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We present a theoretically-exact and stable computed tomography (CT) reconstruction algorithm that is capable of handling interrupted illumination and therefore of using all measured data at arbitrary pitch. This algorithm is based on a differentiated backprojection (DBP) on M-lines. First, we discuss the problem of interrupted illumination and how it affects the DBP. Then we show that it is possible to take advantage of some properties of the DBP to compensate for the effects of interrupted illumination in a mathematically exact way. From there, we have developed an efficient algorithm which we have successfully implemented. We show encouraging preliminary results using both computer-simulated data and real data. Our results show that our method is capable of achieving a substantial reduction of image noise when decreasing the helix pitch compared with the maximum pitch case. We conclude that the proposed algorithm defines for the first time a theoretically-exact and stable reconstruction method that is capable of beneficially using all measured data at arbitrary pitch.