Automatic artery/vein (A/V) classification is one of the important topics in retinal image analysis. It allows the researchers to investigate the association between biomarkers and disease progression on a huge amount of data for arteries and veins separately. Recent proposed methods, which employ contextual information of vessels to achieve better A/V classification accuracy, still rely on the performance of pixel-wise classification, which has received limited attention in recent years. In this paper, we show that these classification methods can be markedly improved. We propose a new normalization technique for extracting four new features which are associated with the lightness reflection of vessels. The accuracy of a linear discriminate analysis classifier is used to validate these features. Accuracy rates of 85.1, 86.9 and 90.6% were obtained on three datasets using only local information. Based on the introduced features, the advanced graph-based methods will achieve a better performance on A/V classification.

We propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations. Additionally, we propose a generalized linear regression framework for the construction of suitable templates using smoothing splines. Here, it is important to recognize a curved geometry on the position-orientation domain, which we identify with the Lie group SE(2): the roto-translation group. Templates are then optimized in a B-spline basis, and smoothness is defined with respect to the curved geometry. We achieve state-of-the-art results on three different applications: detection of the optic nerve head in the retina (99.83 percent success rate on 1,737 images), of the fovea in the retina (99.32 percent success rate on 1,616 images), and of the pupil in regular camera images (95.86 percent on 1,521 images). The high performance is due to inclusion of both intensity and orientation features with effective geometric priors in the template matching. Moreover, our method is fast due to a cross-correlation based matching approach.

Tree-like structures, such as retinal images, are widely studied in computer-aided diagnosis systems for large-scale screening programs. Despite several segmentation and tracking methods proposed in the literature, there still exist several limitations specifically when two or more curvilinear structures cross or bifurcate, or in the presence of interrupted lines or highly curved blood vessels. In this paper, we propose a novel approach based on multi-orientation scores augmented with a contextual affinity matrix, which both are inspired by the geometry of the primary visual cortex (V1) and their contextual connections. The connectivity is described with a 5D kernel obtained as the fundamental solution of the Fokker-Planck equation modeling the cortical connectivity in the lifted space of positions, orientations, curvatures, and intensity. It is further used in a self-tuning spectral clustering step to identify the main perceptual units in the stimuli. The proposed method has been validated on several easy as well as challenging structures in a set of artificial images and actual retinal patches. Supported by quantitative and qualitative results, the method is capable of overcoming the limitations of current state-of-the-art techniques.

Retinal microaneurysms (MAs) are the earliest clinical sign of diabetic retinopathy disease. Detection of MAs is crucial for the early diagnosis of diabetic retinopathy and prevention of blindness. In this paper, a novel and reliable method for automatic detection of MAs in retinal images is proposed. In the first stage of the proposed method, several preliminary microaneurysm candidates are extracted using a gradient weighting technique and an iterative thresholding approach. In the next stage, in addition to intensity and shape descriptors, a new set of features based on local convergence index filters is extracted for each candidate. Finally, the collective set of features is fed to a hybrid sampling/boosting classifier to discriminate the MAs from non-MAs candidates. The method is evaluated on images with different resolutions and modalities (color and scanning laser ophthalmoscope) using six publicly available data sets including the retinopathy online challenges (ROC) data set. The proposed method achieves an average sensitivity score of 0.471 on the ROC data set outperforming state-of-the-art approaches in an extensive comparison. The experimental results on the other five data sets demonstrate the effectiveness and robustness of the proposed MAs detection method regardless of different image resolutions and modalities.

Retinal image analysis is a challenging problem due to the precise quantification required and the huge numbers of images produced in screening programs. This paper describes a series of innovative brain-inspired algorithms for automated retinal image analysis, recently developed for the RetinaCheck project, a large-scale screening program for diabetic retinopathy and other retinal diseases in Northeast China. The paper discusses the theory of orientation scores, inspired by cortical multi-orientation pinwheel structures, and presents applications for automated quality assessment, optic nerve head detection, crossing-preserving enhancement and segmentation of retinal vasculature, arterio-venous ratio, fractal dimension, and vessel tortuosity and bifurcations. Many of these algorithms outperform state-of-the-art techniques. The methods are currently validated in collaborating hospitals, with a rich accompanying base of metadata, to phenotype and validate the quantitative algorithms for optimal classification power.

This paper presents a robust and fully automatic filter-based approach for retinal vessel segmentation. We propose new filters based on 3D rotating frames in so-called orientation scores, which are functions on the Lie-group domain of positions and orientations [Formula: see text]. By means of a wavelet-type transform, a 2D image is lifted to a 3D orientation score, where elongated structures are disentangled into their corresponding orientation planes. In the lifted domain [Formula: see text], vessels are enhanced by means of multi-scale second-order Gaussian derivatives perpendicular to the line structures. More precisely, we use a left-invariant rotating derivative (LID) frame, and a locally adaptive derivative (LAD) frame. The LAD is adaptive to the local line structures and is found by eigensystem analysis of the left-invariant Hessian matrix (computed with the LID). After multi-scale filtering via the LID or LAD in the orientation score domain, the results are projected back to the 2D image plane giving us the enhanced vessels. Then a binary segmentation is obtained through thresholding. The proposed methods are validated on six retinal image datasets with different image types, on which competitive segmentation performances are achieved. In particular, the proposed algorithm of applying the LAD filter on orientation scores (LAD-OS) outperforms most of the state-of-the-art methods. The LAD-OS is capable of dealing with typically difficult cases like crossings, central arterial reflex, closely parallel and tiny vessels. The high computational speed of the proposed methods allows processing of large datasets in a screening setting.

Diffusion Tensor Imaging (DTI) and fiber tracking provide unique insight into the 3D structure of fibrous tissues in the brain. However, the output of fiber tracking contains a significant amount of uncertainty accumulated in the various steps of the processing pipeline. Existing DTI visualization methods do not present these uncertainties to the end-user. This creates a false impression of precision and accuracy that can have serious consequences in applications that rely heavily on risk assessment and decision-making, such as neurosurgery. On the other hand, adding uncertainty to an already complex visualization can easily lead to information overload and visual clutter. In this work, we propose Illustrative Confidence Intervals to reduce the complexity of the visualization and present only those aspects of uncertainty that are of interest to the user. We look specifically at the uncertainty in fiber shape due to noise and modeling errors. To demonstrate the flexibility of our framework, we compute this uncertainty in two different ways, based on (1) fiber distance and (2) the probability of a fiber connection between two brain regions. We provide the user with interactive tools to define multiple confidence intervals, specify visual styles and explore the uncertainty with a Focus+Context approach. Finally, we have conducted a user evaluation with three neurosurgeons to evaluate the added value of our visualization.

Abstract Advancements in the acquisition and modeling of flow fields result in unsteady volumetric flow fields of unprecedented quality. An important example is found in the analysis of unsteady blood‐flow data. Preclinical research strives for a better understanding of correlations between the hemodynamics and the progression of cardiovascular diseases. Modern‐day computer models and MRI acquisition provide time‐resolved volumetric blood‐flow velocity fields. Unfortunately, these fields often remain unexplored, as high‐dimensional data are difficult to conceive. We present a spatiotemporal, i.e., four‐dimensional, hierarchical clustering, yielding a sparse representation of the velocity data. The clustering results underpin an illustrative visualization approach, facilitating visual analysis. The hierarchy allows an intuitive level‐of‐detail selection, largely retaining important flow patterns. The clustering employs dissimilarity measures to construct the hierarchy. We have adapted two existing measures for steady vector fields for use in the spacetime domain. Because of the inherent computational complexity of the multidimensional clustering, we introduce a coarse hierarchical clustering approach, which closely approximates the full hierarchy generation, and considerably improves the performance. The resulting clusters are visualized by representative patharrows, in combination with an illustrative anatomical context. We present various seeding approaches and visualization styles, providing sparse overviews of the unsteady behavior of volumetric flow fields.

