Image Transformation and Classification

Band covariance and principal component analysis

Multispectral data are summarised by statistics that describe how the bands vary together. For two bands x_i and x_j measured over n pixels, the covariance is \text{cov}(x_i,x_j)=\frac{1}{n-1}\sum_{k}(x_{ik}-\bar{x}_i)(x_{jk}-\bar{x}_j), and the variance of one band is the special case \text{var}(x_i)=\text{cov}(x_i,x_i). A variance is therefore the mean of the squared deviations of the pixel values from the band mean, so it measures how widely the values spread about that mean.

The variance-covariance matrix is square and symmetric. Its main diagonal contains the variance of each band, and its off-diagonal elements contain the covariances between pairs of bands. The standard deviation of a band is the square root of its diagonal entry, so two bands with equal diagonal entries have equal standard deviations.

The sign of an off-diagonal element gives the direction of the relation. A positive covariance means positive correlation and a negative covariance means negative correlation. The least-squares slope of band j regressed on band i is \text{cov}(x_i,x_j)/\text{var}(x_i), so it has the same sign as the covariance.

Dividing a covariance by the product of the two standard deviations gives the correlation coefficient r_{ij}=c_{ij}/\sqrt{c_{ii}c_{jj}}, which is bounded between -1 and +1. A value near \pm 1 means the two bands are nearly linearly dependent and carry largely redundant information, whereas a value near zero means they carry largely independent information. Independent bands separate classes better in feature space, so the bands that give good classification are those with low mutual correlation. Low correlation does not make a band pair unusable; it only means that the pair is complementary.

When bands are compared for information content, the usual measure is the spread of their values, so a band with larger variance is taken to convey more information than one with smaller variance. A correlation matrix holds no variance information, so equal standard deviations cannot be inferred from it.

Decorrelation stretch rotates the bands into principal component space, where they are uncorrelated, stretches each component, and rotates back to the original space. This removes the inter-band correlation and makes the colour differences among features more distinct.

Principal component analysis diagonalises the covariance matrix. The eigenvectors are mutually orthogonal, and each principal component is a linear combination of the original bands whose weights are the elements of an eigenvector. The number of components equals the number of original variables, so a two-dimensional data set has at most two. The first component points along the direction of greatest variance and accounts for the largest share of the total variation. The second component is orthogonal to the first and captures the next largest variance; it does not represent the mode of the data.

The eigenvalue \lambda_j of component j equals its variance, so the percentage of total variance it explains is \lambda_j/\sum_i \lambda_i \times 100. To find the minimum number of components that reaches a target such as 99 percent, sort the eigenvalues in descending order and accumulate them until the cumulative sum reaches the target fraction of the total.

Image classification

Supervised classification follows a fixed sequence. The image is first radiometrically and geometrically corrected so that digital numbers are comparable and pixels align with the map. Training samples are then selected from areas of known cover to define the spectral signature of each class. The classifier is applied to the whole image to produce the classified map, and the result is finally checked against independent reference data through accuracy assessment.

Thresholding is applied after each pixel is compared with the training signatures. A threshold on the similarity or probability measure rejects pixels whose best match to every training class is too weak, and such pixels are left unassigned instead of being forced into a class. Thresholding therefore does not correct geometry or remove homogeneous regions.

Divergence is a measure of statistical separability between two classes, computed from their mean vectors and covariance matrices. A larger divergence means the class distributions overlap less, so the pair is easier to separate. Divergence is used to evaluate separability before classification; it does not decorrelate bands, smooth the output or split mixed pixels.

A parametric classifier assumes a specific form for the class distributions, such as the multivariate normal model in maximum likelihood classification. Non-parametric classifiers such as artificial neural networks and fuzzy C-means make no such assumption. An artificial neural network can be supervised, as a multilayer perceptron trained with labelled samples, or unsupervised, as a self-organising map. Fuzzy C-means is an unsupervised clustering method because it groups pixels without training labels, and it assigns each pixel degrees of membership in several clusters.

Decision rules differ in the shape of their class boundaries. Minimum distance to mean assigns each pixel to the nearest class mean, so the boundaries are perpendicular bisectors between means. Maximum likelihood uses means and covariances, so its equal-probability contours are ellipses. Parallelepiped classification sets, for each class and each band, a lower and an upper limit derived from the training mean and standard deviation, typically the mean plus or minus a multiple of the standard deviation. Each class therefore becomes an axis-aligned box in feature space. A pixel inside a box is assigned to that class, and a pixel outside all boxes remains unclassified.

When the best subset of b bands is to be selected from n bands, the number of candidate subsets is the binomial coefficient \binom{n}{b}, because the order of bands does not matter. For six bands taken two at a time, \binom{6}{2}=15 combinations must be evaluated.

The k-nearest neighbour classifier assigns a pixel to the majority class among its k closest training pixels in feature space, usually by Euclidean distance. Choosing an odd k avoids ties between two classes. The analyst counts the classes of the nearest training pixels to the test pixel and assigns the most frequent one.