This tutorial is designed to introduce you to advanced concepts and procedures for analysis of imaging spectrometer data or hyperspectral images . We will use 1995 Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) data from Cuprite, Nevada, to investigate sub-pixel properties of hyperspectral data and advanced techniques for identification and quantification of mineralogy. We will use "Effort" "polished" ATREM-calibrated data, and review matched filter and spectral unmixing results. This tutorial is designed to be completed in two to four hours.
You must have the ENVI TUTORIALS & DATA CD-ROM mounted on your system to access the files used by this tutorial, or copy the files to your disk.
The files used in this tutorial are contained in the C95AVSUB subdirectory of the ENVIDATA directory on the ENVI TUTORIALS & DATA CD-ROM.
The files listed below, along with their associated .hdr files, are required to run this exercise. Optional spectral library files listed below may also be used if more detailed comparisons are desired. Selected data files have been converted to integer format by multiplying the reflectance values by 1000 because of disk space considerations. Values of 1000 in the files represent reflectance values of 1.0.
CUP95EFF.INT "Effort" corrected ATREM apparent reflectance data, 56 bands, 1.96 - 2.51 mm.
CUP95MNF.DAT First 25 MNF bands.
CUP95MNF.ASC MNF eigenvalue spectrum
CUP95PPI.DAT Pixel Purity Index (PPI) Image
CUP95PPI.ROI Region of Interest for PPI values greater than 1750
CUP95PPI.NDV N-D Visualizer Saved State File.
CUP95NDV.ROI ASCII file spectral endmembers corresponding to the N-D Visualizer Saved State File
CUP95_EM.ASC ASCII file of 11 spectral endmembers elected using the PPI threshold, MNF images, and n-D Visualization - used in Matched Filtering and Unmixing
CUP95MAT.DAT Match Filter Results--Mineral Images
CUP95UNM.DAT Unmixing Results--Fractional Abundance images.
JPL1.SLI JPL Spectral Library in ENVI format.
USGS_MIN.SLI USGS Spectral Library in ENVI format.
This is the 1995 AVIRIS ATREM-calibrated apparent reflectance data with the Effort correction applied.
These data will be used for comparison with MNF bands and to extract spectral endmembers for PPI results.
The minimum noise fraction (MNF) transformation is used to determine the inherent dimensionality of image data, to segregate noise in the data, and to reduce the computational requirements for subsequent processing (See Boardman and Kruse, 1994). The MNF transform as modified from Green et al . (1988) and implemented in ENVI is essentially two cascaded Principal Components transformations. The first transformation, based on an estimated noise covariance matrix, decorrelates and rescales the noise in the data. This first step results in transformed data in which the noise has unit variance and no band-to-band correlations. The second step is a standard Principal Components transformation of the noise-whitened data. For the purposes of further spectral processing, the inherent dimensionality of the data is determined by examination of the final eigenvalues and the associated images. The data space can be divided into two parts: one part associated with large eigenvalues and coherent eigenimages, and a complementary part with near-unity eigenvalues and noise-dominated images. By using only the coherent portions, the noise is separated from the data, thus improving spectral processing results.
Figure 1 summarizes the MNF procedure in ENVI. The noise estimate can come from one of three sources; from the dark current image acquired with the data (for example AVIRIS), from noise statistics calculated from the data themselves, or from statistics saved from a previous transform. Both the eigenvalues and the MNF images (eigenimages) are used to evaluate the dimensionality of the data. Eigenvalues for bands that contain information will be an order of magnitude larger than those that contain only noise. The corresponding images will be spatially coherent, while the noise images will not contain any spatial information.
This dataset contains the first 25 MNF bands (floating point) from the 1995 Cuprite Effort data.
Note the "corners" on some MNF scatterplots (Figure 2).
Note when the MNF data stops being "pointy" and begins being "fuzzy" and the relation between scatterplot pixel location and spectral mixing as determined from image color and individual reflectance spectra.
We will now investigate the possibilities of deriving unmixing endmembers from the data using MNF images and the 2-D scatterplot tools.
