M¨OBIUS INVARIANTS IN IMAGE RECOGNITION

. In this paper rational diﬀerential invariants are used to classify various plane shapes as well as plane domains equipped with an additional geometrical object.


1.
Introduction. In this paper we give a survey of applications of Möbius invariants of conformal and projective geometries to analysis of various shapes ( [5], [6]).
Consider a connected and simply connected domain D ⊂CP 1 with smooth boundary γ = ∂D. Sharon E. and Mumford D. ( [15]) suggested the following scheme to study shapes of the curve γ. By the Riemann theorem there are conformal mappings φ i : D →D and φ o : D c → D c of the inner and outer parts of D on the inner and outer part of the unit disk D. Restriction of φ o • φ −1 i to the unit circle gives us a diffeomorphism ψ : S 1 → S 1 which is defined up to PSL 2 (R) × PSL 2 (R)-action on the group of diffeomorphisms Diff S 1 :ψ → A • ψ • B −1 .
In paper ( [15]) some additional conditions on φ o were imposed which lead to the right action of PSL 2 (R) and give the homogeneous space Diff S 1 /PSL 2 (R). In paper ( [5]) these conditions were revised in such a way that they gives another spaces SO (2) \Diff (S 1 )/PSL 2 (R) and PSL 2 (R) \Diff (S 1 )/PSL 2 (R).
The situation becomes to be much more rigid if we consider the domains D ⊂CP 1 equipped with some geometrical object, say set of points, curves, functions etc. We call such domains decorated. Applying once more the Riemann theorem we arrive at a classification problem of the same type objects on the Lobachevsky plane with respect to Lie group isometries PSL 2 (R).
In both these cases we use differential invariants to describe the corresponding orbit spaces. Remark that the actions of the above Lie groups on the corresponding jet bundles are algebraic and it allows us to use the Lie-Tresse theorem ( [10]) to describe the field of rational differential invariants. These fields separate regular orbits in jet bundles and finally we use them to get the smooth classification of the corresponding geometrical objects.
In the first part of the paper we discuss shape spaces and related to them different types of fingerprints. In all cases we give complete description of fields of rational invariants and show their relations to the Hill equations and projective structures on the projective line PR 1 . For the cases of regular orbits these invariants are used to get smooth classification of fingerprints.
The second part of this paper is devoted to the case of decorated domains. We analyze three types of geometrical objects: functions, differential 1-forms and foliations. For all these cases the fields of Möbius invariants are found and used to get smooth classification of corresponding objects.
2. Shape spaces and fingerprints. Let D ⊂ CP 1 be connected and simply connected domain with a smooth boundary γ = ∂D ("shape"). It follows from the Riemann theorem that there is a conformal diffeomorphism φ i : D → D of this domain on the unit disk. For the same reasons there is a conformal diffeomorphism φ o : D c → D c of the outer domain to the complement of the unit disk in CP 1 .
Moreover, two such diffeomorphisms differ on Möbius transformations, preserving of the unit disk, Considering the restriction of φ o • φ −1 i on the unit circle S 1 = ∂D we get an orientation preserving diffeomorphism of the circle (a "fingerprint"): Different choices of the diffeomorphisms φ i and φ o lead us to the action of the group PSL 2 (R) × PSL 2 (R) on the group Diff S 1 of orientation preserving diffeomorphisms of the unit circle.
The isomorphism of the group of matrices a b b a with the standard group SL 2 (R) is given by the adjunction where The corresponding to (2) Lie algebra sl 2 (R)-action given by vector fields: where φ is the angle. It is worth to note that under the stereoprojection diffeomorpfism S 1 → RP 1 the PSL 2 (R)-action (2) becomes to be the action by fractional transformation and the Lie algebra sl 2 (R)-action is given by the vector fields ∂ t , t∂ t , t 2 ∂ t , where t is an affine coordinate.
We'll consider the following specifications of action (1): A-fingerprints: They were introduced by Mumford and Sharon (see, [15]) and they are defined by requirement φ o (∞) = ∞ and φ o (∞) ∈ R. In this case the space of a-fingerprints is the smooth Teichmüller space Diff (S 1 )/PSL 2 (R) and the fingerprints define shapes up to translations and magnifications. B-fingerprints: They were introduced in ([5]) and they are defined by requirement φ o (∞) = ∞. In this case the corresponding space of fingerprints is the double quotient SO (2) \Diff (S) 1 /PSL 2 (R) and the fingerprints define shapes up to similarity. C-fingerprints: They were introduced in ([5]) and they do not impose any requirements on φ o . In this case the corresponding space of fingerprints is the double quotient PSL 2 (R) \Diff (S) 1 /PSL 2 (R) and the fingerprints define shapes up to Möbius transformations.
2.1. Differential invariants. Denote by J k the manifolds of k-jets of preserving orientation diffeomorphisms of the circle. Then action (1) of Lie group PSL 2 (R) × PSL 2 (R) induces the corresponding action on manifolds J k . These manifolds are algebraic and the action is algebraic too. A rational function on manifold J k we call differential invariant (of the corresponding action) of order ≤ k if it is an invariant of the corresponding action.
Due to Lie-Tresse theorem (see, [10]), the fields of invariants have a number of basic invariants and invariant derivations such that any invariant is a rational function of invariant derivatives of the basic invariants. Moreover, these fields separate regular orbits.
2.1.1. Differential invariants for a-fingerprints. This case corresponds to the right A ∈ PSL 2 (R). It has ( [5]) one basic (local) invariant of order 0 : one basic invariant of order 3 (Schwartzian):

