Geographical Raster Processing With Weighted Statistics and Variable Kernel Filtering: Treating Map Projection Distortions In Small Scales

Timofey Samsonov

Lomonosov Moscow State University, Faculty of Geography, Dept. of Cartography and Geoinformatics, Leninskiye Gory 1, Moscow, Russia

Abstract. Filtering is the most common approach to raster processing. In cartography and geographical information science it’s used for numerical generalization of surfaces, calculation of derivatives, edge detection and variety of other tasks concerned with image processing. The kernel that is used for filtering always has fixed size, which is defined once for the whole raster. However, rasters covering large areas contain all the distortions that are introduced by the projection in which the raster is stored. Examples are global digital elevation models, climatological and land cover rasters, mid-dle- and low-resolution space imagery. Processing of such rasters with tra-ditional filtering approach leads to the loss of geographical meaning and incorrect calculation of derivatives, as the same floating window covers different geographical neighbourhoods in different parts of raster. Raster statistics that are derived from the whole matrix of pixels are also biased by areal distortions. We present new approach to raster processing that ap-plies affine transformation to filtering kernel that compensates for local pro-jection distortions.

Keywords: raster filtering, raster generalization, raster statistics, map pro-jection distortions, small-scale mapping.

1. IntroductionRaster data play an extremely important role in mapping and geographical information science. The most widespread areas of raster data model appli-cation are representation of continuous phenomena (various fields such as terrain surface, temperature, population density) and space imagery.

One of the default technologies of raster processing is filtering (McMaster Monmonier 1989). Filtering exploits intrinsic topology of raster model that

guarantees 8 (or 4 depending on the connectivity approach) neighbors for every raster cell except those located at raster boundary. Usually a raster kernel is centered above the pixel being processed and has a symmetric square shape with an odd number of pixels (3, 5, 7 etc.) along X and Y-axes. Sometimes circular, ring or wedge neighborhood is used and in some specific cases (mostly in raster morphology) a kernel can have even num-ber of pixels and thus is positioned asymmetrically above the processed pixel (Li 1994).

De Smith et al. (2013) differentiates between linear and non-linear raster filtering. While linear filtering is based on convolution theory and calculates weighed average in every pixel’s neighborhood, a non-linear filtering is based on algorithmic analysis that cannot be expressed in terms of simple averaging. Several conceptual frameworks for raster processing have been proposed (Peuquet 1979, Li et al. 2001). Filtering has an important role in raster generalization. McMaster and Monmonier (1989) divide raster gener-alization tasks into four categories: a) structural generalization, b) numerical generalization, c) numerical categorization, and d) categorical generaliza-tion. They identified low- and high-pass filtering as means of numerical raster generalization.

Distortions in raster analysis received moderate attention so far. Steinwand et al (1995) revealed the effects of map projection on data quality. Seong and Usery (2001) developed a scale factor model that is based on calcula-tion of projection scale factors. They assesed them for cylindrical equal area, sinusoidal and Mollweide projections. Usery at al (2003) investigated map projection and resampling effects on the tabulation of categorical ar-eas in global raster datasets. They revealed problems in projection engine of some commercial software that do not use the exact projection equa-tions. They also compared areas calculated in spherical and planar coordi-nates and revealed that detailed resolution allows better preservation of areas. Mulcahy (2000) and White (2006) analysed pixel loss and replication patterns in eight projection of world maps. All authors emphasize that sim-plest sinusoidal projection performs very well in minimizing raster resam-pling (error due to the k=1 at all parallels). Finn et al (2012) developed a specialized program for handling map projections of small scale geospatial raster data. However, these investigations are mainly concerned with global effects of distortions.

2. Raster filtering with variable kernel shapesWe offer a variable kernel shape approach in which the convolution kernel is transformed at each pixel according to the local Tissot ellipse of distor-

tion. In this case it is important that the initial kernel size should be defined not in pixels, but in raster projection units (meters).

The idea of using variable window in raster image processing is not new. For example, very high efficiency in image noise reduction can be achieved by adaptive median filtering, in which widow size is varied according to the variance of the values around selected pixel, which helps to filter only highly noised areas (Lin and Willson 1988). Moreover, filtering can be adapted to directionality of the noise (Dong and Xu 2007). Our idea is that raster filtering process could be adaptive to map projection distortions.

