Nonlinear Invariants of Planar Point Clouds Transformed by Matrices
NNONLINEAR INVARIANTS OF
PLANAR POINT CLOUDS
TRANSFORMED BY MATRICES
Stelios Kotsios, Evangelos MelasFaculty of Economics,Department of Mathematics and Computer Science,National and Kapodistrian University of AthensSofokleous 1, Athens 10559, Greece [email protected] , [email protected] Abstract:
The goal of this paper is to present invariants of planarpoint clouds, that is functions which take the same value before and aftera linear transformation of a planar point cloud via a × invertible ma-trix. In the approach we adopt here, these invariants are functions of twovariables derived from the least squares straight line of the planar pointcloud under consideration. A linear transformation of a point cloud in-duces a nonlinear transformation of these variables. The said invariantsare solutions to certain Partial Differential Equations, which are obtainedby employing Lie theory. We find cloud invariants in the general case ofa four − parameter transformation matrix, as well as, cloud invariants ofvarious one − parameter sets of transformations which can be practicallyimplemented. Case studies and simulations which verify our findings arealso provided. a r X i v : . [ m a t h . G M ] M a y eywords: Invariants, Nonlinear Transformations, Lie Theory, PointCloud, OCR, Image Analysis, Computational Geometry.
Analysing level shapes is the key problem in many computer science ar-eas, as image analysis, geometric computing, optical character recognitione.t.c. [1, 2]. Usually, by means of the modern sensing technology, we makedetailed scans of complex plane objects by generating point cloud data,consisting from thousands or millions of points. Then we study the under-lying properties, either by creating appropriate models or by discoveringproperties which remain constant under sets of transformations or underthe influence of noise distortions.In particular, when we deal with planar set of points, a basic approach,which is widely used, is that of determining quantities which can character-ize collectively the behaviour of the whole set, as well as its change, whena transformation is applied to it. In other words, we determine quanti-ties which can represent the planar set of points under consideration, as awhole.One approach along this line is the classical work of Ming − Kuei Hu [3],who introduced the moment invariants methodology, followed in the courseof time by many others [4, 5, 6, 7], to mention but a few. The key ele-ment of their approach was to introduce the so − called moments of planarfigures, in order to identify a planar geometrical figure as a whole, andthen to study their invariants under translation, similitude and orthogo-nal transformations.In the present paper, we consider planar set of points, called henceforthpoint cloud or cloud of points. We advocate a different approach, andin order to characterize collectively the behaviour of the whole cloud ofpoints we introduce two variables M and H . These variables stem from2he least squares line assigned to these points. In fact M is the slope ofthis line, and H is a variation of the y − intercept of this line.Any transformation of the cloud of points, by means of a 2 × M and H . We assume throughout this paper that any 2 × M and H . By the term “invariants”[8] we mean functions which take the same value at the original and atthe transformed values of M and H , when a cloud of points undergoes atransformation with a 2 × − parameter set of transformations. This is the case when the entriesof a transformation matrix are functions of one parameter only. In thiscase a general solution is found by using Lie theory implemented withsymbolic computation.At first sight it might seem that restricting ourselves to a one − parameterset of transformations, useful as it may be, cannot be of great use. How-ever, this is not the case, because as we point out in section 5.2.1 any given matrix belongs to one such one − parameter set of transformations.As a result we find a family of cloud invariants for any given matrix andthis certainly lends itself to practical implementation.By practical implementation we mean that these invariants can be usedas a tool for studying changes of planar figures and for creating propersoftware which monitors and displays these changes in real time. Thiswould have many applications in optical character recognition, as well as,3n image analysis and computer graphics techniques; icons created by thesame “source” will be readily identified.This potential application of our results suggests a direction for future re-search. The cloud of points may come from an icon which has a parabolic − like shape. In this case it is natural to look for cloud invariants which areexpressed in terms of variables which appear as