Affine Differential Invariants of Functions on the Plane

Abstract

A differential invariant is a function defined on the jet space of functions that remains the same under a group action. It is an important concept to solve the equivalence problem. This paper presents an effective method to derive a special type of affine differential invariants. Given some functions defined on the plane and an affine group acting on the plane, there are induced actions of the group on the functions and on the derivative functions of the functions. Affine differential invariants of these functions are useful in many applications. However, there has been little systematic study of this problem at present. No clear and simple results are available for application users to use directly. We propose a direct and simple method to construct affine differential invariants in this situation. Some useful explicit formulas of affine differential invariants of 2D functions are presented.

1. Introduction

The concept of an invariant is ubiquitous in science. It is crucial to solve the equivalence problem. The fundamental equivalence problem is to determine whether two objects of a set of objects are equivalent with respect to a given equivalence relation. An invariant with respect to the equivalence relation is a function defined on the set of objects that is constant on the equivalence classes. Two objects are not equivalent if they have different invariants. The classification problem can be solved by deriving a system of invariants that separates any two equivalence classes. When a set of objects is acted on by a group, invariants can be used to classify the set of objects under the action of the group. In this case the equivalence classes are the orbits of the group.

Classical invariant theory originated in the 19th century with the study of constructing (relative) invariants of a system of forms [1–4]. In this case, the invariants are polynomial functions on the coefficients of a system of forms. Algebraic methods for deriving (relative integral) invariants were developed during this period. Classical invariant theory is closely related to projective geometry. The vanishing of a form in 𝑛 variables corresponds to a hypersurface in an 𝑛−1-dimensional projective space. The invariants are then used to classify hypersurfaces into projective equivalence classes.

Differential invariants were introduced and studied by Lie and Tresse in the late 19th century [5, 6]. A differential invariant is a function defined on a jet space that is invariant under the action of a Lie group. Given 𝑞 functions with 𝑝 independent variables, a jet space is defined by considering the independent variables, the dependent variables, and the derivatives of the dependent variables as functionally independent coordinates of the space. Since differential equations are surfaces in the jet space, differential invariants serve to classify differential equations into equivalence classes. If one differential equation is solved in an equivalence class, then all differential equations in the equivalence class are solved.

The idea of an invariant is important in computer vision [7–9]. Two views of a 3D scene are geometrically related by the epipolar constraint. Two views of a planar object are geometrically related by a projective transformation. Invariants of objects are widely used to match two images that are related by a geometric transformation. Apart from their direct application in object recognition, differential invariants provide a mathematical foundation to design local feature detectors of images.

An elegant tool for deriving differential invariants is the moving frame method [10–22]. The concept of moving frames has a long history. Cartan derived some differential invariants based on his moving frame method [10, 11]. The method was generalized by Fels and Olver for general transformation groups [12, 13]. Differential invariants of one-dimensional manifolds are well studied. There are a few papers devoted to the differential invariants of surfaces [18–22].

This paper presents an effective method to derive a special type of affine differential invariants. Given some functions defined on the plane and an affine group acting on the plane, there are induced actions of the group on the functions and on the derivative functions of the functions. Affine differential invariants of these functions are useful in many applications. However, we have found little systematic study of this problem at present. No clear and simple results are available for application users to use directly. We propose a direct and simple method to construct affine differential invariants in this situation. Some useful formulas of affine differential invariants are presented in explicit form.

The rest of the paper is organized as follows. In Section 2, we define the basic concepts and notation. In Section 3, we present a few theorems on the properties of the affine differential invariants. In Section 4, we present the basic theorem for constructing the affine differential invariants. We give a few explicit formulas of affine differential invariants in Section 5 for ordinary people to user directly. We conclude in Section 6.

2. The Basic Concepts and Notation

In this section, we introduce the basic concepts and define the notation we will use. In what follows, we take the ground field to be ℝ, the field of real numbers. Given a group 𝐺 and a space 𝑆, an action 𝛼 of 𝐺 on 𝑆 is a map 𝛼∶𝐺×𝑆→𝑆 such that 𝛼(𝑒,𝑥)=𝑥 and 𝛼(𝑔,𝛼(ℎ,𝑥))=𝛼(𝑔ℎ,𝑥) for all 𝑔,ℎ∈𝐺 and 𝑥∈𝑆, where 𝑒 is the unit element of 𝐺.

A group 𝐺 acting on a space 𝑆 is called a transformation group if there is a group homomorphism 𝜌∶𝐺→𝛼^𝑋 mapping 𝐺 to the group of invertible maps on 𝑋 induced by the action 𝛼. That is, the action of a transformation group 𝐺 preserves the structure endowed on 𝑆. The name of a transformation group reflects to a certain extent the behavior of the action and the structure endowed on 𝑆.

A 𝐺 invariant of 𝑆 under the action of 𝛼 is a real valued function ℐ defined on 𝑆 such that ℐ𝛼𝑔,𝑥=ℐ(𝑥), (1) for all 𝑔∈𝐺 and 𝑥∈𝑆. That is, ℐ is constant on the orbits of 𝐺.

Given an action 𝛼 of a group 𝐺 on a space 𝑆, there is an induced action 𝜇 on the set of smooth functions 𝒞^∞(𝑆,ℝ) from 𝑆 to ℝ. The action 𝜇 is usually defined as 𝜇∶(𝑔,𝑓)↦𝜇(𝑔,𝑓) for 𝑔∈𝐺 and 𝑓∈𝒞^∞(𝑆,ℝ), where 𝜇(𝑔,𝑓)(𝑥)∶=𝑓(𝛼(𝑔⁻¹,𝑥)) for 𝑥∈𝑆.

In this paper, we will denote a sequence by ⟦^𝑛_𝑖=1𝑎_𝑖. That is, ⟦^𝑛_𝑖=1𝑎_𝑖=𝑎₁,𝑎₂,…,𝑎_𝑛. (2) The 𝑛th order partial derivative of a smooth function 𝑢(𝑥,𝑦) is denoted by 𝑢_{𝑖,𝑛−𝑖}: 𝑢_{𝑖,𝑛−𝑖}=𝜕^𝑛𝑢𝑥,𝑦𝜕𝑥^𝑖𝜕𝑦^𝑛−𝑖. (3)

Given 𝑞 smooth functions ⟦^𝑞_𝑖=1𝑢^𝑖 that depend on 𝑝 independent variables ⟦^𝑝_𝑖=1𝑥^𝑖, the 𝑛th jet space 𝒥^𝑛(𝑋×𝑈^(𝑛)) is an Euclidean space of dimension 𝑝+𝑞(𝑝+𝑛)!𝑝!𝑛!, where 𝑋 is the space whose coordinates are the independent variables ⟦^𝑝_𝑖=1𝑥^𝑖 and 𝑈^(𝑛) is the fiber space whose coordinates are the functions ⟦^𝑞_𝑖=1𝑢^𝑖 and their derivatives up to the order 𝑛. These coordinates are assumed to be functionally independent.

In this paper, we study smooth functions with two variables 𝑥 and 𝑦. In this case a point in the 𝑛th jet space 𝒥 has the following form: Υ=𝑥,𝑦,⟦^𝑞_𝑖=1𝑢^𝑖,⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}. (4)

After defining the action of a group 𝐺 on functions, we can proceed to define the group action on derivatives of functions. This induced action is called the prolonged action. A differential invariant is a function defined on jet space 𝒥, that is, invariant under the prolonged group action.

Let 𝐺𝐿(2,ℝ) denote the general linear group of all invertible matrices of order 2 over ℝ. An affine transformation group 𝐀(2) is the group 𝐺𝐿(2,ℝ)⋉ℝ² acting on ℝ² via the following: 𝛼𝑔,𝑥↦𝐴𝑥+𝑏, (5) where 𝑔=(𝐴,𝑏)∈𝐀(2), 𝐴∈𝐺𝐿(2,ℝ), and 𝑏∈ℝ². In order to study the specific properties of affine differential invariants, we would like to write an affine transformation in the following explicit form: 𝑥=𝑎𝑥+𝑏𝑦+𝑒,𝑦=𝑐𝑥+𝑑𝑦+𝑓, (6) where 𝑎,𝑏,𝑐,𝑑,𝑒,𝑓∈ℝ and (𝑎𝑑−𝑏𝑐)≠0. The point (𝑥,𝑦) is called the transformed point. This convention is not obligatory. From the point of view of transformation, each one of (𝑥,𝑦) and (𝑥,𝑦) is the transformed point of the other.

Under the affine transformation (6), the relation between a smooth function 𝑢(𝑥,𝑦) and the transformed function 𝑢(𝑥,𝑦) is as follows: 𝑢𝑥,𝑦=𝑢𝑎𝑥+𝑏𝑦+𝑒,𝑐𝑥+𝑑𝑦+𝑓=𝑢𝑥,𝑦. (7) Likewise, which one of 𝑢(𝑥,𝑦) and 𝑢(𝑥,𝑦) is called the transformed function is just a point of view. The 𝑛th order partial derivative of the transformed function 𝑢(𝑥,𝑦) is denoted by the following: 𝑢_{𝑖,𝑛−𝑖}=𝜕^𝑛𝑢𝑥,𝑦𝜕𝑥^𝑖𝜕𝑦^𝑛−𝑖. (8)

A relative affine differential invariant with respect to transformation (6) is a polynomial ℐ defined on the jet space 𝒥 such that ℐ⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}=1(𝑎𝑑−𝑏𝑐)^𝑤ℐ⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}, (9) where 𝑤 is called the weight of the differential invariant. The degree of a term in ℐ is the sum of the exponents of the derivatives in the term. From the principles of invariant theory, it suffices to consider only homogeneous polynomials. In this case, all terms in ℐ have the same degree, which will be called the degree of the differential invariant. The order of the differential invariant is the highest order of derivative occurring in ℐ. It is easy to see that an invariant in the sense of (1) can be constructed directly if two relative invariants are known. From now on, when we speak of invariants, we always mean relative invariants in the sense of (9).

