•if the range of values permitted within the grid are limited (e.g. RGB values in the range [0,255]) then computed values that fall outside of this range must be truncated to the range limits. This generates some computational (as opposed to user defined) bias in the processing

With the same data grid as above the filtered value is G=162. More generally, if {Ci} is the set of coefficients of an m=NxN kernel matrix and {Pi} is the set of source grid values within this kernel neighborhood, then the filtered grid value, G, is defined as:

In addition to filters that operate in the spatial domain, pure frequency domain filters may also be applied. These operate on the set of grid values within an image without reference to the spatial pattern. Frequency domain filtering (for example Fast Fourier Transforms or FFTs) is provided in some GIS packages such as TNT/Datum Workstation and TerrSet/Idrisi, but such procedures are not described in this Guide. FFTs are used, for example, to removing striping effects from remotely-sensed imagery.

Most GIS packages provide simple grid filtering functions, whilst those that have a bias towards image processing and grid file handling (e.g. TNT/Datum Workstation, Surfer, TerrSet/Idrisi) tend to provide a broader range of functions. The most commonly provided functions are so-called low-pass and high-pass spatial filters. These are focal functions whose operation is determined by a kernel or neighborhood of NxN cells around each pixel or grid position. Grid cells “covered” by the kernel are multiplied by the matching kernel entry and then the weighted average is calculated and assigned as the value for the central cell, G. A 3x3 symmetric kernel might look something like:

It should be noted that at image or grid edges it is not possible to apply the kernel symmetrically. This effect is more pronounced for larger kernels, but applies in every instance. There are many options for resolving such effects. Common procedures include: leaving the edge pixels unaltered; blanking edge pixels (which causes the grid to shrink); using an asymmetric (i.e. partial) kernel, which can generate artifacts; and mirroring, where grid values inside the kernel are used as surrogates for the missing values outside the kernel.

•multi-band image data filters are applied to each band separately. Hence a sharpening filter applied to an RGB image, for example, would be applied to the R, G and B components

The value at the central grid position G(2,2)=187 is replaced by the weighted average of itself (assigned a weight of 4) and the surrounding 8 cells (each assigned a weight of 1). The adjusted value is 164 after rounding. In the second example if a, b and c are positive with b>a>c the kernel is described as Gaussian, being symmetric but center-weighted, for example:

The extra term, B, here is often set to 0. It is a bias factor that increases or decreases the resulting filter value by a fixed amount.

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•Embossing — for which the kernel matrix elements sum to 0 as per edge detection, but the matrix is asymmetric. It is usual to compute the embossed version of an image on a grayscale version, which is obtained by finding the average value of the bands, e.g. red (r), green (g) and blue (b) components, as x=(r+g+b)/3 for each pixel, and assigning the resulting values in the range [0,255] to a gray scale (where 0=black and 255=white)

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This infrared beam is extremely suitable when you want to be able to see with your nightvision/night rifle scope even when there is little or no ambient light. With this 940nm IR, both the infrared beam and the infrared glow are completely invisible to humans and animals. In addition, you increase the viewing distance up to 600 metres. An IR Illuminator can be used in a wide range of applications, such as tactical applications. But also for hunters, survival or general nature observations this is an indispensable accessory. The human eye cannot detect the IR light, but you can see the 'illuminated' area through night vision viewer.

Edge detecting filters that enhance edges in a selected compass direction to provide an embossed effect. The example here shows a sample north-east kernel

•Edge detection — for which the kernel matrix elements sum to 0 and the matrix exhibits some symmetry. As with sharpening, matrix elements are a mix of positive, negative and zero entries

•Blurring — also known as a smoothing, averaging or low-pass filtering, since this reduces or removes extreme values. In this case the kernel matrix elements sum to >1, entries are normally all positive and the matrix is symmetric

•Sharpening — for which the kernel matrix elements sum to 1 and the matrix is symmetric. Matrix elements are a mix of positive, negative and zero entries

•filtering may be conducted more than once (single pass or multi-pass). Each pass re-applies the same filter or filters to the target grid or image (layer or band)

Applied singly or as a two-pass process. These kernels highlight vertical and horizontal edges. When used in combination they are known as Gradient or Order 1 derivative filters. Typically a=1 and b=1 or 2 and entries sum to 0. A variant known as the Roberts method has row 3 all zero and column 1 zero for the first pass and column 3 zero for the second pass (i.e. a 2x2 filter)

High-pass spatial filters emphasize the difference between the central point of the kernel and the values in its immediate neighborhood. Typically the entries in such kernels have a mix of positive and negative values, but the entries still add up to 1:

Mean effect removal/sharpening filter (focal sum). Provides limited edge detection. Typically entries sum to 1 but may be greater. 3x3 Laplacian kernels typical add to 1. Larger Laplacian kernels (e.g. 7x7) may be more complex and sum to >1

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•user-defined kernels are widely supported, both in image processing packages and within GIS software. Users may be able to select a pre-defined filter and modify this to suit their requirements, or to create their own NxN kernel

In the first of these examples a and b are positive numbers, typically integers. If a=b=1 then the kernel provides a simple averaging or smoothing operation, whereas if a=1 and b=4 more weight would be applied to the grid cell being filtered. In both cases the weighted average is divided by the sum of the kernel elements. No divisor is applied in cases where the kernel sums to 0 or 1. Filters of this type are sometimes described as low-pass spatial filters. For example, consider the filter:

The term spatial filtering is principally associated with digital image processing, although such methods may be applied to almost any type of grid or image. The term is also used, in a related manner, in the area of spatial statistics (see further, Section 5.6.5, Spatial filtering models).

• That you can still see when there is no residual light present • An invisible infrared glow (IR source) and beam for human eye and animals • A significant increase in viewing distance of up to 600 metres • An increase in accuracy in identifying objects • Much better recognition of the smallest details

Values are often used to provide image sharpening prior to further processing. Table 4‑11 provides examples and a summary of typical linear spatial filters used in GIS and image analysis. Basic linear image filtering operations fall into a number of categories:

•all of the examples described involve simple linear operations, essentially weighted averaging (sometimes described as linear convolution). Non-linear filtering is also widely available and is described in more detail in Section 4.6.3

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• Wavelength 940nm: The infrared glow is invisible. In terms of power/viewing distance you do deliver in comparison to 850nm IR • Wavelength 850nm: Maximum power for maximum viewing distance. The red glow is visible at close range

Simple computation of gradient in one of 8 compass directions. east and north directional derivatives are illustrated in the first two examples here. Note that the third example shown corresponds to Horn’s method of derivative calculation for the north/south component (dz/dy) as described in Section 6.1.3. Compares with Edge detection filtering. A non-linear variant of this procedure is to compute all 8 values and select the maximum absolute value of these as the output

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