Scale-Space Theories in Computer Vision

Blurring is not the only way to selectively remove fine spa- tial detail from an image. An alternative is to scramble pixels locally over areas defined by the desired blur circle. We refer to such scram- bled images as “locally disorderly”. Such images have many potentially interesting applications. In this contribution we discuss a formal frame- work for such locally disorderly images. It boils down to a number of intricately intertwined scale spaces, one of which is the ordinary linear scale space for the image. The formalism is constructed on the basis of an operational definition of local histograms of arbitrary bin width and arbitrary support

In a recent work [1], Koenderink and van Doorn consider a family of three intertwined scale-spaces coined the locally orderless image (LOI). The LOI represents the image, observed at inner scale σ, as a local histogram with bin-width β, at each location, with a Gaussian- shaped region of interest of extent α. LOIs form a natural and elegant extension of scale-space theory, show causal consistency and enable the smooth transition between pixels, histograms and isophotes. The aim of this work is to demonstrate the wide applicability and versatility of LOIs. We consider a range of image processing tasks, including variations of adaptive histogram equalization, several methods for noise and scratch removal, texture rendering, classification and segmentation.

Indexing large archives of historical manuscripts, like the pa- pers of George Washington, is required to allow rapid perusal by scholars and researchers who wish to consult the original manuscripts. Presently, such large archives are indexed manually. Since optical character recog- nition (OCR) works poorly with handwriting, a scheme based on match- ing word images called word spotting has been suggested previously for indexing such documents. The important steps in this scheme are seg- mentation of a document page into words and creation of lists containing instances of the same word by word image matching.We have developed a novel methodology for segmenting handwritten document images by analyzing the extent of “blobs” in a scale space representationof the image. We believe this is the first application of scale space to this problem. The algorithm has been applied to around 30 grey level images randomly picked from different sections of the George Washington corpus of 6,400 handwritten document images. An accuracy of 77 – 96 percent was observed with an average accuracy of around 87 percent. The algorithm works well in the presence of noise, shine through and other artifacts which may arise due aging and degradation of the page over a couple of centuries or through the man made processes of photocopying and scanning.

We use an unconditionally stable numerical scheme to im- plement a fast version of the geodesic active contour model. The proposed scheme is useful for object segmentation in images, like tracking moving objects in a sequence of images. The method is based on the Weickert- Romeney-Viergever [33] AOS scheme. It is applied at small regions, mo- tivated by Adalsteinsson-Sethian [1] level set narrow band approach, and uses Sethian’s fast marching method [26] for re-initialization. Experimen- tal results demonstrate the power of the new method for tracking in color movies.

A method for deforming curves in a given image to a desired position in a second image is introduced in this paper. The algorithm is based on deforming the first image toward the second one via a partial differential equation, while tracking the deformation of the curves of in- terest in the first image with an additional, coupled, partial differential equation. The tracking is performed by projecting the velocities of the first equation into the second one. In contrast with previous PDE based approaches, both the images and the curves on the frames/slices of inter- est are used for tracking. The technique can be applied to object tracking and sequential segmentation. The topology of the deforming curve can change, without any special topology handling procedures added to the scheme. This permits for example the automatic tracking of scenes where, due to occlusions, the topology of the objects of interest changes from frame to frame. In addition, this work introduces the concept of project- ing velocities to obtain systems of coupled partial differential equations for image analysis applications. We show examples for object tracking and segmentation of electronic microscopy. We also briefly discuss pos- sible uses of this framework îîfor three dimensional morphing.

Level set methods provide a robust way to implement ge- ometric flows, but they suffer from two problems which are relevant when using smoothing flows to unfold the cortex: the lack of point- correspondence between scales and the inability to implement tangential velocities. In this paper, we suggest to solve these problems by driving the nodes of a mesh with an ordinary differential equation. We state that this approach does not suffer from the known problems of Lagrangian methods since all geometrical properties are computed on the fixed (Eu- lerian) grid. Additionally, tangential velocities can be given to the nodes, allowing the mesh to follow general evolution equations, which could be crucial to achieving the final goal of minimizing local metric distortions. To experiment with this approach, we derive area and volume preserv- ing mean curvature flows and use them to unfold surfaces extracted from MRI data of the human brain.

