The problem of reconstructing signals from their noisy and randomly shifted observations is considered. Our approach is different from existing methods in the choice of slices in the bispectrum domain. Up to now only a single 45 deg slice or multiple parallel-to-axis slices have been used to recover signals. These algorithms either are prone to low signal-to-noise ratio (SNR) estimates or improve SNR of high frequency estimates only. We propose an algorithm that accumulates multiple bispectral radial slices and use the accumulation to recover both the Fourier phase and the magnitude of the signal. Major advantages of the new algorithm are the reduced estimation errors and improvement in SNR in the whole frequency band. A mathematical formulation of the algorithm is given and fast algorithms are presented along with other implementation issues. Performance is evaluated using both simulated data and real radar data. Experimental results show that the proposed algorithm compares favorably with existing methods in terms of mean squared reconstruction error and computational complexity.