Diffusion tensor imaging (DTI) is a magnetic resonance technique used to explore anatomical fibrous structures, like brain white matter. Fiber-tracking methods use the diffusion tensor (DT) field to reconstruct the corresponding fibrous structure. A group of fiber-tracking methods trace geodesics on a Riemannian manifold whose metric is defined as a function of the DT. These methods are more robust to noise than more commonly used methods where just the main eigenvector of the DT is considered. Until now, geodesic-based methods were not able to resolve all geodesics, since they solved the Eikonal equation, and therefore were not able to deal with multivalued solutions. Our algorithm computes multivalued solutions using an Euler–Lagrange form of the geodesic equations. The multivalued solutions become relevant in regions with sharp anisotropy and complex geometries, or when the first arrival time does not describe the geodesic close to the anatomical fibrous structure. In this paper, we compare our algorithm with the commonly used Hamilton–Jacobi (HJ) equation approach. We describe and analyze the characteristics of both methods. In the analysis we show that in cases where, e.g., U-shaped bundles appear, our algorithm can capture the underlying fiber structure, while other approaches will fail. A feasibility study with results for synthetic and real data is shown.

Abstract-Any sufficiently smooth, positive, real-valued function ψ : S(2) → K+ on a sphere S(2) can be expanded by a Laplace expansion into a sum of spherical harmonics. Given the Laplace expansion coefficients, we provide a CPU and GPU-based algorithm that renders the radial graph of ψ in a fast and efficient way by ray-casting the glyph of ψ in the fragment shader of a GPU. The proposed rendering algorithm has proven highly useful in the visualization of high angular resolution diffusion imaging (HARDI) data. Our implementation of the rendering algorithm can display simultaneously thousands of glyphs depicting the local diffusivity of water. The rendering is fast enough to allow for interactive manipulation of large HARDI data sets.

Better understanding of hemodynamics conceivably leads to improved diagnosis and prognosis of cardiovascular diseases. Therefore, an elaborate analysis of the blood-flow in heart and thoracic arteries is essential. Contemporary MRI techniques enable acquisition of quantitative time-resolved flow information, resulting in 4D velocity fields that capture the blood-flow behavior. Visual exploration of these fields provides comprehensive insight into the unsteady blood-flow behavior, and precedes a quantitative analysis of additional blood-flow parameters. The complete inspection requires accurate segmentation of anatomical structures, encompassing a time-consuming and hard-to-automate process, especially for malformed morphologies. We present a way to avoid the laborious segmentation process in case of qualitative inspection, by introducing an interactive virtual probe. This probe is positioned semi-automatically within the blood-flow field, and serves as a navigational object for visual exploration. The difficult task of determining position and orientation along the view-direction is automated by a fitting approach, aligning the probe with the orientations of the velocity field. The aligned probe provides an interactive seeding basis for various flow visualization approaches. We demonstrate illustration-inspired particles, integral lines and integral surfaces, conveying distinct characteristics of the unsteady blood-flow. Lastly, we present the results of an evaluation with domain experts, valuing the practical use of our probe and flow visualization techniques.

High-angular resolution diffusion imaging (HARDI) is a diffusion weighted MRI technique that overcomes some of the decisive limitations of its predecessor, diffusion tensor imaging (DTI), in the areas of composite nerve fiber structure. Despite its advantages, HARDI raises several issues: complex modeling of the data, nonintuitive and computationally demanding visualization, inability to interactively explore and transform the data, etc. To overcome these drawbacks, we present a novel, multifield visualization framework that adopts the benefits of both DTI and HARDI. By applying a classification scheme based on HARDI anisotropy measures, the most suitable model per imaging voxel is automatically chosen. This classification allows simplification of the data in areas with single fiber bundle coherence. To accomplish fast and interactive visualization for both HARDI and DTI modalities, we exploit the capabilities of modern GPUs for glyph rendering and adopt DTI fiber tracking in suitable regions. The resulting framework, allows user-friendly data exploration of fused HARDI and DTI data. Many incorporated features such as sharpening, normalization, maxima enhancement and different types of color coding of the HARDI glyphs, simplify the data and enhance its features. We provide a qualitative user evaluation that shows the potentials of our visualization tools in several HARDI applications.