This paper is concerned with the simulation of the Par- tial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natu- ral choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practi- cal application of the level set method is plagued with such questions as when do we have to “reinitialize” the distance function? How do we “reinitialize” the distance function? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an al- ternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduc- tion of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in two applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces [26], (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface.

Properties of points in images are often measured using con- volution integrals with each convolution kernel associated to a particular scale and perhaps to other parameters, such as an orientation, as well. Assigning to each point the parameter values that yield the maximum value of the convolution integral gives a map from points in the image to the space of parameters by which the given property is measured. The range of this map is the optimal parameter surface. In this paper, we argue that ridge points for the measured quantity are best computed via the pullback metric from the optimal parameter surface. A relatively simple kernel used to measure the property of medialness is explored in detail. For this example, we discuss connectivity of the optimal pa- rameter surface and the possibility of more than one critical scale for medialness at a given point. We demonstrate that medial loci computed as ridges of medialness are in general agreement with the Blum medial axis.

The maximal convexity ridge is not well suited for the anal- ysis of medial functions or, it can be argued, for the analysis of any func- tion that is created via convolution with a kernel based on the Gaussian. In its place one should use the maximal scale ridge, which takes scale’s distinguished role into account. We present the local geometric structure of the maximal scale ridge of smooth and Gaussian blurred functions, a result that complements recent work on scale selection. We also discuss the subdimensional maxima property as it relates to the maximal scale ridge, and we prove that a generalized maximal parameter ridge has the subdimensional maxima property as well.

In this paper we investigate scale space based structural grouping in images. Our strategy is to detect (relative) critical point sets in scale space, which we consider as an extended image representa- tion. In this way the multi-scale behavior of the original image structures is taken into account and automatic scale space grouping and scale se- lection is possible. We review a constructive and efficient topologically based method to detect the (relative) critical points. The method is pre- sented for arbitrary dimensions. Relative critical point sets in a Hessian vector frame provide us with a generalization of height ridges. Auto- matic scale selection is accomplished by a proper reparameterization of the scale axis. As the relative critical sets are in general connected sub- manifolds, it provides a robust method for perceptual grouping with only local measurements.

This paper shows how the performance of feature trackers can be improved by building a view-based object representation consist- ing of qualitative relations between image structures at different scales. The idea is to track all image features individually, and to use the qual- itative feature relations for resolving ambiguous matches and for intro- ducing feature hypotheses whenever image features are mismatched or lost. Compared to more traditional work on view-based object tracking, this methodology has the ability to handle semi-rigid objects and par- tial occlusions. Compared to trackers based on three-dimensional object models, this approach is much simpler and of a more generic nature. A hands-on example is presented showing how an integrated application system can be constructed from conceptually very simple operations.

The method of curve evolution is a popular method for recovering shape boundaries. However isotropic metrics have always been used to induce the flow of the curve and potential steady states tend to be difficult to determine numerically, especially in noisy or low-contrast situations. Initial curves shrink past the steady state and soon vanish. In this paper, anisotropic metrics are considered which remedy the situation by taking the orientation of the feature gradient into account. The problem of shape recovery or segmentation is formulated as the problem of finding minimum cuts of a Riemannian manifold. Approximate methods, namely anisotropic geodesic flows and solution of an eigenvalue problem are discussed.

In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gra- dient. The model is a combination between more classical active contour models using mean curvature motion techniques, and the Mumford-Shah model for segmentation. We minimize an energy which can be seen as a particular case of the so-called minimal partition problem. In the level set formulation, the problem becomes a “mean-curvature flow”-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable.

