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Andrey Kamaev
2012-08-07 13:29:43 +04:00
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commit 5100ca7508
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82
doc/tutorials/imgproc/imgtrans/sobel_derivatives/sobel_derivatives.rst
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@@ -12,8 +12,8 @@ In this tutorial you will learn how to:
.. container:: enumeratevisibleitemswithsquare
* Use the OpenCV function :sobel:`Sobel <>` to calculate the derivatives from an image.
* Use the OpenCV function :scharr:`Scharr <>` to calculate a more accurate derivative for a kernel of size :math:`3 \cdot 3`
* Use the OpenCV function :scharr:`Scharr <>` to calculate a more accurate derivative for a kernel of size :math:`3 \cdot 3`
Theory
========
@@ -29,8 +29,8 @@ Theory
.. image:: images/Sobel_Derivatives_Tutorial_Theory_0.jpg
:alt: How intensity changes in an edge
:align: center
You can easily notice that in an *edge*, the pixel intensity *changes* in a notorious way. A good way to express *changes* is by using *derivatives*. A high change in gradient indicates a major change in the image.
You can easily notice that in an *edge*, the pixel intensity *changes* in a notorious way. A good way to express *changes* is by using *derivatives*. A high change in gradient indicates a major change in the image.
#. To be more graphical, let's assume we have a 1D-image. An edge is shown by the "jump" in intensity in the plot below:
@@ -51,9 +51,9 @@ Theory
Sobel Operator
---------------
#. The Sobel Operator is a discrete differentiation operator. It computes an approximation of the gradient of an image intensity function.
#. The Sobel Operator is a discrete differentiation operator. It computes an approximation of the gradient of an image intensity function.
#. The Sobel Operator combines Gaussian smoothing and differentiation.
#. The Sobel Operator combines Gaussian smoothing and differentiation.
Formulation
^^^^^^^^^^^^
@@ -64,21 +64,21 @@ Assuming that the image to be operated is :math:`I`:
a. **Horizontal changes**: This is computed by convolving :math:`I` with a kernel :math:`G_{x}` with odd size. For example for a kernel size of 3, :math:`G_{x}` would be computed as:
.. math::
G_{x} = \begin{bmatrix}
-1 & 0 & +1 \\
-2 & 0 & +2 \\
-1 & 0 & +1
-1 & 0 & +1
\end{bmatrix} * I
b. **Vertical changes**: This is computed by convolving :math:`I` with a kernel :math:`G_{y}` with odd size. For example for a kernel size of 3, :math:`G_{y}` would be computed as:
.. math::
G_{y} = \begin{bmatrix}
-1 & -2 & -1 \\
0 & 0 & 0 \\
+1 & +2 & +1
+1 & +2 & +1
\end{bmatrix} * I
#. At each point of the image we calculate an approximation of the *gradient* in that point by combining both results above:
@@ -90,7 +90,7 @@ Assuming that the image to be operated is :math:`I`:
Although sometimes the following simpler equation is used:
.. math::
G = |G_{x}| + |G_{y}|
@@ -103,14 +103,14 @@ Assuming that the image to be operated is :math:`I`:
G_{x} = \begin{bmatrix}
-3 & 0 & +3 \\
-10 & 0 & +10 \\
-3 & 0 & +3
\end{bmatrix}
-3 & 0 & +3
\end{bmatrix}
G_{y} = \begin{bmatrix}
-3 & -10 & -3 \\
0 & 0 & 0 \\
+3 & +10 & +3
\end{bmatrix}
+3 & +10 & +3
\end{bmatrix}
You can check out more information of this function in the OpenCV reference (:scharr:`Scharr <>`). Also, in the sample code below, you will notice that above the code for :sobel:`Sobel <>` function there is also code for the :scharr:`Scharr <>` function commented. Uncommenting it (and obviously commenting the Sobel stuff) should give you an idea of how this function works.
@@ -118,12 +118,12 @@ Code
=====
#. **What does this program do?**
* Applies the *Sobel Operator* and generates as output an image with the detected *edges* bright on a darker background.
#. The tutorial code's is shown lines below. You can also download it from `here <http://code.opencv.org/svn/opencv/trunk/opencv/samples/cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp>`_
.. code-block:: cpp
* Applies the *Sobel Operator* and generates as output an image with the detected *edges* bright on a darker background.
