Determination of x‐ray spectra from attenuation data by imposing a priori positiveness and bounded support: Theory and experimental validation

Abstract

The determination of the spectral distribution of an x-ray beam from attenuation measurements in a narrow beam has aroused great interest since it was first proposed by Silberstein in 1932. Since then, it has become clear that this is an ill-conditioned problem. Then, due to the intrinsic difficulties of the problem, care has been taken to improve the mathematical and numerical inversion techniques. Alternative ways to circumvent the explicit inversion by using the attenuation curve itself to calculate directly generalized moments of the solution have also been found. In this work, the explicit reconstruction of the spectral distribution is carried out by an expectation-maximization method that satisfies the a priori condition of positiveness of the solution and allows us to impose the boundness of its support easily, conditions which are known to be fulfilled by physically acceptable solutions. The simulations made and the experimental validation of the proposed method allow us to prove its scope and limitations.

A MOSFET-Based Method for Measuring Peak Kilovoltage (kVp) in Diagnostic X-Ray Beams

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Determinación del espectro de energía de un haz de rayos X terapéutico de kilovoltaje a partir de su curva de atenuación

“…Como alternativa, varios autores han reportado métodos indirectos de determinación del espectro de energía de un haz de rayos X a partir de la curva de atenuación del haz (Dössel & Schlegel, 2010) y del uso de transformadas de Laplace (Delgado, 1999;Delgado, 2007;Mainardi & Bonzi, 2008).…”

Application of an expectation maximization method to the reconstruction of X-ray-tube spectra from transmission data

“…Information about this distribution is recovered by using a series of attenuation measurements in which the spectrum is progressively modified by varying the filtration. A number of different approaches have been proposed for the solution of this problem, including the use of the Laplace transform, least squares methods, singular value decomposition and iterative approaches [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In this work we focus on the application to experimental data of the expectation-maximization method [14,17] to iteratively solve the ill-conditioned linear system that relates the spectral distribution of the X-ray beam to its transmission.…”