Hybrid Diffusion Imaging (HYDI) was one of the first attempts to use multi-shell samplings of the q-space to infer diffusion properties beyond Diffusion Tensor Imaging (DTI) or High Angular Resolution Diffusion Imaging (HARDI). HYDI was intended as a flexible protocol embedding both DTI (for lower b-values) and HARDI (for higher b-values) processing, as well as Diffusion Spectrum Imaging (DSI) when the entire data set was exploited. In the latter case, the spherical sampling of the q-space is re-gridded by interpolation to a Cartesian lattice whose extent covers the range of acquired b-values, hence being acquisition-dependent. The Discrete Fourier Transform (DFT) is afterwards used to compute the corresponding Cartesian sampling of the Ensemble Average Propagator (EAP) in an entirely non-parametric way. From this lattice, diffusion markers such as the Return To Origin Probability (RTOP) or the Mean Squared Displacement (MSD) can be numerically estimated. We aim at re-formulating this scheme by means of a Fourier Transform encoding matrix that eliminates the need for q-space re-gridding at the same time it preserves the non-parametric nature of HYDI-DSI. The encoding matrix is adaptively designed at each voxel according to the underlying DTI approximation, so that an optimal sampling of the EAP can be pursued without being conditioned by the particular acquisition protocol. The estimation of the EAP is afterwards carried out as a regularized Quadratic Programming (QP) problem, which allows to impose positivity constraints that cannot be trivially embedded within the conventional HYDI-DSI. We demonstrate that the definition of the encoding matrix in the adaptive space allows to analytically (as opposed to numerically) compute several popular descriptors of diffusion with the unique source of error being the cropping of high frequency harmonics in the Fourier analysis of the attenuation signal. They include not only RTOP and MSD, but also Return to Axis/Plane Probabilities (RTAP/RTPP), which are defined in terms of specific spatial directions and are not available with the former HYDI-DSI. We report extensive experiments that suggest the benefits of our proposal in terms of accuracy, robustness and computational efficiency, especially when only standard, non-dedicated q-space samplings are available.

AMURA (Apparent Measures Using Reduced Acquisitions) was originally proposed as a method to infer micro-structural information from single-shell acquisitions in diffusion MRI. It reduces the number of samples needed and the computational complexity of the estimation of diffusion properties of tissues by assuming the diffusion anisotropy is roughly independent on the b-value. This simplification allows the computation of simplified expressions and makes it compatible with standard acquisition protocols commonly used even in clinical practice. The present work proposes an extension of AMURA that allows the calculation of general moments of the diffusion signals that can be applied to describe the diffusion process with higher accuracy. We provide simplified expressions to analytically compute a set of scalar indices as moments of arbitrary orders over either the whole 3-D space, particular directions, or particular planes. The existing metrics previously proposed for AMURA (RTOP, RTPP and RTAP) are now special cases of this generalization. An extensive set of experiments is performed on public data and a clinical clase acquired with a standard type acquisition. The new metrics provide additional information about the diffusion processes inside the brain.

Characterization of the noise distribution in magnetic resonance images has multiple applications, including quality assurance and protocol optimization. Noise characterization is particularly important in the presence of parallel imaging acceleration with multi-coil acquisitions, where the noise distribution can contain severe spatial heterogeneities. If the parallel imaging reconstruction is a linear process, an accurate noise analysis can be carried out by taking into account the correlations between all the samples involved. However, for -space-based techniques such as generalized autocalibrating partially parallel acquisition (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 -space to image space. Previously proposed methods avoid this computational burden by formulating 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 sampling of -space (e.g., containing a calibration region). For this reason, in this paper, we develop an accurate characterization of the noise distribution for self-calibrated parallel imaging in the presence of arbitrary Cartesian sampling patterns. By exploiting the symmetries and separability in the noise propagation process, the proposed method is computationally efficient and does not require large matrices. Under the assumption of a fixed reconstruction kernel, this method provides the precise distribution of the noise variance for each coil's image. These coil-by-coil noise maps are subsequently combined according to the coil combination approach used in image reconstruction, and therefore can be applied with both complex coil combination and root-sum-of-squares approaches. In this paper, we present the proposed noise characterization method and compare it to previous techniques using Monte Carlo simulations as well as phantom acquisitions.

The Apparent Diffusion Coefficient (ADC) is a quantitative measure derived from MRI that is able to asses the amount of diffusion within living tissues. It is employed to characterize different diseases and to evaluate response to therapy. For ADC estimation, the diffusion signal needs to be sampled using a small number of values, presenting distortions due to the aliasing and windowing effect. In this work we theoretically study these effects and propose some new robust estimators for ADC based on the Fourier Transform of the signal.

Parallel magnetic resonance imaging (pMRI) techniques have gained a great importance both in research and clinical communities recently since they considerably accelerate the image acquisition process. However, the image reconstruction algorithms needed to correct the subsampling artifacts affect the nature of noise, i.e., it becomes non-stationary. Some methods have been proposed in the literature dealing with the non-stationary noise in pMRI. However, their performance depends on information not usually available such as multiple acquisitions, receiver noise matrices, sensitivity coil profiles, reconstruction coefficients, or even biophysical models of the data. Besides, some methods show an undesirable granular pattern on the estimates as a side effect of local estimation. Finally, some methods make strong assumptions that just hold in the case of high signal-to-noise ratio (SNR), which limits their usability in real scenarios. We propose a new automatic noise estimation technique for non-stationary Rician noise that overcomes the aforementioned drawbacks. Its effectiveness is due to the derivation of a variance-stabilizing transformation designed to deal with any SNR. The method was compared to the main state-of-the-art methods in synthetic and real scenarios. Numerical results confirm the robustness of the method and its better performance for the whole range of SNRs.

Ultrasound (US) imaging exhibits considerable difficulties for medical visual inspection and for development of automatic analysis methods due to speckle, which negatively affects the perception of tissue boundaries and the performance of automatic segmentation methods. With the aim of alleviating the effect of speckle, many filtering techniques are usually considered as a preprocessing step prior to automatic analysis methods or visual inspection. Most of the state-of-the-art filters try to reduce the speckle effect without considering its relevance for the characterization of tissue nature. However, the speckle phenomenon is the inherent response of echo signals in tissues and can provide important features for clinical purposes. This loss of information is even magnified due to the iterative process of some speckle filters, e.g., diffusion filters, which tend to produce over-filtering because of the progressive loss of relevant information for diagnostic purposes during the diffusion process. In this paper, we propose an anisotropic diffusion filter with a probabilistic-driven memory mechanism to overcome the over-filtering problem by following a tissue selective philosophy. In particular, we formulate the memory mechanism as a delay differential equation for the diffusion tensor whose behavior depends on the statistics of the tissues, by accelerating the diffusion process in meaningless regions and including the memory effect in regions where relevant details should be preserved. Results both in synthetic and real US images support the inclusion of the probabilistic memory mechanism for maintaining clinical relevant structures, which are removed by the state-of-the-art filters.

This paper proposes a methodology for the joint alignment of a sequence of images based on a groupwise registration procedure by using a new family of metrics that exploit the expected sparseness of the temporal intensity curves corresponding to the aligned points. Therefore, this methodology is able to tackle the alignment of temporal sequences of images in which the represented phenomenon varies in time. Specifically, we have applied it to the correction of motion in contrast-enhanced first-pass perfusion cardiac magnetic resonance images. The time sequence is elastically registered as a whole by using the aforementioned family of multi-image metrics and jointly optimizing the parameters of the transformations involved. The proposed metrics are able to cope with dynamic changes in the intensity content of corresponding points in the sequence guided by the assumption that these changes allow for a sparse representation in a properly selected frame. Results have shown the statistically significant improvement in the performance of the proposed metric with respect to previous groupwise registration metrics for the problem at hand, which is especially relevant to correct for elastic deformations.

A realistic log-compressed law for ultrasound images based on a real device is proposed. The model takes into account the linear behavior of the logarithmic amplifier for small signal gain which transforms image values in a different way as the classical models do. Additionally, for recovery purposes, a method for the estimation of the compression parameters is also proposed when a realistic log-compressed law is considered. Results with synthetic images show that the proposed method achieved a consistent Rayleigh parameter estimate with a very low error. Experiments with real images show that the inversion method is consistent through the whole acquisition process when parameters of the logarithmic amplifier are assumed constant.

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.

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.

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.

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.

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 method to identify different tissues in ultrasound images is presented. Assuming a Rayleigh model for speckle, the different tissues present in the image will be related to the Rayleigh sigma parameter, with different values for each different tissue. The parameter is locally estimated using well known estimation methods for the Rayleigh distribution, such as the maximum likelihood estimator. A significant increase in the separability of the different tissues present in the image is then achieved. A subsequent clustering process based on the position of the maxima of the estimator distribution allows the classification of each pixel in a tissue class.

In this paper, we focus on the problem of speckle removal by means of anisotropic diffusion and, specifically, on the importance of the correct estimation of the statistics involved. First, we derive an anisotropic diffusion filter that does not depend on a linear approximation of the speckle model assumed, which is the case of a previously reported filter, namely, SRAD. Then, we focus on the problem of estimation of the coefficient of variation of both signal and noise and of noise itself. Our experiments indicate that neighborhoods used for parameter estimation do not need to coincide with those used in the diffusion equations. Then, we show that, as long as the estimates are good enough, the filter proposed here and the SRAD perform fairly closely, a fact that emphasizes the importance of the correct estimation of the coefficients of variation.

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.

A matrix inference method for fuzzy systems (FITM) is used to deal with hierarchical fuzzy systems (HFSs). The contribution of this paper is twofold: on one hand, we show that a HFS (either serial or parallel) can be easily converted into a single fuzzy system, and implemented via FITM. On the other, we show that a serial HFS can be obtained out of a complicated multiple-input single-output fuzzy system. An example of such a conversion for a three-input system is included.

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.

In this paper we propose an alternative implementation of the concept of fuzzy granule. Granules are defined in terms of the degree of overlap with other granules, as opposed to by assigning (somehow arbitrarily) membership values to each and every point of the university of discourse in which the granule is defined. We believe this alternative definition is much closer to the human way of thinking. Two examples of real world applications illustrate this new definition.

We carry out a detailed analysis of a fuzzy version of Reid's classical multiple hypothesis tracking (MHT) algorithm. Our fuzzy version is based on well-known fuzzy feedback systems, but the fact that the system we describe is specialized for likelihood discrimination makes this study particularly novel. We discuss several techniques for rule activation. One of them, namely the sum-product, seems particularly useful for likelihood management and its linearity makes it tractable for further analysis. Our analysis is performed in two stages: 1) we demonstrate that, with appropriately chosen rules, our system can discriminate the correct hypothesis; and 2) the steady-state behavior with a constant input is characterized analytically. This enables us to establish the optimality of the sum-product method and it also gives a simple procedure to predict the system's behavior as a function of the rule base. We believe this fact can be used to devise a simple procedure for fine-tuning the rule base according to the system designer's needs. The application driving our fuzzy MHT implementation and analysis is the tracking of natural language text-based messages. This application is used as an example throughout the paper.

An anisotropic diffusion filter controlled by fuzzy rules is presented. The proposed filter is based in the Perona-Malik technique, using fuzzy reasoning to calculate the diffusion coefficient which controls the whole diffusion. The method has the advantage that it can be used for both smoothing and noise cleaning, as well as edge enhancement. This new approach also allows us to model the diffusion process through a rule base to have a better performance. Some examples are given to illustrate the effectiveness of the proposed technique.