From the beginning of sciences, visual observations have played major roles. 

With the rapid progress in video and computer technology, computers have 

become powerful enough to process image data. As a result, image processing 

techniques are now applied to virtually all natural sciences and technical 

disciplines.  

Mathematical analysis makes image processing algorithms predictable, 

accurate and, in some cases, optimal. New mathematical methods often result 

in novel approaches that can solve previously intractable problems or that 

are much faster or more accurate than previous approaches.   The speed up 

that can be gained by fast algorithm is considerable.  Fast algorithms make 

many image processing techniques applicable and reduce the hardware cost 

considerably.

Wavelet methods are a relatively new mathematical tool that allows us to 

quickly manipulate images, for example, high-resolution image 

reconstructions in some applications, or image compressions in other 

applications.  The wavelet algorithms decompose and arrange an image data 

into strata reflecting their relative importance. This allows a rapid access 

to good coarse resolution of the image while retaining the flexibility for 

increasingly fine representations. It leads to algorithms that give sparse 

and accurate representations of image and medical image for efficient 

computation, analysis, storage, restorations and communication.  In this 

talk, I will illustrate how the wavelet theory is developed and applied to 

various applications in image processing.  Several examples will be given.