Richardson-Lucy Algorithm Research Articles

Source function from two-particle correlation through deblurring

In heavy-ion collisions, low relative-velocity two-particle correlations have been a tool for assessing space-time characteristics of particle emission. Those characteristics may be cast in the form of a relative emission source related to the correlation function through the Koonin-Pratt (KP) convolution formula that involves the relative wave-function for the particles in its kernel. In the literature, the source has been most commonly sought by parametrizing it in a Gaussian form and fitting to the correlation function. At times the source was more broadly imaged from the function, still employing a fitting. Here, we propose the use of the Richardson-Lucy (RL) optical deblurring algorithm for deducing the source from a correlation function. The RL algorithm originally follows from probabilistic Bayesian considerations and relies on the intensity distributions for the optical object and its image, as well as the convolution kernel, being positive definite, which is the case for the corresponding quantities of interest within the KP formula.

Regularized deconvolution for structured illumination microscopy via accelerated linearized ADMM

Due to the ill-posedness of the inverse deconvolution for structured illumination microscopy (SIM), the results of Richardson–Lucy algorithm are not ideal in the presence of noise. Here, we propose an accelerated linearized alternating direction method of multipliers (AL-ADMM) method for solving the regularized SIM deconvolution problem. A modification of the generalized inverse is introduced to overcome the large condition number of the convolution operator. This study shows that regularization or priori knowledge can effectively suppress noise and improve the resolution and contrast of the recovered SIM image. Simulations and experiments demonstrate that the proposed algorithm can efficiently extract higher-frequency information beyond the microscope optical transfer function for the corrupted SIM images to achieve computational super-resolution (SR) without hardware modifications.

Deblurring decay-energy distribution from an invariant-mass measurement

Measured decay-energy spectra from invariant mass spectroscopy can give insights into the shell structure of particle-unbound systems. However, it is challenging to extract the underlying physics from a measured spectrum due to detector resolution and acceptance effects. In this work, we introduce a deblurring method used to restore the decay energy spectrum measured for the three-body decay of 26O. The method utilizes the Richardson-Lucy algorithm, which has proven to be successful in optics, and does not require any prior knowledge about the resonance states in the observed spectrum. It also circumvents the singularity issue by iteratively adjusting a positive definite distribution. The only inputs are the observed energy spectrum and the detector’s response matrix also referred to as the Transfer Matrix (TM). We test the method’s performance on a simulated spectrum using the MoNA-LISA-Sweeper setup and the associated TM. Finally, the approach is applied to the decay energy spectrum of the 26O system measured in an experiment conducted at the National Superconducting Cyclotron Laboratory (NSCL) by the Modular Neutron Array (MoNA) Collaboration. The deblurring approach suggests three peaks in the decay energy spectrum of 26O at 0.15, 1.5 MeV, and a broad peak between 4 and 6 MeV.

Single-Shot 3D Incoherent Imaging Using Deterministic and Random Optical Fields with Lucy–Richardson–Rosen Algorithm

Coded aperture 3D imaging techniques have been rapidly evolving in recent years. The two main directions of evolution are in aperture engineering to generate the optimal optical field and in the development of a computational reconstruction method to reconstruct the object’s image from the intensity distribution with minimal noise. The goal is to find the ideal aperture–reconstruction method pair, and if not that, to optimize one to match the other for designing an imaging system with the required 3D imaging characteristics. The Lucy–Richardson–Rosen algorithm (LR2A), a recently developed computational reconstruction method, was found to perform better than its predecessors, such as matched filter, inverse filter, phase-only filter, Lucy–Richardson algorithm, and non-linear reconstruction (NLR), for certain apertures when the point spread function (PSF) is a real and symmetric function. For other cases of PSF, NLR performed better than the rest of the methods. In this tutorial, LR2A has been presented as a generalized approach for any optical field when the PSF is known along with MATLAB codes for reconstruction. The common problems and pitfalls in using LR2A have been discussed. Simulation and experimental studies for common optical fields such as spherical, Bessel, vortex beams, and exotic optical fields such as Airy, scattered, and self-rotating beams have been presented. From this study, it can be seen that it is possible to transfer the 3D imaging characteristics from non-imaging-type exotic fields to indirect imaging systems faithfully using LR2A. The application of LR2A to medical images such as colonoscopy images and cone beam computed tomography images with synthetic PSF has been demonstrated. We believe that the tutorial will provide a deeper understanding of computational reconstruction using LR2A.

A comparative study of deconvolution techniques for quantum-gas microscope images

Quantum-gas microscopes are used to study ultracold atoms in optical lattices at the single-particle level. In these systems atoms are localised on lattice sites with separations close to or below the diffraction limit. To determine the lattice occupation with high fidelity, a deconvolution of the images is often required. We compare three different techniques, a local iterative deconvolution algorithm, Wiener deconvolution and the Lucy–Richardson algorithm, using simulated microscope images. We investigate how the reconstruction fidelity scales with varying signal-to-noise ratio, lattice filling fraction, varying fluorescence levels per atom, and imaging resolution. The results of this study identify the limits of singe-atom detection and provide quantitative fidelities which are applicable for different atomic species and quantum-gas microscope setups.

Richardson–Lucy Deconvolution with a Spatially Variant Point-spread Function of Chandra: Supernova Remnant Cassiopeia A as an Example

Richardson–Lucy (RL) deconvolution is one of the classical methods widely used in X-ray astronomy and other areas. Amid recent progress in image processing, RL deconvolution still leaves much room for improvement under realistic situations. One direction is to include the positional dependence of a point-spread function (PSF), so-called RL deconvolution with a spatially variant PSF (RLsv). Another is the method of estimating a reliable number of iterations and their associated uncertainties. We developed a practical method that incorporates the RLsv algorithm and the estimation of uncertainties. As a typical example of bright and high-resolution images, the Chandra X-ray image of the supernova remnant Cassiopeia A was used in this paper. RLsv deconvolution enables us to uncover the smeared features in the forward/backward shocks and jet-like structures. We constructed a method to predict the appropriate number of iterations using statistical fluctuation of the observed images. Furthermore, the uncertainties were estimated by error propagation from the last iteration, which was phenomenologically tested with the observed data. Thus, our method is a practically efficient framework to evaluate the time evolution of the remnants and their fine structures embedded in high-resolution X-ray images.

Spatially-variant image deconvolution for photoacoustic tomography.

Photoacoustic tomography (PAT) system can reconstruct images of biological tissues with high resolution and contrast. However, in practice, the PAT images are usually degraded by spatially variant blur and streak artifacts due to the non-ideal imaging conditions and chosen reconstruction algorithms. Therefore, in this paper, we propose a two-phase restoration method to progressively improve the image quality. In the first phase, we design a precise device and measuring method to obtain spatially variant point spread function samples at preset positions of the PAT system in image domain, then we adopt principal component analysis and radial basis function interpolation to model the entire spatially variant point spread function. Afterwards, we propose a sparse logarithmic gradient regularized Richardson-Lucy (SLG-RL) algorithm to deblur the reconstructed PAT images. In the second phase, we present a novel method called deringing which is also based on SLG-RL to remove the streak artifacts. Finally, we evaluate our method with simulation, phantom and in vivo experiments, respectively. All the results show that our method can significantly improve the quality of PAT images.

Deep learning-enhanced fluorescence microscopy via confocal physical imaging model.