coefficients, or variationsthereof, of the parabola which is the best fit for the cloud points we con-sider. Comparison with already existing methodologies, which address thesame questions, via simulations and computational experiments, will bealso the subject of future research.In section 2 we introduce the variables M and H, and we find the trans-formation of these variables which is induced from a transformation witha 2 × − parameter transformationmatrix of the cloud of points under consideration. In section 4 we find afamily of cloud invariants for a general one − parameter set of transforma-tions. In section 5 we find families of cloud invariants for various sets oftransformations. We also find a family of cloud invariants for a “linear”one − parameter set of transformations and we point out that any givenmatrix belongs to such a set. In section 6 we verify our results with sim-ulations and computational experiments in a cloud of 10.000 points. Insection 7 we close the paper with some concluding remarks.4 The Basic Quantities and their Transfor-mations
In this section we present two quantities M and H which characterizecollectively a cloud of points and serve as the independent variables ofthe invariant functions we are going to construct. They originate from theleast squares straight line fitted to the cloud of points under consideration.Let ( x i , y i ), i = 1 , ..., N , be a cloud of points on the plane. We define thequantities: M = N (cid:80) Ni =1 x i y i − (cid:80) Ni =1 x i (cid:80) Ni =1 y i N (cid:80) Ni =1 x i − ( (cid:80) Ni =1 x i ) , and , (1) H = N (cid:80) Ni =1 y i − ( (cid:80) Ni =1 y i ) N (cid:80) Ni =1 x i − ( (cid:80) Ni =1 x i ) . (2) M is the slope of the least squares straight line and H is suggested by thecalculations. We call them the linear coefficients of the cloud. Some-times M is referred as the slope of the cloud and H as the constantterm of the cloud. A transformation of the cloud of points under theaction of a 2 × M and H . This lasttransformation is of prime importance to our construction of invariantfunctions and so we proceed to find it. Firstly, we need a definition. Definition 2.1
Let ( x i , y i ) , i = 1 , . . . , N , be a cloud of points and A = α βγ δ , α, β, γ, δ ∈ R , a given × matrix. Let us suppose thatevery point ( x i , y i ) , i = 1 , . . . , N , of the cloud undergoes a transformation T A : x i y i → ˆ x i ˆ y i according to the rule ˆ x i ˆ y i = A x i y i .We say that the cloud is transformed under the matrix A , and in particularwe say that the cloud (ˆ x i , ˆ y i ) , i = 1 , . . . , N, is the transformation of thecloud ( x i , y i ) , i = 1 , . . . , N, under the matrix A . The transformation of M and H induced by a transformation of the cloudof points via a matrix A is given in the following Theorem.5 heorem 2.1 Let ( x i , y i ) , i = 1 , , . . . , N , be a cloud of points with lin-ear coefficients M and H . Let (ˆ x i , ˆ y i ) , i = 1 , , . . . , N , be the transforma-tion of the cloud ( x i , y i ) , i = 1 , . . . , N, under a matrix A = α βγ δ , α, β, γ, δ ∈ R . Let ˆ M and ˆ H be the linear coefficients of the cloud (ˆ x i , ˆ y i ) , i = 1 , , . . . , N . Then the following relations hold ˆ M = ( αδ + βγ ) M + βδH + αγ αβM + β H + α , (3)ˆ H = 2 γδM + δ H + γ αβM + β H + α . (4) Proof:
Let ( x i , y i ), i = 1 , , . . . , N , be a cloud of points with linear coef-ficients M and H . It is convenient to define the quantities M n , H n , and D , as follows M n = N N (cid:88) i =1 x i y i − N (cid:88) i =1 x i N (cid:88) i =1 y i , (5) H n = N (cid:88) i y i − ( (cid:88) i y i ) , (6) D = N (cid:88) i x i − ( (cid:88) i x i ) . (7)The relations (1) and (2) which define the linear coefficients M and H ofthe cloud of points under consideration can now be written in the followingshorter form: M = M n D , H = H n D . (8)A transformation of the cloud of points under a matrix A = α βγ δ ,α, β, γ, δ ∈ R , induces a transformation to the quantities M n , H n , D. Theinduced transformed values ˆ M n , ˆ H n , ˆ D , which are assigned to the cloud(ˆ x i , ˆ y i ), i = 1 , , . . . , N , are calculated as follows:ˆ M n = N (cid:88) i ˆ x i ˆ y i − (cid:88) i ˆ x i (cid:88) i ˆ y i == N (cid:88) i ( αx i + βy i )( γx i + δy i ) − (cid:88) i ( αx i + βy i ) (cid:88) i ( γx i + δy i ) =6 αγ N (cid:88) x i − (cid:32)(cid:88) i x i (cid:33) + ( αδ + βγ ) (cid:34) N (cid:88) i x i y i − (cid:88) i x i (cid:88) i y i (cid:35) ++ βδ (cid:34) N (cid:88) i y i − (cid:32)(cid:88) i y i (cid:33)(cid:35) = αγD + ( αδ + βγ ) M n + βδH n , (9)ˆ H n = N (cid:88) i ˆ y i − (cid:32)(cid:88) i ˆ y i (cid:33) = N (cid:88) i ( γx i + δy i ) − (cid:34)(cid:88) i ( γx i + δy i ) (cid:35) == γ N (cid:88) i x i − (cid:32)(cid:88) i x i (cid:33) + δ (cid:34) N (cid:88) i y i − (cid:32)(cid:88) i y i (cid:33)(cid:35) + (10)+2 