Map projection distortion elements (Snyder 1987) can be naturally inter-preted as affine transformation parameters: Tissot indicatrix axes a and b for scaling and direction σ for rotation. Considering that the initial kernel parameters are supposed for the point p0 = (x0, y0) with zero distortions, the kernel coordinates u and v for an arbitrary point p = (x, y) can be trans-formed in the following way:

[u 'v ' ]=[ cos σ sin σ−sin σ cos σ ][b 0

0 a] [uv ]where parameters a, b and are calculated at the current pixel using nu-merical or analytical solution. This transformation is only appropriate for symmetric kernels.

If the resulting values should be mapped to the initial coordinates (for ex-ample, to reconstruct the function for the following analysis that requires regular grid of points), inverse affine transformation could be applied.

The following algorithm based on affine transformation of convolution ker-nel is proposed:

1. Define initial shape of the kernel (commonly rectangular or ellipse) and its sizes in both X and Y directions

2. Calculate the extent of the raster in geographical units (degrees).3. Sample raster area by the control points, which are equally spaced

in degrees. Sampling distance is defined by user or can be calcu-lated as a function of raster geographical extent and resolution.

4. Calculate the parameters of distortion ellipse at each control point using projection equations.

5. Using distortion ellipse parameters, define the local matrix of affine transformation.

6. Transform kernel coordinates.7. For each pixel in the initial raster find the closest control point and

assign its number to the pixel.

8. Process the whole raster using kernels from the assigned control points.

3. ImplementationAn algorithm was implemented in a Java application (Figure 1). The user opens raster and begins with setting a projection. The graticule is used to set the grid of control points. The less are the steps between parallels and meridians, the more precise (locally) will be the transformation of the ker-nel. The constructed graticule (in blue) defines the set of control points (at intersections). Each control point has its own zone. Control zones are high-lighted in yellow. Finally, Tissot indicatrix is visualized in red.

Figure 1. Raster processing application that implements variable kernel shape algorithm

4. ResultsIn our first experiment (Raposo and Samsonov 2014) we selected a combi-nation of mean filtering and the Mercator projection. The Mercator projec-tion’s linear distortions are calculated simply as m = n = 1/cos(B) at every point, where B is the latitude, and θ = π/2. Thus the kernel need only be scaled according to value of m. Control points were sampled each degree by latitude.

The source raster was a 5 km resolution DEM covering the European part of Russia. First, we processed the DEM using a 35×35 km (7×7 pixels) square kernel in a traditional, non-variable-kernel approach. Then we ap-plied our variable kernel approach, with size varying from 7×7 pixels at 30° to 67×67 at 84°, always covering approximately the same 35×35 km area. Results of filtering show that our approach produces more geographically sound results than simple filtering. Compare the Kola peninsula in the northern part of the map and the Caucasus in the southern part (Figure 1). Enlarged terrain features at high latitudes appear to be generalized much less then southern territories by simple filtering. In contrast, application of a variable kernel shape smoothes features of the same geographical area, which can be important if the smoothing is performed for geographical anal-ysis applications.

Figure 2. Comparison between source DEM, filtered by fixed 7x7 pixel MEAN ker-nel and by variable shape 35x35 km kernel.

Distortions should also be taken into account for calculation of global raster statistics. While in the case of raster minimum and maximum all pixels should be considered, the mean and standard deviation characteristics should better be taken from a distortion-aware approach. In this case the field of areal distortion can be calculated. The weighted mean of all pixels could be calculated with each weight equal to the inverse of local areal scale factor.

5. ConclusionsThe presented variable kernel shape approach could be used in a raster processing tasks that call for more geographical approach — to cover the similar areas across large raster datasets during focal operations. Such tasks include the search for the geographical features if the same size, cor-rect calculation of slopes and aspects, revealing of raster projection distor-tions through their emphasis (Mulcahy and Clarke 2001), various aniso-tropic filtering opertations. In our future investigations we plan to make an assessment of all distortion parameters as factors of filtering kernel trans-formation and their influence on results and raster statistics.

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