Mixture Model


See how to execute the Mixture Model in the SPRING.


Mixture problems happen in Remote Sensing images because of the sensors spatial resolution that, in general, allow that a scene element (corresponding to an image pixel) includes more than one type of terrain cover. When a sensor observes the scene, the detected radiance is the integration, denominated mixture, of all objects, denominated mixture components, inside the scene element.

In the literature, one may find two different working lines related to the mixture problem:

  • Substitute conventional image classification methods in the total area calculus per theme (class) in a scene, considering that the estimates based on conventional methods are damaged by class mixtures in the boundaries between different targets (e.g. boundary between pasture and culture).
  • Synthetic image generation, that represents the proportions of each mixture component inside the pixels, that is, the number of original bands is reduced to the number of components of the mixture model. (e.g. in a reforesting area, the synthetic three band generation, one representing the vegetation proportion, other of soil proportion and the third one of shadow proportion, in each image pixel).

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 Model

In 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
r 2 = a 21x 1 + a22 x 2 + ... + a 2nx n + e2
...
rm = a m1x1 + am2 x2 + ... + a2n xn + em

that is,

n
ri = S (aij xj ) + ei , i = 1,..., m (number of bands) (1)
j=1 j = 1, ..., n (number of components)

n <= m e
S = summation

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).
    aij : spectral reflectance known of jth component in the ith spectral band.
    xj : value to be estimated of the jth proportion component inside the pixel; and
    ei : estimated error for the ith spectral band.

The xj estimates are related with the following constraints:

n
S xj= 1 (2)
j=1

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 components

The 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 estimate

The 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:

  • Minimum Square with Constraints: the simplest and fast method, applicable when the number of components is equals three. The constraint (3) can or cannot be applied.
  • Weighted Minimum Square: the more general method, it search the solution interactively, trying to satisfy constraint (2) and (3). Optionally, the constraint (3) may not be applied, becoming, in this case, very similar to the method below.
  • Combination between Principal Components and Minimum Square Transformations: this method is used to reduce the number of equations in the system applying initially a transformation of principal components, followed by the Minimum Square estimate method. It has the advantage of fast computation for a number of components different than three. However, constraint (3) can not be applied.

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 generation

Once 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:

  • Model subjected to constraint (3): the pixel values in the n bands are obtained by proportions multiplications xj in each pixel by the scaling factor (255).
  • Model not subjected to constraint (3): the proportion values in the zero to one interval, this is, with physical meaning, are assigned for the 100 to 200 interval. Pixels with negative proportion values, receive the value 0. Pixels with proportion value greater than one, receive the value 255.


Average error computation and error image generation

The 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
ei = ri - S (aij xj ), i = 1, ..., m (number of bands) (4)
j =1

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 Examples

As 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 Model

Executing a mixture model involves some steps user defined. A summary sequence of how to create a model and apply it, can be:

  1. Create a Model;
  2. Define for this model a name and the bands that will be involved;
  3. Define which components of this model and the values for each band;
  4. Save the model;
  5. Specify which input plans, the interest areas, the method, proportions and error;
  6. Apply the model.

See also:
Other Image Processing techniques.
How to execute a Mixture Model.