These will be mapped into the image as colored pixels.
Note that corner pixels on the scatterplots generally make good endmember estimates. However, note also the occurrence of overlapping or repeat ROIs. This is a limitation of examining the data in a pairwise fashion (2-D).
In this portion of the exercise, you will examine the role of convex geometry in determining the relative purity of pixels. Separating purer from more mixed pixels reduces the number of pixels to be analyzed for endmember determination and makes separation and identification of endmembers easier.
The "Pixel-Purity-Index" (PPI) is a means of finding the most "spectrally pure," or extreme, pixels in multispectral and hyperspectral images. See Boardman et al . (1995). The most spectrally pure pixels typically correspond to mixing endmembers. The Pixel Purity Index is computed by repeatedly projecting n -dimensional scatter plots onto a random unit vector. The extreme pixels in each projection are recorded and the total number of times each pixel is marked as extreme is noted. A Pixel Purity Index (PPI) image is created in which the DN of each pixel corresponds to the number of times that pixel was recorded as extreme. Figure 3 summarizes the use of PPI in ENVI.
Brighter pixels represent more spectrally extreme "hits" and indicate pixels that are more spectrally pure. Darker pixels are less spectrally pure.
The PPI image is the result of several thousand iterations of the PPI algorithm discussed above on the MNF data. The values in the image indicate the number of times each pixel was discovered as extreme in some projection. These numbers then indicate the degree of local convexity of the data cloud near each pixel and the proximity of each pixel to the convex hull of the data. In short, the higher values indicate pixels that are nearer "corners" of the n -Dimensional data cloud, and are thus relatively purer than pixels with lower values. Pixels with values of zero were never found to be extreme.
In this configuration, you can examine the spectral profiles of certain pixels as they are selected in the PPI display.
This is a list of pixels where the PPI value is over 1000.
Now try making some of your own thresholded PPI Regions of Interest.
ENVI will determine the number of pixels that meet the selected criteria and post a message.
Only those pixels with values greater than the selected minimum will be included in the ROI built from the PPI image.
This ROI contains the pixel locations of the purest pixels in the image regardless of the endmember to which they correspond. The n -Dimensional Visualizer will be used in the next portion of the exercise to isolate the specific pure endmembers.
Spectra can be thought of as points in an n -dimensional scatter plot, where n is the number of bands. See Boardman et al . (1995). The coordinates of the points in n -space consist of "n" values that are simply the spectral radiance or reflectance values in each band for a given pixel. The distribution of these points in n -space can be used to estimate the number of spectral endmembers and their pure spectral signatures.
ENVI's n -Dimensional Visualizer provides an interactive tool for selecting the endmembers in n -space. In this section, you will examine a Grand Tour of the n -dimensional data. Because the computer method is computer intensive, it only operates well with no more than a few thousand pixels in a few dozen bands. From a theoretical standpoint, it makes little sense to view all the mixed pixels. The most important pixels, those that best suggest the endmember materials, are the purest pixels, previously selected using the PPI thresholding. Figure 4 summarizes the steps involved in using the n -Dimensional Visualizer to select endmember spectra.
Remember that these bands encompass almost all the signal variability and limiting the number of bands will improve the interactive visualization performance. The other, higher-order MNF bands have already been discarded as primarily noise.
If you only have one valid ROI listed in the Regions of Interest Controls dialog, that ROI data will automatically be loaded into the n -Dimensional Visualizer. If you have more than one ROI, choose the ROI derived using the PPI threshold when queried. After a short wait, the n -D scatter plotting window and controls will appear. The numbers 1 through 10 on the n -D Controls dialog refer to the ten spectral bands chosen.
Note the shape of the data clouds. Be sure to examine some of the higher order MNF bands.
Again, note the shape of the data clouds. Be sure to examine some of the higher order MNF bands.
This is an animation of random projections of n -Dimensional space into the scatter plot. In this mode, any number of bands can be examined simultaneously.