The definition of affine differential invariant in (9) does not include 𝑥,𝑦, and 𝑢(𝑥,𝑦). The reason for not including 𝑢(𝑥,𝑦) is that it is always invariant under affine transformation (6). The inclusion of 𝑢(𝑥,𝑦) would be redundant. The reason for not including 𝑥 and 𝑦 is that any nontrivial polynomial of 𝑥 and 𝑦 cannot be an affine invariant. We will prove this in the next section.

3. The Properties of Affine Differential Invariants

In this section, we derive a few properties of the differential invariants of functions defined on the plane under the action of the affine group 𝐀(2). These properties are important for constructing affine differential invariants. We begin with a proof of the proposition we claimed in the last section.

Theorem 1. An affine invariant polynomial ℐ cannot contain 𝑥 or 𝑦.

Proof. It suffices to show that a translation invariant polynomial ℐ cannot contain 𝑥 or 𝑦. Let ℐ(𝑥,𝑦,⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}) be a translation invariant polynomial. After a translation of the following form: 𝑥=𝑥+𝑒,𝑦=𝑦+𝑓, (10) we have the following identities: 𝑢_{𝑖,𝑛−𝑖}=𝑢_{𝑖,𝑛−𝑖}, (11) for all 𝑛≥1 and 0≤𝑖≤𝑛. According to the definition of affine differential invariant, using identities in (11), we have the following: ℐ𝑥,𝑦,⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}=ℐ𝑥,𝑦,⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}, (12) under translation transformation (10). Differentiating both sides of (12) with respect to 𝑒, we have the following: 0=𝜕ℐ𝜕𝑥𝜕𝑥+𝜕𝑒𝜕ℐ𝜕𝑦𝜕𝑦𝜕𝑒+𝜕ℐ𝜕𝑢^𝑘_{𝑖,𝑗−𝑖}𝜕𝑢^𝑘_{𝑖,𝑗−𝑖}.𝜕𝑒 (13) Since 𝜕𝑥𝜕𝑦𝜕𝑒=1,𝜕𝑒=0,𝜕𝑢^𝑘_{𝑖,𝑗−𝑖}𝜕𝑒=0, (14) for all 𝑖,  𝑗, and 𝑘 under consideration, we have the following: 𝜕ℐ𝜕𝑥=0. (15) That is, ℐ does not depend on 𝑥. Similarly, we can prove that ℐ does not depend on 𝑦.

Since 𝑢_{𝑖,𝑛−𝑖}=𝑢_{𝑖,𝑛−𝑖} for all 𝑛≥1 and 0≤𝑖≤𝑛 under translation transformation, we immediately have the following theorem.

Theorem 2. Any polynomial function ℐ(⟦^𝑞_𝑘=1⟦^𝑛_𝑗=1⟦^𝑗_𝑖=0𝑢^𝑘_{𝑖,𝑗−𝑖}) defined on a jet space of two dimensional functions is translation invariant.

Now we give a theorem that is useful for deriving properties of the affine differential invariants.

Theorem 3. Every affine transformation of the form (6) is equivalent to successively performing the following four types of transformations: (),(),(),.𝑥=𝑎𝑥,𝑦=𝑑𝑦,𝑎,𝑑∈ℝ,𝑎𝑑≠0𝑥=𝑥+𝑏𝑦,𝑦=𝑦,𝑏∈ℝ𝑥=𝑥,𝑦=𝑐𝑥+𝑦,𝑐∈ℝ𝑥=𝑥+𝑒,𝑦=𝑦+𝑓,𝑒,𝑓∈ℝ (16)(17)(18)(19)

Proof. An affine transformation of the form (6) is equivalent to a linear transformation followed by a translation. It is proved that every binary linear transformation is a product of linear transformations of the forms (16), (17), and (18) [1]. Thus, every affine transformation is composed of transformations of the forms (16), (17), and (18) plus a translation of the form (19).

We now prove a theorem which describes relations between the original derivative functions and the transformed derivative functions under the 2D affine action.

Theorem 4. Let 𝑢(𝑥,𝑦) be a smooth function defined on the plane, under the action of affine group 𝐀(2) defined in (6): 𝑢_{𝑖,𝑛−𝑖}=^𝑖_𝑗=0^𝑛−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑛−𝑖_𝑗𝑐^𝑖−𝑗𝑏^𝑘×𝑑^{𝑛−𝑖−𝑘}𝑢_{𝑗+𝑘,𝑛−𝑗−𝑘},𝑛≥1,0≤𝑖≤𝑛. (20)

