In many applications such as medical imaging, the measurement data represent counts of photons hitting a detector. Such counts in low-photon settings are often modeled using a Poisson distribution. However, this model assumes that the mean and variance of the signal’s noise distribution are equal. For overdispersed data where the variance is greater than the mean, the negative binomial distribution is a more appropriate statistical model. In this paper, we propose an optimization approach for recovering images corrupted by overdispersed Poisson noise. In particular, we incorporate a weighted anisotropic–isotropic total variation regularizer, which avoids staircasing artifacts that are introduced by a regular total variation penalty. We use an alternating direction method of multipliers, where each subproblem has a closed-form solution. Numerical experiments demonstrate the effectiveness of our proposed approach, especially in very photon-limited settings.

Adam is arguably one of the most commonly used approach in deep learning and machine learning. With good regret bounds and empirical convergence proofs, the approach has produced many state-of-the-art models over a variety of problems. However, the method only uses gradient information at each iterate in addition to some moving averaged gradients and its corresponding moments from the past. In this paper, we propose a method that builds upon Adam and incorporates quasi-Newton matrices for approximating second derivatives. These Hessian approximations satisfy the so-called secant equation, which is the first-order Taylor series expansion of the gradient along the direction of the change in iterates. Judicious choices of quasi-Newton matrices can lead to guaranteed descent in the objective function and improved convergence. In this work, we integrate search directions obtained from using these quasi-Newton Hessian approximations with the Adam optimization algorithm. We provide convergence guarantees and demonstrate improved performance through an extensive experimentation on a variety of applications.

Clustering via representation learning is one of the most promising approaches for self-supervised learning of deep neural networks. It aims at obtaining artificial supervisory signals from unlabeled data. In this paper, we propose an online clustering method called CLOT (Contrastive Learning-Driven and Optimal Transport-Based Clustering) that is based on robust and multiple losses training settings. More specifically, CLOT learns representations by contrasting both the features at the latent space and the cluster assignments. In the first stage, CLOT performs the instance- and cluster-level contrastive learning which is respectively conducted by maximizing the similarities of the projections of positive pairs (views of the same image) while minimizing those of negative ones (views of the rest of images). In the second stage, it extends standard cross-entropy minimization to an optimal transport problem and solves it using a fast variant of the Sinkhorn-Knopp algorithm to produce the cluster assignments. Further, it enforces consistency between the produced assignments obtained from views of the same image. Compared to the state of the arts, the proposed CLOT outperforms eight competitive clustering methods on three challenging benchmarks, namely, CIFAR-100, STL-10, and ImageNet-10 for ResNet-34.

Training a deep learning model with a large annotated dataset is still a dominant paradigm in automatic whole slide images (WSIs) processing for digital pathology. However, obtaining manual annotations is a labor-intensive task, and an error-prone to inter and intra-observer variability. In this study, we offer an online deep learning-based clustering workflow for annotating and analysis of different types of tissues from histopathology images. Inspired by learning and optimal transport theory, our proposed model consists of two stages. In the first stage, our model learns tissue-specific discriminative representations by contrasting the features in the latent space at two levels, the instance- and the cluster-level. This is done by maximizing the similarities of the projections of positive pairs (views of the same image) while minimizing those of negative ones (views of the rest of the images). In the second stage, our framework extends the standard cross-entropy minimization to an optimal transport problem and solves it using the Sinkhorn-Knopp algorithm to produce the cluster assignments. Moreover, our proposed method enforces consistency between the produced assign-ments obtained from views of the same image. Our framework was evaluated on three common histopathological datasets: NCT-CRC, LC2500, and Kather_STAD. Experiments show that our proposed framework can identify different tissues in annotation-free conditions with competitive results. It achieved an accuracy of 0.9364 in human lung patched WSIs and 0.8464 in images of human colorectal tissues outperforming state of the arts contrastive-based methods.

Motion blur in an image results from movements of objects within a scene or from the imaging system itself. In applications such as high-speed license plate recognition, motion blur introduces image artifacts leading to challenges in image classification. In this work, we investigate machine learning techniques for classifying images with motion blur using convolutional neural networks. In particular, we explore how different motion directions and lengths affect the predictive performance of our classification model. We used the MNIST dataset, which contains 70,000 images of handwritten digits, to generate training and testing datasets of images with motion blur using MATLAB. Specifically, we considered motion blurs at various angles and lengths to analyze the effects of different motion blurs on the classification accuracy of images of digits. We found that our model shows very high accuracy when training and testing on datasets with the same type of motion blur. In contrast, training and testing on datasets with different motion blurs result in lower accuracy. We describe how we can improve overall classification performance and offer insights on what additional information can be inferred from the MNIST dataset with motion blur.