and invariant derivation
where d dφ is the total derivation. Applying the Lie-Tresse theorem, we get the following result.
Theorem 2.1. ( [5]) The field of rational differential invariants of PSL 2 (R)-action (6) is generated over the field of rational functions in (cos ψ 0 , sin ψ 0 ) by the differential invariant J 3 of order 3 and by the invariant derivation ∇. That is, any rational differential invariant of order ≤ k is a rational function in All orbits in J k are PSL 2 (R)-regular and the field of rational differential invariants separates them.
2.1.2. Differential invariants for b-fingerprints. This case corresponds to the SO (2) × PSL 2 (R)-action: It is also corresponds to the action of Lie algebra generated by Therefore, differential invariants of this action are invariants of action (6), which are, in addition, invariant with respect to rotation ∂ ψ0 . Then the above theorem gives us the following result.
The field of rational differential invariants of SO (2)×PSL 2 (R) -action (7) is generated by the differential invariant J 3 of order 3 and by the invariant derivation ∇. That is, any rational differential invariant of order ≤ k is a rational function in All orbits in J k are SO (2)×PSL 2 (R) -regular and the field of rational differential invariants separates them.

Differential invariants for c-fingerprints. This case corresponds to the PSL
The corresponding Lie algebra sl 2 ⊕ sl 2 -action is given by the following vector fields: The field of rational differential sl 2 ⊕ sl 2 -invariants is generated by differential invariants of action (6) which are invariants of PSL 2 (R)-action generated by vector fields The field of rational differential invariants of PSL 2 (R) × PSL 2 (R)-action (7) is generated by the differential invariant I 5 of order 5: where θ = J 3 + 2, and θ i = ∇ i (θ), and invariant derivation δ of order 6: That is, any rational differential invariant is a rational function in I 5 , δ(I 5 ), δ 2 (I 5 ), . . . , δ k (I 5 ) . . . This field separates regular orbits.

2.2.
Equivalence. In this section we show how to use differential invariants of the above actions to classify circle diffeomorphisms.

