Opportunities for the Mathematical Sciences

Fuzzy picture of parrotsSharper picture of parrots
Sharpest picture of parrotsOriginal of parrot picture

An illustration of how wavelets permit to construct images by superposing layers of increasing detail. The bottom right image is the orginal; the other three, starting at top left, show a very "broad-brush" approximation, to which first intermediate details (top right) and then fine-scale details are added (bottom left), by using increasingly fine wavelets. Although the original image (bottom right) used 786 kb, the version at bottom left, indistinguishable to the naked eye, is constructed using only 103kb. By combining wavelet decompositions with sophisticated quantization techniques, higher compression rates can be achieved, with little loss of quality . Wavelets have been adopted as the compression method for the JPEG-2000 standard. Wavelet techniques have found applications in many other areas of signal processing, including in medical applications; they are also used for computations when many different scales are present, in chemistry, physics and materials science.In addition, wavelet and multiresolution techniques are increasingly used in computer graphics.

Figure credit: Sinan Gunturk and Ingrid Daubechies.

Back to Summary.