Interpretability continues to be a focus of much research in deep neural network. In this work, we focus on the mathematical interpretability of fully-connected neural networks, especially those that use a rectified linear unit (ReLU) activation function. Our analysis elucidates the difficulty of approximating the reciprocal function. Notwithstanding, using the ReLU activation function halves the error compared with a linear model. In addition, one might have expected the errors to increase only towards the singular point x = 0, but both the linear and ReLU errors are fairly oscillatory and increase near both edge points.

Recurrent neural networks (RNNs) are traditionally used for machine learning applications for temporal sequences such as natural language processing. Its application to image processing is relatively new. In this paper, we apply RNNs to denoise images corrupted by mixed Poisson and Gaussian noise. The motivation for using an RNN comes from viewing the denoising of the Poisson-Gaussian realization as a temporal process. The network then attempts to trace back the steps that create the noisy realization in order to arrive at the noiseless reconstruction. Numerical experiments demonstrate that our proposed RNN approach outperforms convolutional autoen-coder methods for denoising and upsampling low-resolution images from the CIFAR-10 dataset.

In this paper, we propose a novel machine learning technique to detect and correctly classify adversarially modified images. Our approach takes advantage of the adversarial training objective (to influence the prediction with small perturbations to the original image) by searching for large discrepancies in the model’s output relative to the input image. We find clusters in each space, and for an image whose clusters do not agree with each other, we flag the image as potentially adversarial and use the input space cluster’s prediction in place of the attacked model’s output. We have found that this method tends to find adversarial samples which are more likely to be misclassified by the attacked model, and that simply deferring to the input cluster’s prediction on such samples is enough to increase accuracy on adversarial data, even when the attacked model is adversarially trained.

We present a MATLAB implementation of the symmetric rank-one (SC-SR1) method that solves trust-region subproblems when a limited-memory symmetric rank-one (L-SR1) matrix is used in place of the true Hessian matrix, which can be used for large-scale optimization. The method takes advantage of two shape-changing norms [Burdakov and Yuan 2002 ; Burdakov et al. 2017 ] to decompose the trust-region subproblem into two separate problems. Using one of the proposed norms, the resulting subproblems have closed-form solutions. Meanwhile, using the other proposed norm, one of the resulting subproblems has a closed-form solution while the other is easily solvable using techniques that exploit the structure of L-SR1 matrices. Numerical results suggest that the SC-SR1 method is able to solve trust-region subproblems to high accuracy even in the so-called “hard case.” When integrated into a trust-region algorithm, extensive numerical experiments suggest that the proposed algorithms perform well, when compared with widely used solvers, such as truncated conjugate-gradients.

Synthetic aperture radar (SAR) is a remote sensing technique used to obtain high-resolution images, where image classification is a primary application. However, reconstructing SAR data is difficult which makes the classification of these images even more challenging. We present a study that investigates techniques for classifying SAR images using machine learning. In particular, we compare the classification accuracy using three different training data: raw SAR observation data, reconstructions using Kirchoff migration, and reconstructions by learning an approximation to the inverse of the SAR sensing operator. We consider two different architectures, namely a multi-layer perceptron and a convolutional neural network. The training set is composed of 50,000 images from the CIFAR-10 dataset. We find that the images reconstructed using the learned approximate inverse have a higher classification accuracy than that from the SAR measurement and the Kirchoff migration approach.

Genomic variation shared by members of the same species that are longer than a single nucleotide are commonly called structural variants (SVs). Though relatively rare, they represent an increasingly important class of variation as SVs have been associated with diseases and susceptibility to some types of cancer. Computational approaches for detecting SVs often involve parameters that describe certain relevant biological phenomena. In our work, such parameters relate the incidence of inherited and novel SVs to probabilistic models of observing these SVs. In the work presented here, we investigate the sensitivity of our computational framework to these parameters. In particular, we demonstrate the robustness of our method by identifying a wide range of parameter values that lead to high-accuracy SV predictions in simulated data.

In this paper, we implement deep learning methods to recover downsampled noisy signals often present in compressed sensing applications. As an alternative to relying on previously established optimization based algorithms, we implement stacked denoising autoencoders and convolutional neural networks to perform signal reconstructions. Moreover, we propose a Poisson autoencoder inverting network (PAIN) architecture to reconstruct compressed signals imposed with Poisson noise. We observe less computational costs associated with this method while improving on reconstructions from a traditional stacked denoising autoencoder and remaining competitive with a more complex architecture in terms of Mean Squared Error (MSE). We train all proposed architectures on the MNIST dataset and establish deep neural networks as a reconstruction method.

