Bulletin Spring‧Summer 1998

Research Focus Figure 1: An imageofan oceanreconnaissance satellite observedby aground-basedimaging system Figure 2: Therestoredimageafter matrix inversion distorting effects of the earth's atmosphere can be d i m i n i s h ed a nd a clearer image of the celestial object is available (Fig. 2). The second stage of the proces s ma y seem like a s t r a i g h t f o r wa r d ma t r ix inversio n problem. Wh at makes the difference is the order of the Toeplit z matrix, or the value of n. The value of n ma y range f r om 60,000 for l o w resolutio n images to over a m i l l i o n for h i g h r e s o l u t i on i ma g e s. A s a t mo s p h e r ic effects are changing almos t every second, the i n v e r s i o ns of ma t r i c es h a ve to be d o ne continuously. G i v en that a generic ma t r ix inversion a l g o r i t hm fo r a ma t r ix of order n requires about n 3 operations, z i l l i o n s a n d zillions of computations w i l l have to be done and done fast i n order to deblur the images. But this is b e y o n d the capability of even th e fastest compute r available. Prof. Chan a n d his p a r t n e r s h a ve d e v e l o p e d a specific fast a l g o r i t hm for these Toeplitz matrices that reduces the a m o u n t of c o m p u t a t i o n to a ma n a g e a b l e l e v e l, as a r e s u lt of w h i c h reasonably high-spee d computers ca n solve the problems w i t h i n a short time. Telescopes with Adaptive Optics E q u i p p e d w i t h these t e c h n i q u e s , telescopes are able to see 50 to 150 time s more clearly. The researchers envisag e that the mathematics of adaptiv e optics w i l l one d a y a l l ow ground-based telescopes to possess the s ame i m a g i n g p o w e r as t h e H u b b l e . A d a p t i v e optics can also be u s ed to kee p better tabs o n spy satellites, protec t space shuttle crew s a n d satellites f r om o r b i t i ng space j u n k s , a nd p r o d u ce h i g h l y accurate laser-guided weapons . I n a f o r ma l hearing organized b y the US Ho u s e A p p r o p r i a t i on Comm i t t ee i n to the 1997 b u d g et requests, P r o f . P l e mm o n s e x p l a i n ed t he researc h Stargazing Through the Mathematical Telescope 31