Theory¶

How OrientationJ measures a direction: the tensor it builds around every pixel, and the features it reads from it. The two settings, scale and gradient, are explained in how to use.

Gradient Structure Tensor¶

The aim is to characterize the orientation and isotropy properties of a local area of interest (Region of Interest, ROI) in an image. To that end, we first define the weighted inner product

\[ \langle f, g \rangle_w = \iint_{\mathbb{R}^2} w(x,y)\, f(x,y)\, g(x,y)\, \mathrm{d}x\, \mathrm{d}y , \]

where \(w(x,y) \geq 0\) is a weighting function that specifies the area of interest. It is typically a normalized square window of size \(L\) centered on a location of interest \((x_0, y_0)\), or a Gaussian window of standard deviation \(\sigma_w\). The norm associated with this inner product is \(\lVert f \rVert_w = \sqrt{\langle f, f \rangle_w}\). Next, we consider the derivative in the direction specified by the unit vector \(\mathbf{u}_\theta = (\cos\theta, \sin\theta)^\top\), which is given by

\[ D_{\mathbf{u}_\theta} f(x,y) = \mathbf{u}_\theta^\top\, \nabla f(x,y) , \]

where \(\nabla f = (f_x, f_y)^\top\) is the gradient of the image under consideration. We are now interested in finding the direction \(\mathbf{u}\) along which the directional derivative is maximized over the ROI:

\[ \mathbf{u}_{\max} = \arg\max_{\lVert\mathbf{u}\rVert = 1}\ \lVert D_{\mathbf{u}} f \rVert_w^2 . \]

A standard inner-product manipulation then yields

\[ \lVert D_{\mathbf{u}} f \rVert_w^2 = \langle \mathbf{u}^\top \nabla f,\ \nabla f^\top \mathbf{u} \rangle_w = \mathbf{u}^\top \mathbf{J}\, \mathbf{u} , \]

\[ \mathbf{J} = \langle \nabla f, \nabla f^\top \rangle_w = \begin{bmatrix} \langle f_x, f_x \rangle_w & \langle f_x, f_y \rangle_w \\ \langle f_x, f_y \rangle_w & \langle f_y, f_y \rangle_w \end{bmatrix}, \]

where \(\mathbf{J}\) is the so-called structure tensor, a \(2 \times 2\) symmetric positive-semidefinite matrix. The solution of the optimization problem is obtained by setting the derivative of \(\mathbf{u}^\top \mathbf{J} \mathbf{u} + \lambda (1 - \mathbf{u}^\top \mathbf{u})\) with respect to \(\mathbf{u}\) to zero, which yields the eigenvector equation

\[ \mathbf{J}\, \mathbf{u} = \lambda\, \mathbf{u} . \]

This implies that the first eigenvector \(\mathbf{e}_1\) of \(\mathbf{J}\) gives the direction of maximal gradient energy; the corresponding eigenvalue is \(\lambda_1 = \max \lVert D_{\mathbf{u}} f \rVert_w^2\). Conversely, the directional derivative is minimized along the second eigenvector \(\mathbf{e}_2\), with \(\lambda_2 = \min \lVert D_{\mathbf{u}} f \rVert_w^2\). Since the gradient is perpendicular to the local structures, the visible structures (fibers, edges) are aligned with \(\mathbf{e}_2\); this is the orientation reported by OrientationJ. The structure tensor therefore contains all the relevant directional information of the ROI.

Features and invariants¶

Principle¶

The tensor of a region is a symmetric 2×2 matrix, so it carries exactly three numbers. Everything the plugin reports is a function of its two eigenvalues \(\lambda_1 \ge \lambda_2\) and of the direction of the first eigenvector: the orientation is that direction, and the features below are combinations of the eigenvalues — their difference over their sum, their sum, their squared difference. Choosing between them is choosing what to be sensitive to, not changing the measurement.

Features and tensor invariants¶

With the eigenvalues \(\lambda_1 \geq \lambda_2 \geq 0\), the mean \(\bar\lambda = \tfrac12 \operatorname{tr}(\mathbf{J})\), and the deviator \(\mathbf{s} = \mathbf{J} - \tfrac12 \operatorname{tr}(\mathbf{J})\, \mathbf{I}\), OrientationJ computes the following features.

Orientation — the direction of the structures themselves, that of the eigenvector of the smaller eigenvalue, along which the intensity varies least. In \([-90°, +90°]\): an angle, not a vector, since a structure and its opposite are the same structure.

\[ \theta = \frac{1}{2} \arctan\!\left( \frac{2 \langle f_x, f_y \rangle_w}{\langle f_y, f_y \rangle_w - \langle f_x, f_x \rangle_w} \right) \in [-\pi/2,\ \pi/2] \]

Energy — how much gradient there is in the window, whatever its direction. Zero on a flat region, large on a contrasted one; unbounded, and proportional to the square of the image contrast, so it is a relative quantity, comparable only between images acquired alike.

\[ E = \operatorname{tr}(\mathbf{J}) = \lambda_1 + \lambda_2 \in [0, \infty) \]

Coherency — how well defined that orientation is: \(C = 1\) where a single orientation dominates, \(C = 0\) where the neighborhood is isotropic and the angle means nothing. Bounded in \([0, 1]\) and independent of contrast, it is the quantity to threshold on, and the one coherence-enhancing methods build upon (Weickert, 1999).

\[ C = \frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2} = \frac{\sqrt{\bigl( \langle f_y, f_y \rangle_w - \langle f_x, f_x \rangle_w \bigr)^2 + 4 \langle f_x, f_y \rangle_w^2}}{\langle f_x, f_x \rangle_w + \langle f_y, f_y \rangle_w} \in [0, 1] \]

Directionality — the second invariant of the deviator, the von Mises invariant. It grows with contrast and with alignment at once, being the product of the two: unnormalized and unbounded, useful to rank regions within one image rather than across images.

\[ J_2 = \tfrac12 \operatorname{tr}(\mathbf{s}^2) = \tfrac14 (\lambda_1 - \lambda_2)^2 = \tfrac14\, C^2 E^2 \in [0, \infty) \]

Fractional anisotropy — the same information as the coherency, normalized by the Frobenius norm of the tensor instead of its trace, in the usage established by diffusion-tensor imaging (Basser & Pierpaoli, 1996). Also in \([0, 1]\), and in one-to-one correspondence with \(C\), so the choice between them is one of habit.

\[ \mathrm{FA} = \frac{\lambda_1 - \lambda_2}{\sqrt{\lambda_1^2 + \lambda_2^2}} = \frac{\sqrt{2}\, C}{\sqrt{1 + C^2}} \in [0, 1] \]

Summary of features and invariants¶

Two independent scalars fix the tensor up to rotation; complete sets include \((\lambda_1, \lambda_2)\), \((I_1, J_2)\), \((E, C)\) and \((E, \mathrm{FA})\). The gradient structure tensor is \(\mathbf{J} = \langle \nabla I\, \nabla I^\top \rangle_w\) with eigenvalues \(\lambda_1 \geq \lambda_2 \geq 0\), mean \(\bar\lambda = \tfrac12 I_1\), and \(g = \lVert \nabla I \rVert\).

