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doc/tutorials/imgproc/anisotropic_image_segmentation/anisotropic_image_segmentation.markdown

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+Anisotropic image segmentation by a gradient structure tensor {#tutorial_anisotropic_image_segmentation_by_a_gst}
+==========================
+
+@tableofcontents
+
+@prev_tutorial{tutorial_motion_deblur_filter}
+@next_tutorial{tutorial_periodic_noise_removing_filter}
+
+|    |    |
+| -: | :- |
+| Original author | Karpushin Vladislav |
+| Compatibility | OpenCV >= 3.0 |
+
+Goal
+----
+
+In this tutorial you will learn:
+
+-   what the gradient structure tensor is
+-   how to estimate orientation and coherency of an anisotropic image by a gradient structure tensor
+-   how to segment an anisotropic image with a single local orientation by a gradient structure tensor
+
+Theory
+------
+
+@note The explanation is based on the books @cite jahne2000computer, @cite bigun2006vision and @cite van1995estimators. Good physical explanation of a gradient structure tensor is given in @cite yang1996structure. Also, you can refer to a wikipedia page [Structure tensor].
+@note A anisotropic image on this page is a real world  image.
+
+### What is the gradient structure tensor?
+
+In mathematics, the gradient structure tensor (also referred to as the second-moment matrix, the second order moment tensor, the inertia tensor, etc.) is a matrix derived from the gradient of a function. It summarizes the predominant directions of the gradient in a specified neighborhood of a point, and the degree to which those directions are coherent (coherency). The gradient structure tensor is widely used in image processing and computer vision for 2D/3D image segmentation, motion detection, adaptive filtration, local image features detection, etc.
+
+Important features of anisotropic images include orientation and coherency of a local anisotropy. In this paper we will show how to estimate orientation and coherency, and how to segment an anisotropic image with a single local orientation by a gradient structure tensor.
+
+The gradient structure tensor of an image is a 2x2 symmetric matrix. Eigenvectors of the gradient structure tensor indicate local orientation, whereas eigenvalues give coherency (a measure of anisotropism).
+
+The gradient structure tensor \f$J\f$ of an image \f$Z\f$ can be written as:
+
+\f[J = \begin{bmatrix}
+J_{11} & J_{12}  \\
+J_{12} & J_{22}
+\end{bmatrix}\f]
+
+where \f$J_{11} = M[Z_{x}^{2}]\f$, \f$J_{22} = M[Z_{y}^{2}]\f$, \f$J_{12} = M[Z_{x}Z_{y}]\f$ - components of the tensor, \f$M[]\f$ is a symbol of mathematical expectation (we can consider this operation as averaging in a window w), \f$Z_{x}\f$ and \f$Z_{y}\f$ are partial derivatives of an image \f$Z\f$ with respect to \f$x\f$ and \f$y\f$.
+
+The eigenvalues of the tensor can be found in the below formula:
+\f[\lambda_{1,2} = \frac{1}{2} \left [ J_{11} + J_{22} \pm \sqrt{(J_{11} - J_{22})^{2} + 4J_{12}^{2}} \right ] \f]
+where \f$\lambda_1\f$ - largest eigenvalue, \f$\lambda_2\f$ - smallest eigenvalue.
+
+### How to estimate orientation and coherency of an anisotropic image by gradient structure tensor?
+
+The orientation of an anisotropic image:
+\f[\alpha = 0.5arctg\frac{2J_{12}}{J_{22} - J_{11}}\f]
+
+Coherency:
+\f[C = \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\f]
+
+The coherency ranges from 0 to 1. For ideal local orientation (\f$\lambda_2\f$ = 0, \f$\lambda_1\f$ > 0) it is one, for an isotropic gray value structure (\f$\lambda_1\f$ = \f$\lambda_2\f$ \> 0) it is zero.
+
+Source code
+-----------
+
+You can find source code in the `samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp` of the OpenCV source code library.
+
+@add_toggle_cpp
+    @include cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp
+@end_toggle
+
+@add_toggle_python
+    @include samples/python/tutorial_code/imgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.py
+@end_toggle
+
+Explanation
+-----------
+An anisotropic image segmentation algorithm consists of a gradient structure tensor calculation, an orientation calculation, a coherency calculation and an orientation and coherency thresholding:
+
+@add_toggle_cpp
+    @snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp main
+@end_toggle
+
+@add_toggle_python
+    @snippet samples/python/tutorial_code/imgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.py main
+@end_toggle
+
+A function calcGST() calculates orientation and coherency by using a gradient structure tensor. An input parameter w defines a window size:
+
+@add_toggle_cpp
+    @snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp calcGST
+@end_toggle
+
+@add_toggle_python
+    @snippet samples/python/tutorial_code/imgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.py calcGST
+@end_toggle
+
+
+The below code applies a thresholds LowThr and HighThr to image orientation and a threshold C_Thr to image coherency calculated by the previous function. LowThr and HighThr define orientation range:
+
+@add_toggle_cpp
+    @snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp thresholding
+@end_toggle
+
+@add_toggle_python
+    @snippet samples/python/tutorial_code/imgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.py thresholding
+@end_toggle
+
+
+And finally we combine thresholding results:
+
+@add_toggle_cpp
+    @snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp combining
+@end_toggle
+
+@add_toggle_python
+    @snippet samples/python/tutorial_code/imgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.py combining
+@end_toggle
+
+
+Result
+------
+
+Below you can see the real anisotropic image with single direction:
+![Anisotropic image with the single direction](images/gst_input.jpg)
+
+Below you can see the orientation and coherency of the anisotropic image:
+![Orientation](images/gst_orientation.jpg)
+![Coherency](images/gst_coherency.jpg)
+
+Below you can see the segmentation result:
+![Segmentation result](images/gst_result.jpg)
+
+The result has been computed with w = 52, C_Thr = 0.43, LowThr = 35, HighThr = 57. We can see that the algorithm selected only the areas with one single direction.
+
+References
+------
+- [Structure tensor] - structure tensor description on the wikipedia
+
+<!-- invisible references list -->
+[Structure tensor]: https://en.wikipedia.org/wiki/Structure_tensor