Ultrasound (US) beamforming is the process of reconstructing an image from acquired echo traces on several transducer elements. Typical beamforming approaches, such as Delay-And-Sum, perform simple projection operations, while techniques using statistical information also exist, e.g. adaptive, phase-coherence, Delay-Multiply-And-Sum, and sparse coding approaches. Inspired by the feasibility and success of inverse problem formulations in several image reconstruction problems, such as computed tomography, we herein devise an inverse problem approach for US beamforming. We define a linear forward model for the synthesis of the beamformed image, and solve its inverse problem thanks to several intuitive and physics-based constraints and regularization terms proposed. These reflect the prior knowledge about the spectra of carrier signal and spatial coherence of modulated signal. These constraints admit effective formulation through sparse representations. Our proposed method was evaluated for plane-wave imaging (PWI) that transmits unfocused waves, enabling high frame-rates with large field of view at the expense of much lower image quality with conventional beamforming techniques. Results are evaluated in numerical simulations, as well as tissue-mimicking phantoms and in-vivo data provided by Plane-wave Imaging Challenge in Medical UltraSound (PICMUS). The best results achieved by our proposed techniques are 0.39 mm full-width at half-maximum for spatial resolution and 16.3 dB contrast-to-noise ratio, using a single plane-wave transmit.