Chapter 12 Spatial Sharpening of Spectral Image Data

Spatially enhanced unmixing

HSI cube Pan-MS * spatial convolution * spectral convolution Pan-MS superpixel His DC y x Superpixel contains s sharpening band Subpixels corresponding to an HSI pixel (s=9 for this example) λ Linear Histogram transform yields m and b Transforms high resolution spatial sharpening data into common radiometry with the low resolution HSI derived sharpening bands. R p (λ) Figure A

To define the radiometric relationship between the HSI data and the sharpening data we need to derive transform coefficients This can be expressed as or its numerical equivalent (1) (2)

spatially convolve the sharpening bands down to the HSI superpixel size (3)

Schott 1997 suggests that the differences in the atmosphere, atmospheric correction errors, instrument calibration, sensor gain and bias are to first order linear such that the aggregate relation due to any combination can be expressed as: (4)

use these values to transform the sharpening band(s) images into common radiometry with the HSI synthesized bands and more importantly, the synthesized end members using (5) (6) (7) For synthesized and corresponding region of spatially degraded band p

We are going to use the subpixel data to unmix down to the subpixel level or where Is the observed element column vector for the j th pixel made up of the radiometrically equalized sharpening band values (note: if only one sharpening band is used, this is simply the normalized pan band brightness for the j th pixel. (8) (9)

spectral response functions for the sharpening bands is the fraction of the n th end member in the j th pixel and is the vector containing the error in the model for the j th pixel

Is the matrix of end member vectors and Is a column vector containing the fraction of each member in the j th sub pixel

consistency constraints require that the subpixel fractional average must equal the superpixel fractions (10)

(11) squared error metric

Gross and Schott Figures 3,4

Gross+Schott 96 fig 5

Figure 6

Figure 7

Figure 8

Spatial Fusion (12) Single sharpening band, output spectral brightness is computed for each pixel in a superpixel

For poorly correlated bands, a regression equation is solved of the following form Once the coefficients are computed, they are applied at the subpixel level according to: (13) (14)

Compute image wide single band correlation's between each hyperspectral band and the pan superpixel brightnesses. -Select bands with correlation's r 2 >T (threshold ) -Generate hybrid images at highest resolution for all highly correlated bands using simple ratio method (Equation 11) -Note, if multiple sharpening bands are available, the correlation with each sharpening bands is computed and the most highly correlated band is used for that hyperspectral band. 1) 2) Classify HSI scene using an unsupervised classifier (e.g.. K means) Figure B: fusion scheme

3)Use image wide (global) classification to support 2 band regressions -Compute image wide residuals for all two band regressions using each of the remaining HSI bands as dependent variable and the sharpening band(s) or the hybrid bands from step 1 as the independent variables. For all bands where the residuals are below a threshold (R) use a two band regression to compute the hybrid brightness -Compute regression coefficients by class from the entire scene using supervised brightness values and the band combinations with the lowest residuals. -Apply the regression coefficients at the subpixel level using the sharpening bands(s) or previously computed hybrid bands (i.e. from step 1) Fusion scheme B con’t

4) Repeat step 3 using three independent variables. To date this has been done with no threshold just accepting the lowest residuals. However higher order regressions could be used and step 3 repeated again. 5) The output from this process is a hybrid image cube at the spatial resolution of the sharpening band(s). B con’t: Fusion steps 4 and 5

Variations A) step 1 could be implemented using alternate ratios at the subpixel level (i.e. allow for mixed pixels). If σ pan > threshold assume a mixed pixel and use the ratio of the adjacent or target superpixel with the closest mean pan brightness B)The unsupervised image classification can be conducted at the subpixel level and the regression coefficients selected at the subpixel level. This is done using pixel replication to generate HSI pixels at the spatial scale of the sharpening bands. The pixel replicated HSI spectral vectors are then augmented with the sharpening band(s) and the unsupervised classifier run at the highest resolution. B con’t: Fusion variations

Figure C TM false color IR Fused false color IRSpot panchromatic Plate 8.16 Fusion of multispectral TM data with geometrically registered and resampled SPOT data.

Figure 1. Original synthetic scene (band 2) Figure 2. Sample spectra used for synthetic scene Robinson figures 1,2

Robinson figures 3,4 Figure 3. Original DAEDALUS scene (band 4) Figure 4. Sample spectra for DAEDALUS scene

Robinson figure 5 Figure 5. Test plan overview

Robinson figure 7 Figure 7. Fraction maps for DAEDALUS image. Top ¯¯ key; middle ¯¯ stepwise unmix/sharpen; bottom-fuse/unmix

Robinson figures 6,8 Figure 6. Image enhancement of synthetic scene at various scale factors Figure 8. Image enhancement of DAEDALUS scene at two scale factors