Two-dimensional phase unwrapping using robust derivative estimation and adaptive integration

The adaptive integration (ADI) method for two-dimensional (2-D) phase unwrapping is presented. The method uses an algorithm for noise robust estimation of partial derivatives, followed by a noise robust adaptive integration process. The ADI method can easily unwrap phase images with moderate noise levels, and the resulting images are congruent modulo 2pi with the observed, wrapped, input images. In a quantitative evaluation, both the ADI and the BLS (Strand et al.) methods were better than the least-squares methods of Ghiglia and Romero (GR), and of Marroquin and Rivera (MRM). In a qualitative evaluation, the ADI, the BLS, and a conjugate gradient version of the MRM method (MRMCG), were all compared using a synthetic image with shear, using 115 magnetic resonance images, and using 22 fiber-optic interferometry images. For the synthetic image and the interferometry images, the ADI method gave consistently visually better results than the other methods. For the MR images, the MRMCG method was best, and the ADI method second best. The ADI method was less sensitive to the mask definition and the block size than the BLS method, and successfully unwrapped images with shears that were not marked in the masks. The computational requirements of the ADI method for images of nonrectangular objects were comparable to only two iterations of many least-squares-based methods (e.g., GR). We believe the ADI method provides a powerful addition to the ensemble of tools available for 2-D phase unwrapping.