Multiscale segmentation respectful of the visual perception is an important issue of Computer Vision. We present an image model derived from the level sets representation which offers most of the prop- erties sought to a good segmentation: the borders are located at the per- ceptual edges; they are invariant by affine map and by contrast change; they are sorted according to their perceptual significance using a scale parameter. At last, a compact version of this model has been developed to be used in a progressive, and artifact-free, image compression scheme.

It is well known that a conveniently rescaled iterated convo- lution of a linear positive kernel converges to a Gaussian. Therefore, all iterative linear smoothing methods of a signal or an image boils down to the application to the signal of the Heat Equation. In this survey, we explain how a similar analysis can be performed for image iterative smoothing by contrast invariant monotone operators. In particular, we prove that all iterated affine and contrast invariant monotone opera- tors are equivalent to the unique affine invariant curvature motion. We also prove that under very broad conditions, weighted median filters are equivalent to the Mean Curvature Motion Equation.

We decompose images into “shapes”, based on connected components of level sets, which can be put in a tree structure. This tree contains the purely geometric information present in the image, sepa- rated from the contrast information. This structure allows to suppress easily some shapes without affecting the others, which yields a peculiar kind of scale-space, where the information present at each scale is already present in the original image.

A morphological scale-space representation is presented based on a morphological strong filter, the levelings. The scale-properties are analysed and illustrated. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharp and perfectly localised. This paper is followed by a companion paper on pde formula- tions of levelings.

The partial differential equations describing the propagation of (wave) fronts in space are closely connected with the morphological erosion and dilation. Strangely enough this connection has not been ex- plored in the derivation of numerical schemes to solve the differential equations. In this paper the morphological facet model is introduced in which an analytical function is locally fitted to the data. This function is then dilated analytically with an infinitesimal small structuring element. These sub-pixel dilationsform the core of the numerical solution schemes presented in this paper. One of the simpler morphological facet models leads to a numerical scheme that is identical with a well known classical upwind finite difference scheme. Experiments show that the morpholog- ical facet model provides stable numerical solution schemes for these partial differential equations.

We show that regularization methods can be regarded as scale-spaces where the regularization parameter serves as scale. In analogy to nonlinear diffusion filtering we establish continuity with respect to scale, causality in terms of a maximum/minimum principle, simplifica- tion properties by means of Lyapunov functionals and convergence to a constant steady-state. We identify nonlinear regularization with a single implicit time step of a diffusion process. This implies that iterated regu- larization with small regularization parameters is a numerical realization of a diffusion filter. Numerical experiments in two and three space dimen- sions illustrate the scale-space behaviour of regularization methods.

Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. On the other hand, compression is an important topic in image process- ing as well. Here a method is presented which combines the two aspects in an efficient way. It is based on a semi-implicit Finite Element im- plementation of nonlinear diffusion. Error indicators guide a successive coarsening process. This leads to locally coarse grids in areas of resulting smooth image intensity, while enhanced edges are still resolved on fine grid levels. Special emphasis has been put on algorithmical aspects such as storage requirements and efficiency. Furthermore, a new nonlinear anisotropic diffusion method for vector field visualization is presented.

This paper presents an interpretation of a classic optical flow method by Nagel and Enkelmann as a tensor-driven anisotropic diffusion approach in digital image analysis. We introduce an improvement into the model formulation, and we establish well-posedness results for the resulting system of parabolic partial differential equations. Our method avoids linearizations in the optical flow constraint, and it can recover displacement fields which are far beyond the typical one-pixel limits that are characteristic for many differential methods for optical flow recovery. A robust numerical scheme is presented in detail. We avoid convergence to irrelevant local minima by embedding our method into a linear scale- space framework and using a focusing strategy from coarse to fine scales. The high accuracy of the proposed method is demonstrated by means of a synthetic and a real-world image sequence.

This paper introduces a new method for analyzing scaling phenomena in natural images, and draws some consequences as to whether natural images belong to the space of functions with bounded variation.