γδ (cid:34) N (cid:88) i x i y i − (cid:88) i x i (cid:88) i y i (cid:35) = γ D + δ H n + 2 γδM n , and,ˆ D = N (cid:88) i ˆ x i − (ˆ x i ) = N (cid:88) i ( αx i + βy i ) − (cid:34)(cid:88) i ( αx i + βy i ) (cid:35) == α N (cid:88) i x i − (cid:32)(cid:88) i x i (cid:33) + β (cid:34) N (cid:88) i y i − (cid:32)(cid:88) i y i (cid:33)(cid:35) + (11)+2 αβ (cid:34) N (cid:88) i x i y i − (cid:88) i x i (cid:88) i y i (cid:35) = α D + β H n + 2 αβM n . From relations (8), (9), (10), and (11), we conclude that the linear coef-ficients ˆ M and ˆ H of the cloud (ˆ x i , ˆ y i ), i = 1 , , . . . , N , are given by therelations (3), (4) and the theorem has been proved. (cid:3) We denote the set of transformations (3) and (4) by T ( A ) . These aretransformations of the formˆ M = M ( M, H, α, β, γ, δ ) , ˆ H = H ( M, H, α, β, γ, δ ) . (12)If A is a matrix with entries ( α, β ; γ, δ ), then we can associate to it anelement of the set T ( A ) , namely the transformation given by (3) and (4).We denote this transformation by T ( A ) ( α,β,γ,δ ) . The following remarks arein order regarding this association: 7
The set of transformations T ( A ) form a Lie group, under the usualcomposition of transformations, if and only if the set of matrices A form also a Lie group, under the usual multiplication of matrices,namely the group GL (2), i.e., the group of 2 × • This association is not one − to − one. Indeed, one can easily checkthat T ( A ) ( α,β,γ,δ ) = T ( A ) ( κα,κβ,κγ,κδ ) , κ ∈ R, κ (cid:54) = 0 . Therefore all matrices κA are associated to the same element T ( A ) ( α,β,γ,δ ) of T ( A ) . • We can make the association between the sets A and T ( A ) one − to − oneby assigning arbitrarily a fixed non − zero value to any of the entries( α, β ; γ, δ ) of the matrices of the set A . • In this paper we prefer not to make this association one − to − onebecause this may give the false impression that restrictions are im-posed to the set of transformations A which act on the cloud ofpoints under consideration. • Needless to say that the results are identical regardless of whetherwe make or we do not make the association between the sets A and T ( A ) one − to − one.We note that in the search for invariants we do not need to restrict tothe case where A , and therefore T ( A ), is a group. As it will becomeevident from the proof in the next section, and as it will be demonstratedin the example given in subsection 5.2.1, what it is really necessary isthat the set A , and subsequently the set T ( A ), must contain the identityelement. The identity elements of both A and T ( A ) are obtained when α = 1 , β = 0 , γ = 0 , and , δ = 1. For brevity we write e = (1 , , ,
1) andwe denote by e i , i = 1 , , , , its components.8 Invariants
A main objective in cloud of points theory is that of finding invariants.These are quantities which remain unchanged whenever a cloud of pointsis transformed under the action of a 2 × M and H . Therefore, we are looking for invariants which are functions of thesetwo quantities. This is formalized in the following definition: Definition 3.1
Let ( x i , y i ) , i = 1 , ..., N, be a cloud of points on the planewith linear coefficients M and H. Let (ˆ x i , ˆ y i ) , i = 1 , . . . , N, be the transfor-mation of the cloud ( x i , y i ) , i = 1 , . . . , N, under a matrix A . Let ˆ M , ˆ H bethe linear coefficients of the cloud (ˆ x i , ˆ y i ) , i = 1 , . . . , N. ˆ M and ˆ H are thetransformed values of M and H under the induced set of transformations T ( A ) . We say that a function I : R → R is a cloud invariant if and onlyif I ( ˆ M , ˆ H ) = I ( M, H ) . (13)The next theorem is the key result in our study because it provides us witha mechanism for finding cloud invariants. Its proof, is the proof in ourcase, of Lie’s theory fundamental result [10] that the nonlinear condition(13) is equivalent to a linear condition provided that the invariant functionis properly analytic. The details are as follows: Theorem 3.1
Let ( x i , y i ) , i = 1 , ..., N, be a cloud of points on the planewith linear coefficients M and H, and let (ˆ x i , ˆ y i ) , i = 1 , ..., N, be thetransformation of the cloud ( x i , y i ) , i = 1 , ..., N, under a matrix A = α βγ δ . A function : R → R , analytic in the parameters α, β, γ, δ , is a cloud invariantif and only if the following equations hold simultaneously ξ α ∂I∂M + ξ α ∂I∂H = 0 , (14) ξ β ∂I∂M + ξ β ∂I∂H = 0 , (15) ξ γ ∂I∂M + ξ γ ∂I∂H = 0 , (16) ξ δ ∂I∂M + ξ δ ∂I∂H = 0 , (17) where, ξ Q = (cid:32) ∂ ˆ M∂Q (cid:33) e = ∂ M ( M, H, , , , ∂Q , ξ Q = (cid:32) ∂ ˆ H∂Q (cid:33) e = ∂ H ( M, H, , , , ∂Q ,Q = α, β, γ, δ. Proof:
Let ( x i , y i ), i = 1 , ..., N, be a cloud of points on the plane withlinear coefficients M and H. Let (ˆ x i , ˆ y i ), i = 1 , . . . , N, be the transforma-tion of the cloud ( x i , y i ), i = 1 , . . . , N, under a matrix A = α βγ δ .Let ˆ M , ˆ H be the linear coefficients of the cloud (ˆ x i , ˆ y i ), i = 1 , . . . , N. ˆ M and ˆ H are the transformed values of M and H under the induced set oftransformations T ( A ) , given by equations (3) and (4). Let I ( ˆ M , ˆ H ) be areal − valued function analytic in the parameters α, β, γ, and δ. The Taylorexpansion of I ( ˆ M , ˆ H ) , with center e , reads: I ( ˆ M , ˆ H ) = I ( M, H ) + ( α − (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e + β (cid:32) ∂I ( ˆ M , ˆ H ) ∂β (cid:33) e + γ (cid:32) ∂I ( ˆ M , ˆ H ) ∂γ (cid:33) e + ( δ − (cid:32) ∂I ( ˆ M , ˆ H ) ∂δ (cid:33) e + 12! (cid:0) ( α − (cid:32) ∂ I ( ˆ M , ˆ H ) ∂α (cid:33) e + β (cid:32) ∂ I ( ˆ M , ˆ H ) ∂β (cid:33) e + γ (cid:32) ∂ I ( ˆ M , ˆ H ) ∂γ (cid:33) e +( δ − (cid:32) ∂ I ( ˆ M , ˆ H ) ∂δ (cid:33) e + 2( α − β (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e + · · · γ ( δ − (cid:32) ∂I ( ˆ M , ˆ H ) ∂γ (cid:33) e (cid:32) ∂I ( ˆ M , ˆ H ) ∂δ (cid:33) e (cid:33) + · · · . (18)10he form of the functional dependence of I ( ˆ M , ˆ H ) on the parameters α, β, γ, δ allows to simplify (18) in a way which suggests the conclusionof the theorem. To illustrate the point at hand we use the chain rule toevaluate the derivative (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e : (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e = (cid:32) ∂I ( ˆ M , ˆ H ) ∂ ˆ M ∂ ˆ M∂α + ∂I ( ˆ M , ˆ H ) ∂ ˆ H ∂ ˆ H∂α (cid:33) e . (19) (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e = (cid:32) ∂I ( ˆ M , ˆ H ) ∂ ˆ M (cid:33) e (cid:32) ∂ ˆ M∂α (cid:33) e + (cid:32) ∂I ( ˆ M , ˆ H ) ∂ ˆ H (cid:33) e (cid:32) ∂ ˆ H∂α (cid:33) e . (20)By introducing the quantities: ξ Q = (cid:32) ∂ ˆ M∂Q (cid:33) e , ξ Q = (cid:32) ∂ ˆ H∂Q (cid:33) e , Q = α, β, γ, δ, (21)and by noting (cid:32) ∂I ( ˆ M , ˆ H ) ∂ ˆ M (cid:33) e = ∂I ( M, H ) ∂M , and , (cid:32) ∂ I ( ˆM , ˆH ) ∂ ˆH (cid:33) e = ∂ I ( M , H ) ∂ H , (22)we can rewrite equation (20) in the following shorter form (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e = ξ α ∂I ( M, H ) ∂M + ξ α ∂I ( M, H ) ∂H . (23)By introducing the operator X α = ξ α ∂∂M + ξ α ∂∂H , (24)we rewrite equation (23) as (cid:32) ∂I ( ˆ M , ˆ H ) ∂α (cid:33) e = X α I, (25)where for short we wrote I instead of I ( M, H ) . For the second order deriva-tive (cid:16) ∂ I ( ˆ M, ˆ H ) ∂α (cid:17) e we have: (cid:32) ∂ I ( ˆ M , ˆ H ) ∂α (cid:33) e = ∂ (cid:16) ∂I ( ˆ M, ˆ H ) ∂α (cid:17) ∂α e = ξ α ∂ (cid:16) ∂I ( ˆ M, ˆ H ) ∂α (cid:17) e ∂M + ξ α ∂ (cid:16) ∂I ( ˆ M, ˆ H ) ∂α (cid:17) e ∂H = X α ( X α I ) . (26)11 similar analysis applies to the derivatives of all orders in the Taylorexpansion (18). As a result the Taylor expansion (18) reads: I ( ˆ M , ˆ H ) = I ( M, H ) + (cid:88) i =1 ( Q i − e i ) ( X Q i I ) +12! (cid:88) i,j =1 ( Q i − e i )( Q j − e j ) X Q i (cid:0) X Q j I (cid:1) +13! (cid:88) i,j,k =1 ( Q i − e i )( Q j − e j )( Q k − e k ) X Q i (cid:0) X Q j ( X Q k I ) (cid:1) + · · · . (27)For convenience, by Q we denote the vector ( α, β, γ, δ ) , and by Q i , i =1 , , , , its components.From equation (27) we conclude that when the linear infinitesimal condi-tions X Q i I = 0 , Q i = α, β, γ, δ, (28)are satisfied then I ( ˆ M , ˆ H ) = I ( M, H ). Therefore I is a cloud invariant.Conversely, when I is a cloud invariant, then I ( ˆ M , ˆ H ) = I ( M, H ) , andequation (27) gives: (cid:88) i =1 ( Q i − e i ) ( X Q i I ) + 12! (cid:88) i,j =1 ( Q i − e i )( Q j − e j ) X Q i (cid:0) X Q j I (cid:1) +13! (cid:88) i,j,k =1 ( Q i − e i )( Q j − e j )( Q k − e k ) X Q i (cid:0) X Q j ( X Q k I ) (cid:1) + · · · = 0 . (29)For every pair of values M and H equation (29) becomes a polynomialin the variables α, β, γ, δ . Consequently equation (29) can only hold if forevery pair of values M and H the coefficients of the polynomial vanish,i.e., if the following relations hold X Q i I = X Q i (cid:0) X Q j I (cid:1) = X Q i (cid:0) X Q j ( X Q k I ) (cid:1) = · · · = 0 , i, j, k ∈ { , , , } , (30)for every pair of values M and H. If X Q i I = 0 , Q i = α, β, γ, δ, the restof the relations (30) follow. Equations (28), X Q i I = 0 , Q i = α, β, γ, δ, are nothing but equations (14), (15), (16), and (17), respectively.12his completes the proof. (cid:3) Sophus Lie’s great advance was to replace the complicated, nonlinear finiteinvariance condition (13) by the vastly more useful linear infinitesimalcondition (28) and to recognize that if a function satisfies the infinitesimalcondition then it also satisfies the finite condition, and vice versa, providedthat the function is analytic in the parameters α, β, γ, and δ. It is to be noted that in the proof of Lie’s main Theorem (3.1) we usedthe following:1. The assumption that cloud invariant I is a function analytic in theparameters α, β, γ, and δ.
