 |
| Titel |
Non-negative Matrix Factorisation in Context |
| VerfasserIn |
Albrecht Schmidt, Erwan Treguier, Frédéric Schmidt, Saïd Moussaoui, Nicolas Dobigeon |
| Konferenz |
EGU General Assembly 2011
|
| Medientyp |
Artikel
|
| Sprache |
Englisch
|
| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 13 (2011) |
| Datensatznummer |
250053763
|
|
|
|
|
|
| Zusammenfassung |
| Various methods which implement non-negative matrix factorisation have been
used successfully in recent years for analysing hyperspectral images.
Assuming linear mixings, the methods estimate the source spectrograms which
are used to approximate the individual pixels of the hyperspectral image,
given the number of sources one expects to find in the data. This
presentation will try to put matrix factorisation techniques into context and
compare them to other unsupervised data analysis methods and concepts, such as
nearest neighbours, k-means, principal components, and convex hulls.
The general setting of the work is framed as follows: we consider P pixels of
a hyperspectral image which are acquired at L frequency bands and which are
represented as a PxL data matrix X. Each row of this matrix contains a
measured spectrum at a pixel with spatial index p=1..P, i.e., the original
topology is disregarded. Since we assume linear mixing, the p-th spectrum,
1<=p<=P, can be expressed as a linear combination of r, 1<=r<=R, pure spectra
of the surface components. Thus, X=AxS+E, E being an error matrix, which
should be minimised, and X, A, and S only have non-negative entries. The rows
of matrix S are the estimated surface spectra of the R components, and each
entry of A expresses the strength of the r-th component in the pixel with
spatial index p.
A distinguishing feature of matrix factorisation techniques is that they allow
a constructive interpretation of matrices A and S: the rows of S can be
interpreted as the physical source spectrograms that would appear in some
pixels if instrumental and geometrical limitations allowed this. The rows of A can
be seen as quantitative descriptions of how the various rows of S are mixed to
obtain the original image. Thus, the underlying processes are abstracted and
fully described by the above mathematical representation. Note that all operations
take place in the original space which remains untransformed throughout the
computation.
We use both theorical examples and case-studies from the planetary
spectrometers to illustrate the potential of the techniques and the physical
implications they imply. We will also try to illustrate how to optimise
computations by taking advantage of well-known concepts such as nearest
neighbours and convex hulls to accelerate the computations. |
|
|
|
|
|
|