A-fingerprints, Hill equations and projective structures.
Projective structures.
A projective structure on an interval U ⊂ RP 1 consists of a covering U by charts (U α , t α ), where t α is a local coordinate on U α , such that on intersections U α ∩ U β these coordinates are connected by fractional transformations: The standard projective structure on RP 1 has the group SL 2 (R) as the group of projective transformations with Lie algebra sl 2 generated by vector fields: where t is an affine coordinate.
On the other hand, due to Sophus Lie theorem, any Lie algebra of vector fields on an interval U , which isomorphic to sl 2 , is locally isomorphic to Lie algebra ∂ s , s∂ s , s 2 ∂ s , for some local coordinate s. Moreover, any two such realizations are connected by fractional transformation of the corresponding local parameters. Therefore, the above definition of the projective structure could be reformulated in the following way: for each α we define the Lie algebra g α of vector fields on interval U α : Then, the requirement that t α and t β on the interval U α ∩ U β are connected by a fractional transformation simply means that In other words, a projective structure could be defined by a covering U α together with simple Lie algebras g a of vector fields on U α such that dim g 3 = 3 and g a = g β on U α ∩ U β .
Hill equations. Consider a Lie algebra g of vector fields on an interval U , which is isomorphic to sl 2 .
Let A, B, H be the Chevalley basis in g: and let Then commutation relations (9) takes the form: Multiplying the first equation by h, the second equation by (−b), and the last one by a, we get h 2 = 2ab. It could be shown (see, ( [7]) for more details), that functions a and b can be chosen to be positive on the interval. Hence, for some smooth functions f and g. In this case system (10) of differential equations is equivalent to a single equation: Putting we get the following result.
) Any subalgebra g sl 2 of vector fields on an interval U has the form where f and g is a fundamental system of solution of an equation with Wronskian f g − f g = 1.
• Functions f 2 , 2f g, g 2 also give a fundamental system of solutions for differential equation Thus, elements of Lie algebra g are vector fields z (t) ∂ t , where z (t) are solutions of (12). • The standard projective structure on RP 1 corresponds to the trivial potential x transforms the standard projective structure on RP 1 to projective structure on the circle that corresponds to Lie algebra (5).
Consider an interval U equipped with the Lie algebra g = f 2 ∂ t , 2f g∂ t , g 2 ∂ t of vector fields.
Let's choose the following covering of the interval by charts of two types: Therefore, the potential W defines a projective structure on the interval. Let now f : S 1 → S 1 be an orientation preserving diffeomorphism, and let for some periodic functions x and y.
Then, because f > 0, one could choose a function λ (φ) > 0, such that functions X = λx, Y = λy satisfy relation Then, X X = Y Y and we get the following result. Lemma 2.5. Any orientation preserving diffeomorphism f : S 1 → S 1 can be presented in the form where u (φ) and v (φ) is a fundamental solution of a Hill equation on the circle, with Wronskian u v − uv = 1.
Change of the fundamental system corresponds to PSL 2 (R)-action where A ∈ PSL 2 (R).

Equivalence of A-fingerprints.
In this section we use the representation of the circle diffeomorphisms, given in the above lemma, to get a classification of a-fingerprints.
Let's consider an orientation preserving diffeomorphism f : and let the inverse diffeomorphism be of the form: where functions u and v is a fundamental solution of some Hill equation with Wronskian equals 1.
Then the straightforward computations show that the value of differential invariant J 3 on f is proportional to the potential: Therefore, given diffeomorphism f , we'll find the potential W (ψ), using formula (16).
This potential defines the inverse f −1 up to PSL 2 (R)-action (14): or defines the diffeomorphism f itself up to PSL 2 (R)-action (6). Summarizing, we get the following result.
Theorem 2.6. The potential W , given by formula (16), defines a-fingerprints up to PSL 2 (R)-action (6). That is, two PSL 2 (R)-equivalent diffeomorphisms have the same potentials and if two diffeomorphisms have the same potential then they are PSL 2 (R)-equivalent.
Remark also that the functions W , which we call potentials of the corresponding diffeomorphisms, are not arbitrary. First of all, the corresponding Hill equations should have the trivial monodromy. Secondly, mappings (15) are local diffeomorphisms S 1 → S 1 , for monodromy free potentials, and to have diffeomorphism, one should require that their degree equals one. Equivalence of B-fingerprints. Any diffeomorphism f : S 1 → S 1 defines a map where J 4 (f ) = ∇ (J 3 (f )), J 5 (f ) = ∇ (J 4 (f )) , . . ., etc.