Edges are viewed as statistical outliers with respect to local image gradient magnitudes. Within local image regions we compute a robust statistical measure of the gradient variation and use this in an anisotropic diffusion framework to determine a spatially varying “edge- stopping” parameter σ. We show how to determine this parameter for two edge-stopping functions described in the literature (Perona-Malik and the Tukey biweight). Smoothing of the image is related the local texture and in regions of low texture, small gradient values may be treated as edges whereas in regions of high texture, large gradient magni- tudes are necessary before an edge is preserved. Intuitively these results have similarities with human perceptual phenomena such as masking and “popout”. Results are shown on a variety of standard images.

Fractal Brownian motions have been introduced as a statis- tical description of natural images. We analyze the Gaussian scale-space scaling of derivatives of fractal images. On the basis of this analysis we propose a method for estimation of the fractal dimension of images and scale-space normalisation used in conjunction with automatic scale se- lection assuming either constant energy over scale or self similar energy scaling.

The lattice Boltzmann method has attracted more and more attention as an alternative numerical scheme to traditional numerical methods for solving partial differential equations and modeling physical systems. The idea of the lattice Boltzmann method is to construct a simplified discrete microscopic dynamics to simulate the macroscopic model described by the partial differential equations. In this paper, we present the lattice Boltzmann models for nonlinear diffusion filtering. We show that image feature selective smoothing can be achieved by making the relaxation parameter in the lattice Boltzmann equation be image feature and direction dependent. The models naturally lead to the numerical algorithms that are easy to implement. Experimental results on both synthetic and real images are described.

We merge techniques developed in the Beltrami framework to deal with multi-channel, i.e. color images, and the Mumford-Shah func- tional for segmentation. The result is a color image enhancement and segmentation algorithm. The generalization of the Mumford-Shah idea includes a higher dimension and codimension and a novel smoothing mea- sure for the color components and for the segmenting function which is introduced via the I-convergence approach. We use the I-convergence technique to derive, through the gradient descent method, a system of coupled PDEs for the color coordinates and for the segmenting function.

We present a supervised classification model based on a vari- ational approach. This model is devoted to find an optimal partition com- pound of homogeneous classes with regular interfaces. We represent the regions of the image defined by the classes and their interfaces by level set functions, and we define a functional whose minimum is an optimal partition. The coupled Partial Differential Equations (PDE) related to the minimization of the functional are considered through a dynamical scheme. Given an initial interface set (zero level set), the different terms of the PDE’s are governing the motion of interfaces such that, at con- vergence, we get an optimal partition as defined above. Each interface is guided by internal forces (regularity of the interface), and external ones (data term, no vacuum, no regions overlapping). Several experiments were conducted on both synthetic an real images.

The behaviour of critical points of Gaussian scale-space im- ages is mainly described by their creation and annihilation. In existing literature these events are determined in so-called canonical coordinates. A description in a user-defined Cartesian coordinate system is stated, as well as the results of a straightforward implementation. The location of a catastrophe can be predicted with subpixel accuracy. An example of an annihilation is given. Also an upper bound is derived for the area where critical points can be created. Experimental data of an MR, a CT, and an artificial noise image satisfy this result.

Since the work by Osher and Sethian on level-sets algorithms for numerical shape evolutions, this technique has been used for a large number of applications in numerous fields. In medical imaging, this nu- merical technique has been successfully used for example in segmentation and cortex unfolding algorithms. The migration from a Lagrangian im- plementation to an Eulerian one via implicit representations or level-sets brought some of the main advantages of the technique, mainly, topology independence and stability. This migration means also that the evolution is parametrization free, and therefore we do not know exactly how each part of the shape is deforming, and the point-wise correspondence is lost. In this note we present a technique to numerically track regions on sur- faces that are being deformed using the level-sets method. The basic idea is to represent the region of interest as the intersection of two implicit surfaces, and then track its deformation from the deformation of these surfaces. This technique then solves one of the main shortcomings of the very useful level-sets approach. Applications include lesion localization in medical images, region tracking in functional MRI visualization, and geometric surface mapping.

