Noise Adjusted Principal Component Transform (NAPC)

1. Noise Adjusted Principal Component Transform (NAPC) Data are first preprocessed to remove system bias by subtracting dark signal. This subtraction is also applied to a noise image. The spectral covariance of the noise image is then computed. The spectral covariance of the image is also computed. NAPC

2. Noise Adjusted Principal (cont’d) If the noise and signal components of the image data are uncorrelated, then where VS is the covariance of the signal, VR is the covariance of the bias subtracted image data, and VN is the covariance of the bias subtracted noise image. = - V V V (1) s R N NAPC

3. Noise Adjusted Principal (cont’d) v ¢ w We seek vectors that maximize the SNR we also want the bands to be uncorrelated so that i ¢ ¢ v v v v w V w w V w = = - i S i i R i SNR 1 ¢ ¢ v v v v w V w w V w i N i i N i ¢ v v = w V w 0 i R j NAPC

4. Noise Adjusted Principal (cont’d) If E is the eigen vector matrix of the noise such that where DN is the diagonal matrix of eigen values of VN such that if we normalize by DN-1/2 ¢ ¢ = = E V E D and E E I N N - - 1 2 1 ¢ ¢ = = = F ED then F V F I and F F D N N N NAPC

5. Noise Adjusted Principal (cont’d) i.e., operation by F will produce bands with the same noise in each band. Introduce a set of vectors such that and substitute into the equation for SNR v Z v v - v v 1 = = Z F w or F Z w i i i i ¢ ¢ ¢ ¢ ¢ v v v v v v ( Z F ) V ( F Z ) Z F V F Z Z F V F Z i R i i R i i R i + = = = SNR 1 ¢ ¢ ¢ ¢ ¢ v v v v v v Z F V F Z Z ( F V F ) Z Z Z i N i i N i i i NAPC

6. Noise Adjusted Principal (cont’d) We still wish to select vectors that maximize (SNR+1) subject to constraint . The principal components of FVRF will do this (i.e., the are the principal components of (FVRF) and the eigen values of the transformed matrix are equal to the SNR+1. Z i ¢ = Z Z 0 i j Z i NAPC

7. Noise Adjusted Principal (cont’d) The utility of this approach is shown in Figure 1 where noise is shown to be correlated between bands which simple noise scaling will not account for (N.B. residuals are pixel values after initial calibration). Figure 3 shows how NAPC eigen values approach 1 (SNR+1) after several NAPC bands as expected and images show spatial detail in first N images with little spatial noise. NAPC

8. Noise Adjusted Principal (cont’d) Fig. 1. The noise covariance matrix for residuals of the Coppin Gap GER scanner data set. NAPC

9. Noise Adjusted Principal (cont’d) Fig. 3. Eigenvalues of the residuals noise covariance matrix (equation (5)), and the NAPC covariance matrix (equation (9)). Indicates significant spatial detail in the associated image (Fig. 5(a)). NAPC

10. Noise Adjusted Principal (cont’d) Fig. 4. The six major eigenvectors of the residuals noise covariance matrix NAPC

11. Noise Adjusted Principal (cont’d) Note in some NAPC bands (3 and 4 in this analysis) show edge effects due to spectral misregistration (attributed to timing in adjacent bands) this is not removed by this process since the processing is spectral, not spatial. Note how in Figure 6 it shows up as a saw tooth in the eigen value weights in the original spectral space . v ( F Z ) i NAPC

12. Noise Adjusted Principal (cont’d) Fig. 6. Eigenvectors of the noise-adjusted data covariance matrix transformed into the spectral coordinates. (a) The six major eigenvectors. (b) The six most minor eigenvectors. NAPC

13. Noise Adjusted Principal (cont’d) Fig. 7. The six major eigenvectors of the data covariance matrix. NAPC

14. Noise Adjusted Principal (cont’d) Note, similarities between the early noise eigen vector weights Figure 4 and last eigen vector weights of the NAPC (Figure 6) and the first PC weights of the unprocessed image data (Figure 7). Also note, the saw tooth in bands 3 and 4 of NAPC transform. Spectral smoothing of back transform by eliminating is shown in Figure 9. NAPC

15. Noise Adjusted Principal (cont’d) Fig. 9. Pixel spectra for a set of nine adjacent pixels from the Coppin Gap data set. (a) Residuals spectra NAPC

16. Noise Adjusted Principal (cont’d) Fig. 9. Pixel spectra for a set of ninc adjacent pixels from the cappin data set. (b) Spectra reconstructed from NAPC’s 1-9. NAPC

17. Noise Adjusted Principal (cont’d) Fig. 9. Pixel spectra set of ninc pixels from the Coppin Gap set. (C) Spectra reconstructed from NAPC’s 1, 2, and 5-9. NAPC

18. Noise Adjusted Principal (cont’d) Note figure 9b has reduced random noise but still has fixed pattern noise associated with NAPC components 3 and 4 (i.e., misregistration noise). This is eliminated in Figure 9c. Note the importance of looking at weights on original bands where they have physical meaning!! Std. PC problems. NAPC