主元分析是一种广泛应用的多元统计技术.在处理高维数据时,其结果的统计一致性与物理可解释性难以保证.引入以变量选择为目标的稀疏性约束,可有效缓解上述困难.基于最近10年的研究进展,本文阐述了稀疏性的基本概念和罚函数的设计标准,介绍了经典的稀疏性约束lasso及其多个变种:融合lasso、成组lasso、自适应lasso、弹性网等等.Lasso及其变种均可用作主元分析的约束,构建稀疏主元分析框架,但关键在于如何将稀疏主元转化为凸优化问题并快速求解.本文比较了稀疏主元的多种转化形式:奇异值分解、稀疏回归、低阶秩逼近、罚矩阵分解和半正定松弛.分析了基于最小角回归算法的一般lasso及广义lasso问题的求解方法.此外还初步探讨了函数型数据的稀疏主元分析问题.

提出了一种有效地结构化字典生成算法以及图像双重稀疏表示方法。在Rubinstein等提出的图像双重稀疏表示模型的基础上，引入小波零树结构，将同一空间位置对应的同方向跨尺度小波基函数的线性组合作为新的基函数，并通过K-SVD学习算法得到线性组合系数，由此得到了一种更加切合图像方向特征的结构化字典学习算法。在此基础上提出了相应的图像分解与重构算法。遥感图像M项逼近实验以及压缩仿真实验表明，本文提出的结构化字典比已有的字典具有更好的图像稀疏表示效果。

Multi-modal imaging requires image fusion to combine advantages of different types of sensors and requires super-resolution (SR) because of limited spatial resolution of source images. In this study, a novel framework is proposed for unification of image SR and the fusion process to obtain a high-resolution (HR)-fused image from a set of low-resolution (LR) multi-modal images. The jointly trained dictionaries of LR patches and corresponding HR patches are used for sparse representation of LR source image patches and HR-fused image patches, respectively. The sparse coefficients vectors for corresponding patches of source LR images are determined by using orthogonal matching pursuit and a local information content-based metric is employed to fuse these sparse coefficients. The corresponding HR-fused image patch is obtained by combining elements of the HR dictionary as per the fused coefficients of the LR image patches. The experimental results on sets of multi-modal images exhibited that the proposed method outperformed the existing fusion and SR techniques in terms of visual quality and image fusion quality metrics.

如何设计更加高效并能保持图像几何和纹理结构的多幅图像超分辨模型和算法是目前该领域有待解决的难点问题.针对图像的几何、纹理结构形态,分别建立符合类内强稀疏而类间强不相干的几何结构和纹理分量稀疏表示子成份字典,形成图像的多形态稀疏表示模型,进而提出一种新的基于多形态稀疏性正则化的多帧图像超分辨凸变分模型,模型中的正则项刻画了理想图像在多成份字典下的稀疏性先验约束,保真项度量其在退化模型下与观测信号的一致性,采用交替迭代法对该多变量优化问题进行数值求解,每一子问题采用前向后向的算法分裂法进行快速求解.针对可见光与红外图像序列进行了数值仿真,实验结果验证了本文模型与数值算法的有效性.

In this paper, we present a novel spatial and spectral fusion model (SASFM) that uses sparse matrix factorization to fuse remote sensing imagery with different spatial and spectral properties. By combining the spectral information from sensors with low spatial resolution (LSaR) but high spectral resolution (HSeR) (hereafter called HSeR sensors), with the spatial information from sensors with high spatial resolution (HSaR) but low spectral resolution (LSeR) (hereafter called HSaR sensors), the SASFM can generate synthetic remote sensing data with both HSaR and HSeR. Given two reasonable assumptions, the proposed model can integrate the LSaR and HSaR data via two stages. In the first stage, the model learns from the LSaR data a spectral dictionary containing pure signatures, and in the second stage, the desired HSaR and HSeR data are predicted using the learned spectral dictionary and the known HSaR data. The SASFM is tested with both simulated data and actual Landsat 7 Enhanced Thematic Mapper Plus (ETM+) and Terra Moderate Resolution Imaging Spectroradiometer (MODIS) acquisitions, and it is also compared to other representative algorithms. The experimental results demonstrate that the SASFM outperforms other algorithms in generating fused imagery with both the well-preserved spectral properties of MODIS and the spatial properties of ETM+. Generated imagery with simultaneous HSaR and HSeR opens new avenues for applications of MODIS and ETM+.

To take advantage of the wide swath width of Landsat Thematic Mapper (TM)/Enhanced Thematic Mapper Plus (ETM+) images and the high spatial resolution of Syste me Pour l'Observation de la Terre 5 (SPOT5) images, we present a learning-based super-resolution method to fuse these two data types. The fused images are expected to be characterized by the swath width of TM/ETM+ images and the spatial resolution of SPOT5 images. To this end, we first model the imaging process from a SPOT image to a TM/ETM+ image at their corresponding bands, by building an image degradation model via blurring and downsampling operations. With this degradation model, we can generate a simulated Landsat image from each SPOT5 image, thereby avoiding the requirement for geometric coregistration for the two input images. Then, band by band, image fusion can be implemented in two stages: 1) learning a dictionary pair representing the high- and low-resolution details from the given SPOT5 and the simulated TM/ETM+ images; 2) super-resolving the input Landsat images based on the dictionary pair and a sparse coding algorithm. It is noteworthy that the proposed method can also deal with the conventional spatial and spectral fusion of TM/ETM+ and SPOT5 images by using the learned dictionary pairs. To examine the performance of the proposed method of fusing the swath width of TM/ETM+ and the spatial resolution of SPOT5, we illustrate the fusion results on the actual TM images and compare with several classic pansharpening methods by assuming that the corresponding SPOT5 panchromatic image exists. Furthermore, we implement the classification experiments on both actual images and fusion results to demonstrate the benefits of the proposed method for further classification applications.

