sobel_derivatives.markdown 6.8 KB
Sobel Derivatives {#tutorial_sobel_derivatives}
@prev_tutorial{tutorial_copyMakeBorder} @next_tutorial{tutorial_laplace_operator}
| | | | -: | :- | | Original author | Ana Huamán | | Compatibility | OpenCV >= 3.0 |
Goal
In this tutorial you will learn how to:
- Use the OpenCV function Sobel() to calculate the derivatives from an image.
- Use the OpenCV function 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]
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 **Scharr()** function. This is as fast
but more accurate than the standard 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]
You can check out more information of this function in the OpenCV reference - **Scharr()** .
Also, in the sample code below, you will notice that above the code for **Sobel()** function
there is also code for the **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.
@add_toggle_cpp You can also download it from here @include samples/cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp @end_toggle
@add_toggle_java You can also download it from here @include samples/java/tutorial_code/ImgTrans/SobelDemo/SobelDemo.java @end_toggle
@add_toggle_python You can also download it from here @include samples/python/tutorial_code/ImgTrans/SobelDemo/sobel_demo.py @end_toggle
Explanation
Declare variables
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp variables
Load source image
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp load
Reduce noise
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp reduce_noise
Grayscale
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp convert_to_gray
Sobel Operator
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp sobel
We calculate the "derivatives" in x and y directions. For this, we use the function Sobel() as shown below: 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.
Convert output to a CV_8U image
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp convert
Gradient
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp blend
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).
Show results
@snippet cpp/tutorial_code/ImgTrans/Sobel_Demo.cpp display
Results
-# Here is the output of applying our basic detector to lena.jpg:
