Downscaling in remote sensing

International Journal of Applied Earth Observation and Geoinformation 22 (2013) 106–114

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Downscaling in remote sensing Peter M. Atkinson Global Environmental Change and Earth Observation Research Group, Geography and Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK

a r t i c l e

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Article history: Received 1 November 2011 Accepted 18 April 2012 Keywords: Downscaling Super-resolution mapping Area-to-point prediction Area-to-point kriging

a b s t r a c t Downscaling has an important role to play in remote sensing. It allows prediction at a finer spatial resolution than that of the input imagery, based on either (i) assumptions or prior knowledge about the character of the target spatial variation coupled with spatial optimisation, (ii) spatial prediction through interpolation or (iii) direct information on the relation between spatial resolutions in the form of a regression model. Two classes of goal can be distinguished based on whether continua are predicted (through downscaling or area-to-point prediction) or categories are predicted (super-resolution mapping), in both cases from continuous input data. This paper reviews a range of techniques for both goals, focusing on area-to-point kriging and downscaling cokriging in the former case and spatial optimisation techniques and multiple point geostatistics in the latter case. Several issues are discussed including the information content of training data, including training images, the need for model-based uncertainty information to accompany downscaling predictions, and the fundamental limits on the representativeness of downscaling predictions. The paper ends with a look towards the grand challenge of downscaling in the context of time-series image stacks. The challenge here is to use all the available information to produce a downscaled series of images that is coherent between images and, thus, which helps to distinguish real changes (signal) from noise. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Downscaling refers to an increase in spatial resolution. Conversely, upscaling refers to a decrease or coarsening of spatial resolution (Atkinson and Tate, 2000). Both goals are required for comparison and integration of disparate datasets and for calibration and validation of models in a range of applications. In the context of remote sensing, downscaling refers to a decrease in the pixel size of remotely sensed images. This task, which amounts to an implicit assumption that the information content of the downscaled imagery will increase, has been attempted through a variety of different approaches and techniques. These approaches are reviewed in this paper. Within the overarching goal of downscaling, several sub-goals can be defined. These include area-to-point prediction, which involves predicting the same continuous variable as is input to the downscaling process, but with a finer spatial resolution than the input, and super-resolution mapping, a term coined to describe the process of downscaling whilst at the same time transforming the input variable to a categorical variable, usually representing land cover class. Both of these main sub-goals are described in this paper. Before reviewing the techniques used

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in these two broad sub-goals, the rationale and justification for downscaling are considered in the next section. 1.1. Why scaling? Scaling is important in a range of fields. In ecology and biogeography scaling is central to some of the core theories such as island biogeography (MacArthur and Wilson, 1967). Moreover, core concepts such as patches, ecosystems, populations and biomes are all scale-dependent definitions. Ecologists are particularly interested to understand how biodiversity scales between ˛-, ˇ- and -diversity. In climate science, downscaling has become commonplace in the production of regional-scale climate circulation models from global circulation models (GCMs) (e.g., Huth, 2002). The techniques used for this important endeavour are not always the most sophisticated, with serious implications for the quality of the results. For example, a common approach to forecasting future fine resolution scenarios is to take a present day spatially distributed scenario (e.g., rainfall), average it to a larger GCM cell, and adjust the distributed values by a factor equal to the difference between the present day average and some climate forecast for the same GCM cell. However, more sophisticated approaches are being developed (Deidda, 2000). Scaling has a long history of importance in geomorphology and hydrology. For example, Thornes (1973) examined the

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scale differences in rills and gullies. In both geomorphology and hydrology, downscaling is often used to prepare data for input to process models. Goodchild (2011) provides a review of scale issues in GIS within the context of geomorphology, focusing on both measurement concepts and process representations. Downscaling has also found application in population mapping. For example, several regression-type models have been developed to downscale the population from census enumeration areas to grids for the European Union (Gallego and Bamps, 2008) and globally (Hay et al., 2005; Balk et al., 2006; Tatem et al., 2011). Distributing population on grids facilitates integration and comparison with other gridded data sets, although uncertainty in the predictions remains a concern (Martin et al., 2011). Population downscaling approaches are described further below in relation to the modifiable areal unit problem (MAUP). Scaling has also gained prominence in the field of remote sensing. Many examples now exist in which data are upscaled, for example, ground data are averaged to provide a more coherent match with image pixels (e.g., Atkinson et al., 2000). Similarly, examples now exist in which data are downscaled. The benefits of downscaling are explained below in relation to area-to-point prediction and super-resolution mapping. 1.2. Measurement concepts It is useful to distinguish between processes and states. Processes are represented by process models, which in a geographical context are usually spatially distributed to some extent. Such process models are scale-dependent in their construction, that is, they are defined at a particular resolution. Moreover, a key concern in testing models for application to new sites is to ensure “grid-independence”, that is, that the results of the model are independent of resolution within a defined range (Atkinson and Tate, 2000). On the other hand, states may be investigated directly through measurement processes. Within this measurement process, two aspects are important. The first is sampling and the second is measurement uncertainty. The sampling process is defined by the following parameters: the support (the space over which each observation is defined) and the spatial extent. In turn, the spatial extent can be decomposed into the sampling scheme (e.g., random, stratified), the number of observations and the density of the sample (Atkinson and Tate, 2000). Measurement error is incurred at the scale of the observation as follows:



