Super-resolution land cover mapping with indicator geostatistics

Abstract

Many satellite images have a coarser spatial resolution than the extent of land cover patterns on the ground, leading to mixed pixels whose composite spectral response consists of responses from multiple land cover classes. Spectral unmixing procedures only determine the fractions of such classes within a coarse pixel without locating them in space. Super-resolution or sub-pixel mapping aims at providing a fine resolution map of class labels, one that displays realistic spatial structure (without artifact discontinuities) and reproduces the coarse resolution fractions. In this paper, existing approaches for super-resolution mapping are placed within an inverse problem framework, and a geostatistical method is proposed for generating alternative synthetic land cover maps at the fine (target) spatial resolution; these super-resolution realizations are consistent with all the information available.

More precisely, indicator coKriging is used to approximate the probability that a pixel at the fine spatial resolution belongs to a particular class, given the coarse resolution fractions and (if available) a sparse set of class labels at some informed fine pixels. Such Kriging-derived probabilities are used in sequential indicator simulation to generate synthetic maps of class labels at the fine resolution pixels. This non-iterative and fast simulation procedure yields alternative super-resolution land cover maps that reproduce: (i) the observed coarse fractions, (ii) the fine resolution class labels that might be available, and (iii) the prior structural information encapsulated in a set of indicator variogram models at the fine resolution. A case study is provided to illustrate the proposed methodology using Landsat TM data from SE China.

Introduction

Sensors with spatial resolution larger than the extent of classes on the ground yield mixed pixels, i.e., pixels whose spectral signature is a composite of signatures of different classes. Spectral unmixing is the procedure of determining the fractions of such classes that occupy any coarse pixel; see, for example, Richards and Jia (1999) or Tso and Mather (2001) for a survey of unmixing methods. Spectral unmixing, however, provides only class fractions without locating the constituent classes, or for that matter the end members used for unmixing, within any coarse pixel. That additional task lies in the realm of super-resolution mapping, also termed sub-pixel mapping, or downscaling; see, for example, Atkinson (2001).

In this paper, we view super-resolution mapping from an inverse problem perspective (Bertero and Boccacci, 1998, Menke, 1989, Tarantola, 2005): that of reconstructing a fine resolution map of class labels from a set of coarse class fractions. The forward problem of computing coarse fractions from a fine resolution map of class labels is trivial. The inverse problem, however, is under-determined, in that it has multiple plausible solutions: many fine resolution class maps can lead to an equally good reproduction of the available coarse fractions. In order to solve such an under-determined inverse problem, one needs to invoke prior information that will resolve the inherent ambiguity. This prior information should pertain to the fine (target) spatial resolution, so that it constrains the “space” of possible spatial patterns of classes that can occur at that resolution. In what follows, we will occasionally refer to this fine resolution prior information as a model of spatial structure, structural model, or structural information, which, in this work, can also be loosely regarded as pertaining to texture (Tso & Mather, 2001).

The most primitive form of spatial structure is the rather unrealistic assumption of classes randomly distributed at the fine spatial resolution. Another particular form of prior information is the assumption of maximum class auto-correlation at the target resolution, which underpins several approaches for super-resolution mapping (Atkinson, 2001, Atkinson et al., 1997, Mertens et al., 2003, Tatem et al., 2001, Verhoeye and De Wulf, 2001). This prior structural model might be appropriate when the extent of spatial patterns on the ground is much larger than that of the coarse pixel, a scenario termed H-resolution by Jupp et al. (1988). Such a model, however, is too rigid, in that it cannot be adapted to generic scenes.

Moving away from the rigid notion of maximum spatial auto-correlation, prior information has been specified implicitly or explicitly in the form of parametric indicator variogram models. Atkinson (2001) and Makido and Shortridge (2005) used an iterative class swapping procedure to generate super-resolution maps, whereby a parametric function of distance (e.g., exponential decay) was used as a proxy for a formal indicator variogram model to determine the benefit of changing a simulated class label at any particular pixel in each iteration. Tatem et al., 2002, Tatem et al., 2003 explicitly used indicator variogram models as objective function components within an iterative optimization framework. Prior information has also been specified in the form of interactions between predefined groups of pixels (cliques) linked to the parametric energy function of a Markov Random Field model (Kasetkasem et al., 2005, Tso and Mather, 2001). Alternatively, that prior information could be extracted from analog images, by computing directly without any parametrization the probability of occurrence of different spatial patterns, i.e., sets of class labels over groups of pixels (Strebelle, 2002). This latter approach to prior information specification can account for complex spatial patterns (e.g., meandering objects) that cannot be adequately characterized by two-point statistics, such as indicator variograms.

In this work, we assume knowledge of the coarse resolution fractions; we do not address how such fractions were derived from the original satellite reflectance values. Moreover, we assume that these fractions are exact measurements with no error or inherent uncertainty; the possibility of relaxing this assumption is not pursued here due to space limitations. In addition, and contrary to many existing approaches to super-resolution mapping, we also consider the case whereby the analyst has access to a typically small set of class labels at the fine resolution, possibly obtained via ground surveys. Such sparse, with respect to the abundant coarse fractions, fine resolution data provide information on the actual location of classes at the target resolution, and hence should be reproduced exactly in the final maps.

