Our analysis of images is hampered by their numbers and size. Many of the pixels in each image represent sky or otherwise similar areas: In other words – the ‘degrees of freedom’ of the image is probably less (much less) than the number of pixels in the image. In order to explore the potential of methods that operate not in the image plane but in various transform planes I investigated just how many singular values are needed to satisfactorily reconstruct a typical lunar image from an SVD – a singular value decomposition. First I just tried to reconstruct an image completely, using all 512 singular values:

At upper left is the original (equalized) image. At upper right is the reconstructed image, and at lower left is the difference image. At 512 singular values the RMSE in % was very low – 0.00219 %. The RMSE fell below 0.1% at 420 singular values. The usual plot of the singular values sorted by size shows a ‘break’ near 100 singular values, suggesting that by using more than 100 SVs it is mainly noise that is being reconstructed. Inspection of the spatial reconstruction error as function of increasing number of SVs showed that at 80 SVs the added effort goes mainly into reconstructing the sky – i.e. noise.