Lecture 13: Spectral Mixture Analysis

Tuesday 16 February 2010 Lecture 13: Spectral Mixture Analysis Reading Ch 7.7 – 7.12 Smith et al. Vegetation in deserts (class website) Last lecture: framework for viewing image processing and details about some standard algorithms

NDVINormalized Difference Vegetation Index DN4-DN3is a measure of how much chlorophyll absorption is present, but it is sensitive to cos(i) unless the difference is divided by the sum DN4+DN3.

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. 19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade

Some materials are commonly found together. These are mixed. 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 19.1% Trees 43.0% Road 24.7% Grass/GV 13.2% Shade

Spectral Mixtures 100 Reflectance 0 Wavelength 100 Reflectance 0 Wavelength

d Linear vs. Non-Linear Mixing • Linear Mixing (additive) • Non-Linear Mixing • Intimate mixtures, Beer’s Law r = fg·rg+ rs·(1- fg) r = rg+ rs·(1- rg)·exp(-kg·d) · (1-rg)·exp(-kg·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 spectralendmembers. 0 1 2 Wavelength, μm

Spectral Mixtures 25% Green Vegetation (GV) 75% Soil 60 100% GV 100% Soil 40 75% GV 25% GV TM Band 4 50% GV 20 0 0 0 20 40 60 TM Band 3

Spectral Mixtures 25% Green Vegetation 70% Soil 5% Shade 60 100% GV 100% Soil 40 TM Band 4 20 100% Shade 0 0 20 40 60 TM Band 3

r f mix,b em r em,b Linear Spectral Mixtures There can be at most m=n+1 endmembers or else you cannot solve for the fractions f uniquely = 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 eb 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. http://en.wikipedia.org/wiki/Singular_value_decomposition

Landsat TM image of part of the Gifford Pinchot National Forest

Old growth Burned Mature regrowth Shadow Immature regrowth Broadleaf Deciduous Clearcut Grasses

Spectral mixture analysis from the Gifford Pinchot National Forest In fraction images, light tones indicate high abundance Green vegetation NPV R = NPV G = green veg. B = shade Shade

Spectral Mixture Analysis - North Seattle Blue – concrete/asphalt Green - green vegetation Red - dry grass

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.

Foreground / Background Analysis (FBA) • 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. DNk X ▫ ▪ F ▪ ▪ DNj B ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ ▪ DNi

Foreground: Background: Vector w is defined as a projection in hyperspace of all foreground DNs (DNF) as 1 and all background DNs as (DNB) 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. http://en.wikipedia.org/wiki/Singular_value_decomposition

Mixing analysis is useful because – • It makes fraction pictures that are closer to what you want to know about abundance of physically meaningful scene components • 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

Lecture 13: Spectral Mixture Analysis