Principal Components Transformations

See how to execute the Principal Components Transformations

Notice that frequently the individual bands of a multispectral image are highly correlated, that is, the bands are visually and numerically similar.

This correlation comes from the shadows effect resulting from the topography, from the overlap of the spectral windows between adjacent bands and from the object spectral behavior.

The analyses of individual spectral bands can be inefficient because of the redundant information present in each of these bands.

The principal components generation is an enhancing technique that reduces or removes the spectral redundancy, that is, generates a new set of images such that the individual bands present information not available in other bands.

This transformation is based on the covariance matrix between the bands and generates a new set of images where each "pixel" value is a linear combination of the original values. The number of principal components is the same as the number of spectral bands used and they are ordered according to the gray level variance decrease. The first principal component has the highest variance (greater contrast) and the last, the smallest variance.

The figure shows that the transformation of principal components into two dimensions corresponds to the rotation of the coordinate original axis to coincide with the maximum and minimum data variance direction.

In this process the correlation coefficient is used, or the covariance, to determine a set of quantities called autovalues.

The autovalues represent the axis length of the principal components of an image and they are measured in variance units. Associated to each autovalue, there is a vector of unitary module called autovector. The autovector represents the principal components axis directions. They are weighting factors that defines a contribution of each original band to a principal component, in an additive and linear combination.

To facilitate the perception of these contributions, one has to transform the autovectors in percentages. Knowing the signal of each autovector coefficient, it is possible to compare the percentages with the spectral curves of known materials (for example: vegetation, water, different soil types), determining, in this way, in which principal component(s) the desired spectral information is concentrated.

The SPRING allows the user to analyze the autovalues and autovectors data (statistical parameters). Next, an example showing how this data is provided is presented.

Bands Averages Variance Components % Information
B1 40.08 209.79 P1 64.76
B2 48.81 273.13 P2 35.24

Autovectors Matrix

+ 0.5271 + 0.8498

+ 0.8498 - 0.5271

In the example, the first principal component (P1) presents an autovalue of 64.76, that is 64.76% of the information from B1 and B2 are in P1, and 35.24% are in P2.

Analyzing the autovector matrix one gets:

  • P1 = B1 x (+ 0.5271) + B2 x (+ 0.8498)
  • P2 = B1 x (+ 0.8498) + B2 x (- 0.5271)

In this way, it is understood that for P1 band 2 (B2) is contributing with more information. The same reasoning process can be adopted for the "n" principal components.

The first principal component has the brightness information associated to topographic shadows and the greater general spectral reflectance variations from the bands. This principal component has the largest part of the data total variance, concentrating information, previously diluted, in several dimensions.

The second and the subsequent principal components, present gradually less contrast between the targets and they are unprovided of topological information, because of the missing shadowing.

The third and fourth principal components typically have less image structure and more noise than the first two, indicating the data compression in the first channels.

The last principal component represents basically the existing noise in the original data.

The figures below show the three transformation components with three bands (3, 4 and 5) of the Landsat 5.

ima_br_b5_tcp1.gif - 34619 Bytes First Component
ima_br_b5_tcp1.gif - 34619 Bytes Second Component
ima_br_b5_tcp1.gif - 34619 Bytes Third Component

The principal components images can be combined in colors, as any other. When compared with any original channel combination, the colored combination of the principal components presents an enhancing in the color distribution, once there is no correlation between bands.

A colored composition of the principal components image may present only pure spectral colors and saturated intensities, not presenting gray levels (indicating correlation).

Before executing the Principal Components function, you can analyze the statistical parameters of the selected bands. The user can see these parameters, referring to the whole image area or for a portion selected by the cursor.

See also:

About other Image Processing techniques.

Image Processing Options Menu.