Proof. We prove the theorem by induction on 𝑛. For the base case 𝑛=1, since 𝑢₁₀=𝑎𝑢₁₀+𝑐𝑢₀₁,𝑢₀₁=𝑏𝑢₁₀+𝑑𝑢₀₁, (21) the statement is true. Now suppose that the theorem holds for 𝑛=𝑚>1. Then we have the following: 𝑢_{𝑖,𝑚+1−𝑖}=𝜕𝑢𝜕𝑦^{𝑖,𝑚−𝑖}=^𝑖_𝑗=0^𝑚−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗𝑐^𝑖−𝑗𝑏^𝑘+1×𝑑^{𝑚+1−𝑖−(𝑘+1)}𝑢_{𝑗+𝑘+1,𝑚+1−𝑗−(𝑘+1)}+^𝑖_𝑗=0^𝑚−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗𝑐^𝑖−𝑗𝑏^𝑘×𝑑^{𝑚+1−𝑖−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}=^𝑖_𝑗=0^{𝑚+1−𝑖}_𝑘=1𝑖𝑗𝑎𝑚−𝑖𝑘−1^𝑗𝑐^𝑖−𝑗𝑏^𝑘×𝑑^{𝑚+1−𝑖−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}+^𝑖_𝑗=0^𝑚−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗𝑐^𝑖−𝑗𝑏^𝑘×𝑑^{𝑚+1−𝑖−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}=^𝑖_𝑗=0^{𝑚+1−𝑖}_𝑘=0𝑖𝑗𝑘𝑎𝑚+1−𝑖_𝑗𝑐^𝑖−𝑗𝑏^𝑘×𝑑^{𝑚+1−𝑖−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘},𝑢_{𝑖+1,𝑚−𝑖}=𝜕𝑢𝜕𝑥^{𝑖,𝑚−𝑖}=^𝑖_𝑗=0^𝑚𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗+1𝑐^{𝑖+1−(𝑗+1)}𝑏^𝑘×𝑑^{𝑚+1−(𝑖+1)−𝑘}𝑢_{𝑗+𝑘+1,𝑚+1−(𝑗+1)−𝑘}+^𝑖_𝑗=0^𝑚−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗𝑐^{𝑖+1−𝑗}𝑏^𝑘×𝑑^{𝑚+1−(𝑖+1)−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}=^𝑖+1_𝑗=1^𝑚−𝑖_𝑘=0𝑖𝑘𝑎𝑗−1𝑚−𝑖_𝑗𝑐^{𝑖+1−𝑗}𝑏^𝑘×𝑑^{𝑚+1−(𝑖+1)−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}+^𝑖_𝑗=0^𝑚−𝑖_𝑘=0𝑖𝑗𝑘𝑎𝑚−𝑖_𝑗𝑐^{𝑖+1−𝑗}𝑏^𝑘×𝑑^{𝑚+1−(𝑖+1)−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}=^𝑖+1_𝑗=0^𝑚−𝑖_𝑘=0𝑗𝑘𝑎𝑖+1𝑚−𝑖_𝑗𝑐^{𝑖+1−𝑗}𝑏^𝑘×𝑑^{𝑚+1−(𝑖+1)−𝑘}𝑢_{𝑗+𝑘,𝑚+1−𝑗−𝑘}. (22) So, the theorem is true for 𝑛=𝑚+1. We conclude from the principle of mathematical induction that the theorem is true for all positive integer 𝑛.

Next, we proceed to prove several theorems that describe properties of the affine differential invariants.

Theorem 5. Let ℐ=𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘} (23) be a homogeneous and isobaric polynomial function defined on the jet space, where 𝒞_𝐼 are coefficients indexed by 𝐼. A necessary and sufficient condition for the function ℐ to be an invariant with respect to the transformation (16) is ^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑘𝑙_{𝐼𝑖𝑗𝑘}=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑗−𝑘𝑙_{𝐼𝑖𝑗𝑘}. (24)

Proof. Let ℐ=𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘} (25) be an affine differential invariant with respect to the linear transformation (16). From (20), we have the following: 𝑢_{𝑖,𝑛−𝑖}=𝑎^𝑖𝑑^𝑛−𝑖𝑢_{𝑖,𝑛−𝑖}. (26) Substituting (26) into (25), we have the following: ℐ=𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}=𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑎^𝑘𝑑^𝑗−𝑘𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}. (27) From the definition in (9), for ℐ to be an affine differential invariant, there must be an identity of the following form: 𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}=1(𝑎𝑑)^𝜆𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}. (28) From (27) and (28), we have 𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑎^𝑘𝑑^𝑗−𝑘𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}=(𝑎𝑑)^𝜆𝒞_𝐼^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}^{𝑙𝐼𝑖𝑗𝑘}. (29) Comparing the degrees of 𝑎 and 𝑑 on both sides of (29), we conclude that 𝜆=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑗−𝑘𝑙_{𝐼𝑖𝑗𝑘}=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑘𝑙_{𝐼𝑖𝑗𝑘}. (30) This ends the proof.