Confocal microscopy is one of the most widely used tools for high-resolution cellular, tissue imaging and industrial inspection. Micrograph reconstruction based on deep learning has become an effective tool for modern microscopy imaging techniques. While most deep learning methods neglect the imaging process mechanism, which requires a lot of work to solve the multi-scale image pairs aliasing problem. We show that these limitations can be mitigated via an image degradation model based on Richards-Wolf vectorial diffraction integral and confocal imaging theory. The low-resolution images required for network training are generated by model degradation from their high-resolution counterparts, thereby eliminating the need for accurate image alignment. The image degradation model ensures the generalization and fidelity of the confocal images. By combining the residual neural network with a lightweight feature attention module with degradation model of confocal microscopy ensures high fidelity and generalization. Experiments on different measured data report that compared with the two deconvolution algorithms, non-negative least squares algorithm and Richardson-Lucy algorithm, the structural similarity index between the network output image and the real image reaches a high level above 0.82, and the peak signal-to-noise ratio can be improved by more than 0.6 dB. It also shows good applicability in different deep learning networks.

Synthesized Image Reconstruction for Post-Reconstruction Resolution Recovery.

Resolution recovery (RR) techniques in positron emission tomography (PET) imaging aim to mitigate spatial resolution losses and related inaccuracies in quantification by using a model of the system's point spread function (PSF) during reconstruction or post-processing. However, including PSF modeling in fully 3-D image reconstruction is far from trivial as access to the scanner-specific forward and back-projectors is required, along with access to the 3-D sinogram data. Hence, post-reconstruction RR methods, such as the Richardson-Lucy (RL) algorithm, can be more practical. However, the RL method leads to relatively rapid noise amplification in early image iterations, giving inferior image quality compared to iterates obtained by placing the PSF model in the reconstruction algorithm. We propose a post-reconstruction RR method by synthesizing PET data by a forward projection of an initial real data reconstruction (such reconstructions are usually available via a scanner's standard reconstruction software). The synthetic PET data are then used to reconstruct an image, but crucially now including a modeled PSF within the system model used during reconstruction. Results from simulations and real data demonstrate the proposed method improves image quality compared to the RL algorithm, whilst avoiding the need for scanner-specific projectors and raw sinogram data (as required by standard PSF modeling within reconstruction).

Improved Classification of Blurred Images with Deep-Learning Networks Using Lucy-Richardson-Rosen Algorithm

Pattern recognition techniques form the heart of most, if not all, incoherent linear shift-invariant systems. When an object is recorded using a camera, the object information is sampled by the point spread function (PSF) of the system, replacing every object point with the PSF in the sensor. The PSF is a sharp Kronecker Delta-like function when the numerical aperture (NA) is large with no aberrations. When the NA is small, and the system has aberrations, the PSF appears blurred. In the case of aberrations, if the PSF is known, then the blurred object image can be deblurred by scanning the PSF over the recorded object intensity pattern and looking for pattern matching conditions through a mathematical process called correlation. Deep learning-based image classification for computer vision applications gained attention in recent years. The classification probability is highly dependent on the quality of images as even a minor blur can significantly alter the image classification results. In this study, a recently developed deblurring method, the Lucy-Richardson-Rosen algorithm (LR2A), was implemented to computationally refocus images recorded in the presence of spatio-spectral aberrations. The performance of LR2A was compared against the parent techniques: Lucy-Richardson algorithm and non-linear reconstruction. LR2A exhibited a superior deblurring capability even in extreme cases of spatio-spectral aberrations. Experimental results of deblurring a picture recorded using high-resolution smartphone cameras are presented. LR2A was implemented to significantly improve the performances of the widely used deep convolutional neural networks for image classification.

An Adaptive Richardson-Lucy Algorithm for Medical Image Restoration

Image restoration is the process of estimating the original image content from a degraded picture. In this paper, the Richardson-Lucy iterative algorithm was developed to improve the quality of degraded medical images. It has been assumed that medical images are exposed to two types of degradation. The first type is the blur function in the Gaussian form with different widths, i.e. σ = 1 , 2, and 3. The second type of degradation was assumed to be of the independent white Gaussian noise type with different signal-to-noise ratio values: SNR = 10, 50 , and 100. The results obtained from the adaptive filter are compared, quantitatively, with different conventional filters: inverse, Wiener, and constraint least square, by applying different measures, such as: power signal to noise ratio (PSNR), structural similarity index (SSID), and root mean square error (RMSE). The comparison showed that the adaptive recovery filter achieves better results.