Structural variants (SVs) - novel adjacencies in an individual's genome - lead to genomic diversity across all organisms. When DNA fragments of an unknown genome are compared to a reference genome, errors in sequencing and mapping obscure true genomic rearrangements. When the sequencing coverage is low, this may lead to high false positive rates in predicted SVs. In this paper, we propose a novel maximum likelihood approach to SV prediction incorporating low-coverage sequencing data and coverage distribution. Specifically, we address mean and variance assumptions proposed by Poisson models and develop a Negative Binomial framework which reflects a more accurate representation of DNA fragments in an individual's genome. We incorporate both sparsity and inheritance in our model with an ℓ1penalty and linear constraints, respectively. We validate our model on both simulated and real genomic data of related individuals. Moreover, our results indicate an improvement on thresholding observations of candidate variants.

Deep learning algorithms often require solving a highly non-linear and nonconvex unconstrained optimization problem. Generally, methods for solving the optimization problems in machine learning and in deep learning specifically are restricted to the class of first-order algorithms, like stochastic gradient descent (SGD). The major drawback of the SGD methods is that they have the undesirable effect of not escaping saddle-points. Furthermore, these methods require exhaustive trial-and-error to fine-tune many learning parameters. Using the second-order curvature information to find the search direction can help with more robust convergence for the non-convex optimization problem. However, computing the Hessian matrix for the large-scale problems is not computationally practical. Alternatively, quasi-Newton methods construct an approximate of Hessian matrix to build a quadratic model of the objective function. Quasi-Newton methods, like SGD, require only first-order gradient information, but they can result in superlinear convergence, which makes them attractive alternatives. The limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) approach is one of the most popular quasi-Newton methods that construct positive-definite Hessian approximations. Since the true Hessian matrix is not necessarily positive definite, an extra initialization condition is required to be introduced when constructing the L-BFGS matrices to avoid false negative curvature information. In this paper, we propose various choices for initialization methods of the L-BFGS matrices within a trust-region framework. We provide empirical results on the classification task of the MNIST digits dataset to compare the performance of the trust-region algorithm with different L-BFGS initialization methods.

Recent advances in high-throughput sequencing technologies, have led to the collection of vast quantities of genomic data., Structural variants (SVs) - rearrangements of the genome, larger than one letter such as inversions, insertions, deletions, and duplications - are an important source of genetic, variation and have been implicated in some genetic diseases., However, inferring SVs from sequencing data has proven to, be challenging because true SVs are rare and are prone to, low-coverage noise. In this paper, we attempt to mitigate the, deleterious effects of low-coverage sequences by following a, maximum likelihood approach to SV prediction. Specifically, we model the noise using Poisson statistics and constrain, the solution with a sparsity-promoting ℓ1penalty since SV, instances should be rare. In addition, because offspring SVs, inherit SVs from their parents, we incorporate familial relationships, in the optimization problem formulation to increase, the likelihood of detecting true SV occurrences. Numerical, results are presented to validate our proposed approach.

Critical to accurate reconstruction of sparse signals from low-dimensional low-photon count observations is the solution of nonlinear optimization problems that promote sparse solutions. In this paper, we explore recovering high-resolution sparse signals from low-resolution measurements corrupted by Poisson noise using a gradient-based optimization approach with non-convex regular-ization. In particular, we analyze zero-finding methods for solving the p-norm regularized minimization subproblems arising from a sequential quadratic approach. Numerical results from fluorescence molecular tomography are presented.

In photon-limited image reconstruction, the behavior of noise at the detector end is more accurately modeled as a Poisson point process than the common choice of a Gaussian distribution. As such, to recover the original signal more accurately, a penalized negative Poisson log-likelihood function - and not a least-squares function - is minimized. In many applications, including medical imaging, additional information on the signal of interest is often available. Specifically, its maximum and minimum amplitudes might be known a priori. This paper describes an approach that incorporates this information into a sparse photon-limited recovery method by the inclusion of upper and lower bound constraints. We demonstrate the effectiveness of the proposed approach on two different low-light deblurring examples.

Critical to accurate reconstruction of sparse signals from low-dimensional Poisson observations is the solution of nonlinear optimization problems that promote sparse solutions. Theoretically, non-convex ℓp-norm minimization (0 ≤ p1-norm relaxation commonly used in sparse signal recovery. In this paper, we propose an extension to the existing SPIRAL-ℓ1algorithm based on the Generalized Soft-Thersholding (GST) function to better recover signals with mostly nonzero entries from Poisson observations. This approach is based on iteratively minimizing a sequence of separable subproblems of the nonnegatively constrained, ℓp-penalized negative Poisson log-likelihood objective function using the GST function. We demonstrate the effectiveness of the proposed method, called SPIRAL-ℓp, through numerical experiments.