2. The assumption that the set of transformations T ( A ), and subse-quently the set of transformations A , contain the identity element,which is obtained when α = 1 , β = 0 , γ = 0 , and δ = 1 .
3. The chain rule for the differentiation of composite functions.Nowhere in the proof of Lie’s main Theorem (3.1) is the assumption madethat the set of transformations T ( A ) is closed under the usual compositionof transformations, or equivalently, that the associated set of matrices A is closed under the usual matrix multiplication. This will become evidentand exemplified in subsection 5.2.1 where we find cloud invariants I undera set of transformations T ( A ) which are such that the associated set ofmatrices A are not closed under the usual multiplication of matrices. As a first application of Theorem (3.1) we find the cloud invariants undera general matrix A. This is the content of the next Corollary.
Corollary 3.1
If a cloud of points is transformed via a matrix A = α βγ δ , then the only cloud invariants are: . The constant functions I ( M, H ) = c , c ∈ R.
2. The level curve I ( M, H ) = 0 of the function I ( M, H ) = HM − . Proof:
According to Theorem (3.1) a function I ( ˆ M , ˆ H ) , analytic in theparameters α, β, γ, δ, is a cloud invariant if and only if it satisfies thesystem of PDEs (14), (15), (16), and (17), which read: ξ α ( x ) ∂I∂M + ξ α ( x ) ∂I∂H = − M ∂I∂M − H ∂I∂H = 0 , (31) ξ β ( x ) ∂I∂M + ξ β ( x ) ∂I∂H = ( H − M ) ∂I∂M − HM ∂I∂H = 0 , (32) ξ γ ( x ) ∂I∂M + ξ γ ( x ) ∂I∂H = ∂I∂M − M ∂I∂H = 0 , (33) ξ δ ( x ) ∂I∂M + ξ δ ( x ) ∂I∂H = M ∂I∂M + 2
H ∂I∂H = 0 . (34)We easily find that the only solutions to the last system of equationsare:1. The constant functions I ( M, H ) = c , c ∈ R.
2. The level curve I ( M, H ) = 0 of the function I ( M, H ) = HM − (cid:3) The second invariant implies in particular that when the values of M andH are such that H = M , then their transformed values ˆ H and ˆ M aresuch that ˆ H = ˆ M . − Parameter Case
The cloud invariants under a general matrix A, given in Corollary 3.1, donot lend themselves to practical implementation. This leads us to exam-ining particular cases of A. We start by considering the one − parametercase, i.e. the case where the entries of the matrix A are analytic functionsof a single parameter ϕ . Interestingly enough it turns out that in this14ase we can find cloud invariants in closed form which can be practicallyimplemented. The first step to prove this assertion is the next Corollary. Corollary 4.1
Let ( x i , y i ) , i = 1 , ..., N, be a cloud of points on the planewith linear coefficients M and H, and let (ˆ x i , ˆ y i ) , i = 1 , ..., N, be thetransformation of the cloud ( x i , y i ) , i = 1 , ..., N, under a matrix A ( ϕ ) = α ( ϕ ) β ( ϕ ) γ ( ϕ ) δ ( ϕ ) , where α ( ϕ ) , β ( ϕ ) , γ ( ϕ ) , and δ ( ϕ ) , are real analyticfunctions of a parameter ϕ ∈ R . We assume that there exists a value of ϕ , denoted by ϕ ∗ , such that A ( ϕ ∗ ) = I , I the × identity matrix. Ananalytic function I : R → R is a cloud invariant if and only if the nextequation holds: [( H − M ) β (cid:48) ( ϕ ∗ ) − δM + γ (cid:48) ( ϕ ∗ )] ∂I∂M +2[ γ (cid:48) ( ϕ ∗ ) M − β (cid:48) ( ϕ ∗ ) HM − δH ] ∂I∂H = 0 , (35) where, δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) . Proof:
In order to find the cloud invariants we apply Theorem 3.1. Thekey point is that in the case under consideration all the entries of the ma-trix A ( ϕ ) are functions of a single parameter ϕ. This implies in particularthat equations (14), (15), (16), and (17), whose solution space are thecloud invariants, reduce to one equation: ξ ϕ ∂I∂M + ξ ϕ ∂I∂H = 0 , (36) ξ ϕ = (cid:32) ∂ ˆ M∂ϕ (cid:33) e = ∂ M ( M, H, , , , ∂ϕ , ξ ϕ = (cid:32) ∂ ˆ H∂ϕ (cid:33) e = ∂ H ( M, H, , , , ∂ϕ . Differentiation is now with respect to ϕ, that is Q = ϕ. Consequentlycloud invariants in the one − parameter case are solutions to equation (36)which reads:[( H − M ) β (cid:48) ( ϕ ∗ ) − δM + γ (cid:48) ( ϕ ∗ )] ∂I∂M +2[ γ (cid:48) ( ϕ ∗ ) M − β (cid:48) ( ϕ ∗ ) HM − δH ] ∂I∂H = 0 , where, δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) . This completes the proof. (cid:3)
15t is difficult to obtain in closed form the whole set of solutions of equation(35). However we can find, in closed form, a wide subclass of solutions ofequation (35). This is the content of the following Theorem.
