Radontranszformáció

Radontranszformáció refers to a mathematical operation that transforms a function defined on a space into another function defined on a related space. In its most common form, it involves integrating a function over a family of curves or surfaces. The Radon transform is a fundamental tool in various scientific fields, particularly in image reconstruction and inverse problems.

The concept was first introduced by Johann Radon in 1917. In two dimensions, the standard Radon transform of a function f(x, y) takes a line in the xy-plane and maps it to a point in a parameter space. Specifically, if a line is defined by its distance p from the origin and its angle $\theta$ with the x-axis, the Radon transform $Rf(\rho, \theta)$ is the integral of f(x, y) along that line. This can be expressed as $Rf(\rho, \theta) = \int_{-\infty}^{\infty} f(x, y) ds$, where ds is the element of arc length along the line.

The inverse Radon transform aims to reconstruct the original function f(x, y) from its transformed version $Rf(\rho, \theta)$. This is crucial in applications like computed tomography (CT) scans, where X-ray projections taken from various angles are used to reconstruct a cross-sectional image of the body. Other applications include seismic tomography, where seismic wave travel times are used to image the Earth's interior, and particle physics, where reconstructive techniques are employed to understand particle interactions. The mathematical properties of the Radon transform, including its invertibility and various stability results, are active areas of research.