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In this work we analyse the evolution of the critical points of an image by the curvature and affine morphological scale spaces. We define the notions of circular and elliptic extremum and show that an extremum becomes circular by the curvature scale space and elliptic by the affine morphological scale space. The evolution of a saddle point by the curvature scale space is also described. And we show how these properties can lead to numerical methods for the simulation of the curvature scale space. [ABSTRACT FROM AUTHOR]

We introduce a short time kernel for the Beltrami image enhancing flow. The flow is implemented by ‘convolving' the image with a space dependent kernel in a similar fashion to the implementation of the heat equation by a convolution with a gaussian kernel. The expression for the kernel shows, yet again, the connection between the Beltrami flow and the Bilateral filter. The kernel is calculated by measuring distances on the image manifold by an efficient variation of the fast marching method. The kernel, thus obtained, can be used for arbitrary large time steps in order to produce adaptive smoothing and/or a new scale-space. We apply it to gray scale and color images to demonstrate its flow like behavior. [ABSTRACT FROM AUTHOR]

We consider α scale spaces, a parameterized class (α ∈ (0, 1]) of scale space representations beyond the well-established Gaussian scale space, which are generated by the α-th power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundary condition ensures that there is no grey-value flux through the boundary. Thereby no artificial grey-values from outside the image affect the evolution proces, which is the case for the α scale spaces on an unbounded domain. Moreover, the connection between the α scale spaces which is not trivial in the unbounded domain case, becomes straightforward: The generator of the Gaussian semigroup extends to a compact, self-adjoint operator on the Hilbert space L2(Ω) and therefore it has a complete countable set of eigen functions. Taking the α-th power of the Gaussian generator simply boils down to taking the α-th power of the corresponding eigenvalues. Consequently, all α scale spaces have exactly the same eigen-modes and can be implemented simultaneously as scale dependent Fourier series. The only difference between them is the (relative) contribution of each eigen-mode to the evolution proces. By introducing the notion of (non-dimensional) relative scale in each α scale space, we are able to compare the various α scale spaces. The case α = 0.5, where the generator equals the square root of the minus Laplace operator leads to Poisson scale space, which is at least as interesting as Gaussian scale space and can be extended to a (Clifford) analytic scale space. [ABSTRACT FROM AUTHOR]

This work presents a new approach to texture classification, in which orientation histograms and multiscale analysis have been combined to achieve a reliable method. From the outputs of a set of filters, the orientation and magnitude of the gradient in every point of a texture are estimated. By combining the orientations and relative magnitudes of the gradient, we build an orientation histogram for each texture. We have used Fourier analysis to measure the similarity between the histograms of different textures, considering the effects of a change in the size or orientation of the image to make our method invariant under these phenomena. Since different textures may generate very similar histograms, we have analyzed the evolution of these histograms at different scales, extracting a scale factor for each couple of compared textures to adjust the filters which are applied to them when the multiscale analysis is carried out. [ABSTRACT FROM AUTHOR]

A minimal variance reconstruction scheme is derived using derivatives of the Gaussian as filters. A closed form mixed correlation matrix for reconstructions from multiscale points and their local derivatives up to the second order is presented. With the inverse of this mixed correlation matrix, a reconstruction of the image can be easily calculated. Some interesting results of reconstructions from multiscale critical points are presented. The influence of limited calculation precision is considered, using the condition number of the mixed correlation matrix. [ABSTRACT FROM AUTHOR]

In this paper we present a method for the regularization of a set of unstructured 3D points obtained from a sequence of stereo images. This method takes into account the information supplied by the disparity maps computed between pairs of images to constraint the regularization of the set of 3D points. We propose a model based on an energy which is composed of two terms: an attachment term that minimizes the distance from 3D points to the projective lines of camera points, and a second term that allows for the regularization of the set of 3D points by preserving discontinuities presented on the disparity maps. We embed this energy in a 2D finite element method. After minimizing, this method results in a large system of equations that can be optimized for fast computations. We derive an efficient implicit numerical scheme which reduces the number of calculations and memory allocations. [ABSTRACT FROM AUTHOR]

