Images recognition and classification require an extraction technique of feature vectors of these images. These vectors must be invariant to the three geometric transformations: rotation, translation and scaling. Several authors used the theory of orthogonal moments to extract the feature vectors of images. Jacobi moments are orthogonal moments, which have been widely applied in imaging and pattern recognition. However, the invariance to rotation of Cartesian Jacobi moments is very difficult to obtain. In this paper, we obtain at first a set of transformed orthogonal Jacobi polynomials, called “Adapted Jacobi polynomials”. Based on these polynomials, a set of orthogonal moments is presented, named adapted Jacobi moments (AJMs). These moments are orthogonal on the rectangle \(\left[ {0, N\left] { \times } \right[0, M} \right],\) where \(N \times M\) is the size of the described image. We also provide a new series of feature vectors of images based on adapted Jacobi orthogonal invariants moments, which are a linear combination of geometric moment invariants, where the latest ones are invariant under rotation, translation and scaling of the described image. Based on k-NN algorithm, we apply a new 2D image classification system. We introduce a set of experimental tests in pattern recognition. The obtained results express the efficiency of our method. The performance of these feature vectors is compared with someones extracted from Hu, Legendre and Tchebichef invariant moments using three different 2D image databases: MPEG7-CE shape database, Columbia Object Image Library (COIL-20) database and ORL database. The results of the comparative study show the performance and superiority of our orthogonal invariant moments.

图像识别和分类需要这些图像的特征向量的提取技术。这些向量必须对于三个几何变换是不变的：旋转，平移和缩放。几位作者使用正交矩理论提取图像的特征向量。Jacobi矩是正交矩，已广泛应用于成像和模式识别。然而，笛卡尔雅可比矩的旋转不变性很难获得。在本文中，我们首先获得了一组变换的正交Jacobi多项式，称为“ Adapted Jacobi多项式”。基于这些多项式，提出了一组正交矩，称为自适应雅可比矩（AJM）。这些矩在矩形上是正交的\（\ left [{0，N \ left] {\ times} \ right [0，M} \ right]，\）其中\（N \ times M \）是所描述图像的尺寸。我们还基于适应的Jacobi正交不变量提供了一系列新的图像特征向量，这些正交向量是几何矩不变量的线性组合，其中最新的不变量在所述图像的旋转，平移和缩放下是不变的。基于k-NN算法，我们应用了新的2D图像分类系统。我们介绍了模式识别中的一组实验测试。所得结果表明了我们方法的有效性。使用三个不同的2D图像数据库：MPEG7-CE形状数据库，哥伦比亚对象图像库（COIL-20）数据库和ORL数据库，将这些特征向量的性能与从Hu，Legendre和Tchebichef不变矩中提取的特征进行比较。