
the deviation
largest ⎧ Minimizing root-meansquare deviation the tetrahedron faces from rootMinimizing the meansquare of deviation coordinate of point 2 =∑︀ → ,, min ( + + + )2 2 + 2 + 2 2 = ∑︀
( − (, ))2 → mi0n one (,
) is the lowest value calcu- ⎪⎪ tions (2) lated for each of the system equa- ⎪⎪⎨ ∑︀
Types of three-dimensional reconstruction optimization problems ⎪⎪ 3 + 3 + 3 + 3 = 0 ⎪ ⎪⎩ 4 + 4 + 4 + 4 = 0 ⎪
1 + 1 + 1 + 1 = 0 ⎪ ⎪ ⎧ ⎪ ⎪ = 0 2 2 2 + = 0 22 +
= 0 + 4 +
Experimental designing and the 3D model orientation determinantion
Various afine transformations generally performed in homogeneous coordinates are used to manipulate the model. The weights of points afect the object center of mass, hence changing them
may lead to an inordinary displacement trajectory of the center of
mass and a change in the orientation of the object. It is necessary to draw a straight
spatial line to determine the position and orientation of the 3D object [
=1 ︀∑ , ︀∑ =1 ︀∑ =1 ,
=1 ︃) ︀∑ =1 , 2 =
We introduce where (,¯ ,¯ ¯) are the object centroid coordinates. The position line forms angles , , with reference axes OX, OY, OZ and its equation can be written as −¯ = −¯ = −¯ , where: , ,
are the angular coeficients of a straight line in space to be determined as a result of the problem solution; , ,
are the coordinates of an arbitrary object point.
The distance from the object point to the desired line is calculated by [( − ¯) − ( − ¯) ]2 + [( − ¯) − ( − ¯) ]2 + [( − ¯) − ( − ¯) ]2 2 + 2 + 2 . = ∑︀ =1( − ¯)2 , = ∑︀
=1( − ¯)2 , = ∑︀ =1( − ¯)2 , = ∑︀
=1( − ¯)( − ¯) , = ∑︀ =1( − ¯)( − ¯) , = ∑︀ =1( − ¯)( − ¯) . variables , , the following system:
Let A, ..., F be called moments of inertia and be subsequently used as constant coeficients.To solve the problem, we have to find partial derivatives of function S of and equate them to zero. After the necessary transformations we get ⎧ ⎪ ⎩ ⎨ ⎪ ((
+ ) − (( + ) (( + ) − − − − − ) = ((
+ ) ) = ((
+ ) − ) = (( + ) − −
− − − ), ), ).
These equations can be rewritten as follows:
Herein,(×), (·) are the signs of vector and scalar products. Thus, the system can be rewritten as where =
︁( some number.
︁) , = ⎜⎝ ⎛ × = 0, | | = 1, ⎞ ⎠ orthogonal and correspond to the directing vectors of the ellipsoid determining the object orientation in space. The lengths of the ellipsoid axes correspond to the eigenvalues. Moreover, the maximum characteristic number corresponds to the desired direction of the position line. The vector corresponding to the second largest eigenvalue determines the object’s rotation direction; the third eigenvector determines the rotation by an angle around the main axis.
4.
Let (, ,
) be the function describing the brightness value of points with coordinates (, ,
) in a 3D space. It is required to build moments invariant to the group of afine
transformations for correct correlation of images [
is an image pixel coordinate determination area; (¯,¯,¯) is the object’s
centroid. In accordance with the analysis carried out by using diferent sources, the
following moments were selected [
1 = 200 + 020 + 002, 2 = 200 020 + 200 002 + 020 002 − 2101 − 2110 − 2011, 3 = 200 020 002 − 002 1210 − 020 1201 − 200 0211+ 1 = 2003 + 6 2012 + 6 2021 + 6 2030 + 6 2102 + 15 2111 − 3 2102 120 + 6 2120− −3 021 201 + 6 2201 − 3 003( 021 + 201) − 3 030 210 + 6 2210−
The sensitivity of 3D invariants to linear image distortions is established. An evaluation of the theoretical sensitivity of 3D invariants to geometric distortions is presented in Table 3
The sensitivity of 3D invariants Moment 1 2 3 1 2 3 4 5 Sensitivity 2 4 6 6 4 6 6 6
The results can be used to obtain invariants for scaling. To do this, we can use: = √︀ 200 + 020 + 002 and perform the valuation by dividing moments 1, ..., 5 by the corresponding values of the coeficients from Table 3.
Proposition 4. Normalized moments 1, ..., 5 are 3D invariants of the operations of rotate, tranlate and scale.
Table 4 exemplifies the sensitivity values of the invariants and the scaling coeficients for the chosen 3D face model.
An experimental study of as the scaling facto
0.5 0.25
All the calculations were carried out by using the software of the Matlab modeling system. The study of the moments’ properties shows that it is possible to solve human face recognition problems utilizing their geometric invariants.
5.
Stereo pair-based evaluations of the stability of three-dimensional image reconstruction against fluctuations are obtained provided that the fundamental matrix is initially well conditioned. The statements of various optimization problems of three-dimensional reconstruction based on quality criteria are given. A system of 3D invariants is defined and their stability against image fluctuations is investigated. The developed algorithms are expedient for using as a part of systems searching for faces from photos.
The publication has been prepared with the support of the “RUDN University Program 5-100”.