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In diffusion MRI, the Ensemble Average diffusion Propagator (EAP) provides relevant micro-structural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like Diffusion Tensor Imaging (DTI). The direct estimation of the EAP, however, requires a dense sampling of the Cartesian q-space involving a huge amount of samples (diffusion gradients) for proper reconstruction. A collection of more efficient techniques have been proposed in the last decade based on parametric representations of the EAP, but they still imply acquiring a large number of diffusion gradients with different b-values (shells). Paradoxically, this has come together with an effort to find scalar measures gathering all the q-space micro-structural information probed in one single index or set of indices. Among them, the return-to-origin (RTOP), return-to-plane (RTPP), and return-to-axis (RTAP) probabilities have rapidly gained popularity. In this work, we propose the so-called "Apparent Measures Using Reduced Acquisitions" (AMURA) aimed at computing scalar indices that can mimic the sensitivity of state of the art EAP-based measures to micro-structural changes. AMURA drastically reduces both the number of samples needed and the computational complexity of the estimation of diffusion properties by assuming the diffusion anisotropy is roughly independent from the radial direction. This simplification allows us to compute closed-form expressions from single-shell information, so that AMURA remains compatible with standard acquisition protocols commonly used even in clinical practice. Additionally, the analytical form of AMURA-based measures, as opposed to the iterative, non-linear reconstruction ubiquitous to full EAP techniques, turns the newly introduced apparent RTOP, RTPP, and RTAP both robust and efficient to compute.

A novel diffusion anisotropy metric is presented. It is based on dissimilarity between the acquired diffusion signal and its isotropic equivalent. Using the inner product of signals, a closed form expression is obtained, which allows its computation using spherical harmonics from a reduced set of acquired data, compatible with most popular diffusion MRI acquisition protocols. Results show that the proposed metric (1) is able to discriminate among different microstructure scenarios; (2) shows a robust behaviour in clinical studies.

Purpose To characterize the noise distributions in 3D-MRI accelerated acquisitions reconstructed with GRAPPA using an exact noise propagation analysis that operates directly in k-space. Theory and methods We exploit the extensive symmetries and separability in the reconstruction steps to account for the correlation between all the acquired k-space samples. Monte Carlo simulations and multi-repetition phantom experiments were conducted to test both the accuracy and feasibility of the proposed method; a high-resolution in-vivo experiment was performed to assess the applicability of our method to clinical scenarios. Results Our theoretical derivation shows that the direct k-space analysis renders an exact noise characterization under the assumptions of stationarity and uncorrelation in the original k-space. Simulations and phantom experiments provide empirical support to the theoretical proof. Finally, the high-resolution in-vivo experiment demonstrates the ability of the proposed method to assess the impact of the sub-sampling pattern on the overall noise behavior. Conclusions By operating directly in the k-space, the proposed method is able to provide an exact characterization of noise for any Cartesian pattern sub-sampled along the two phase-encoding directions. Exploitation of the symmetries and separability into independent blocks through the image reconstruction procedure allows us to overcome the computational challenges related to the very large size of the covariance matrices involved.

In diffusion MRI, the Ensemble Average diffusion Propagator (EAP) provides relevant microstructural information and meaningful descriptive maps of the white matter previously obscured by traditional techniques like the Diffusion Tensor. The direct estimation of the EAP, however, requires a dense sampling of the Cartesian q-space. Due to the huge amount of samples needed for an accurate reconstruction, more efficient alternative techniques have been proposed in the last decade. Even so, all of them imply acquiring a large number of diffusion gradients with different b-values. In order to use the EAP in practical studies, scalar measures must be directly derived, being the most common the return-to-origin probability (RTOP) and the return-to-plane and return-to-axis probabilities (RTPP, RTAP).In this work, we propose the so-called “Apparent Measures Using Reduced Acquisitions” (AMURA) to drastically reduce the number of samples needed for the estimation of diffusion properties. AMURA avoids the calculation of the whole EAP by assuming the diffusion anisotropy is roughly independent from the radial direction. With such an assumption, and as opposed to common multi-shell procedures based on iterative optimization, we achieve closed-form expressions for the measures using information from one single shell. This way, the new methodology remains compatible with standard acquisition protocols commonly used for HARDI (based on just one b-value). We report extensive results showing the potential of AMURA to reveal microstructural properties of the tissues compared to state of the art EAP estimators, and is well above that of Diffusion Tensor techniques. At the same time, the closed forms provided for RTOP, RTPP, and RTAP-like magnitudes make AMURA both computationally efficient and robust.

In this manuscript we derive the Cramer-Rao Lower Bound (CRLB) of the monoexponential diffusion-weighted signal model under a realistic noise assumption, and propose a formulation to obtain optimized sets of b-values that maximize the noise performance of the Apparent Diffusion Coefficient (ADC) maps given a target ADC and a signal-to-noise ratio. Therefore, for various sets of parameters (S0 and ADC), signal-to-noise ratios (SNR) and noise distribution, we computed optimized sets of b-values using CRLB-based analysis in two different ways: (i) through a greedy algorithm where b-values from a pool of candidates were added iteratively to the set, and (ii) through a two b-value search algorithm were all two b-value combinations of the pool of candidates were tested. Further, optimized sets of b-values were computed from synthetic data, phantoms, and in-vivo liver diffusion-weighted imaging (DWI) experiments to validate the CRLB-based analysis. The optimized sets of b-values obtained through the proposed CRLB-based analysis showed good agreement with the optimized sets obtained experimentally from synthetic, phantoms, and in-vivo liver data. The variance of the ADC maps decreased when employing the optimized set of b-values compared to various sets of b-values proposed in the literature for in-vivo liver DWI, although differences of notable magnitude between noise models and optimization strategies were not found. In addition, the higher b-values decreased for lower SNR under the Rician noise distribution. Optimization of the set b-values is critical to maximize the noise performance (i.e., maximize the precision and minimize the variance) of the estimated ADC maps in diffusion-weighted MRI. Hence, the proposed approach may help to optimize and standardize liver diffusion-weighted MRI acquisitions by computing optimized set of b-values for a given set of parameters.

The Ensemble Average Propagator (EAP) provides a compact theoretical framework to explore the underlying microstructural properties of the tissues with diffusion magnetic resonance imaging. To model tissue characteristics, it is usually required to fit a functional basis to a densely sampled q-space data and then retrieve the EAP-related maps. In this work, we analytically derive a new closed-form formula to calculate one of the EAP features the Return-To-the-Origin Probability (RTOP) map directly from the data leaving aside the EAP estimation step. Our RTOP estimation approach exploits only single-shell data and additionally handles noise-induced bias using a non-stationary log-Rician statistics. We validated our proposal using an in vivo Human Connectome Project database achieving an increased accuracy of the method when subsampling of the q-space was considered and strong correlations to multiple-shell state-of-the-art methods.

A number of computational techniques have been lately devised to image the Ensemble Average Propagator (EAP) within the white matter of the brain, propelled by the deployment of multi-shell acquisition protocols and databases: approaches like Mean Apparent Propagator Imaging (MAP-MRI) and its Laplacian-regularized version (MAPL) aim at describing the low frequency spectrum of the EAP (limited by the maximum b-value acquired) and afterwards computing scalar indices that embed useful descriptions of the white matter, e. g. the Return-to-Origin, Plane, or Axis Probabilities (RTOP, RTPP, RTAP). These methods resort to a non-parametric, bandwidth limited representation of the EAP that implies fitting a set of 3-D basis functions in a large-scale optimization problem. We propose a semi-parametric approach inspired by signal theory: the EAP is approximated as the spherical convolution of a Micro-Structure adaptive Gaussian kernel with a non-parametric orientation histogram, which aims at representing the low-frequency response of an ensemble of coherent sets of fiber bundles at the white matter. This way, the optimization involves just the 2 to 3 parameters that describe the kernel, making our approach far more efficient than the related state of the art. We devise dual Fourier domains Integral Transforms to analytically compute RTxP-like scalar indices as moments of arbitrary orders over either the whole 3-D space, particular directions, or particular planes. The so-called MiSFIT is both time efficient (a typical multi-shell data set can be processed in roughly one minute) and accurate: it provides estimates of widely validated indices like RTOP, RTPP, and RTAP comparable to MAPL for a wide variety of white matter configurations.

Purpose In this work, we have proposed a methodology for the estimation of the apparent diffusion coefficient in the body from multiple breath hold diffusion weighted images, which is robust to two preeminent confounding factors: noise and motion during acquisition. Methods We have extended a method for the joint groupwise multimodal registration and apparent diffusion coefficient estimation, previously proposed by the authors, in order to correct the bias that arises from the non-Gaussianity of the data and the registration procedure. Results Results show that the proposed methodology provides a statistically significant improvement both in robustness for displacement fields calculation and in terms of accuracy for the apparent diffusion coefficient estimation as compared with traditional sequential approaches. Reproducibility has also been measured on real data in terms of the distribution of apparent diffusion coefficient differences obtained from different b-values subsets. Conclusions Our proposal has shown to be able to effectively correct the estimation bias by introducing additional computationally light procedures to the original method, thus providing robust apparent diffusion coefficient maps in the liver and allowing an accurate and reproducible analysis of the tissue.

In this chapter, we will review some of the methods proposed in literature to remove the speckle pattern from ultrasound data based on nonlinear processing. Note that some filters that can also be considered as nonlinear are left aside since they will be deeply treated in other chapters. That is the case of the wavelet-based methods and Bayesian methods. Other methods also treated in other chapters, like diffusion-based schemes are only briefly reviewed in order to place them inside the global partial differential equation (PDE) classification. On the other hand, note that some of the filters here reviewed are not initially proposed for ultrasound imaging but derived for Synthetic Aperture Radar (SAR) images, where noise can be modeled similarly. In those images, the multiplicative model for speckle holds and therefore, many of the methods defined in literature for SAR can be easily extrapolated to ultrasound denoising. This is the case of some of the most popular speckle filters. In addition, we would like to remark that the effectiveness of many of these schemes lays on a proper modeling of the speckle statistics. For some purposes, a simple multiplicative model will suffice, while for some specific applications, more accurate models must be used. Finally, as we stated in the previous chapter, the filtering method must be selected following the specific needs of the problem. There is no all-purpose filter that, with the same configuration parameters, could perform excellent in all situations.

Noise characterization in MRI has multiple applications, including quality assurance and protocol optimization. It is particularly important in the presence of parallel imaging acceleration, where the noise distribution can contain severe spatial heterogeneities. If the parallel imaging reconstruction is a linear process, an exact noise analysis is possible by taking into account the correlations between all the samples involved. However, for k-space based techniques like GRAPPA, the exact analysis has been considered computationally prohibitive due to the very large size of the noise covariance matrices required to characterize the noise propagation from k-space to image-space. Previous methods avoid this computational burden by approximating the GRAPPA reconstruction as a pixel-wise linear operation performed in the image-space. However, these methods are not exact in the presence of non-uniform k-space undersampling (e.g.: containing a calibration region). For this reason, in this work we develop an exact characterization of the noise distribution for self-calibrated parallel imaging in the presence of arbitrary Cartesian undersampling patterns. By exploiting the symmetries and separability in the noise propagation process, the proposed method is computationally efficient and does not require large matrices. In this manuscript, we present the proposed noise characterization method and compare it to previous techniques using Monte-Carlo simulations as well as phantom acquisitions., 24 pages, 6 figures

Purpose The purpose of this work is to develop a groupwise elastic multimodal registration algorithm for robust ADC estimation in the liver on multiple breath hold diffusion weighted images. Methods We introduce a joint formulation to simultaneously solve both the registration and the estimation problems. In order to avoid non-reliable transformations and undesirable noise amplification, we have included appropriate smoothness constraints for both problems. Our metric incorporates the ADC estimation residuals, which are inversely weighted according to the signal content in each diffusion weighted image. Results Results show that the joint formulation provides a statistically significant improvement in the accuracy of the ADC estimates. Reproducibility has also been measured on real data in terms of the distribution of ADC differences obtained from different b-values subsets. Conclusions The proposed algorithm is able to effectively deal with both the presence of motion and the geometric distortions, increasing accuracy and reproducibility in diffusion parameters estimation.

In order to accelerate the acquisition process in multiple-coil Magnetic Resonance scanners, parallel techniques were developed. These techniques reduce the acquisition time via a sub-sampling of the k-space and a reconstruction process. From a signal and noise perspective, the use of a acceleration techniques modify the structure of the noise within the image. In the most common algorithms, like SENSE, the final magnitude image after the reconstruction is known to follow a Rician distribution for each pixel, just like single coil systems. However, the noise is spatially non-stationary, i.e. the variance of noise becomes x-dependent. This effect can also be found in magnitude images due to other processing inside the scanner. In this work we propose a method to adapt well-known noise filtering techniques initially designed to deal with stationary noise to the case of spatially variant Rician noise. The method copes with inaccurate estimates of variant noise patterns in the image, showing its robustness in realistic cases. The method employs a consensus strategy in conjunction with a set of aggregation functions and a penalty function. Multiple possible outputs are generated for each pixel assuming different unknown input parameters. The consensus approach merges them into a unique filtered image. As a filtering technique, we have selected the Linear Minimum Mean Square Error (LMMSE) estimator for Rician data, which has been used to test our methodology due to its simplicity and robustness. Results with synthetic and in vivo data confirm the good behavior of our approach.

Strain and related tensors play a major role in cardiac function characterization, so correct estimation of the local phase in tagged images is crucial for quantitative myocardial motion studies. We propose an Harmonic Phase related procedure that is adaptive in the spatial and the spectral domains: as for the former, we use an angled-steered analysis window prior to the Fourier Transform; as for the latter, the bandpass filter is also angle-adaptive. Both of them are narrow in the modulation direction and wide in the orthogonal direction. Moreover, no parameters are manually set since their values are partially based on the information available at the DICOM headers and additional information is estimated from data. The procedure is tested in terms of accuracy (on synthetic data) and reproducibility (on real data) of the deformation gradient tensor, measured by means of the distribution of the Frobenius norm differences between two tensor datasets.

The different approaches to estimate \(\sigma ^2\) out of Rician magnitude MR images are reviewed and classified, assuming the noise to be stationary, i.e., \(\sigma \) does not depend on the position. Most of the approaches to noise estimation in MRI in the literature are precisely focused on this kind of noise. The different methods are systematized and classified attending their nature. As a case study, a special kind of estimators, those based on the calculation of local moments, is deeply studied. This case serves as an illustration of the different considerations that must be taken into account when implementing an estimation procedure. In the last part of the chapter, the advantages and drawbacks of the different methods are analyzed through synthetic and real data controlled experiments. The estimators for Rician noise of here reviewed are the basis for many of the estimators proposed in the following chapters, which can be seen as extensions of these.

In this preliminary chapter, the origin of the problem of noise in MR acquisitions is introduced and set in context. The three main issues dealt along the book are presented: noise modeling, noise estimation and noise filtering. The different parts and chapters of the book are reviewed.

Parallel imaging (pMRI) methods allow increasing the acquisition rate in MRI via subsampling of the k-space. As a consequence of the subsampling image in the x-space presents some artifacts that must be corrected with the so-called pMRI methods. Among them, SENSE and GRAPPA are two of the most popular. In most pMRI methods, the reconstruction process yields to a value of the variance of noise in the final image that depends on the position. Hence, traditional noise estimation methods, based on a single noise level for the whole image, fail. In this chapter, we review a methodology to estimate the spatial dependent pattern of the variance of noise in SENSE and GRAPPA reconstructed images. The estimation follows a parametric methodology: the models for noise are known and some parameters of those models are assumed to be known beforehand: the sensitivity maps of each receiver coil, the reconstruction weights, and the correlation among coils. Although we only focus on two pMRI methods, the results here presented are easily extrapolated to other parallel algorithms.

Some basic concepts about MRI acquisition are reviewed. The foundations here described will be necessary to understand the different assumptions about signal and noise used along the book; the statistical models defined in following chapters are based on the premises here set. We especially focus on modeling the k-space and x-space from a signal processing point of view. Sequences and specifics on acquisition modalities will be left aside to confine us to an upper level modeling of the acquired signal. First, we review the formation process in single-coil systems and then we extrapolate the results to multiple-coil systems. For the sake of illustration, two methods are considered in order to fuse the information from the different antennae into one single magnitude image: the sum of squares (SoS) and the spatial matched filter (SMF). The chapter concludes with the analysis of some parallel MRI methods, focusing on two of the most used ones, SENSE and GRAPPA.

The counterpart of the acceleration capability of parallel MRI (pMRI) techniques, is that the artifact correction techniques affect the nature of noise. As a consequence of the different algorithms, noise in the reconstructed signal becomes non-stationary, i.e., the variance of noise \(\sigma ^2(\mathbf{x})\) becomes dependent on the position and, sometimes, also on the signal. Some methods have been proposed in the previous chapters to deal with non-stationary noise in MRI. However, their performance depends on information not usually available, such as multiple acquisitions, the receiver covariance matrix, the sensitivity coil profiles, reconstruction coefficients, or even biophysical models of the data. In this chapter, a different paradigm is considered: blind estimation. The estimation is carried out only assuming a statistical model for the data. No specifics about the transformation, nor extra parameters are needed. The main difficulty of this kind of analysis is that a single value of \(\sigma ^2\) no longer characterizes the whole image, on the contrary, a value for each position x must be calculated. We review the different proposals in literature for blind non-stationary noise estimation are reviewed and validated, assuming three different noise models: Gaussian, Rician, and nc-\(\chi \). A deeper attention is paid to the homomorphic. Finally, the methods are tested through several synthetic and real data sets.

Many image-processing applications within MRI are grounded on stochastic methods based on the prior knowledge on the statistics of noise. In this chapter, the noise models for the different acquisitions modalities reviewed in Chap. 2 are presented. The noise in MRI has been traditionally modeled as a stationary process governed by a Rician distribution with constant noise power at each voxel. Modern MRI systems turn this model questionable, making it necessary to develop into more complex patterns. We aim at comprehensively reviewing the main statistical rationales and formulations for the noise in MRI lately found in the literature. The starting point is the complex Gaussian model for the signal acquired in each coil. From there, the different processing and reconstruction schemes are analyzed to generate the models of noise on the final composite magnitude signals. Gaussian, Rician, and noncentral-\(\chi \) distributions will be considered, as well as stationary and non-stationary models. Finally, we explore the applicability of the surveyed models to some MRI protocols commonly used. Whereas many parallel and nonparallel acquisitions like GRAPPA and SENSE may be fitted into one of the existing models, other nonlinear reconstruction procedures are lacking a proper noise characterization. The chapter is complemented with some practical examples using synthetic and real data.

Least Squares (LS) and its minimum variance counterpart, Weighted Least Squares (WLS), have become very popular when estimating the Diffusion Tensor (DT), to the point that they are the standard in most of the existing software for diffusion MRI. They are based on the linearization of the Stejskal–Tanner equation by means of the logarithmic compression of the diffusion signal. Due to the Rician nature of noise in traditional systems, a certain bias in the estimation is known to exist. This artifact has been made patent through some experimental set-ups, but it is not clear how the distortion translates in the reconstructed DT, and how important it is when compared to the other source of error contributing to the Mean Squared Error (MSE) in the estimate, i.e. the variance. In this paper we propose the analytical characterization of log-Rician noise and its propagation to the components of the DT through power series expansions. We conclude that even in highly noisy scenarios the bias for log-Rician signals remains moderate when compared to the corresponding variance. Yet, with the advent of Parallel Imaging (pMRI), the Rician model is not always valid. We make our analysis extensive to a number of modern acquisition techniques through the study of a more general Non Central-Chi (nc-χ) model. Since WLS techniques were initially designed bearing in mind Rician noise, it is not clear whether or not they still apply to pMRI. An important finding in our work is that the common implementation of WLS is nearly optimal when nc-χ noise is considered. Unfortunately, the bias in the estimation becomes far more important in this case, to the point that it may nearly overwhelm the variance in given situations. Furthermore, we evidence that such bias cannot be removed by increasing the number of acquired gradient directions. A number of experiments have been conducted that corroborate our analytical findings, while in vivo data have been used to test the actual relevance of the bias in the estimation.

The characterization of the distribution of noise in the magnitude MR image is a very important problem within image processing algorithms. The Rician noise assumed in single-coil acquisitions has been the keystone for signal-to-noise ratio estimation, image filtering, or diffusion tensor estimation for years. With the advent of parallel protocols such as sensitivity encoding or Generalized Autocalibrated Partially Parallel Acquisitions that allow accelerated acquisitions, this noise model no longer holds. Since Generalized Autocalibrated Partially Parallel Acquisitions reconstructions yield the combination of the squared signals recovered at each receiving coil, noncentral Chi statistics have been previously proposed to model the distribution of noise. However, we prove in this article that this is a weak model due to several artifacts in the acquisition scheme, mainly the correlation existing between the signals obtained at each coil. Alternatively, we propose to model such correlations with a reduction in the number of degrees of freedom of the signal, which translates in an equivalent nonaccelerated system with a minor number of independent receiving coils and, consequently, a lower signal-to-noise ratio. With this model, a noncentral Chi distribution can be assumed for all pixels in the image, whose effective number of coils and effective variance of noise can be explicitly computed in a closed form from the Generalized Autocalibrated Partially Parallel Acquisitions interpolation coefficients. Extensive experiments over both synthetic and in vivo data sets have been performed to show the goodness of fit of out model.

The filtering of the Diffusion Weighted Images (DWI) prior to the estimation of the diffusion tensor or other fiber Orientation Distribution Functions (ODF) has been proved to be of paramount importance in the recent literature. More precisely, it has been evidenced that the estimation of the diffusion tensor without a previous filtering stage induces errors which cannot be recovered by further regularization of the tensor field. A number of approaches have been intended to overcome this problem, most of them based on the restoration of each DWI gradient image separately. In this paper we propose a methodology to take advantage of the joint information in the DWI volumes, i.e., the sum of the information given by all DWI channels plus the correlations between them. This way, all the gradient images are filtered together exploiting the first and second order information they share. We adapt this methodology to two filters, namely the Linear Minimum Mean Squared Error (LMMSE) and the Unbiased Non-Local Means (UNLM). These new filters are tested over a wide variety of synthetic and real data showing the convenience of the new approach, especially for High Angular Resolution Diffusion Imaging (HARDI). Among the techniques presented, the joint LMMSE is proved a very attractive approach, since it shows an accuracy similar to UNLM (or even better in some situations) with a much lighter computational load.

In this paper, we propose a new method for collision handling between 3-D deformable (organs) and rigid (surgical tools) objects valid for nonstructured interaction scenes and, specifically, for laparoscopic surgery simulators. During the simulation step, vertices of the organ detected as collided must be accurately shifted out of the tool to elude a visual interpenetration. The proposed approach obtains the new position of each collided vertex of the organ taking into account both kinematic information of the surgical tool and geometric information of the organ surface that surrounds the vertex under analysis. Three parameters inferred from a fuzzy feedback system weigh the nature of the tool motion with respect to the organ. Experimental results show that the solution proposed in this paper is able to avoid the interpenetration among the multiple colliding points efficiently with physically and spatially coherent results.

In this paper, we focus on the problem of automatic noise parameter estimation for additive and multiplicative models and propose a simple and novel method to this end. Specifically we show that if the image to work with has a sufficiently great amount of low-variability areas (which turns out to be a typical feature in most images), the variance of noise (if additive) can be estimated as the mode of the distribution of local variances in the image and the coefficient of variation of noise (if multiplicative) can be estimated as the mode of the distribution of local estimates of the coefficient of variation. Additionally, a model for the sample variance distribution for an image plus noise is proposed and studied. Experiments show the goodness of the proposed method, specially in recursive or iterative filtering methods.

This paper introduces and analyzes a linear minimum mean square error (LMMSE) estimator using a Rician noise model and its recursive version (RLMMSE) for the restoration of diffusion weighted images. A method to estimate the noise level based on local estimations of mean or variance is used to automatically parametrize the estimator. The restoration performance is evaluated using quality indexes and compared to alternative estimation schemes. The overall scheme is simple, robust, fast, and improves estimations. Filtering diffusion weighted magnetic resonance imaging (DW-MRI) with the proposed methodology leads to more accurate tensor estimations. Real and synthetic datasets are analyzed.

A matrix procedure to obtain a hierarchical decomposition of a multiple-input-single-output (MISO) fuzzy system is proposed. It is based on the fast inference using transition matrices (FITM) framework recently described. The resulting hierarchical system is totally equivalent to the original one in terms of input-output relation. A study of the rule-reduction capability of this system is carried out. Additionally, this paper describes a rule-base definition procedure of each of the systems resulting within the hierarchy. This procedure is based on closed-form analytical expressions and it does not need any manual interaction.

The reconstruction process in multiple coil MRI scanners makes the noise features in the final magnitude image become non-stationary, i.e. the variance of noise becomes position-dependent. Therefore, most noise estimators proposed in the literature cannot be used in multiple-coil acquisitions. This effect is augmented when parallel imaging methods, such as GRAPPA, are used to increase the acquisition rate. In this work we propose a new technique that allows the estimation of the spatially variant maps of noise from the GRAPPA reconstructed signal when only one single image is available and no additional information is provided. Other estimators in the literature need extra information that is not always available, which has supposed an important limitation in the usage of noise models for GRAPPA. The proposed approach uses a homomorphic separation of the spatially variant noise in two terms: a stationary noise term and one low frequency signal that correspond to the x-dependent variance of noise. The non-stationary variance of noise is estimated by a low pass filtering. The noise term is obtained via prior wavelet decomposition. Results in real and synthetic experiments evidence the suitability of the simplification used and the good performance of the proposed methodology.

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. This work was supported by the Catalonian Government (2014SGR1573), several grants from the Plan Nacional de I+D+i and co-funded by the Instituto de Salud Carlos III-Subdireccion General de Evaluacion y Fomento de la Investigacion and the European Regional Development Fund (FEDER): Miguel Servet Research Contracts (CP10/00596 to EP-C, CP13/00018 to RS and CP14/00041 to JR) and Research Project Grants (PI14/00292 to JR, PI14/01148 to EP-C and PI14/01151 to RS). The funding organizations played no role in the study design, data collection and analysis, or manuscript approval.

Fast inference using transition matrices (FITMs) was recently presented as a new fast algorithm for performing inferences in fuzzy systems. However, this algorithm only applies to the standard additive model (SAM) using the sum-product inference composition. In this paper, we show how this methodology carries over to more general Mamdani fuzzy systems, i.e., those using t-norms and t-conorms that satisfy a requirement of distributivity. In addition, FITM, as originally described, requires inputs to be fuzzy sets. We here extend this methodology to handle crisp inputs as well.

Fast inference using transition matrices (FITM) is a new fast algorithm for performing inferences in fuzzy systems. It is based on the assumption that fuzzy inputs can be expressed as a linear composition of the fuzzy sets used in the rule base. This representation let us interpret a fuzzy set as a vector, so we can just work with the coordinates of it instead of working with the whole set. The inference is made using transition matrices. The key of the method is the fact that a lot of operations can be precomputed offline to obtain the transition matrices, so actual inferences are reduced to a few online matrix additions and multiplications. The algorithm is designed for the standard additive model using the sum-product inference composition.

GRAPPA is a well-known parallel imaging method that recovers the MR magnitude image from aliasing by using a weighted interpolation of the data in k-space. To estimate the optimal reconstruction weights, GRAPPA uses a band along the center of the k-space where the signal is sampled at the Nyquist rate, the so-called autocalibrated (ACS) lines. However, while the subsampled lines usually belong to the medium- to high-frequency areas of the spectrum, the ACS lines include the low-frequency areas around the DC component. The use for estimation and reconstruction of areas of the k-space with very different features may negatively affect the final reconstruction quality. We propose a simple, yet powerful method to eliminate reconstruction artifacts, based on the discrimination of the low-frequency spectrum. The proposal to improve the estimation of the weights lays on a proper selection of the coefficients within the ACS lines, which advises discarding those points around the DC component. A simple approach is the elimination of a square window in the center of the k-space, although more developed approaches can be used. The method is tested using real multiple-coil MRI acquisitions. We empirically show this approach achieves great enhancement rates, while keeping the same complexity of the original GRAPPA and reducing the g-factor. The reconstruction is even more accurate when combined with other reconstruction methods. Improvement rates of 35 % are achieved for 32 ACS and acceleration rate of 3. The method proposed highly improves the accuracy of the GRAPPA coefficients and therefore the final image reconstruction. The method is fully compatible with the original GRAPPA formulation and with other optimization methods proposed in literature, and it can be easily implemented into the commercial scanning software.

Low contrast in ultrasound images caused by the granular pattern (speckle) makes difficult computational analysis and diagnosis. Thus, speckle filtering is a common step before computed automatic analysis. However, speckle depends on the inner echo-morphology of tissue and it should be removed without over-filtering relevant details for diagnostic purposes. Some methods were proposed to preserve important details by means of anisotropic diffusion schemes. However, they require to solve an evolutionary partial differential equation, which needs considerable computation time thatmakes this kind of filters impractical for real-time scenarios. Additionally, there is no rational criteria to select the optimal stop criteria. Some other detail preserving filters are based in the Non-Local means philosophy, however they involves an even higher computational cost.

Recently, some methods have been proposed for filtering multi-coil MRI acquisitions with correlation between coils. Those methods are based on statistical models of noise to develop a Linear Minimum Mean Square Error (LMMSE) filter. The advantage of LMMSE-based filters stems from their simplicity and robustness. However, they exhibit some drawbacks: their performance strongly depends on the underlying statistical model and on the way the local moments are estimated. The first problem can be avoided when considering effective values provided by recent studies on the models of noise in multi-coil systems with correlation between coils. However, the local moments are estimated in square neighborhoods which can include different kinds of tissues. Thus, the local variance is biased towards upper values, which results in an inaccurate estimate in regions close to tissue boundaries. In this work we propose to overcome this problem by introducing an anisotropic diffusion step in the LMMSE estimate for correlated multi-coil systems which improves the estimation of the signal in regions where other LMMSE methods fail. Results demonstrate the better behavior in different noisy scenarios.

The Funk-Radon Transform allows to approximate the central section integral of the diffusion signal to compute probabilistic Orientation Distribution Functions (ODF) of fiber populations, provided the behavior for the b-values not acquired is known. To relax the latter demand, it has been proposed to compute instead the integral inside the disk defined in the central section by the b-value acquired. The price to pay is low-pass filtering the ODF, cropping the information it provides. We propose a method to palliate this problem with negligible computational overload. Not only it improves the accuracy in the determination of the diffusion directions, but it also refines the crossing angles that can be resolved.

Diffusion tensor imaging (DTI) constitutes the most used paradigm among the diffusion-weighted magnetic resonance imaging (DW-MRI) techniques due to its simplicity and application potential. Recently, real-time estimation in DW-MRI has deserved special attention, with several proposals aiming at the estimation of meaningful diffusion parameters during the repetition time of the acquisition sequence. Specifically focusing on DTI, the underlying model of the noise present in the acquired data is not taken into account, leading to a suboptimal estimation of the diffusion tensor. In this paper, we propose an optimal real-time estimation framework for DTI reconstruction in single-coil acquisitions. By including an online estimation of the time-changing noise variance associated to the acquisition process, the proposed method achieves the sequential best linear unbiased estimator. Results on both synthetic and real data show that our method outperforms those so far proposed, reaching the best performance of the existing proposals by processing a substantially lower number of diffusion images.

Modern magnetic resonance (MR) imaging protocols based on multiple-coil acquisitions have carried on a new attention to noise and signal statistical modeling, as long as most of the existing techniques for data processing are model based. In particular, nonaccelerated multiple-coil and GeneRalized Autocalibrated Partially Parallel Acquisitions (GRAPPA) have brought noncentral-χ (nc-χ) statistics into stake as a suitable substitute for traditional Rician distributions. However, this model is only valid when the signals received by each coil are roughly uncorrelated. The recent literature on this topic suggests that this is often not the case, so nc-χ statistics are in principle not adequate. Fortunately, such model can be adapted through the definition of a set of effective parameters, namely, an effective noise power (greater than the actual power of thermal noise in the Radio Frequency receiver) and an effective number of coils (smaller than the actual number of RF receiving coils in the system). The implications of these artifacts in practical algorithms have not been discussed elsewhere. In the present paper, we aim to study their actual impact and suggest practical rules to cope with them. We define the main noise parameters in this context, introducing a new expression for the effective variance of noise which is of capital importance for the two image processing problems studied: first, we propose a new method to estimate the effective variance of noise from the composite magnitude signal of MR data when correlations are assumed. Second, we adapt several model-based image denoising techniques to the correlated case using the noise estimation techniques proposed. We show, through a number of experiments with synthetic, phantom, and in vivo data, that neglecting the correlated nature of noise in multiple-coil systems implies important errors even in the simplest cases. At the same time, the proper statistical characterization of noise through effective parameters drives to improved accuracy (both qualitatively and quantitatively) for both of the problems studied.

A new method to estimate the variance of noise from the composite magnitude signal of GRAPPA reconstructed images is presented. Parallel imaging methods allow to increase the acquisition rate via subsampled acquisitions of the k-space. However, the reconstruction process yields to a variance of noise value which is dependent on the position within the image. The proposed method uses information of the GRAPPA reconstruction coefficients and assumes a final non-central chi distribution to recover the spatial pattern of noise.

The influence of the cartesian interpolation of ultrasound data over the final image statistical model is studied. When fully formed speckle is considered and no compression of the data is done, we show that the interpolated final image can be modeled following a Gamma distribution, which is a good approximation for the weighted sum of Rayleigh variables. The importance of taking into account the interpolation stage to statistically model ultrasound images is pointed out. The interpolation model here proposed can be easily extended to more complex distributions.

In geometric active contour algorithms, a re-initialization step must be performed by the level set function to remain close to a signed distance function, in order to avoid numerical instabilities. We propose a new re-initialization method that may be employed as a standalone method to recover the signed distance condition, or may be embedded directly into a variational framework as an additional term for the energy functional. Its purpose is to make the pixels near the propagating contour be less affected by the re-initialization. Experimental results show that whereas previous approaches change the position of the zero level set, our method keeps it virtually unchanged.

A novel anisotropic diffusion filter is proposed in this work with application to cardiac ultrasonic images. It includes probabilistic models which describe the probability density function (PDF) of tissues and adapts the diffusion tensor to the image iteratively. For this purpose, a preliminary study is performed in order to select the probability models that best fit the stastitical behavior of each tissue class in cardiac ultrasonic images. Then, the parameters of the diffusion tensor are defined taking into account the statistical properties of the image at each voxel. When the structure tensor of the probability of belonging to each tissue is included in the diffusion tensor definition, a better boundaries estimates can be obtained instead of calculating directly the boundaries from the image. This is the main contribution of this work. Additionally, the proposed method follows the statistical properties of the image in each iteration. This is considered as a second contribution since state-of-the-art methods suppose that noise or statistical properties of the image do not change during the filter process.

A new filtering method to remove Rician noise from magnetic resonance images is presented. This filter relies on a robust estimation of the standard deviation of the noise and combines local linear minimum mean square error filters and partial differential equations for MRI, as the speckle reducing anisotropic diffusion did for ultrasound images. The parameters of the filter are automatically chosen from the estimated noise. This property improves the convergence rate of the diffusion while preserving contours, leading to more robust and intuitive filtering. The partial derivative equation of the filter is extended to a new matrix diffusion filter which allows a coherent diffusion based on the local structure of the image and on the corresponding oriented local standard deviations. This new filter combines volumetric, planar, and linear components of the local image structure. The numerical scheme is explained and visual and quantitative results on simulated and real data sets are presented. In the experiments, the new filter leads to the best results.

A model for the distribution of the sample local variance (SLV) of magnetic resonance data is proposed. It is based on a bimodal Gamma distribution, whose maxima are related to the signal and background areas of the image. The model is valid for single- and multiple-coil systems. The proposed distribution allows us to characterize some signal/background properties in MR data. As an example, the model is used to study the effect of the background size over noise estimation techniques and a method to test the validity of background-based noise estimators is presented.

Noise estimation is a challenging task in magnetic resonance imaging (MRI), with applications in quality assessment, filtering or diffusion tensor estimation. Main noise estimators based on the Rician model are revisited and classified in this article, and new useful methods are proposed. Additionally, all the surveyed estimators are extended to the noncentral chi model, which applies to multiple-coil MRI and some important parallel imaging algorithms for accelerated acquisitions. The proposed new noise estimation procedures, based on the distribution of local moments, show better performance in terms of smaller variance and unbiased estimation over a wide range of experiments, with the additional advantage of not needing to explicitly segment the background of the image.

One known issue in Q---Ball imaging is the blurring in the radial integral defining the Orientation Distribution Function of fiber bundles, due to the computation of the Funk---Radon Transform (FRT). Three novel techniques to overcome this problem are presented, all of them based upon different assumptions about the behavior of the attenuation signal outside the sphere densely sampled from HARDI data sets. A systematic study with synthetic data has been carried out to show that the FRT blurring is not as important as the error introduced by some unrealistic assumptions, and only one of the three techniques (the one with the less restrictive assumption) improves the accuracy of Q---Balls.

Least Squares (LS) and its weighted version are standard techniques to estimate the Diffusion Tensor (DT) from Diffusion Weighted Images (DWI). They require to linearize the problem by computing the logarithm of the DWI. For the single-coil Rician noise model it has been shown that this model does not introduce a significant bias, but for multiple array coils and parallel imaging, the noise cannot longer be modeled as Rician. As a result the validity of LS approaches is not assured. An analytical study of noise statistics for a multiple coil system is carried out, together with the Weighted LS formulation and noise analysis for this model. Results show that the bias in the computation of the components of the DT may be comparable to their variance in many cases, stressing the importance of unbiased filtering previous to DT estimation.

A new method for noise filtering in images that follow a Rician model-with particular attention to magnetic resonance imaging-is proposed. To that end, we have derived a (novel) closed-form solution of the linear minimum mean square error (LMMSE) estimator for this distribution. Additionally, a set of methods that automatically estimate the noise power are developed. These methods use information of the sample distribution of local statistics of the image, such as the local variance, the local mean, and the local mean square value. Accordingly, the dynamic estimation of noise leads to a recursive version of the LMMSE, which shows a good performance in both noise cleaning and feature preservation. This paper also includes the derivation of the probability density function of several local sample statistics for the Rayleigh and Rician model, upon which the estimators are built.

We propose a new methodology for Linear Minimum Mean Square Error (LMMSE) filtering of Diffusion Weighted Imaging (DWI). We consider each voxel as an N-dimensional vector that comprises all the DWI volumes, and then compute the LMMSE estimator for the whole DWI data set jointly, taking into account the underlying tensor model. Our experiments, both with phantom and real data, show that this is a more convenient approach compared to the separate processing of each DWI, that translates to better removal of noise and preservation of structural information. Besides, our model has a simple algebraic formulation which makes the overall computational complexity very close to that of the scalar case, and it does not need multiple samples per DWI.

This paper introduces an outlier rejection and signal reconstruction method for high angular resolution diffusion weighted imaging. The approach is based on the thresholding of Laplacian measurements over the sphere of the apparent diffusion coefficient profiles defined for a given set of gradient directions. Exemplary results are presented.

A method to estimate the magnitude MR data from several noisy samples is presented. It is based on the Linear Minimum Mean Squared Error (LMMSE) estimator for the Rician noise model when several scanning repetitions are available. This method gives a closed-form analytical solution that takes into account the probability distribution of the data as well as the existing level of noise, showing a better performance than methods such as the average or the median.

Acquisition parameters play a crucial role in Diffusion Tensor Imaging (DTI), as they have a major impact on the values of scalar measures such as Fractional Anisotropy (FA) or Mean Diffusivity (MD) that are usually the focus of clinical studies based on white matter analysis. This paper presents an analysis on the impact of the variation of several acquisition parameters on these scalar measures with a novel double focus. First, a tractography-based approach is employed, motivated by the significant number of clinical studies that are carried out using this technique. Second, the consequences of simultaneous changes in multiple parameters are analyzed: number of gradient directions, b-value and voxel resolution. Results indicate that the FA is most affected by changes in the number of gradients and voxel resolution, while MD is specially influenced by variations in the b-value. Even if the choice of a tractography algorithm has an effect on the numerical values of the final scalar measures, the evolution of these measures when acquisition parameters are modified is parallel.