KONOVENKO NADIIA AND LYCHAGIN VALENTIN
The differential of this map is non trivial if J 2 4 (f ) + J 2 5 (f ) = 0, and mapping σ f is embedding in this case.
We'll assume in addition that σ f is a diffeomorphism of S 1 on Σ f = Imσ f . Then the following result holds ( [5]).

Equivalence of C-fingerprints.
In this case we associate mapping σ f : S 1 → R 2 with a diffeomorphism f : S 1 → S 1 as follows: where I 6 (f ) = δ (I 5 (f )), I 7 (f ) = δ (I 6 (f )) , . . . , etc. For this case the differential of this map is non trivial if I 2 6 (f ) + I 2 7 (f ) = 0, and we assume in addition that σ f is a diffeomorphism of S 1 on Σ f = Im σ f . Theorem 2.8.
3. Decorated shapes. In the previous section we've studied plane simply connected domains with smooth boundaries up to conformal diffeomorphisms. In this section we study decorated domains, i.e. connected and simply connected domains with smooth boundaries equipped with some geometrical objects such as set of points, function,differential form, foliation, etc. We'll consider such domains up to conformal equivalence too. First of all, by using the Riemann conformal mapping theorem, we establish conformal isomorphism this domain with the unit disk. Secondly, two such conformal diffeomorphisms differ on conformal transformation that preserve the unit disk, i.e. on transformations from the Möbius group PSL 2 (R), and, therefore, conformal classification of decorated domains transforms to the classification problem of the corresponding geometrical objects on the Lobachevsky plane with respect to Möbius group PSL 2 (R), or the group of isometries of the Lobachevsky plane.
For example, if we consider finite sets of points, then the classification problem becomes to be a part of the classical problem on classification of binary forms ( [1]).
After such a reformulation we use the Lie-Tresse theorem to find (we call them Möbius) rational differential invariants of the PSL 2 (R)-actions. These fields of Möbius differential invariants separates regular PSL 2 (R)-orbits in the corresponding jet spaces and give us complete solutions of the of the equivalence problems in regular cases.

