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A new algorithm is proposed for reconstructing raw RF data into ultrasound images. Previous delay-and-sum beamforming reconstruction algorithms are essentially one- dimensional, because a sum is performed across all receiving elements. In contrast, the present approach is two-dimensional, potentially allowing any time point from any receiving element to contribute to any pixel location. Computer-intensive matrix inversions are performed once, in advance, to create a reconstruction matrix that can be reused indefinitely for a given probe and imaging geometry. Individual images are generated through a single matrix multiplication with the raw RF data, without any need for separate envelope detection or gridding steps. Raw RF data sets were acquired using a commercially available digital ultrasound engine for three imaging geometries: a 64-element array with a rectangular field-of-view (FOV), the same probe with a sector-shaped FOV, and a 128-element array with rectangular FOV. The acquired data were reconstructed using our proposed method and a delay-and-sum beamforming algorithm for comparison purposes. Point spread function (PSF) measurements from metal wires in a water bath showed that the proposed method was able to reduce the size of the PSF and its spatial integral by about 20 to 38%. Images from a commercially available quality-assurance phantom had greater spatial resolution and contrast when reconstructed with the proposed approach.