Uniqueness theorem of reconstruction of preimage by its image under degenerate mapping

DOI:

, Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. One of the urgent problems in the construction of computer vision systems is the problem of determining the spatial orientation and spatial position of an object from a photograph. For example, this task is especially important for autonomous driving systems, where the positioning of nearby vehicles is a key issue for autonomous vehicles in an urban environment. In this regard, for example, in 2019, the Baidu Robotics and Autonomous Driving Laboratory together with Peking University, set an appropriate task for the Kaggle community and provided more than 60,000 copies of three-dimensional cars marked from 5,277 real images. In this article, we formulate some results that are the mathematical basis for substantiating methods for solving the above-mentioned reconstruction problems in computer vision systems.

We choose the finite subset P C R 3 as a 3D model of objects in space. Let n : R ³ ^ R 2 be central projection which is defined by formulas

n(x,y,z) = (x/z,y/z\ z = 0.

It is clear that the set P can’t be reconstructed by its image P ‘ = n(P). But if we introduce some restrictions on the structure of finite set P we can hope that this reconstruction will be possible. Note, that earlier we considered some problems which are closed to the considered here:

• •    the problem of determination the plane of the circle in R ³ by its projection [1];

• •    the problem of determination the spatial triangle with defined angles by plane triangle of its projection [6];

• •    the problem of the surface reconstruction for objects in space by its image [2];

• •    the problem of determination the profile function of the visible part of the rotation surface by its projection [5].

In addition, we developed machine learning models using artificial Convolutional Neural Network (CNN) and performed a series of experiments for determination of the spatial orientations vehicles on the photographs [3; 4].

The goal of this article to give mathematical foundations for explanation of possibility the solving of the reconstruction problem. The key statement to solve this problem is uniqueness theorem. In other words, we propose that structure of the set P is described by the admissible transformations F : R ³ ^ R ³ of this set. Uniqueness theorem presuppose that if n(P) = n(F(P)) then P = F(P ).

Now we give strict mathematical formulations. Let P C R ³ be some finite set. Consider some family Ф of differentiable transformations of R ³ . What the conditions we set for this family which imply the fact: the equality n(P) = n(F (P ‘ )) for some transformation F : R ³ ^ R ³ , F E Ф leads to P ‘ = P . We shall consider general case of degenerate mappings.

Let / : R ” ^ R ^m , 1 < m < n be some differentiable mapping. We consider Lie group G with unit e E G of differentiable transformations of R ” and let Л о be its Lie algebra. It is well known that for each X E Л о it corresponds vector field X * in R ” such that exp(tX ) be local 1-parameter of local transformations of the vector field X * (see, for example, Proposition 4.1 in [7]). We define the following set for each X E Л о

M _x = { ж E R ” : X * (ж) E Ker d/ } .

Example. Let G be a group of parallel transformations in R ” . Then Л о is isomorphic to R ” . Then if X E Л о , then X * is constant vector field X * (ж) = X.

Hausdorff distance is defined by the formula dn(Л, B) = max{sup inf |a — b\, sup inf |a — b\}.

_aE A ^ ^ ^в        ь е в ^{a E A}

1.    Uniqueness theorem

Consider some finite set P C R ” such that V X E Ag the set P is not subset of M _x .

Theorem 1. For all g ₀ E G there is some neighborhood U C G of g ₀ such that if for some g E U it holds

/ (g(P)) = / (g o (P))

then g = g o .

For the proof of theorem 1 we use the following result.

Lemma 1. Let d _H (A, В ) denotes Hausdorff distance between subsets A, В C R ^m . Consider curve Y t = exp(tX) for X E A _G . We put

h(t) = d _H (/ (P ),/ (Y t (P))).

Then

dh(t) dt t=o

= max ^{| d/(X} * (p)) | .

P E P

Proof. We note that there exists point p E P such that for all sufficient small t > 0 we have dH(/(P),/(Yt(P))) = |/(p) -/(Yt(p))|.

It follows from the fact that the nearest point in the set / (Y _t (P ) for P E P is the point / (Y t (p)). We set

q^(t) = ^/(p) ^{- /}W,  ^^(t) = ША

It is clear that ^(0) = 0 and g‘(0) = lim AX -d/ "„'    )= -d/(X•).