We explain how a discrete grey level image can be numer- ically translated into a completely pixel independent geometric struc- ture made of oriented curves with grey levels attached to them. For that purpose, we prove that the Affine Morphological Scale Space of an image can be geometrically computed using a level set decomposi- tion/reconstruction and a well adapted curve evolution scheme. Such an algorithm appears to be much more accurate than classical pixel-based ones, and allows continuous deformations of the original image.

The classical morphological segmentation paradigm is based on the watershed transform, constructed by flooding the gradient im- age seen as a topographic surface. For flooding a topographic surface, a topographic distance is defined from which a minimum distance algo- rithm is derived for the watershed. In a continuous formulation, this is modeled via the eikonal PDE, which can be solved using curve evolution algorithms. Various ultrametric distances between the catchment basins may then be associated to the flooding itself. To each ultrametric dis- tance is associated a multiscale segmentation; each scale being the closed balls of the ultrametric distance.

In this paper we develop partial differential equations (PDEs) that model the generation of a large class of morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks rang- ing from enhancement, to geometric feature detection, to segmentation. The developed PDEs are nonlinear functions of the first spatial deriva- tives and model these nonlinear filters as the limit of a controlled growth starting from an initial seed signal. This growth is of the multiscale di- lation or erosion type and the controlling mechanism is a switch that reverses the growth when the difference between the current evolution and a reference signal switches signs. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution and corresponding filter implementation, and provide insights via several ex- periments. Finally, we outline the use of these PDEs for improving the Gaussian scale-space by using the latter as initial seed to generate mul- tiscale levelings that have a superior preservation of image edges and boundaries.

We present an extension of the scale space idea to surfaces, with the aim of extending ideas like Gaussian derivatives to function on curved spaces. This is done by using the fact, also valid for normal images, that among the continuous range of scales at which one can look at an image, or surface, there is a infinite discrete subset which has a natural geometric interpretation. We call them “proper scales” as they are defined by eigenvalues of an elliptic partial differential operator associated with the image, or shape. The computations are performed using the Finite Element technique.

Electron tomography is a powerful tool for investigating the threedimensional (3D) structure of biological objects at a resolution in the nanometer range. However, visualization and interpretation of the resulting volumetric data is a very difficult task due to the extremely low signal to noise ratio (<0dB). In this paper, an approach for noise reduction in volumetric data is presented, based on nonlinear anisotropic diffusion, using a hybrid of the edge enhancing and the coherence enhancing techniques. When applied to both, artificial or real data sets, the method turns out to be superior to conventional filters. In order to assess noise reduction and structure preservation experimentally, resolution tests commonly used in structure analysis are applied to the data in the frequency domain.

We propose a simple approach to evolution of polygonal curves that is specially designed to fit discrete nature of curves in digi- tal images. It leads to simplification of shape complexity with no blur- ring (i.e., shape rounding) effects and no dislocation of relevant features. Moreover, in our approach the problem to determine the size of discrete steps for numerical implementations does not occur, since our evolution method leads in a natural way to a finite number of discrete evolution steps which are just the iterations of a basic procedure of vertex deletion.

Multiresolution techniques are often used to shorten the ex- ecution times of dynamic programming based deformable contour op- timization methods by decreasing the image resolution. However, the speedup comes at the expense of contour optimality due to the loss of details and insufficient usage of the external energy in decreased res- olutions. In this paper, we present a new scale-space based technique for deformable contour optimization, which achieves faster optimization times and performs better than the current multiresolution methods. The technique employs a multiscale representation of the underlying images to analyze the behavior of the external energy of the deformable contour with respect to the change in the scale dimension. The result of this anal- ysis, which involves information theoretic comparisons between scales, is used in segmentation of the original images. Later, an exhaustive search on these segments is carried out by dynamic programming to optimize the contour energy. A novel gradient descent algorithm is employed to find optimal internal energy for large image segments, where the external energy remains constant due to segmentation.We present the results of our contour tracking experiments performed on medical images. We also demonstrate the efficiency and the performance of our system by quantitatively comparing the results with the multires- olution methods, which confirm the effectiveness and the accuracy of our method.