Insight into the dynamics of blood-flow considerably improves the understanding of the complex cardiovascular system and its pathologies. Advances in MRI technology enable acquisition of 4D blood-flow data, providing quantitative blood-flow velocities over time. The currently typical slice-by-slice analysis requires a full mental reconstruction of the unsteady blood-flow field, which is a tedious and highly challenging task, even for skilled physicians. We endeavor to alleviate this task by means of comprehensive visualization and interaction techniques. In this paper we present a framework for pre-clinical cardiovascular research, providing tools to both interactively explore the 4D blood-flow data and depict the essential blood-flow characteristics. The framework encompasses a variety of visualization styles, comprising illustrative techniques as well as improved methods from the established field of flow visualization. Each of the incorporated styles, including exploded planar reformats, flow-direction highlights, and arrow-trails, locally captures the blood-flow dynamics and may be initiated by an interactively probed vessel cross-section. Additionally, we present the results of an evaluation with domain experts, measuring the value of each of the visualization styles and related rendering parameters.

High angular resolution diffusion imaging (HARDI) is an emerging magnetic resonance imaging (MRI) technique that overcomes some decisive limitations of its predecessor diffusion tensor imaging (DTI). HARDI can resolve locally more than one direction in the diffusion pattern of water molecules and thereby opens up the opportunity to display and track crossing fibers. Showing the local structure of the reconstructed, angular probability profiles in a fast, detailed, and interactive way can improve the quality of the research in this area and help to move it into clinical application. In this paper we present a novel approach for HARDI glyph visualization or, more generally, for the visualization of any function that resides on a sphere and that can be expressed by a Laplace series. Our GPU-accelerated glyph rendering improves the performance of the traditional way of HARDI glyph visualization as well as the visual quality of the reconstructed data, thus offering interactive HARDI data exploration of the local structure of the white brain matter in-vivo. In this paper we exploit the capabilities of modern GPUs to overcome the large, processor-intensive and memory-consuming data visualization.

Abstract The heart consists of densely packed muscle fibres. The orientation of these fibres can be acquired by using Diffusion Tensor Imaging (DTI) ex vivo. A good way to visualize the fibre structure in a cross section of the heart is by showing short line segments originating from the cross section and aligned with the local direction of the fibres. If the line segments are placed dense enough, one can see how the fibre orientations change. However, generation of the line segments takes time and thus the user has to wait for new geometry to be generated when the plane defining the cross section is changed. We present a new direct rendering method for the visualization of the 3D vector field in a 2D user‐definable cross section of a heart. On the intersection of the plane with the vector field, the full 3D vectors are rendered as 3D line segments with a local ray casting approach. No preprocessing of the data is needed and no geometry is generated. This technique allows a fast inspection of the data to identify interesting areas where further analysis is necessary (e.g. quantification or generation of streamlines). We also show how the technique is generalized to other glyph shapes than line segments by implementing ellipsoids.

Fiber tracking of Diffusion Tensor Imaging (DTI) data offers a unique insight into the three-dimensional organisation of white matter structures in the living brain. However, fiber tracking algorithms require a number of user-defined input parameters that strongly affect the output results. Usually the fiber tracking parameters are set once and are then re-used for several patient datasets. However, the stability of the chosen parameters is not evaluated and a small change in the parameter values can give very different results. The user remains completely unaware of such effects. Furthermore, it is difficult to reproduce output results between different users. We propose a visualization tool that allows the user to visually explore how small variations in parameter values affect the output of fiber tracking. With this knowledge the user cannot only assess the stability of commonly used parameter values but also evaluate in a more reliable way the output results between different patients. Existing tools do not provide such information. A small user evaluation of our tool has been done to show the potential of the technique.

In this paper we address the problem of object retrieval based on scale-space interest points, namely top-points. The original retrieval algorithm can only cope with scale-Euclidean transformations. We extend the algorithm to the case of non-rigid transformations like affine and perspective transformations and investigate its robustness. The proposed algorithm is proven to be highly robust under various degrading factors, such as noise, occlusion, rendering artifacts, etc. and can deal with multiple occurrences of the object.

In this paper we describe a method for estimating curvature of elongated structures in images. The curvature estimation is performed on an invertible orientation score, which is a 3D entity obtained from a 2D image by convolution with a rotating kernel. By considering the group structure we can define left-invariant derivatives, which are essential to construct operations on the orientation score that amount to rotationally invariant operations on the corresponding image. The problem of estimating curvature of an oriented structure is stated as a minimization problem, which can be solved by eigenvector analysis of a matrix constructed from the non-symmetric Hessian matrix. The experiments show the method performs well for a wide range of curvatures and noise levels. The method clearly outperforms a related curvature estimation method by Van Ginkel et al. that tends to give estimates that are too small. We show how we can incorporate the curvature estimate in our method for coherence-enhancing diffusion in orientation scores. This method has superior performance in enhancing crossing contours, which is demonstrated on medical images.

An active shape model segmentation scheme is presented that is steered by optimal local features, contrary to normalized first order derivative profiles, as in the original formulation [Cootes and Taylor, 1995, 1999, and 2001]. A nonlinear kNN-classifier is used, instead of the linear Mahalanobis distance, to find optimal displacements for landmarks. For each of the landmarks that describe the shape, at each resolution level taken into account during the segmentation optimization procedure, a distinct set of optimal features is determined. The selection of features is automatic, using the training images and sequential feature forward and backward selection. The new approach is tested on synthetic data and in four medical segmentation tasks: segmenting the right and left lung fields in a database of 230 chest radiographs, and segmenting the cerebellum and corpus callosum in a database of 90 slices from MRI brain images. In all cases, the new method produces significantly better results in terms of an overlap error measure (p < 0.001 using a paired T-test) than the original active shape model scheme.

A fully automatic method is presented to detect abnormalities in frontal chest radiographs which are aggregated into an overall abnormality score. The method is aimed at finding abnormal signs of a diffuse textural nature, such as they are encountered in mass chest screening against tuberculosis (TB). The scheme starts with automatic segmentation of the lung fields, using active shape models. The segmentation is used to subdivide the lung fields into overlapping regions of various sizes. Texture features are extracted from each region, using the moments of responses to a multiscale filter bank. Additional "difference features" are obtained by subtracting feature vectors from corresponding regions in the left and right lung fields. A separate training set is constructed for each region. All regions are classified by voting among the k nearest neighbors, with leave-one-out. Next, the classification results of each region are combined, using a weighted multiplier in which regions with higher classification reliability weigh more heavily. This produces an abnormality score for each image. The method is evaluated on two databases. The first database was collected from a TB mass chest screening program, from which 147 images with textural abnormalities and 241 normal images were selected. Although this database contains many subtle abnormalities, the classification has a sensitivity of 0.86 at a specificity of 0.50 and an area under the receiver operating characteristic (ROC) curve of 0.820. The second database consist of 100 normal images and 100 abnormal images with interstitial disease. For this database, the results were a sensitivity of 0.97 at a specificity of 0.90 and an area under the ROC curve of 0.986.

We propose a computational method for segmenting topological subdimensional point-sets in scalar images of arbitrary spatial dimensions. The technique is based on calculating the homotopy class defined by the gradient vector in a subdimensional neighborhood around every image point. This neighborhood is defined as the linear envelope spawned over a given subdimensional vector frame. In the simplest case where the rank of this frame is maximal, we obtain a technique for localizing the critical points. We consider, in particular, the important case of frames formed by an arbitrary number of the first largest by absolute value principal directions of the Hessian. The method then segments positive and and negative ridges as well as other types of critical surfaces of different dimensionalities. The signs of the eigenvalues associated to the principal directions provide a natural labeling of the critical subsets. The result, in general, is a constructive definition of a hierarchy of point-sets of different dimensionalities linked by inclusion relations. Because of its explicit computational nature, the method gives a fast way to segment height ridges or edges in different applications. The defined topological point-sets are connected manifolds and, therefore, our method provides a tool for geometrical grouping using only local measurements. We have demonstrated the grouping properties of our construction by presenting two different cases where an extra image coordinate is introduced.

The traditional chest radiograph is still ubiquitous in clinical practice, and will likely remain so for quite some time. Yet, its interpretation is notoriously difficult. This explains the continued interest in computer-aided diagnosis for chest radiography. The purpose of this survey is to categorize and briefly review the literature on computer analysis of chest images, which comprises over 150 papers published in the last 30 years. Remaining challenges are indicated and some directions for future research are given.

We propose a Bayesian grouping approach for recognition and segmentation of large-scale structures representing objects in images. It is based on detection of local image properties, extraction of simple geometrical primitives, and grouping these primitives according to probability rules and prior models. As opposed to the various template matching techniques, our method does not rely on a fixed set of input data to generate the prior with a maximum likelihood. Instead, it selects a list of subsets of the local primitives and finds the optimum set of model priors that maximizes the likelihood of the model samples representing the selected subsets. In contrast with global recognition methods that classify the whole image, our approach aims at solving the recognition task together with the segmentation task. As an illustration we give a medical data example of feature grouping in 2D images involving vessel detection from local ridges.

In direct area-based stereo problems occur if there is a significant local deformation between the two images, and at borders of objects. We propose a correlation method with affine deformable windows in which an optimal window size is selected using model-based reasoning. Two criteria for the window size selection are studied and the resulting algorithms are applied to a random dot pair, an autostereogram and Hoff's fruit pair. The results are found to be both accurate and robust.

Nonlinear diffusion filtering in image processing is usually performed with explicit schemes. They are only stable for very small time steps, which leads to poor efficiency and limits their practical use. Based on a discrete nonlinear diffusion scale-space framework we present semi-implicit schemes which are stable for all time steps. These novel schemes use an additive operator splitting (AOS), which guarantees equal treatment of all coordinate axes. They can be implemented easily in arbitrary dimensions, have good rotational invariance and reveal a computational complexity and memory requirement which is linear in the number of pixels. Examples demonstrate that, under typical accuracy requirements, AOS schemes are at least ten times more efficient than the widely used explicit schemes.

In this paper implicit representations of deformable models for medical image enhancement and segmentation are considered. The advantage of implicit models over classical explicit models is that their topology can be naturally adapted to objects in the scene. A geodesic formulation of implicit deformable models is especially attractive since it has the energy minimizing properties of classical models. The aim of this paper is twofold. First, a modification to the customary geodesic deformable model approach is introduced by considering all the level sets in the image as energy minimizing contours. This approach is used to segment multiple objects simultaneously and for enhancing and segmenting cardiac computed tomography (CT) and magnetic resonance images. Second, the approach is used to effectively compare implicit and explicit models for specific tasks. This shows the complementary character of implicit models since in case of poor contrast boundaries or gaps in boundaries e.g. due to partial volume effects, noise, or motion artifacts, they do not perform well, since the approach is completely data-driven.

A hierarchical segmentation model is obtained by using linear scale evolution of gray-scale images. At each scale segments are generated as Voronoi diagrams with a distance measure defined on the image landscape. The set of centers of the Voronoi cells is the set of local extrema of the gray-scale image. This set is localized by using the winding number distribution of the gradient vector field. Scale evolution induces hierarchical structure of embedded segments. Objects defined at coarser scales "decompose" into sub-objects at finer scales. The process is naturally described in terms of singularity catastrophes within the smooth scale evolution. Alternatively, we present a purely topological segmentation procedure, based on singular isophotes. The last are generated by the set of saddle points in the image which are detected also with the topological winding-number method.

In most cases nonlinear diffusion filtering is implemented by means of explicit finite difference schemes. These algorithms are not very efficient, since they are only stable for small time steps. We address this problem by presenting unconditionally stable semi-implicit schemes which are based on an additive operator splitting (AOS). They are very efficient since they can be implemented by recursive filtering, and their separability allows a straightforward implementation in any dimension. We analyse their behaviour on a parallel computer and demonstrate that parallel AOS schemes on a modern shared-memory multiprocessor system with 8 processors allow a speed-up of two orders of magnitude in comparison to the widely-used explicit scheme on a single processor.

In optic flow based velocity estimation the image brightness constraint equation is used. However, for measurements performed at a certain scale, the brightness constraint equation does not apply. We therefore use a recently developed approach which reconciles optic flow and scale space theory. It specifically incorporates the scale (aperture) of image measurements, leading to a scheme which is essentially different from existing approaches. To obtain a unique velocity field, the data-derived information has to be augmented with physical knowledge. By keeping a strict separation between data-derived and external information, we can locally adapt or modify the user-supplied information without affecting the image-derived information. The two free scale parameters in time and space can be used for attentive vision (selecting particular velocities or objects) and to improve the reliability of velocity estimates.

Many image processing applications require one to solve problems such as denoising with edge enhancement, preprocessing for segmentation, or the completion of interrupted lines. This may be accomplished by applying a suitable nonlinear anisotropic diffusion process to the image. Its diffusion tensor is adapted to the differential structure of the underlying image. Although being image enhancement tools, filters of this class are well-posed. The temporal evolution of the diffusion process creates a scale-space whose causality properties can be understood in a deterministic, stochastic, information theory based and Fourier based way. The well-posedness and scale-space results carry over to the discrete setting, which gives rise to reliable algorithms preserving all properties of the continuous framework.

In recent years several nonlinear diffusion schemes have been introduced. We discuss the numerical implementation of a number of current nonlinear evolution schemes, using the notion of well-posed differentiation by Gaussian kernels. The infinitesimal change of an image when increasing scale depends on the local differential invariants evaluated at the scale of the image considered, i.e. on terms of the local jet (the set of all spatial partial derivatives at that point). All these differential terms can be obtained in a well-posed fashion by a convolution of the original image with the family of the Gaussian and its derivatives. The nonlinear partial differential evaluation can thus be numerically approximated by an iterative calculation of the appropriate terms in the local jet. Examples are given for medical images.>

Differentiation is known to be ill-posed in the sense of Hadamard. The theory of regular tempered distributions and the concept of Gaussian convolution filters open the way to a well-posed differentiation process, thereby introducing the notion of scale (or: inverse resolution). There is no a priori fundamental limit to the order of differentiation of images provided they are calculated on a sufficiently high scale (relative to pixel scale and noise correlation width), and provided we have a sufficient dynamic range of intensity values. Constraints in resolution (both in the spatial and in the intensity domain) enforce a scale-dependent restriction to the accuracy with which Gaussian kernels G/sub n/(x; /spl sigma/) can be represented in a physical sense. So at a given scale /spl sigma/ (e.g. in units of the sampling scale) and a given measure of inaccuracy /spl alpha/ there is a maximal order n above which the margin /spl alpha/ is exceeded. In this paper we quantify this relation.>

Presents a method of multi-resolution image analysis that gives the most concise set of local orthogonally invariant features of 2-dimensional input images. Solving the equivalence problem corresponding to a local jet and the group of orthogonal transformations of the cartesian coordinate frame the authors find a complete and irreducible set of local algebraic invariants that may describe any local orthogonally invariant feature of the jet.>