OpenCV  4.5.1
Open Source Computer Vision
Anisotropic image segmentation by a gradient structure tensor

Prev Tutorial: Motion Deblur Filter
Next 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 [jahne2000computer], [bigun2006vision] and [van1995estimators]. Good physical explanation of a gradient structure tensor is given in [yang1996structure]. Also, you can refer to a wikipedia page Structure tensor.
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 \(J\) of an image \(Z\) can be written as:

\[J = \begin{bmatrix} J_{11} & J_{12} \\ J_{12} & J_{22} \end{bmatrix}\]

where \(J_{11} = M[Z_{x}^{2}]\), \(J_{22} = M[Z_{y}^{2}]\), \(J_{12} = M[Z_{x}Z_{y}]\) - components of the tensor, \(M[]\) is a symbol of mathematical expectation (we can consider this operation as averaging in a window w), \(Z_{x}\) and \(Z_{y}\) are partial derivatives of an image \(Z\) with respect to \(x\) and \(y\).

The eigenvalues of the tensor can be found in the below formula:

\[\lambda_{1,2} = \frac{1}{2} \left [ J_{11} + J_{22} \pm \sqrt{(J_{11} - J_{22})^{2} + 4J_{12}^{2}} \right ] \]

where \(\lambda_1\) - largest eigenvalue, \(\lambda_2\) - smallest eigenvalue.

How to estimate orientation and coherency of an anisotropic image by gradient structure tensor?

The orientation of an anisotropic image:

\[\alpha = 0.5arctg\frac{2J_{12}}{J_{22} - J_{11}}\]

Coherency:

\[C = \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\]

The coherency ranges from 0 to 1. For ideal local orientation ( \(\lambda_2\) = 0, \(\lambda_1\) > 0) it is one, for an isotropic gray value structure ( \(\lambda_1\) = \(\lambda_2\) > 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.

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:

A function calcGST() calculates orientation and coherency by using a gradient structure tensor. An input parameter w defines a window size:

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:

And finally we combine thresholding results:

Result

Below you can see the real anisotropic image with single direction:

Below you can see the orientation and coherency of the anisotropic image:

Below you can see the segmentation result:

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

cv::imread
CV_EXPORTS_W Mat imread(const String &filename, int flags=IMREAD_COLOR)
Loads an image from a file.
cv::bitwise_and
void bitwise_and(InputArray src1, InputArray src2, OutputArray dst, InputArray mask=noArray())
computes bitwise conjunction of the two arrays (dst = src1 & src2) Calculates the per-element bit-wis...
cv::NORM_MINMAX
@ NORM_MINMAX
flag
Definition: base.hpp:207
cv::Sobel
void Sobel(InputArray src, OutputArray dst, int ddepth, int dx, int dy, int ksize=3, double scale=1, double delta=0, int borderType=BORDER_DEFAULT)
Calculates the first, second, third, or mixed image derivatives using an extended Sobel operator.
cv::inRange
void inRange(InputArray src, InputArray lowerb, InputArray upperb, OutputArray dst)
Checks if array elements lie between the elements of two other arrays.
cv::normalize
void normalize(const SparseMat &src, SparseMat &dst, double alpha, int normType)
cv::threshold
double threshold(InputArray src, OutputArray dst, double thresh, double maxval, int type)
Applies a fixed-level threshold to each array element.
cv::waitKey
int waitKey(int delay=0)
Waits for a pressed key.
cv::Mat::convertTo
void convertTo(OutputArray m, int rtype, double alpha=1, double beta=0) const
Converts an array to another data type with optional scaling.
cv::Size
Size2i Size
Definition: types.hpp:347
CV_32F
#define CV_32F
Definition: interface.h:78
cv::Mat::empty
bool empty() const
Returns true if the array has no elements.
imgcodecs.hpp
cv::IMREAD_GRAYSCALE
@ IMREAD_GRAYSCALE
If set, always convert image to the single channel grayscale image (codec internal conversion).
Definition: imgcodecs.hpp:71
cv::boxFilter
void boxFilter(InputArray src, OutputArray dst, int ddepth, Size ksize, Point anchor=Point(-1,-1), bool normalize=true, int borderType=BORDER_DEFAULT)
Blurs an image using the box filter.
cv::dnn::print
static void print(const MatShape &shape, const String &name="")
Definition: shape_utils.hpp:198
cv::divide
void divide(double scale, InputArray src2, OutputArray dst, int dtype=-1)
cv::Scalar
Scalar_< double > Scalar
Definition: types.hpp:669
cv::Mat
n-dimensional dense array class
Definition: mat.hpp:798
cv::imshow
void imshow(const String &winname, const ogl::Texture2D &tex)
Displays OpenGL 2D texture in the specified window.
cv
"black box" representation of the file storage associated with a file on disk.
Definition: affine.hpp:52
cv::phase
void phase(InputArray x, InputArray y, OutputArray angle, bool angleInDegrees=false)
Calculates the rotation angle of 2D vectors.
imgproc.hpp
cv::sqrt
Quat< S > sqrt(const Quat< S > &q, QuatAssumeType assumeUnit=QUAT_ASSUME_NOT_UNIT)
cv::multiply
void multiply(InputArray src1, InputArray src2, OutputArray dst, double scale=1, int dtype=-1)
Calculates the per-element scaled product of two arrays.
cv::imwrite
CV_EXPORTS_W bool imwrite(const String &filename, InputArray img, const std::vector< int > &params=std::vector< int >())
Saves an image to a specified file.
cv::divide
void divide(InputArray src1, InputArray src2, OutputArray dst, double scale=1, int dtype=-1)
Performs per-element division of two arrays or a scalar by an array.
cv::normalize
static Vec< _Tp, cn > normalize(const Vec< _Tp, cn > &v)