A simple derivation of properties of a normal white noise random field in linear scale-space is presented. The central observation is that the random field has a scaling invariance property. From this invariance it is easy to derive the scaling behaviour of measurements made on normal white noise random fields.

In this paper we summarize the main features of a new time dependent model to approximate the solution to the nonlinear total vari- ation optimization problem for deblurring and noise removal introduced by Rudin, Osher and Fatemi. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on an ENO Hamilton-Jacobi version of Roe’s scheme. We show numerical evidence of the speed, resolution and stability of this simple explicit procedure in two representative 1D and 2D numerical examples.

The maxima of Curvature Scale Space (CSS) image have already been used to represent 2-D shapes under affine transforms. Since the CSS image employs the arc length parametrisation which is not affine invariant, we expect some deviation in the maxima of the CSS image under general affine transforms.In this paper we examine the advantage of using affine length rather than arc length to parametrise the curve prior to computing its CSS image. The parametrisation has been proven to be invariant under affine trans- formation and has been used in many affine invariant shape recognition methods.The CSS representation with affine length parametrisation has been used to find similar shapes from a large prototype database.

The notion of a stochastic scale space has been introduced through a stochastic approximation to the Perona-Malik equation. The approximate solution has been shown to preserve scale-space causality and is well-posed in an expected sense. The algorithm also converges to a (unique) constant image.

In this paper we address two important problems in motion analysis: the detection of moving objects and their localization. Statis- tical and level set approaches are adopted in order to formulate these problems. For the change detection problem, the inter-frame difference is modeled by a mixture of two zero-mean Laplacian distributions. At first, statistical tests using criteria with negligible error probability are used for labeling as many as possible sites as changed or unchanged. All the connected components of the labeled sites are seed regions, which give the initial level sets, for which velocity fields for label propagation are provided.We introduce a new multi-label fast marching algorithm for expanding competitive regions. The solution of the localization problem is based on the map of changed pixels previously extracted. The bound- ary of the moving object is determined by a level set algorithm, which is initialized by two curves evolving in converging opposite directions. The sites of curve contact determine the position of the object boundary. For illustrating the efficiency of the proposed approach, experimental results are presented using real video sequences.

In this paper we present and briefly describe a Windows user- friendly system designed to assist with the analysis of images in general, and biomedical images in particular. The system, which is being made publicly available to the research community, implements basic 2D image analysis operations based on partial differential equations (PDE’s). The system is under continuous development, and already includes a large number of image enhancement and segmentation routines that have been tested for several applications.

Segmentation based on color, instead of intensity only, pro- vides an easier distinction between materials, on the condition that ro- bustness against irrelevant parameters is achieved, such as illumination source, shadows, geometry and camera sensitivities. Modeling the phys- ical process of the image formation provides insight into the effect of different parameters on object color.In this paper, a color differential geometry approach is used to detect material edges, invariant with respect to illumination color and imaging conditions. The performance of the color invariants is demonstrated by some real-world examples, showing the invariants to be successful in discounting shadow edges and illumination color.

This paper introduces a new approach for computing a hi- erarchical aspect graph of curved objects using multiple range images. Characteristic deformations occur in the neighborhood of a cusp point as viewpoint moves. We analyze the division types of viewpoint space in scale space in order to generate aspect graphs from a limited number of viewpoints. Moreover, the aspect graph can be automatically generated using an algorithm of the minimization criteria.