Theorem 6. A necessary and sufficient condition for a polynomial function ℐ to be an affine differential invariant with respect to the linear transformation (17) is 𝒪ℐ=0, (31) where 𝒪=^𝑞𝑛_𝑖=1𝑗=1^𝑗−1_𝑘=0𝑗−𝑘𝑢^𝑖_{𝑘+1,𝑗−𝑘−1}𝜕𝜕𝑢^𝑖_{𝑘,𝑗−𝑘}. (32)

Proof. With respect to the linear transformation (17), we have the following: 𝑢_{𝑖,𝑛−𝑖}=^𝑛−𝑖_𝑘=0𝑘𝛽𝑛−𝑖_𝑘𝑢_{𝑖+𝑘,𝑛−𝑖−𝑘}, (33) where 𝛽=−𝑏. Differentiating both sides of (33) with respect to 𝛽, we have the following: 𝑑𝑢𝑑𝛽^{𝑖,𝑛−𝑖}=^𝑛−𝑖_𝑘=1𝑘𝑛−𝑖𝑘𝛽^𝑘−1𝑢_{𝑖+𝑘,𝑛−𝑖−𝑘}=^{𝑛−𝑖−1}_𝑘=0()𝛽𝑛−𝑖𝑘+1𝑘+1^𝑘𝑢_{𝑖+𝑘+1,𝑛−𝑖−𝑘−1}=^{𝑛−𝑖−1}_𝑘=0(𝑛−𝑖)!()!()!×()𝛽𝑘+1𝑛−𝑖−𝑘−1𝑘+1^𝑘𝑢_{𝑖+𝑘+1,𝑛−𝑖−𝑘−1}=(𝑛−𝑖)^{𝑛−𝑖−1}_𝑘=0𝑘𝛽𝑛−𝑖−1_𝑘𝑢_{𝑖+𝑘+1,𝑛−𝑖−𝑘−1}=(𝑛−𝑖)𝑢_{𝑖+1,𝑛−𝑖−1}. (34) With respect to the linear transformation (17), an affine differential invariant has to make the following equation: ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}=ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}. (35) Holding in the 𝛽. Differentiating both sides of (35) with respect to 𝛽, we have the following: 𝜕ℐ⟦^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}𝜕𝑢^𝑖_{𝑘,𝑗−𝑘}𝑑𝑢^𝑖_{𝑘,𝑗−𝑘}𝑑𝛽=0. (36) Substituting (34) into (36), we have the following: 𝑗−𝑘𝑢^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0^𝑖_{𝑘+1,𝑗−𝑘−1}𝜕ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}𝜕𝑢^𝑖_{𝑘,𝑗−𝑘}=0. (37) This ends the proof.

Similarly, we can prove the following theorem.

Theorem 7. A necessary and sufficient condition for a polynomial function ℐ to be an affine differential invariant with respect to the linear transformation (18) is 𝒟ℐ=0, (38) where 𝒟=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=1𝑘𝑢^𝑖_{𝑘−1,𝑗−𝑘+1}𝜕𝜕𝑢^𝑖_{𝑘,𝑗−𝑘}. (39)

Combining Theorems 2, 5, 6, and 7, we have the following.

Theorem 8. The necessary and sufficient conditions for a homogeneous and isobaric polynomial function ℐ to be an affine differential invariant with respect to the affine transformation (6) is 𝒪ℐ=0,𝒟ℐ=0,_{𝑞𝑛𝑖=1𝑗𝑗=1𝑘=0}𝑘𝑙_{𝐼𝑖𝑗𝑘}=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑗−𝑘𝑙_{𝐼𝑖𝑗𝑘}. (40)

4. The Generating Operator of Affine Differential Invariants

We present direct and simple methods to construct the differential invariants in this section. We begin with two special cases.

Let 𝑢¹(𝑥,𝑦) and 𝑢²(𝑥,𝑦) be two smooth functions in variables 𝑥 and 𝑦. The Jacobian𝑢𝐽=det¹¹⁰𝑢¹₀₁𝑢²₁₀𝑢²₀₁ (41) of the two functions is invariant with respect to the affine transformation (6). This proposition can be proved through direct calculation: 𝑢det¹¹⁰𝑢¹₀₁𝑢²₁₀𝑢²₀₁𝑎𝑢=det¹¹⁰+𝑐𝑢¹₀₁𝑏𝑢¹₁₀+𝑑𝑢¹₀₁𝑎𝑢²₁₀+𝑐𝑢²₀₁𝑏𝑢²₁₀+𝑑𝑢²₀₁=()𝑢𝑎𝑑−𝑏𝑐det¹¹⁰𝑢¹₀₁𝑢²₁₀𝑢²₀₁. (42)

