Nonnegative matrix factorization in polynomial feature space

Plenty of methods have been proposed in order to discover latent variables (features) in data sets. Such approaches include the principal component analysis (PCA), independent component analysis (ICA), factor analysis (FA), etc., to mention only a few. A recently investigated approach to decompose a data set with a given dimensionality into a lower dimensional space is the so-called nonnegative matrix factorization (NMF). Its only requirement is that both decomposition factors are nonnegative. To approximate the original data, the minimization of the NMF objective function is performed in the Euclidean space, where the difference between the original data and the factors can be minimized by employing L(2)-norm. In this paper, we propose a generalization of the NMF algorithm by translating the objective function into a Hilbert space (also called feature space) under nonnegativity constraints. With the help of kernel functions, we developed an approach that allows high-order dependencies between the basis images while keeping the nonnegativity constraints on both basis images and coefficients. Two practical applications, namely, facial expression and face recognition, show the potential of the proposed approach.     

概率二维主分量分析

二维主分量分析是一种直接面向图像矩阵表达方式的特征抽取与降维方法. 提出了一个基于二维主分量分析的概率模型. 首先, 通过对此产生式概率模型参数的最大似然估计得到主分量(矢量); 然后, 考虑到缺失数据问题, 利用期望最大化算法迭代估计模型参数和主分量. 混合概率二维主分量分析模型在人脸聚类问题上的应用表明概率二维主分量分析模型能作为图像矩阵的密度估计工具. 含有缺失值的人脸图像重构实验阐述了此模型及迭代算法的有效性.     

Color face recognition based on quaternion matrix representation

There are several methods to recognize and reconstruct a human face image. The principal component analysis (PCA) is a successful approach because of its effective extraction of the global feature and excellent reconstruction of face image. However, the crucial shortcomings of PCA are its low recognition rate and overfitting of feature extraction which leads to the dependence of training data on training samples. In this paper, a modified two-dimension principal component analysis (2DPCA) and bidirectional principal component analysis (BDPCA) methods based on quaternion matrix are proposed to recognize and reconstruct a color face image. In these methods, the spatial distribution information of color images is used to represent a color face, and the 2DPCA or BDPCA feature of color face image is extracted by reducing the dimensionality in both column and row directions. A method obtaining orthogonal eigenvector set of quaternion matrix is proposed. Numerous experiments show that the present approach based on quaternion matrix can effectively smooth the overfitting issue and substantially enhance the recognition rate.     

基于矩阵分解的海洋SAR图像舰船检测

针对海洋原始图像与低秩和稀疏矩阵分解模型数据结构不一致的问题,本文提出一种新的基于矩阵分解的海洋SAR图像舰船检测方法。首先该方法需对结构化相似的海洋SAR图像进行重组;然后根据重组矩阵特性适应性设计一个分解精度更高、分解速度更快的新矩阵分解模型,并利用增广拉格朗日乘子法求解模型,在不依赖任何杂波模型和检测统计量的前提下,实现代表舰船目标的稀疏成分的提取;最后利用形态学处理进行优化,实现海洋SAR图像舰船目标的检测。基于高分三号SAR卫星数据的实验结果表明,相比已有的基于鲁棒主成分分析的舰船检测方法,本文方法在处理复杂海况时,能更快速度地以较好的形状从海杂波中准确提取舰船目标,具有更好的鲁棒性。     

