@ARTICLE\{IMM2001-0287, author = "A. A. Nielsen", title = "Spectral Mixture Analysis: Linear and Semi-parametric Full and Iterated Partial Unmixing in Multi- and Hyperspectral Image Data", year = "2001", keywords = "{LS} regression, spectral angle mapping (SAM), orthogonal subspace projection (OSP), iterated constrained energy minimization (CEM), target constrained interference minimized filter (TCIMF), non-linear semi-parametric unmixing (SPU)", pages = "17-37", journal = "Journal of Mathematical Imaging and Vision", volume = "15", editor = "", number = "1-2", publisher = "", note = "The description of iterated {CEM} should say that iterations should give weight to the background, i.e., it should focus on low and not high values of the projection w'r.", url = "http://www.kluweronline.com/issn/0924-9907/", abstract = "As a supplement or an alternative to classification of hyperspectral image data linear and semi-parametric mixture models are considered in order to obtain estimates of abundance of each class or end-member in pixels with mixed membership. Full unmixing based on both ordinary least squares (OLS) and non-negative least squares (NNLS), and the partial unmixing methods orthogonal subspace projection (OSP), constrained energy minimization (CEM) and an eigenvalue formulation alternative are dealt with. The solution to the eigenvalue formulation alternative proves to be identical to the {CEM} solution. The matrix inversion involved in {CEM} can be avoided by working on (a subset of) orthogonally transformed data such as signal maximum autocorrelation factors, MAFs, or signal minimum noise fractions, MNFs. This will also cause the partial unmixing result to be independent of the noise isolated in the {MAF}/MNFs not included in the analysis. {CEM} and the eigenvalue formulation alternative enable us to perform partial unmixing when we know one desired end-member spectrum only and not the full set of end-member spectra. This is an advantage over full unmixing and {OSP}. The eigenvalue formulation of {CEM} inspires us to suggest an iterated {CEM} scheme. Also the target constrained interference minimized filter (TCIMF) is described. Spectral angle mapping (SAM) is briefly described. Finally, semi-parametric unmixing (SPU) based on a combined linear and additive model with a non-linear, smooth function to represent end-member spectra unaccounted for is introduced. An example with two generated bands shows that both full unmixing, the {CEM,} the iterated {CEM} and {TCIMF} methods perform well. A case study with a 30 bands subset of {AVIRIS} data shows the utility of full unmixing, {SAM,} {CEM} and iterated {CEM} to more realistic data. Iterated {CEM} seems to suppress noise better than {CEM}. A study with {AVIRIS} spectra generated from real spectra shows (1) that ordinary least squares in this case with one unknown spectrum performs better than non-negative least squares, and (2) that although not fully satisfactory the semi-parametric model gives better estimates of end-member abundances than the linear model.", isbn_issn = "DOI:10.1023/A:1011269530293" }