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Building Spatial Databases – Theory / Edge detectors

Learners guide

Summary

The edge detecting is an important part of the image processing, which plays role in geoinformatics, remote sensing and in the raster-vector conversion processes. The technical details will be shown in this unit.

Requirement

To fill in the self-test successfully

Edge detectors

Edge detectors are local-based filters on a given point with the help of the derivative intensity values of neighbours (Figure 67). Our goal is to emphasize the edges and to remove pixels having similar properties.

Figure 67. Maximum value on first derivative over the edges, the second derivative has a zero value.

Gradient filter

By using the image as surface points, we take the derivative with $x$ and $y$ gradient vector approximated by the differential coefficient.

The filter reacts to large intensity changes, resulting 0 for homogeneous areas. Its kernel is defined in the following:

The "p" value may be freely selected, though the used values are usually p=2, p=3, p=2+sqrt(2). With p=2 it is a Prewitt operator, with p=3 it is a Sobel operator, and with p=2+sqrt(2) is an isotropic operator (Figure 68). The result must be normalized with p.

Figure 68. From left to right. Original image. An x and y vector graident approximation with isotropic gradient filter. Edge emphasis calculated from the two images.

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Laplace filter

Operator definition of a Laplace filter:

The following equation is a result of a secondary derivative at (x,y) point if the Laplace operator has been discretized:

Therefore, the Laplace filtered value at (x,y) point can be calculated with the following convolution kernel:

The Laplace filter in practice is the difference between the smoothened and the original image. It reacts very efficiently to large intensity value fluctuations, therefore it reacts well for errors as well. Without smoothening it is not effective at all.

Figure 69. From left to right. The original image. Laplace filtered image. LoG filtered image. Emboss filtered image set for detection of edges having North-East to South-West directions.

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LoG filter

We may achieve a good edge detection filter by applying Gauss smoothening before a Laplace filter (Figure 69). Due to the fact that convolution is an associative operation, it is possible to convolute the kernels of the Laplace and Gauss filters and use the resulted matrix. This type of filter is called "Laplacian of Gaussian" (LoG). LoG reacts sensitively to noises and is good for edge detection. Its kernel matrix looks as the following:

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Emboss filter

The goal of Emboss filters is to detect edges in different directions (Figure 69). A special kernel has been used, in which +1 and -1 values are on the opposite corners of the matrix. The filter reacts sensitively to edges perpendicular to the values on the diagonals. An example of the kernel of an emboss filter:

The kernel will detect North-East to South-West direction edges, while edges perpendicular to it will be neglected.

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A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszirozásával valósul meg.

A Társadalominformatika: moduláris tananyagok, interdiszciplináris tartalom- és tudásmenedzsment rendszerek fejlesztése az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával, az ELTE TÁMOP 4.1.2.A/1-11/1-2011-0056 projekt keretében valósult meg.
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