Zˆ u (x) =

1 |u(x)|





Z• (y)dy + εu(x)

(1)

u(x)

where, Zˆ u (x) is the observation (the only information that we can gain about reality), Z• (y) is the underlying process defined on a point support, the left hand term on the right hand side of the equation is the integral of the underlying process over the space of observation and εu(x) is a measurement error term. The support is the parameter of interest in relation to downscaling. In geostatistics, the support has been of long-standing interest. For example, Clark (1977) and Journel and Huijbregts (1978) developed some of the earliest models for regularising (increasing the support of) the semivariogram, a function describing the character of spatial variation (Atkinson, 1999). This geostatistical “operation” of regularisation (also known as convolution in the frequency domain literature), allows one to change the support (or measurement scale) of the semivariogram function without requiring new data on the new support. The importance of regularisation in the context of remote sensing, and in relation to measurement error, was explored by Atkinson (1993, 1997a) and Atkinson et al. (1996). Issues of scale in remote sensing have also led researchers to attempt to define an optimal pixel size (Woodcock and Strahler,

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1987; Atkinson, 1997b; Atkinson and Curran, 1995, 1997; Curran and Atkinson, 1999), including for harmonisation of time series imagery (Tarnavsky et al., 2008). The support has also been the subject of much research on the so-called modifiable areal unit problem (MAUP) (Openshaw, 1984) which has confounded researchers using census data on irregular census units such as Wards until recently (Liu et al., 2008; Yoo et al., 2010). The MAUP comprises two “problems”; the zonation problem and the aggregation problem, the former (despite the confusing name) essentially being a change of support problem. Recent developments in geostatistics have provided a principled and efficient solution to this problem. The support is also gaining in interest in relation to data fusion. Images and other data are now provided frequently in massive volumes, at often low cost from heterogeneous sources. For example, remotely sensed images may be provided at different view angles, from different positions, from different sensors with different spatial resolutions etc. Data fusion approaches that infer an underlying model from these different sources of data depend on knowledge of the measurement process and critically the effect of the support. In many ways, it is not surprising that the support has become the focus of attention in remote sensing. Satellite and airborne remote sensing provide data in the form of an image, usually in several wavebands of the electromagnetic spectrum. Such images provide complete coverage of an area on the surface of the Earth, synoptically, at a particular spatial resolution. Thus, given that the support is usually approximated by the pixel size, the sampling process is defined by the pixel size and the number of pixels in x and y only. In short, for remotely sensed images, the support has a special significance in defining the entire sampling strategy (Curran and Atkinson, 1999; Atkinson and Tate, 2000). The number of images produced by remote sensing satellite and airborne sensors has increased steadily over the last four decades since the launch of the first Landsat satellite in 1972. However, the spatial resolution continues to provide a limit on what is achievable in terms of information provided by an image. Indeed, there is a trade-off between spatial resolution, temporal revisit frequency and spatial coverage. The moderate resolution imaging spectrometer (MODIS) onboard the Terra and Aqua satellites provides data at 250 m, 500 m and 1 km spatial resolutions on a daily basis over scenes that are 2330 km across. Sensors such as Thematic Mapper onboard the Landsat series of satellites, provide data with a spatial resolution of 30 m, but with a limited revisit capability of 16 days and over scenes that are 185 km by 170 km. Typically, due to the limitations of the available sensor technology, fine resolution means poor coverage and revisit frequency. Thus, for global and daily monitoring, MODIS currently provides amongst the finest spatial resolutions available at 250 m. It is small wonder then that researchers have devoted so much attention to the goal of downscaling such imagery. The motivation for downscaling in the context of remote sensing also comes from an admission and understanding that complete coverage as provided through remotely sensed images does not equate to complete information. Just as increasing the number of points in a soil survey leads to potentially greater information, so increasing the resolution in remote sensing leads to potentially greater detail, despite the “complete coverage” at all resolutions. Moreover, there exists a more general desire to understand the effects of the support on data acquired through measurement. The support is a filter on reality that is not limited to remote sensing, but rather applies to all knowledge acquired through measurement (Eq. (1)). Thus, the support acts as a fundamental limit on what can be known about the real world, and as such it deserves significant attention. In remote sensing, most research on techniques for downscaling has fallen into one of two broad sub-goals as given in

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the introduction. These are area-to-point prediction and superresolution mapping. These are now discussed in detail in turn. 2. Downscaling continua 2.1. Regression approaches Several regression-type approaches have been developed for downscaling continua in remote sensing. These have typically been focused on downscaling thermal infrared waveband imagery using relations between fine resolution thermal data and coarse resolution images in other wavebands, and several recent papers have emerged in this field. For example, Stathopoulou and Cartalis (2009) developed a regression-type approach to downscale 1 km resolution land surface temperature image data provided by the Advanced Very High Resolution Radiometer (AVHRR) sensor. Pouteau et al. (2011) developed multiple regression and boosted regression tree approaches to downscale MODIS 1 km resolution maps of frost occurrence to 100 m. Zhan et al. (2011) developed regression and modulation approaches for downscaling LST data and combined these within an assimilation framework. They simulated 960 m resolution data from landsat TM imagery to provide an artificial test case and they included an intercomparison of the three approaches. The TsHARP model developed by Kustas et al. (2003) generates fine spatial resolution thermal data from coarse spatial resolution (≥1 km) data on the basis of an anticipated inverse linear relationship between the normalised difference vegetation index (NDVI) at fine spatial resolution and land surface temperature at coarse spatial resolution. Recently, Jeganathan et al. (2011) developed the TsHARP model by localising the model fitting. In a somewhat different approach, Zurita-Milla et al. (2009) extended downscaling based on spectral mixture analysis to a time-series of images. The above approaches to downscaling depend on direct measurement of the fine resolution target and some coarse resolution covariate. As such, they are limited in that they do not attempt to provide a formal model for the effect of the support and also do not attempt to characterise the desired pattern of spatial variation at the fine resolution. 2.2. Area-to-point prediction Area-to-point prediction (ATPP) refers to the downscaling of continua through interpolation. In area-to-point prediction, the input variable (at a coarse resolution) is the same as the output variable (but at a fine resolution) and both are generally continuous variables. The downscaling of continua is interesting as a prediction goal because it is an interpolation task, but it is different to the common interpolation task of predicting between sparsely distributed points (or quasi-points). Thus, downscaling continua presents a relatively new interpolation task. The challenge with downscaling continua is to take into account explicitly the support of the data, and more specifically to account explicitly for the change of support, in the statistical model used for prediction. This makes downscaling continua different to the common interpolation task and attractive and challenging to researchers, especially those involved in the development of geostatistics. It is, nevertheless de facto an interpolation task. The implicit assumption in all of the techniques reviewed here is that they use interpolation to provide a solution, not new sources of data. This means that geostatistical techniques are readily applicable and that the solution is inferred through a model, not resolved through the provision of new, fine resolution data. Kyriakidis (2004) first popularised the technique of area-topoint kriging (ATPK) following the earlier works of Cressie (1996)

and Gotway and Young (2002, 2004). ATPK differs from the standard forms of kriging in the following ways. Punctual kriging refers to prediction on the same support as that of the original data. Block kriging refers to prediction on a support that is much larger than that of the original data (Journel and Huijbregts, 1978). ATPK refers to prediction on a support that is smaller than that of the original data. As with punctual kriging, block kriging is achieved through use of the sample semivariogram estimated from data on the original support. However, ATPK requires estimation and use of the punctual semivariogram, that is, the semivariogram defined on a point support. Clearly, for satellite remote sensing is not possible to measure on a strictly point support, although in some fields (e.g., soil survey) observation on quasi-point supports that are much smaller than the support of interest is fairly common. Thus, in remote sensing the punctual semivariogram must be estimated through a de-regularisation or deconvolution procedure. There are several available approaches for this task including analytical approaches (Journel and Huijbregts, 1978), iteratively re-weighted generalised least-squares (Gotway and Young, 2004), and iterative empirical procedures (Goovaerts, 2008). Since this inverse problem is ill-posed, one can never be sure that the estimated function is sufficiently representative or accurate, although one can obtain reassurance about regularisations (convolutions) of the punctual functions within a limited range of positive sizes of support. This remains a problem with ATPK and all its variants. The two properties of predicting on a smaller support and the requirement to use the punctual semivariogram combine to define the ATPK algorithm. ATPK requires integration of the punctual semivariogram across two supports; that of the data and that of the predicted value, both having positive dimensions. This integration is often achieved through numerical approximation, but other approaches are also possible (Kyriakidis, 2004). Following elaboration of the technique for the generic case of census data, Kyriakidis and Yoo (2005) developed the technique further for application directly to remote sensing imagery, of direct relevance to this review. In fact, the remote sensing case involving regular supports (pixels of the same size and shape, at least in definition) is simpler than the census case which involves irregularly defined supports (e.g., Wards). Thus, efficiencies in processing were possible. Kyriakidis and Yoo (2005) demonstrated that the method reproduced the point histogram, the point semivariogram and the coarse resolution data (perfect coherence between scales). Yoo and Kyriakidis (2006) developed the ATPK method further to account for inequality constraints and Liu et al. (2008) applied the ATPK method for population downscaling. An interesting theoretical paper by Yoo et al. (2010) compares ATPK with Tobler’s pycnophylactic interpolation method. The differences were shown to be small, with ATPK superior in terms of the ability to report the uncertainty associated with the prediction using the model itself. Yoo and Kyriakidis (2008) introduced some boundary conditions into the ATPK model, whilst Yoo and Kyriakidis (2009) applied ATPK to the downscaling of hedonic house prices. Although the basic model is readily applicable to remote sensing data, the number of applications of ATPK in remote sensing has been rather limited. Following the work of Monestiez et al. (2005, 2006), Goovaerts (2006a,b) developed a new variant of ATPK called ATP Poisson Kriging (or ATPPK) that allowed prediction of rare diseases. The use of the Poisson model in geostatistics is equivalent to generalised linear modelling in relation to regression. Goovaerts (2006a,b) applied his new model to cancer data in the USA, with visually excellent results. Goovaerts (2008) extended this work, focusing on the problem of deconvolution of the semivariogram and the problems therein, whilst applying the model to both population downscaling and health data downscaling problems. Although yet to be applied to remote sensing data, the Poisson extension represents a significant