求解低秩矩阵融合高动态范围图像

目的 利用低秩矩阵恢复方法可从稀疏噪声污染的数据矩阵中提取出对齐且线性相关低秩图像的优点,提出一种新的基于低秩矩阵恢复理论的多曝光高动态范围（HDR）图像融合的方法,以提高HDR图像融合技术的抗噪声与去伪影的性能。方法 以部分奇异值（PSSV）作为优化目标函数,可构建通用的多曝光低动态范围（LDR）图像序列的HDR图像融合低秩数学模型。然后利用精确增广拉格朗日乘子法,求解输入的多曝光LDR图像序列的低秩矩阵,并借助交替方向乘子法对求解算法进行优化,对不同的奇异值设置自适应的惩罚因子,使得最优解尽量集中在最大奇异值的空间,从而得到对齐无噪声的场景完整光照信息,即HDR图像。结果 本文求解方法具有较好的收敛性,抗噪性能优于鲁棒主成分分析（RPCA）与PSSV方法,且能适用于多曝光LDR图像数据集较少的场合。通过对经典的Memorial Church与Arch多曝光LDR图像序列的HDR图像融合仿真结果表明,本文方法对噪声与伪影的抑制效果较为明显,图像细节丰富,基于感知一致性（PU）映射的峰值信噪比（PSNR）与结构相似度（SSIM）指标均优于对比方法：对于无噪声的Memorial Church图像序列,RPCA方法的PSNR、SSIM值分别为28.117 dB与0.935,而PSSV方法的分别为30.557 dB与0.959,本文方法的分别为32.550 dB与0.968。当为该图像序列添加均匀噪声后,RPCA方法的PSNR、SSIM值为28.115 dB与0.935,而PSSV方法的分别为30.579 dB与0.959,本文方法的为32.562 dB与0.967。结论 本文方法将多曝光HDR图像融合问题与低秩最优化理论结合,不仅可以在较少的数据量情况下以较低重构误差获取到HDR图像,还能有效去除动态场景伪影与噪声的干扰,提高融合图像的质量,具有更好的鲁棒性,适用于需要记录场景真实光线变化的场合。     

Structural two-dimensional principal component analysis for image recognition

In this paper, a new technique called structural two-dimensional principal component analysis (S2DPCA) is proposed for image recognition. S2DPCA is a subspace learning method that identifies the structural information for discrimination. Different from conventional two-dimensional principal component analysis (2DPCA) that only reflects within-row information of images, the goal of S2DPCA is to discover structural discriminative information contained in both within-row and between-row of the images. By contrast with 2DPCA, S2DPCA is directly based on the augmented images encoding corresponding row membership, and the projection directions of S2DPCA are obtained by solving an eigenvalue problem of the augmented image covariance matrix. Computationally, S2DPCA is straightforward and comparative with 2DPCA. Like 2DPCA, the singularity problem is completely avoided in S2DPCA. Experiments on face recognition and handwritten digit recognition are presented to show the effectiveness of the proposed approach.     

Max-margin non-negative matrix factorization with flexible spatial constraints based on factor analysis

Non-negative matrix factorization (NMF) is a popular feature encoding method for image understanding due to its non-negative properties in representation, but the learnt basis images are not always local due to the lack of explicit constraints in its objective. Various algebraic or geometric local constraints are hence proposed to shape the behaviour of the original NMF. Such constraints are usually rigid in the sense that they have to be specified beforehand instead of learning from the data. In this paper, we propose a flexible spatial constraint method for NMF learning based on factor analysis. Particularly, to learn the local spatial structure of the images, we apply a series of transformations such as orthogonal rotation and thresholding to the factor loading matrix obtained through factor analysis. Then we map the transformed loading matrix into a Laplacian matrix and incorporate this into a max-margin non-negative matrix factorization framework as a penalty term, aiming to learn a representation space which is non-negative, discriminative and localstructure- preserving. We verify the feasibility and effectiveness of the proposed method on several real world datasets with encouraging results.     

基于描述特征的人脸识别研究

针对基于主成分分析识别人脸存在计算复杂、不能准确地估计训练图像的协方差矩阵等问题,提出了一种基于描述特征的人脸识别算法(Expressive feature face recognitionalgorithm, EFFRA).该算法用训练图像的右奇异向量代替PCA求解的子空间的基向量,避免了将人脸图像转换成图像向量,明显降低了计算复杂性.进一步研究发现,EFFRA提取的每一个主成分向量中含有冗余,在此基础上,利用PCA实现了EFFRA的简化算法(MEFFRA),在ORL和Essex数据库上的实验结果表明,EFFRA及MEFFRA明显优于特征脸算法,MEFFRA的识别精度略好于EFFRA,但明显减少了对存储空间的需求.     

Subjective principal component analysis

In this study we deal with the problem of finding subjective principal components for a given set of variables in a data matrix. The principal components are not determined by maximizing their variances; they are specified by a user, who can maximize the absolute values of the correlations between principal components and the variables important to him. The correlation matrix of the variables is the basic information needed in the analysis.The problem is formulated as a multiple criteria problem and solved by using an interactive procedure. The procedure is convenient to use and easy to implement. We have implemented an experimental version on an APPLE III microcomputer. A graphical display is used as an aid in finding the principal components. An illustrative application is presented, too.     

