EM (Expectation Maximization)算法是统计学中的核心算法,也是本校近代数理统计课程教学过程中的一个重难点。论文采用案例式、启发式、研讨式教学方法,以基于高斯混合模型(GMM)的轴承退化阶段划分问题为例,引导学生发现隐变量模型极...EM (Expectation Maximization)算法是统计学中的核心算法,也是本校近代数理统计课程教学过程中的一个重难点。论文采用案例式、启发式、研讨式教学方法,以基于高斯混合模型(GMM)的轴承退化阶段划分问题为例,引导学生发现隐变量模型极大似然估计(MLE)存在的困难,设计问题链启发学生探寻参数估计的数值方法,并总结出EM算法的一般过程。基于matlab编程可视化EM算法下的GMM模型参数更新过程,对比MLE目标函数和EM迭代目标函数,分析EM算法的内涵思想并结合图形进行直观展示,并且挖掘其中蕴含的思政元素,在知识传授的同时实现价值塑造。Expectation maximization (EM) algorithm is a core algorithm in statistics and also a key and difficult point in the teaching process of modern mathematical statistics courses in our school. The paper adopts a case-based and heuristic teaching method, taking the Gaussian Mixture Model (GMM) based bearing degradation stage division problem as an example, guiding students to discover the difficulties of maximum likelihood estimation (MLE) in the latent variable model, designing a problem chain to inspire students to explore numerical methods for parameter estimation, and summarizing the general process of EM algorithm. Based on Matlab programming, the parameter update process of GMM based on EM algorithm is visualized. Comparing the MLE objective function and EM iteration objective function, the intrinsic thought of EM algorithm is analyzed and visually displayed with graphics. The ideological and political elements are also explored, so as to achieve value shaping while knowledge transmission.展开更多
高光谱图像(Hyperspectral Images, HSIs)在遥感和医学成像等领域具有广泛的应用,但在采集过程中容易受到各种噪声的干扰。尽管目前已有多种去噪方法应用于高光谱图像处理,但这些方法在应对复杂光谱特征和复杂或非均匀噪声分布时仍面临...高光谱图像(Hyperspectral Images, HSIs)在遥感和医学成像等领域具有广泛的应用,但在采集过程中容易受到各种噪声的干扰。尽管目前已有多种去噪方法应用于高光谱图像处理,但这些方法在应对复杂光谱特征和复杂或非均匀噪声分布时仍面临挑战,且部分方法可能导致图像细节的丢失,降低光谱数据的真实性和有效性。为克服这些不足,文章提出了一种基于多元高斯混合模型的高光谱图像去噪方法。该方法将噪声建模为多元高斯混合模型,通过多模态特征表示图像的复杂光谱结构,以更好地适应不同光谱分布。我们采用变分贝叶斯(Variational Bayes, VB)方法进行参数估计,从而改善了传统期望最大化(EM)算法易于陷入局部最优的局限性,提高了参数估计的稳定性和模型收敛效率。实验结果表明,本文方法在多个数据集和多种噪声情况下均表现出优异的去噪效果,还能更好地保持光谱特性和图像结构的一致性,验证了其在高光谱图像去噪任务中的有效性。Hyperspectral Images (HSIs) have widespread applications in fields such as remote sensing and medical imaging, but they are often subject to various types of noise during the acquisition process. Despite the availability of numerous denoising methods for HSI processing, these methods still face challenges in handling complex spectral features and non-uniform noise distributions, which may lead to the loss of image details and compromise the authenticity and effectiveness of spectral data. To address these shortcomings, this paper proposes a hyperspectral image denoising method based on a multivariate Gaussian mixture model. The proposed method models noise using a multivariate Gaussian mixture model, employing multimodal features to represent the complex spectral structure of images, thereby better adapting to diverse spectral distributions. We adopt the Variational Bayes (VB) method for parameter estimation to overcome the limitations of the traditional Expectation-Maximization (EM) algorithm, which tends to get stuck in local optima, thus improving the stability of parameter estimation and convergence efficiency. Experimental results demonstrate that the proposed method achieves superior denoising performance across multiple datasets and various noise levels, while also better preserving spectral characteristics and image structure, validating its effectiveness for HSI denoising tasks.展开更多