Lobachevskian geometry.
Let D ⊂ C, D = {x 2 + y 2 < 1}, be the unit disk. The Möbius group PSL 2 (R) acts on D in the following way: where a, b ∈ C, |a| 2 − |b| 2 = 1. The corresponding representation of Lie algebra sl 2 is given by the following vector fields: where t = 1 − x 2 − y 2 . It is well known that the Möbius transformations of PSL 2 (R) are isometries of the metric and preserve, in addition, the symplectic form Ω = dx ∧ dy t 2 , and complex structure 3.1. Functions. Denote by J k the manifolds of k-jets of smooth functions, defined on the unit disk D, and by (x, y, u, u 10 , u 01 , u 11 , u 12 , u 22 , . . .) we denote the standard canonical coordinates in J k . The value of u ij at the k-jet [f ] k a of function f at point a ∈ D is equal to ∂ i+j f ∂x i ∂y j (a). Remark that the Möbius group PSL 2 (R) is algebraic, manifolds of k-jets J k are algebraic and irreducible, and PSL 2 (R)-action in J k is algebraic too.
By Möbius differential invariant of order ≤ k we mean rational function defined on J k and invariant under induced PSL 2 (R)-action. Due to Lie-Tresse theorem the fields of Möbius differential invariants separate regular orbits in the jet spaces, and the field of all Möbius differential invariants is generated by some number basic invariants and invariant derivations.
By invariant derivation here we mean a total derivation of the form which commutes with PSL 2 (R)-action.
Here A, B are rational functions on some k-jet space J k and d dx , d dy are the total derivations in x and y respectively. Secondly, using invariant complex structure I , (18), we get another differential 1-form on the 1-jet space: Iρ = −u 01 dx + u 10 dy, which is a conformal invariant. Therefore, horizontal differential 1-forms where d is the total differential and is a second order Möbius invariant. The frame, dual to coframe (ω 1 , ω 2 ), is formed by the following PSL 2 (R)-invariant derivations: These derivations are satisfied the following commutation relation: 3.1.2. Möbius invariants for functions. Two invariants of order 0 and 1 are obvious. They are J 0 = u, J 1 = g (ω 1 , ω 1 ) = 1 − x 2 − y 2 2 u 2 10 + u 2 01 . Therefore, dimensions of regular orbits in J k , when k ≥ 1, equal 3 and we expect (k + 1) independent invariants of pure order k, when k ≥ 2.
All other invariants we can get by taking invariant derivatives of invariants J 1 and J 2 . Thus in order k, we get k invariants by taking invariant derivatives of J 1 of order (k − 1) and (k − 1) invariants by takins invariant derivatives of J 2 of order (k − 2). Therefore, we get (2k − 1) − (k + 1) = k − 2 syzygies in pure order k. In particular, we have only one syzygy in order 3.
Hence, all syzygy could be obtained by taking invariant derivatives of the syzygy of order 3.
Theorem 3.1. The field of Möbius differential invariants for the PSL 2 (R)-action on the jet spaces of functions on the unit disk is generated by invariants J 0 , J 1 , J 2 and their invariant derivatives. This field separates orbits, where u 2 10 + u 2 01 = 0.