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extension of functionality, especially for application to health data. Goovaerts (2009, 2010) developed a new approach called area-andpoint-kriging (AAPK) in which he attempted to combine categorical secondary data and some primary continuous variable in a single model. The multivariate alternative to ATPK is called downscaling cokriging (DSCK). This approach involves the same considerations as for ATPK except that now two or more variables are involved and the model requires definition of the punctual semivariograms of both the primary and secondary variables, as well as the crossvariogram(s) between the primary and secondary variables. The set of semivariograms and cross-variograms that is accessible empirically through the available data is limited and the remainder must be inferred through deconvolution processes as for ATPK. Pardo-Iguzquiza et al. (2006) first introduced the DSCK concept in a remote sensing context, using a Landsat TM image to provide an example. Pardo-Iguzquiza and Atkinson (2007) focused on a novel method for deconvolving the empirical semivariogram to estimate the punctual function. In both of these studies, the focus was on image fusion. That is, the objective was to downscale an image in one waveband (e.g., visible waveband, coarse resolution, primary variable) using an image in another waveband (e.g., panchromatic waveband, fine resolution, secondary variable). However, the DSCK technique is entirely generic and Atkinson et al. (2008) demonstrated the technique for downscaling to a resolution that is finer than that of either, or any, of the input images. As for ATPK, the technique leads to perfect coherence with the coarse resolution input imagery. Pardo-Iguzquiza et al. (2011) extended the technique to include non-stationarity in the semivariograms and cross-variograms, with a marginal increase in prediction precision. The software for DSCK is freely available via the Computers and Geosciences website (Pardo-Iguzquiza et al. (2010)).

3. Super-resolution mapping It should be made clear from the outset that the approach referred to here as super-resolution mapping is different from image super-resolution established in the computer science and computer vision literature. The latter refers to the super-position of different views (e.g., multi-temporal image sequences) of the same scene to increase the actual resolution of the data (see, e.g., Park et al., 2003; Pickup et al., 2009; Zhu et al., 2010; Huang and He, 2011). Further applications may be found in medical imaging (Greenspan, 2009), and even in remote sensing (Shen et al., 2007). Super-resolution mapping refers to the use of interpolation based on a variety of methods to increase the apparent spatial resolution of a single input image (or set of input images) whilst also transforming it to a classification, usually of land cover. Thus, in super-resolution mapping, the input data are often a single image (in several wavebands) and the task is to “zoom in” through interpolation. Although the goal of super-resolution mapping is to predict on a support that is finer than that of the original data, as for ATPK and DSCK, it is different to ATPK and DSCK in some quite fundamental ways. Most importantly, super-resolution mapping involves transformation of a continuous variable as input (e.g., image of reflectance in a given waveband) to a categorical variable as output (e.g., land cover map). Most early approaches for super-resolution mapping relied on a two-step process whereby (i) an image of reflectance (or radiance, brightness etc.) is converted into an image of land cover class proportions (at the original resolution) and, subsequently, (ii) the image of land cover proportions is converted into an image of hard land cover classes (at a finer resolution). The first stage is usually achieved via a standard soft classifier or area proportions prediction technique such as a spectral mixture model

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(Settle and Drake, 1993). Then, stage two involves super-resolution techniques. An important point is that the representation of land cover proportions at the original pixel size is sub-optimal in several ways. First, although proportions are predicted, the data representation is still dominated by the original support. This leads to an unnatural and blocky appearance. Second, the proportions are known not to be coherent with our understanding of the way that the real world (and remotely sensed scenes in particular) is organised. In particular, the spatial character of the image is unlike that of reality. Finally, to represent k classes requires k proportions images. Where k is large this can lead to a set of images that is very difficult to interpret (Atkinson et al., 1997). For these reasons, the superresolution product, which equates to a single image fine resolution thematic land cover map, is preferable to a proportions representation at coarse resolution. However, additionally, super-resolution predictions are generally more accurate on a per-sub-pixel basis than the equivalent soft classified image data. The proportions image stage ((i) above) provides a further difference with the ATPK and DSCK approaches. Because land cover proportions are represented within pixels, the original proportions can be used as a constraint on the super-resolution mapping solution, giving a level of coherence between scales depending on the goal and the confidence that one wishes to place in the original proportions. This is similar to the ATPK and DSCK approaches where coherence is guaranteed either to the original data at coarse resolution or to a noise-filtered version of the original data if noise is present (Kerry et al., 2012). Having said this, it should be made clear that not all super-resolution techniques require a two stage mapping. The alternatives are discussed below. Super-resolution mapping techniques can be broadly differentiated based on the source of prior knowledge that provides the basis for interpolation (the “magic” that allows an apparent increase in resolution). In the first set, the goal is to maximise the spatial correlation between neighbouring sub-pixels (pixels at the finer, output resolution). This amounts to an assumption of maximal clustering between sub-pixel classes, and it is only appropriate where the “objects” in the scene of interest are large relative to the pixel size (Woodcock and Strahler’s (1987) H-resolution case). It also invokes assumptions about the nature of the boundaries between classes (i.e., that they are simple and convex for the objects relative to the background class). In the second and more general set, the prior knowledge comes from a model or training data. The model could be a semivariogram characterising the spatial character of variation, or the training image could be a real remotely sensed image or toy image drawn by the investigator. In either case, the use of such prior information implies a desire to match the character of the output super-resolution map to that of the training data. This set of techniques is appropriate more generically to all object sizes, but it has generally been applied for small objects relative to the pixel size (Woodcock and Strahler’s (1987) L-resolution case). The range of techniques adopted and developed for superresolution mapping is very broad, in part because the interpolation task can be solved in a variety of ways, most notably through spatial optimisation and regression for which a range of pre-existing techniques exist. This range of techniques is reviewed below. 3.1. Early approaches The concept of super-resolution mapping using the two step procedure described in Section 3 above was first proposed based on a simple sub-pixel-swapping procedure (Atkinson, 1997a,b,c). Several early approaches for super-resolution mapping emerged shortly afterwards. Tatem et al. (2001a) developed the concept of super-resolution mapping using a Hopfield neural network (HNN) for the first time. A suite of papers followed, which are described in

