核能谱单能峰快速高精度高斯函数拟合算法研究

摘要

在核能谱分析中，高斯函数最小二乘拟合法是计算单能峰净峰面积常用的方法，该方法精度较高，但噪声敏感性较强，导致拟合出的高斯函数在峰位附近的残差向量较大。针对该问题，对高斯函数最小二乘拟合法进行了详细推导，分析了峰位附近残差向量较大的原因，提出了一种基于高斯函数最小二乘拟合法的高斯函数加权最小二乘拟合法，即在高斯函数最小二乘拟合法的基础上，引入了权重因子。该权重因子与取对数后数据权重削弱趋势相反或与数据本身趋势相符，以减小噪声敏感性。由于在求解高斯函数参数的过程中涉及到求逆矩阵运算，计算量较大，耗时较长，为了提高实时性，将求逆矩阵的运算过程转换为了简单的方程组运算，并给出了高斯函数的幅值、中心及方差参数的快速求解公式。将这两种方法用于55 Fe 的特征 X 射线单能峰的实际拟合中，结果表明，高斯函数加权最小二乘拟合法效果均较好，这说明该方法降低了噪声敏感性，减小了高斯函数在峰位附近的残差向量，进一步提高了拟合精度。另外，使用快速求解公式，也减小了运算量，增强了实时性，为在便携式设备中的有效使用提供了可能。%In nuclear spectrum,Gaussian function least square fitting is a commonly used method.Usually the method has high precision,but it is very much sensitive to noise,which causes that the residual vector is larger near the peak in the Gaussian function.To solve the problem,Gaussian function least square fitting was deduced particularly,and the causes are analyzed.As a result,Gaussian function weighted least square fitting is proposed,i.e.,a weight factor,which had an opposite tendency to the data weight reduction tendency after taking logarithm,or it had the same tendency to the origin data.This was introduced based on Gaussian function least square fitting to reduce noise sensitivity.In the process of solving Gaussian parameter,to im-prove the real-time performance,the solution process of inverse matrix was transferred to the solution process of simple equa-tions because the computation of inverse matrix was time consuming.Gaussian function parameter,amplitude,center value and variance,were given with the fast calculation formulas.By applying these two methods to the practical fitting of 5 5 Fe character-istic X-ray single spectrum peak,respectively,the results show that Gaussian function weighted least square fitting is more satis-factory.It indicates the proposed method can decrease the noise sensitivity and reduce the residual vector near the peak;in addi-tion,the fitting precision is also improved.What's more,the real-time performance is improved by applying fast calculation for-mulas,which makes it possible to apply the proposed method to portable equipment efficiently.