Search

Recently, deep learning approaches have become the main research frontier for biological image reconstruction and enhancement problems thanks to their high performance, along with their ultra-fast inference times. However, due to the difficulty of obtaining matched reference data for supervised learning, there has been increasing interest in unsupervised learning approaches that do not need paired reference data. In particular, self-supervised learning and generative models have been successfully used for various biological imaging applications. In this paper, we overview these approaches from a coherent perspective in the context of classical inverse problems, and discuss their applications to biological imaging, including electron, fluorescence and deconvolution microscopy, optical diffraction tomography and functional neuroimaging.

Recent proposals of deep learning-based beamformers for ultrasound imaging (US) have attracted significant attention as computational efficient alternatives to adaptive and compressive beamformers. Moreover, deep beamformers are versatile in that image post-processing algorithms can be readily combined. Unfortunately, with the existing technology, a large number of beamformers need to be trained and stored for different probes, organs, depth ranges, operating frequency, and desired target 'styles', demanding significant resources such as training data, etc. To address this problem, here we propose a switchable and tunable deep beamformer that can switch between various types of outputs such as DAS, MVBF, DMAS, GCF, etc., and also adjust noise removal levels at the inference phase, by using a simple switch or tunable nozzle. This novel mechanism is implemented through Adaptive Instance Normalization (AdaIN) layers, so that distinct outputs can be generated using a single generator by merely changing the AdaIN codes. Experimental results using B-mode focused ultrasound confirm the flexibility and efficacy of the proposed method for various applications.

Metal artifact reduction (MAR) is one of the most important research topics in computed tomography (CT). With the advance of deep learning approaches for image reconstruction, various deep learning methods have been suggested for metal artifact reduction, among which supervised learning methods are most popular. However, matched metal-artifact-free and metal artifact corrupted image pairs are difficult to obtain in real CT acquisition. Recently, a promising unsupervised learning for MAR was proposed using feature disentanglement, but the resulting network architecture is so complicated that it is difficult to handle large size clinical images. To address this, here we propose a simple and effective unsupervised learning method for MAR. The proposed method is based on a novel β -cycleGAN architecture derived from the optimal transport theory for appropriate feature space disentanglement. Moreover, by adding the convolutional block attention module (CBAM) layers in the generator, we show that the metal artifacts can be more focused so that it can be effectively removed. Experimental results confirm that we can achieve improved metal artifact reduction that preserves the detailed texture of the original image.

Multispectral satellite imaging sensors acquire various spectral band images and have a unique spectroscopic property in each band. Unfortunately, image artifacts from imaging sensor noise often affect the quality of scenes and have a negative impact on applications for satellite imagery. Recently, deep learning approaches have been extensively explored to remove noise in satellite imagery. Most deep learning denoising methods, however, follow a supervised learning scheme, which requires matched noisy image and clean image pairs that are difficult to collect in real situations. In this article, we propose a novel unsupervised multispectral denoising method for satellite imagery using a wavelet directional cycle-consistent adversarial network (WavCycleGAN). The proposed method is based on an unsupervised learning scheme using adversarial loss and cycle-consistency loss to overcome the lack of paired data. Moreover, in contrast to the standard image-domain cycleGAN, we introduce a wavelet directional learning scheme for effective denoising without sacrificing high-frequency components such as edges and detailed information. Experimental results for the removal of vertical stripes and wave noise in satellite imaging sensors demonstrate that the proposed method effectively removes noise and preserves important high-frequency features of satellite images.

Recently, deep learning approaches have been successfully used for ultrasound (US) image artifact removal. However, paired high-quality images for supervised training are difficult to obtain in many practical situations. Inspired by the recent theory of unsupervised learning using optimal transport driven CycleGAN (OT-CycleGAN), here, we investigate the applicability of unsupervised deep learning for US artifact removal problems without matched reference data. Two types of OT-CycleGAN approaches are employed: one with the partial knowledge of the image degradation physics and the other with the lack of such knowledge. Various US artifact removal problems are then addressed using the two types of OT-CycleGAN. Experimental results for various unsupervised US artifact removal tasks confirmed that our unsupervised learning method delivers results comparable to supervised learning in many practical applications.

Optical diffraction tomography measures the three-dimensional refractive index map of a specimen and visualizes biochemical phenomena at the nanoscale in a non-destructive manner. One major drawback of optical diffraction tomography is poor axial resolution due to limited access to the three-dimensional optical transfer function. This missing cone problem has been addressed through regularization algorithms that use a priori information, such as non-negativity and sample smoothness. However, the iterative nature of these algorithms and their parameter dependency make real-time visualization impossible. In this article, we propose and experimentally demonstrate a deep neural network, which we term DeepRegularizer, that rapidly improves the resolution of a three-dimensional refractive index map. Trained with pairs of datasets (a raw refractive index tomogram and a resolution-enhanced refractive index tomogram via the iterative total variation algorithm), the three-dimensional U-net-based convolutional neural network learns a transformation between the two tomogram domains. The feasibility and generalizability of our network are demonstrated using bacterial cells and a human leukaemic cell line, and by validating the model across different samples. DeepRegularizer offers more than an order of magnitude faster regularization performance compared to the conventional iterative method. We envision that the proposed data-driven approach can bypass the high time complexity of various image reconstructions in other imaging modalities.

Time-resolved MR angiography (tMRA) has been widely used for dynamic contrast enhanced MRI (DCE-MRI) due to its highly accelerated acquisition. In tMRA, the periphery of the ${\textit k}$ -space data are sparsely sampled so that neighbouring frames can be merged to construct one temporal frame. However, this view-sharing scheme fundamentally limits the temporal resolution, and it is not possible to change the view-sharing number to achieve different spatio-temporal resolution trade-offs. Although many deep learning approaches have been recently proposed for MR reconstruction from sparse samples, the existing approaches usually require matched fully sampled ${\textit k}$ -space reference data for supervised training, which is not suitable for tMRA due to the lack of high spatio-temporal resolution ground-truth images. To address this problem, here we propose a novel unpaired training scheme for deep learning using optimal transport driven cycle-consistent generative adversarial network (cycleGAN). In contrast to the conventional cycleGAN with two pairs of generator and discriminator, the new architecture requires just a single pair of generator and discriminator, which makes the training much simpler but still improves the performance. Reconstruction results using in vivo tMRA and simulation data set confirm that the proposed method can immediately generate high quality reconstruction results at various choices of view-sharing numbers, allowing us to exploit better trade-off between spatial and temporal resolution in time-resolved MR angiography.

Optical diffraction tomography (ODT) produces a three-dimensional distribution of the refractive index (RI) by measuring scattering fields at various angles. Although the distribution of the RI is highly informative, due to the missing cone problem stemming from the limited-angle acquisition of holograms, reconstructions have very poor resolution along the axial direction compared to the horizontal imaging plane. To solve this issue, we present a novel unsupervised deep learning framework that learns the probability distribution of missing projection views through an optimal transport-driven CycleGAN. The experimental results show that missing cone artifacts in ODT data can be significantly resolved by the proposed method.

Conebeam CT using a circular trajectory is quite often used for various applications due to its relative simple geometry. For conebeam geometry, Feldkamp, Davis and Kress algorithm is regarded as the standard reconstruction method, but this algorithm suffers from so-called conebeam artifacts as the cone angle increases. Various model-based iterative reconstruction methods have been developed to reduce the cone-beam artifacts, but these algorithms usually require multiple applications of computational expensive forward and backprojections. In this paper, we develop a novel deep learning approach for accurate conebeam artifact removal. In particular, our deep network, designed on the differentiated backprojection domain, performs a data-driven inversion of an ill-posed deconvolution problem associated with the Hilbert transform. The reconstruction results along the coronal and sagittal directions are then combined using a spectral blending technique to minimize the spectral leakage. Experimental results show that our method outperforms the existing iterative methods despite significantly reduced runtime complexity., Comment: This paper is accepted for IEEE Trans. Medical Imaging

The nineteen papers in this special section focus on domain enriched learning for medical imaging. In recent years, learning based methods have emerged to complement traditional model and feature based methods for a variety of medical imaging problems such as image formation, classification and segmentation, quality enhancement etc. In the case of deep neural networks, many solutions have achieved unprecedented performance gains and have defined a new state of the art. Despite the progress, compelling open challenges remain. One such key challenge is that many learning frameworks (notably deep learning) are purely data-driven approaches and their performance depends strongly on the quantity and quality of training image data available. When training is limited or noisy, the performance drops sharply. Deep neural networks based approaches additionally face the challenge of often not being straightforward to interpret. Fortunately, exciting recent progress has emerged in enriching learning frameworks with domain knowledge and signal structure. As a couple of representative examples: in image reconstruction problems, this may involve using statistical/structural image priors; for image segmentation, shape and anatomical knowledge (conveyed by an expert) may be leveraged, etc. This special issue brings together contributions that combine signal, image priors and other flavors of domain knowledge with machine learning methods for solving many diverse medical imaging problems.

In ultrasound (US) imaging, various types of adaptive beamforming techniques have been investigated to improve the resolution and contrast-to-noise ratio of the delay and sum (DAS) beamformers. Unfortunately, the performance of these adaptive beamforming approaches degrade when the underlying model is not sufficiently accurate and the number of channels decreases. To address this problem, here we propose a deep learning-based beamformer to generate significantly improved images over widely varying measurement conditions and channel subsampling patterns. In particular, our deep neural network is designed to directly process full or sub-sampled radio-frequency (RF) data acquired at various subsampling rates and detector configurations so that it can generate high quality ultrasound images using a single beamformer. The origin of such input-dependent adaptivity is also theoretically analyzed. Experimental results using B-mode focused ultrasound confirm the efficacy of the proposed methods., This is a significantly extended version of the original paper in arXiv:1901.01706. This paper is accepted for IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control

Under the global pandemic of COVID-19, the use of artificial intelligence to analyze chest X-ray (CXR) image for COVID-19 diagnosis and patient triage is becoming important. Unfortunately, due to the emergent nature of the COVID-19 pandemic, a systematic collection of the CXR data set for deep neural network training is difficult. To address this problem, here we propose a patch-based convolutional neural network approach with a relatively small number of trainable parameters for COVID-19 diagnosis. The proposed method is inspired by our statistical analysis of the potential imaging biomarkers of the CXR radiographs. Experimental results show that our method achieves state-of-the-art performance and provides clinically interpretable saliency maps, which are useful for COVID-19 diagnosis and patient triage., Accepted for IEEE Trans. on Medical Imaging Special Issue on Imaging-based Diagnosis of COVID-19

Deconvolution microscopy has been extensively used to improve the resolution of the wide-field fluorescent microscopy, but the performance of classical approaches critically depends on the accuracy of a model and optimization algorithms. Recently, the convolutional neural network (CNN) approaches have been studied as a fast and high performance alternative. Unfortunately, the CNN approaches usually require matched high resolution images for supervised training. In this article, we present a novel unsupervised cycle-consistent generative adversarial network (cycleGAN) with a linear blur kernel, which can be used for both blind- and non-blind image deconvolution. In contrast to the conventional cycleGAN approaches that require two deep generators, the proposed cycleGAN approach needs only a single deep generator and a linear blur kernel, which significantly improves the robustness and efficiency of network training. We show that the proposed architecture is indeed a dual formulation of an optimal transport problem that uses a special form of the penalized least squares cost as a transport cost. Experimental results using simulated and real experimental data confirm the efficacy of the algorithm.

Recent state-of-the-art image segmentation algorithms are mostly based on deep neural networks, thanks to their high performance and fast computation time. However, these methods are usually trained in a supervised manner, which requires large number of high quality ground-truth segmentation masks. On the other hand, classical image segmentation approaches such as level-set methods are formulated in a self-supervised manner by minimizing energy functions such as Mumford-Shah functional, so they are still useful to help generation of segmentation masks without labels. Unfortunately, these algorithms are usually computationally expensive and often have limitation in semantic segmentation. In this paper, we propose a novel loss function based on Mumford-Shah functional that can be used in deep-learning based image segmentation without or with small labeled data. This loss function is based on the observation that the softmax layer of deep neural networks has striking similarity to the characteristic function in the Mumford-Shah functional. We show that the new loss function enables semi-supervised and unsupervised segmentation. In addition, our loss function can be also used as a regularized function to enhance supervised semantic segmentation algorithms. Experimental results on multiple datasets demonstrate the effectiveness of the proposed method., Comment: Accepted for IEEE Transactions on Image Processing

The articles in this special section focus on computational magnetic resonance imaging (MRI) using compressed sensing applications. Presents recent developments in computational MRI. These developments are pushing the frontier of computational imaging beyond CS. Similar to CS, most of these algorithms rely on image representation in one form or another. However, the common recent thread is the departure from handcrafted image representations to learning-based image representations. These learned representations are seamlessly combined with clever measurement strategies to significantly advance the state of the art in a number of areas. Several exciting applications including significantly improved spatial and temporal resolution, a considerable reduction in scan time, measurement of biophysical parameters directly from highly undersampled data, and direct measurement of very high-dimensional data are reviewed in this special issue of SPM. This issue describes key ideas underlying the computational approaches used in MRI. These approaches range from CS algorithms that rely on fixed transforms or dictionaries, to adaptive or shallow-learning algorithms that adapt the image representation to the data to recent deep-learning methods that learn a highly nonlinear representation from exemplar data. The articles provide insight into the capabilities of the current algorithms, their limitations, and their utility in challenging MRI problems.

