Spectral Angle Mapper Digital Imaging and Remote Sensing

Spectral Angle Mapper Digital Imaging and Remote Sensing Laboratory Spectral Concepts 1

Spectral Angle Mapper Definitions: é a 1 ù ê ú v ê a 2 ú v A= , Bu = ê Mú ê ú ëan û v B vector A, vector BU of unit magnitude, i. e. , a unit vector Digital Imaging and Remote Sensing Laboratory Spectral Concepts 2

Spectral Angle Mapper (cont’d) Vector Projection and Angle Calculations: Recall that: v v A × B = a 1 b 1 + a 2 b 2 +. . . an bn v v v 1/ 2 Is the dot product and is the A = (A × A ) v magnitude (length) of A Digital Imaging and Remote Sensing Laboratory Spectral Concepts 3

Spectral Angle Mapper (cont’d) Also recall: v v A × B = A B cosq v v -1 é A × B ù q = cos ê ú ë ABû so: Digital Imaging and Remote Sensing Laboratory Spectral Concepts 4

Spectral Angle Mapper (cont’d) The projection of B on A is: v v A×B v v = = Au × B B cos q = B AB A Digital Imaging and Remote Sensing Laboratory Spectral Concepts 5

Spectral Angle Mapper (cont’d) v v i. e. , the projection of onto is the DOT product B A v v of with the unit vector in direction of B A Digital Imaging and Remote Sensing Laboratory Spectral Concepts 6

Spectral Angle Mapper (cont’d) Spectral angle is defined as, , the angle between two spectral vectors Digital Imaging and Remote Sensing Laboratory Spectral Concepts 7

Spectral Angle Mapper (cont’d) Insensitive to illumination and shadow fraction because the magnitude is irrelevant only direction matters. v v For every end member and unknown pixel Ei X Compute Ei. X and assign X to the class i for which Ei. X is a minimum. Can assign a threshold such that if Ei. X is not below the threshold, do not assign. Digital Imaging and Remote Sensing Laboratory Spectral Concepts 8

Spectral Concepts (cont’d) X 3 1 E 1 X E 2 X 2 E 3 X Digital Imaging and Remote Sensing Laboratory Spectral Concepts 9

Spectral Concepts (cont’d) SAM requires spectra of interest. These need to be spectrally different and ideally spectral extremes to provide good spectral separation. They can come from library spectra if we have one in reflectance units or from the image itself using expert selection or PPI. Digital Imaging and Remote Sensing Laboratory Spectral Concepts 10