Structure tensor

Summary

In mathematics, the structure tensor, also referred to as the second-moment matrix, is a matrix derived from the gradient of a function. It describes the distribution of the gradient in a specified neighborhood around a point and makes the information invariant respect the observing coordinates. The structure tensor is often used in and computer vision. The 2D structure tensor Continuous version For a function I of two variables p = (x, y), the structure tensor is the 2×2 matrix S_w(p) = \begin{bmatrix} \int w(r) (I_x(p-r))^2,dr & \int w(r) I_x(p-r)I_y(p-r),dr \[10pt] \int w(r) I_x(p-r)I_y(p-r),dr & \int w(r) (I_y(p-r))^2,dr \end{bmatrix} where I_x and I_y are the partial derivatives of I with respect to x and y; the integrals range over the plane \mathbb{R}^2; and w is some fixed "window function" (such as a Gaussian blur), a distribution on two variables. Note t

Official source

https://en.wikipedia.org/wiki/Structure_tensor

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Digital Phase Reconstruction via Iterative Solutions of Transport-of-Intensity Equation

Emrah Bostan, Emmanuel Augustin Julien Froustey, Stamatios Lefkimmiatis, Michaël Unser

We develop a variational algorithm for reconstructing phase objects from a series of bright field micrographs. Our mathematical model is based on the transport-of-intensity equation (TIE), which links the phase of a complex field to the axial derivative of its intensity. To reduce reconstruction artifacts, we formulate TIE in a regularized fashion by introducing a family of penalty functionals based on the eigenvalues of the structure tensor. To solve the arising optimization problem, we propose an algorithm based on the alternating direction method of multipliers (ADMM). We apply our method on simulated data and illustrate improved performance compared to the conventional methods such as Tikhonov and total variation (TV) regularizations. We further demonstrate the applicability of the proposed approach by applying it to experimentally-acquired bright field data.

Ieee2014

Digital Phase Reconstruction via Iterative Solutions of Transport-of-Intensity Equation

Emrah Bostan, Emmanuel Augustin Julien Froustey, Stamatios Lefkimmiatis, Michaël Unser

IEEE2014

We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as structure tensor total variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the total variation seminorm and vectorial extensions of it. Meanwhile, thanks to the structure tensor's ability to capture first-order information around a local neighborhood, the STV functionals can provide more robust measures of image variation. Further, we prove that the STV regularizers are convex while they also satisfy several invariance properties w.r.t. image transformations. These properties qualify them as ideal candidates for imaging applications. In addition, for the discrete version of the STV functionals we derive an equivalent definition that is based on the patch-based Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative definition allow us to derive a dual problem formulation. The duality of the problem paves the way for employing robust tools from convex optimization and enables us to design an efficient and parallelizable optimization algorithm. Finally, we present extensive experiments on various inverse imaging problems, where we compare our regularizers with other competing regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.

SIAM2015