Sparse deconvolution for background noise suppression with total variation regularization in light field microscopy.

In this Letter, we present a method aiming at background noise removal in the 3D reconstruction of light field microscopy (LFM). Sparsity and Hessian regularization are taken as two prior knowledges to process the original light field image before 3D deconvolution. Due to the noise suppression function of total variation (TV) regularization, we add the TV regularization term to the 3D Richardson-Lucy (RL) deconvolution. By comparing the light field reconstruction results of our method with another state-of-the-art method that is also based on RL deconvolution, the proposed method shows improved performance in terms of removing background noise and detail enhancement. This method will be beneficial to the application of LFM in biological high-quality imaging.

Enhanced µCT Imaging Protocol to Enable High-Resolution 3D Visualization of Microdamage in Rat Vertebrae

Contrast-enhanced μCT imaging has been used to provide non-destructive 3D images of microdamage, but at a lower quality than found in histology and 2D backscatter electron (BSE) imaging. This study aimed to quantify potential improvements in microdamage characterization by enhancing µCT scanning parameters. Eleven slides from 9 rat vertebrae (healthy = 3, osteolytic metastases = 3, mixed metastases = 3) previously stained for microdamage with BaSO4 and analyzed with BSE imaging (2μm voxel spacing) were used in this study. μCT imaging conducted under varying protocols (x-ray voltage, tube current, frame averaging) demonstrated enhanced scan parameters at 90 kVp, 44 µA, 0.5 mm aluminum filter, 8 times frame averaging, and 4.9 µm voxel spacing. Post-processing with Richardson-Lucy deconvolution further deblurred the μCT images. Labeled microdamage in the baseline, enhanced and deblurred μCT images were segmented and spatially quantified vs. BSE-labeled microdamage using a probability-based correlation metric at six inflation radii. Enhanced μCT scan parameters improved damage visualization and increased spatial correlation probability with BSE images. Deblurring improved the sharpness of stain boundaries but did not significantly improve spatial correlation probabilities in comparison to the enhanced scans. This enhanced μCT protocol facilitates 3D visualization of microdamage, an indicator of bone quality important to bone damage mechanics.

PAM, not spam: Towards quantitative, reproducible, and energy-preserving cavitation imaging

Since the development of the original passive acoustic mapping (PAM) algorithm for cavitation imaging in 2008, which employed a conventional beamformer adapted from seismology (time exposure acoustics), sequential algorithmic improvements have sought to enhance spatial resolution, reduce computation time and enable increasingly setup-independent reporting of cavitation activity. These enhancements have included the use of frequency-domain rather than time-domain implementations to improve computational efficiency, the introduction of the Robust Capon Beamformer to reduce artefacts, sub-aperture processing to further increase frame rate and the implementation of corrections for frequency-dependent attenuation and for array sensitivity and diffraction to account for probe- and depth-related variability. Once these corrections are taken into account, current PAM algorithms produce accurate estimates of cavitation source strength at a single location, but not in the surrounding region, leading to inaccurate estimates of the total cavitation energy over the treatment volume. We present an energy-preserving and computationally efficient PAM algorithm based on Lucy-Richardson deconvolution, that enables the calculation of a set-up independent Cavitational Radiated Energy Density (CRED) that ensures cavitation images accurately reflect the total energy of radiated acoustic emissions within the imaging domain.