Note how the rotations seem different when more than three bands are included. This is the result of dimensions greater than 3 being "folded" in upon themselves in the projection. This should convince you that the data is truly high dimensional and why 2-D scatterplots are inadequate for dealing with hyperspectral data.
The colored pixels in the visualizer represent previously selected endmembers.
Pay particular attention to similar spectra and the positions of painted clusters.
You can view reflectance spectra for specific endmembers while you're painting endmember and rotating the scatterplot. This allows you to preview spectra before finalizing spectral classes
Once you have identified a few endmembers, it may be difficult to find additional endmembers, even though you run the grand tour or many different 2-D projections of the n-Dimensional data.
Class Collapsing in ENVI is a means of simplifying this endmember determination problem by allowing you to group the endmembers you have already found into one group representing the background. The net result is that mixing features that were previously hidden become visible, and can be selected using ENVI's ROI drawing in the n-D Visualizer window.
Note that the bands selected are now listed in red and only two are chosen. Additionally, a MNF plot appears, estimating the dimensionality of the data, and the number of endmembers remaining to be found. Repeat the endmember selection and class collpsing process until there are no new endmembers found.
You can also export mean spectra for the selected endmembers.
Matched filtering is based on well-known signal processing methodologies. It maximizes the response of a known endmember and suppresses the response of the composite unknown background, thus "matching" the known signature (Chen and Reed, 1987; Stocker et al., 1990; Yu et al., 1993; Harsanyi and Chang, 1994). It provides a rapid means of detecting specific minerals based on matches to specific library or image endmember spectra. This technique produces images similar to the unmixing as described below, but with significantly less computation. Matched filter results are presented as gray-scale images with values from 0 to 1.0, which provide a means of estimating relative degree of match to the reference spectrum (where 1.0 is a perfect match).
Open Files and Compare Matched Filter Results
A grayscale image of the matched filter results for Kaolinite will be displayed. The brightest pixels represent the best match to the endmember spectrum.
Close all Displays and Other Windows
Natural surfaces are rarely composed of a single uniform material. Spectral mixing occurs when materials with different spectral properties are represented by a single image pixel. Several researchers have investigated mixing scales and linearity. Singer and McCord (1979) found that if the scale of the mixing is large (macroscopic), mixing occurs in a linear fashion (Figure 7) For microscopic or intimate mixtures, the mixing is generally nonlinear (Nash and Conel, 1974; Singer, 1981).
The linear model assumes no interaction between materials. If each photon only "sees" one material, these signals add (a linear process). Multiple scattering involving several materials can be thought of as cascaded multiplications (a non-linear process). The spatial scale of the mixing and the physical distribution of the materials governs the degree of non-linearity. Large-scale aerial mixing is very linear. Small-scale intimate mixtures are slightly non-linear. In most cases, the non-linear mixing is a second order effect. Many surface materials mix in non-linear fashions but linear unmixing techniques, while at best an approximation, appear to work well in many circumstances (Boardman and Kruse, 1994). While abundances determined using the linear techniques are not as accurate as those determined using non-linear techniques, to the first order they appear to adequately represent conditions at the surface.
A variety of factors interact to produce the signal received by the imaging spectrometer
The simplest model of a mixed spectrum is a linear one, in which the spectrum is a linear combination of the "pure" spectra of the materials located in the pixel area, weighted by their fractional abundance (Figure 8).
This simple model can be formalized in three ways: a physical model a mathematical model, and a geometric model. The physical model as discussed above includes the Ground Instantaneous Field of View (GIFOV) of the pixels, the incoming irradiance, the photon-material interactions, and the resulting mixed spectra. A more abstract mathematical model is required to simplify the problem and to allow inversion, or unmixing (Figure 9).
A spectral library forms the initial data matrix for the analysis. The ideal spectral library contains endmembers that when linearly combined can form all other spectra. The mathematical model is a simple one. The observed spectrum (a vector) is considered to be the product of multiplying the mixing library of pure endmember spectra (a matrix) by the endmember abundances (a vector). An inverse of the original spectral library matrix is formed by multiplying together the transposes of the orthogonal matrices and the reciprocal values of the diagonal matrix (Boardman, 1989). A simple vector-matrix multiplication between the inverse library matrix and an observed mixed spectrum gives an estimate of the abundance of the library endmembers for the unknown spectrum.
