Lecture 13: Spectral Mixture Analysis Tuesday 16 February 2010 Last lecture: framework for viewing image processing and details about some standard algorithms Reading Ch 7.7 – 7.12 Smith et al. Vegetation in deserts (class website)
19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade
19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade Spectral images measure mixed or integrated spectra over a pixel
19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade Each pixel contains different materials, many with distinctive spectra.
Some materials are commonly found together. These are mixed. 19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade
19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade Others are not. They may be rare, or may be pure at multi-pixel scales
Wavelength Reflectance Spectral Mixtures Wavelength 0 Reflectance 100
Linear vs. Non-Linear Mixing Linear Mixing (additive) Non-Linear Mixing –Intimate mixtures, Beer’s Law r = f g ·r g + r s ·(1- f g ) r = r g + r s ·(1- r g )·exp(-k g ·d) · (1-r g )· exp(-kg·d) +……. d
Spectral Mixture Analysis works with spectra that mix together to estimate mixing fractions for each pixel in a scene. The extreme spectra that mix and that correspond to scene components are called spectral endmembers Wavelength, μm
Spectral Mixtures 25% Green Vegetation (GV) 75% Soil TM Band 4 TM Band % GV 50% GV 25% GV 100% GV 100% Soil
Spectral Mixtures 25% Green Vegetation 70% Soil 5% Shade TM Band 4 TM Band % GV 100% Shade 100% Soil
Linear Spectral Mixtures r mix,b f em r em,b = Reflectance of observed (mixed) image spectrum at each band b = Fraction of pixel filled by endmember em = Reflectance of each endmember at each band = Reflectance in band b that could not be modeled = number of image bands, endmembers bb There can be at most m=n+1 endmembers or else you cannot solve for the fractions f uniquely n,m
In order to analyze an image in terms of mixtures, you must somehow estimate the endmember spectra and the number of endmembers you need to use Endmember spectra can be pulled from the image itself, or from a reference library (requires calib- ration to reflectance). To get the right number and identity of endmembers, trial-and-error usually works. Almost always, “shade” will be an endmember “shade”: a spectral endmember (often the null vector) used to model darkening due to terrain slopes and unresolved shadows
Inverse SMA (“unmixing”) The point of spectral mixture analysis (SMA) is usually to solve the inverse problem to find the spectral endmember fractions that are proportional to the amount of the physical endmember component in the pixel. Since the mixing equation (two slides ago) should be underdetermined – more bands than endmembers – this is a least-squares problem solved by “singular value decomposition” in ENVI.
Landsat TM image of part of the Gifford Pinchot National Forest
Burned Mature regrowth Old growth Immature regrowth Broadleaf Deciduous Clearcut Grasses Shadow
Green vegetation NPV Shade Spectral mixture analysis from the Gifford Pinchot National Forest R = NPV G = green veg. B = shade In fraction images, light tones indicate high abundance
Blue – concrete/asphalt Green - green vegetation Red - dry grass Spectral Mixture Analysis - North Seattle
As a rule of thumb, the number of useful endmembers in a cohort is 4-5 for Landsat TM data. It rises to about 8-10 for imaging spectroscopy. There are many more spectrally distinctive components in many scenes, but they are rare or don’t mix, so they are not useful endmembers. A beginner’s mistake is to try to use too many endmembers.
Objective: Search for known material against a complex background “Mixture Tuned Matched Filter™” in ENVI is a special case of FBA in which the background is the entire image (including the foreground) Geometrically, FBA may be visualized as the projection of a DN data space onto a line passing through the centroids of the background and foreground clusters The closer mystery spectrum X plots to F, the greater the confidence that the pixel IS F. Mixed pixels plot on the line between B & F. Foreground / Background Analysis (FBA) ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ B F DN i DN j DN k ▫ X
Foreground:Background: Vector w is defined as a projection in hyperspace of all foreground DNs (DN F ) as 1 and all background DNs as (DN B ) 0. n is the number of bands and c is a constant. The vector w and constant c are simultaneously calculated from the above equations using singular-value decomposition.
Mixing analysis is useful because – 1)It makes fraction pictures that are closer to what you want to know about abundance of physically meaningful scene components 2)It helps reduce dimensionality of data sets to manageable levels without throwing away much data 3) By isolating topographic shading, it provides a more stable basis for classification and a useful starting point for GIS analysis
Next lecture – Image classification