In portable, three dimensional, and ultra-fast ultrasound imaging systems, there is an increasing demand for the reconstruction of high quality images from a limited number of radio-frequency (RF) measurements due to receiver (Rx) or transmit (Xmit) event sub-sampling. However, due to the presence of side lobe artifacts from RF sub-sampling, the standard beamformer often produces blurry images with less contrast, which are unsuitable for diagnostic purposes. Existing compressed sensing approaches often require either hardware changes or computationally expensive algorithms, but their quality improvements are limited. To address this problem, here we propose a novel deep learning approach that directly interpolates the missing RF data by utilizing redundancy in the Rx-Xmit plane. Our extensive experimental results using sub-sampled RF data from a multi-line acquisition B-mode system confirm that the proposed method can effectively reduce the data rate without sacrificing image quality., The title has been changed. This version will appear in IEEE Trans. on Medical Imaging

Localization microscopy, such as STORM / PALM, can reconstruct super-resolution images with a nanometer resolution through the iterative localization of fluorescence molecules. Recent studies in this area have focused mainly on the localization of densely activated molecules to improve temporal resolutions. However, higher density imaging requires an advanced algorithm that can resolve closely spaced molecules. Accordingly, sparsitydriven methods have been studied extensively. One of the major limitations of existing sparsity-driven approaches is the need for a fine sampling grid or for Taylor series approximation which may result in some degree of localization bias toward the grid. In addition, prior knowledge of the point-spread function (PSF) is required. To address these drawbacks, here we propose a true grid-free localization algorithm with adaptive PSF estimation. Specifically, based on the observation that sparsity in the spatial domain implies a low rank in the Fourier domain, the proposed method converts source localization problems into Fourier-domain signal processing problems so that a truly gridfree localization is possible. We verify the performance of the newly proposed method with several numerical simulations and a live-cell imaging experiment.

The widespread application of artificial intelligence in health research is currently hampered by limitations in data availability. Distributed learning methods such as federated learning (FL) and shared learning (SL) are introduced to solve this problem as well as data management and ownership issues with their different strengths and weaknesses. The recent proposal of federated split task-agnostic (FeSTA) learning tries to reconcile the distinct merits of FL and SL by enabling the multi-task collaboration between participants through Vision Transformer (ViT) architecture, but they suffer from higher communication overhead. To address this, here we present a multi-task distributed learning using ViT with random patch permutation. Instead of using a CNN based head as in FeSTA, p-FeSTA adopts a randomly permuting simple patch embedder, improving the multi-task learning performance without sacrificing privacy. Experimental results confirm that the proposed method significantly enhances the benefit of multi-task collaboration, communication efficiency, and privacy preservation, shedding light on practical multi-task distributed learning in the field of medical imaging., 10 pages

X-ray computed tomography (CT) using sparse projection views is a recent approach to reduce the radiation dose. However, due to the insufficient projection views, an analytic reconstruction approach using the filtered back projection (FBP) produces severe streaking artifacts. Recently, deep learning approaches using large receptive field neural networks such as U-Net have demonstrated impressive performance for sparse- view CT reconstruction. However, theoretical justification is still lacking. Inspired by the recent theory of deep convolutional framelets, the main goal of this paper is, therefore, to reveal the limitation of U-Net and propose new multi-resolution deep learning schemes. In particular, we show that the alternative U- Net variants such as dual frame and the tight frame U-Nets satisfy the so-called frame condition which make them better for effective recovery of high frequency edges in sparse view- CT. Using extensive experiments with real patient data set, we demonstrate that the new network architectures provide better reconstruction performance., This will appear in IEEE Transaction on Medical Imaging, a special issue of Machine Learning for Image Reconstruction

Parallel MRI (pMRI) and compressed sensing MRI (CS-MRI) have been considered as two distinct reconstruction problems. Inspired by recent k-space interpolation methods, an annihilating filter based low-rank Hankel matrix approach (ALOHA) is proposed as a general framework for sparsity-driven k-space interpolation method which unifies pMRI and CS-MRI. Specifically, our framework is based on the fundamental duality between the transform domain sparsity in the primary space and the low-rankness of weighted Hankel matrix in the reciprocal space, which converts pMRI and CS-MRI to a k-space interpolation problem using structured matrix completion. Using theoretical results from the latest compressed sensing literatures, we showed that the required sampling rates for ALOHA may achieve the optimal rate. Experimental results with in vivo data for single/multi-coil imaging as well as dynamic imaging confirmed that the proposed method outperforms the state-of-the-art pMRI and CS-MRI.