#. The tutorial code's is shown lines below. You can also download it from `here <http://code.opencv.org/projects/opencv/repository/revisions/master/raw/samples/cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp>`_
.. code-block:: cpp
#include "opencv2/imgproc/imgproc.hpp"
#include "opencv2/highgui/highgui.hpp"
@@ -137,7 +137,7 @@ Code
{
Mat src, src_gray;
Mat grad;
Mat grad;
char* window_name = "Sobel Demo - Simple Edge Detector";
int scale = 1;
int delta = 0;
@@ -162,15 +162,15 @@ Code
/// Generate grad_x and grad_y
Mat grad_x, grad_y;
Mat abs_grad_x, abs_grad_y;
/// Gradient X
//Scharr( src_gray, grad_x, ddepth, 1, 0, scale, delta, BORDER_DEFAULT );
Sobel( src_gray, grad_x, ddepth, 1, 0, 3, scale, delta, BORDER_DEFAULT );
Sobel( src_gray, grad_x, ddepth, 1, 0, 3, scale, delta, BORDER_DEFAULT );
convertScaleAbs( grad_x, abs_grad_x );
/// Gradient Y
/// Gradient Y
//Scharr( src_gray, grad_y, ddepth, 0, 1, scale, delta, BORDER_DEFAULT );
Sobel( src_gray, grad_y, ddepth, 0, 1, 3, scale, delta, BORDER_DEFAULT );
Sobel( src_gray, grad_y, ddepth, 0, 1, 3, scale, delta, BORDER_DEFAULT );
convertScaleAbs( grad_y, abs_grad_y );
/// Total Gradient (approximate)
@@ -192,7 +192,7 @@ Explanation
.. code-block:: cpp
Mat src, src_gray;
Mat grad;
Mat grad;
char* window_name = "Sobel Demo - Simple Edge Detector";
int scale = 1;
int delta = 0;
@@ -203,12 +203,12 @@ Explanation
.. code-block:: cpp
src = imread( argv[1] );
if( !src.data )
{ return -1; }
#. First, we apply a :gaussian_blur:`GaussianBlur <>` to our image to reduce the noise ( kernel size = 3 )
.. code-block:: cpp
GaussianBlur( src, src, Size(3,3), 0, 0, BORDER_DEFAULT );
@@ -220,27 +220,27 @@ Explanation
cvtColor( src, src_gray, CV_RGB2GRAY );
#. Second, we calculate the "*derivatives*" in *x* and *y* directions. For this, we use the function :sobel:`Sobel <>` as shown below:
.. code-block:: cpp
Mat grad_x, grad_y;
Mat abs_grad_x, abs_grad_y;
/// Gradient X
Sobel( src_gray, grad_x, ddepth, 1, 0, 3, scale, delta, BORDER_DEFAULT );
/// Gradient Y
Sobel( src_gray, grad_y, ddepth, 0, 1, 3, scale, delta, BORDER_DEFAULT );
Sobel( src_gray, grad_x, ddepth, 1, 0, 3, scale, delta, BORDER_DEFAULT );
/// Gradient Y
Sobel( src_gray, grad_y, ddepth, 0, 1, 3, scale, delta, BORDER_DEFAULT );
The function takes the following arguments:
* *src_gray*: In our example, the input image. Here it is *CV_8U*
* *grad_x*/*grad_y*: The output image.
* *src_gray*: In our example, the input image. Here it is *CV_8U*
* *grad_x*/*grad_y*: The output image.
* *ddepth*: The depth of the output image. We set it to *CV_16S* to avoid overflow.
* *x_order*: The order of the derivative in **x** direction.
* *y_order*: The order of the derivative in **y** direction.
* *x_order*: The order of the derivative in **x** direction.
* *y_order*: The order of the derivative in **y** direction.
* *scale*, *delta* and *BORDER_DEFAULT*: We use default values.
Notice that to calculate the gradient in *x* direction we use: :math:`x_{order}= 1` and :math:`y_{order} = 0`. We do analogously for the *y* direction.
Notice that to calculate the gradient in *x* direction we use: :math:`x_{order}= 1` and :math:`y_{order} = 0`. We do analogously for the *y* direction.
#. We convert our partial results back to *CV_8U*:
@@ -248,7 +248,7 @@ Explanation
convertScaleAbs( grad_x, abs_grad_x );
convertScaleAbs( grad_y, abs_grad_y );
#. Finally, we try to approximate the *gradient* by adding both directional gradients (note that this is not an exact calculation at all! but it is good for our purposes).
@@ -268,7 +268,7 @@ Results
========
#. Here is the output of applying our basic detector to *lena.jpg*:
.. image:: images/Sobel_Derivatives_Tutorial_Result.jpg
:alt: Result of applying Sobel operator to lena.jpg