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Alexander Alekhin
6 years ago
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Anisotropic image segmentation by a gradient structure tensor {#tutorial_anisotropic_image_segmentation_by_a_gst} |
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========================== |
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Goal |
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---- |
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In this tutorial you will learn: |
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- what the gradient structure tensor is |
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- how to estimate orientation and coherency of an anisotropic image by a gradient structure tensor |
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- how to segment an anisotropic image with a single local orientation by a gradient structure tensor |
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Theory |
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------ |
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@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]. |
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@note A anisotropic image on this page is a real world image. |
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### What is the gradient structure tensor? |
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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. |
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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. |
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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). |
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The gradient structure tensor \f$J\f$ of an image \f$Z\f$ can be written as: |
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\f[J = \begin{bmatrix} |
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J_{11} & J_{12} \\ |
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J_{12} & J_{22} |
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\end{bmatrix}\f] |
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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$y\f$. |
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The eigenvalues of the tensor can be found in the below formula: |
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\f[\lambda_{1,2} = J_{11} + J_{22} \pm \sqrt{(J_{11} - J_{22})^{2} + 4J_{12}^{2}}\f] |
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where \f$\lambda_1\f$ - largest eigenvalue, \f$\lambda_2\f$ - smallest eigenvalue. |
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### How to estimate orientation and coherency of an anisotropic image by gradient structure tensor? |
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The orientation of an anisotropic image: |
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\f[\alpha = 0.5arctg\frac{2J_{12}}{J_{22} - J_{11}}\f] |
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Coherency: |
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\f[C = \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\f] |
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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. |
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Source code |
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----------- |
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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. |
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@include cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp |
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Explanation |
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An anisotropic image segmentation algorithm consists of a gradient structure tensor calculation, an orientation calculation, a coherency calculation and an orientation and coherency thresholding: |
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@snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp main |
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A function calcGST() calculates orientation and coherency by using a gradient structure tensor. An input parameter w defines a window size: |
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@snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp calcGST |
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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: |
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@snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp thresholding |
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And finally we combine thresholding results: |
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@snippet samples/cpp/tutorial_code/ImgProc/anisotropic_image_segmentation/anisotropic_image_segmentation.cpp combining |
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Result |
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------ |
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Below you can see the real anisotropic image with single direction: |
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Below you can see the orientation and coherency of the anisotropic image: |
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Below you can see the segmentation result: |
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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. |
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References |
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- [Structure tensor] - structure tensor description on the wikipedia |
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<!-- invisible references list --> |
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[Structure tensor]: https://en.wikipedia.org/wiki/Structure_tensor |
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