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  1. Abstract Purpose

    Synthetic digital mammogram (SDM) is a 2D image generated from digital breast tomosynthesis (DBT) and used as a substitute for a full‐field digital mammogram (FFDM) to reduce the radiation dose for breast cancer screening. The previous deep learning‐based method used FFDM images as the ground truth, and trained a single neural network to directly generate SDM images with similar appearances (e.g., intensity distribution, textures) to the FFDM images. However, the FFDM image has a different texture pattern from DBT. The difference in texture pattern might make the training of the neural network unstable and result in high‐intensity distortion, which makes it hard to decrease intensity distortion and increase perceptual similarity (e.g., generate similar textures) at the same time. Clinically, radiologists want to have a 2D synthesized image that feels like an FFDM image in vision and preserves local structures such as both mass and microcalcifications (MCs) in DBT because radiologists have been trained on reading FFDM images for a long time, while local structures are important for diagnosis. In this study, we proposed to use a deep convolutional neural network to learn the transformation to generate SDM from DBT.

    Method

    To decrease intensity distortion and increase perceptual similarity, a multi‐scale cascaded network (MSCN) is proposed to generate low‐frequency structures (e.g., intensity distribution) and high‐frequency structures (e.g., textures) separately. The MSCN consist of two cascaded sub‐networks: the first sub‐network is used to predict the low‐frequency part of the FFDM image; the second sub‐network is used to generate a full SDM image with textures similar to the FFDM image based on the prediction of the first sub‐network. The mean‐squared error (MSE) objective function is used to train the first sub‐network, termed low‐frequency network, to generate a low‐frequency SDM image. The gradient‐guided generative adversarial network's objective function is to train the second sub‐network, termed high‐frequency network, to generate a full SDM image with textures similar to the FFDM image.

    Results

    1646 cases with FFDM and DBT were retrospectively collected from the Hologic Selenia system for training and validation dataset, and 145 cases with masses or MC clusters were independently collected from the Hologic Selenia system for testing dataset. For comparison, the baseline network has the same architecture as the high‐frequency network and directly generates a full SDM image. Compared to the baseline method, the proposed MSCN improves the peak‐to‐noise ratio from 25.3 to 27.9 dB and improves the structural similarity from 0.703 to 0.724, and significantly increases the perceptual similarity.

    Conclusions

    The proposed method can stabilize the training and generate SDM images with lower intensity distortion and higher perceptual similarity.

     
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  2. Abstract

    We study the use of deep learning techniques to reconstruct the kinematics of the neutral current deep inelastic scattering (DIS) process in electron–proton collisions. In particular, we use simulated data from the ZEUS experiment at the HERA accelerator facility, and train deep neural networks to reconstruct the kinematic variables$$Q^2$$Q2andx. Our approach is based on the information used in the classical construction methods, the measurements of the scattered lepton, and the hadronic final state in the detector, but is enhanced through correlations and patterns revealed with the simulated data sets. We show that, with the appropriate selection of a training set, the neural networks sufficiently surpass all classical reconstruction methods on most of the kinematic range considered. Rapid access to large samples of simulated data and the ability of neural networks to effectively extract information from large data sets, both suggest that deep learning techniques to reconstruct DIS kinematics can serve as a rigorous method to combine and outperform the classical reconstruction methods.

     
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  3. We consider a minimization problem whose objective function is the sum of a fidelity term, not necessarily convex, and a regularization term defined by a positive regularization parameter [Formula: see text] multiple of the [Formula: see text] norm composed with a linear transform. This problem has wide applications in compressed sensing, sparse machine learning and image reconstruction. The goal of this paper is to understand what choices of the regularization parameter can dictate the level of sparsity under the transform for a global minimizer of the resulting regularized objective function. This is a critical issue but it has been left unaddressed. We address it from a geometric viewpoint with which the sparsity partition of the image space of the transform is introduced. Choices of the regularization parameter are specified to ensure that a global minimizer of the corresponding regularized objective function achieves a prescribed level of sparsity under the transform. Results are obtained for the spacial sparsity case in which the transform is the identity map, a case that covers several applications of practical importance, including machine learning, image/signal processing and medical image reconstruction. 
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    Free, publicly-accessible full text available July 1, 2024
  4. Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as −log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m+1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N2) or O(N2−1m+1) computational complexity, where N is the length of the time series analyzed. When N is big, the computational costs of these algorithms are large. We propose a super fast algorithm to estimate sample entropy based on Monte Carlo, with computational costs independent of N (the length of the time series) and the estimation converging to the exact sample entropy as the number of repeating experiments becomes large. The convergence rate of the algorithm is also established. Numerical experiments are performed for electrocardiogram time series, electroencephalogram time series, cardiac inter-beat time series, mechanical vibration signals (MVS), meteorological data (MD), and 1/f noise. Numerical results show that the proposed algorithm can gain 100–1000 times speedup compared to the kd-tree and assisted sliding box algorithms while providing satisfactory approximate accuracy. 
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  5. Abstract The goal of this study is to develop a new computed tomography (CT) image reconstruction method, aiming at improving the quality of the reconstructed images of existing methods while reducing computational costs. Existing CT reconstruction is modeled by pixel-based piecewise constant approximations of the integral equation that describes the CT projection data acquisition process. Using these approximations imposes a bottleneck model error and results in a discrete system of a large size. We propose to develop a content-adaptive unstructured grid (CAUG) based regularized CT reconstruction method to address these issues. Specifically, we design a CAUG of the image domain to sparsely represent the underlying image, and introduce a CAUG-based piecewise linear approximation of the integral equation by employing a collocation method. We further apply a regularization defined on the CAUG for the resulting ill-posed linear system, which may lead to a sparse linear representation for the underlying solution. The regularized CT reconstruction is formulated as a convex optimization problem, whose objective function consists of a weighted least square norm based fidelity term, a regularization term and a constraint term. Here, the corresponding weighted matrix is derived from the simultaneous algebraic reconstruction technique (SART). We then develop a SART-type preconditioned fixed-point proximity algorithm to solve the optimization problem. Convergence analysis is provided for the resulting iterative algorithm. Numerical experiments demonstrate the superiority of the proposed method over several existing methods in terms of both suppressing noise and reducing computational costs. These methods include the SART without regularization and with the quadratic regularization, the traditional total variation (TV) regularized reconstruction method and the TV superiorized conjugate gradient method on the pixel grid. 
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  6. Shawe-Taylor, John (Ed.)
    Learning a function from a finite number of sampled data points (measurements) is a fundamental problem in science and engineering. This is often formulated as a minimum norm interpolation (MNI) problem, a regularized learning problem or, in general, a semi-discrete inverse problem (SDIP), in either Hilbert spaces or Banach spaces. The goal of this paper is to systematically study solutions of these problems in Banach spaces. We aim at obtaining explicit representer theorems for their solutions, on which convenient solution methods can then be developed. For the MNI problem, the explicit representer theorems enable us to express the infimum in terms of the norm of the linear combination of the interpolation functionals. For the purpose of developing efficient computational algorithms, we establish the fixed-point equation formulation of solutions of these problems. We reveal that unlike in a Hilbert space, in general, solutions of these problems in a Banach space may not be able to be reduced to truly finite dimensional problems (with certain infinite dimensional components hidden). We demonstrate how this obstacle can be removed, reducing the original problem to a truly finite dimensional one, in the special case when the Banach space is ℓ1(N). 
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