After a prior model of spatial structure has been specified, a probabilistic formulation of inverse problems seeks to determine the conditional probability distribution of the unknowns given the available data; that probability distribution encapsulates our uncertainty about the unknown attribute values given the current level of information (Tarantola, 2005). In the context of super-resolution mapping, this amounts to determining the multivariate or multi-pixel conditional probability of obtaining any particular spatial combination of class labels at the target resolution, given the abundant coarse fractions and possibly some sparsely sampled class labels at that target resolution. That conditional distribution is typically complex with multiple modes and may not be analytically tractable. Instead of determining a single summary measure from the conditional distribution, such as its mean or mode, that distribution is “explored” by generating multiple samples or realizations from it (Kaipio and Somersalo, 2004, Mosegaard and Tarantola, 1995, Sambridge and Mosegaard, 2002, Tarantola, 2005). In our setting, this amounts to generating alternative simulated super-resolution maps of class labels that are consistent with all the information available; that is, the prior structural model, the coarse fractions, and the fine class labels if available. These simulated super-resolution maps can then be used to determine the likelihood of occurrence of patterns of classes over various groups or templates of pixels, by calculating the frequency of occurrence of such class patterns over the realizations.

More importantly, by using these synthetic super-resolution maps as inputs to a process simulation model, e.g., in a wildfire propagation simulator, one can build a probability distribution for the model outputs in a Monte Carlo framework (Bachmann & Allgöwer, 2002). By fixing other input variables to a nominal value or set of values, one could also assess via Monte Carlo simulation the sensitivity of that model to an uncertain land cover map; see Crosetto et al. (2001) for a discussion of uncertainty and sensitivity analysis techniques in a remote sensing context. In our case, that sensitivity would pertain to the lack of class labels at the appropriate model resolution.

One of the earliest approaches to super-resolution land cover mapping is that of Verhoeye and De Wulf (2001), who proposed a deterministic solution based on linear programming. This approach, however, does not acknowledge the existence of multiple super-resolution maps due to multiple optima in a linear programming formulation. In general, most existing algorithms for super-resolution land cover mapping are iterative in nature, and (rightfully so) yield different such maps depending on the initial map used in the iteration procedure. Mertens et al. (2003) adopted the same objective function as Verhoeye and De Wulf (2001), but used a genetic algorithm to search for plausible super-resolution maps. Tatem et al. (2001) trained a Hopfield neural network to optimize an initial super-resolution map (used for further iterations) with the simultaneous objectives of coarse fraction reproduction and spatial auto-correlation maximization; that method was successfully tested on an actual case study (Tatem et al., 2003). Mertens et al. (2004) used wavelets to account for the resolution difference between fine class labels and coarse fractions. At the fine scale, a neural network was trained to estimate the wavelet coefficients, from which a super-resolution map was reconstructed. Tatem et al. (2002) extended their neural network approach to account for indicator variogram models. Atkinson (2001) and Makido and Shortridge (2005) adopted a swapping algorithm, as used in spatial simulated annealing, to construct plausible super-resolution maps. In these latter works, the coarse fractions were matched exactly by construction. This was achieved by applying the swapping algorithm to an initial purely random super-resolution map comprised of the correct class fractions within each coarse pixel.

The common concern with the above iterative approaches is their rate of convergence and computational burden, associated with the repetitive evaluation of mismatch between simulated and observed coarse fractions, and most importantly between simulated and expected spatial structure. Other more complex sampling methods, such as Markov Chain Monte Carlo methods and simulated annealing are also iterative, and can become computationally prohibitive due to slow convergence; see, for example, Kaipio and Somersalo (2004). In addition, none of the existing approaches for super-resolution mapping accounts for fine resolution data in the form of a sparse set of class labels at informed fine pixels. The recently developed probability perturbation method of Caers and Hoffman (2006), although still iterative, appears to be less computational expensive, and warrants further attention in the context of super-resolution mapping.

In this paper, we propose a novel approach for super-resolution land cover mapping based on the geostatistical methods of indicator Kriging (Journel, 1983) and indicator stochastic simulation (Journel & Alabert, 1989), accounting explicitly for the resolution difference between the available coarse fractions and the sought-after class labels. The proposed approach: (i) is non-iterative and computationally inexpensive, (ii) offers exact, within round-off errors, reproduction of coarse resolution fractions, (iii) ensures exact reproduction of observed class labels at informed fine resolution pixels that might be available, and (iv) closely reproduces a set of indicator variogram models linked to transition probabilities of class labels from one fine pixel to another. Since we explicitly acknowledge that there are multiple solutions to super-resolution land cover mapping, the end product of our method is a set of alternative realizations or maps of class labels having the properties listed above.

Section 2.1 describes the links between the spatial statistics of the fine resolution class labels and the coarse resolution class fractions. These links are exploited in Section 2.2 to derive, via indicator coKriging, the conditional probability of class occurrence at any fine resolution pixel, given the neighboring coarse fractions and possibly some fine resolution sample class labels. In Section 2.3, the coKriging formulation is used within a modified sequential indicator simulation framework to generate alternative realizations of fine resolution class labels with the properties listed above. A case study is provided in Section 3, illustrating the applicability of our proposed methodology for super-resolution land cover mapping using data from a Landsat TM scene over SE China. Lastly, we offer some discussion and recommendations for future work in Section 4.