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more detail below. A group in Belgium led by Prof. De Wulf developed several distinct approaches for super-resolution mapping including an integrated mixture modelling approach (Verhoeye and De Wulf, 2002), a genetic algorithm approach (Mertens et al., 2003) and an approach that combined artificial neural networks and wavelets (Mertens et al., 2004). The genetic algorithm approach is very similar to the sub-pixel-swapping advocated by Atkinson (1997c), whereas the ANN approach is really a regression approach at heart and, thus, fundamentally different, requiring training data. Atkinson (2005) refined the earlier sub-pixel swapping approach of Atkinson (1997c) in a remote sensing context. Thornton et al. (2006, 2007) extended the sub-pixel swapping algorithm to the downscaling of linear features, something that was known to be problematic in standard algorithms. The sub-pixel swapping approach was developed recently within a remote sensing context by Ge et al. (2009), amongst others. Atkinson (2004) extended the pixel-swapping approach to spatial simulated annealing based on transition probabilities. This latter algorithm was a pattern matching algorithm in which the goal was to recreate the patterns of small objects within larger pixels. This algorithm differed to the pixel-swapping algorithm in that it required training in the form of the two-point histogram. An approach based on transition probabilities that is not dissimilar to Atkinson (2004) was developed recently by Cao et al. (2009, 2011). A recent development has focused on the use of super-resolution mapping for border segmentation (Cipolletti et al., 2012). 3.2. Hopfield neural network Tatem et al. (2001a) were the first to develop the HNN for super-resolution mapping. The HNN is a flexible spatial optimisation algorithm in the form of a recurrent, fully connected neural network (Hopfield and Tank, 1985). The HNN is easy to program and is readily adapted to new problems. The basic formulation is as an energy function comprised of a goal and constraints: E = k1 G + k1 C + b

(2)

where, E is the energy to be minimised, G is the goal, C is a constraint, the coefficients k1 and k2 are weights and b is a bias term. In Tatem et al. (2001a) the goal G was to maximise the spatial clustering between sub-pixels, whilst the constraint C was to maintain the proportions in the input imagery. The weights k1 = 1 − k2 can be adjusted to obtain the optimal balance between G and C in terms of reducing the energy E. The architecture of the HNN was such that each neuron represented a sub-pixel, and each output from a neuron was connected as an input to every other neuron. The algorithm is run iteratively until E reaches a minimum or some stopping criterion is reached. Since the goal was to maximise clustering, the algorithm is suitable for application in the H-resolution case only. Moreover, the algorithm was demonstrated for the binary case of a target on a background only. Tatem et al. (2001b) extended the 2001a algorithm to the case of multiple land cover classes. This was achieved by adding a layer of neurons (representing sub-pixels) in the HNN, one for each class to be represented, and constraining the sum of each set of neurons per sub-pixel to equal one. Tatem et al. (2002) extended the 2001a algorithm to the Lresolution case, that is, for objects that are smaller than a pixel. Indeed, the algorithm in Tatem et al. (2002) is designed for pattern matching rather than point prediction. The algorithm requires, as input, prior knowledge in the form of an empirical semivariogram. The algorithm is formulated as follows: E = k1 S1 + k2 S2 + k3 S3 + k4 S4 + k5 C + b

(3)