Remote sensing image fusion can integrate the spatial detail of panchromatic (PAN) image and the spectral information of a low-resolution multispectral (MS) image to produce a fused MS image with high spatial resolution. In this paper, a remote sensing image fusion method is proposed with sparse representations over learned dictionaries. The dictionaries for PAN image and low-resolution MS image are learned from the source images adaptively. Furthermore, a novel strategy is designed to construct the dictionary for unknown high-resolution MS images without training set, which can make our proposed method more practical. The sparse coefficients of the PAN image and low-resolution MS image are sought by the orthogonal matching pursuit algorithm. Then, the fused high-resolution MS image is calculated by combining the obtained sparse coefficients and the dictionary for the high-resolution MS image. By comparing with six well-known methods in terms of several universal quality evaluation indexes with or without references, the simulated and real experimental results on QuickBird and IKONOS images demonstrate the superiority of our method.

嫦娥探月工程二期嫦娥 3号的着陆任务需要高分辨率影像数据的支持。因此，提出了一种基于压缩感知的超分辨率图像重建方法，实现了虹湾地区月球卫星影像的超分辨率重建。通过从美国阿波罗计划获取的月球影像、嫦娥 1、2号卫星影像和嫦娥工程二期试验中获取的图像中提取了大量训练图块，完成了高、低分辨率过完备字典对 Ah和 Al的联合训练，采用正则正交基追踪算法求解优化问题，获得关于低分辨率图块的稀疏表示，并将表示系数用于 Ah以生成对应的高分辨率图块，得到满足重构约束的高分辨率图像。实验验证了算法的有效性，表明其在视觉效果及 PSNR和 RMSE指标上均优于传统方法。

针对图像高分辨率重建过程中稀疏解的存在性和唯一性问题以及超分辨率图像的边缘特征和平滑噪声的关系进行了研究,提出了局部正则化参数自适应选取的方法。结合联合构造字典的算法,在重建过程中动态调整正则化参数。通过对图像的超分辨率实验证明,改进的算法具有较高的可行性,能有效平衡超分辨率图像的边缘特征和平滑噪声两者的关系,与传统的超分辨率重建算法相比,有更高的峰值信噪比。

<p align="justify">在基于稀疏表示的幻觉脸重建过程中，由于冗余的过完备字典会降低稀疏编码的效率和精度，提出用紧的聚类子字典来表示人脸图像的不同结构对象。由高分辨率（high resolution， HR）/低分辨率（low resolution， LR）的人脸图像样本集进行K-均值聚类，为使紧的聚类子字典能够表达图像块的整体特征，对各聚类子集采用主成分分析（principal component analysis， PCA）方法构造字典。得到同构的HR/LR的聚类字典后，对于输入的LR人脸图像块，经自适应选择合适的子字典后，对稀疏编码添加正则化项，采用集中式稀疏编码，以使稀疏表示系数更逼近要重建的HR人脸图像块。由此稀疏表示系数与HR字典的线性组合得到HR人脸图像块，将此图像块与近似结果进行合成，从而得到最终的人脸图像。经仿真实验，并与其他的方法进行比较，实验结果验证了所提方法的有效性。</p>

<p align="justify">针对多数传统分类算法应用于高光谱分类都存在运算速度慢、精度比较低和难以收敛等问题，从稀疏表示基本理论出发建立了一个基于自适应稀疏表示的高光谱分类模型。利用训练样本构建字典，聚类每一步迭代所产生的余项，将聚类中心作为新的字典原子，然后将测试样本看成冗余字典中训练样本的线性组合，令字典能够更适应于样本的稀疏表示。利用华盛顿地区的HYDICE高光谱遥感数据进行试验，并且与主成分分析、线性鉴别分析、支持向量机、神经网络算法进行比较，结果表明，该算法的总体分类精度比其他算法提高了约12%，有效提高了高光谱影像的分类精度。</p>

针对多光谱图像与全色图像的融合,提出一种结合小波变换和稀疏表示的融合算法。该算法充分利用小波变换具有保持光谱信息这一优势,首先对多光谱图像进行IHS(intensity-hue-satuation)变换,然后对亮度分量和全色图像进行单层小波变换,得到对应的高低频系数。分析高低频系数的特征,对于不能认为是"稀疏"的低频系数采用稀疏表示进行融合;对于可以认为是"稀疏"的高频系数采用图像信息融合规则进行融合。最后进行小波逆变换和IHS逆变换得到融合结果。实验结果表明,该算法最大限度地保留了光谱信息,并提高了空间分辨率。

为提高融合图像保留源图像的信息量和边缘特征，提出了非负矩阵分解和新轮廓波变换的图像融合算法。以具有尖锐频率局部化特征的新轮廓波对循环平移后的源图像进行分解；运用非负矩阵分解实现低通子带融合，采用能量方差测度函数和匹配度函数实现带通子带融合；对各子带信号重构并逆循环平移，得到融合图像。实验结果分析表明，该方法保留了更多的信息量和边缘细节特征，应用效果较好。