Common Names: Sobel, also related is Prewitt Gradient Edge Detector
The Sobel operator performs a 2-D spatial gradient measurement on an image and so emphasizes regions of high spatial gradient that correspond to edges. Typically it is used to find the approximate absolute gradient magnitude at each point in an input greyscale image.
In theory at least, the operator consists of a pair of 3×3 convolution masks as shown in Figure 1. One mask is simply the other rotated by 90°. This is very similar to the Roberts Cross operator.
Figure 1 Sobel convolution masks
These masks are designed to respond maximally to edges running vertically and horizontally relative to the pixel grid, one mask for each of the two perpendicular orientations. The masks can be applied separately to the input image, to produce separate measurements of the gradient component in each orientation (call these Gx and Gy). These can then be combined together to find the absolute magnitude of the gradient at each point and the orientation of that gradient. The gradient magnitude is given by:
Although typically, an approximate magnitude is computed using:
which is much faster to compute.
The angle of orientation of the edge (relative to the pixel grid) giving rise to the spatial gradient is given by:
In this case, orientation 0 is taken to mean that the direction of maximum contrast from black to white runs from left to right on the image, and other angles are measured anti-clockwise from this.
Often, this absolute magnitude is the only output the user sees --- the two components of the gradient are conveniently computed and added in a single pass over the input image using the pseudo-convolution operator shown in Figure 2.
Figure 2 Pseudo-convolution masks used to quickly compute approximate gradient magnitude
Using this mask the approximate magnitude is given by:
The Sobel operator is slower to compute than the Roberts Cross operator, but its larger convolution mask smooths the input image to a greater extent and so makes the operator less sensitive to noise. The operator also generally produces considerably higher output values for similar edges compared with the Roberts Cross.
As with the Roberts Cross operator, output values from the operator can easily overflow the maximum allowed pixel value for image types that only support smallish integer pixel values (e.g. 8-bit integer images). When this happens the standard practice is to simply set overflowing output pixels to the maximum allowed value. The problem can be avoided by using an image type that supports pixel values with a larger range.
Natural edges in images often lead to lines in the output image that are several pixels wide due to the smoothing effect of the Sobel operator. Some thinning may be desirable to counter this. Failing that, some sort of hysteresis ridge tracking could be used as in the Canny operator.
shows the results of applying the Sobel operator to
. Compare this with the equivalent Roberts cross output
. Note that the spurious noise that afflicted the
Roberts cross output image is still present in this image, but its
intensity relative to the genuine lines has been reduced, and so there
is a good chance that we can get rid of this entirely by thresholding.
Also notice that the lines corresponding to edges have become thicker
compared with the Roberts cross output due to the increased smoothing
of the Sobel operator.
shows a simpler scene containing just a single flat
dark object against a lighter background. Applying the Sobel operator
produces 
. Note that the lighting has been
carefully set up to ensure that the edges of the object are nice and
sharp and free of shadows.
The Sobel edge detector can also be applied to range images like
. The corresponding edge image is
. All edges in the image have been detected and can
be nicely separated from the background using a
threshold of 150, as can be seen in
.
Although the Sobel operator is not as sensitive to noise
as the Roberts Cross operator, it still amplifies high
frequencies. 
is the result of adding Gaussian
noise with a standard deviation of 15 to the original
image. Applying the Sobel operator yields 
and
thresholding the result at a value of 150 produces
. We can see that the noise has increased during
the edge detection and it is not possible anymore to find a threshold
which removes all noise pixels and at the same time retains the
edges of the objects.
The object in the previous example contains sharp edges and its
surface is rather smooth. Therefore, we could (in the noise free case)
easily detect the boundary of the object without getting any erroneous
pixels. A more difficult task is to detect the boundary of
, because it contains many fine depth variations (i.e.
resulting in intensity changes in the image) on its surface. Applying
the Sobel operator straightforwardly yields 
. We can
see that the intensity of many pixels on the surface is as high as
along the actual edges. One reason is that the output of many edge
pixels is greater than the maximum pixel value and therefore they are `cut
off' at 255. To avoid this overflow we scale the
range image by a factor 0.25 prior to the edge detection and than
normalize the output, as can be seen in
. Although the result improved significantly, we
still cannot find a threshold so that a closed line along the boundary
remains and all the noise disappears. Compare this image with the
results obtained with the Canny edge detector.
A related operator is the Prewitt gradient edge detector (not to be confused with the Prewitt compass edge detector). This works in a very similar way to the Sobel operator but uses slightly different masks, as shown in Figure 3. This mask produces similar results to the Sobel, but is not as isotropic in its response.
Figure 3 Masks for the Prewitt gradient edge detector.
and see if you can
use thresholding to detect the edges of the object
without obtaining noise. Compare the results with the one obtained
with the Roberts cross operator.
R. Gonzalez and R. Woods Digital Image Processing, Addison Wesley, 1992, pp 414 - 428.
R. Boyle and R. Thomas Computer Vision: A First Course, Blackwell Scientific Publications, 1988, pp 48 - 50.
E. Davies Machine Vision: Theory, Algorithms and Practicalities Academic Press, 1990, Chap 5.
D. Vernon Machine Vision, Prentice-Hall, 1991, Chap 5.
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