Here the i = 1, 2,. . ., n (n = 4 in Eq. (3)) semivariance goals Si now replace the clustering goal G meaning that the outcome of the algorithm is very different to Eq. (2). Specifically, S1 is the semivariance

at a lag of one sub-pixel, S2 is the semivariance at a lag of two sub-pixels and so on. The proportion constraint remains as before. The results showcased the ability of the algorithm to represent the pattern of objects that were much smaller than a pixel, with potential applications in, for example, providing realistic data as input to spatial simulation models. Finally, Tatem et al. (2003) applied the multivariate version of the clustering algorithm in Eq. (2) to real imagery with the goal of predicting a large number of land cover classes to provide a robust test. The results were impressive, although the inability to represent linear features well was evident. Minh et al. (2005) extended the HNN to include multivariate data inputs, specifically adding LiDAR imagery at an intermediate resolution to optical imagery at a coarse resolution such as to predict a set of land cover classes at a fine resolution. Minh et al. (2011) extended this to the case of fusing optical input imagery with panchromatic imagery at an intermediate resolution, a common case in remote sensing where the panchromatic waveband is available at a finer resolution than the multispectral data. The concept underlying the algorithm was novel. It involved adding a forward model that took the predicted land cover class per sub-pixel as input and, using end member spectra, predicted the multispectral values per sub-pixel. These multispectral values were then converted to intermediate resolution panchromatic values through spatial and spectral integration operations. The predicted panchromatic values were compared to the observed input panchromatic image and the error used to adjust the HNN. Minh et al. (2006) then extended the model in Eq. (3) to supplement optical data inputs with a panchromatic image at the intermediate spatial resolution, but where the panchromatic image had been fused with the optical data at a coarser spatial resolution. The HNN approach has seen limited application outside the above body of work. Collins and De Jong (2004) corrected some details in the original Tatem et al. (2001a) formulation such as to provide a speed-up in processing time. Muad and Foody (2010) extended the HNN to the case of multiple time-series inputs from MODIS and Landsat TM. 3.3. Geostatistics Geostatistics has seen limited application to super-resolution mapping, presumably in part because of the dependence of earlier approaches on a two-stage mapping and, in part, because the problem appears to require non-linear spatial optimisation between sub-pixels rather than linear regression-type prediction if one is to avoid smoothing effects. Boucher and Kyriakidis (2006) developed a geostatistical approach based on Indicator Cokriging (CoIK). They applied their method to the case of super-resolution mapping in remote sensing. Boucher and Kyriakidis (2007) then extended their method to include point conditioning. The method is further explained by Boucher (2008). Boucher et al. (2008) were one of the few to compare techniques for super-resolution mapping; they compared CoIK with a relatively new approach called multiple point geostatistics (MPG), which is described further below. The first to introduce multiple point statistics (MPS) to the geostatistical community were Guardiano and Srivastava (1993) and Strebelle (2002). In MPS, spatial prediction and spatial simulation are undertaken based on prior information from a training image (Zhang et al., 2005). The training image, thus, replaces the role of the semivariogram in classical geostatistics. Indeed, the training image replaces the role of the formal random function model, which is central to inference in classical geostatistics (Journel and Huijbregts, 1978; Atkinson, 1999). The advantage of using a training image is that one is not limited to two-point statistics as with the semivariogram or spatial covariance function. Because of this, one is able to capture complex patterns (e.g., connectivity) and reproduce them in the realised image. Zhang et al. (2005) were one of the

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first to introduce MPG, whilst Okabe and Blunt (2005) used MPS for pore space reconstruction and Alcolea and Renard (2010) applied MPS to hydrogeological data. Liu (2006) developed software called SNESIM for MPG, which is now widely available. Boucher (2008, 2009) applied MPG for downscaling in remote sensing, providing an alternative new framework to challenge the spatial optimisation algorithms developed previously. Specifically, this framework can be seen as an alternative to the pattern matching algorithms developed by Tatem et al. (2002) based on the semivariogram and Atkinson (2004) based on the twopoint histogram. Others have focused on issues of how to provide representative training images, and how to model non-stationarity in training images in particular (Van den Boogaart, 2006; de Vries et al., 2009; Mirowski et al., 2009). Mariethoz et al. (2010) has developed a fascinating cut-down version of MPG called direct sampling that seems to work well in a range of test conditions. Given its simplicity, it seems bound to find application in downscaling in remote sensing. It is based on the concept of self-similarity and, thus, resembles the approaches for downscaling from a single image based on fractal concepts (e.g., Hu et al., 2009; Glasner et al., 2009; Saito and Graselli, 2010). The Saito and Graselli (2010) approach involves application of spatial simulated annealing whilst accounting for texture and trying to minimise the prediction error. 3.4. MRF models Kasetkasem et al. (2005) were the first to suggest a Markov random field (MRF) approach for super-resolution mapping. The approach has been developed further by Tolpekin and Stein (2009), with a particular focus on optimal choice of the smoothness parameter. Ardila et al. (2010) applied the algorithm for the identification of individual trees in an urban setting. One advantage of the approach is that it involves a one-stage mapping, taking as input the raw image of reflectance (or radiance, brightness etc.) and not the proportions image, and produces the downscaled land cover map in one transform. 4. Discussion 4.1. Training data and training information Certain algorithms require training data on the character or pattern of spatial variation to recreate in the super-resolution classified image. Examples include the HNN pattern-matching algorithm of Tatem et al. (2002), the spatial simulated annealing approach of Atkinson (2004) and the multiple point geostatistics approaches of Boucher (2008, 2009). It is interesting to consider the differences in the training information provided in these three cases. Tatem et al. (2002) and Atkinson (2004) considered only the two-class case, although the algorithms are generalisable, whereas the MPG approaches of Boucher (2008, 2009) consider the multiple class case. The more important point relates to the information content of the functions chosen for a fixed data input. Tatem et al. (2002) used the empirical semivariogram to provide information from a training image to the algorithm, with surprisingly good results. Atkinson (2004) used the two-point histogram. This function contains exactly twice the potential information content as the semivariogram because the directionality of differences (e.g., A to B, B to A, A to A, B to B) is included as well as the differences (e.g., different class, same class) themselves. However, both characterisations are severely limited in relation to an entire training image. Whereas both the semivariogram and the two-point histogram are based on two-point statistics, a training image is able to impart information based on multiple-point comparisons. Information exists