文摘EM (Expectation Maximization)算法是统计学中的核心算法,也是本校近代数理统计课程教学过程中的一个重难点。论文采用案例式、启发式、研讨式教学方法,以基于高斯混合模型(GMM)的轴承退化阶段划分问题为例,引导学生发现隐变量模型极大似然估计(MLE)存在的困难,设计问题链启发学生探寻参数估计的数值方法,并总结出EM算法的一般过程。基于matlab编程可视化EM算法下的GMM模型参数更新过程,对比MLE目标函数和EM迭代目标函数,分析EM算法的内涵思想并结合图形进行直观展示,并且挖掘其中蕴含的思政元素,在知识传授的同时实现价值塑造。Expectation maximization (EM) algorithm is a core algorithm in statistics and also a key and difficult point in the teaching process of modern mathematical statistics courses in our school. The paper adopts a case-based and heuristic teaching method, taking the Gaussian Mixture Model (GMM) based bearing degradation stage division problem as an example, guiding students to discover the difficulties of maximum likelihood estimation (MLE) in the latent variable model, designing a problem chain to inspire students to explore numerical methods for parameter estimation, and summarizing the general process of EM algorithm. Based on Matlab programming, the parameter update process of GMM based on EM algorithm is visualized. Comparing the MLE objective function and EM iteration objective function, the intrinsic thought of EM algorithm is analyzed and visually displayed with graphics. The ideological and political elements are also explored, so as to achieve value shaping while knowledge transmission.
文摘高光谱图像(Hyperspectral Images, HSIs)在遥感和医学成像等领域具有广泛的应用,但在采集过程中容易受到各种噪声的干扰。尽管目前已有多种去噪方法应用于高光谱图像处理,但这些方法在应对复杂光谱特征和复杂或非均匀噪声分布时仍面临挑战,且部分方法可能导致图像细节的丢失,降低光谱数据的真实性和有效性。为克服这些不足,文章提出了一种基于多元高斯混合模型的高光谱图像去噪方法。该方法将噪声建模为多元高斯混合模型,通过多模态特征表示图像的复杂光谱结构,以更好地适应不同光谱分布。我们采用变分贝叶斯(Variational Bayes, VB)方法进行参数估计,从而改善了传统期望最大化(EM)算法易于陷入局部最优的局限性,提高了参数估计的稳定性和模型收敛效率。实验结果表明,本文方法在多个数据集和多种噪声情况下均表现出优异的去噪效果,还能更好地保持光谱特性和图像结构的一致性,验证了其在高光谱图像去噪任务中的有效性。Hyperspectral Images (HSIs) have widespread applications in fields such as remote sensing and medical imaging, but they are often subject to various types of noise during the acquisition process. Despite the availability of numerous denoising methods for HSI processing, these methods still face challenges in handling complex spectral features and non-uniform noise distributions, which may lead to the loss of image details and compromise the authenticity and effectiveness of spectral data. To address these shortcomings, this paper proposes a hyperspectral image denoising method based on a multivariate Gaussian mixture model. The proposed method models noise using a multivariate Gaussian mixture model, employing multimodal features to represent the complex spectral structure of images, thereby better adapting to diverse spectral distributions. We adopt the Variational Bayes (VB) method for parameter estimation to overcome the limitations of the traditional Expectation-Maximization (EM) algorithm, which tends to get stuck in local optima, thus improving the stability of parameter estimation and convergence efficiency. Experimental results demonstrate that the proposed method achieves superior denoising performance across multiple datasets and various noise levels, while also better preserving spectral characteristics and image structure, validating its effectiveness for HSI denoising tasks.
