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Reconstruction of missing data in remote sensing images using conditional stochastic optimization with global geometric constraints

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Abstract

This paper addresses the issue of missing data reconstruction for partially sampled, two-dimensional, rectangular grid images of differentiable random fields. We introduce a stochastic gradient–curvature (GC) reconstruction method, which is based on the concept of a random field model defined by means of local interactions (constraints). The GC reconstruction method aims to match the gradient and curvature constraints for the entire grid with those of the sample using conditional Monte Carlo simulations that honor the sample values. The GC reconstruction method does not assume a parametric form for the underlying probability distribution of the data. It is also computationally efficient and requires minimal user input, properties that make it suitable for automated processing of large data sets (e.g. remotely sensed images). The GC reconstruction performance is compared with established classification and interpolation methods for both synthetic and real world data. The impact of various factors such as domain size, degree of thinning, discretization, initialization, correlation properties, and noise on GC reconstruction performance are investigated by means of simulated random field realizations. An assessment of GC reconstruction performance on real data is conducted by removing randomly selected and contiguous groups of points from satellite rainfall data and an image of the lunar surface.

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Notes

  1. Ergodicity implies that the spatial average of an observable is equal to the respective ensemble average. In contrast, we assume that the spatial average over the entire grid equals the sample average. For example, for a grid with length comparable to the random field correlation length, ergodicity is not satisfied. Nevertheless, the gradient and curvature constraints over the grid have definite values (not necessarily identical to the ensemble values); our assumption is that the sample drawn from the grid gives accurate estimates of the constraints over the grid.

  2. By interacting grid nodes we imply the pairs and triplets of adjacent nodes that contribute to the gradient and curvature constraints given by (2) and (3), respectively.

  3. A circular stencil would correspond to the Euclidean distance based KNN classification algorithm, k being the number of sampling points inside the stencil.

  4. Other definitions of misclassification rate can also be used based on smoother distance functions between class values.

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Acknowledgments

This research project has been supported in part by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme under contract number MTKD-CT-2004-014135. We also acknowledge support for a short visit by Prof. Žukovič at TUC that helped to finalize this paper from the Hellenic Ministry of Education—Department of Inter-University Relations, the State Scholarships Foundation of Greece and the Slovak Republic’s Ministry of Education through the Bilateral Programme of Educational Exchanges between Greece and Slovakia. We are grateful to Prof. M. Kanevski (Université de Lausanne, Switzerland) for providing us with the full version of the GeoSVM software. Rainfall data used in this study were produced with the Giovanni online data system, developed and maintained by the NASA Goddard Earth Sciences (GES) Data and Information Services Center (DISC). We also acknowledge the TRMM mission scientists and associated NASA personnel for the production of the rainfall data used in this research effort.

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Authors and Affiliations

  1. Institute of Physics, Faculty of Science, Pavol Jozef Šafárik University, Park Angelinum 9, Košice, 041 54, Slovakia

    Milan Žukovič

  2. SORS Research a.s., Moyzesova 38, Košice, 040 01, Slovakia

    Milan Žukovič

  3. Geostatistics Research Unit, Department of Mineral Resources Engineering, Technical University of Crete, Chania, 73100, Greece

    Dionissios T. Hristopulos

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  1. Milan Žukovič
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Correspondence to Dionissios T. Hristopulos.

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Žukovič, M., Hristopulos, D.T. Reconstruction of missing data in remote sensing images using conditional stochastic optimization with global geometric constraints. Stoch Environ Res Risk Assess 27, 785–806 (2013). https://doi.org/10.1007/s00477-012-0618-5

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