Existing Bayesian spatial priors for functional magnetic resonance imaging (fMRI) data correspond to stationary isotropic smoothing filters that may oversmooth at anatomical boundaries. We propose two anatomically informed Bayesian spatial models for fMRI data with local smoothing in each voxel based on a tensor field estimated from a T1-weighted anatomical image. We show that our anatomically informed Bayesian spatial models results in posterior probability maps that follow the anatomical structure.

Numerous applications in diffusion MRI involve computing the orientationally-averaged diffusion-weighted signal. Most approaches implicitly assume, for a given b-value, that the gradient sampling vectors are uniformly distributed on a sphere (or shell), computing the orientationally-averaged signal through simple arithmetic averaging. One challenge with this approach is that not all acquisition schemes have gradient sampling vectors distributed over perfect spheres. To ameliorate this challenge, alternative averaging methods include: weighted signal averaging; spherical harmonic representation of the signal in each shell; and using Mean Apparent Propagator MRI (MAP-MRI) to derive a three-dimensional signal representation and estimate its isotropic part. Here, these different methods are simulated and compared under different signal-to-noise (SNR) realizations. With sufficiently dense sampling points (61 orientations per shell), and isotropically-distributed sampling vectors, all averaging methods give comparable results, (MAP-MRI-based estimates give slightly higher accuracy, albeit with slightly elevated bias as b-value increases). As the SNR and number of data points per shell are reduced, MAP-MRI-based approaches give significantly higher accuracy compared with the other methods. We also apply these approaches to in vivo data where the results are broadly consistent with our simulations. A statistical analysis of the simulated data shows that the orientationally-averaged signals at each b-value are largely Gaussian distributed.

This paper presents new methods for use of dense motion fields for motion compensation of interlaced video. The motion estimation is based on previously decoded field-images. The motion is then temporally predicted and used for motion compensated prediction of the field-image to be coded. The motion estimation algorithm is phase-based and uses two or three field-images to achieve motion estimates with sub-pixel accuracy. To handle non-constant motion and the specific characteristics of the field-image to be coded, the initially predicted image is refined using forward motion compensation, based on block-matching. Tests show that this approach achieves higher PSNR than forward block-based motion estimation, when coding the residual with the same coder. The subjective performance is also better.

The use of temporal redundancy is of vital importance for a successful video coding algorithm. An effective approach is the hybrid video coder where motion estimation is used for prediction of the next image frame and code the prediction error, and the motion field. The standard method for motion estimation is block matching as in MPEG-2, typically resulting in block artifacts. In this paper a perception based velocity estimator and its use for pixel based motion compensated prediction of interlaced video is presented.

This paper presents new methods for use of dense motion fields for motion compensation of interlaced video. The motion is estimated using previously decoded field-images. An initial motion compensated prediction is produced using the assumption that the motion is predictable in time. The motion estimation algorithm is phase-based and uses two or three field-images to achieve motion estimates with sub-pixel accuracy. To handle non-constant motion and the specific characteristics of the field-image to be coded, the initially predicted image is refined using forward motion compensation, based on block-matching. Tests show that this approach achieves higher PSNR than forward block-based motion estimation, when coding the residual with the same coder. The subjective performance is also better.

The problem of signal estimation for sparsely and irregularly sampled signals is dealt with using continuous normalized convolution. Image values on real-valued positions are estimated using integration of signals and certainties over a neighbourhood employing a local model of both the signal and the used discrete filters. The result of the approach is that an output sample close to signals with high certainty is interpolated using a small neighbourhood. An output sample close to signals with low certainty is spatially predicted from signals in a large neighbourhood.

This paper introduce multiple hierarchical motion estimation to achieve motion estimates with high spatial resolution. The approach is based on phase-based motion estimation. Results show that the algorithm deal with the smooth motion field of hierarchical motion estimation while keeping the advantages of such an approach.

This paper presents a novel method for performing fast estimation of data samples on a desired output grid from samples on an irregularly sampled grid. The output signal is estimated using integration of signals over a neighbourhood employing a local model of the signal using discrete filters. The strength of the method is demonstrated in motion compensation examples by comparing to traditional techniques.

The multilinear least-squares (MLLS) problem is an extension of the linear leastsquares problem. The difference is that a multilinear operator is used in place of a matrix-vector product. The MLLS is typically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows for moving from one local minimizer to a better one. The efficiency of this strategy is illustrated by results of numerical experiments performed for some problems related to the design of filter networks.

The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinearoperator is used in place of a matrix-vector product. The MLLS istypically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows formoving from one local minimizer to a better one. The efficiencyof this strategy isillustrated by results of numerical experiments performed forsome problems related to the design of filter networks.

Filter networks is a powerful tool used for reducing the image processing time, while maintaining its reasonably high quality.They are composed of sparse sub-filters whose low sparsity ensures fast image processing.The filter network design is related to solvinga sparse optimization problem where a cardinality constraint bounds above the sparsity level.In the case of sequentially connected sub-filters, which is the simplest network structure of those considered in this paper, a cardinality-constrained multilinear least-squares (MLLS) problem is to be solved. If to disregard the cardinality constraint, the MLLS is typically a large-scale problem characterized by a large number of local minimizers. Each of the local minimizers is singular and non-isolated.The cardinality constraint makes the problem even more difficult to solve.An approach for approximately solving the cardinality-constrained MLLS problem is presented.It is then applied to solving a bi-criteria optimization problem in which both thetime and quality of image processing are optimized. The developed approach is extended to designing filter networks of a more general structure. Its efficiency is demonstrated by designing certain 2D and 3D filter networks. It is also compared with the existing approaches.

Filter networks are used as a powerful tool used for reducing the image processing time and maintaining high image quality.They are composed of sparse sub-filters whose high sparsity ensures fast image processing.The filter network design is related to solvinga sparse optimization problem where a cardinality constraint bounds above the sparsity level.In the case of sequentially connected sub-filters, which is the simplest network structure of those considered in this paper, a cardinality-constrained multilinear least-squares (MLLS) problem is to be solved. Even when disregarding the cardinality constraint, the MLLS is typically a large-scale problem characterized by a large number of local minimizers, each of which is singular and non-isolated.The cardinality constraint makes the problem even more difficult to solve.

An approach for approximately solving the cardinality-constrained MLLS problem is presented.It is then applied to solving a bi-criteria optimization problem in which both thetime and quality of image processing are optimized. The developed approach is extended to designing filter networks of a more general structure. Its efficiency is demonstrated by designing certain 2D and 3D filter networks. It is also compared with the existing approaches.

The aim of this project is to keep the x-ray exposure of the patient as low as reasonably achievable while improving the diagnostic image quality for the radiologist. The means to achieve these goals is to develop and evaluate an efficient adaptive filtering (denoising/image enhancement) method that fully explores true 4D image acquisition modes.

The proposed prototype system uses a novel filter set having directional filter responses being monomials. The monomial filter concept is used both for estimation of local structure and for the anisotropic adaptive filtering. Initial tests on clinical 4D CT-heart data with ECG-gated exposure has resulted in a significant reduction of the noise level and an increased detail compared to 2D and 3D methods. Another promising feature is that the reconstruction induced streak artifacts which generally occur in low dose CT are remarkably reduced in 4D.

Tensors and tensor fields are commonly used in multidimensional signal processing to represent the local structure of the signal. This paper focuses on the case where the sampling on the original signal is anisotropic, e.g when the resolution of the multidimensional image varies depending on the direction which is common e.g. in medical imaging devices. To obtain a geometrically correct description of the local structure there are mainly two possibilities. To resample the image prior to the computation of the local structure tensor field or to compute the tensor field on the original grid and transform the result to obtain a correct geometry of the local structure. This paper deals with the latter alternative and contains an in depth theoretical analysis establishing the appropriate rules for tensor transformations induced by changes in space-time geometry with emphasis on velocity and motion estimation.

A method of imaging a blood vessel in a body using X-rays and an injectable contrast medium is described. The contrast medium is injected into the body, and signals constituted by an X-ray image sequence depicting X-ray attenuation values is recorded. The X-ray attenuated values in each spaced-time neighborhood are combined in a way that is dependent on the processed image sequence and separately established for each neighborhood, and separating, from background and vessel signals, flow signals having energy contributions mainly in an area of frequency domain bounded by surfaces corresponding to threshold velocities separately established for each neighborhood, which surfaces are shifted a specified amount along a temporal frequency axis.

The aim of this medical image science project is to increase patient safety in terms of improved image quality and reduced exposure to ionizing radiation in CT. The means to achieve these goals is to develop and evaluate an efficient adaptive filtering (denoising/image enhancement) method that fully explores true 4D image acquisition modes. Four-dimensional (4D) medical image data are captured as a time sequence of image volumes. During 4D image acquisition, a 3D image of the patient is recorded at regular time intervals. The resulting data will consequently have three spatial dimensions and one temporal dimension. Increasing the dimensionality of the data impose a major increase the computational demands. The initial linear filtering which is the cornerstone in all adaptive image enhancement algorithms increase exponentially with the dimensionality. On the other hand the potential gain in Signal to Noise Ratio (SNR) also increase exponentially with the dimensionality. This means that the same gain in noise reduction that can be attained by performing the adaptive filtering in 3D as opposed to 2D can be expected to occur once more by moving from 3D to 4D. The initial tests on on both synthetic and clinical 4D images has resulted in a significant reduction of the noise level and an increased detail compared to 2D and 3D methods. When tuning the parameters for adaptive filtering is extremely important to attain maximal diagnostic value which not necessarily coincide with an an eye pleasing image for a layman. Although this application focus on CT the resulting adaptive filtering methods will be beneficial for a wide range of 3D/4D medical imaging modalities e.g. shorter acquisition time in MRI and improved elimination of noise in 3D or 4D ultrasound datasets.

Three-dimensional data processing is becoming more and more common. Typical operations are for example estimation of optical flow in video sequences and orientation estimation in 3-D MR images. This paper proposes an efficient approach to robust low level feature extraction for 3-D image analysis. In contrast to many earlier algorithms the methods proposed in this paper support the use of relatively complex models at the initial processing steps. The aim of this approach is to provide the means to handle complex events at the initial processing steps and to enable reliable estimates in the presence of noise. A limited basis filter set is proposed which forms a basis on the unit sphere and is related to spherical harmonics. From these basis filters, different types of orientation selective filters are synthesized. An interpolation scheme that provides a rotation as well as a translation of the synthesized filter is presented. The purpose is to obtain a robust and invariant feature extraction at a manageable computational cost.

Three-dimensional data processing is becoming more and more common. Typical operations are for example estimation of optical flow in video sequences and orientation estimation in 3-D MR images. This paper proposes an efficient approach to robust low level feature extraction for 3-D image analysis. In contrast to many earlier algorithms the methods proposed in this paper support the use of relatively complex models at the initial processing steps. The aim of this approach is to provide the means to handle complex events at the initial processing steps and to enable reliable estimates in the presence of noise. A limited basis filter set is proposed which forms a basis on the unit sphere and is related to spherical harmonics. From these basis filters, different types of orientation selective filters are synthesized. An interpolation scheme that provides a rotation as well as a translation of the synthesized filter is presented. The purpose is to obtain a robust and invariant feature extraction at a manageable computational cost.

This paper describes a new algorithm for local orientation estimation. The proposed algorithm detects and separates interfering events in ambiguous neighbourhoods and produces robust estimates of the two most dominant events. A representation suitable for simultaneous representation of two orientations is introduced. The main purpose of this representation is to make averaging of outputs for neigbourhoods containing two orientations possible. The feature extraction is performed by a set of quadrature filters. A method to obtain a large set of quadrature filter responses from a limited basis filter set is introduced. The estimation of the neighbourhood and the separation of the present events are based upon the quadrature responses in terms of local magnitude and phase. The performance of the algorithm is demonstrated using test images.

This paper presents a new and efficient approach for optimization and implementation of filter banks e.g. velocity channels, orientation channels and scale spaces. The multi layered structure of a filter network enable a powerful decomposition of complex filters into simple filter components and the intermediary results may contribute to several output nodes. Compared to a direct implementation a filter network uses only a fraction of the coefficients to provide the same result. The optimization procedure is recursive and all filters on each level are optimized simultaneously. The individual filters of the network, in general, contain very few non-zero coefficients, but there are are no restrictions on the spatial position of the coefficients, they may e.g. be concentrated on a line or be sparsely scattered. An efficient implementation of a quadrature filter hierarchy for generic purposes using sparse filter components is presented.

A recursive method to condense general multidimensional FIR-filters into a sequence of simple kernels with mainly one dimensional extent has been worked out. Convolver networks adopted for 2, 3 and 4D signals is presented and the performance is illustrated for spherically separable quadrature filters. The resulting filter responses are mapped to a non biased tensor representation where the local tensor constitutes a robust estimate of both the shape and the orientation (velocity) of the neighbourhood. A qualitative evaluation of this General Sequential Filter concept results in no detectable loss in accuracy when compared to conventional FIR (Finite Impulse Response) filters but the computational complexity is reduced several orders in magnitude. For the examples presented in this paper the attained speed-up is 5, 25 and 300 times for 2D, 3D and 4D data respectively The magnitude of the attained speed-up implies that complex spatio-temporal analysis can be performed using standard hardware, such as a powerful workstation, in close to real time. Due to the soft implementation of the convolver and the tree structure of the sequential filtering approach the processing is simple to reconfigure for the outer as well as the inner (vector length) dimensionality of the signal. The implementation was made in AVS (Application Visualization System) using modules written in C.

A computationally efficient data-driven method for exploratory analysis of functional MRI data is presented. The basic idea is to reveal underlying components in the fMRI data that have maximum autocorrelation. The tool for accomplishing this task is Canonical Correlation Analysis. The proposed method is more robust and much more computationally efficient than independent component analysis, which previously has been applied in fMRI.

A robust, general and computationally simple reinforcement learning system is presented. It uses a channel representation which is robust and continuous. The accumulated knowledge is represented as a reward prediction function in the outer product space of the input- and output channel vectors. Each computational unit generates an output simply by a vector-matrix multiplication and the response can therefore be calculated fast. The response and a prediction of the reward are calculated simultaneously by the same system, which makes TD-methods easy to implement if needed. Several units can cooperate to solve more complicated problems. A dynamic tree structure of linear units is grown in order to divide the knowledge space into a sufficiently number of regions in which the reward function can be properly described. The tree continuously tests split- and prune criteria in order to adapt its size to the complexity of the problem.

This paper presents a novel algorithm that uses CCA and phase analysis to detect the disparity in stereo images. The algorithm adapts filters in each local neighbourhood of the image in a way which maximizes the correlation between the filtered images. The adapted filters are then analysed to find the disparity. This is done by a simple phase analysis of the scalar product of the filters. The algorithm can even handle cases where the images have different scales. The algorithm can also handle depth discontinuities and give multiple depth estimates for semitransparent images.

This paper presents a novel algorithm that uses CCA and phase analysis to detect the disparity in stereo images. The algorithm adapts filters in each local neighbourhood of the image in a way which maximizes the correlation between the filtered images. The adapted filters are then analyzed to find the disparity. This is done by a simple phase analysis of the scalar product of the filters. The algorithm can even handle cases where the images have different scales. The algorithm can also handle depth discontinuities and give multiple depth estimates for semi-transparent images.

A stereo algorithm that can estimate multiple depths in semi-transparent images is presented. The algorithm is based on a combination of phase analysis and canonical correlation analysis. The algorithm adapts filters in each local neighbourhood of the image in a way which maximizes the correlation between the filtered images. The adapted filters are then analysed to find the disparity. This is done by a simple phase analysis of the scalar product of the filters. For images with different but constant depths, a simple reconstruction procedure is suggested.

This paper presents a general strategy for designing efficient visual operators. The approach is highly task oriented and what constitutes the relevant information is defined by a set of examples. The examples are pairs of images displaying a strong dependence in the chosen feature but are otherwise independent. Particularly important concepts in the work are mutual information and canonical correlation. Visual operators learned from examples are presented, e.g. local shift invariant orientation operators and image content invariant disparity operators. Interesting similarities to biological vision functions are observed.

This paper presents a novel algorithm for analysis of stochastic processes. The algorithm can be used to find the required solutions in the cases of principal component analysis (PCA), partial least squares (PLS), canonical correlation analysis (CCA) or multiple linear regression (MLR). The algorithm is iterative and sequential in its structure and uses on-line stochastic approximation to reach an equilibrium point. A quotient between two quadratic forms is used as an energy function and it is shown that the equilibrium points constitute solutions to the generalized eigenproblem.

We present a method that finds edges between certain image features, e.g. gray-levels, and disregards edges between other features. The method uses a channel representation of the features and performs normalized convolution using the channel values as certainties. This means that areas with certain features can be disregarded by the edge filter. The method provides an important tool for finding tissue specific edges in medical images, as demonstrated by an MR-image example

We present a novel method that finds edges between certain image features, e.g. gray-levels, and disregards edges between other features. The method uses a channel representation of the features and performs normalized convolution using the channel values as certainties. This means that areas with certain features can be disregarded by the edge filter. The method provides an important new tool for finding tissue specific edges in medical images, as demonstrated by an MR-image example.

The Riemannian exponential map, and its inverse the Riemannian logarithm map, can be used to visualize metric tensor fields. In this chapter we first derive the well-known metric sphere glyph from the geodesic equations, where the tensor field to be visualized is regarded as the metric of a manifold. These glyphs capture the appearance of the tensors relative to the coordinate system of the human observer. We then introduce two new concepts for metric tensor field visualization: geodesic spheres and geodesically warped glyphs. These additions make it possible not only to visualize tensor anisotropy, but also the curvature and change in tensorshape in a local neighborhood. The framework is based on the exp maps, which can be computed by solving a second order Ordinary Differential Equation (ODE) or by manipulating the geodesic distance function. The latter can be found by solving the eikonal equation, a non-linear Partial Differential Equation (PDE), or it can be derived analytically for some manifolds. To avoid heavy calculations, we also include first and second order Taylor approximations to exp and log. In our experiments, these are shown to be sufficiently accurate to produce glyphs that visually characterize anisotropy, curvature and shape-derivatives in smooth tensor fields. 

In this paper we present a framework for unsupervised segmentation of white matter fiber traces obtained from diffusion weighted MRI data. Fiber traces are compared pairwise to create a weighted undirected graph which is partitioned into coherent sets using the normalized cut (Ncut) criterion. A simple and yet effective method for pairwise comparison of fiber traces is presented which in combination with the Ncut criterion is shown to produce plausible segmentations of both synthetic and real fiber trace data. Segmentations are visualized as colored stream-tubes or transformed to a segmentation of voxel space, revealing structures in a way that looks promising for future explorative studies of diffusion weighted MRI data.

This chapter describes a framework for storage of tensor array data, useful to describe regularly sampled tensor fields. The main component of the framework, called Similar Tensor Array Core (STAC), is the result of a collaboration between research groups within the SIMILAR network of excellence. It aims to capture the essence of regularly sampled tensor fields using a minimal set of attributes and can therefore be used as a “greatest common divisor” and interface between tensor array processing algorithms. This is potentially useful in applied fields like medical image analysis, in particular in Diffusion Tensor MRI, where misinterpretation of tensor array data is a common source of errors. By promoting a strictly geometric perspective on tensor arrays, with a close resemblance to the terminology used in differential geometry, (STAC) removes ambiguities and guides the user to define all necessary information. In contrast to existing tensor array file formats, it is minimalistic and based on an intrinsic and geometric interpretation of the array itself, without references to other coordinate systems.

Imagine an ant walking around on the curved surface of a plant, a radio amateur planning to broadcast to a distant location across the globe or a pilot taking o from an airport - all of them are helped by egocentric maps of the world around them that shows directions and distances to various remote places. It is not surprising that this idea has already been used in cartography, where it is known as Azimuthal Equidistant Projection (AEP). If Earth is approximated by a sphere, distances and directions between two places are computed from arcs along great circles. In physics and mathematics, the same idea is known as Riemannian Normal Coordinates (RNC). It has been given a precise and general denition for surfaces (2-D), curved spaces (3-D) and generalized to smooth manifolds (N-D). RNC are the Cartesian coordinates of vectors that index points on the surface (or manifold) through the so called exponential map, which is a well known concept in dierential geometry. They are easily computed for a particular point if the inverse of the exponential map, the logarithm map, is known. Recently, RNC and similar coordinate systems have been used in computer graphics, visualization and related areas of research. In Fig. 1 for instance, RNC are used to produce a texture on the Stanford bunny through decal compositing. Given the growing use of RNC, which is further elaborated on in the next section, it is meaningful to develop accurate and reproducible techniques to compute this parameterization. In this paper, we describe a technique to compute RNC for surfaces represented by triangular meshes, which is the predominant representation of surfaces in computer graphics. The method that we propose has similarities to the Logmap framework, which has previously been developed for dimension reduction of unorganized point clouds in high-dimensional spaces, a.k.a. manifold learning. For this reason we sometimes refer to it as "Logmap for triangular meshes" or simply Logmap.

We propose a novel post processing method for visualization of fiber traces from DT-MRI data. Using a recently proposed non-linear dimensionality reduction technique, Laplacian eigenmaps [3], we create a mapping from a set of fiber traces to a low dimensional Euclidean space. Laplacian eigenmaps constructs this mapping so that similar traces are mapped to similar points, given a custom made pairwise similarity measure for fiber traces. We demonstrate that when the low-dimensional space is the RGB color space, this can be used to visualize fiber traces in a way which enhances the perception of fiber bundles and connectivity in the human brain.

We present a one-pass framework for filtering vector-valued images and unordered sets of data points in an N-dimensional feature space. It is based on a local Bayesian framework, previously developed for scalar images, where estimates are computed using expectation values and histograms. In this paper we extended this framework to handle N-dimensional data. To avoid the curse of dimensionality, it uses importance sampling instead of histograms to represent probability density functions. In this novel computational framework we are able to efficiently filter both vector-valued images and data, similar to e.g. the well-known bilateral, median and mean shift filters.

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.

We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra’s algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.

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