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only in the differences between data and a training image contains a lot more differences and, thus, information than the semivariogram or two-point histogram. Thus, it is recommended that where available, approaches based on training images be used in place of simpler functions. Attention has been turning in recent years to the best ways to obtain training images and how to manipulate them to obtain the desired characteristics. For example, training images can be obtained in the remote sensing context from (i) fine resolution classified maps from other areas of similar character or (ii) fine resolution images of the same area from a previous airborne campaign or satellite overpass. However, fine resolution digital remote sensing is not the only possible source of training data. Alternative sources include land cover maps, topographic maps, aerial photographs, Google Earth and so on. Indeed, there is a strong argument for allowing users to create artificial training images using computer technology without empirical basis. This latter approach amounts to a form of capturing of expert knowledge. Where empirical training data are obtained, for example, in the form of a fine resolution land cover map from a nearby area to the target area, attention has turned recently to how such data can be manipulated to give the desired characteristics for the site in question. Of particular interest is the ability to turn a stationary field into a non-stationary (Atkinson, 2001) one through warping and other transformations of the geographical space (Van den Boogaart, 2006; de Vries et al., 2009; Mirowski et al., 2009). 4.2. Generalisation and smoothing Supposing that training data are provided in the form of the semivariogram, it is clear that the range of possible patterns that can be reproduced is extremely limited. Specifically, the detail obtained at the sub-pixel level, constrained by the coarse resolution parent pixel, is likely to be limited and highly generalised. This link between the quality of training data and the level of generalisation in the predicted map, imposed as a limit or restriction, is critical. It is not possible to move beyond this restriction given a specific coarse resolution input and limited training data (e.g., semivariogram). It is in the difference between this generalisation level, and the spatial character of the real scene of interest imaged at fine resolution, that effort should be directed in improving superresolution and downscaling algorithms. The basis for increasing the detail and removing this restriction of generalisation should be in either the quality of training data (e.g., multiple point, appropriateness, handling non-stationarity) or in providing further data to the process (e.g., Minh et al., 2006). However, whilst interesting, the latter approaches are a kind of cheating; the ultimate expression of this “fusion” approach is to measure directly at the fine resolution. Thus, it is not surprising that attention is currently being directed at ways to improve the quality of training data. The training data express our ability to represent the spatial character of reality. 4.3. Uncertainty Most of the early approaches for super-resolution were developed without concern for the ability to estimate the uncertainty associated with downscaled predictions based on the prediction model itself. For example, the HNN approach of Tatem et al. (2001a,b, 2002) does not include an estimate of prediction uncertainty. However, estimating the uncertainty associated with downscaled predictions should be of great concern for the following reasons. Despite the enticing statement at the beginning of this article about increases in information, one strictly cannot obtain more information than one starts with. Thus, the initial information content of a coarse spatial resolution image limits what can be achieved in terms of downscaling, both for continua through ATPP