This paper introduces a new approach for generating the hi- erarchical description of a non-rigid density object. Scale-space is useful for the hierarchical analysis. Process-grammar which describes the defor- mation process between a circle and the shape is proposed. We use these two approaches to track the deformation of a non-rigid density object. We extend 2D process-grammar to 3D density process-grammar. We an- alyze the event that the topology of zero-crossing surface of scale-space changes, as a density object changes smoothly. The analysis is useful to generate 3D density process-grammar from a limited number of obser- vations. This method can be used for tracking a non-rigid object using MRI data.

In this paper, we introduce the linear scale-space theory for functions on finite graphs. This theory permits us to derive a discrete version of the mean curvature flow. This discrete version yields a defor- mation procedure for polyhedrons. The adjacent matrix and the degree matrix of a polyhedral graph describe the system equation of this poly- hedral deformation. The spectral thepry of graphs derive the stability condition of the polyhedral deformation.

We propose a method for contour figure approximation which does not assume the shape of primitives for contours. By smoothing out only local details by curvature flow process, a given contour figure is ap- proximated. The amount of smoothing is determined adaptively based on the sizes of the local details. To detect local details and to determine the amount of the local smoothing, the method uses the technique of the scale space analysis. Experimental results show that this approxima- tion method has preferable properties for contour figure recognition, e.g. only finite number of approximations are obtained from a given contour figure.

A one-dimensional deconvolution problem is discretized and certain multilevel preconditioned iterative methods are applied to solve the resulting linear system. The numerical results suggest that multilevel multiplicative preconditioners may have no advantage over two-level mul- tiplicative preconditioners. In fact, in the numerical experiments they perform worse than comparable two-level preconditioners.

Using the registration of remote imagery as an example do- main, this work describes an efficient approach to the structural matching of multi-resolution representations where the scale difference, rotation and translation are unknown. The matching process is posed within an optimisation framework in which the parameter space is the probabil- ity hyperspace of all possible matches. In this application, searching for corresponding features at all scales generates a parameter space of enor- mous dimensions - typically 1-10 million. In this work we use feature’s hierarchical relationships to decompose the parameter space into a series of smaller subspaces over which optimisation is computationally feasible.

We introduce a 3D tracing method based on differential ge- ometry in Gaussian blurred images. The line point detection part of the tracing method starts with calculation of the line direction from the eigenvectors of the Hessian matrix. The sub-voxel center line position is estimated from a second order Taylor approximation of the 2D intensity profile perpendicular to the line. The line diameter is obtained at a single scale using the theoretical scale dependencies of the 0-th and 2nd order Gaussian derivatives at the line center. Experiments on synthetic images reveal that the localization of the centerline is mainly affected by line curvature. The diameter measurement is accurate for diameters as low as 4 voxels.

We develop on estimation method, for the derivative field of an image based on Bayesian approach which is formulated in a geometric way. The Maximum probability configuration of the derivative field is found by a gradient descent method which leads to a non-linear diffusion type equation with added constraints. The derivatives are assumed to be piecewise smoothe and the Beltrami framework is used in the development of an adaptive smoothing process.

Automatic segmentation is performed using watersheds of the gradient magnitude and compression techniques. Linear Scale-Space is used to discover the neighbourhood structure and catchment basins are locally merged with Minimum Description Length. The algorithm can form a basis for a large range of automatic segmentation algorithms based on watersheds, scale-spaces, and compression.

In this paper binocular stereo in a linear scale-space setting is studied. A theoretical extension of previous work involving the optic flow constraint equation is obtained, which is embedded in a robust top- down algorithm. The method is illustrated by some examples.

In this paper, classical nonlinear diffusion methods of ma- chine vision are revisited in the light of recent results in nonlinear sta- bility analysis. Global exponential convergence rates are quantified, and suggest specific choices of nonlinearities and image coupling terms. In particular, global stability and exponential convergence can be guaran- teed for nonlinear filtering of time-varying images.