Multidimensional inverse problems in ultrasonic imaging

This chapter describes the reconstruction of reflectivity profiles from ultrasonic back-scattered echoes as an inverse problem involving incomplete and noisy measurements. Given pulse-echo measurements from a synthetically created one-dimensional or two-dimensional array of sensors, the chapter highlights that the maximum likelihood estimate of the reflectivity profile requires the pseudoinverse of the array point spread function (PSF). As a consequence of the physical properties, the point spread function is shown to be highly structured block Toeplitz linear operator and is a function of the sensor and array characteristics. The chapter reviews the literature in the field of ultrasonic digital signal processing and image reconstruction. It provides an imaging model based on the physical structure of the imaging system and the discretised problem is stated in terms of a linear discrete multidimensional convolution. The inverse problem is posed and issues such as ill-posedness, ill-conditioned systems and regularization are treated in a practical way. © 1995, Elsevier Inc.