We investigate Lp(γ)-Lq(γ) off-diagonal estimates for the Ornstein-Uhlenbeck semigroup (etL)t>0. For sufficiently large t (quantified in terms of p and q), these estimates hold in an unrestricted sense, while, for sufficiently small t, they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.@en

This dissertation consists out of two rather disjoint parts. One part concerns some results on Gaussian harmonic analysis and the other on an optimization problem in optics. In the first part we study the Ornstein–Uhlenbeck process with respect to the Gaussian measure. We focus on two areas. One is on “Gaussian” analogues of classical results in harmonic analysis, and in the second area we study the higher time-derivatives of the integral kernels associated to the Ornstein–Uhlenbeck operator. After introducing the necessary preliminaries on Hermite polynomials, we look at the non-tangential maximal function for the Ornstein–Uhlenbeck semigroup and we prove analogues to classical results. An important distinction here with the classical case is that the maximal function result for the Laplacian allows for t > 0, whereas our result only holds for certain 0 < t < 1 where the exact range depends on the position in space. Next, we compute an explicit formula for the higher time-derivatives of the integral kernel related to the Ornstein–Uhlenbeck operator. As an application we show several kernel bounds using our formula. Finally, as far as the mathematical part is concerned, we study off-diagonal estimates related to the Ornstein–Uhlenbeck operator. It is well-known that these hold for the Laplacian with respect to the Lebesgue measure, but do these hold for the Ornstein–Uhlenbeck operator with respect to the Gaussian measure? It is known that we always would have L2-L2 bounds, even for all t > 0, but in applications one often wants L2-L1 bounds. Even though these do hold for the Laplacian, we show that these cannot hold for the Ornstein–Uhlenbeck operator even for small 0 < t < 1. Moreover, our proof shows that letting the maximal t depend on the position in space will not work either. In the final part of this dissertation we study a problem in theoretical optics. Here we study the optimization of the electric field induced by light as a plane wave in a disk with given radius. We study the electric fields in the lens pupil and the focal region for several radii in the orde of magnitude of the wave length of the light used.@en

Digital breast tomosynthesis is rapidly replacing digital mammography as the basic x-ray technique for evaluation of the breasts. However, the sparse sampling and limited angular range gives rise to different artifacts, which manufacturers try to solve in several ways. In this study we propose an extension of the Learned Primal- Dual algorithm for digital breast tomosynthesis. The Learned Primal-Dual algorithm is a deep neural network consisting of several a€reconstruction blocks', which take in raw sinogram data as the initial input, perform a forward and a backward pass by taking projections and back-projections, and use a convolutional neural network to produce an intermediate reconstruction result which is then improved further by the successive reconstruction block. We extend the architecture by providing breast thickness measurements as a mask to the neural network and allow it to learn how to use this thickness mask. We have trained the algorithm on digital phantoms and the corresponding noise-free/noisy projections, and then tested the algorithm on digital phantoms for varying level of noise. Reconstruction performance of the algorithms was compared visually, using MSE loss and Structural Similarity Index. Results indicate that the proposed algorithm outperforms the baseline iterative reconstruction algorithm in terms of reconstruction quality for both breast edges and internal structures and is robust to noise.@en

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL  etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N  LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit. @en

Computer-aided detection aims to improve breast cancer screening programs by helping radiologists to evaluate digital mammography (DM) exams. DM exams are generated by devices from different vendors, with diverse characteristics between and even within vendors. Physical properties of these devices and postprocessing of the images can greatly influence the resulting mammogram. This results in the fact that a deep learning model trained on data from one vendor cannot readily be applied to data from another vendor. This paper investigates the use of tailored transfer learning methods based on adversarial learning to tackle this problem. We consider a database of DM exams (mostly bilateral and two views) generated by Hologic and Siemens vendors. We analyze two transfer learning settings: 1) unsupervised transfer, where Hologic data with soft lesion annotation at pixel level and Siemens unlabelled data are used to annotate images in the latter data; 2) weak supervised transfer, where exam level labels for images from the Siemens mammograph are available. We propose tailored variants of recent state-of-the-art methods for transfer learning which take into account the class imbalance and incorporate knowledge provided by the annotations at exam level. Results of experiments indicate the beneficial effect of transfer learning in both transfer settings. Notably, at 0.02 false positives per image, we achieve a sensitivity of 0.37, compared to 0.30 of a baseline with no transfer. Results indicate that using exam level annotations gives an additional increase in sensitivity.@en

Computer-aided detection or decision support systems aim to improve breast cancer screening programs by helping radiologists to evaluate digital mammography (DM) exams. Commonly such methods proceed in two steps: Selection of candidate regions for malignancy, and later classification as either malignant or not. In this study, we present a candidate detection method based on deep learning to automatically detect and additionally segment soft tissue lesions in DM. A database of DM exams (mostly bilateral and two views) was collected from our institutional archive. In total, 7196 DM exams (28294 DM images) acquired with systems from three different vendors (General Electric, Siemens, Hologic) were collected, of which 2883 contained malignant lesions verified with histopathology. Data was randomly split on an exam level into training (50%), validation (10%) and testing (40%) of deep neural network with u-net architecture. The u-net classifies the image but also provides lesion segmentation. Free receiver operating characteristic (FROC) analysis was used to evaluate the model, on an image and on an exam level. On an image level, a maximum sensitivity of 0.94 at 7.93 false positives (FP) per image was achieved. Similarly, per exam a maximum sensitivity of 0.98 at 7.81 FP per image was achieved. In conclusion, the method could be used as a candidate selection model with high accuracy and with the additional information of lesion segmentation.@en