Theorem 4.1
A class of solutions of equation (35), and hence a family ofinvariants of a cloud of points ( x i , y i ) , i = 1 , ..., N, when it is transformedunder a matrix A ( ϕ ) = α ( ϕ ) β ( ϕ ) γ ( ϕ ) δ ( ϕ ) , is given by: I ( M, H ) = F (cid:18) M − H ( Hβ (cid:48) ( ϕ ∗ ) − γ (cid:48) ( ϕ ∗ ) + δM ) (cid:19) , (37) where F ( . ) , is an arbitrary real valued function and δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) .We assume that α ( ϕ ) , β ( ϕ ) , γ ( ϕ ) , and δ ( ϕ ) , are real analytic functionsof a single parameter ϕ ∈ R . We also assume that there exists a value of ϕ , denoted by ϕ ∗ , such that A ( ϕ ∗ ) = I , I being the × identity matrix. Proof:
In order to find solutions of equation (35) we use the undeterminedcoefficients method. This method consists in seeking for solutions I ( M, H )of the form: F (cid:32) (cid:80) ni =1 (cid:80) mj =1 a ij M i H j (cid:80) ni =1 (cid:80) mj =1 b ij M i H j (cid:33) , (38)where a ij and b ij are unknown coefficients to be determined. By substi-tuting this particular form of the solution into equation (35) we obtainthat a polynomial in the two variables M and H is equal to zero. Theresulting condition, the coefficients of the polynomial are equal to zero,gives solution (37). This completes the proof. (cid:3) A Corollary of the previous theorem is that we can find cloud invariants,when the cloud is transformed under a matrix, provided the matrix is anelement of the one − parameter set of transformations A ( ϕ ) considered inthis Theorem. Corollary 4.2
Let ( x i , y i ) , i = 1 , ..., N, be a cloud of points and let thiscloud be transformed under a matrix A = a a a a , aij ∈ R . If here exist real valued, analytic, functions α ( ϕ ) , β ( ϕ ) , γ ( ϕ ) , δ ( ϕ ) and val-ues ϕ ∗ , ϕ , such that:1. α ( ϕ ∗ ) = 1 , β ( ϕ ∗ ) = 0 , γ ( ϕ ∗ ) = 0 , δ ( ϕ ∗ ) = 1 α ( ϕ ) = a , β ( ϕ ) = a , γ ( ϕ ) = a , δ ( ϕ ) = a then, the quantity I ( M, H ) = F (cid:18) M − H ( Hβ (cid:48) ( ϕ ∗ ) − γ (cid:48) ( ϕ ∗ ) + δM ) (cid:19) , where F ( . ) is an arbitrary real valued function and δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) , isa cloud invariant. Proof:
This is an immediate consequence of Theorem 4.1. (cid:3)
In this section, by using Theorem theorem 4.1, we find cloud invariantswhen a cloud is transformed under various sets of transformations A ( ϕ ) . As we pointed out in section 2 it is not necessary the set A ( ϕ ) to form agroup under the usual multiplication of matrices. Firstly we consider setsof transformations A ( ϕ ) which do form a group and then we consider aset A ( ϕ ) which does not form a group. Finally, in the last subsection, byusing the previous findings, we find cloud invariants for any given matrix. A ( ϕ ) which form a group We start with simpler sets of transformations A ( ϕ ) and gradually proceedto more general cases. A ( ϕ ) is a diagonal matrix We start by assuming that A ( ϕ ) is diagonal and has the form: A ( ϕ ) = ϕ , (39)17here ϕ ∈ R . In this case we can easily see that ϕ ∗ = 1 and that β (cid:48) ( ϕ ∗ ) = 0 , γ (cid:48) ( ϕ ∗ ) = 0 , δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) = − . Therefore, accordingto Theorem 4.1, a family of cloud invariants is: F (cid:18) M − H ( − M ) (cid:19) = h (cid:18) HM (cid:19) , (40)where F ( · ) and h ( · ) are arbitrary real valued functions. A ( ϕ ) is an upper triangular matrix We assume that A ( ϕ ) is upper triangular and has the form: A ( ϕ ) = ϕ , (41) ϕ ∈ R . In this case we easily verify that ϕ ∗ = 0 and that β (cid:48) ( ϕ ∗ ) =1 , γ (cid:48) ( ϕ ∗ ) = 0 , δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) = 0 . Consequently, according to Theo-rem 4.1, a family of cloud invariants is: F (cid:18) M − H ( H · (cid:19) = F (cid:18) M − HH (cid:19) , (42)where F ( · ) is an arbitrary real valued function. A ( ϕ ) is a lower triangular matrix We assume that A ( ϕ ) is lower triangular and has the form: A ( ϕ ) = ϕ , (43) ϕ ∈ R . In this case we easily find that ϕ ∗ = 0 and that β (cid:48) ( ϕ ∗ ) =0 , γ (cid:48) ( ϕ ∗ ) = 1 , δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) = 0 . Consequently, according to Theo-rem 4.1, a family of cloud invariants is: F (cid:18) M − H ( − (cid:19) = h ( H − M ) , (44)where F ( · ) and h ( · ) are arbitrary real valued functions.18 .1.4 A ( ϕ ) is a rotation matrix Finally, we assume that A ( ϕ ) is a rotation matrix and has the form: A ( ϕ ) = cos ϕ sin ϕ − sin ϕ cos ϕ , (45) ϕ ∈ R . In this case we easily obtain that ϕ ∗ = 0 and that β (cid:48) ( ϕ ∗ ) =1 , γ (cid:48) ( ϕ ∗ ) = − , δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) = 0 . Consequently, according toTheorem 4.1, a family of cloud invariants is: F (cid:18) M − H ( H + 1) (cid:19) , (46)where F ( · ) is an arbitrary real valued function. A ( ϕ ) which does not form a group A set of transformations A ( ϕ ) which subsumes the sets of transforma-tions considered in subsections 5.1.1, 5.1.2, and 5.1.3 is the set of “linear”matrices A ( ϕ ) = a a a a + ϕ b b b b = a + b ϕ a + b ϕa + b ϕ a + b ϕ , (47) where a ij , b ij ∈ R , and ϕ is a real free parameter. The set of matrices A ( ϕ ∗ ) does not, in general, form a group under matrix multiplication.However, we assume that there exists a value of ϕ , denoted by ϕ ∗ , suchthat A ( ϕ ∗ ) = I , I being the 2 × ϕ ∗ exists if and only if the entries a ij , b ij satisfy one ofthe following conditions: b (cid:54) = 0 ∧ a = ( a − b b ∧ a = ( a − b b ∧ a = a b − b + b b , (48)or b = 0 ∧ a = 1 ∧ b (cid:54) = 0 ∧ a = a b b ∧ a = a b + b b , (49)19r b = 0 ∧ b = 0 ∧ a = 1 ∧ a = 0 ∧ b (cid:54) = 0 ∧ a = a b + b b , (50)or b = 0 ∧ b = 0 ∧ b = 0 ∧ a = 1 ∧ a = 0 ∧ a = 0 ∧ b (cid:54) = 0 . (51)In this case we easily find that β (cid:48) ( ϕ ∗ ) = b , γ (cid:48) ( ϕ ∗ ) = b , δ = α (cid:48) ( ϕ ∗ ) − δ (cid:48) ( ϕ ∗ ) = b − b . According to Theorem 4.1, a family of cloud invariantsis: F (cid:18) M − H ( Hb − b + ( b − b ) M ) (cid:19) , (52)where F ( · ) is an arbitrary real valued function. For any given matrix there always exists a set of transformations A ( ϕ ) , of the form (47), which contains this matrix. In fact one can easily provethat there exists a two parameter family of such sets A ( ϕ ) . Accordingto Corollary 4.2, a family of cloud invariants, when a cloud of points istransformed under this matrix, is given by relation (52). As a case studywe consider the matrix: M = . − . − .
05 0 . . (53)Let A ( ϕ ) be the set of transformations: A ( ϕ ) = − − − / / + ϕ
12 81 2 = − ϕ − ϕ − / ϕ / ϕ . (54)One can easily check that A (0 .
2) = M . We have b = 8 , b = 1 , b − b = 10 . According to Corollary 4.2, a family of cloud invariants is: F (cid:18) M − H (8 H − M ) (cid:19) , (55)20igure 1: The Original Schemewhere F ( · ) is an arbitrary real valued function. Since there exists a twoparameter family of sets A ( ϕ ) , of the form (47), which contain M , thereexists a two parameter family of cloud invariants of the form (55), whena cloud is transformed via M . However, the explicit form of this twoparameter family of cloud invariants is not needed here.
To see how the above theory works in practise, we consider a cloud of10000 points forming the scheme of Figure 1. Using relations (1) and(2), we calculate the linear coefficients M and H of the cloud. We find M = 1 . H = 2 . A = to act on the cloud.Both the initial and the transformed schemes are depicted in Figure 2.Using relations (3) and (4) we calculate the linear coefficients of the newcloud and we obtain ˆ M = 3 . H = 9 . A, is I ( M, H ) = F ( H/M ) , where F ( · ) is an arbitrary real valued function. Indeed, for the initial and thetransformed linear coefficients, we find H/M = ˆ H/ ˆ M = 1 . . Itfollows that we have I ( M, H ) = I ( ˆ M , ˆ H ) for any real valued function F ( · ). 21igure 2: A Diagonal TransformationFigure 3: An Upper Triangular TransformationAs a second example of transformation, we let the upper triangular matrix B = .
70 1 to act on the cloud. The result of this transformation isgiven in Figure 3. The transformation of the cloud we consider under B has linear coefficients ˆ M = 0 . H = 0 . B, is I ( M, H ) = F (( M − H ) /H ) , where F ( · ) is an arbitrary real valued function. Indeed, we have ( M − H ) /H =( ˆ M − ˆ H ) / ˆ H = − . . Consequently we have I ( M, H ) = I ( ˆ M , ˆ H )for any real valued function F ( · ).As a third example of transformation, we act on the cloud of points witha rotation matrix C = cos π π − sin π π . The initial and the rotatedclouds are shown in Figure 4. The linear coefficients of the rotated cloudare ˆ M = − . H = 0 . . We found in subsection 5.1.4,that a family of cloud invariants, when a cloud is transformed with therotation matrix C, is I ( M, H ) = F (cid:18) M − H ( H + 1) (cid:19) , where F ( · ) is an ar-22igure 4: A RotationFigure 5: An Arbritrary Matrixbitrary real valued function. Indeed, we have M − H ( H + 1) = ˆ M − ˆ H ( ˆ H + 1) = − . . Henceforth we have I ( M, H ) = I ( ˆ M , ˆ H ) , for any real valuedfunction F ( · ).Finally, we act on the cloud of points with the matrix D = . − . − .
