In computed tomography, a whole scan of the object may be impossible, generally because the object is larger than the scanner field-of-view. Such a set up leads to truncated projections. Using differentiated backprojection, the reconstruction problem can be reduced to a set of one-dimensional problems consisting of the inversion of the Hilbert transform. When the object partly overlaps the scanner field-of-view, this problem is commonly referred to as the "one-sided truncated Hilbert transform". Our work investigates this situation and proposes a novel approach to address it. Using differentiated backprojection, and the object extent supposedly known a priori, a pseudo-inverse of the truncated Hilbert transform is computed by truncated singular value decomposition, and its truncated singular values are replaced by a simple estimation. The estimation is calculated using the singular value decomposition of the known convex hull filled with a constant value per line computed from the corresponding projection in the direction of the Hilbert transform. Experiments illustrate the image quality improvements resulting from this approach compared to a simple truncation of the singular values and the reconstruction speed improvement compared to two-dimensional iterative reconstruction solving penalized least squares with the conjugate gradient algorithm.