AI 中文总结
研究二维形状和图像高阶泽尼克矩精确计算问题，提出基于边缘公式，通过格林定理转换积分消除误差，适用于多种图像类型，推导递推关系，实验表明该方法在低阶与经典方法精度相当，高阶时稳定。
AI 中文摘要
泽尼克矩是形状和图像分析中广泛使用的旋转不变描述符，但其标准计算依赖于基于像素的求积法，将每个像素视为位于其中心的点质量。这种近似会引入空间混叠，随着矩阶数增加而加剧，降低图像重建质量和高阶矩的判别能力。我们提出一种基于边缘的公式，通过应用格林定理将定义泽尼克矩的二维面积积分转换为沿图像边界的一维积分之和，消除误差源。该框架适用于多边形形状、二值图像、灰度图像和彩色图像。我们推导了所需径向基元的递推关系，表明变换后的边缘被积函数是多项式函数，可使用Clenshaw-Curtis求积法精确求值。数值实验表明，该公式在低阶时与经典方法精度匹配，在基于像素的矩存在显著混叠和数值退化的高阶时保持稳定。
英文摘要
Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green's theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw--Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixel-based approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.
Comments17 pages, 9 figures
