Deblurring Images:  I Can See Clearly Now by David Austin, Professor of Mathematics, Grand Valley State University

ABSTRACT:  When the Hubble Space Telescope was launched in 1990, the first images returned were blurry and showed that the primary mirror had been ground to the wrong shape.  While NASA planned their eventual brilliant and daring rescue mission, mathematicians devised several techniques for deblurring the images to improve their quality.  We'll describe one of those techniques in this talk and see how it's related to what has become a standard technique in data science. 

As we'll discover, the blurring process can be described by matrix multiplication.  If  x  is the true image, then the blurred image is formed as A x = b for some matrix A.   So if we have the blurred image b, we just need to solve A x = b to recover the true image, right?  Well, it's not quite so simple for an interesting reason involving some simple linear algebra that we'll explore.  We'll eventually learn about the regularization technique that forms the basis of ridge regression in data science.

DAVID AUSTIN received the prestigious Haimo Distinguished Teaching Award from the Mathematics Association of America in 2021.  A gifted communicator of mathematics, David has contributed to the Feature Column of the American Mathematical Society since 2005.  Committed to ensuring opportunity for studying mathematics, he published his textbook Understanding Linear Algebra as an open-access book, and he is a coauthor on several other open-access textbooks.  A member of the Choctaw Nation of Oklahoma, David is active in building connections and providing mathematical opportunities for Indigenous students that respect Indigenous culture.  He also teaches in the Hope-Western Prison Education Program.