In this paper, we present shift-invariant filtered backprojection (FBP) cone-beam image reconstruction algorithms for a cone-beam CT system based on a clinical C-arm gantry. The source trajectory consists of two concentric arcs which is complete in the sense that the Tuy data sufficiency condition is satisfied. This scanning geometry is referred to here as a CC geometry (each arc is shaped like the letter " C"). The challenge for image reconstruction for the CC geometry is that the image volume is not well populated by the familiar doubly measured (DM) lines. Thus, the well-known DM-line based image reconstruction schemes are not appropriate for the CC geometry. Our starting point is a general reconstruction formula developed by Pack and Noo which is not dependent on the existence of DM-lines. For a specific scanning geometry, the filtering lines must be carefully selected to satisfy the Pack-Noo condition for mathematically exact reconstruction. The new points in this paper are summarized here. (1) A mathematically exact cone-beam reconstruction algorithm was formulated for the CC geometry by utilizing the Pack-Noo image reconstruction scheme. One drawback of the developed exact algorithm is that it does not solve the long-object problem. (2) We developed an approximate image reconstruction algorithm by deforming the filtering lines so that the long object problem is solved while the reconstruction accuracy is maintained. (3) In addition to numerical phantom experiments to validate the developed image reconstruction algorithms, we also validate our algorithms using physical phantom experiments on a clinical C-arm system.