We present a novel variational approach to dense motion estimation of highly non-rigid structures in image sequences. Our representation of the motion vector field is based on the extended Helmholtz Decomposition into its principal constituents: The laminar flow and two potential functions related to the solenoidal and irrotational flow, respectively. The potential functions, which are of primary interest for flow pattern analysis in numerous application fields like remote sensing or fluid mechanics, are directly estimated from image sequences with a variational approach. We use regularizers with derivatives up to third order to obtain unbiased high-quality solutions. Computationally, the approach is made tractable by means of auxiliary variables. The performance of the approach is demonstrated with ground-truth experiments and real-world data. [ABSTRACT FROM AUTHOR]

With a mind towards achieving means of image comprehension by computer, we intend to convey the benefits of (1) characterizing the geometry of object complexes in the real world as contrasted with the geometric conformation of their images, and (2) describing populations of object complexes probabilistically. We show how a multi-scale description of inter-scale residues of geometric features provides a set of efficiently trainable probability distributions via a Markov random field approach, and specifies the location and scale of geometric differences between populations. These ideas and methods are illustrated using medial representations for 3D objects, depending on their properties (1) that local descriptors have an associated coordinate frame and distance metric, and (2) that continuous geometric random variables can be used to describe all members of a population of object complexes with a common structure and the variation among those members. We demonstrate with respect to the following object-complex-relative discrete scale levels: a whole object complex, individual objects, various object parts and sections, and fine boundary details. Using this illustrative framework, we show how to build Markov random field (MRF) models on the geometry scale space based on the statistics of shape residues across scales and between neighboring geometric entities at the level of locality given by its scale. In this paper, we present how to design and estimate MRF models on two scale levels, namely boundary displacement and object sections. [ABSTRACT FROM AUTHOR]

When reconstructing surfaces from image data, reflections on specular surfaces are usually viewed as a nuisance that should be avoided. In this paper a different view is taken. Noting that such reflections contain information about the surface, this information could and should be used when estimating the shape of the surface. Specifically, assuming that the position of the light source and the cameras (i.e. the motion) are known, the reflection from a specular surface in a given image constrain the surface normal with respect to the corresponding camera. Here the constraints on the normals, given by the reflections, are used to formulate a partial differential equation (PDE) for the surface. A smoothness term is added to this PDE and it is solved using a level set framework, thus giving a "shape from specularity" approach. The structure of the PDE also allows other properties to be included, e.g. the constraints from PDE based stereo. The proposed PDE does not fit naturally into a level set framework. To address this issue it is proposed to couple a force field to the level set grid. To demonstrate the viability of the proposed method it has been applied successfully to synthetic data. [ABSTRACT FROM AUTHOR]

We propose a novel variational approach based on a level set formulation of the Mumford-Shah functional and shape priors. We extend the functional by a labeling function which indicates image regions in which the shape prior is enforced. By minimizing the proposed functional with respect to both the level set function and the labeling function, the algorithm selects image regions where it is favorable to enforce the shape prior. By this, the approach permits to segment multiple independent objects in an image, and to discriminate familiar objects from unfamiliar ones by means of the labeling function. Numerical results demonstrate the performance of our approach. [ABSTRACT FROM AUTHOR]

Several types of interest point detectors have been proposed for spatial images. This paper investigates how this notion can be generalised to the detection of interesting events in space-time data. Moreover, we develop a mechanism for spatio-temporal scale selection and detect events at scales corresponding to their extent in both space and time. To detect spatio-temporal events, we build on the idea of the Harris and Förstner interest point operators and detect regions in space-time where the image structures have significant local variations in both space and time. In this way, events that correspond to curved space-time structures are emphasised, while structures with locally constant motion are disregarded. To construct this operator, we start from a multi-scale windowed second moment matrix in space-time, and combine the determinant and the trace in a similar way as for the spatial Harris operator. All spacetime maxima of this operator are then adapted to characteristic scales by maximising a scale-normalised space-time Laplacian operator over both spatial scales and temporal scales. The motivation for performing temporal scale selection as a complement to previous approaches of spatial scale selection is to be able to robustly capture spatio-temporal events of different temporal extent. It is shown that the resulting approach is truly scale invariant with respect to both spatial scales and temporal scales. The proposed concept is tested on synthetic and real image sequences. It is shown that the operator responds to distinct and stable points in space-time that often correspond to interesting events. The potential applications of the method are discussed. [ABSTRACT FROM AUTHOR]

