Fuzzy spectral signatures.
Traditional spectral signatures consist of sequential plots of mean
reflectance, for a given land cover class, in a series of spectral bands.
They are derived from ground areas which are known to be pure examples of those land covers. In the graphs below, although data are only available for the 6 spectral bands under consideration, the line drawn between these points helps to visualise the characteristic signatures.
Since these signatures are based on mean values, the standard deviations of the data can also be plotted as bounds around the signature. These outer bounds and centre line correspond to the centre and outer boundary of the hard landcover cluster used to classify unknown pixels in a thematic classification.
This simple signature (for the class 'grass') actually represents data in 6 dimensions, and plotting the signature of a novel pixel over it corresponds to plotting that pixel in the 6-dimensional feature space used for multispectral classification. According to traditional classifications, a pixel (assumed to be entirely composed of one landcover type) is either 'grass' or 'not-grass'.
In a hard classification, only pixels which fall within the bounds of this landcover cluster (see left) will be classified as belonging to this class. The pixel whose signature is plotted on the right is too different, and would be classified as 'not-grass'.
The fuzzy classification approach, on the other hand, attempts to model
these landcover clusters as having softer boundaries, so that membership
in a cluster is not Boolean (yes or no), but can be a matter of degree.
Thus a pixel near the edge of a cluster can still have a low membership
for that landcover type. The spectral signatures above have been modified
for the FLIERS project, to visualise these fuzzy, 6-dimensional clusters
in a simple 2D graph.
A fuzzy spectral signature looks something like this;
The principles by which this graph is produced are described here.
Essentially, an array of statistics is calculated, which summarises the
distribution of the membership values in each landcover class, and how
they relate to the spectral values in the same pixels. These statistics
are summarised to produce a plot of mean memberships, for this class, in
every spectral band, and then 'slices' are taken across this plot, to produce
the colour bands shown above. In the above plot, the values shown in the
key represent the mean memberships in class 'unknown' which are found at
the spectral scores shown on the y-axis.
There is a central line, which represents the reflectance, in each band, at which the maximum memberships tend to be found. The bands around this line represent the spread of memberships, and it can be seen that, in some spectral bands, the pixels are quite tightly centred around a narrow range of reflectances, while others are more spread out.
In order to better visualise this spread, for each spectral band, the
mean membership plot which was 'sliced' to produce these divisions can
be viewed by clicking over the band required.
For each band clicked, a graph will be produced showing the plot of mean fuzzy memberships against reflectance in that band, and the percentile values which have been used to produce the coloured bands of the graph. The x axis of the new graph is equivalent to the y axis of the original fuzzy spectral signature. These individual graphs are equivalent to fuzzy membership plots; they are a rough approximation to the fuzzy membership function for this landcover, for this spectral band.
A set of fuzzy spectral signatures can be produced using the 'Fuzzy spectral signature' option. The example file 'Urban.fli' contains all the statistical summary data necessary.
If fuzzy plots have not previously been produced from a raster set,
then calculation of the statistical summaries can take some time, as, for
every pixel, each fuzzy layer is compared to each spectral layer.
On selection of the 'Fuzzy spectral signature' option, if statistical summaries do not already exist, then they are calculated as follows;
For each layer defined as containing 'fuzzy memberships', where there are non-zero values, a fuzzy spectral signature like the one above will be produced.
To view all the membership plots which were 'sliced' to generate these spectral signatures, use the 'Fuzzy membership plot' button, and select all layers, to retrieve a line graph, for each spectral band, like the one below;
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