1 / 18

Chapter 11

Chapter 11. Stepwise Unmixing. Recall that we can express conventional linear unmixing as the solution to an expression of the form. (1). Figure A. x = A f. are the end member vectors. The partially constrained case would add the additional constraint

milos
Download Presentation

Chapter 11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 11 Stepwise Unmixing

  2. Recall that we can express conventional linear unmixing as the solution to an expression of the form (1)

  3. Figure A x = Af are the end member vectors

  4. The partially constrained case would add the additional constraint that the fractions must sum to unity i.e. (2)

  5. The stepwise metric Equation 1 can be solved using the pseudo inverse according to: (3) i.e. (4) (5) where is the pseudo inverse of

  6. (n-1) reduced order model (6)

  7. Total variation (about origin) (7) note that and a variation about the regression (the residual error) expressed as: (8) (9)

  8. Analysis of variance (10) where the number of degrees of freedom is n. The mean square error of residuals is (11) where is the number of degrees of freedom

  9. Finally, the value of including the extra term in the regression uses the mean square value for the extra term expressed as: (12) Note that this term represents the difference between the amount of variation explained by the model with a term added to the variation explained by the previous model. The ratio of the MSextra to MSE is distributed with degrees of freedom [n-(n-1)]=1 and and we can test to see if the augmented model is significant i.e. if (13)

  10. Figure B Where Stepwise Logic Step A: Test all end members to find the best single end member model (n=1) Select Fi max Step B: Test all remaining end members to find the best augmented model (n=2) yes Does max Fi exceed threshold Yes add ei to model No proceed to step E ... no Step E

  11. Figure B con’t Stepwise Logic Continued Step C • Find the best augmented model (n>2) by adding • each remaining end member yes Does max Fi exceed threshold Yes add ei to model No proceed to step E ... no Step E

  12. Figure B con’t Stepwise Logic Continued Step D • Test a reduced model for all combinations of the included • end members by testing each as if it were the • most recent to enter Does Fi (the model with the ith end member removed) fail to exceed the threshold. Yes remove ei from the model and proceed to Step C No return to Step C • Finalize solutions by solving for fractions using the desired • unmixing model (e.g. unconstrained, partially • constrained or fully constrained). Note if • unconstrained is desired no further calculations are • required Step E

  13. Synthetic approach allows “truth” fractions (14) where N is the number of Pixels in the test.

  14. Gross+Schott 96 fig 5

  15. Robinson(5) fig 1,2 Figure 1. Original synthetic scene (band 2) Figure 2. Sample spectra used for synthetic scene

More Related