Image Reconstruction With Ridgelets
Po‐Shen Loh, Emmanuel J. Candès · 2003
When we capture or transmit a digital image of an object, we sometimes lose a fraction of the pixels to noise or error. It is therefore of interest to develop methods that can reconstruct the original image. Mathematicians have developed bases for image data that tend to yield sparse decompositions, which in turn yield good heuristics for image recovery. These are just heuristics, however, and do not guarantee accuracy. We investigated a hypothesis concerning the new Ridgelet basis, which was pioneered by Emmanuel Candes; there exists a low threshold (about 1% for a 128 128 image) for which we can guarantee perfect reconstruction of an image consisting of a single basis element as long as of the pixels are visible. This is signican t because it extends to a related result concerning images that are superpositions of several basis elements. Many geometric images can be closely approximated by superimposing relatively few elements, so this means that they can be accurately reconstructed even if most of the pixels are missing.