Mathematical approach to horizontal and vertical magnification factors in rotational panoramic radiography--with attention to redundant shadows

Rotational panoramic radiographs consist of two different images, the tomographic image and redundant shadows. Many authors have reported magnification factors for the tomographic image which include the term "real image", as proposed by McDavid et al. in 1983. However, a mathematical formula has never been proposed for ghost images, which redundant shadows projected only when the object is located between the center of rotation and the X-ray source. The purpose of this study was to propose a new mathematical approach for calculating the horizontal and the vertical magnification factors for ghost images. Using this approach, we investigated the effect of variations in the position of the object, its length, and its placement angle with respect to the X-ray beam. Since the object was assumed to be small, an approximated expression for film speed was integrated to determine the length of the object on the film in the horizontal dimension. The approximated and the geometric expressions proposed by Wakoh in 1988 were used to calculate the vertical dimension. Differences between the scanning and the film speeds were assumed to affect the horizontal magnification factors of the ghost image. In addition, differences in the setting position, length, and angle of the object affected the calculated values. Vertical magnification factors were considerably influenced by the Z coordinate of the position, even though the length and angle remained constant.