结构正则化多视图非负矩阵分解

现存非负矩阵分解（non-negative matrix factorization,NMF）研究多考虑单一视图分解数据,忽略了数据信息的全面性。此外,NMF限制其获取数据的内在几何结构。针对以上问题,提出一个结构正则化多视图非负矩阵分解算法（structure regularized multi-view nonnegative matrix factorization,SRMNMF）。首先,通过主成分分析来对数据进行全局结构的判别式学习;其次,利用流形学习来捕获数据的局部结构;然后,通过利用多视图数据的多样性和差异性来学习表征。模型提升了算法聚类的整体性能,更加有效地挖掘数据的结构信息。此外,采用高效的交替迭代算法优化目标函数得到最优的因子矩阵。在六个数据集上与现存的代表性方法比较,所提出的SRMNMF的准确率、NMI和Purity分别最大提高4.4%、6.1%和4.05%。     

一种基于块共同特征值的人脸识别方法

主成分分析与线性判别分析是人脸识别的重要识别方法,它们都通过求解特征值问题实现特征提取,但由于维数灾难会导致小样本和奇异性问题。提出了一种简单的人脸识别方法,无需进行奇异值分解,能有效地降低计算代价。首先将图像划分成块,然后计算多项式系数,得到友阵用于特征提取。基于两张不同图像的多项式系数友阵来计算对称阵。最后通过计算对称阵的零空间的零化度识别相似的人脸图像。为验证提出方法的有效性,在ORL、Yale和FERET人脸数据库上进行了实验。结果表明,该方法对于有较大姿态与光照变化的人脸识别具有较高的识别性能。     

基于变量相关性的多元时间序列特征表示

针对高维特性对多元时间序列数据挖掘过程和结果的影响,以及传统主成分分析方法在多元时间序列数据特征表示上的局限性,提出一种基于变量相关性的多元时间序列数据特征表示方法。通过协方差矩阵描述每个多元时间序列的分布特征和变量相关关系,利用主成分分析方法对综合协方差矩阵进行主元分析,进而实现多元时间序列的数据降维和特征表示。实验结果表明,所提出的方法不仅能提高多元时间序列数据挖掘的质量,还可以对不等长多元时间序列进行快速有效的挖掘。     

基于分块PCA与端元提取的壁画线条增强研究

线状特征是壁画中的重要元素。然而受到自然及人为因素的影响,壁画的部分线条常常变得模糊,人眼难以辨别。因此,提出一种利用高光谱影像分块主成分分析(PCA)与端元提取相结合的线状特征增强方法。首先,利用支持向量机(SVM)对壁画的合成真彩色影像进行分类,根据分类结果得到壁画标签数据,实现高光谱影像同质区域的分块数据。其次,对各分块影像进行顶点成分分析(VCA)得到候选端元集,通过构造投影矩阵合并相似端元确定最终端元集。然后,利用非负最小二乘算法解混得到线条丰度图。最后,将分块PCA的第一主成分影像归一化后与线条丰度图进行波段加权平均获取线状特征增强影像,将其与合成真彩色影像进行HSV图像融合得到线状特征融合影像。以瞿昙寺壁画局部高光谱影像为例进行了验证,结果表明,该算法能增强壁画中的线状特征,且较PCA增强法效果更好。     

基于特征量积与PCA的小波遥感图像融合

在遥感图像融合中,传统PCA算法会损失部分有用信息,从而使得融合结果的光谱分辨率受到较大影响,针对这种情况,借助小波变换优良的时频分析特性,利用特征量积来融合多光谱图像的第一主成分,实现了一种基于特征量积与PCA的小波遥感图像融合算法。通过对来自不同场景不同卫星的多光谱和全色图像进行融合实验,结果表明,该算法无论在主观视觉还是在客观统计数据上,均具有比其他方法较佳的融合效果。     

分块多线性主成分分析及其在人脸识别中的应用研究

主成分分析(Principal Component Analysis,PCA)是人脸识别中一个经典的算法,但PCA方法在特征提取时考虑的是图像的整体信息,并没有考虑图像的局部信息,而分块PCA(Modular Principal Component Analysis,Modular PCA)则可以有效地提取图像中重要的局部信息,所以在人脸识别实验中获得了比传统PCA更好的识别效果。但PCA和Modular PCA都要进行图像的矢量化,这会破坏原始数据的空间结构,也有可能会导致"维数灾难"。多线性主成分分析(Multilinear Principal Component Analysis,Multilinear PCA)作为PCA在高维数据上的扩展,直接使用矩阵或者高阶的张量来获得有效特征,既可以避免"维数灾难",又可以体现直接将张量数据作为处理对象时保留原始数据较好基本结构信息的优点。在研究Modular PCA和Multilinear PCA的基础上,提出了分块多线性主成分分析(Modular Multilinear Principal Component Analysis,M2PCA)算法,用于识别人脸。在Yale、XM2VTS和JAFFE人脸数据库上进行了人脸识别实验,结果表明,在同等的分块条件下,所提出的方法的识别效果要优于Modular PCA的方法。     

Process identification based on last principal component analysis

A simple linear identification algorithm is presented in this paper. The last principal component (LPC), the eigenvector corresponding to the smallest eigenvalue of a non-negative symmetric matrix, contains an optimal linear relation of the column vectors of the data matrix. This traditional, well-known principal component analysis is extended to the generalized last principal component analysis (GLPC). For processes with colored measurement noise or disturbances, consistency of the GLPC estimator is achieved without involving iteration or non-linear numerical optimization. The proposed algorithm is illustrated by a simulated example and application to a pilot-scale process.     

Joint weighted nuclear norm and total variation regularization for hyperspectral image denoising

A hyperspectral image is typically corrupted by multiple types of noise including Gaussian noise and impulse noise. On the other hand, a hyperspectral image possesses a high correlation in its spectral dimensions, and its Casorati matrix has a very low rank. Inspired by the recent development of robust principal component analysis, which can be used to remove sparse and arbitrarily large noise from a low-rank matrix, we propose a joint weighted nuclear norm and total variation regularization method to denoise a hyperspectral image data. First, weighted nuclear norm regularization is constructed for sparse noise removal. Total variation regularization is then imposed on each band of the hyperspectral image to further remove the Gaussian noise. A concrete optimization algorithm is developed to implement the two-stage regularization. The combined approach is expected to effectively denoise hyperspectral images even with varying data structures and under varying imaging conditions. Extensive experiments on both simulated and real data sets validate the performance of our proposed method.     

Stability and dimensionality of Karhunen-Loêve multispectral image expansions

The Karhunen-Loêve (K.L.) expansion is a useful tool for the representation, pre-processing and orthogonal coding of multispectral imagery: Each spatial pixel is analysed independently as the K.L. transform is taken in the spectral dimension, i.e. along the various N spectral channels. The eigenvectors are those of the covariance matrix. The (principal) eigenimages are thus “false color” images, which can be viewed without decoding as the spatial topology is unchanged, and the higher order principal images present a strong contrast enhancement.⁽¹⁾ These principal images are also uncorrelated, a very desirable feature for many applications including clustering.⁽²⁾

The source dependency of the eigenvectors, however, introduces “instability” in the form of pronounced statistical noise on some principal images. This paper gives the results of a numerical study carried out on a 7 channel Daedalus Multispectral Scene. The uncertainties of the eigenvalues and eigenvectors are evaluated from two “drawings” of the pixels of the raw data.

Both the numerical results of the study and the direct viewing of the principal images show that three out of the seven have so much noise that they do not yield any useful information. Only the first two principal images have excellent stability, and they contain most of the total contrast variance of the scene. Two other principal images of lower order are also stable, but, contribute very little to the total contrast variance. These images carry texture information rather than homogeneous zone clustering information.     

The generation and interpretation of false-colour composite principal component images

The presentation of multichannel image information in terms of a false-colour composite can, in general, be achieved with less sacrifice of data if the three leading principal component images are used rather than the three conventional raw channels

A hardwired electronic system which has been developed is described in outline; it implements principal components analysis on 256 x 256 four-channel array of pixels in a few seconds. The system is under the control of an APPLE II microcomputer, and can generate colour hard-copy output in the form of composites of the principal component images

The use of this system is illustrated by reference to a LANDSAT MSS subframe of southern Spain. A standard false-colour composite of this scene contains 73 per cent of the available image variance whereas the principal component composite contains 98 per cent. The necessity of contrast stretching the minor principal component images in order to generate a visually effective colour composite is demonstrated. Some consideration is given to the choice of the most appropriate colour coding in the principal component composite images.     

基于改进2DPCA的快速交通标志识别法

为了提高交通标志的识别速度,提出了一种样本类中间值结合均值的的2DPCA.该方法采用每类训练样本中间值的均值代替训练样本的平均值,以此重建散布矩阵从而得到最优投影矩阵.在已经建立的两个交通标志图像集上用该方法进行了实验,结果表明新方法识别速度得到了大幅提升并且取得了较高的识别率.