Coherence Filtering to Enhance the Mandibular Canal in Cone-Beam CT data

Segmenting the mandibular canal from cone beam CT data, is difficult due to low edge contrast and high image noise. We introduce 3D coherence filtering as a method to close the interrupted edges and denoise the structure of the mandibular canal. Coherence Filtering is an anisotropic non-linear tensor based diffusion algorithm for edge enhancing image filtering. We test different numerical schemes of the tensor diffusion equation, non-negative, standard discretization and also a rotation invariant scheme of Weickert [1]. Only the\ud scheme of Weickert did not blur the high spherical images frequencies on the image diagonals of our test volume. Thus this scheme is chosen to enhance the small curved mandibular canal structure. The best choice of the diffusion equation parameters c1 and c2, depends on the image noise. Coherence filtering on the CBCT-scan works well, the noise in the mandibular canal is gone and the edges are connected. Because the algorithm is tensor based it cannot deal with edge joints or splits, thus is less fit for more complex image structures

A flexible space-variant anisotropic regularisation for image restoration with automated parameter selection

We propose a new space-variant anisotropic regularisation term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally according to a bivariate generalised Gaussian distribution. The highly flexible variational structure of the corresponding regulariser encodes several free parameters which hold the potential for faithfully modelling the local geometry in the image and describing local orientation preferences. For an automatic estimation of such parameters, we design a robust maximum likelihood approach and report results on its reliability on synthetic data and natural images. For the numerical solution of the corresponding image restoration model, we use an iterative algorithm based on the Alternating Direction Method of Multipliers (ADMM). A suitable preliminary variable splitting together with a novel result in multivariate non-convex proximal calculus yield a very efficient minimisation algorithm. Several numerical results showing significant quality-improvement of the proposed model with respect to some related state-of-the-art competitors are reported, in particular in terms of texture and detail preservation

Study of Computational Image Matching Techniques: Improving Our View of Biomedical Image Data

Image matching techniques are proven to be necessary in various fields of science and engineering, with many new methods and applications introduced over the years. In this PhD thesis, several computational image matching methods are introduced and investigated for improving the analysis of various biomedical image data. These improvements include the use of matching techniques for enhancing visualization of cross-sectional imaging modalities such as Computed Tomography (CT) and Magnetic Resonance Imaging (MRI), denoising of retinal Optical Coherence Tomography (OCT), and high quality 3D reconstruction of surfaces from Scanning Electron Microscope (SEM) images. This work greatly improves the process of data interpretation of image data with far reaching consequences for basic sciences research. The thesis starts with a general notion of the problem of image matching followed by an overview of the topics covered in the thesis. This is followed by introduction and investigation of several applications of image matching/registration in biomdecial image processing: a) registration-based slice interpolation, b) fast mesh-based deformable image registration and c) use of simultaneous rigid registration and Robust Principal Component Analysis (RPCA) for speckle noise reduction of retinal OCT images. Moving towards a different notion of image matching/correspondence, the problem of view synthesis and 3D reconstruction, with a focus on 3D reconstruction of microscopic samples from 2D images captured by SEM, is considered next. Starting from sparse feature-based matching techniques, an extensive analysis is provided for using several well-known feature detector/descriptor techniques, namely ORB, BRIEF, SURF and SIFT, for the problem of multi-view 3D reconstruction. This chapter contains qualitative and quantitative comparisons in order to reveal the shortcomings of the sparse feature-based techniques. This is followed by introduction of a novel framework using sparse-dense matching/correspondence for high quality 3D reconstruction of SEM images. As will be shown, the proposed framework results in better reconstructions when compared with state-of-the-art sparse-feature based techniques. Even though the proposed framework produces satisfactory results, there is room for improvements. These improvements become more necessary when dealing with higher complexity microscopic samples imaged by SEM as well as in cases with large displacements between corresponding points in micrographs. Therefore, based on the proposed framework, a new approach is proposed for high quality 3D reconstruction of microscopic samples. While in case of having simpler microscopic samples the performance of the two proposed techniques are comparable, the new technique results in more truthful reconstruction of highly complex samples. The thesis is concluded with an overview of the thesis and also pointers regarding future directions of the research using both multi-view and photometric techniques for 3D reconstruction of SEM images

A PDE Approach to Data-driven Sub-Riemannian Geodesics in SE(2)

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub-Riemannian (SR) geodesics in the roto-translation group $SE(2) = \mathbb{R}^2 \rtimes S^1$ with a metric tensor depending on a smooth external cost $\mathcal{C}:SE(2) \to [\delta,1]$, $\delta>0$, computed from image data. The method consists of a first step where a SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman (HJB) system derived via Pontryagin's Maximum Principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For $\mathcal{C}=1$ we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case $\mathcal{C}=1$. Regarding image analysis applications, tracking of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the sub-Riemannian geometry.Comment: Extended version of SSVM 2015 conference article "Data-driven Sub-Riemannian Geodesics in SE(2)

Resonant nonlinear magneto-optical effects in atoms

In this article, we review the history, current status, physical mechanisms, experimental methods, and applications of nonlinear magneto-optical effects in atomic vapors. We begin by describing the pioneering work of Macaluso and Corbino over a century ago on linear magneto-optical effects (in which the properties of the medium do not depend on the light power) in the vicinity of atomic resonances, and contrast these effects with various nonlinear magneto-optical phenomena that have been studied both theoretically and experimentally since the late 1960s. In recent years, the field of nonlinear magneto-optics has experienced a revival of interest that has led to a number of developments, including the observation of ultra-narrow (1-Hz) magneto-optical resonances, applications in sensitive magnetometry, nonlinear magneto-optical tomography, and the possibility of a search for parity- and time-reversal-invariance violation in atoms.Comment: 51 pages, 23 figures, to appear in Rev. Mod. Phys. in Oct. 2002, Figure added, typos corrected, text edited for clarit

Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers

Anisotropic Diffusion Partial Differential Equations in Multi-Channel Image Processing : Framework and Applications

We review recent methods based on diffusion PDE's (Partial Differential Equations) for the purpose of multi-channel image regularization. Such methods have the ability to smooth multi-channel images anisotropically and can preserve then image contours while removing noise or other undesired local artifacts. We point out the pros and cons of the existing equations, providing at each time a local geometric interpretation of the corresponding processes. We focus then on an alternate and generic tensor-driven formulation, able to regularize images while specifically taking the curvatures of local image structures into account. This particular diffusion PDE variant is actually well suited for the preservation of thin structures and gives regularization results where important image features can be particularly well preserved compared to its competitors. A direct link between this curvature-preserving equation and a continuous formulation of the Line Integral Convolution technique (Cabral and Leedom, 1993) is demonstrated. It allows the design of a very fast and stable numerical scheme which implements the multi-valued regularization method by successive integrations of the pixel values along curved integral lines. Besides, the proposed implementation, based on a fourth-order Runge Kutta numerical integration, can be applied with a subpixel accuracy and preserves then thin image structures much better than classical finite-differences discretizations, usually chosen to implement PDE-based diffusions. We finally illustrate the efficiency of this diffusion PDE's for multi-channel image regularization - in terms of speed and visual quality - with various applications and results on color images, including image denoising, inpainting and edge-preserving interpolation