Virtually every area of ultrasonic imaging research requires accurate estimation of the spatiotemporal impulse response of the instrument, and yet accurate measurements are difficult to achieve. The impulse response can also be difficult to predict numerically for a specific device because small unknown perturbations in array properties can generate significant changes in predicted pulse-echo field patterns. A typical measurement for a 1-D array transducer employs a line scatterer oriented perpendicular to the scan plane. Echoes from line scatterers located throughout the field of view constitute estimates of shift-varying line response functions. We propose an inverse-problem approach to the reconstruction of point-spread functions from line-spread functions. A collection of echoes recorded for a range of line-scatterer rotation angles are treated as projections of sound pressure onto the transducer array surface. Although the reconstruction is mathematically equivalent to filtered backprojection, it provides significant advantages with respect to interpolation that confound straightforward implementations. Field II predictions used to model measurements made on commercial systems suggest the reconstruction accuracy is with 0.32% for noiseless echo data. Application of the method to data acquired from a commercial system are evaluated from the perspective of deconvolution.