As an emerging near-infrared molecular imaging modality, fluorescence molecular tomography (FMT) has great potential in resolving the molecular and cellular processes in 3D objects through the reconstruction of the injected fluorescence probe concentration. In practice, when a charge-coupled device (CCD) camera is used to obtain FMT measurements, the observations are corrupted by noise which follows a Poisson distribution. To reconstruct the original concentration, the standard least-squares function for data-fitting is not a suitable objective function to minimize since this model assumes measurement noise which follows a Gaussian distribution. Rather, in this paper, we minimize a negative log-likelihood function to more accurately model the CCD camera shot noise. Furthermore, we exploit the presence of the flourescence in only small regions of the 3D object by introducing a non-convex penalty term that promotes sparsity in the reconstruction. This paper proposes a method to solve the FMT reconstruction problem from low-dimensional and low-mean photon count measurements. Using simulated data, we validate the effectiveness of the proposed non-convex Poisson-based reconstruction method for FMT inverse problems.

A MATLAB implementation of the Moré-Sorensen sequential (MSS) method is presented. The MSS method computes the minimizer of a quadratic function defined by a limited-memory BFGS matrix subject to a two-norm trust-region constraint. This solver is an adaptation of the Moré-Sorensen direct method into an L-BFGS setting for large-scale optimization. The MSS method makes use of a recently proposed stable fast direct method for solving large shifted BFGS systems of equations [Erway and Marcia 2012; Erway et al. 2012] and is able to compute solutions to any user-defined accuracy. This MATLAB implementation is a matrix-free iterative method for large-scale optimization. Numerical experiments on the CUTEr [Bongartz et al. 1995; Gould et al. 2003]) suggest that using the MSS method as a trust-region subproblem solver can require significantly fewer function and gradient evaluations needed by a trust-region method as compared with the Steihaug-Toint method.

The recovery of sparse images from noisy, blurry, and potentially low-dimensional observations can be accomplished by solving an optimization problem that minimizes the least-squares error in data fidelity with a sparsity-promoting regularization term (the so-called ℓ2- ℓ1minimization problem). This paper focuses on the reconstruction of a video sequence of images where known pixel-intensity bounds exist at each video frame. It has been established that the ℓ2- ℓ1minimization problem can be solved efficiently using gradient projection, which was recently extended to solve general bound-constrained ℓ2- ℓ1minimization problems. Furthermore, the video reconstruction can be made more efficient by exploiting similarities between consecutive frames. In this paper, we propose a method for reconstructing a video sequence that takes advantage of the inter-frame correlations while constraining the solution to satisfy known a priori bounds, offering a higher potential for increasingly accurate reconstructions. To demonstrate the effectiveness of this approach, we have included the results of our numerical experiments.

Observations in many applications consist of counts of discrete events, such as photons hitting a detector, which cannot be effectively modeled using an additive bounded or Gaussian noise model, and instead require a Poisson noise model. As a result, accurate reconstruction of a spatially or temporally distributed phenomenon (f*) from Poisson data (y) cannot be effectively accomplished by minimizing a conventional penalized least-squares objective function. The problem addressed in this paper is the estimation of f* from y in an inverse problem setting, where the number of unknowns may potentially be larger than the number of observations and f* admits sparse approximation. The optimization formulation considered in this paper uses a penalized negative Poisson log-likelihood objective function with nonnegativity constraints (since Poisson intensities are naturally nonnegative). In particular, the proposed approach incorporates key ideas of using separable quadratic approximations to the objective function at each iteration and penalization terms related to l1 norms of coefficient vectors, total variation seminorms, and partition-based multiscale estimation methods.

This paper describes a methodology for detecting anomalies from sequentially observed and potentially noisy data. The proposed approach consists of two main elements: 1) filtering, or assigning a belief or likelihood to each successive measurement based upon our ability to predict it from previous noisy observations and 2) hedging, or flagging potential anomalies by comparing the current belief against a time-varying and data-adaptive threshold. The threshold is adjusted based on the available feedback from an end user. Our algorithms, which combine universal prediction with recent work on online convex programming, do not require computing posterior distributions given all current observations and involve simple primal-dual parameter updates. At the heart of the proposed approach lie exponential-family models which can be used in a wide variety of contexts and applications, and which yield methods that achieve sublinear per-round regret against both static and slowly varying product distributions with marginals drawn from the same exponential family. Moreover, the regret against static distributions coincides with the minimax value of the corresponding online strongly convex game. We also prove bounds on the number of mistakes made during the hedging step relative to the best offline choice of the threshold with access to all estimated beliefs and feedback signals. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality, as well as on the Enron email dataset.

The l2-l1sparse signal minimization problem can be solved effi ciently by gradient projection. In many applications, the signal to be estimated is known to lie in some range of values. With these additional constraints on the estimate, the resultingconstrained min imization problem is more challenging to solve. In previous work, we proposed a gradient projection approach for solving this type of minimization problem with nonnegativity constraints. In this paper, we generalize this approach to solve any bound-constrained l2-l1minimization problem. Our method is based on solving the Lagrangian dual problem, and we show that by constraining the solution to known a priori bounds within the optimization method, we can obtain a more accurate estimate than simply thresholding the solution from the unconstrained minimization problem. Numerical results are presented to demonstrate the effectiveness of this approach.

This paper concerns the reconstruction of a temporally-varying scene from a video sequence of noisy linear projections. Assuming that each video frame is sparse or compressible in some basis, this inverse problem can be formulated as an ℓ2-ℓ1minimization problem, which can be solved efficiently using gradient projection. Since the signal of interest corresponds to nonnegative pixel intensities, additional nonnegativity constraints are included in the minimization problem, rendering the optimization problem more difficult to solve but with a greater potential for more accurate reconstructions. In this paper, we propose a method for reconstructing a video sequence that incorporates nonnegativity constraints and exploits inter-frame correlations to improve upon the naive approach of solving each frame independently. We present numerical experiments to demonstrate the effectiveness of this approach.

The ℓ2-ℓ1compressed sensing minimization problem can be solved efficiently by gradient projection. In imaging applications, the signal of interest corresponds to nonnegative pixel intensities; thus, with additional nonnegativity constraints on the reconstruction, the resulting constrained minimization problem becomes more challenging to solve. In this paper, we propose a gradient projection approach for sparse signal recovery where the reconstruction is subject to nonnegativity constraints. Numerical results are presented to demonstrate the effectiveness of this approach.

This paper describes an optimization framework for reconstructing nonnegative image intensities from linear projections contaminated with Poisson noise. Such Poisson inverse problems arise in a variety of applications, ranging from medical imaging to astronomy. A total variation regularization term is used to counter the ill-posedness of the inverse problem and results in reconstructions that are piecewise smooth. The proposed algorithm sequentially approximates the objective function with a regularized quadratic surrogate which can easily be minimized. Unlike alternative methods, this approach ensures that the natural nonnegativity constraints are satisfied without placing prohibitive restrictions on the nature of the linear projections to ensure computational tractability. The resulting algorithm is computationally efficient and outperforms similar methods using wavelet-sparsity or partition-based regularization.

This paper considers the problem of sequential assignment of probabilities (likelihoods) to elements of an individual sequence using an exponential family of probability distributions. We draw upon recent work on online convex programming to devise an algorithm that does not require computing posterior distributions given all current observations, involves simple primal-dual parameter updates, and achieves minimax per-round regret against slowly varying product distributions with marginals drawn from the same exponential family. We validate the theory on synthetic data drawn from a time-varying distribution over binary vectors of high dimensionality.

Recent work in the emerging field of compressive sensing indicates that, when feasible, judicious selection of the type of distortion induced by measurement systems may dramatically improve our ability to perform reconstruction. The basic idea of this theory is that when the signal of interest is very sparse (i.e., zero-valued at most locations) or compressible, relatively few incoherent observations are necessary to reconstruct the most significant non-zero signal components. However, applying this theory to practical imaging systems is challenging in the face of several measurement system constraints. This paper describes the design of coded aperture masks for super- resolution image reconstruction from a single, low-resolution, noisy observation image. Based upon recent theoretical work on Toeplitz- structured matrices for compressive sensing, the proposed masks are fast and memory-efficient to compute. Simulations demonstrate the effectiveness of these masks in several different settings.

Many infrared optical systems in wide-ranging applications such as surveillance and security frequently require large fields of view. Often this necessitates a focal plane array (FPA) with a large number of pixels, which, in general, is very expensive. In this paper, we propose a method for increasing the field of view without increasing the pixel resolution of the FPA by superimposing the multiple subimages within a scene and disambiguating the observed data to reconstruct the original scene. This technique, in effect, allows each subimage of the scene to share a single FPA, thereby increasing the field of view without compromising resolution. To disambiguate the subimages, we develop wavelet regularized reconstruction methods which encourage sparsity in the solution. We present results from numerical experiments that demonstrate the effectiveness of this approach.