Perpframes, Aligners and Hangers
Linear Systems, Pseudo-Inverse
Matrix Norm, Rank One
|From the really great book by David Kahaner,
Cleve Moler and Stephen Nash -
"Numerical Methods and Software", Prentice-Hall, 1989:
"Suppose that a satellite in space is taking photographs of Jupiter to be sent back to earth.
The satellite digitizes the picture by subdividing it into tiny squares called pixels or picture elements.
Each pixel is represented by a single number that records the average light intensity in that square.
If each photograph were divided into 500 x 500 pixels, it would have to send 250,000 numbers to earth for each picture.
This would take a great deal of time and would limit the number of photographs that could be transmitted.
It [is sometimes] possible to approximate this matrix with a matrix which requires less storage."
The image is stored as a 256 x 264 matrix M with entries between 0 and 1.
The matrix M has rank 256.
Here's a plot of the singular values for M.
Since the singular values are decaying so rapidly, you can expect that there will be a good lower rank approximation to M.
That is, for a relatively small k, you should have
Have a look at various lower rank approximations to M.
What's the advantage of using a lower rank approximation for M?
To send the matrix M you need to send 256 x 264 = 67584 numbers.
To send the rank 36 approximation to M you need only send
So in total you need to send only 36(1+256+264)=18756 numbers.