A multiple measurement vector problem (MMV) is a generalization of the compressed sensing problem that addresses the recovery of a set of jointly sparse signal vectors. One of the important contributions of this paper is to show that the seemingly least related state-of-the-art MMV joint sparse recovery algorithms—the M-SBL (multiple sparse Bayesian learning) and subspace-based hybrid greedy algorithms—have a very important link. More specifically, we show that replacing the $\log\det(\,{\cdot}\,)$ term in the M-SBL by a rank surrogate that exploits the spark reduction property discovered in the subspace-based joint sparse recovery algorithms provides significant improvements. In particular, if we use the Schatten- $p$ quasi-norm as the corresponding rank surrogate, the global minimizer of the cost function in the proposed algorithm becomes identical to the true solution as $p \rightarrow 0$ . Furthermore, under regularity conditions, we show that convergence to a local minimizer is guaranteed using an alternating minimization algorithm that has closed form expressions for each of the minimization steps, which are convex. Numerical simulations under a variety of scenarios in terms of SNR and the condition number of the signal amplitude matrix show that the proposed algorithm consistently outperformed the M-SBL and other state-of-the art algorithms.

In this paper, we propose a patch-based image inpainting method using a low-rank Hankel structured matrix completion approach. The proposed method exploits the annihilation property between a shift-invariant filter and image data observed in many existing inpainting algorithms. In particular, by exploiting the commutative property of the convolution, the annihilation property results in a low-rank block Hankel structure data matrix, and the image inpainting problem becomes a low-rank structured matrix completion problem. The block Hankel structured matrices are obtained patch-by-patch to adapt to the local changes in the image statistics. To solve the structured low-rank matrix completion problem, we employ an alternating direction method of multipliers with factorization matrix initialization using the low-rank matrix fitting algorithm. As a side product of the matrix factorization, locally adaptive dictionaries can be also easily constructed. Despite the simplicity of the algorithm, the experimental results using irregularly subsampled images as well as various images with globally missing patterns showed that the proposed method outperforms existing state-of-the-art image inpainting methods.

Diffuse optical tomography (DOT) is a non-invasive imaging technique to reconstruct optical properties of biological tissues using near-infrared light, and it has been successfully used to measure functional brain activities via changes in cerebral blood volume and cerebral blood oxygenation. However, DOT presents a severely ill-posed inverse problem, so various types of regularization should be incorporated to overcome low spatial resolution and lack of depth sensitivity. Another limitation of the conventional DOT reconstruction methods is that an inference step is separately performed after the reconstruction, so complicated interaction between reconstruction and regularization is difficult to analyze. To overcome these technical difficulties, we propose a unified sparse recovery framework using a random effect model whose termination criterion is determined by the statistical inference. Both numerical and experimental results confirm that the proposed method outperforms the conventional approaches.

In a multiple measurement vector problem (MMV), where multiple signals share a common sparse support and are sampled by a common sensing matrix, we can expect joint sparsity to enable a further reduction in the number of required measurements. While a diversity gain from joint sparsity had been demonstrated earlier in the case of a convex relaxation method using an $l_1/l_2$ mixed norm penalty, only recently was it shown that similar diversity gain can be achieved by greedy algorithms if we combine greedy steps with a MUSIC-like subspace criterion. However, the main limitation of these hybrid algorithms is that they often require a large number of snapshots or a high signal-to-noise ratio (SNR) for an accurate subspace as well as partial support estimation. One of the main contributions of this work is to show that the noise robustness of these algorithms can be significantly improved by allowing sequential subspace estimation and support filtering, even when the number of snapshots is insufficient. Numerical simulations show that a novel sequential compressive MUSIC (sequential CS-MUSIC) that combines the sequential subspace estimation and support filtering steps significantly outperforms the existing greedy algorithms and is quite comparable with computationally expensive state-of-art algorithms.

