Left-invariant diffusions on R^3 x S^2 and their application to crossing-preserving smoothing on HARDI-images

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this article we study left-invariant diffusion on the 3D Euclidean motion group SE(3) and its application to crossing-preserving smoothing of high angular resolution diffusion imaging (HARDI), which is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on the space R3 o S2 of positions and orientations embedded in SE(3) and can be solved by R3 o S2-convolution with the corresponding Green’s functions. We provide analytic approximation formulae and explicit sharp Gaussian estimates for these Green’s functions. In our design and analysis for appropriate (non-linear) convection-diffusions on HARDI-data we put emphasis on the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan-connection. This Cartan-connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI-data containing crossing-fibers

Diffusion on the 3D Euclidean motion group for enhancement of HARDI data

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study left-invariant diffusion on the 3D Euclidean motion group SE(3), which is useful for processing three-dimensional data. In particular, it is useful for the processing of High Angular Resolution Diffusion Imaging (HARDI) data, since these data can be considered as orientation scores directly, without the need to transform the HARDI data to a different form. In principle, all theory of the 2D case can be mapped to the 3D case. However, one of the complicating factors is that all practical 3D orientation scores are not functions on the entire group SE(3), but rather on a coset space of the group. We will show how we can still conceptually apply processing on the entire group by requiring the operations to preserve the introduced notion of alpha-right-invariance of such functions on SE(3). We introduce left-invariant derivatives and describe how to estimate tangent vectors that locally fit best to the elongated structures in the 3D orientation score. We propose generally applicable techniques for smoothing and enhancing functions on SE(3) using left-invariant diffusion on the group. Finally, we will discuss implementational issues and show a number of results for linear diffusion on artificial HARDI data

Sensitizing thermochemotherapy with a PARP1-inhibitor

Cis-diamminedichloroplatinum(II) (cisplatin, cDDP) is an effective chemotherapeutic agent that induces DNA double strand breaks (DSBs), primarily in replicating cells. Generally, such DSBs can be repaired by the classical or backup non-homologous end joining (c-NHEJ/b-NHEJ) or homologous recombination (HR). Therefore, inhibiting these pathways in cancer cells should enhance the efficiency of cDDP treatments. Indeed, inhibition of HR by hyperthermia (HT) sensitizes cancer cells to cDDP and in the Netherlands this combination is a standard treatment option for recurrent cervical cancer after previous radiotherapy. Additionally, cDDP has been demonstrated to disrupt c-NHEJ, which likely further increases the treatment efficacy. However, if one of these pathways is blocked, DSB repair functions can be sustained by the Poly-(ADP-ribose)-polymerase1 (PARP1)-dependent b-NHEJ. Therefore, disabling b-NHEJ should, in principle, further inhibit the repair of cDDP-induced DNA lesions and enhance the toxicity of thermochemotherapy. To explore this hypothesis, we treated a panel of cancer cell lines with HT, cDDP and a PARP1-i and measured various end-point relevant in cancer treatment. Our results demonstrate that PARP1-i does not considerably increase the efficacy of HT combined with standard, commonly used cDDP concentrations. However, in the presence of a PARP1-i, ten-fold lower concentration of cDDP can be used to induce similar cytotoxic effects. PARP1 inhibition may thus permit a substantial lowering of cDDP concentrations without diminishing treatment efficacy, potentially reducing systemic side effects

Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images

HARDI (High Angular Resolution Diffusion Imaging) is a recent magnetic resonance imaging (MRI) technique for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. In this article we study left-invariant diffusion on the group of 3D rigid body movements (i.e. 3D Euclidean motion group) SE(3) and its application to crossing-preserving smoothing of HARDI images. The linear left-invariant \mbox{(convection-)diffusions} are forward Kolmogorov equations of Brownian motions on the space of positions and orientations in 3D embedded in SE(3) and can be solved by $\R^3 \rtimes S^{2}$-convolution with the corresponding Green's functions. We provide analytic approximation formulas and explicit sharp Gaussian estimates for these Green's functions. In our design and analysis for appropriate (nonlinear) convection-diffusions on HARDI data we explain the underlying differential geometry on SE(3). We write our left-invariant diffusions in covariant derivatives on SE(3) using the Cartan connection. This Cartan connection has constant curvature and constant torsion, and so have the exponential curves which are the auto-parallels along which our left-invariant diffusion takes place. We provide experiments of our crossing-preserving Euclidean-invariant diffusions on artificial HARDI data containing crossing-fibers

Crossing-preserving coherence-enhancing diffusion on invertible orientation scores

Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherence-enhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence-enhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore, the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings

Diffusion on the 3D Euclidean motion group for enhancement of HARDI data

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study left-invariant diffusion on the 3D Euclidean motion group SE(3), which is useful for processing three-dimensional data. In particular, it is useful for the processing of High Angular Resolution Diffusion Imaging (HARDI) data, since these data can be considered as orientation scores directly, without the need to transform the HARDI data to a different form. In principle, all theory of the 2D case can be mapped to the 3D case. However, one of the complicating factors is that all practical 3D orientation scores are not functions on the entire group SE(3), but rather on a coset space of the group. We will show how we can still conceptually apply processing on the entire group by requiring the operations to preserve the introduced notion of alpha-right-invariance of such functions on SE(3). We introduce left-invariant derivatives and describe how to estimate tangent vectors that locally fit best to the elongated structures in the 3D orientation score. We propose generally applicable techniques for smoothing and enhancing functions on SE(3) using left-invariant diffusion on the group. Finally, we will discuss implementational issues and show a number of results for linear diffusion on artificial HARDI data

Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part II: Non-linear left-invariant diffusions on invertible orientation scores

By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard "coherence enhancing diffusion" equations on images as they can enhance images with crossing contours. Here we employ differential geometry on SE(2) to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions