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This paper is the sequel of Part I [1], where we showed how to use the so-called Malliavin calculus in order to devise efficient Monte-Carlo (numerical) methods for Finance. First, we return to the formulas developed in [1] concerning the "greeks" used in European options, and we answer to the question of optimal weight functional in the sense of minimal variance. Then, we investigate the use of Malliavin calculus to compute conditional expectations. The integration by part formula provides a powerful tool when used in the framework of Monte Carlo simulation. It allows to compute everywhere, on a single set of trajectories starting at one point, solution of general options related PDEs.Our final application of Malliavin calculus concerns the use of Girsanov transforms involving anticipating drifts. We give an example in numerical Finance of such a transform which gives reduction of variance via importance sampling.Finally, we include two appendices that are concerned with the PDE interpretation of the formulas presented in [1] for the delta of a European option and with the connections between the functional dependence of some random variables and their Malliavin derivatives.

Purpose: Modern computed tomography (CT) scanners have an extended field-of-view (eFoV) for reconstructing images up to the bore size, which is relevant for patients with higher BMI or nonisocentric positioning due to fixation devices. However, the accuracy of the image reconstruction in eFoV is not well known since truncated data are used. This study introduces a new deep learningbased algorithm for extended field-of-view reconstruction and evaluates the accuracy of the eFoV reconstruction focusing on aspects relevant for radiotherapy. Methods: A life-size three-dimensional (3D) printed thorax phantom, based on a patient CT for which eFoV was necessary, was manufactured and used as reference. The phantom has holes allowing the placement of tissue mimicking inserts used to evaluate the Hounsfield unit (HU) accuracy. CT images of the phantom were acquired using different configurations aiming to evaluate geometric and HU accuracy in the eFoV. Image reconstruction was performed using a state-of-the-art reconstruction algorithm (HDFoV), commercially available, and the novel deep learning-based approach (HDeepFoV). Five patient cases were selected to evaluate the performance of both algorithms on patient data. There is no ground truth for patients so the reconstructions were qualitatively evaluated by five physicians and five medical physicists. Results: The phantom geometry reconstructed with HDFoV showed boundary deviations from 1.0 to 2.5 cm depending on the volume of the phantom outside the regular scan field of view. HDeepFoV showed a superior performance regardless of the volume of the phantom within eFOV with a maximum boundary deviation below 1.0 cm. The maximum HU (absolute) difference for soft issue inserts is below 79 and 41 HU for HDFoV and HDeepFoV, respectively. HDeepFoV has a maximum deviation of À18 HU for an inhaled lung insert while HDFoV reached a 229 HU difference. The qualitative evaluation of patient cases shows that the novel deep learning approach produces images that look more realistic and have fewer artifacts. Conclusion: To be able to reconstruct images outside the sFoV of the CT scanner there is no alternative than to use some kind of extrapolated data. In our study, we proposed and investigated a new deep learning-based algorithm and compared it to a commercial solution for eFoV reconstruction. The deep learning-based algorithm showed superior performance in quantitative evaluations based on phantom data and in qualitative assessments of patient data.

During a typical cardiac short scan, the heart can move several millimeters. As a result, the corresponding CT reconstructions may be corrupted by motion artifacts. Especially the assessment of small structures, such as the coronary arteries, is potentially impaired by the presence of these artifacts. In order to estimate and compensate for coronary artery motion, this manuscript proposes the deep partial angle-based motion compensation (Deep PAMoCo). Methods: The basic principle of the Deep PAMoCo relies on the concept of partial angle reconstructions (PARs), that is, it divides the short scan data into several consecutive angular segments and reconstructs them separately. Subsequently, the PARs are deformed according to a motion vector field (MVF) such that they represent the same motion state and summed up to obtain the final motioncompensated reconstruction. However, in contrast to prior work that is based on the same principle, the Deep PAMoCo estimates and applies the MVF via a deep neural network to increase the computational performance as well as the quality of the motion compensated reconstructions. Results: Using simulated data, it could be demonstrated that the Deep PAMoCo is able to remove almost all motion artifacts independent of the contrast, the radius and the motion amplitude of the coronary artery. In any case, the average error of the CT values along the coronary artery is about 25 HU while errors of up to 300 HU can be observed if no correction is applied. Similar results were obtained for clinical cardiac CT scans where the Deep PAMoCo clearly outperforms state-of-the-art coronary artery motion compensation approaches in terms of processing time as well as accuracy. Conclusions: The Deep PAMoCo provides an efficient approach to increase the diagnostic value of cardiac CT scans even if they are highly corrupted by motion.

Purpose Dual‐source computed tomography (DSCT) uses two source‐detector pairs offset by about 90°. In addition to the well‐known forward scatter, a special issue in DSCT is cross‐scattered radiation from X‐ray tube A detected in the detector of system B and vice versa. This effect can lead to artifacts and reduction of the contrast‐to‐noise ratio of the images. The purpose of this work is to present and evaluate different deep learning‐based methods for scatter correction in DSCT. Methods We present different neural network‐based methods for forward and cross‐scatter correction in DSCT. These deep scatter estimation (DSE) methods mainly differ in the input and output information that is provided for training and inference and in whether they operate on two‐dimensional (2D) or on three‐dimensional (3D) data. The networks are trained and validated with scatter distributions obtained by our in‐house Monte Carlo simulation. The simulated geometry is adapted to a realistic clinical setup. Results All DSE approaches reduce scatter‐induced artifacts and lead to superior results than the measurement‐based scatter correction. Forward scatter, under the presence of cross‐scatter, is best estimated either by our network that uses the current projection and a couple of neighboring views (fDSE 2D few views) or by our 3D network that processes all projections simultaneously (fDSE 3D). Cross‐scatter, under the presence of forward scatter, is best estimated using xSSE XDSE 2D, with xSSE referring to a quick single scatter estimate of cross scatter, or by xDSE 3D that uses all projections simultaneously. By using our proposed networks, the total scatter error in dual could be reduced from about 18 HU to approximately 3 HU. Conclusions Deep learning‐based scatter correction can reduce scatter artifacts in DSCT. To achieve more accurate cross‐scatter estimations, the use of a cross‐scatter approximation improves the results. Also, the ability to leverage across different projection angles improves the precision of the algorithm.