Extended object reconstruction in adaptive-optics imaging: the multiresolution approach

We propose the application of multiresolution transforms, such as wavelets (WT) and curvelets (CT), to the reconstruction of images of extended objects that have been acquired with adaptive optics (AO) systems. Such multichannel approaches normally make use of probabilistic tools in order to distinguish significant structures from noise and reconstruction residuals. Furthermore, we aim to check the historical assumption that image-reconstruction algorithms using static PSFs are not suitable for AO imaging. We convolve an image of Saturn taken with the Hubble Space Telescope (HST) with AO PSFs from the 5-m Hale telescope at the Palomar Observatory and add both shot and readout noise. Subsequently, we apply different approaches to the blurred and noisy data in order to recover the original object. The approaches include multi-frame blind deconvolution (with the algorithm IDAC), myopic deconvolution with regularization (with MISTRAL) and wavelets- or curvelets-based static PSF deconvolution (AWMLE and ACMLE algorithms). We used the mean squared error (MSE) and the structural similarity index (SSIM) to compare the results. We discuss the strengths and weaknesses of the two metrics. We found that CT produces better results than WT, as measured in terms of MSE and SSIM. Multichannel deconvolution with a static PSF produces results which are generally better than the results obtained with the myopic/blind approaches (for the images we tested) thus showing that the ability of a method to suppress the noise and to track the underlying iterative process is just as critical as the capability of the myopic/blind approaches to update the PSF.Comment: In revision in Astronomy & Astrophysics. 19 pages, 13 figure

Convolutional Deblurring for Natural Imaging

In this paper, we propose a novel design of image deblurring in the form of one-shot convolution filtering that can directly convolve with naturally blurred images for restoration. The problem of optical blurring is a common disadvantage to many imaging applications that suffer from optical imperfections. Despite numerous deconvolution methods that blindly estimate blurring in either inclusive or exclusive forms, they are practically challenging due to high computational cost and low image reconstruction quality. Both conditions of high accuracy and high speed are prerequisites for high-throughput imaging platforms in digital archiving. In such platforms, deblurring is required after image acquisition before being stored, previewed, or processed for high-level interpretation. Therefore, on-the-fly correction of such images is important to avoid possible time delays, mitigate computational expenses, and increase image perception quality. We bridge this gap by synthesizing a deconvolution kernel as a linear combination of Finite Impulse Response (FIR) even-derivative filters that can be directly convolved with blurry input images to boost the frequency fall-off of the Point Spread Function (PSF) associated with the optical blur. We employ a Gaussian low-pass filter to decouple the image denoising problem for image edge deblurring. Furthermore, we propose a blind approach to estimate the PSF statistics for two Gaussian and Laplacian models that are common in many imaging pipelines. Thorough experiments are designed to test and validate the efficiency of the proposed method using 2054 naturally blurred images across six imaging applications and seven state-of-the-art deconvolution methods.Comment: 15 pages, for publication in IEEE Transaction Image Processin

From Symmetry to Geometry: Tractable Nonconvex Problems

As science and engineering have become increasingly data-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from the signal and data acquisition to modeling and prediction. The optimization problems encountered in practice are often nonconvex. While challenges vary from problem to problem, one common source of nonconvexity is nonlinearity in the data or measurement model. Nonlinear models often exhibit symmetries, creating complicated, nonconvex objective landscapes, with multiple equivalent solutions. Nevertheless, simple methods (e.g., gradient descent) often perform surprisingly well in practice. The goal of this survey is to highlight a class of tractable nonconvex problems, which can be understood through the lens of symmetries. These problems exhibit a characteristic geometric structure: local minimizers are symmetric copies of a single "ground truth" solution, while other critical points occur at balanced superpositions of symmetric copies of the ground truth, and exhibit negative curvature in directions that break the symmetry. This structure enables efficient methods to obtain global minimizers. We discuss examples of this phenomenon arising from a wide range of problems in imaging, signal processing, and data analysis. We highlight the key role of symmetry in shaping the objective landscape and discuss the different roles of rotational and discrete symmetries. This area is rich with observed phenomena and open problems; we close by highlighting directions for future research.Comment: review paper submitted to SIAM Review, 34 pages, 10 figure

Efficient Bayesian-based Multi-View Deconvolution

Light sheet fluorescence microscopy is able to image large specimen with high resolution by imaging the sam- ples from multiple angles. Multi-view deconvolution can significantly improve the resolution and contrast of the images, but its application has been limited due to the large size of the datasets. Here we present a Bayesian- based derivation of multi-view deconvolution that drastically improves the convergence time and provide a fast implementation utilizing graphics hardware.Comment: 48 pages, 20 figures, 1 table, under review at Nature Method

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Seismic sparse-spike deconvolution via Toeplitz-sparse matrix factorization

We have developed a new sparse-spike deconvolution (SSD) method based on Toeplitz-sparse matrix factorization (TSMF), a bilinear decomposition of a matrix into the product of a Toeplitz matrix and a sparse matrix, to address the problems of lateral continuity, effects of noise, and wavelet estimation error in SSD. Assuming the convolution model, a constant source wavelet, and the sparse reflectivity, a seismic profile can be considered as a matrix that is the product of a Toeplitz wavelet matrix and a sparse reflectivity matrix. Thus, we have developed an algorithm of TSMF to simultaneously deconvolve the seismic matrix into a wavelet matrix and a reflectivity matrix by alternatively solving two inversion subproblems related to the Toeplitz wavelet matrix and sparse reflectivity matrix, respectively. Because the seismic wavelet is usually compact and smooth, the fused Lasso was used to constrain the elements in the Toeplitz wavelet matrix. Moreover, due to the limitations of computer memory, large seismic data sets were divided into blocks, and the average of the source wavelets deconvolved from these blocks via TSMF-based SSD was used as the final estimation of the source wavelet for all blocks to deconvolve the reflectivity; thus, the lateral continuity of the seismic data can be maintained. The advantages of the proposed deconvolution method include using multiple traces to reduce the effect of random noise, tolerance to errors in the initial wavelet estimation, and the ability to preserve the complex structure of the seismic data without using any lateral constraints. Our tests on the synthetic seismic data from the Marmousi2 model and a section of field seismic data demonstrate that the proposed method can effectively derive the wavelet and reflectivity simultaneously from band-limited data with appropriate lateral coherence, even when the seismic data are contaminated by noise and the initial wavelet estimation is inaccurate

A Primal-Dual Proximal Algorithm for Sparse Template-Based Adaptive Filtering: Application to Seismic Multiple Removal

Unveiling meaningful geophysical information from seismic data requires to deal with both random and structured "noises". As their amplitude may be greater than signals of interest (primaries), additional prior information is especially important in performing efficient signal separation. We address here the problem of multiple reflections, caused by wave-field bouncing between layers. Since only approximate models of these phenomena are available, we propose a flexible framework for time-varying adaptive filtering of seismic signals, using sparse representations, based on inaccurate templates. We recast the joint estimation of adaptive filters and primaries in a new convex variational formulation. This approach allows us to incorporate plausible knowledge about noise statistics, data sparsity and slow filter variation in parsimony-promoting wavelet frames. The designed primal-dual algorithm solves a constrained minimization problem that alleviates standard regularization issues in finding hyperparameters. The approach demonstrates significantly good performance in low signal-to-noise ratio conditions, both for simulated and real field seismic data

In Situ Two-Thermocouple Sensor Characterisation using Cross-Relation Blind Deconvolution with Signal Conditioning for Improved Robustness

Thermocouples are one of the most widely used temperature measurement devices due to their low cost, ease of manufacture and robustness. However, their robustness is obtained at the expense of limited sensor bandwidth. Consequently, in many applications signal compensation techniques are needed to recover the true temperature from the attenuated measurements. This, is turn, necessitates in situ thermocouple characterisation. Recently the authors proposed a novel characterisation technique based on the cross-relation method of blind deconvolution applied to the output of two thermocouples simultaneously measuring the same temperature. This offers a number of advantages over competing methods including low estimation variance and no need for a priori knowledge of the time constant ratio. A weakness of the proposed method is that it yields biased estimates in the presence of measurement noise. In this paper we propose the inclusion of a signal conditioning step in the characterisation algorithm to improve the robustness to noise. The enhanced performance of the resulting algorithm is demonstrated using both simulated and experimental data