05 0 . .The result of this transformation is shown in Figure 5. The linear coeffi-cients of the transformed cloud are ˆ M = − . H = 27 . . Wefound in subsection 5.2.1, that a family of cloud invariants, when a cloudis transformed with the matrix D, is I ( M, H ) = F (cid:18) M − H (8 H − M ) (cid:19) , where F ( · ) is an arbitrary real valued function. Indeed, we have M − H (8 H − M ) =ˆ M − ˆ H (8 ˆ H − M ) = − . . Consequently we obtain I ( M, H ) = I ( ˆ M , ˆ H ) , for any real valued function F ( · ).We note that the results we obtained by considering the aforemen-tioned cloud of points verify our findings in section 5.23 Concluding Remarks
We have studied transformations, with 2 × α βγ δ , α, β, γ, δ ∈ R , of planar set of points, called clouds of points for convenience. Our aimin this paper is to find cloud invariants, i.e. functions which take the samevalue when they are evaluated for the initial and for the transformed cloudof points. It is natural the cloud invariants to be functions of variableswhich carry information for the whole cloud. The cloud invariants we findare functions of two such variables M and H.M and H are functions of the coordinates of the points of the cloud.As a result we find that any transformation of a cloud of points bya 2 × M, H ) → ( ˆ M , ˆ H ) , ˆ M = M ( M, H, α, β, γ, δ ) , ˆ H = H ( M, H, α, β, γ, δ ) , given explicitly byequations (3) and (4), of the variables M and H.M and H originate from the best fitting straight line through the cloud ofpoints under consideration. This straight line is determined by the leastsquares fitting technique. Henceforth by definition a cloud invariant is anyfunction I ( M, H ) which satisfies the relation (13), I ( M, H ) = I ( ˆ M , ˆ H ) , where ˆ M and ˆ H are the values of the variables M and H for the trans-formed cloud.We find cloud invariants by using Lie theory. Lie theory replaces thecomplicated, nonlinear finite invariance condition (13) by the more usefuland tractable linear infinitesimal condition (28) provided that the function I ( ˆ M , ˆ H ) is analytic in the parameters α, β, γ, and δ. Linear condition (28)is a set of linear PDEs. Any solution to this system of PDEs gives a cloudinvariant.Cloud invariants can be practically implemented in various fields, e.g. inoptical character recognition, in image analysis and computer graphicstechniques, by providing the necessary tools in order to identify iconscreated by the same “source”. The cloud invariants we find for the gen-24ral four − parameter case, when a cloud is transformed with a matrix α βγ δ , cannot be practically implemented.However, the cloud invariants we find for various one − parameter groupsof transformations can be practically implemented. In particular we findcloud invariants for a group consisting of diagonal matrices, for a groupconsisting of upper triangular matrices, for a group consisting of lowertriangular matrices, and for the group of rotations SO (2) . More importantly, for the practical implementation of our findings, we findcloud invariants for any given matrix. We find these cloud invariants bynoticing that any given matrix belongs to a one − parameter “linear” setof transformations of the form A + B ϕ, A and B are given 2 × ϕ ∈ R . Our findings are verified by examples and simulations in acloud of 10.000 points.We expressed the cloud invariants in terms of the variables M and H. M and H are essentially the coefficients of the straight line which is the bestfit for the cloud of points under consideration. This provides a naturalguide for future research. With a view to apply our results in fields such ascharacter recognition, the next logical step is to consider the case wherethe cloud of points originates from an icon which has a parabolic − likeshape.In this case we will look for cloud invariants which are expressed in terms ofvariables which appear as coefficients, or variations thereof, of the parabolawhich is the best fit for the cloud of points under consideration. For similarreasons, subsequently, we will look for new cloud invariants expressed interms of variables which appear as coefficients in third or higher degreecurves. We will compare our findings, with simulations and computationalexperiments, with those acquired by other approaches.25 Acknowledgement
The first author would like to express his thanks to Mr. Koutsoulis Nikos,for his attempts to face the problem initially.
References [1] Flusser, J. Moment Invariants in Image Analysis,
International Jour-nal of Computer, Electrical, Automation, Control and InformationEngineering,
V.1, N.11, 2007.[2] Kalouptsidis, N.
Signal Processing Systems, Theory and Design,
JohnWiley and Sons, 1997.[3] Hu, M.K. Visual Pattern Recognition by Moment Invariants,
IRETransactions of Information Theory.,
V.8, N.2, 1962, pp.179 − Computational Intelligence,
V.14,N.4, 1998, p.p. 461 − IEEE Transactions on Pattern Analysis and Machine Intelligence,
V.20, N.6, 1998, p.p. 590 − SoCG’03 Proceedings of the nineteenth annual symposium on Com-putational geometry,
ACM New York, NY, 2003, p.p. 322 − Advances in Pat-tern Recognition ICAPR 2001,
V. 2013 of the series Lecture Notesin Computer Science, p.p. 361 − Invariant Theory . Student Mathematical Library. 36.Providence, R.I.: American Mathematical Society, 2007.[9] Gilmore, R.
Lie Groups,Physics, and Geometry , Cambridge Univer-sity Press, New York, 2008.[10] Cantwell, B.J.