Sobel Derivatives {#tutorial_sobel_derivatives}
=================
Goal
----
In this tutorial you will learn how to:
- Use the OpenCV function @ref cv::Sobel to calculate the derivatives from an image.
- Use the OpenCV function @ref cv::Scharr to calculate a more accurate derivative for a kernel of
size \f$3 \cdot 3\f$
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:
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:
-# The edge "jump" can be seen more easily if we take the first derivative (actually, here appears
as a maximum)
-# 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 \f$I\f$:
-# We calculate two derivatives:
-# **Horizontal changes**: This is computed by convolving \f$I\f$ with a kernel \f$G_{x}\f$ with odd
size. For example for a kernel size of 3, \f$G_{x}\f$ would be computed as:
\f[G_{x} = \begin{bmatrix}
-1 & 0 & +1 \\
-2 & 0 & +2 \\
-1 & 0 & +1
\end{bmatrix} * I\f]
-# **Vertical changes**: This is computed by convolving \f$I\f$ with a kernel \f$G_{y}\f$ with odd
size. For example for a kernel size of 3, \f$G_{y}\f$ would be computed as:
\f[G_{y} = \begin{bmatrix}
-1 & -2 & -1 \\
0 & 0 & 0 \\
+1 & +2 & +1
\end{bmatrix} * I\f]
-# At each point of the image we calculate an approximation of the *gradient* in that point by
combining both results above:
\f[G = \sqrt{ G_{x}^{2} + G_{y}^{2} }\f]
Although sometimes the following simpler equation is used:
\f[G = |G_{x}| + |G_{y}|\f]
@note
When the size of the kernel is `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 @ref cv::Scharr function. This is as fast
but more accurate than the standar Sobel function. It implements the following kernels:
\f[G_{x} = \begin{bmatrix}
-3 & 0 & +3 \\
-10 & 0 & +10 \\
-3 & 0 & +3
\end{bmatrix}\f]\f[G_{y} = \begin{bmatrix}
-3 & -10 & -3 \\
0 & 0 & 0 \\
+3 & +10 & +3
\end{bmatrix}\f]
@note
You can check out more information of this function in the OpenCV reference (@ref cv::Scharr ).
Also, in the sample code below, you will notice that above the code for @ref cv::Sobel function
there is also code for the @ref cv::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](https://github.com/opencv/opencv/tree/master/samples/cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp)
@include samples/cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp
Explanation
-----------
-# First we declare the variables we are going to use:
@code{.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;
@endcode
-# As usual we load our source image *src*:
@code{.cpp}
src = imread( argv[1] );
if( !src.data )
{ return -1; }
@endcode
-# First, we apply a @ref cv::GaussianBlur to our image to reduce the noise ( kernel size = 3 )
@code{.cpp}
GaussianBlur( src, src, Size(3,3), 0, 0, BORDER_DEFAULT );
@endcode
-# Now we convert our filtered image to grayscale:
@code{.cpp}
cvtColor( src, src_gray, COLOR_RGB2GRAY );
@endcode
-# Second, we calculate the "*derivatives*" in *x* and *y* directions. For this, we use the
function @ref cv::Sobel as shown below:
@code{.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 );
@endcode
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: \f$x_{order}= 1\f$ and
\f$y_{order} = 0\f$. We do analogously for the *y* direction.
-# We convert our partial results back to *CV_8U*:
@code{.cpp}
convertScaleAbs( grad_x, abs_grad_x );
convertScaleAbs( grad_y, abs_grad_y );
@endcode
-# 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{.cpp}
addWeighted( abs_grad_x, 0.5, abs_grad_y, 0.5, 0, grad );
@endcode
-# Finally, we show our result:
@code{.cpp}
imshow( window_name, grad );
@endcode
Results
-------
-# Here is the output of applying our basic detector to *lena.jpg*:
