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The left image shows a usual X-ray: An X-ray field parallels are emitted from a uniform network on a plane. All rays start with the same intensity, and after passing through the subject, these intensities are reduced at different rates depending on the density of the material (muscle, bone and other tissues) by The rays pass, as a result, the plate with the sensitive material which contains the X-ray shows a very good approximation inside the subject, but that seems inadequate, for example when searching for an evil purpose that may be very deeply located between objects of greater density.

The second image is the result of computerized axial tomography (better known in the medical world by the acronym CAT) on the same subject. In this case, the parallel X-ray beam is located on a uniform network segment. This segment is rotated about an axis on the subject slowly until complete 360 \u200b\u200bdegrees, and each pause, take the intensities received after the broadcast. As a result, we get a picture square, each of its columns providing information on emissions at different angles. In the picture above is collected 180 breaks, and the parallel beam consisted of 100 bulbs.

But this image is not easy to interpret, is where one has to use the power of mathematics, more specifically techniques An & # 225; advanced mathematical analysis and numerical analysis addressing the reversal of the Radon transform. After a series of numerical calculations (based on algorithms that after more a century are still being perfected), the doctor will get the following picture: analysis of patients is much more ; s direct and accurate after viewing these images, which have become part of the usual techniques of treating diseases in Western medicine today .

But overall investment Radon transform is a very costly process, mainly by the volume of information accumulated-note that this operation must be repeated as many times as necessary down and up the front axle in order to obtain an accurate three-dimensional image inside the patient. Not all medical institutions can afford the maintenance adquisicióny CT equipment to obtain the best possible definition, and the scanners used in most health centers offer some approximation errors & # 243; n derived from these technological deficiencies. One possible solution to this problem was proposed in the early of 90. Instead of investing the Radon transform, we can proceed to use a linear combination operator "lambda" and its inverse, which are nothing more than the ra & # 237; z square Laplacian and Riesz transform. These operators offer the possibility of recovering, not already a picture of the inside of the subject, but an idea of \u200b\u200bthe location of the edges of the various parts that are divided each of the cuts. Sometimes, the latter leading to better analysis, visualization and understanding of the internal anatomy of the subjects, and has the added advantage of being much more stable your computer, and much more fa ; easy to store electronically, as they require less data.

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