Multi-spectral Imaging of La Casa de las Golondrinas Rock Paintings

Clustering by Spectral Shape

When choosing a distortion measure for use in either vector quantization or cluster analysis, it is important to choose a measure that is relevant to the data. In the case of multispectral data, particularly data collected under non-uniform lighting conditions, an ideal distortion measure will be sensitive to differences in spectral shape, while being relatively insensitive to variations in the overall light intensity. This is because the objective in multispectral cluster analysis is to find groups of pixels that come from similar materials in the imaged subject. Because different materials generally may be uniquely characterized by their spectral reflectance, pixels representing a given material in the subject would be expected to have similar spectral reflectance vectors. Variations in illumination intensity would tend to raise and lower this shape as a whole, but the relative shape would remain constant for each material.

Thinking of multispectral data vectors as position vectors in n-dimensional Euclidean space, each possible spectral shape corresponds to a specific direction in that space (n corresponds to the number of elements in each reflectance vector). Increasing the magnitude of the illumination, then, would increase each spectral component by a constant factor, which would simply change the length of the resulting vector without affecting its direction. With this thought in mind, given a multispectral data vector x and a cluster prototype y, the difference (or distortion) between these two vectors in the n-dimensional spectral reflectance space may be computed in terms of the angle between their directions as

d (,) = 1 – cos(θ),                                      (1)

where θ is the angle between the vectors and . With this particular formulation, vectors that have the same shape (i.e., lie along the same direction) will have zero distortion, regardless of differences in illumination, while those that are reflections of each other will have a maximum distortion of two.

Because of its relationship to the angle between the two vectors in spectral reflectance space some have dubbed this distortion measure the "spectral-angle measure," (SAM) (Kruse, 2000). In addition to its strong intuitive appeal for use in clustering multispectral data, this distortion measure is also quite simple computationally. Using vector analysis, a simpler computational form of Eq. (1) may be stated as

d (,) = 1 – · ,                                       (2)

where and ŷ are unit vectors obtained by dividing each vector by its respective magnitude. Thus, in preparing multispectral image data for cluster analysis using the spectral-angle measure in the LBG algorithm, the last preprocessing step is to normalize each pixel’s spectrum vector by dividing by its magnitude. In turn, each time the codebook vectors are recomputed in the LBG algorithm, they must be re-normalized to ensure they remain unit vectors as well. Incorporating the spectral angle measure into the LBG algorithm produces the proposed clustering algorithm, which we shall refer to hereafter as the SAM/LBG algorithm.

Because it uses an index array to keep track of the partitioning of the training data throughout the VQ design process, the LBG algorithm can provide this array as an output that tells the cluster to which each pixel is assigned. By assigning a unique color to each cluster index, this index array can be used to create a false-color image of the clustering results, which is useful in assessing the value of the results obtained.

Previous Page  |  Table of Contents  |  Next Page

Return to top of page