Richardson-Lucy Research Articles

A Novel Richardson-Lucy Model with Dictionary Basis and Spatial Regularization for Isolating Isotropic Signals.

Diffusion-weighted magnetic resonance imaging is a non-invasive imaging method that has been increasingly used in neuroscience imaging over the last decade. Partial volume effects (PVEs) exist in sampling signal for many physical and actual reasons, which lead to inaccurate fiber imaging. We overcome the influence of PVEs by separating isotropic signal from diffusion-weighted signal, which can provide more accurate estimation of fiber orientations. In this work, we use a novel response function (RF) and the correspondent fiber orientation distribution function (fODF) to construct different signal models, in which case the fODF is represented using dictionary basis function. We then put forward a new index Piso, which is a part of fODF to quantify white and gray matter. The classic Richardson-Lucy (RL) model is usually used in the field of digital image processing to solve the problem of spherical deconvolution caused by highly ill-posed least-squares algorithm. In this case, we propose an innovative model integrating RL model with spatial regularization to settle the suggested double-models, which improve noise resistance and accuracy of imaging. Experimental results of simulated and real data show that the proposal method, which we call iRL, can robustly reconstruct a more accurate fODF and the quantitative index Piso performs better than fractional anisotropy and general fractional anisotropy.

Quantitative PET image reconstruction employing nested expectation-maximization deconvolution for motion compensation

Bulk body motion may randomly occur during PET acquisitions introducing blurring, attenuation-emission mismatches and, in dynamic PET, discontinuities in the measured time activity curves between consecutive frames. Meanwhile, dynamic PET scans are longer, thus increasing the probability of bulk motion. In this study, we propose a streamlined 3D PET motion-compensated image reconstruction (3D-MCIR) framework, capable of robustly deconvolving intra-frame motion from a static or dynamic 3D sinogram. The presented 3D-MCIR methods need not partition the data into multiple gates, such as 4D MCIR algorithms, or access list-mode (LM) data, such as LM MCIR methods, both associated with increased computation or memory resources. The proposed algorithms can support compensation for any periodic and non-periodic motion, such as cardio-respiratory or bulk motion, the latter including rolling, twisting or drifting. Inspired from the widely adopted point-spread function (PSF) deconvolution 3D PET reconstruction techniques, here we introduce an image-based 3D generalized motion deconvolution method within the standard 3D maximum-likelihood expectation-maximization (ML-EM) reconstruction framework. In particular, we initially integrate a motion blurring kernel, accounting for every tracked motion within a frame, as an additional MLEM modeling component in the image space (integrated 3D-MCIR). Subsequently, we replaced the integrated model component with a nested iterative Richardson-Lucy (RL) image-based deconvolution method to accelerate the MLEM algorithm convergence rate (RL-3D-MCIR). The final method was evaluated with realistic simulations of whole-body dynamic PET data employing the XCAT phantom and real human bulk motion profiles, the latter estimated from volunteer dynamic MRI scans. In addition, metabolic uptake rate Ki parametric images were generated with the standard Patlak method. Our results demonstrate significant improvement in contrast-to-noise ratio (CNR) and noise-bias performance in both dynamic and parametric images. The proposed nested RL-3D-MCIR method is implemented on the Software for Tomographic Image Reconstruction (STIR) open-source platform and is scheduled for public release.

SU-G-IeP2-01: A Method to Reduce Information Loss in JPEG Compressed Medical Images Using Richardson-Lucy Deconvolution

Purpose: To minimize the information loss of medical images compressed by JPEG using Richardson-Lucy (RL) deconvolution. Methods: We first generated reference images having three different noise structures; low-passed noise, high-passed noise, and white noise, which mimic noise structures for different medical imaging modalities. Each reference image was filtered by a Gaussian function and compressed/decompressed by JPEG 2000 algorithm. After decompression, RL deconvolution was applied to recover the reference image using the same Gaussian function. The goal of the RL deconvolution is to restore the reference image from the decompressed image as similar as possible. For each reference image, the optimal kernel size of the Gaussian function and iteration number of the RL deconvolution were found by brute-force searching. The optimal parameters were chosen when the Structural Similarity Index Metric (SSIM) value between reference image and restored image by the proposed method was maximized. We also compared the SSIM value of decompressed conventional JPEG image. Results: For each reference image, proposed algorithm produces a higher SSIM value than conventional JPEG image, demonstrating the information loss by JPEG can be reduced by a simple deconvolution technique. Conclusion: In this work, we present a simple method to prevent the information loss caused by JPEG algorithm in medical images. We only tested our method using a simple Gaussian function, but using different functions optimized for different noise structures would improve the performance of the proposed method, which will be a subject of our future research. This research was supported by the MSIP(Ministry of Science, ICT and Future Planning), Korea, under the IT Consilience Creative Programs (IITP-2015-R0346-15-1008) supervised by the IITP (Institute for Information & Communications Technology Promotion), Basic Science Research Program through the National Research Foundation of Korea (NRF) unded by the MSIP(2015R1C1A1A01052268) and framework of international cooperation program managed by NRF(NRF- 2015K2A1A2067635).

Spherical Deconvolution of Multichannel Diffusion MRI Data with Non-Gaussian Noise Models and Spatial Regularization.

Spherical deconvolution (SD) methods are widely used to estimate the intra-voxel white-matter fiber orientations from diffusion MRI data. However, while some of these methods assume a zero-mean Gaussian distribution for the underlying noise, its real distribution is known to be non-Gaussian and to depend on many factors such as the number of coils and the methodology used to combine multichannel MRI signals. Indeed, the two prevailing methods for multichannel signal combination lead to noise patterns better described by Rician and noncentral Chi distributions. Here we develop a Robust and Unbiased Model-BAsed Spherical Deconvolution (RUMBA-SD) technique, intended to deal with realistic MRI noise, based on a Richardson-Lucy (RL) algorithm adapted to Rician and noncentral Chi likelihood models. To quantify the benefits of using proper noise models, RUMBA-SD was compared with dRL-SD, a well-established method based on the RL algorithm for Gaussian noise. Another aim of the study was to quantify the impact of including a total variation (TV) spatial regularization term in the estimation framework. To do this, we developed TV spatially-regularized versions of both RUMBA-SD and dRL-SD algorithms. The evaluation was performed by comparing various quality metrics on 132 three-dimensional synthetic phantoms involving different inter-fiber angles and volume fractions, which were contaminated with noise mimicking patterns generated by data processing in multichannel scanners. The results demonstrate that the inclusion of proper likelihood models leads to an increased ability to resolve fiber crossings with smaller inter-fiber angles and to better detect non-dominant fibers. The inclusion of TV regularization dramatically improved the resolution power of both techniques. The above findings were also verified in human brain data.

Block-iterative Richardson-Lucy methods for image deblurring

In this paper, we extend the Richardson-Lucy (RL) method to block-iterative versions, separated BI-RL, and interlaced BI-RL, for image deblurring applications. We propose combining algorithms for separated BI-RL to form block artifact-free output images from separately deblurred block images. For interlaced BI-RL to accelerate the iteration, we propose an interlaced block-iteration algorithm on down-sampled blocks of the observed image. Simulation studies show that separated BI-RL and interlaced BI-RL achieve desired goals in Gaussian and diagonal deblurrings.

Deconvolution for three-dimensional acoustic source identification based on spherical harmonics beamforming

Spherical Harmonics Beamforming (SHB) with solid spherical arrays has become a particularly attractive tool for doing acoustic sources identification in cabin environments. However, it presents some intrinsic limitations, specifically poor spatial resolution and severe sidelobe contaminations. This paper focuses on overcoming these limitations effectively by deconvolution. First and foremost, a new formulation is proposed, which expresses SHB’s output as a convolution of the true source strength distribution and the point spread function (PSF) defined as SHB’s response to a unit-strength point source. Additionally, the typical deconvolution methods initially suggested for planar arrays, deconvolution approach for the mapping of acoustic sources (DAMAS), nonnegative least-squares (NNLS), Richardson–Lucy (RL) and CLEAN, are adapted to SHB successfully, which are capable of giving rise to highly resolved and deblurred maps. Finally, the merits of the deconvolution methods are validated and the relationships of source strength and pressure contribution reconstructed by the deconvolution methods vs. focus distance are explored both with computer simulations and experimentally. Several interesting results have emerged from this study: (1) compared with SHB, DAMAS, NNLS, RL and CLEAN all can not only improve the spatial resolution dramatically but also reduce or even eliminate the sidelobes effectively, allowing clear and unambiguous identification of single source or incoherent sources. (2) The availability of RL for coherent sources is highest, then DAMAS and NNLS, and that of CLEAN is lowest due to its failure in suppressing sidelobes. (3) Whether or not the real distance from the source to the array center equals the assumed one that is referred to as focus distance, the previous two results hold. (4) The true source strength can be recovered by dividing the reconstructed one by a coefficient that is the square of the focus distance divided by the real distance from the source to the array center. (5) The reconstructed pressure contribution is almost not affected by the focus distance, always approximating to the true one. This study will be of great significance to the accurate localization and quantification of acoustic sources in cabin environments.

A new method to measure directional modulation transfer function using sphere phantoms in a cone beam computed tomography system.

We propose a new method to measure directional modulation transfer function (MTF) using sphere phantoms in a cone beam computed tomography (CBCT) system. To measure spatially varying 3-D MTFs, we model FDK reconstruction in local regions and calculate the plane integrals of an ideal sphere phantom and reconstructed sphere phantoms. Then, we modify the Richardson-Lucy (RL) deconvolution method to relax the non-negativity constraint in RL deconvolution and apply it to estimate the directional plane spread functions (PlSFs). Directional MTFs are calculated by taking the modulus of the Fourier transform of the estimated directional PlSFs. To validate the proposed method, we simulate ideal 3-D MTFs and compare them with directional MTFs measured by simulation and experimental data along three major axes. For quantitative evaluation, we compare full-width at half-maximum (FWHM) and full-width at tenth-maximum (FWTM) of measured and ideal directional MTFs. The measured directional MTFs from simulation and experimental data show excellent agreement with the ideal directional MTFs, demonstrating the effectiveness of the proposed method to estimate directional MTFs in a CBCT system.

Spatially varying regularization of deconvolution in 3D microscopy.

Confocal microscopy has become an essential tool to explore biospecimens in 3D. Confocal microcopy images are still degraded by out-of-focus blur and Poisson noise. Many deconvolution methods including the Richardson-Lucy (RL) method, Tikhonov method and split-gradient (SG) method have been well received. The RL deconvolution method results in enhanced image quality, especially for Poisson noise. Tikhonov deconvolution method improves the RL method by imposing a prior model of spatial regularization, which encourages adjacent voxels to appear similar. The SG method also contains spatial regularization and is capable of incorporating many edge-preserving priors resulting in improved image quality. The strength of spatial regularization is fixed regardless of spatial location for the Tikhonov and SG method. The Tikhonov and the SG deconvolution methods are improved upon in this study by allowing the strength of spatial regularization to differ for different spatial locations in a given image. The novel method shows improved image quality. The method was tested on phantom data for which ground truth and the point spread function are known. A Kullback-Leibler (KL) divergence value of 0.097 is obtained with applying spatially variable regularization to the SG method, whereas KL value of 0.409 is obtained with the Tikhonov method. In tests on a real data, for which the ground truth is unknown, the reconstructed data show improved noise characteristics while maintaining the important image features such as edges.

Single image motion deblurring: An accurate PSF estimation and ringing reduction

In this paper we have proposed a single image motion deblurring algorithm that is based on a novel use of dual Fourier spectrum combined with bit plane slicing algorithm and Radon transform (RT) for accurate estimation of PSF parameters such as, blur length and blur angle. Even after very accurate PSF estimation, the deconvolution algorithms tend to introduce ringing artifacts at boundaries and near strong edges. To prevent this post deconvolution effect, a post processing method is also proposed in the framework of traditional Richardson–Lucy (RL) deconvolution algorithm. Experimental results evaluated on the basis of both qualitative and quantitative (PSNR, SSIM) metrics, verified on the dataset of both grayscale and color blurred images show that the proposed method outperforms the existing algorithms for removal of uniform blur. A comparison with state-of-the-art methods proves the usefulness of the proposed algorithm for deblurring real-life images/photographs.

A Turbulence-Degraded Image Restoration Algorithm Using the Maximum Likelihood Estimation Method

The RL(Richardson-Lucy) algorithm is an important method for restoration of turbulence-degraded images. However, the shortcoming of this method is that it tends to amplify the noise and exsits excessive smoothing in the iterative procedure. This paper discusses the RL algorithm and its improving methods focusing on turbulence-degraded images restoration.Firstly, a short exposure atmospheric turbulence-degraded model is established and a numerical computing method is proposed for random phase screen. Secondly, the essential principle and computational formula are deduced. To restore the object image effectively from the turbulence-degraded image, a new double-circulation iterative Richardson-Lucy restoration algorithm using TV-regularized method is proposed. This new algorithm introduces the total variation restraint and estimates the object image and the point spread function based on the inner and outer double-circulation iteration, which can use the inherent relation between the object image and the point spread function adequately. Simulation experiments show that the proposed algorithm can effectively preserve the details and edges of the image and its restoration effect is obviously better than the traditional RL algorithm.

Improvement of image deblurring for opto-electronic joint transform correlator under projective motion vector estimation

In this paper we propose an efficient algorithm to improve the performance of image deblurring based on opto-electronic joint transform correlator (JTC) that is capable of detecting the motion vector of a space camera. Firstly, the motion vector obtained from JTC is divided into many sub-motion vectors according to the projective motion path, which represents the degraded image as an integration of the clear scene under a sequence of planar projective transforms. Secondly, these sub-motion vectors are incorporated into the projective motion Richardson–Lucy (RL) algorithm to improve deblurred results. The simulation results demonstrate the effectiveness of the algorithm and the influence of noise on the algorithm performance is also statically analyzed.

Richardson–Lucy Deconvolution as a General Tool for Combining Images with Complementary Strengths

We use Richardson-Lucy (RL) deconvolution to combine multiple images of a simulated object into a single image in the context of modern fluorescence microscopy techniques. RL deconvolution can merge images with very different point-spread functions, such as in multiview light-sheet microscopes,1, 2 while preserving the best resolution information present in each image. We show that RL deconvolution is also easily applied to merge high-resolution, high-noise images with low-resolution, low-noise images, relevant when complementing conventional microscopy with localization microscopy. We also use RL deconvolution to merge images produced by different simulated illumination patterns, relevant to structured illumination microscopy (SIM)3, 4 and image scanning microscopy (ISM). The quality of our ISM reconstructions is at least as good as reconstructions using standard inversion algorithms for ISM data, but our method follows a simpler recipe that requires no mathematical insight. Finally, we apply RL deconvolution to merge a series of ten images with varying signal and resolution levels. This combination is relevant to gated stimulated-emission depletion (STED) microscopy, and shows that merges of high-quality images are possible even in cases for which a non-iterative inversion algorithm is unknown.

R-L Method and BLS-GSM Denoising for Penumbra Image Reconstruction

When neutron yield is very low, reconstruction of coding penumbra image is rather difficult. In this paper, low-yield (109) 14 MeV neutron penumbra imaging was simulated by Monte Carlo method. The Richardson Lucy (R-L) iteration method was proposed to incorporated with Bayesian least square-Gaussian scale mixture model (BLS-GSM) wavelet denoising for the simulated image. Optimal number of R-L iterations was gotten by a large number of tests. The results show that compared with Wiener method and median filter denoising, this method is better in restraining background noise, the correlation coefficient Rsr between the reconstructed and the real images is larger, and the reconstruction result is better.

Image Restoration with Lucy-Richardson Algorithm

Motion blur is a phenomenon common to nearly all photographs. Moving objects in the scene or camera movement during the exposure progress lead to blurred images. Lucy Richardson (LR) algorithm has been used to process image restoration of motion blur. In this paper LR algorithm would be discussed. The simulation experiment result shows the PSF plays an important role in the image restoration with LR iterative algorithm.

An improved deconvolution method for X-ray coded imaging in inertial confinement fusion

In inertial confinement fusion (ICF), X-ray coded imaging is considered as the most potential means to diagnose the compressed core. The traditional Richardson—Lucy (RL) method has a strong ability to deblur the image where the noise follows the Poisson distribution. However, it always suffers from over-fitting and noise amplification, especially when the signal-to-noise ratio of image is relatively low. In this paper, we propose an improved deconvolution method for X-ray coded imaging. We model the image data as a set of independent Gaussian distributions and derive the iterative solution with a maximum-likelihood scheme. The experimental results on X-ray coded imaging data demonstrate that this method is superior to the RL method in terms of anti-overfitting and noise suppression.

RL Algorithm for Passive Millimeter Wave Imaging Based on BM3D

In a passive millimeter wave (PMMW) imaging system, the resolution of the acquired image is limited by the antenna size. The Richardson—Lucy (RL) algorithm is a simple and nonlinear method, which can improve the resolution of the image. However, when the noise can not be neglected, it is difficult for RL algorithm to get good restoration of the corrupted image. To the best of our knowledge, the block-matching with 3D transform domain collaborative filtering (BM3D) algorithm achieves very good performance in image de-noising. In order to improve the resolution of passive millimeter wave images, a RL imaging algorithm for passive millimeter wave based on BM3D is proposed in this paper. The modified algorithm effectively reduces the influence of noise on RL algorithm by using de-noise algorithm based on BM3D. Experimental results demonstrate that the proposed algorithm improves the performance of RL algorithm. Furthermore, the algorithm can be easily implemented for passive millimeter wave imaging．

Non-blind image deconvolution using natural image gradient prior

Since non-blind image deconvolution is inherently ill-posed, the results of unregularized methods are often contaminated by noise and ringing artifacts. To reach a stable solution, we adopt the natural image gradient prior to regularize the latent image and obtain an improved version of the Richardson–Lucy (RL) algorithm. We use both synthetic and real world blurred images to test the proposed method. Experimental results show that the negative artifacts are significantly suppressed and the restored images are of high quality.

A convergent blind deconvolution method for post-adaptive-optics astronomical imaging

In this paper, we propose a blind deconvolution method which applies to data perturbed by Poisson noise. The objective function is a generalized Kullback–Leibler (KL) divergence, depending on both the unknown object and unknown point spread function (PSF), without the addition of regularization terms; constrained minimization, with suitable convex constraints on both unknowns, is considered. The problem is non-convex and we propose to solve it by means of an inexact alternating minimization method, whose global convergence to stationary points of the objective function has been recently proved in a general setting. The method is iterative and each iteration, also called outer iteration, consists of alternating an update of the object and the PSF by means of a fixed number of iterations, also called inner iterations, of the scaled gradient projection (SGP) method. Therefore, the method is similar to other proposed methods based on the Richardson–Lucy (RL) algorithm, with SGP replacing RL. The use of SGP has two advantages: first, it allows one to prove global convergence of the blind method; secondly, it allows the introduction of different constraints on the object and the PSF. The specific constraint on the PSF, besides non-negativity and normalization, is an upper bound derived from the so-called Strehl ratio (SR), which is the ratio between the peak value of an aberrated versus a perfect wavefront. Therefore, a typical application, but not a unique one, is to the imaging of modern telescopes equipped with adaptive optics systems for the partial correction of the aberrations due to atmospheric turbulence. In the paper, we describe in detail the algorithm and we recall the results leading to its convergence. Moreover, we illustrate its effectiveness by means of numerical experiments whose results indicate that the method, pushed to convergence, is very promising in the reconstruction of non-dense stellar clusters. The case of more complex astronomical targets is also considered, but in this case regularization by early stopping of the outer iterations is required. However, the proposed method, based on SGP, allows generalization to the case of differentiable regularization terms added to the KL divergence, even if this generalization is outside the scope of this paper.

Determination of particle-size distributions from light-scattering measurement using Lucy-Richardson algorithm

A classical iterative Lucy-Richardson (LR) inversion algorithm used for recovering particle-size distributions (PSD) from light-scattering data is proposed. The convergence of iteration is validated in the numerical simulation for three different distributions: the gamma, the log-normal, and the Rosin-Rammler. The accuracy of the inversion is checked graphically against the exact distribution with good results, even for the synthesized intensity data of a signal-noise-ratio smaller than 20 dB. Finally, an experiment with linear charge coupled device as the detector is carried out, and the PSD is recovered successfully by the LR inversion method.

An improved image deconvolution approach using local constraint

Conventional deblurring approaches such as the Richardson–Lucy (RL) algorithm will introduce strong noise and ringing artifacts, though the point spread function (PSF) is known. Since it is difficult to estimate an accurate PSF in real imaging system, the results of those algorithms will be worse. A spatial weight matrix (SWM) is adopted as local constraint, which is incorporated into image statistical prior to improve the RL approach. Experiments show that our approach can make a good balance between preserving image details and suppressing ringing artifacts and noise.