.. _sobel_derivatives:
Sobel Derivatives
******************
Goal
=====
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`
Theory
========
.. note::
The explanation below belongs to the book **Learning OpenCV** by Bradski and Kaehler.
#. In the last two tutorials we have seen applicative examples of convolutions. One of the most important convolutions is the computation of derivatives in an image (or an approximation to them).
#. Why may be important the calculus of the derivatives in an image? Let's imagine we want to detect the *edges* present in the image. For instance:
.. 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.
#. 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:
.. image:: images/Sobel_Derivatives_Tutorial_Theory_Intensity_Function.jpg
:alt: Intensity Plot for an edge
:align: center
#. The edge "jump" can be seen more easily if we take the first derivative (actually, here appears as a maximum)
.. image:: images/Sobel_Derivatives_Tutorial_Theory_dIntensity_Function.jpg
:alt: First derivative of Intensity - Plot for an edge
:align: center
#. So, from the explanation above, we can deduce that a method to detect edges in an image can be performed by locating pixel locations where the gradient is higher than its neighbors (or to generalize, higher than a threshold).
#. More detailed explanation, please refer to **Learning OpenCV** by Bradski and Kaehler
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 combines Gaussian smoothing and differentiation.
Formulation
^^^^^^^^^^^^
Assuming that the image to be operated is :math:`I`:
#. We calculate two derivatives:
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
\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
\end{bmatrix} * I
#. At each point of the image we calculate an approximation of the *gradient* in that point by combining both results above:
.. math::
G = \sqrt{ G_{x}^{2} + G_{y}^{2} }
Although sometimes the following simpler equation is used:
.. math::
G = |G_{x}| + |G_{y}|
.. note::
When the size of the kernel is :math:`3`, the Sobel kernel shown above may produce noticeable inaccuracies (after all, Sobel is only an approximation of the derivative). OpenCV addresses this inaccuracy for kernels of size 3 by using the :scharr:`Scharr <>` function. This is as fast but more accurate than the standar Sobel function. It implements the following kernels:
.. math::
G_{x} = \begin{bmatrix}
-3 & 0 & +3 \\
-10 & 0 & +10 \\
-3 & 0 & +3
\end{bmatrix}
G_{y} = \begin{bmatrix}
-3 & -10 & -3 \\
0 & 0 & 0 \\
+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.
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 `_
.. code-block:: cpp
#include "opencv2/imgproc/imgproc.hpp"
#include "opencv2/highgui/highgui.hpp"
#include
#include
using namespace cv;
/** @function main */
int main( int argc, char** argv )
{
Mat src, src_gray;
Mat grad;
char* window_name = "Sobel Demo - Simple Edge Detector";
int scale = 1;
int delta = 0;
int ddepth = CV_16S;
int c;
/// Load an image
src = imread( argv[1] );
if( !src.data )
{ return -1; }
GaussianBlur( src, src, Size(3,3), 0, 0, BORDER_DEFAULT );
/// Convert it to gray
cvtColor( src, src_gray, CV_BGR2GRAY );
/// Create window
namedWindow( window_name, CV_WINDOW_AUTOSIZE );
/// 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 );
convertScaleAbs( grad_x, abs_grad_x );
/// 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 );
convertScaleAbs( grad_y, abs_grad_y );
/// Total Gradient (approximate)
addWeighted( abs_grad_x, 0.5, abs_grad_y, 0.5, 0, grad );
imshow( window_name, grad );
waitKey(0);
return 0;
}
Explanation
=============
#. First we declare the variables we are going to use:
.. code-block:: cpp
Mat src, src_gray;
Mat grad;
char* window_name = "Sobel Demo - Simple Edge Detector";
int scale = 1;
int delta = 0;
int ddepth = CV_16S;
#. As usual we load our source image *src*:
.. 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 );
#. Now we convert our filtered image to grayscale:
.. code-block:: cpp
cvtColor( src, src_gray, CV_BGR2GRAY );
#. 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 );
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.
* *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.
* *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.
#. We convert our partial results back to *CV_8U*:
.. code-block:: cpp
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).
.. code-block:: cpp
addWeighted( abs_grad_x, 0.5, abs_grad_y, 0.5, 0, grad );
#. Finally, we show our result:
.. code-block:: cpp
imshow( window_name, grad );
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
:width: 300pt
:align: center