Revisiting a theorem of L. A. Shepp on optimal stopping

Using a bondholder who seeks to determine when to sell his bond as our motivating example, we revisit one of Larry Shepp’s classical theorems on optimal stopping. We offer a novel proof of Theorem 1 from from [7]. Our approach is that of guessing the optimal control function and proving its optimality with martingales

Towards segmentation and spatial alignment of the human embryonic brain using deep learning for atlas-based registration

We propose an unsupervised deep learning method for atlas based registration to achieve segmentation and spatial alignment of the embryonic brain in a single framework. Our approach consists of two sequential networks with a specifically designed loss function to address the challenges in 3D first trimester ultrasound. The first part learns the affine transformation and the second part learns the voxelwise nonrigid deformation between the target image and the atlas. We trained this network end-to-end and validated it against a ground truth on synthetic datasets designed to resemble the challenges present in 3D first trimester ultrasound. The method was tested on a dataset of human embryonic ultrasound volumes acquired at 9 weeks gestational age, which showed alignment of the brain in some cases and gave insight in open challenges for the proposed method. We conclude that our method is a promising approach towards fully automated spatial alignment and segmentation of embryonic brains in 3D ultrasound

Efficient Bayesian-based Multi-View Deconvolution

Light sheet fluorescence microscopy is able to image large specimen with high resolution by imaging the sam- ples from multiple angles. Multi-view deconvolution can significantly improve the resolution and contrast of the images, but its application has been limited due to the large size of the datasets. Here we present a Bayesian- based derivation of multi-view deconvolution that drastically improves the convergence time and provide a fast implementation utilizing graphics hardware.Comment: 48 pages, 20 figures, 1 table, under review at Nature Method

On Exceptional Times for generalized Fleming-Viot Processes with Mutations

If $\mathbf Y$ is a standard Fleming-Viot process with constant mutation rate (in the infinitely many sites model) then it is well known that for each $t>0$ the measure $\mathbf Y_t$ is purely atomic with infinitely many atoms. However, Schmuland proved that there is a critical value for the mutation rate under which almost surely there are exceptional times at which $\mathbf Y$ is a finite sum of weighted Dirac masses. In the present work we discuss the existence of such exceptional times for the generalized Fleming-Viot processes. In the case of Beta-Fleming-Viot processes with index $\alpha\in\,]1,2[$ we show that - irrespectively of the mutation rate and $\alpha$ - the number of atoms is almost surely always infinite. The proof combines a Pitman-Yor type representation with a disintegration formula, Lamperti's transformation for self-similar processes and covering results for Poisson point processes

Spatiotemporal PET reconstruction using ML-EM with learned diffeomorphic deformation

Patient movement in emission tomography deteriorates reconstruction quality because of motion blur. Gating the data improves the situation somewhat: each gate contains a movement phase which is approximately stationary. A standard method is to use only the data from a few gates, with little movement between them. However, the corresponding loss of data entails an increase of noise. Motion correction algorithms have been implemented to take into account all the gated data, but they do not scale well, especially not in 3D. We propose a novel motion correction algorithm which addresses the scalability issue. Our approach is to combine an enhanced ML-EM algorithm with deep learning based movement registration. The training is unsupervised, and with artificial data. We expect this approach to scale very well to higher resolutions and to 3D, as the overall cost of our algorithm is only marginally greater than that of a standard ML-EM algorithm. We show that we can significantly decrease the noise corresponding to a limited number of gates

Source localization of reaction-diffusion models for brain tumors

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures