Radon transform

The Radon transform is an integral transform of a function of two variables. There the integral of the function along all lines of the plane is determined. For each of these lines one can imagine the Radon transform of as a projection of the function on the perpendicular to this line. The Radon transform in two dimensions is a generalization of the Abel transform and the special case of the Hough transform.

The Radon transform is named after the Austrian mathematician Johann Radon. He led them in 1917 in the publication concerning the definition of functions by their integral values ​​along certain manifolds one.

Definition

Is continuous and outside a circle of finite radius is identical to zero, and a straight line which is defined by the angle to the y- axis and its distance r to the origin. Then, the Radon transform is given by the line integral of f ( x, y) along.

The straight line can be parameterized as. This can also write the line integral as

Inverse transformation

The inverse transformation can be performed using the filtered back-projection or indirectly via the Fourier transform.

The problem of inverse transformation is an ill-posed problem because the solution is not a continuous function of the input data. However, to solve the problem sufficiently precise regularization techniques or iterative methods can be applied.

Applying the Radon transform

In tomography, the integrals of a function are determined by straight lines and calculated by inverse Radon projection from images. For example, in computed tomography X-radiation absorption of the radiation along a straight line from the X-ray source to a detector, that is the integral of the absorbance is determined. The measurement is carried out for a large number of such straight lines in a plane ( many detectors and many positions of the x-ray source is moved around the patient ). It is the Radon transform of the X-ray absorption ( even if only for a finite number of values ​​of the two parameters). From these values ​​can win the two-dimensional image using the inverse transformation. The juxtaposition of several such two-dimensional " slice images " results in a three-dimensional image.