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X-ray scatter poses a significant limitation to image quality in cone-beam CT (CBCT), resulting in contrast reduction, image artifacts, and lack of CT number accuracy. We report the performance of a simple scatter correction method in which scatter fluence is estimated directly in each projection from pixel values near the edge of the detector behind the collimator leaves. The algorithm operates on the simple assumption that signal in the collimator shadow is attributable to x-ray scatter, and the 2D scatter fluence is estimated by interpolating between pixel values measured along the top and bottom edges of the detector behind the collimator leaves. The resulting scatter fluence estimate is subtracted from each projection to yield an estimate of the primary-only images for CBCT reconstruction. Performance was investigated in phantom experiments on an experimental CBCT bench-top, and the effect on image quality was demonstrated in patient images (head, abdomen, and pelvis sites) obtained on a preclinical system for CBCT-guided radiation therapy. The algorithm provides significant reduction in scatter artifacts without compromise in contrast-to-noise ratio (CNR). For example, in a head phantom, cupping artifact was essentially eliminated, CT number accuracy was restored to within 3%, and CNR (breast-to-water) was improved by up to 50%. Similarly in a body phantom, cupping artifact was reduced by at least a factor of 2 without loss in CNR. Patient images demonstrate significantly increased uniformity, accuracy, and contrast, with an overall improvement in image quality in all sites investigated. Qualitative evaluation illustrates that soft-tissue structures that are otherwise undetectable are clearly delineated in scatter-corrected reconstructions. Since scatter is estimated directly in each projection, the algorithm is robust with respect to system geometry, patient size and heterogeneity, patient motion, etc. Operating without prior information, analytical modeling, or Monte Carlo, the technique is easily incorporated as a preprocessing step in CBCT reconstruction to provide significant scatter reduction.