The multiple measurement vector (MMV) problem addresses the identification of unknown input vectors that share common sparse support. Even though MMV problems have been traditionally addressed within the context of sensor array signal processing, the recent trend is to apply compressive sensing (CS) due to its capability to estimate sparse support even with an insufficient number of snapshots, in which case classical array signal processing fails. However, CS guarantees the accurate recovery in a probabilistic manner, which often shows inferior performance in the regime where the traditional array signal processing approaches succeed. The apparent dichotomy between the probabilistic CS and deterministic sensor array signal processing has not been fully understood. The main contribution of the present article is a unified approach that revisits the link between CS and array signal processing first unveiled in the mid 1990s by Feng and Bresler. The new algorithm, which we call compressive MUSIC, identifies the parts of support using CS, after which the remaining supports are estimated using a novel generalized MUSIC criterion. Using a large system MMV model, we show that our compressive MUSIC requires a smaller number of sensor elements for accurate support recovery than the existing CS methods and that it can approach the optimal -bound with finite number of snapshots even in cases where the signals are linearly dependent.

Diffuse optical tomography (DOT) is a sensitive and relatively low cost imaging modality that reconstructs optical properties of a highly scattering medium. However, due to the diffusive nature of light propagation, the problem is severely ill-conditioned and highly nonlinear. Even though nonlinear iterative methods have been commonly used, they are computationally expensive especially for three dimensional imaging geometry. Recently, compressed sensing theory has provided a systematic understanding of high resolution reconstruction of sparse objects in many imaging problems; hence, the goal of this paper is to extend the theory to the diffuse optical tomography problem. The main contributions of this paper are to formulate the imaging problem as a joint sparse recovery problem in a compressive sensing framework and to propose a novel noniterative and exact inversion algorithm that achieves the l(0) optimality as the rank of measurement increases to the unknown sparsity level. The algorithm is based on the recently discovered generalized MUSIC criterion, which exploits the advantages of both compressive sensing and array signal processing. A theoretical criterion for optimizing the imaging geometry is provided, and simulation results confirm that the new algorithm outperforms the existing algorithms and reliably reconstructs the optical inhomogeneities when we assume that the optical background is known to a reasonable accuracy.

Recent theory of compressed sensing informs us that near-exact recovery of an unknown sparse signal is possible from a very limited number of Fourier samples by solving a convex L1 optimization problem. The main contribution of the present letter is a compressed sensing-based novel nonparametric shape estimation framework and a computational algorithm for binary star shape objects, whose radius functions belong to the space of bounded-variation functions. Specifically, in contrast with standard compressed sensing, the present approach involves directly reconstructing the shape boundary under sparsity constraint. This is done by converting the standard pixel-based reconstruction approach into estimation of a nonparametric shape boundary on a wavelet basis. This results in an L1 minimization under a nonlinear constraint, which makes the optimization problem nonconvex. We solve the problem by successive linearization and application of one-dimensional L1 minimization, which significantly reduces the number of sampling requirements as well as the computational burden. Fourier imaging simulation results demonstrate that high quality reconstruction can be quickly obtained from a very limited number of samples. Furthermore, the algorithm outperforms the standard compressed sensing reconstruction approach using the total variation norm.

Optical diffusion tomography is a technique for imaging a highly scattering medium using measurements of transmitted modulated light. Reconstruction of the spatial distribution of the optical properties of the medium from such data is a difficult nonlinear inverse problem. Bayesian approaches are effective, but are computationally expensive, especially for three-dimensional (3-D) imaging. This paper presents a general nonlinear multigrid optimization technique suitable for reducing the computational burden in a range of nonquadratic optimization problems. This multigrid method is applied to compute the maximum a posteriori (MAP) estimate of the reconstructed image in the optical diffusion tomography problem. The proposed multigrid approach both dramatically reduces the required computation and improves the reconstructed image quality.

We introduce confidence region techniques for analyzing and visualizing the performance of two-dimensional parametric shape estimators. Assuming an asymptotically normal and efficient estimator for a finite parameterization of the object boundary, Cramer-Rao bounds are used to define an asymptotic confidence region, centered around the true boundary. Computation of the probability that an entire boundary estimate lies within the confidence region is a challenging problem, because the estimate is a two-dimensional nonstationary random process. We derive lower bounds on this probability using level crossing statistics. The same bounds also apply to asymptotic confidence regions formed around the estimated boundaries, lower-bounding the probability that the entire true boundary lies within the confidence region. The results make it possible to generate asymptotic confidence regions for arbitrary prescribed probabilities. These asymptotic global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated object, and facilitate geometric inferences. Numerical simulations suggest that the new bounds are quite tight.

There are a few corrections for the above titled paper (IEEE Trans. Inf. Theory, vol. 58, no. 1, pp. 278-301, Jan. 2012). They are presented here.