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Wavelets are used in signal processing. A high-resolution digital image is composed of a huge number of tiny pixels a pixel is the smallest controllable element of a picture represented on a screen , each of them a small square of one constant colour or grey level if the image is not in colour.
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Numbers can quantify the exact colour or grey level of the pixels, so that the image corresponds to an enormous array of numbers. Yet most pixels in an image are very similar to their neighbours; although of course there are also many places in the image wherever something interesting happens where a pixel differs appreciably from at least some if the neighbouring pixels, these locations are still in the minority.
A wavelet decomposition of an image exploits that aspect.
Wavelets decompose the image into building blocks of different scale that, together, describe what's going on in the image. Simplifying a lot, this approach tells you where you need to put lots of detail because pixels differ a lot from their neighbours and where not, in image analysis. What exactly do you do in the application of image analysis to art conservation?
French Mathematician Yves Meyer Wins Top Prize for 'Wavelet Theory' | Live Science
We analyse old paintings with imaging techniques, to provide artists and people who restore old paintings with tools to understand the nature of the masterpiece from yet another point of view than what they are already using. The first step is talking to people to find out what kind of questions they have. The interesting thing is that we often do not have ready-made solutions for them, so their questions also pose interesting challenges for us.
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We recently applied digital image-processing to the famous Ghent Altarpiece, a polyptych located in the Saint Bavo Cathedral in Ghent, Belgium, which is composed of 12 panels, eight of which are hinged shutters. Dating back to the 14th century, it is considered not only a masterpiece, but also a key element in understanding the art history of that period in Western Europe. Like most paintings of its age and materials, it has many small cracks or breaks in the paint layer, caused by differences in the way the paint layers and the underlying wood support react to temperature and humidity changes.
We carry out virtual restorations of high-resolution digital versions of paintings by automatically detecting and removing cracks, which turns out to be remarkable challenging. In the case of the Ghent Altarpiece we combined three detection methods to make a map of all the cracks. Then we virtually inpainted these cracks, thus reconstructing a sharper view of, in this particular case, the letters in a medieval book depicted in the background on one of the panels.
This made it possible for paleographers to decipher many more words than the 2 they could read before; as a result they can now unambiguously identify not only that the painter referred to a particular text by Thomas of Aquinas about the Annunciation, which turns out to be especially appropriate for the panel in question. Basically everywhere.
In other projects connecting image analysis and art restoration, we develop techniques that will relieve art conservators from some of their more tedious tasks; after we develop the algorithms, we passed them on to other software experts, to turn them into user-friendly packages for professional art renovators. This has led to new interdisciplinary collaborations that are giving interesting results. Is maths an important subject for scientists from developing countries?
What is your experience on this?
Mathematics is very popular in developing countries — it has a great appeal because it is so neat — you literally solve problems and build approaches by just the power of thought. What it is sometimes surprising to me is that people prefer theoretical to applied mathematics — I think it is best to always develop both: applied mathematics helps in building strong STEM education, and will benefit other sciences and engineering; theoretical mathematics is the essential foundation, on which everything is built and that makes it possible to extend ideas useful in one direction to apparently completely different frameworks.
The good news is that mathematics is spreading thanks to the Internet and to the attitude that mathematicians have to make their work available about a quarter of all papers are now available on Internet even before they are accepted for publication. I think this is great because it makes literature accessible to all. Also, maths is cheaper to do than other sciences, as it needs less investment in terms of labs and equipment. As far as my experience with developing countries, I lived in Madagascar for three months, which is not very long, but nevertheless was an eye-opening experience.
I still have links with people there, and I'm trying to help the local department of mathematics to further develop. I think TWAS is wonderful in what it does for developing countries, especially the fellowships programme. Skip to main content.
Varied life and interests
Maths is also for women. A common prejudice holds that women can't match the strength of men in mathematics. As the IMU president you must be in a powerful position. How does it feel? Did you develop this passion for mathematics early in your life?
Can you make an example? Where does maths come in, in this case?
What is your impression of TWAS and its role in the developing world? Ingrid Daubechies. Ordering on the AMS Bookstore is limited to individuals for personal use only. Advanced search. Author s Product display : Martin J. Abstract: These notes introduce the central concepts surrounding wavelets and their applications.
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