A genetic algorithm for phase unwrapping errors correction

Differential Synthetic Aperture Radar Interferometry (DInSAR) is a remote sensing technique with important applications for geological and entropic processes studies by benefiting of a very spatially dense ground displacement maps with centimeter to millimeter accuracy. In particular, starting from an appropriate sequence of DInSAR interferograms, it is possible to study the temporal evolution of the detected displacements through advanced approaches. An effective way to investigate this temporal behaviour is via the generation of deformation time-series. In this context, one of the most difficult task for the retrieval of displacement information is represented by the Phase Unwrapping (PhU) step, especially in presence of fast deformation phenomena or in area affected by decorrelation phenomena [1]. In this paper we present an innovative technique to correct the phase unwrapping errors in a generic deformation time-series retrieval process based on a guided scanning by means of a genetic algorithm (GA) [2]. This algorithm permits to manage sets of integers, representing the multiples of 2 pi that have to be added to the wrapped phase value, resulting in a solution coherent with the discrete nature of the phase unwrapping itself. Once the unwrapped interferograms are inverted to generate the corresponding displacement time series, then the wrapped interferograms can be easily reconstructed from the time series itself. The phase difference between the wrapped interferograms and the reconstructed ones is higher for interferograms where a phase unwrapping error has occurred. The presented algorithm considers a first inversion step based on the L1-norm minimization [3], which guarantees a more sharp and accurate identification of the interferograms that may need a correction with respect to the most common L1-norm inversion. For every single considered pixel, the algorithm compares the unwrapped interferograms reconstructed from the L1-norm inversion with the starting phase unwrapping solution. When the difference between these two solutions is higher than a fixed threshold, the corresponding interferogram is considered in the correction procedure as a gene of the GA chromosome. The GA creates a large number of chromosomes, starting from the first one evaluated from the L1 inversion taken as reference. The rest of the chromosomes are creating by mutations of single genes and crossing over among two parent chromosomes. The former is based on a Poissonian distribution, since the allowed values of the genes are integers to be considered as 2pi multiples for the phase unwrapping. Every generation ranks and saves the most performing chromosomes from whom a new generation is created until convergence. Convergence is reached when a chromosome keep on staying in the first position of the rank for a certain number of generations, whose value depends on the chromosome length[4]. The objective function for the rank of the GA procedure is given by the temporal coherence [5] of the pixel, which is evaluated after the L2-norm inversion of the new unwrapped solution of the GA procedure. We realised that a pixel with an already low value of temporal coherence, which entails an high presence of noise and PhU errors, is not so sensitive to be corrected, or, in a worse case, can be corrected by GA procedure toward an algebraic solution with no physical meaning. So we decided to apply the GA only to pixel with a temporal coherence higher than a fixed threshold. In other words we consider pixels where the number of wrong unwrapped solutions is not so high. In principles, the presented algorithm can be applied to more generic procedure of unwrapping, once the fitting function for the algorithm is identified. In our case, we apply the correction technique to a solution obtained by the Small-Baseline Subset (SBAS) DInSAR which retrieve the deformation time series by relying on a least squares inversion (so based on a L2 norm) of an overdetermined system of equations representing the interferometric pairs involved in the processing [5]. The phase unwrapping is carried out through the well known EMCF (Extended Minimum Cost Flow) approach [6], based on a temporal and spatial phase unwrapping procedure. The above mentioned correction is applied at the end of the whole phase unwrapping procedure. The parameters of the GA have been tuned according with the SBAS DinSAR technique, after a long testing phase, which included also old reconstructions and simulated data. We recommend a redefinitions of the parameters for the application of the GA to different cases. The results we obtained show that in case of strong deformations the algorithm maximise the temporal coherence resulting in an increasing of the deformation dynamics, according with the known effect of underestimation risk for the phase unwrapping algorithms [7]. Another important result is given by the preservation of spatial consistence of the corrections estimated by GA procedure, according with the phase unwrapping error behaviour. It is important to highlight that the space consistency is not intrinsically required by the GA procedure, but it emerges as a consequence. To conclude, we observe that, comparing our correction technique with respect to a standard L1-norm corrected solution for retrieving phase unwrapping errors, the algorithm reduce the false alarm probability from 8% to 2%, based on simulated data. In addition, the developed algorithm has been build up with a parallelization strategy characterized by a very high granularity level (pixel level) thus entailing an high scalability of the algorithm itself and corresponding reduced processing time. References [1] H. A. Zebker and J. Villasenor. Decorrelation in interferometric radar echoes, IEEE Transactions on Geoscience and Remote Sensing, vol 30, issue 5, pp. 950-959, September 1992. [2] Mitchell, Melanie. An Introduction to Genetic Algorithms, Cambridge, MA: MIT Press, 1996. ISBN 9780585030944. [3] Qin Lyu Zhouchen Lin, Yiyuan She, and Chao Zhang. A comparison of typical lp minimization algorithms, Neurocomputing, vol. 119, pp. 413-424, 2013 [4] Holland, John. Adaptation in Natural and Artificial Systems. Cambridge, MA: MIT Press. ISBN 978-0262581110. [5] P. Berardino, G. Fornaro, R. Lanari, and E. Sansosti. A new algorithm for surface deformation monitoring based on small baseline differential car interferograms, IEEE Trans. Geosci. Remote Sent., vol. 40, no. 11, pp.2375-2383, 2002 [6] A. Pepe, R. Lanari. On the Extension of the Minimum Cost Flow Algorithm for Phase Unwrapping of Multitemporal Differential SAR Interferograms, IEEE Trans. Geosci. Remote Sent., vol. 44, no. 9, pp.2374-2383, 2006 [7] D. Just, R. Bamler. Phase statistics of interferograms with applications to synthetic aperture radar, Applied Optics, Vol. 33, Issue 20, pp. 4361-4368, 1994