Roberts Cross
Encyclopedia
The Roberts' Cross operator is used in image processing
and computer vision
for edge detection
. It was one of the first edge detectors and was initially proposed by Lawrence Roberts
in 1963. As a differential operator, the idea behind the Robert's Cross operator is to approximate the gradient
of an image through discrete differentiation which is achieved by computing the sum of the squares of the differences between diagonally adjacent pixels.
where x is the initial intensity value in the image, z is the computed derivative and i,j represent the location in the image.
The results of this operation will highlight changes in intensity in a diagonal direction. One of the most appealing aspects of this operation is its simplicity; the kernel is small and contains only integers. However with the speed of computers today this advantage is negligible and the Roberts cross suffers greatly from sensitivity to noise.
the original image, with the following two kernels:
Let
be a point in the original image and 
be a point in an image formed by convolving with the first kernel and 
be a point in an image formed by convolving with the second kernel. The gradient can then be defined as:
The direction of the gradient can also be defined as follows:
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...
and computer vision
Computer vision
Computer vision is a field that includes methods for acquiring, processing, analysing, and understanding images and, in general, high-dimensional data from the real world in order to produce numerical or symbolic information, e.g., in the forms of decisions...
for edge detection
Edge detection
Edge detection is a fundamental tool in image processing and computer vision, particularly in the areas of feature detection and feature extraction, which aim at identifying points in a digital image at which the image brightness changes sharply or, more formally, has discontinuities...
. It was one of the first edge detectors and was initially proposed by Lawrence Roberts
Lawrence Roberts (scientist)
Lawrence G. Roberts received the Draper Prize in 2001 and the Principe de Asturias Award in 2002 "for the development of the Internet" along with Leonard Kleinrock, Robert Kahn, and Vinton Cerf....
in 1963. As a differential operator, the idea behind the Robert's Cross operator is to approximate the gradient
Gradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
of an image through discrete differentiation which is achieved by computing the sum of the squares of the differences between diagonally adjacent pixels.
Motivation
According to Roberts, an edge detector should have the following properties: the produced edges should be well-defined, the background should contribute as little noise as possible, and the intensity of edges should correspond as close as possible to what a human would perceive. With these criteria in mind and based on then prevailing psychophysical theory Roberts proposed the following equations:where x is the initial intensity value in the image, z is the computed derivative and i,j represent the location in the image.
The results of this operation will highlight changes in intensity in a diagonal direction. One of the most appealing aspects of this operation is its simplicity; the kernel is small and contains only integers. However with the speed of computers today this advantage is negligible and the Roberts cross suffers greatly from sensitivity to noise.
Formulation
In order to perform edge detection with the Roberts operator we first convolveConvolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
the original image, with the following two kernels:
Let
be a point in the original image and be a point in an image formed by convolving with the first kernel and be a point in an image formed by convolving with the second kernel. The gradient can then be defined as:The direction of the gradient can also be defined as follows:
Example comparisons
Here, four different gradient operators are used to estimate the magnitude of the gradient of the test image. |
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See also
- Digital image processingDigital image processingDigital image processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing...
- Feature detection (computer vision)
- Feature extractionFeature extractionIn pattern recognition and in image processing, feature extraction is a special form of dimensionality reduction.When the input data to an algorithm is too large to be processed and it is suspected to be notoriously redundant then the input data will be transformed into a reduced representation...
- Sobel operator
- Prewitt operator