X-ray scatter results in a significant degradation of image quality in computed tomography (CT), representing a major limitation in cone-beam CT (CBCT) and large field-of-view diagnostic scanners. In this work, a novel scatter estimation and correction technique is proposed that utilizes peripheral detection of scatter during the patient scan to simultaneously acquire image and patient-specific scatter information in a single scan, and in conjunction with a proposed compressed sensing scatter recovery technique to reconstruct and correct for the patient-specific scatter in the projection space. The method consists of the detection of patient scatter at the edges of the field of view (FOV) followed by measurement based compressed sensing recovery of the scatter through-out the projection space. In the prototype implementation, the kV x-ray source of the Varian TrueBeam OBI system was blocked at the edges of the projection FOV, and the image detector in the corresponding blocked region was used for scatter detection. The design enables image data acquisition of the projection data on the unblocked central region of and scatter data at the blocked boundary regions. For the initial scatter estimation on the central FOV, a prior consisting of a hybrid scatter model that combines the scatter interpolation method and scatter convolution model is estimated using the acquired scatter distribution on boundary region. With the hybrid scatter estimation model, compressed sensing optimization is performed to generate the scatter map by penalizing the L1 norm of the discrete cosine transform of scatter signal. The estimated scatter is subtracted from the projection data by soft-tuning, and the scatter-corrected CBCT volume is obtained by the conventional Feldkamp-Davis-Kress algorithm. Experimental studies using image quality and anthropomorphic phantoms on a Varian TrueBeam system were carried out to evaluate the performance of the proposed scheme. The scatter shading artifacts were markedly suppressed in the reconstructed images using the proposed method. On the Catphan©504 phantom, the proposed method reduced the error of CT number to 13 Hounsfield units, 10% of that without scatter correction, and increased the image contrast by a factor of 2 in high-contrast regions. On the anthropomorphic phantom, the spatial nonuniformity decreased from 10.8% to 6.8% after correction. A novel scatter correction method, enabling unobstructed acquisition of the high frequency image data and concurrent detection of the patient-specific low frequency scatter data at the edges of the FOV, is proposed and validated in this work. Relative to blocker based techniques, rather than obstructing the central portion of the FOV which degrades and limits the image reconstruction, compressed sensing is used to solve for the scatter from detection of scatter at the periphery of the FOV, enabling for the highest quality reconstruction in the central region and robust patient-specific scatter correction.