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2014-11-14T00:50:05Z

Legal applications

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	~~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_(scientist)\|Lawrence Roberts]] in 1963.<ref>[http://www.packet.cc/files/mach-per-3D-solids~~.~~html Machine Perception Of Three-Dimensional Solids]</ref>~~ As a ~~[[Difference operator\|differential operator]], the idea behind the Roberts 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~~.

	~~==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:~~

	~~:<math>~~
	~~y_{i,j} = \sqrt{x_{i,j}}~~
	~~</math>~~

	~~:<math>~~
	~~z_{i,j} = \sqrt{(y_{i,j} - y_{i+1,j+1})^2 + (y_{i+1,j} - y_{i, j+1})^2 }~~
	~~</math>~~

	~~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.<ref>LS. Davis, "A survey of edge detection techniques", Computer Graphics~~ and ~~Image Processing, vol 4, no~~. ~~3, pp 248-260~~, ~~1975</ref>~~

	~~==Formulation==~~
	~~In order~~ to ~~perform edge detection with the Roberts operator we first [[convolution\|convolve]] the original image, with the following two kernels~~:

	:~~<math>~~
	~~\begin{bmatrix}~~
	~~+1 & 0 \\~~
	~~0 & -1\\~~
	~~\end{bmatrix}~~
	~~\quad \mbox{and} \quad~~
	~~\begin{bmatrix}~~
	~~0 & +1 \\~~
	~~-1 & 0 \\~~
	~~\end{bmatrix}.~~
	</~~math>~~

	~~Let <math>I(x,y)<~~/~~math> be a point in the original image and <math>G_x(x,y)<~~/~~math> be a point in an image formed by convolving with the first kernel and~~ <~~math~~>~~G_y(x,y)~~<~~/math~~> ~~be a point in an image formed by convolving with the second kernel. The gradient can then be defined as~~:

	~~:<math>~~
	~~\nabla I(x,y) = G(x,y) = \sqrt{ G_x^2 + G_y^2 }.~~
	</~~math>~~

	~~The direction of the gradient can also be defined as follows:~~

	~~:<math>~~
	~~\Theta(x,y) = \arctan{\left(\frac{G_y(x,y)}{G_x(x,y)}\right)}.~~
	</~~math>~~

	~~== Example comparisons ==~~
	~~Here, four different gradient operators are used to estimate the magnitude of the gradient of the test image.~~

	{\|
	~~\|[[Image:Bikesgray.jpg\|thumb\|200px\|Grayscale test image of brick wall and bike rack]]~~

	~~\|[[Image:Bikesgray_roberts~~.~~JPG\|thumb\|200px\|Gradient magnitude from Roberts cross operator]]~~

	~~\|[[Image:Bikesgray_sobel.JPG\|thumb\|200px\|Gradient magnitude from [[Sobel operator]]]]~~
	\|-
	~~\|[[Image:Bikesgray-scharr.png\|thumb\|200px\|Gradient magnitude from Scharr operator]]~~

	~~\|[[Image:Bikesgray_prewitt.JPG\|thumb\|200px\|Gradient magnitude from [[Prewitt operator]]]]~~
	\|}

	~~==See also==~~
	* [[Digital image processing]]
	* [[Feature detection (computer vision)]]
	* [[Feature extraction]]
	* [[Sobel operator]]
	* [[Prewitt operator]]

	~~==References==~~
	~~<references~~/>

	~~[[Category:Feature detection]~~]

en>Seonookim: Added {{technical}} tag to article (TW)

2013-10-25T07:55:53Z

Added {{technical}} tag to article (TW)

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{{FeatureDetectionCompVisNavbox}}

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_(scientist)|Lawrence Roberts]] in 1963.<ref>[http://www.packet.cc/files/mach-per-3D-solids.html Machine Perception Of Three-Dimensional Solids]</ref> As a [[Difference operator|differential operator]], the idea behind the Roberts 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.

==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:

:<math>
y_{i,j} = \sqrt{x_{i,j}}
</math>

:<math>
z_{i,j} = \sqrt{(y_{i,j} - y_{i+1,j+1})^2 + (y_{i+1,j} - y_{i, j+1})^2 }
</math>

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.<ref>LS. Davis, "A survey of edge detection techniques", Computer Graphics and Image Processing, vol 4, no. 3, pp 248-260, 1975</ref>

==Formulation==
In order to perform edge detection with the Roberts operator we first [[convolution|convolve]] the original image, with the following two kernels:

:<math>
\begin{bmatrix}
+1 & 0 \\
0 & -1\\
\end{bmatrix}
\quad \mbox{and} \quad
\begin{bmatrix}
0 & +1 \\
-1 & 0 \\
\end{bmatrix}.
</math>

Let <math>I(x,y)</math> be a point in the original image and <math>G_x(x,y)</math> be a point in an image formed by convolving with the first kernel and <math>G_y(x,y)</math> be a point in an image formed by convolving with the second kernel. The gradient can then be defined as:

:<math>
\nabla I(x,y) = G(x,y) = \sqrt{ G_x^2 + G_y^2 }.
</math>

The direction of the gradient can also be defined as follows:

:<math>
\Theta(x,y) = \arctan{\left(\frac{G_y(x,y)}{G_x(x,y)}\right)}.
</math>

== Example comparisons ==
Here, four different gradient operators are used to estimate the magnitude of the gradient of the test image.

{|
|[[Image:Bikesgray.jpg|thumb|200px|Grayscale test image of brick wall and bike rack]]

|[[Image:Bikesgray_roberts.JPG|thumb|200px|Gradient magnitude from Roberts cross operator]]

|[[Image:Bikesgray_sobel.JPG|thumb|200px|Gradient magnitude from [[Sobel operator]]]]
|-
|[[Image:Bikesgray-scharr.png|thumb|200px|Gradient magnitude from Scharr operator]]

|[[Image:Bikesgray_prewitt.JPG|thumb|200px|Gradient magnitude from [[Prewitt operator]]]]
|}

==See also==
* [[Digital image processing]]
* [[Feature detection (computer vision)]]
* [[Feature extraction]]
* [[Sobel operator]]
* [[Prewitt operator]]

==References==
<references/>

[[Category:Feature detection]]

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14.200.68.118: /* Legal applications */

en>Seonookim: Added {{technical}} tag to article (TW)