/ > 0 t            d dt t=o)

Therefore

p‘(0) = lim t>o

< 9^(tW^(t) ) ^^(t)

k‘(0) | 2

H ^z (0) ’

and finitely p ‘ (0) = | g^/(0) | = | d/(X * ) | .

Proof of Theorem 1. It is obviously that it is sufficient to prove theorem for g ₀ = e. We put h x (t) = d H (/(P),/ (Y t (P))).

From lemma 1 it follows that for all X E Ag there exists t _x >  0 such that h' _x (t) > 0 for t E [0, t _x ]. We define

U = {g E G : g = exp(tX), t E [0,t _x ] } .

Now we note that if g = exp(sX) E U for some X E A _G and s E [0, t _x ] such that / (g(P )) = /(P) then h _x (0) = h _x (s) = 0. Therefore h' _x (t * ) = 0 for some t * E [0,t _x ]. It contradicts to definition of U . Theorem is proved.

Corollary 1. Let M C R 3 some polyhedron and P = V (M ) is the set of its vertices. Consider some rotation R : R 3 ^ R 3 by sufficiently small angle a >  0 . If п о R(P ) = n(P) then a = 0 .

2.    Stability Theorem

In addition to the uniqueness theorem, to substantiate the possibility of reconstructing objects, we formulate and prove a stability theorem. This theorem assumes that with a small deformation of the projection of the object, the result of the reconstruction also undergoes small deformation. Let us move on to the exact wording.

Let / : Rn ^ Rn be some smooth mapping such that detd/ > 0

everywhere. We fix some positive A > 0. Let D = d/ • d/ ^T and 0 < A ₁ <  ... <  A _n , E ₁ , ...,E _n eigenvalues and eigenvectors of matrix D. Here ( • ) ^T denotes a transposed matrix of d/ . We suppose that it exists 1 <  к <  n such that A <  A _k+1 <  ... <  A _n . We denote by L _k -subspace spanned by vectors E ₁ , ...,E _k .

For each X E Ag and for 0 <  0 <  n/2 we define the following set

M x (0) = { ж E R ” : value of angle Z (X * ,L _k ) < 0 } .

For X E Ag we put g = exp(X) and Y t = exp(tX) t E [0,1]. Let r(p) = { y _t (p),t E E [0,1] } be integral curve of the vector field X * which joins the points p and g(p). We define the following values

L(g,P) = max |r(p)|, Z(g,p) = max |/(r(p))|, PEP                     PEP where |r(p)| and |/(r(p))| denote the length of the curves.

Theorem 2. Let X E Ag, g = exp(X). We suppose that for some 0 < 0 <  n/2 and for all t E [0,1] it holds

Y t (P ) H M x (0) = 0 .

Then

^L(g^,P’ <7^ .

• V A sin 0

Lemma 2. If ж E M _x (0) then

| d/(X * ) | > | X * |V Asin0.

Proof. Indeed, we have

| d/(X * ) | ² = ( d/(X * ),d/(X * ) ) = ( d/ ^T • d/ (X * ),X * ) = ( D(X * ),X * ) .

Let a ,г = 1, 2, ...,n be the angles between vector X * and eigenvectors E _j . Then

( D(X * ),X * ) = | X * | ² ^2 A j cos ² a j .

г=1

We denote by a the angle between vector X* and subspace Lk. It is clear that k cos2 a = 52 cos2 Oj. г=1

Therefore

{D(X * ),X * ) = | X * | 2

(£ i=1

X _i cos ²

П a + Xi cos2

i=k+1

O i

≥

> | X * | ²

(£ i=1

(X _i — A) cos ² a + A

≥

> | X * | ² ( 1 — A) cos ² a + A ) > | X * | ² Asin ² a > | X * | ² Asin ² 0.

Proof of Theorem 2. Let p E P be point such that

| Г(р) | = L(g,P ).

Using Lemma 2 we conclude

l(g,P) >\J(Г(р))| = / AlL(p2) dl dt =

j ^{| d}f ⁽Yt⁽PУ^{) |}

dt > V A sin 0 J

M(p) | dt = VAsin 0 | f (Г(р)) | = VAsin 0L(g,P ).

NOTE

• ¹    This work was supported by the Ministry of Education and Science of Russia (the project “Development of Virtual 3D Reconstruction of Historical Objects Technique”, scientific theme code 2019-0920, project number in the research management system FZUU-0633-2020-0004.