Analysis of geometrical distortion and statistical variance in length, area, and volume in a linearly scanned 3-D ultrasound image

A linearly scanned three-dimensional (3-D) ultrasound imaging system is considered. The transducer array is initially oriented along the a: axis and aimed in the y direction. After being tilted by an angle theta about the x axis, and then swiveled by an angle phi about the y axis, it is translated in the z direction, in steps of size d, to acquire a series of parallel two-dimensional (2-D) images. From these, the 3-D image is reconstructed, using the nominal values of the parameters (phi, theta, d). Thus, any systematic or random errors in these, relative to their actual values (phi(0), theta(0), d(0)), will respectively cause distortions or variances in length, area, and volume in the reconstructed 3-D image, relative to the 3-D object. Here, we analyze these effects, Compact linear approximations are derived for the relative distortions as functions of the parameter errors, and hence, for the relative variances as functions of the parameter variances. Also, exact matrix formulas for the relative distortions are derived for arbitrary values of (phi, theta, d) and (phi(0), theta(0) d(0)). These were numerically compared to the linear approximations and to measurements from simulated 3-D images of a cubical object and real 3-D images of a wire phantom. In every case tested, the theory was confirmed within experimental error (0.5%).