## 1 Introduction

Researchers have found that the geometric deformation of the object, caused by the change of viewpoint, is an important factor leading to the object to be misidentified. In order to solve this problem, various methods have been proposed in order to get image features which are robust to the geometric deformation. Moments and moment invariants are one of them.

The concepts of moment and moment invariants were first proposed by Hu in 1962 [1]. He employed the theory of algebraic invariants, which was studied in 19th century[2]

, and defined geometric moments. Then he constructed seven geometric moment invariants which are invariant to the similarity transform. This set of invariants was widely used in various fields of pattern recognition, like

[3]. But the similarity transform can’t represent all geometric deformations. When the distance between the camera and the object is much larger than the size of the object itself, the geometric deformation of the object can be represented by the affine transform. The landmark work of affine moment invariants(AMIs) was proposed by Flusser and Suk in 1993 [4]. They used geometric moments to construct several low-order and low-degree AMIs, which were more effective to practical applications, for example, image registration [5]. In order to obtain more AMIs, Suk and Flusser proposed the graph method which can generate AMIs of every order and degree [6]. Xu and Li [7]derived moment invariants in the intuitive way by multiple integrals of invariant geometric primitives like distance, area and volume. This method not only simplified the construction of AMIs, but also made them have a clear geometric meaning. Recently, Li

improved the method of geometric primitives [8]. They found a way to further simplify geometric primitives and used dot-product and cross-product of vectors to generate invariants. Meanwhile, researchers were also constantly expanding the definition of moments. Due to the lack of orthogonality, information redundancy of geometric moments became inevitable and image reconstruction from geometric moments was very difficult

[8]. Therefore, various orthogonal polynomials were used to define new moments. The first one proposed in 1980, Teague introduced orthogonal Legendre and Zernike moments [9]. Then pseudo-Zernike moments [10], Fourier-Mellin moments [11], Chebyshey-Fourier moments [12], pseudo-Jacobi-Fourier moments [13] and Gaussian Hermite moments [14] were proposed. However, it was very difficult to obtain affine moment invariants of these orthogonal moments. This weakness greatly limited the use of orthogonal moments. Additionally, previous studies have shown that low-order and low-degree moment invariants have better performance, such as stability, than high-order and high-degree moment invariants. But the number of low-order and low-degree moment invariants was very limited. So, it s very useful to get more low-order and low-degree moment invariants.The studies of local differential invariants are another area, which need to be concerned. Olver generalized the moving frame method and got differential invariants for general transformation groups [15]. He defined the affine gradient by using local affine differential invariants [16]. Ge [17] presented a local feature descriptor under color affine transformation by using the affine gradient. Wang [18] proposed an effective method to derive a special type of affine differential invariants. Given some functions defined on the plane and affine group acting on the plane. However, they didn’t explain how to use these local affine differential invariants in practical applications and how to improve the numerical accuracy of partial derivatives on discrete image.

In this paper, we use the frame of geometric moments and partial derivatives to define a kind of weighted moments, which can be named as differential moments(DMs). According to the definition of DMs and local affine differential invariants, affine weighted moment invariants(AWMIs) can be obtained easily, which use both global and local information. The experimental results show that AWMIs have good stability and distinguishability. Also, they can improve the accuracy of image retrieval.

## 2 Some Basic Definitions and Theorems

In order to understand the construction frame of AWMIs more clearly, we first introduce some basic definitions and theorems.

### 2.1 The Definition of Geometric Moments

The geometric moment of the image is defined by

(1) |

where , , is the order of . In order to eliminate the effect of translation, central geometric moments are usually used. The central moment of the order is defined by

(2) |

where

(3) |

### 2.2 Coordinate Transformation under the Affine Transform

Suppose the image is transformed into another image by the affine transform A and the translation T. is the corresponding point of . We can get the following relationship

(4) |

where A is a nonsingular matrix.

### 2.3 The Construction of Affine Moments Invariants

For the image , let and be two arbitrary points in the domain of . The geometric primitive proposed in [7] can be defined by

(5) |

Suppose the image is transformed into another image by Eq.(4). in are the corresponding points of in . Then, there is a relation

(6) |

where is the determinant of A. Therefore, using N points in , can be defined by

(7) |

where , , . represents the number of point in all geometric primitives, .

Let in be corresponding points of in . It’s obviously that

(8) |

where

(9) |

Finally, using , can be defined by

(10) |

In [7], Xu and Li proved that Eq.(10) didn’t change when the image was transformed by Eq.(4). Eq.(10) is the general form of AMIs. In fact, this multiple integral can be expressed as polynomials of central geometric moments.

(11) |

where represents the number of multiplicative items in this expansion, represents the coefficient of the j-th multiplicative item. In general, is named as the degree of Eq.(10), is named as the order of Eq.(10). They are determined by .

### 2.4 Local Differential Invariants under the Affine Transform

For the differentiable function , Olver [15] used the contact-invariant coframe to obtain local differential invariants under the affine transform. The first and second order local differential invariants of were defined by:

(12) |

(13) |

(14) |

(15) |

(16) |

Among them, and are pure differential invariants, which don’t contain or . and are absolute differential invariants. , and are relative differential invariants, which meanS

(17) |

where and are local differential invariants of . and satisfy the relationship shown in Eq.(4).

In addition, Olver indicated that differential invariants shown in were not independent [15]. The relationship of them was defined by

(18) |

## 3 The Construction Frame of AWMIs

### 3.1 The Definition of DMs

Definition 1.

Let be the differentiable function. The first-order DMs are defined by:

(19) |

where .

The second-order differential moments are defined by:

(20) |

where .

Similarly, we can construct higher-order differential moments. But considering their convenience and the accuracy of calculation, we only define the first-order and second-order DMs. Compared with the definition of geometric central moments in Eq(2), DMs are constructed by using the polynomial functions and derivative functions of . Thus, they can represent internal information of images better.

### 3.2 The Construction of AWMIs

Definition 2.

Suppose be the differentiable function, the first-order AWMIs constructed by the the first-order DMs are defined by

(21) |

where

(22) |

Note that we assume and . Then, we can get the following theorem.

Theorem 1.

Suppose the image is transformed into another image by Eq.(4). in are corresponding points of in . The following equation are established.

(23) |

where

(24) |

The proof of Eq.(23) is the same as that of Eq.(10) proved in [7] .

### 3.3 The Instances of AWMIs

In [6], Flusser and Suk proved that there were seven kinds of AMIs when the degree and the order . They are listed in the Table 1. It is important to note that and are always zero. Thus, and can’t be used as invariants. That means there are only two AMIs, {}, when the degree and the order .

No. | Core | AMI |
---|---|---|

0 | ||

0 | ||

0 | ||

But now, we can use Eq.(21) to construct many AWMIs. When , and , there are 8 kinds of DCores which can be constructed AWMIs. They are list in the Table 2.

No. | DCore |
---|---|

In the Table 3, we list AWMIs constructed by Dcores in the Table 2. They are all constructed by first-order DMs. It is worth noting that we have removed Dcores of which expansions are always 0 or contain . In fact, using a similar definition to Eq.(23), we can get AWMIs constructed by the second-order DMs. But here, we give a new definition. We want to point out that there are many different methods to construct AWMIs.

No. | AWMI |
---|---|