Independent Component Analysis

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Any given remote sensing image can be decomposed into several features. The term ‘feature’ here refers to remote sensing scene objects (for example, vegetation types or urban materials) with similar spectral characteristics. Therefore, the main objective of a feature extraction technique is to accurately retrieve these features. The extracted features can be subsequently utilized for improving the performance of various remote sensing applications (for example, classification, target detection, unmixing, and so forth).

Independent component analysis (ICA) is a high order feature extraction technique. Principal Components Analysis (PCA) and Minimum Noise Fraction (MNF) model the data using a Gaussian distribution, however most remote sensing data do not follow a Gaussian distribution. ICA exploits the higher order statistical characteristics of multispectral and hyperspectral imagery such as skewness and kurtosis. Skewness is a measure of asymmetry in an image histogram. Kurtosis is a measure of peakedness or flatness of an image histogram (that is, departure from a Normal Distribution).

Unlike Principal Component axes obtained by PCA that exhibit a precedence in their order, (that is, the direction of the first principal axis denotes a representation with maximum data variance, second principal axis with second highest data variance and so forth), the labeling of Independent Component axes does not imply any order. Additionally, note that these axes are not restricted to being orthogonal and thus lead to transformed data that is not only uncorrelated (second order statistics) but also independent (higher order statistics).

ICA performs a linear transformation of the spectral bands such that the resulting components are decorrelated and independent. Each independent component (IC) will contain information corresponding to a specific feature in the original image. For a detailed description on the mathematical formulation of ICA refer to Shah, C. A., 2003 and Common, P., 1994.

Component Ordering

It is always desirable to have the number of components greater than the number of features in order to ensure that all the features are recovered as ICs. In cases where the number of features happens to be smaller than the number of components, the additional components will contain very little feature information and will resemble a noisy image.

In ERDAS IMAGINE, you can order the ICs so that the noisy images are the last few components and can be easily eliminated from further analysis. The available options for component ordering are:

  • None
  • Correlation Coefficient
  • Skewness
  • Kurtosis
  • Combinations of the above
  • Entropy
  • Negentropy

None

No ordering is applied to the components; as a result, the ICs appear in an arbitrary order. Noisy components may occur at any position in the image stack.

Three basic statistical measures provided for component ordering are as follows

Correlation Coefficient

Correlation coefficient (Rxy) between two images X and Y is defined as:

ii_spectral_correlation_coef

Where:

P = total number of image pixels

The correlation coefficient is a measure of similarity between two images; the higher the correlation the greater is the similarity between their pixel values. The ICs are ordered based on their correlation with the spectral bands. ICs with low correlation correspond to noisy images and are therefore lower in the image stack (that is, higher band numbers).

Skewness

Skewness is a measure of asymmetry in an image histogram. The skewness of image X is defined as:

iispectral_skewness

Output bands are ordered by increasing asymmetry.

Kurtosis

Kurtosis is a measure of peakedness or flatness of an image histogram (that is, departure from Normal Distribution). The kurtosis of image X is defined as:

iispectral_kurtosis

An image with a normally distributed histogram has zero skewness and kurtosis.

Combinations

Two combinations of the above three measures are also options for component ordering in ERDAS IMAGINE. These are the absolute values of the following equations:

iispectral_correlationxskewxkurtosis

iispectral_skewnessxkurtosis=skewxkur

Entropy

Entropy is a measure of image information content. Entropy of image X is defined as:

iispectral_entropy

Where:

G = number of gray levels

k = gray level

P = probability

Negentropy

Negentropy is a measure of how far the distribution of a particular image departs from a normal distribution. Negentropy of image X is defined as:

iispectral_negentropy

and is proportional to its skewness and kurtosis.

Lower values of skewness/kurtosis/negentropy correspond to ICs resembling noisy images.

Band Generation for Multispectral Imagery

It must be noted that the number of desired components cannot exceed the number of spectral bands in the imagery. This should not be of concern when processing hyperspectral imagery, where the number of spectral bands is significantly higher compared to the number of features in the scene. However, feature extraction from multispectral imagery necessitates the generation of additional spectral bands as explained below.

A linear combination of the original spectral bands will not lead to additional information required for ICA feature extraction from multispectral imagery. Therefore, you should generate additional spectral bands through non-linear operations such as

enhance_ica_band_generation

Where Xi is a multispectral band and .

Remote Sensing Applications for ICs

More information and examples of Spectral Unmixing, Shadow Removal, and Classification may be found in Shah, et al, 2007.

  • Visual analysis

ICs can be used to improve the visual interpretability through component color coding. A similar approach can be used for enhanced visual interpretability of hyperspectral imagery employing ICs.

  • Resolution merge

Improved integration of imagery at different spatial resolution can be attained by substituting a high spatial resolution image for an IC followed by an inverse transformation.

  • Spectral unmixing

ICA can be employed for linear spectral unmixing when you have no prior information regarding the spectral response of features present in the scene. The formulation of each band in the case of three features can be expressed as

enhance_ica_spectral_band_equation

where enhance_ica_a_i_wavelength_symbol is the sensor response to feature i at wavelength enhance_ica_wavelength_symbol.

enhance_ica_a_i_wavelength_symbol and the ith feature estimated by ICA correspond to the spectral response and abundance respectively of the ith feature in spectral band at wavelength enhance_ica_wavelength_symbol. Hence, ICA extracted features can be further analyzed for identifying the proportion of each feature in a pixel.

  • Shadow detection

High spatial resolution multispectral images necessitate shadow detection to facilitate improved feature analysis. By employing band combinations (for example, band ratio) for band generation, the spectral difference between the shadow and the shadow occluded features can be enhanced. ICs obtained from these bands would recover the shadow in one of the components.

  • Land use/ land cover classification

ICs can be further analyzed based on their spectral, textural, and contextual information in order to obtain an improved thematic map.

  • Multi temporal data analysis

Since feature based change detection techniques necessitate extraction of features with high accuracy, ICs are well suited in the analysis of multi temporal data.

  • Anomaly/Target detection

In cases where there is no prior information regarding material of the target features present in the scene, spectra from libraries can not be used for detecting them. ICA, however, when employed for such applications, will remove the anomalous features (that is, features with spectral response significantly different from other features present in the scene). Those anomalous features are contained in the independent components. These components can be further analyzed for improved anomaly/target detection.

Tips and Tricks

  • Band generation

Use of meaningful band combinations for non-linear band generation in case of multispectral imagery improves the performance of ICA in extracting features. For example, employing a Normalized Differential Vegetation Index (NDVI) as an additional band would certainly enhance the performance of ICA in extracting vegetation features.

  • Desired number of components

A typical scene imaged by a hyperspectral sensor would not contain more than 10 to 15 features. Hence, the number of desired components should be restricted to less than 20 in order to ensure accurate and efficient results.

  • Image background with zero pixel values

In cases where the images have background with zero values, use a subset of that image by eliminating background pixels for ICA feature extraction.

  • Visual inspection of ICs

In addition to using any of the component ordering techniques, the ICs may be visually inspected to ensure that they do not resemble noisy images.

Also, the number of desired components impacts the result of ICA. Let us consider the following two scenarios: First, ICA is performed on an imagery and the desired number of components is 3. Next, the ICA is performed on the same imagery and this time the number of desired components is 4. The 3 ICs obtained in the first case would not be identical to any of the 4 ICs in the second case.