Feature	Components	Eigenvalues	Correspondence	Interpretation
Tensor \(\mathbf{J}\)	\(\begin{bmatrix} J_{xx} & J_{xy} \\ J_{xy} & J_{yy} \end{bmatrix}\)	\(\begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}\)	—	gradient structure tensor (Bigün 1987); positive-semidefinite
Orientation \(\theta\)	\(\frac12 \arctan\!\left( \frac{2 J_{xy}}{J_{yy} - J_{xx}} \right)\)	\(\mathbf{e}_2\)	—	principal direction, nematic director; radians, \([-\pi/2, \pi/2]\)
Energy \(E\)	\(J_{xx} + J_{yy}\)	\(\lambda_1 + \lambda_2\)	\(E = I_1\)	gradient energy (Jähne 1997); units \(g^2\)
Coherency \(C\)	\(\frac{\sqrt{(J_{yy} - J_{xx})^2 + 4 J_{xy}^2}}{J_{xx} + J_{yy}}\)	\(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\)	\(C = \frac{2\sqrt{J_2}}{I_1}\)	alignment index, nematic order parameter; 0 isotropic, 1 fiber
Deviator \(\mathbf{s}\)	\(\mathbf{J} - \tfrac12 \operatorname{tr}(\mathbf{J})\, \mathbf{I}\)	\(\begin{bmatrix} \lambda_1 - \bar\lambda & 0 \\ 0 & \lambda_2 - \bar\lambda \end{bmatrix}\)	—	deviatoric part of \(\mathbf{J}\); \(\operatorname{tr}(\mathbf{s}) = 0\)
First invariant \(I_1\)	\(\operatorname{tr}(\mathbf{J})\)	\(\lambda_1 + \lambda_2\)	\(I_1 = E\)	first invariant
Directionality \(J_2\)	\(\tfrac14 (J_{xx} - J_{yy})^2 + J_{xy}^2\)	\(\tfrac14 (\lambda_1 - \lambda_2)^2\)	\(J_2 = \tfrac14 C^2 E^2\)	second deviatoric invariant (von Mises 1913); units \(g^4\)
Distortion energy \(\sigma_d\)	\(\sqrt{2}\, \lVert \mathbf{s} \rVert\)	\(\lambda_1 - \lambda_2\)	\(\sigma_d = 2\sqrt{J_2} = I_1\, \mathrm{RA}\)	equivalent uniaxial magnitude; units \(g^2\)
Relative anisotropy \(\mathrm{RA}\)	\(\frac{\lVert \mathbf{s} \rVert}{\sqrt{2}\, \bar\lambda}\)	\(\frac{\lambda_1 - \lambda_2}{\lambda_1 + \lambda_2}\)	\(\mathrm{RA} = C\)	coefficient of variation of the \(\lambda_i\) (Basser 1996)
Fractional anisotropy \(\mathrm{FA}\)	\(\frac{\sqrt{2}\, \lVert \mathbf{s} \rVert}{\lVert \mathbf{J} \rVert}\)	\(\frac{\lambda_1 - \lambda_2}{\sqrt{\lambda_1^2 + \lambda_2^2}}\)	\(\mathrm{FA} = \frac{\sqrt{2}\, C}{\sqrt{1 + C^2}}\)	degree of anisotropy (Basser 1996)

Typical cases¶

Every feature evaluated on canonical eigenvalue pairs \((\lambda_1, \lambda_2)\), from the ideal oriented case \((1, 0)\) to the isotropic case \((1, 1)\):

\((\lambda_1, \lambda_2)\)	Structure	\(E = I_1\)	\(J_2\)	\(\sigma_d\)	\(C\)	\(\mathrm{RA}\)	\(\mathrm{FA}\)
(1, 0)	ideal oriented	1.000	0.250	1.000	1.000	1.000	1.000
(5, 0.2)	strong	5.200	5.760	4.800	0.923	0.923	0.959
(3, 1)	oriented	4.000	1.000	2.000	0.500	0.500	0.632
(2, 1)	moderate	3.000	0.250	1.000	0.333	0.333	0.447
(1, 0.5)	weak	1.500	0.062	0.500	0.333	0.333	0.447
(1, 0.9)	near-isotropic	1.900	0.002	0.100	0.053	0.053	0.074
(1, 1)	isotropic	2.000	0.000	0.000	0.000	0.000	0.000
(0, 0)	flat	0.000	0.000	—	—	—	—

Implementation notes¶

In OrientationJ, the weighting function \(w\) is a Gaussian window whose standard deviation \(\sigma_w\) ("local window") is set in the dialog; the gradient is computed by cubic-spline interpolation by default. A small regularization \(\epsilon\) is added to the denominators of \(C\) and \(\mathrm{FA}\) to avoid division by zero in flat regions.

Coherency and fractional anisotropy are dimensionless and naturally bounded in \([0, 1]\); they are displayed as computed. Energy and directionality are unbounded (units \(g^2\) and \(g^4\)); since version 2.1.0, they are either linearly rescaled to \([0, 1]\) for display (option Scale [0..1], the default) or shown with their raw values (option No scale). The same option applies to the channels used to build the HSB/RGB color survey, where channel values are clamped to \([0, 1]\).

References¶

J. Bigün and G. H. Granlund, "Optimal orientation detection of linear symmetry," Proceedings of the First IEEE International Conference on Computer Vision, London, 1987.
B. Jähne, Digital Image Processing, Springer, 1997.
J. Weickert, "Coherence-enhancing diffusion filtering," International Journal of Computer Vision, vol. 31, 1999.
R. von Mises, "Mechanik der festen Körper im plastisch-deformablen Zustand," Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 1913.
P. J. Basser and C. Pierpaoli, "Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI," Journal of Magnetic Resonance, Series B, vol. 111, 1996.
Z. Püspöki, M. Storath, D. Sage, and M. Unser, "Transforms and operators for directional bioimage analysis: A survey," Advances in Anatomy, Embryology and Cell Biology, vol. 219, Focus on Bio-Image Informatics, Springer, 2016.