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and for categorical predictions through super-resolution mapping. For super-resolution mapping the increase in number of data and apparent increase in information content occur through (i) the transformation of multiple waveband imagery into a single waveband at a finer resolution, (ii) knowledge of the spatial character of image scenes in nature and (iii) training data injected into the process that represent succinctly the nature of spatial variation at the target fine spatial resolution. For downscaling through ATPP (i) is omitted, unless multiple covariate wavebands are used in DSCK. Thus, as discussed above, the quality of training data is critical. Further, it is important to use the model for downscaling to provide an estimate of uncertainty where this is possible. Geostatistical and regression-based approaches more generally allow some internal estimate of prediction precision to be made, for example, the kriging variance, which can be useful as an estimate of local uncertainty (Lloyd and Atkinson, 2001). However, the overall character of the uncertainty for downscaled predictions is likely to be determined by the interpolation goal, the method adopted and by the data available for training, as discussed above. In particular, some algorithms are likely to lead to predictions with a level of spatial generalisation in the output that may be greater than that desired, for example, that represented in the training data. Stochastic simulation is used in geostatistics to characterise the spatial uncertainty, that is, the uncertainty due to spatial generalisation induced through interpolation (Stein et al., 1998). Such stochastic simulation approaches allow multiple realisations of the random function, each of which recreates the desired level of spatial generalisation, but at the expense of being less certain than the kriged prediction (which can be reconstructed for a given location as the mean of the realisations). Since the approaches described above differ in their ability to capture the desired level of spatial generalisation, it is important that assessment of the quality of downscaled maps includes a comparison of its spatial character with that of the target, as well as estimates of sub-pixel prediction precision. 4.4. Time-series It was pointed out at the beginning of this review that superresolution mapping is distinguished from super-resolution in signal processing which has to do with the super-position of multiple look or shifted images in order to resolve the object or information of interest. In remote sensing, attention has shifted in recent years from mapping to monitoring using multiple images in sequence. Monitoring was, of course, the early goal of missions such as Landsat. However, due to difficulties with geometric and atmospheric correction, as well as other difficulties such as cloud removal, intersensor calibration and the relatively short temporal sequences of images, the monitoring goal was not realised in the early days of remote sensing. However, these issues have been resolved to a level that operational monitoring over long time periods is now undertaken routinely. The free availability since 2006 of the entire Landsat archive from 1972 to the present has enabled the compilation of long time-series of “image stacks” (Wulder et al., 2008). Moreover, sensors such as MODIS which cover the Earth on a daily basis have now been in operation for more than 10 years. The question for super-resolution mapping is how can the space–time data cubes provided by such image stacks be converted into finer resolution data sets that are coherent from one date to another in terms of the objects that are represented within the image scenes. It is evident that an approach for super-resolution mapping based on a single date image can be enhanced by borrowing information from images before it and after it in time. Related approaches are employed in space–time geostatistics where interpolation is based on the space–time semivariogram (De Cesare et al., 2002). However, what is required is an approach that blends together the time-series of images to produce a super-resolution

mapping sequence that is coherent from one image to the other. That is, where constraints are placed on the ability of an object, or in the sub-pixel sense, the mapping of a class within a pixel, to move within a pixel from one image to the next. In this sense, the borrowing of information between images is a means by which to separate real change from noise. The result of such a process should be a smoothly varying set of sub-pixel mappings (or objects) and the ability to pin-point and quantify real changes. 5. Summary ATPP and super-resolution mapping represent novel approaches for extracting the maximum information from multiple waveband remotely sensed imagery and for presenting it in a visual manner that is easy to assimilate by the human brain. The objective of downscaling in remote sensing represents a significant challenge and it has drawn out some of the most innovative approaches in geostatistics and related fields in recent years, including the application of MPG. The ability to downscale depends on a clear model of both the nature of spatial variation in reality and of sampling and support effects, both of which are fundamentally important to our conception and understanding of the real world. Recent developments in downscaling have included ATPK, DSCK and MPG. There is a need for a classification of the various methods that have been employed in downscaling, to allow greater understanding of their limitations and their appropriate application. There is also a need to undertake a wider range of empirical comparisons. I have previously advocated a spatial inter-comparison competition based around super-resolution mapping both to provide a greater empirical evidence base and to draw out greater innovation in this area (Atkinson, 2009). The same would now also be useful for ATPP techniques. Future research is likely to focus on handling time-series data and in particular, the goal of providing a coherent space–time cube of downscaled predictions. For ATPP this is readily achievable using ATPK in space–time. For super-resolution mapping a wide range of possibilities exist. Acknowledgements This paper was given as a keynote presentation at the first conference on Spatial Statistics held in Enschede in 2011. The author thanks Prof. Heuvelink, Prof. Alfred Stein and Prof. Edzer Pebesma for the invitation to contribute to the conference and Prof. Gerard Heuvelink for his patience whilst this manuscript was written. References Alcolea, A., Renard, P., 2010. Blocking moving window algorithm: conditioning multiple-point simulations to hydrogeological data. Water Resources Research 46, 1–18. Ardila, J., Tolpekin, V., Bijker, W., Stein, A., 2010. Markov random field based superresolution mapping for identification of urban trees in VHR images. ISPRS Journal of Photogrammetry and Remote Sensing 66, 762–775. Atkinson, P.M., 1993. The effect of spatial resolution on the experimental variogram of airborne MSS imagery. International Journal Remote Sensing 14, 1005–1011. Atkinson, P.M., 1997c. Mapping sub-pixel boundaries from remotely sensed images. Innovations in GIS IV, 166–180. Atkinson, P.M., 1997a. On estimating measurement error in remotely-sensed images with the variogram. International Journal of Remote Sensing 18, 3075–3084. Atkinson, P.M., 1997b. Selecting the spatial resolution of airborne MSS imagery for small-scale agricultural mapping. International Journal of Remote Sensing 18, 1903–1917. Atkinson, P.M., 1999. Spatial statistics. In: Stein, A., van der Meer, F., Gorte, B. (Eds.), Spatial Statistics for Remote Sensing. Kluwer Academic Publishers, Dordrecht, pp. 57–81. Atkinson, P.M., 2001. Geographical information science: geocomputation and nonstationarity. Progress in Physical Geography 25, 111–122. Atkinson, P.M., 2004. Super-resolution land cover classification using the twopoint histogram. In: Sánchez-Vila, X., Carrera, J., Gómez-Hernández, J. (Eds.),

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