Möbius equivalence of functions.
Tresse derivatives. Assume that given two functions F 1 and F 2 on the k-jet space, such that their total differentials are linear independent in a domain O in J k+1 : Then, for the total differential dG of any function on jet space J l , one has decomposition, These functions λ i , which are functions on J m , where m = max(k + 1, l + 1), are called Tresse derivatives and will be denoted as DG DF i .
Remark that under condition that functions F 1 , F 2 and G are invariants their Tresse derivatives are invariants also. In other words, derivations In our case we have also two invariant derivations ∇ 1 and ∇ 2 . Therefore Tresse derivatives D DJ0 , D DJ1 are linear combinations of ∇ 1 , ∇ 2 , and vice versa.
One has and in domain where J 1 = 0 and J 12 = 0. The above theorem, therefore, could be reformulated in terms of Tresse derivatives in the following way. We say that a function f on the unit disk is regular if values of differential invariants J 1 (f ) and J 12 (f ) do not take zero values.
It is easy to check that for regular functions the mapping is a local diffeomorphism. Let Σ f = Imσ f . Therefore, for regular functions, value of invariants of the second order, J 2 , J 11 , J 12 , are functions of (J 0 (f ) , J 1 (f )): (20) Let's consider system (20) as a system of differential equations of the second order: (21) This is overdeterminated system of the Frobeneus type, having symmetry group PSL 2 (R) and the function f is one of its solutions.
We show that (20) is an automorphic system in the sense that the symmetry group acts in a transitive way on the solution space.
First of all remark, that system (20) is completely integrable, and the integrability conditions are exactly the syzygy for differential invariants, which are satisfied by the construction of functions A, B, C.
The solution space of this system could be identified with space J 1 a of 1-jets of functions at a point a ∈ D.
Assume that g is another regular function and solution of (21) and let Σ f = Σ g . Then, up to transformation from PSL 2 (R), we can assume that f (a) = g (a) and Stationary subgroup of the point a acts in a transitive way on T * a D \ {0} and therefore there exist an element of group PSL 2 (R) that transforms f to g. 3.1.4. Möbius invariants for differential 1-forms. Here we discuss the conformal equivalence of simply connected domains D ⊂C equipped with differential 1-forms θ ∈ Ω 1 (D).
As above this problem is equivalent to the problem of classification of differential 1-forms defined on the unit disk D with respect to the Möbius group PSL 2 (R).
Thus differential 1-form θ = a (x, y) dx + b (x, y) dy defines a section S θ : (x, y) → (u = a (x, y) , v = b (x, y)) of the bundle τ * and its k-jet prolongation u ij = ∂ i+j a ∂x i ∂y j (x, y), v ij = ∂ i+j b ∂x i ∂y j (x, y). The universal Liouville form ρ = udx + vdy ∈ Ω 1 (T * D) gives the isomorphism between sections of the cotangent bundle τ * and differential forms on the unit disk D: θ = S * θ (ρ). The PSL 2 (R)-action on the unit disk has the natural prolongation to PSL 2 (R)action on the cotangent bundle T * D.
The correspondent realization of this action by vector fields is given by the following Hamiltonian vector fields: where vectors fields (X, Y, Z) correspond to PSL 2 (R)-action (17) on D.
As above, using the Liouville form ρ and its image Iρ we get PSL 2 (R)-invariant coframe on T * D : The structure equations for this coframe are They give us two Möbius invariants of order one: In addition to them we have also obvious invariant of order zero: This invariant separates regular orbits in J 0 τ * . The frame, dual to coframe (22), has the form Dimensions of regular orbits in J 0 τ * equal dim PSL 2 (R) = 3. Therefore, in pure order 1 we expect 4 independent invariants and they are J 11 , J 12 , δ 1 (J 0 ) , δ 2 (J 0 ) .
Applying the invariant derivations to them we get 7 invariants of pure order 2 and therefore one syzygy in order 2.
Straightforward computations show that this syzygy is In order k ≥ 3, we have 2 (k + 1) invariants of pure order k. Differentiations of invariant J 0 of k times give us (k + 1) invariants and differentiations of invariants J 11 , J 12 of (k − 1) times give us 2k invariants of pure order k.
Finally, we get the following result.
Theorem 3.4. The field of Möbius invariants for differential 1-forms on the unit disk is generated by invariants J 0 , J 11 , J 12 and derivations δ 1 , δ 2 . All syzygies are generated by (24). The field separates orbits where J 0 = 0.
Using functions (A (θ) , B (θ)) as coordinates on the disk, we get relations: Then, as above, consider (25) as a system of differential equations for differential 1-forms on the disk. By the construction this is completely integrable PSL 2 (R)-invariant system with 3 dimensional space (each solution is determined by its 1-jet at a point) of solutions that are images of θ under the PSL 2 (R)-action.
Summarising, we arrive at the following result.
3.1.6. Möbius invariants for foliations. In the last section we consider conformal classification of oriented foliations defined in simply connected domains, or equally PSL 2 (R)-classification of oriented foliations on the unit disk. Remark, that any such a foliation is defined by some differential 1-form θ of the length 1 : g (θ, θ) = 1. Denote by T * 1 (D) ⊂ T * (D) the subbundle of covectors, having length 1, and let π : T * 1 (D) → D be the corresponding bundle. Sections of this bundle could be identified with oriented foliations.
Denote also by J k (π) the bundles of k-jets of sections of π. Thus, they are jets of oriented foliations.
The PSL 2 (R)-action on D in a natural way could be lifted on the bundle π, and prolong to actions on the bundles of k-jets. Rational invariants of these actions we call Möbius invariants of foliations.
Remark that J k (π) ⊂ J k (τ * ) are subbundles with PSL 2 (R)-action induced by the PSL 2 (R)-action on the orbit J 0 = 1. This subbundle contains regular orbits and therefore Möbius invariants of foliations could be found as the restrictions of invariants differential 1-forms.
Denote by (x, y, w, w 10 , w 01 , ...) the standard coordinates in the jet spaces J k (π), where w is the angle.
Then, the restriction on the regular orbit J 0 = 1, gives us the following: