中文摘要：

引入一种按邻点对的相似性权值计算次数来归类Laplacian的思想，并从理论上证明了包含多次相似性权值计算的Laplacian构造比只计算一次或两次相似性权值的Laplacian构造更能精细地刻画数据局部几何结构．据此提出了一种新的更能胜任自然图像景物提取任务的Laplacian构造方法．该方法通过任意一对相邻像素在不同局部邻域内建立一个线性学习模型来重构不同的相似性权值．结合用户提供的部分前、背景标记约束，导出求解景物提取的半监督二次优化目标函数．当考虑通过对前、背景抽样来估计未知像素的颜色值时，优化目标可以迭代求解．更有意义的是，该迭代方法可以成功地将原来构造的其他Laplacian推广应用于只提供稀疏指示条带的景物提取问题中．理论分析与实验结果均证实，所构造的Laplacian能够更充分地表达图像像素间的内在结构，能以更精细的方式约束传播前、背景的成分比例而不仅仅是标号，从而获得更优的景物提取效果．

英文摘要：

A scheme of categorizing Laplacians is introduced in this paper based on the computation times of similarity weights for each pair of adjacent data points. It is also theoretically proven that the Laplacian construction with multiple computations of similarity weights for each pair of adjacent points can better capture the local intrinsic structure of data than those methods with only one or two such computations. A novel Laplacian construction method is then proposed, which is more suitable for natural image matting task. In this method, all the different similarity weights for any pair of adjacent pixels are reconstructed by using a local linear model in the neighborhoods they fall into. By combining the user-provided constraints which specify some pixels as foreground or background, a quadratic objective function for matting based on semi-supervised learning is formed. When estimating the colors of unknown pixels by sampling foreground and background colors, this optimization problem is reformulated and solved in an iterative manner. What＇s more, this iterative scheme can also be successfully generalized and applied into other previously constructed Laplacians for image matting tasks with only sparse label scribbles. Both the theoretical analysis and experimental results validate that the proposed Laplacian construction approach can better capture the intrinsic structure between image pixels, and can propagate the finer ingredients of an image foreground and background rather than just their labels, and thus the mattes of higher quality are obtained.