Let 𝑢(𝑥,𝑦) be a smooth function in variables 𝑥 and 𝑦. The Hessian𝑢𝐻=det²⁰𝑢₁₁𝑢₁₁𝑢₀₂ (43) of the function is invariant with respect to the affine transformation (6). This can be verified by direct calculation also: 𝑢det²⁰𝑢₁₁𝑢₁₁𝑢₀₂𝑐=det²𝑢₀₂+2𝑎𝑐𝑢₁₁+𝑎²𝑢₂₀𝑐𝑑𝑢₀₂+𝑐𝑏𝑢₁₁+𝑎𝑑𝑢₁₁+𝑎𝑏𝑢₂₀𝑐𝑑𝑢₀₂+𝑐𝑏𝑢₁₁+𝑎𝑑𝑢₁₁+𝑎𝑏𝑢₂₀𝑑²𝑢₀₂+2𝑏𝑑𝑢₁₁+𝑏²𝑢₂₀𝑢=det𝑎𝑐𝑏𝑑det²⁰𝑢₁₁𝑢₁₁𝑢₀₂=()det𝑎𝑏𝑐𝑑𝑎𝑑−𝑏𝑐²𝑢det²⁰𝑢₁₁𝑢₁₁𝑢₀₂. (44)

The Jacobian and Hessian are two special cases of a general method for constructing affine differential invariants from two smooth functions. The method is presented in the following theorem.

Theorem 9. Let 𝑢¹(𝑥,𝑦) and 𝑢²(𝑥,𝑦) be two smooth functions in variables 𝑥 and 𝑦. The following function ^𝑟_𝑠=0(−1)^𝑠𝑟𝑠𝑢¹_{𝑟−𝑠,𝑠}𝑢²_{𝑠,𝑟−𝑠} (45) is invariant with respect to the affine transformation (6), where r is a positive integer.

Proof. From Theorem 2, (45) is invariant under translation transformation. Now we have to show that (45) is also invariant under linear transformations (16), (17), and (18). Under the linear transformation (16), we have the following: ^𝑟_𝑠=0(−1)^𝑠𝑟𝑠𝑢¹_{𝑟−𝑠,𝑠}𝑢²_{𝑠,𝑟−𝑠}=(𝑎𝑑)^𝑟𝑟_𝑠=0(−1)^𝑠𝑟𝑠𝑢¹_{𝑟−𝑠,𝑠}𝑢²_{𝑠,𝑟−𝑠}. (46) The theorem is true in this case. With respect to the linear transformation (17), since 𝑢^𝜏_{𝑖,𝑛−𝑖}=^𝑛−𝑖_𝑘=0𝑘𝑏𝑛−𝑖_𝑘𝑢^𝜏_{𝑖+𝑘,𝑛−𝑖−𝑘},𝜏=1,2, (47) we have the following: ^𝑟_𝑠=0(−1)^𝑠𝑟𝑠𝑢¹_{𝑟−𝑠,𝑠}𝑢²_{𝑠,𝑟−𝑠}=^𝑟𝑠_𝑠=0𝑗=0^𝑟−𝑠_𝑘=0(−1)^𝑠𝑏^𝑗+𝑘𝑟𝑠𝑠𝑗×𝑘𝑢𝑟−𝑠_{1𝑟−𝑠+𝑗,𝑠−𝑗}𝑢²_{𝑠+𝑘,𝑟−𝑠−𝑘}=^𝑟𝑟_{𝑗=0𝑠=𝑗}^𝑟−𝑠_𝑘=0(−1)^𝑠𝑏^𝑗+𝑘𝑟𝑠𝑠𝑗×𝑘𝑢𝑟−𝑠_{1𝑟−𝑠+𝑗,𝑠−𝑗}𝑢²_{𝑠+𝑘,𝑟−𝑠−𝑘}=^𝑟_𝑗=0^𝑟−𝑗_𝑠=0^{𝑟−𝑗−𝑠}_𝑘=0(−1)^𝑠+𝑗𝑏^𝑗+𝑘𝑟𝑗×𝑘𝑢𝑠+𝑗𝑠+𝑗𝑟−𝑠−𝑗_{1𝑟−𝑠,𝑠}𝑢²_{𝑠+𝑗+𝑘,𝑟−𝑠−𝑗−𝑘}=^𝑟_𝑠=0^𝑟−𝑠_𝑗=0^{𝑟−𝑗−𝑠}_𝑘=0(−1)^𝑠+𝑗𝑏^𝑗+𝑘𝑟𝑗×𝑘𝑢𝑠+𝑗𝑠+𝑗𝑟−𝑠−𝑗_{1𝑟−𝑠,𝑠}𝑢²_{𝑠+𝑗+𝑘,𝑟−𝑠−𝑗−𝑘}=^𝑟_𝑠=0^𝑟−𝑠_𝑗=0^𝑟−𝑠_𝑘=𝑗(−1)^𝑠+𝑗𝑏^𝑘𝑟𝑗×𝑢𝑠+𝑗𝑠+𝑗𝑟−𝑠−𝑗𝑘−𝑗^{1𝑟−𝑠,𝑠}𝑢²_{𝑠+𝑘,𝑟−𝑠−𝑘}=^𝑟_𝑠=0^𝑟−𝑠_𝑘=0(−1)^𝑠𝑏^𝑘𝑟!()𝑢𝑘!𝑟−𝑠−𝑘!𝑠!^{1𝑟−𝑠,𝑠}𝑢²_{𝑠+𝑘,𝑟−𝑠−𝑘}×^𝑘_𝑗=0(−1)^𝑗𝑘𝑗=^𝑟_𝑠=0^𝑟−𝑠_𝑘=0(−1)^𝑠𝑏^𝑘𝑟!()𝑢𝑘!𝑟−𝑠−𝑘!𝑠!^{1𝑟−𝑠,𝑠}𝑢²_{𝑠+𝑘,𝑟−𝑠−𝑘}0^𝑘=^𝑟_𝑠=0(−1)^𝑠𝑟𝑠𝑢¹_{𝑟−𝑠,𝑠}𝑢²_{𝑠,𝑟−𝑠}. (48) So, the theorem is true in this case. Similarly, we can prove that the theorem is true in the case of (18). From the principles of group action, we conclude that the theorem is true.

