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The condition number of the mixing matrices is 6. In both panels the values on the diagonal are set to zero.

Independent Component Analysis 3

Figure 11 Dendrogram for Fig. Figure 12 Upper panel: scatter plot of the two Gaussian sources with different spectra. The nearly horizontal curve shows the behavior without, the sinusoidal one the result with using delay embedding. Here the actual mixing angle is 0. Figure 14 Test problem B of Sec. The gray bars on the right panel show the full MI given in Eq.

Figure 18 Dendrogram for Fig. Figure 20 Short segment from the original ECG a , of the mother and fetus contributions estimated without delay embedding b,c , and of the two contributions estimated with delay embedding d,e.

Blind source separation dependent component analysis - DRO

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A Phys. B Phys. C Phys. D Phys. E Phys. Fluids Phys. Materials Phys. Applied Phys. Beams Phys. Items in Repository are protected by copyright, with all rights reserved, unless otherwise indicated. Show full item record Give your opinion. Unmixing hyperspectral data: independent and dependent component analysis. Hyperspectral Data Exploitation: Theory and Applications. ISBN Chapter 6. The development of high spatial resolution airborne and spaceborne sensors has improved the capability of ground-based data collection in the fields of agriculture, geography, geology, mineral identification, detection [2, 3], and classification [4—8].


The signal read by the sensor from a given spatial element of resolution and at a given spectral band is a mixing of components originated by the constituent substances, termed endmembers, located at that element of resolution. This chapter addresses hyperspectral unmixing, which is the decomposition of the pixel spectra into a collection of constituent spectra, or spectral signatures, and their corresponding fractional abundances indicating the proportion of each endmember present in the pixel [9, 10].

Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. The linear mixing model holds when the mixing scale is macroscopic [13]. The nonlinear model holds when the mixing scale is microscopic i. The linear model assumes negligible interaction among distinct endmembers [16, 17].

The nonlinear model assumes that incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [18].


Under the linear mixing model and assuming that the number of endmembers and their spectral signatures are known, hyperspectral unmixing is a linear problem, which can be addressed, for example, under the maximum likelihood setup [19], the constrained least-squares approach [20], the spectral signature matching [21], the spectral angle mapper [22], and the subspace projection methods [20, 23, 24]. Orthogonal subspace projection [23] reduces the data dimensionality, suppresses undesired spectral signatures, and detects the presence of a spectral signature of interest.

The basic concept is to project each pixel onto a subspace that is orthogonal to the undesired signatures. As shown in Settle [19], the orthogonal subspace projection technique is equivalent to the maximum likelihood estimator. This projection technique was extended by three unconstrained least-squares approaches [24] signature space orthogonal projection, oblique subspace projection, target signature space orthogonal projection.

Other works using maximum a posteriori probability MAP framework [25] and projection pursuit [26, 27] have also been applied to hyperspectral data.

Blind Source Separation and Independent Component Analysis

In most cases the number of endmembers and their signatures are not known. Independent component analysis ICA is an unsupervised source separation process that has been applied with success to blind source separation, to feature extraction, and to unsupervised recognition [28, 29]. ICA consists in finding a linear decomposition of observed data yielding statistically independent components. Given that hyperspectral data are, in given circumstances, linear mixtures, ICA comes to mind as a possible tool to unmix this class of data.

In fact, the application of ICA to hyperspectral data has been proposed in reference 30, where endmember signatures are treated as sources and the mixing matrix is composed by the abundance fractions, and in references 9, 25, and 31—38, where sources are the abundance fractions of each endmember.

In the first approach, we face two problems: 1 The number of samples are limited to the number of channels and 2 the process of pixel selection, playing the role of mixed sources, is not straightforward. In the second approach, ICA is based on the assumption of mutually independent sources, which is not the case of hyperspectral data, since the sum of the abundance fractions is constant, implying dependence among abundances. This dependence compromises ICA applicability to hyperspectral images. In addition, hyperspectral data are immersed in noise, which degrades the ICA performance.

IFA [39] was introduced as a method for recovering independent hidden sources from their observed noisy mixtures. IFA implements two steps. First, source densities and noise covariance are estimated from the observed data by maximum likelihood.