A Richardson-Lucy Algorithm Based Blind Deconvolution to Decouple the Two Unknown Spatial-Temporal SRAM Margin Variations

A Richardson-Lucy algorithm (RLA) based blind deconvolution with a new filter design technique was successfully developed. This allowed decoupling and retrieving of the information of the two unknown factors at once for the Random Temporal Variation (RTV) and the Random Spatial Variation (RSV). The variations of SRAM operating voltage (VDDMIN) are caused by the two coupled factors of RTV and RSV. Thus, this decoupling technique is prerequisite for the margin analysis. The designs of the device characteristics and the screening test conditions based on the analysis results were briefly demonstrated. The proposed filter design technique for the RLA based blind deconvolution allowed to circumvent the ill-conditioning ringing phenomena. This gives to enjoy the advantage in convergence speed and simplicity of the RLA. The relative errors of the blind-deconvolution for the RSV and RTV distributions were reduced to less than 0.3% within only 300-iteration cycles compared with the conventional one.

Automatic Choroidal Segmentation in Optical Coherence Tomography Images Based on Curvelet Transform and Graph Theory.

Automatic segmentation of the choroid on optical coherence tomography (OCT) images helps ophthalmologists in diagnosing eye pathologies. Compared to manual segmentations, it is faster and is not affected by human errors. The presence of the large speckle noise in the OCT images limits the automatic segmentation and interpretation of them. To solve this problem, a new curvelet transform-based K-SVD method is proposed in this study. Furthermore, the dataset was manually segmented by a retinal ophthalmologist to draw a comparison with the proposed automatic segmentation technique. In this study, curvelet transform-based K-SVD dictionary learning and Lucy-Richardson algorithm were used to remove the speckle noise from OCT images. The Outer/Inner Choroidal Boundaries (O/ICB) were determined utilizing graph theory. The area between ICB and outer choroidal boundary was considered as the choroidal region. The proposed method was evaluated on our dataset and the average dice similarity coefficient (DSC) was calculated to be 92.14% ± 3.30% between automatic and manual segmented regions. Moreover, by applying the latest presented open-source algorithm by Mazzaferri et al. on our dataset, the mean DSC was calculated to be 55.75% ± 14.54%. A significant similarity was observed between automatic and manual segmentations. Automatic segmentation of the choroidal layer could be also utilized in large-scale quantitative studies of the choroid.

Improved bias and reproducibility of coronary artery calcification features using deconvolution.

Our long-range goal is to improve whole-heart CT calcium scores by extracting quantitative features from individual calcifications. Here, we perform deconvolution to improve bias/reproducibility of small calcification assessments, which can be degraded at the normal CT calcium score image resolution. We analyzed features of individual calcifications on repeated standard (2.5mm) and thin (1.25mm) slice scans from QRM-Cardio phantom, cadaver hearts, and CARDIA study participants. Preprocessing to improve the resolution involved of Lucy-Richardson deconvolution with a measured point spread function (PSF) or three-dimensional blind deconvolution in which the PSF was iteratively optimized on high detail structures such as calcifications in images. Using QRM with inserts having known mg-calcium, we determined that both blind and conventional deconvolution improved mass measurements nearly equally well on standard images. Further, deconvolved thin images gave an excellent recovery of actual mass scores, suggesting that such processing could be our gold standard. For CARDIA images, blind deconvolution greatly improved results on standard slices. Bias across 33 calcifications (without, with deconvolution) was (23%, 9%), (18%, 1%), and ( , ) for Agatston, volume, and mass scores, respectively. Reproducibility was (0.13, 0.10), (0.12, 0.08), and (0.11, 0.06), respectively. Mass scores were more reproducible than Agatston scores or volume scores. For many other calcification features, blind deconvolution improved reproducibility in 21 out of 24 features. Cadaver images showed similar improvements in bias/reproducibility and slightly better results with a measured PSF. Deconvolution improves bias and reproducibility of multiple features extracted from individual calcifications in CT calcium score exams. Blind deconvolution is useful for improving feature assessments of coronary calcification in archived datasets.

Super-resolution reconstruction algorithm for optical-resolution photoacoustic microscopy images based on sparsity and deconvolution.