The geometric mixing model provides an alternate, intuitive means to understand spectral mixing. Mixed pixels are visualized as points in n -dimensional scatter-plot space (spectral space), where n is the number of bands. In two dimensions, if only two endmembers mix, then the mixed pixels will fall in a line (Figure 10a). The pure endmembers will fall at the two ends of the mixing line. If three endmembers mix, then the mixed pixels will fall inside a triangle (Figure 10b). Mixtures of endmembers "fill in" between the endmembers.
All mixed spectra are "interior" to the pure endmembers, inside the simplex formed by the endmember vertices, because all the abundances are positive and sum to unity. This "convex set" of mixed pixels can be used to determine how many endmembers are present and to estimate their spectra. The geometric model is extensible to higher dimensions where the number of mixing endmembers is one more than the inherent dimensionality of the mixed data.
Two very different types of unmixing are typically used: Using "known" endmembers and using "derived" endmembers.
Using known endmembers, one seeks to derive the apparent fractional abundance of each endmember material in each pixel, given a set of "known" or assumed spectral endmembers. These known endmembers can be drawn from the data (averages of regions picked using previous knowledge), drawn from a library of pure materials by interactively browsing through the imaging spectrometer data to determine what pure materials exist in the image, or determined using expert systems as described above or other routines to identify materials.
The mixing endmember matrix is made up of spectra from the image or a reference library. The problem can be cast in terms of an overdetermined linear least squares problem. The mixing matrix is inverted and multiplied by the observed spectra to get least-squares estimates of the unknown endmember abundance fractions. Constraints can be placed on the solutions to give positive fractions that sum to unity. Shade and shadow are included either implicitly (fractions sum to 1 or less) or explicitly as an endmember (fractions sum to 1).
The second unmixing method uses the imaging spectrometer data themselves to "derive" the mixing endmembers (Boardman and Kruse, 1994). The inherent dimensionality of the data is determined using a special orthogonalization procedure related to principal components:
Spectral unmixing is one of the most promising hyperspectral analysis research areas. Analysis procedures using the convex geometry approach already developed for AVIRIS data have produced quantitative mapping results for a a variety of materials (geology, vegetation, oceanography) without a priori knowledge. Combination of the unmixing approach with model-based data calibration and expert system identification capabilities could potentially result in an end-to-end quantitative yet automated analysis methodology.
In this section, you will examine the results of unmixing using the means of the ROIs restored above and applied to the first ten MNF bands. You will then run your own unmixing using endmember suites of your own choosing.
Brighter areas correspond to higher abundances.
Note the occurrence of non-primary colors (not R, G, or B).
Note areas where unreasonable results were obtained (eg. fractions greater than one or less than zero).
You can create your own endmembers from single pixels, ROI means, or import them from spectral libraries.
Using the drag and drop method from spectral plots, you can build up your own spectral suite of mixing endmembers.
Think about how the results depend entirely on the endmember selection.
Boardman, J. W., 1993, Automated spectral unmixing of AVIRIS data using convex geometry concepts: in Summaries, Fourth JPL Airborne Geoscience Workshop, JPL Publication 93-26, v. 1, p. 11 - 14.
Boardman J. W., and Kruse, F. A., 1994, Automated spectral analysis: A geologic example using AVIRIS data, north Grapevine Mountains, Nevada: in Proceedings, Tenth Thematic Conference on Geologic Remote Sensing, Environmental Research Institute of Michigan, Ann Arbor, MI, p. I-407 - I-418.
Boardman, J. W., Kruse, F. A., and Green, R. O., 1995, Mapping target signatures via partial unmixing of AVIRIS data: in Summaries, Fifth JPL Airborne Earth Science Workshop, JPL Publication 95-1, v. 1, p. 23-26.
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