Although the method presented in Theorem 9 is very general and can produce infinite number of affine differential invariants, there are affine differential invariants that cannot be constructed in this way. We need a method that can derive completely all affine differential invariants. Classical invariant theory of binary forms is a well-studied field. There are methods to derive a complete set of invariants of a system of binary forms under the action of the general linear group. The following theorem establishes a relation between classical invariant theory of binary forms and affine differential invariants of 2D functions.

Theorem 10. Given a system of binary forms ⟦^𝑞_𝑖=1⟦^𝑛𝑗_𝑗=1𝑘=0𝑗𝑘𝑎^𝑖_𝑘,𝑗𝑋^𝑗−𝑘𝑌^𝑘 (49) and a set of smooth functions in variables x and y⟦^𝑞_𝑖=1𝑢^𝑖𝑥,𝑦, (50) the homogeneous and isobaric polynomial function ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗 (51) is an invariant of the system of binary forms in (49) with respect to the following linear transformation: 𝑌𝑋=𝜆𝑋+𝜅𝑌,𝑌=𝜂𝑋+𝜇 (52) if and only if the homogeneous and isobaric polynomial function ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘} (53) is an affine differential invariant with respect to affine transformation (6).

Proof. From Theorem 8, the necessary and sufficient conditions for the homogeneous and isobaric polynomial function ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘} (54) to be an affine differential invariant are as follows: 𝒪ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}⟦=0,𝒟ℐ^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}=0,_{𝑞𝑛𝑖=1𝑗𝑗=1𝑘=0}𝑘𝑙_{𝐼𝑖𝑗𝑘}=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑗−𝑘𝑙_{𝐼𝑖𝑗𝑘}. (55) According to theorems of classical invariant theory (see book Sections I.4 and I.9 in [1], cf. chap.I and chap.II in [3]), the necessary and sufficient conditions for ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗 (56) to be an invariant of the system of binary forms in (49) are as follows: 𝒪ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗=0,𝒟ℐ⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗=0,_{𝑞𝑛𝑖=1𝑗𝑗=1𝑘=0}𝑘𝑙_{𝐼𝑖𝑗𝑘}=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=0𝑗−𝑘𝑙_{𝐼𝑖𝑗𝑘}, (57) where 𝒪=^𝑞𝑛_𝑖=1𝑗=1^𝑗−1_𝑘=0𝑗−𝑘𝑎^𝑖_𝑘+1,𝑗𝜕𝜕𝑎^𝑖_𝑘,𝑗,𝒟=^𝑞𝑛_𝑖=1^𝑗_𝑗=1𝑘=1𝑘𝑎^𝑖_{𝑘−1,𝑗}𝜕𝜕𝑎^𝑖_𝑘,𝑗. (58) It is easy to see that, after a replacement of the corresponding variables, formula 𝒪ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}) is the same as formula 𝒪ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗) and formula 𝒟ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘}) is the same as formula 𝒟ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑎^𝑖_𝑘,𝑗). This means that (55) and (57) have the same truth values. This ends the proof of the theorem.

5. Affine Differential Invariants of Low Orders

By the methods provided in the previous section, it is easy to construct affine differential invariants of smooth functions up to any order. A general method for constructing affine differential invariants of smooth functions uses Theorem 8 directly. To construct all the differential invariants of degree 𝑑 and order 𝑛, we can enumerate all homogeneous and isobaric polynomial functions of derivatives with unknown coefficients such that ∑^𝑞_𝑖=1∑^𝑛_𝑗=1∑^𝑗_𝑘=0𝑘𝑙_{𝐼𝑖𝑗𝑘}=∑^𝑞_𝑖=1∑^𝑛_𝑗=1∑^𝑗_𝑘=0(𝑗−𝑘)𝑙_{𝐼𝑖𝑗𝑘}. We then use the conditions 𝒪ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘})=0, 𝒟ℐ(⟦^𝑞_𝑖=1⟦^𝑛_𝑗=1⟦^𝑗_𝑘=0𝑢^𝑖_{𝑘,𝑗−𝑘})=0 to solve these unknown coefficients, cf. [1]. However, since there is a 1-1 correspondence between affine differential invariants and invariants of binary forms by Theorem 10, we do not need to derive affine differential invariants again. Low-degree invariants of binary forms are known completely [1–4]. There are general methods in the classical theory such as the transvection method which can derive complete sets of invariants up to any order.