The lateral resolution of the optical-resolution photoacoustic microscopy (OR-PAM) system depends on the focusing diameter of the probe beam. By increasing the numerical aperture (NA) of optical focusing, the lateral resolution of OR-PAM can be improved. However, the increase in NA results in smaller working distances, and the entire imaging system becomes very sensitive to small optical imperfections. The existing deconvolution-based algorithms are limited by the image signal-to-noise ratio when improving the resolution of OR-PAM images. In this paper, a super-resolution reconstruction algorithm for OR-PAM images based on sparsity and deconvolution is proposed. The OR-PAM image is sparsely reconstructed according to the constructed loss function, which utilizes the sparsity of the image to combat the decrease in the resolution. The gradient accelerated Landweber iterative algorithm is used to deconvolve to obtain high-resolution OR-PAM images. Experimental results show that the proposed algorithm can improve the resolution of mouse retinal images by approximately 1.7 times without increasing the NA of the imaging system. In addition, compared to the Richardson-Lucy algorithm, the proposed algorithm can further improve the image resolution and maintain better imaging quality, which provides a foundation for the development of OR-PAM in clinical research.

Gaussian convolution decomposition for non-Gaussian shaped pulsed LiDAR waveform

The full waveform decomposition technique is significant for LiDAR ranging. It is challenging to extract the parameters from non-Gaussian shaped waveforms accurately. Many parametric models (e.g. the Gaussian distribution, the lognormal distribution, the generalized normal distribution, the Burr distribution, and the skew-normal distribution) were proposed to fit sharply-peaked, heavy-tailed, and negative-tailed waveforms. However, these models can constrain the shape of the waveform components. In this article, the Gaussian convolution model is established. Firstly, a set of Gaussian functions is calculated to characterize the system waveform so that asymmetric and non-Gaussian system waveforms can be included. The convolution result of the system waveform and the target response is used as the model for fitting the overlapped echo. Then a combination method of the Richardson–Lucy deconvolution, layered iterative, and Gaussian convolution is introduced to estimate the initial parameters. The Levenberg–Marquardt algorithm is used for the optimization fitting. Through experiments on synthetic data and practical recorded coding LiDAR data, we compare the proposed method with two decomposition approaches (Gaussian decomposition and skew-normal decomposition). The experiment results revealed that the proposed method could precisely decompose the overlapped non-Gaussian heavy-tailed waveforms and provide the best ranging accuracy, component fitting accuracy, and anti-noise performance. However, the traditional Gaussian and skew-normal decomposition methods can not fit the components well, resulting in inaccurate range estimates.

Luminosity rectified blind Richardson-Lucy deconvolution for single retinal image restoration

Background and objectiveDue to imperfect imaging conditions, retinal images can be degraded by uneven/insufficient illumination, blurriness caused by optical aberrations and unintentional motions. Degraded images reduce the effectiveness of diagnosis by an ophthalmologist. To restore the image quality, in this research we propose the luminosity rectified Richardson-Lucy (LRRL) blind deconvolution framework for single retinal image restoration. MethodsWe established an image formation model based on the double-pass fundus reflection feature and developed a differentiable non-convex cost function that jointly achieves illumination correction and blind deconvolution. To solve this non-convex optimization problem, we derived the closed-form expression of the gradients and used gradient descent with Nesterov-accelerated adaptive momentum estimation to accelerate the optimization, which is more efficient than the traditional half quadratic splitting method. ResultsThe LRRL was tested on 1719 images from three public databases. Four image quality matrixes including image definition, image sharpness, image entropy, and image multiscale contrast were used for objective assessments. The LRRL was compared against the state-of-the-art retinal image blind deconvolution methods. ConclusionsOur LRRL corrects the problematic illumination and improves the clarity of the retinal image simultaneously, showing its superiority in terms of restoration quality and implementation efficiency. The MATLAB code is available on Github.