Ultrasonic Reflectivity Tomography: Reconstruction with Circular Transducer Arrays

An analysis is presented of backprojection methods for reconstructing cross-sectional images of ultrasonic reflectivity from scattering measurements. A circular array of transducer elements is considered, using three basic modes of data acquisition and image reconstruction: (1) the same element serves as transmitter and receiver and data is backprojected along circular paths centered at the element (2) distinct transmitter and receiver with fixed separation and backprojection along elliptical paths with the elements at the foci and (3) distinct transmitter and receiver with varying separations and backprojection along corresponding elliptical paths. The point spread function (PSF) for each of these cases is derived and is shown to depend on the shape of the acoustic pulse used. PSF's are evaluated for three different pulses — a narrowband, wideband, and an analytically-derived optimum pulse which yields the best sidelobe response and a mainlobe width equal to 0.3Λ_C, where Λ_C is the wavelength corresponding to the cutoff frequency of the pulse. When backprojection is performed along elliptical paths, the mainlobe width is shown to be broadened by a factor proportional to the cosine of half the angle subtending the transmit and receive elements at the center of the array. The close analogy between the techniques used here to reconstruct reflectivity and the convolution/backprojection method used in computerized x-ray tomography is discussed in detail. Salient properties predicted by the analytically-derived PSF's are confirmed in computer simulations. The characteristics of the PSF's are also examined as a function of the number of array elements, the location of a reflecting point in the object and the shape of the ultrasonic pulse.