We present a few affine differential invariants of low orders in this section. The invariants are in explicit form. This is important since application users may not be familiar with the mathematical background to derive the invariants.

There are two functional independent affine differential invariants of order two: ℐ²_2,1=𝑢₂₀𝑢₀₂−𝑢²₁₁,ℐ²_2,2=𝑢₂₀𝑢²₀₁−2𝑢₁₁𝑢₀₁𝑢₁₀+𝑢₀₂𝑢²₁₀. (59)

There are four functional independent affine differential invariants of order three. Two of them have weight three: ℐ³_3,1=𝑢₃₀𝑢³₀₁−3𝑢₂₁𝑢₁₀𝑢²₀₁+3𝑢₁₂𝑢²₁₀𝑢₀₁−𝑢₀₃𝑢³₁₀,ℐ³_3,2=𝑢₃₀𝑢₀₂𝑢₀₁−2𝑢₂₁𝑢₁₁𝑢₀₁−𝑢₂₁𝑢₀₂𝑢₁₀+𝑢₁₂𝑢₂₀𝑢₀₁+2𝑢₁₂𝑢₁₁𝑢₁₀−𝑢₀₃𝑢₂₀𝑢₁₀. (60) One of them has weight four: ℐ⁴_3,3=𝑢₃₀𝑢₁₂𝑢₀₂−𝑢₃₀𝑢₀₃𝑢₁₁−𝑢²₂₁𝑢₀₂+𝑢₂₁𝑢₁₂𝑢₁₁+𝑢₂₁𝑢₀₃𝑢₂₀−𝑢²₁₂𝑢₂₀. (61)

Another third-order affine differential invariant has weight six: ℐ⁶_3,4=𝑢²₃₀𝑢²₀₃−6𝑢₃₀𝑢₂₁𝑢₁₂𝑢₀₃+4𝑢₃₀𝑢³₁₂−3𝑢²₂₁𝑢²₁₂+4𝑢³₂₁𝑢₀₃. (62)

There are five functional independent affine differential invariants of order four. One of them has weight six: ℐ⁶_4,1=𝑢₄₀𝑢₂₂𝑢₀₄−𝑢₄₀𝑢²₁₃+2𝑢₃₁𝑢₂₂𝑢₁₃−𝑢²₃₁𝑢₀₄−𝑢³₂₂. (63) Four of them have weight four: ℐ⁴_4,2=𝑢₄₀𝑢₀₄−4𝑢₃₁𝑢₁₃+3𝑢²₂₂,ℐ⁴_4,3=𝑢₄₀𝑢⁴₀₁−4𝑢₃₁𝑢₁₀𝑢³₀₁+6𝑢₂₂𝑢²₁₀𝑢²₀₁−4𝑢₁₃𝑢³₁₀𝑢₀₁+𝑢₀₄𝑢⁴₁₀,ℐ⁴_4,4=𝑢₄₀𝑢²₀₂−4𝑢₃₁𝑢₁₁𝑢₀₂+2𝑢₂₂𝑢₂₀𝑢₀₂+4𝑢₂₂𝑢²₁₁−4𝑢₂₀𝑢₁₃𝑢₁₁+𝑢²₂₀𝑢₀₄,ℐ⁴_4,5=𝑢₄₀𝑢₀₃𝑢₀₁−𝑢₃₁3𝑢₁₂𝑢₀₁+𝑢₀₃𝑢₁₀+3𝑢₂₂𝑢₂₁𝑢₀₁+𝑢₁₂𝑢₁₀−𝑢₁₃𝑢₃₀𝑢₀₁+3𝑢₂₁𝑢₁₀+𝑢₀₄𝑢₃₀𝑢₁₀. (64)

Needless to say, the presented invariants all satisfy the properties (40). There are certainly other forms of invariants. The presented invariants are supposed to be the simplest ones.

6. Conclusion

We have presented an effective method to construct affine differential invariants of functions defined on the plane. The method is clear and simple. It is possible to generalize this method to construct differential invariants of functions defined on higher dimensional spaces. The proposed differential invariants may be applied in applications such as computer vision. These are the directions of future research.

Acknowledgments

This work is supported by the National Science Foundation for Distinguished Young Scholars of China under Grant no. 61225012; the National Natural Science Foundation of China under Grants no. 61070162, no. 61073062, no. 71071028 and no. 70931001; the Specialized Research Fund of the Doctoral Program of Higher Education for the Priority Development Areas under Grant no. 20120042130003; the Specialized Research Fund for the Doctoral Program of Higher Education under Grants no. 20100042110025 and no. 20110042110024; the Specialized Development Fund for the Internet of Things from the Ministry of Industry and Information Technology of the China; and the Fundamental Research Funds for the Central Universities under Grant no. N110204003.