Parcellation of anatomically segregated cortical and subcortical brain regions is required in diffusion MRI (dMRI) analysis for region-specific quantification and better anatomical specificity of tractography. Most current dMRI parcellation approaches compute the parcellation from anatomical MRI (T1- or T2-weighted) data, using tools such as FreeSurfer or CAT12, and then register it to the diffusion space. However, the registration is challenging due to image distortions and low resolution of dMRI data, often resulting in mislabeling in the derived brain parcellation. Furthermore, these approaches are not applicable when anatomical MRI data is unavailable. As an alternative we developed the Deep Diffusion Parcellation (DDParcel), a deep learning method for fast and accurate parcellation of brain anatomical regions directly from dMRI data. The input to DDParcel are dMRI parameter maps and the output are labels for 101 anatomical regions corresponding to the FreeSurfer Desikan-Killiany (DK) parcellation. A multi-level fusion network leverages complementary information in the different input maps, at three network levels: input, intermediate layer, and output. DDParcel learns the registration of diffusion features to anatomical MRI from the high-quality Human Connectome Project data. Then, to predict brain parcellation for a new subject, the DDParcel network no longer requires anatomical MRI data but only the dMRI data. Comparing DDParcel's parcellation with T1w-based parcellation shows higher test-retest reproducibility and a higher regional homogeneity, while requiring much less computational time. Generalizability is demonstrated on a range of populations and dMRI acquisition protocols. Utility of DDParcel's parcellation is demonstrated on tractography analysis for fiber tract identification.

Joint relaxation-diffusion measurements can provide new insight about the tissue microstructural properties. Most recent methods have focused on inverting the Laplace transform to recover the joint distribution of relaxation-diffusion. However, as is well-known, this problem is notoriously ill-posed and numerically unstable. In this work, we address this issue by directly computing the joint moments of transverse relaxation rate and diffusivity, which can be robustly estimated. To zoom into different parts of the joint distribution, we further enhance our method by applying multiplicative filters to the joint probability density function of relaxation and diffusion and compute the corresponding moments. We propose an approach to use these moments to compute several novel scalar indices to characterize specific properties of the underlying tissue microstructure. Furthermore, for the first time, we propose an algorithm to estimate diffusion signals that are independent of echo time based on the moments of the marginal probability density function of diffusion. We demonstrate its utility in extracting tissue information not contaminated with multiple intra-voxel relaxation rates. We compare the performance of four types of filters that zoom into tissue components with different relaxation and diffusion properties and demonstrate it on an in-vivo human dataset. Experimental results show that these filters are able to characterize heterogeneous tissue microstructure. Moreover, the filtered diffusion signals are also able to distinguish fiber bundles with similar orientations but different relaxation rates. The proposed method thus allows to characterize the neural microstructure information in a robust and unique manner not possible using existing techniques.

In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works on the Riemannian framework for diffusion MRI. Using data from the Human Connectome Project, as well as population data from an autism spectrum disorder study, we demonstrate that this new Finsler metric, together with the new connectivity measures, results in connectivity maps that are much closer to known tract anatomy compared to previous geodesic connectivity methods. Our implementation can be used to compute geodesic distance and connectivity maps for segmented areas and is publicly available.

The ensemble average diffusion propagator (EAP) obtained from diffusion MRI (dMRI) data captures important structural properties of the underlying tissue. As such, it is imperative to derive an accurate estimate of the EAP from the acquired diffusion data. In this work, we propose a novel method for estimating the EAP by representing the diffusion signal as a linear combination of directional radial basis functions scattered in q-space. In particular, we focus on a special case of anisotropic Gaussian basis functions and derive analytical expressions for the diffusion orientation distribution function (ODF), the return-to-origin probability (RTOP), and mean-squared-displacement (MSD). A significant advantage of the proposed method is that the second and the fourth order moment tensors of the EAP can be computed explicitly. This allows for computing several novel scalar indices (from the moment tensors) such as mean-fourth-order-displacement (MFD) and generalized kurtosis (GK)-which is a generalization of the mean kurtosis measure used in diffusion kurtosis imaging. Additionally, we also propose novel scalar indices computed from the signal in q-space, called the q-space mean-squared-displacement (QMSD) and the q-space mean-fourth-order-displacement (QMFD), which are sensitive to short diffusion time scales. We validate our method extensively on data obtained from a physical phantom with known crossing angle as well as on in-vivo human brain data. Our experiments demonstrate the robustness of our method for different combinations of b-values and number of gradient directions.

We propose a novel probabilistic framework to merge information from diffusion weighted imaging tractography and resting-state functional magnetic resonance imaging correlations to identify connectivity patterns in the brain. In particular, we model the interaction between latent anatomical and functional connectivity and present an intuitive extension to population studies. We employ the EM algorithm to estimate the model parameters by maximizing the data likelihood. The method simultaneously infers the templates of latent connectivity for each population and the differences in connectivity between the groups. We demonstrate our method on a schizophrenia study. Our model identifies significant increases in functional connectivity between the parietal/posterior cingulate region and the frontal lobe and reduced functional connectivity between the parietal/posterior cingulate region and the temporal lobe in schizophrenia. We further establish that our model learns predictive differences between the control and clinical populations, and that combining the two modalities yields better results than considering each one in isolation.

In this paper, we present a non-linear (multi-affine) registration algorithm based on a local polynomial expansion model. We generalize previous work using a quadratic polynomial expansion model. Local affine models are estimated using this generalized model analytically and iteratively, and combined to a deformable registration algorithm. Experiments show that the affine parameter calculations derived from this quadratic model are more accurate than using a linear model. Experiments further indicate that the multi-affine deformable registration method can handle complex non-linear deformation fields necessary for deformable registration, and a faster convergent rate is verified from our comparison experiment.

Particle systems have gained importance as a methodology for sampling implicit surfaces and segmented objects to improve mesh generation and shape analysis. We propose that particle systems have a significantly more general role in sampling structure from unsegmented data. We describe a particle system that computes samplings of crease features (i.e. ridges and valleys, as lines or surfaces) that effectively represent many anatomical structures in scanned medical data. Because structure naturally exists at a range of sizes relative to the image resolution, computer vision has developed the theory of scale-space, which considers an n-D image as an (n+1)-D stack of images at different blurring levels. Our scale-space particles move through continuous four-dimensional scale-space according to spatial constraints imposed by the crease features, a particle-image energy that draws particles towards scales of maximal feature strength, and an inter-particle energy that controls sampling density in space and scale. To make scale-space practical for large three-dimensional data, we present a spline-based interpolation across scale from a small number of pre-computed blurrings at optimally selected scales. The configuration of the particle system is visualized with tensor glyphs that display information about the local Hessian of the image, and the scale of the particle. We use scale-space particles to sample the complex three-dimensional branching structure of airways in lung CT, and the major white matter structures in brain DTI.

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 presents a segmentation method that extends geodesic active region methods by the incorporation of a statistical classifier trained using feature selection. The classifier provides class probability maps based on class representative local features, and the geodesic active region formulation enables the partitioning of the image according to the region information. We demonstrate automatic segmentation results of the myocardium in cardiac late gadolinium-enhanced magnetic resonance imaging (CE-MRI) data using coupled level set curve evolutions, in which the classifier is incorporated both from a region term and from a shape term from particle filtering. The results show potential for clinical studies of scar tissue in late CE-MRI data.

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.

We introduce a versatile framework for characterizing and extracting salient structures in three-dimensional symmetric second-order tensor fields. The key insight is that degenerate lines in tensor fields, as defined by the standard topological approach, are exactly crease (ridge and valley) lines of a particular tensor invariant called mode. This reformulation allows us to apply well-studied approaches from scientific visualization or computer vision to the extraction of topological lines in tensor fields. More generally, this main result suggests that other tensor invariants, such as anisotropy measures like fractional anisotropy (FA), can be used in the same framework in lieu of mode to identify important structural properties in tensor fields. Our implementation addresses the specific challenge posed by the non-linearity of the considered scalar measures and by the smoothness requirement of the crease manifold computation. We use a combination of smooth reconstruction kernels and adaptive refinement strategy that automatically adjust the resolution of the analysis to the spatial variation of the considered quantities. Together, these improvements allow for the robust application of existing ridge line extraction algorithms in the tensor context of our problem. Results are proposed for a diffusion tensor MRI dataset, and for a benchmark stress tensor field used in engineering research.

We propose a manifold learning approach to fiber tract clustering using a novel similarity measure between fiber tracts constructed from dual-rooted graphs. In particular, to generate this similarity measure, the chamfer or Hausdorff distance is initially employed as a local distance metric to construct minimum spanning trees between pairwise fiber tracts. These minimum spanning trees are effective in capturing the intrinsic geometry of the fiber tracts. Hence, they are used to capture the neighborhood structures of the fiber tract data set. We next assume the high-dimensional input fiber tracts to lie on low-dimensional non-linear manifolds. We apply Locally Linear Embedding, a popular manifold learning technique, to define a low-dimensional embedding of the fiber tracts that preserves the neighborhood structures of the high-dimensional data structure as captured by the method of dual-rooted graphs. Clustering is then performed on this low-dimensional data structure using the k-means algorithm. We illustrate our resulting clustering technique on both synthetic data and on real fiber tract data obtained from diffusion tensor imaging.

For data samples in Rn, the mean is a well known estimator. When the data set belongs to an embedded manifold M in Rn, e.g. the unit circle in R2, the definition of a mean can be extended and constrained to M by choosing either the intrinsic Riemannian metric of the manifold or the extrinsic metric of the embedding space. A common view has been that extrinsic means are approximate solutions to the intrinsic mean problem. This paper study both means on the unit circle and reveal how they are related to the ML estimate of independent samples generated from a Brownian distribution. The conclusion is that on the circle, intrinsic and extrinsic means are maximum likelihood estimators in the limits of high SNR and low SNR respectively.

We introduce a segmentation framework which extends spatially varying classification to not only incorporate anatomical localization from shape estimation, but to also encode certainty of the localization by local shape variability. The method iterates between a classification step where a statistical classifier learned from feature selection is extended with anatomical localization features, and a shape estimation step where, given the class probability maps, shape is inferred by particle filtering using a level set shape model that accounts for local degrees of anatomical variability. The spatially varying classification is embedded in a geodesic active region framework which allows for local deviations from the inferred shape using an iteratively updated classification based region term. The method is evaluated on late gadolinium enhanced cardiac MRI and is to our knowledge the first automatic segmentation method demonstrated on this type of data.

In this paper we demonstrate an integrated registration and clustering algorithm to compute an atlas of fiber-bundles from a set of multi-subject diffusion weighted MR images. We formulate a maximum likelihood problem which the proposed method solves using a generalized Expectation Maximization (EM) framework. Additionally, the algorithm employs an outlier rejection and denoising strategy to produce sharp probabilistic maps of certain bundles of interest. This map is potentially useful for making diffusion measurements in a common coordinate system to identify pathology related changes or developmental trends.

Ultrasound imaging systems provide the clinician with noninvasive, low-cost, and real-time images that can help them in diagnosis, planning, and therapy. However, although the human eye is able to derive the meaningful information from these images, automatic processing is very difficult due to noise and artifacts present in the image. The speckle reducing anisotropic diffusion filter was recently proposed to adapt the anisotropic diffusion filter to the characteristics of the speckle noise present in the ultrasound images and to facilitate automatic processing of images. We analyze the properties of the numerical scheme associated with this filter, using a semi-explicit scheme. We then extend the filter to a matrix anisotropic diffusion, allowing different levels of filtering across the image contours and in the principal curvature directions. We also show a relation between the local directional variance of the image intensity and the local geometry of the image, which can justify the choice of the gradient and the principal curvature directions as a basis for the diffusion matrix. Finally, different filtering techniques are compared on a 2-D synthetic image with two different levels of multiplicative noise and on a 3-D synthetic image of a Y-junction, and the new filter is applied on a 3-D real ultrasound image of the liver.

Guided by empirically established connections between clinically important tissue properties and diffusion tensor parameters, we introduce a framework for decomposing variations in diffusion tensors into changes in shape and orientation. Tensor shape and orientation both have three degrees-of-freedom, spanned by invariant gradients and rotation tangents, respectively. As an initial demonstration of the framework, we create a tunable measure of tensor difference that can selectively respond to shape and orientation. Second, to analyze the spatial gradient in a tensor volume (a third-order tensor), our framework generates edge strength measures that can discriminate between different neuroanatomical boundaries, as well as creating a novel detector of white matter tracts that are adjacent yet distinctly oriented. Finally, we apply the framework to decompose the fourth-order diffusion covariance tensor into individual and aggregate measures of shape and orientation covariance, including a direct approximation for the variance of tensor invariants such as fractional anisotropy.

We propose a new white matter atlas creation method that learns a model of the common white matter structures present in a group of subjects. We demonstrate that our atlas creation method, which is based on group spectral clustering of tractography, discovers structures corresponding to expected white matter anatomy such as the corpus callosum, uncinate fasciculus, cingulum bundles, arcuate fasciculus, and corona radiata. The white matter clusters are augmented with expert anatomical labels and stored in a new type of atlas that we call a high-dimensional white matter atlas. We then show how to perform automatic segmentation of tractography from novel subjects by extending the spectral clustering solution, stored in the atlas, using the Nystrom method. We present results regarding the stability of our method and parameter choices. Finally we give results from an atlas creation and automatic segmentation experiment. We demonstrate that our automatic tractography segmentation identifies corresponding white matter regions across hemispheres and across subjects, enabling group comparison of white matter anatomy.

Endovascular therapy provides a minimally invasive solution for the treatment of brain aneurysms. Interventional neuroradiologists navigate microcathers, platinum coils and stents from the femoral artery to treat the pathological target vessel under fluoroscopic guidance. The procedure involves a clinically significant radiation dose to the patient and the clinical team. We have developed a multi-modality navigation platform providing real-time visualization of the position of endovascular tools without radiation exposure. The system integrates magnetic tracking of surgical devices, and 3D visualization of their position inside a reconstructed model of the vasculature. Preliminary phantom tests have demonstrated an accuracy of 3.8 mm. By merging imaging and localization data, this prototype can provide 3D real-time navigation in brain arteries.

White matter fiber bundles in the human brain can be located by tracing the local water diffusion in diffusion weighted magnetic resonance imaging (MRI) images. In this paper, a novel Bayesian modeling approach for white matter tractography is presented. The uncertainty associated with estimated white matter fiber paths is investigated, and a method for calculating the probability of a connection between two areas in the brain is introduced. The main merits of the presented methodology are its simple implementation and its ability to handle noise in a theoretically justified way. Theory for estimating global connectivity is also presented, as well as a theorem that facilitates the estimation of the parameters in a constrained tensor model of the local water diffusion profile.

A common goal of multivariate visualization is to enable data inspection at discrete points, while also illustrating larger-scale continuous structures. In diffusion tensor visualization, glyphs are typically used to meet the first goal, and methods such as texture synthesis or fiber tractography can address the second. We adapt particle systems originally developed for surface modeling and anisotropic mesh generation to enhance the utility of glyph-based tensor visualizations. By carefully distributing glyphs throughout the field (either on a slice, or in the volume) into a dense packing, using potential energy profiles shaped by the local tensor value, we remove undue visual emphasis of the regular sampling grid of the data, and the underlying continuous features become more apparent. The method is demonstrated on a DT-MRI scan of a patient with a brain tumor.

Ultrasound images provide the clinician with non-invasive, low cost, and real-time images that can help them in diagnosis, planning and therapy. However, although the human eye is able to derive the meaningful information from these images, automatic processing is very difficult because of the noise and artefacts present in the image. In this work, we propose to extend the current anisotropic diffusion technique to deal with the speckle noise present in the Ultrasound images. To this end, we use a previously derived model of the noise, and we write the restoration scheme as a energy minimization constrained by the noise model and parameters. This approach leads to a new data attachment term whose optimal weight can be automatically estimated.

Averaging, filtering and interpolation of 3-D object orientation data is important in both computer vision and computer graphics, for instance to smooth estimates of object orientation and interpolate between keyframes in computer animation. In this paper we present a novel framework in which the non-linear nature of these problems is avoided by embedding the manifold of 3-D orientations into a 16-dimensional Euclidean space. Linear operations performed in the new representation can be shown to be rotation invariant, and defining a projection back to the orientation manifold results in optimal estimates with respect to the Euclidean metric. In other words, standard linear filters, interpolators and estimators may be applied to orientation data, without the need for an additional machinery to handle the non-linear nature of the problems. This novel representation also provides a way to express uncertainty in 3-D orientation, analogous to the well known tensor representation for lines and hyperplanes.

This paper presents an extension of the phase correlation image alignment method to N-dimensional data sets. By the Fourier shift theorem, the motion model for translational shifts between N-dimensional images can be represented as a rank-one tensor. Through use of a high-order singular value decomposition, the phase correlation between two N-dimensional data sets can be decomposed to independently identify translational displacements along each dimension with subpixel resolution. Using three-dimensional MRI data sets, we demonstrate the effectiveness of this approach relative to other N-dimensional image registration methods.

This method is an extension of the phase correlation method for image registration to multidimensional data sets. Through use of a high-order singular value decomposition, phase correlation can be used to identify translational displacements independently along each dimension with subpixel resolution. The validity of this approach is demonstrated using multiple 3D MRI data sets.

We propose a new method for artifact reduction of upsampled multidimensional signals. These artifacts are evident near edges and they are due to the spectral narrowing associated with upsampling. The method is based on first upsampling the signal with conventional optimal sinc interpolation and then applying a local filter that reduces the high frequencies associated with the ringing artifacts along the edges while leaving unchanged the directions orthogonal to them. The method is specially suitable when dealing with medical images that contain small structures, such as thin bones in CT.

Curve evolution schemes for segmentation, implemented with level set methods, have become an important approach in computer vision. Previous work has modeled evolving contours which are curves in 2D or surfaces in 3D. Our objective is to explore recent mathematical work enabling the evolution of manifolds of higher co-dimension. We consider 1D curves in 3D (codimension-two) for the application of automatically, segmenting blood vessels in volumetric magnetic resonance angiography (MRA) images. This paper describes the theoretical foundations of our system, CURVES, then provides segmentation results compared against segmentation obtained interactively by a neurosurgeon. Segmentation of bronchi in lung computed tomography (CT) scans are also presented. The new experiments, comparisons to manual segmentation, and sample comparison to the use of a codimension-one regularization force are the primary contributions of this report.

This paper describes a novel approach to tissue classification using three-dimensional (3D) derivative features in the volume rendering pipeline. In conventional tissue classification for a scalar volume, tissues of interest are characterized by an opacity transfer function defined as a one-dimensional (1D) function of the original volume intensity. To overcome the limitations inherent in conventional 1D opacity functions, we propose a tissue classification method that employs a multidimensional opacity function, which is a function of the 3D derivative features calculated from a scalar volume as well as the volume intensity. Tissues of interest are characterized by explicitly defined classification rules based on 3D filter responses highlighting local structures, such as edge, sheet, line, and blob, which typically correspond to tissue boundaries, cortices, vessels, and nodules, respectively, in medical volume data. The 3D local structure filters are formulated using the gradient vector and Hessian matrix of the volume intensity function combined with isotropic Gaussian blurring. These filter responses and the original intensity define a multidimensional feature space in which multichannel tissue classification strategies are designed. The usefulness of the proposed method is demonstrated by comparisons with conventional single-channel classification using both synthesized data and clinical data acquired with CT (computed tomography) and MRI (magnetic resonance imaging) scanners. The improvement in image quality obtained using multichannel classification is confirmed by evaluating the contrast and contrast-to-noise ratio in the resultant volume-rendered images with variable opacity values.

In this paper we focus on using local 3D structure for segmentation. A tensor descriptor is estimated for each neighbourhood, i.e. for each voxel in the data set. The tensors are created from a combination of the outputs form a set of 3D quadrature filters. The shape of the tensors describe locally the structure of the neighbourhood in terms of how much it is like a plane, a line, and a sphere. We apply this to segmentation of bone from Computer Tomography data (CT). Traditional methods are based purely on gray-level value discrimination and have difficulties in recovering thin bone structures due to so called partial voluming, a problem which is present in all such sampled data. We illuminate the partial voluming problem by showing that thresholding creates complicated artifacts even if the signal is densely enough sampled and can be perfectly reconstructed. The unwanted effects of thresholding can be reduced by a change of the signal basis. We show that by using additional local structure information can significantly reduce the degree of sampling artifacts. Evaluation of the method on a clinical case is presented, the segmentation of a human skull from a CT volume. The method shows that many of the thin bone structures which disappear in a pure thresholding can be recovered.

Focus of attention mechanisms for robot vision are discussed. A new method for neglecting low level filter responses from already modelled structures is presented. The method is based on a filtering technique termed normalized convolution. In one experiment, the robot is continuously moving its arm in the scene while tracking other objects. It is shown how the arm can be made "invisible" so that only the moving object of interest is detected. This makes tracking of objects much simpler. In another experiment, the attention of the system is shifted between objects by simply cancelling the mask of the object to be attended to. With this strategy the low level processes do not need to know the difference between a new object entering the scene and a mask being cancelled, and thus a complex communication structure between high and low levels is avoided.

This paper establishes an algebraic relation between two methods recently reported; normalized convolution and normalized differential convolution. These are general methods for filtering incomplete or uncertain data and are based on the separation of both data and operator into a signal part and a certainty part. General filtering can be performed without preprocessing input data with an interpolation step. The methods allow both data and operators to be scalars, vectors or tensors of higher order. Normalized differential convolution has been used in a wide range of applications. Examples are estimation of gradient estimation in irregularly sampled data, estimation of differential invariants in sparse image flow fields and image edge effect reduction. It was previously shown that normalized convolution produces a description of the neighbourhood which is optimal in a least square sense. The algebraic relation to normalized differential convolution presented in this paper proves that the latter method is also optimal in the same sense as well.>

This paper describes a robust algorithm for estimation of local signal frequency and bandwidth. The method is based on combining local estimates of instantaneous frequency over a large number of scales. The filters used are a set of lognormal quadrature wavelets. A novel feature is that an estimate of local frequency bandwidth can be obtained. The bandwidth can be used to produce a measure of certainty for the estimated frequency. The algorithm is applicable to multidimensional data and examples of the performance of the method are demonstrated for one-dimensional and two-dimensional signals.>

This paper presents a method for computation of two-dimensional motion vector fields from an image sequence. The magnitudes from a set of spatio-temporal quadrature filters are combined into a tensor description. The shape of the tensors describe locally the structure of the spatio-temporal neighbourhood and provides information about local velocity and if true flow or only normal flow is present. It is shown how normal flow estimates are combined into a true flow using linear filtering on this tensor field description.>

An approach for processing incomplete or uncertain data based on subspace methods is presented in this paper. The paper addresses the problem of how a subset of a parameter vector describing a signal can be estimated. The term parameter vector refers to the coefficients in the linear combination of basis functions describing a local image neighbourhood. Images are normally described locally using simple basis functions. Low order local momentums such as order 0 (the local DC component), 1 and 2 are commonly used. Low order differentiations are also useful descriptors. In densely regularly sampled images, these descriptors are easily computed using standard convolution. However, when working with irregularly sampled data or incomplete data the signal model has to be of higher order than the signal variations of interest. This is the case where only a part of the parameter vector is to be estimated. If possible, only this part of the parameter should be calculated explicitly as opposed to calculating the whole parameter vector. This paper describes such a method based on partitioning the model subspace into two parts.

It is shown how false operator responses due to missing or uncertain data can be significantly reduced or eliminated. It is shown how operators having a higher degree of selectivity and higher tolerance against noise can be constructed using simple combinations of appropriately chosen convolutions. The theory is based on linear operations and is general in that it allows for both data and operators to be scalars, vectors or tensors of higher order. Three new methods are represented: normalized convolution, differential convolution and normalized differential convolution. All three methods are examples of the power of the signal/certainty-philosophy, i.e., the separation of both data and operator into a signal part and a certainty part. Missing data are handled simply by setting the certainty to zero. In the case of uncertain data, an estimate of the certainty must accompany the data. Localization or windowing of operators is done using an applicability function, the operator equivalent to certainty, not by changing the actual operator coefficients. Spatially or temporally limited operators are handled by setting the applicability function to zero outside the window.>