![]() Mixture Model
In the literature, one may find two different working lines related to the mixture problem:
The model mixture tool in the SPRING takes this second working line. The next sections present the theoretical basis required to use the Mixture Model adequately, application examples to the forest area and instructions about the tool manipulation. For more details, see Aguiar [1991] and Shimabukuru [1987].
Mixture Linear ModelIn a Mixture Linear Model, the pixel value in any spectral band is considered as a linear combination of the answer for each component inside the pixel. The model can be expressed as: r 1 = a11 x 1 + a12 x 2
+ ... + a 1nx n + e1 that is, n n <= m e where: ri : spectral reflectance in the ith spectral band of a
pixel (i.e., pixel value in the band i, converted for a reflectance value). The xj estimates are related with the following constraints: n 0 <= xj <= 1 (3) This constraints are imposed because the xj represent area proportions inside a scene element. However, the (3) constraint is optional, as it will be described in the next sections.
Analyzing Spectral signatures and obtaining mixture componentsThe spectral signature selection of the considered elements as mixture components is critical for the correct estimation of the proportions. In the current version of the SPRING tool, the spectral signature values can be digitized or obtained with the cursor, over pixels considered "pure". In future versions it is planned to implement interfaces with spectral curves library. In the model description above, the spectral signature and pixels values are described with reflectance values. This is indicated when the spectral curves are obtained in an external library or through a work previously performed. In the current software version, if this is the case, the original image has to be converted to reflectance values, assigned to the [0,...,255] interval. Unfortunately, the SPRING does not have this functionality, which has to be performed externally. To compute the proportions estimate using the original image without a previous conversion, may cause estimate errors. If the values were obtained from the image, through pure pixels, the conversion will not be necessary. The average error computation in the proportions estimate process and the error images generation, are indicators of the selected components fitness and its signatures. Additionally, there is the possibility of not applying restriction (3). In this case, the negative proportion values or superiors to one, obtained in certain pixels, although without a physical meaning, can be considered as spatial indicators of the mixture model inadequacy adopted for a certain scene.
Methods for proportions estimateThe methods implemented in the SPRING to estimate proportions inside a pixel try to select the proportions such that the combination of the components spectral signature has the best approximation of the observed pixel value. The methods are based on the Minimum Square criteria, where the goal is to estimate the proportions xj minimizing the error square sum ei, subject to the constraint given by equation (2) and, optionally, subject to equation (3). The following methods are available:
The results obtained by these methods are similar; the method selection most adequate has to be based, on the number of components of the mixture and on the decision about constraint (3) application. Proportions of synthetic bands generationOnce the proportions xj were obtained, n (number of components) proportion bands are generated. The proportion bands generated belongs to the Image Model, are included in the same Project of the original bands, and are stored in the GRIB format as 8 bits images. The values attributed to these image pixels depend on the application or not of constraint (3), as described next:
Average error computation and error image generationThe computation of the error indicators described in this section will help the adequate analysis of the mixture model to a certain scene. For each pixel in the image, after estimating the proportions by one of the methods described above, it is possible to compute the estimate error for each band. For each band i, the error ei is given by: n j = 1, ..., n (number of components) Taking these error values by pixel, the average error can be computed by band and total. Additionally, it is possible to generate the error images, which presents the error spatial distribution. The value of these images are obtained by multiplication of the absolute values of the ei by the scale factor 255. Normally, the error values are too low, thus it is suggested the contrast enhancing application in these images to facilitate the error spatial distribution visualization.
Usage ExamplesAs in Shimabukuro [1987], in forested areas it is found, mainly, three components: the tree tops, soil and shadow. Adams et al. [1990] describes types of soil usage found in the Amazon region in terms of four components: vegetation, soil, shadow and wood. Shimabukuro propose the image usage formed by the shadow proportion in each pixel as indicator of forest structure variations, that is, the shadow proportion estimate indicates age variations, type and shape of tree tops. As in Aguiar [1991], the Mixture Model usage can be considered as an alternative method of conventional techniques of the attributes space reduction, either as input for automatic classification methods by the Maximum Likelihood, compared to traditional methods, or as Visual Interpretation. In this case, the mixture model presents as an advantage the fact that the information in the generated images represent physical concepts, that is, the components proportion, easier assimilated than the target spectral signature.
To execute the Mixture ModelExecuting a mixture model involves some steps user defined. A summary sequence of how to create a model and apply it, can be:
![]() |