This paper focuses on the computation of stationary curves, which are sometimes called fingerprints for one dimensional real signals in the linear scale space. Images for the analysis in the linear scale space are expressed as digital images for each quantized scale. Therefore, we develop a discrete version of the linear scale space analysis, employing the results of digital image analysis. For the application of linear scale space analysis to the time-varying images and objects, our method has advantages, because our method is based on the digital geometry on a plane which is suitable for the computation in digital computers. [ABSTRACT FROM AUTHOR]

In this paper we discuss how to define a scale space suitable for temporal measurements. We argue that such a temporal scale space should possess the properties of: temporal causality, linearity, continuity, positivity, recursitivity as well as translational and scaling covariance. It is shown that these requirements imply a one parameter family of convolution kernels. Furthermore it is shown that these measurements can be realized in a time recursive way, with the current data as input and the temporal scale space as state, i.e. there is no need for storing earlier input. This family of measurement processes contains the diffusion equation on the half line (that represents the temporal scale) with the input signal as boundary condition on the temporal axis. The diffusion equation is unique among the measurement processes in the sense that it is preserves positivity (in the scale domain) and is infinitesimally generated. A numerical scheme is developed and relations to other approaches are discussed. [ABSTRACT FROM AUTHOR]

Since the introduction of the slope transform by Dorst/van den Boomgaard and Maragos as the morphological equivalent of the Fourier transform, people have been surprised about the almost logarithmic relation between linear and morphological system theory. This article gives an explanation by revealing that morphology in essence is linear system theory in a specific algebra. While classical linear system theory uses the standard (+, ×)-algebra, the morphological system theory is based on the idempotent (max, +)-algebra and the (min, +)-algebra. We identify the nonlinear operations of erosion and dilation as linear convolutions *e and *d induced by these idempotent algebras. The slope transform in the (max, +)-algebra, however, corresponds to the logarithmic multivariate Laplace transform in the (+, ×)-algebra. We study relevant properties of this transform and its links to convex analysis. This leads to the definition of the so-called Cramer transform as the Legendre-Fenchel transform of the logarithmic Laplace transform. Originally known from the theory of large deviations in stochastics, the Cramer transform maps standard convolution to *e-convolution, and it maps Gaussians to quadratic functions. The article is a step towards the unification of linear and morphological system theories on the basis of a general linear system theory in an appropriate algebra. [ABSTRACT FROM AUTHOR]

Morphological operations are simple mathematical constructs, which have led to effective solution for many problems in image processing and computer vision. These solutions employ discrete operators and are applied to digitized images. The mathematics behind the morphological operators also exists in the continuous domain, the domain where the images came from. We observed that the discrete operators cannot reproduce the results obtained by the continuous operators. The reason for this is that neither the operator (the structuring element) nor the result of the operation are band-limited, and thus cannot be represented by equidistant samples without loss of information. The differences between continuous-domain and discrete-domain morphology are best shown by the dependency of the discrete morphology on sub-pixel translations and rotations of the images before digitization. This article describes an algorithm that applies continuous-domain morphology to properly sampled images. We implemented the dilation for one-dimensional images (signals), and with it constructed the erosion, the closing and the opening. We provide a discussion on a possible extension to higher-dimensional images. [ABSTRACT FROM AUTHOR]

We construct an algorithm to split an image into a sum u + v of a bounded variation component and a component containing the textures and the noise. This decomposition is inspired from arecent work of Y. Meyer. We find this decomposition by minimizing a convex functional which depends on the two variables u and v, alternatively in each variable. Each minimization is based on a projection algorithm to minimize the total variation. We carry out the mathematical study of our method. We present some numerical results. In particular, we show how the u component can be used in nontextured SAR image restoration. [ABSTRACT FROM AUTHOR]

In this paper it is argued that the Brownian image model is the least committed, scale invariant, statistical image model which describes the second order statistics of natural images. Various properties of three different types of Gaussian image models (white noise, Brownian and fractional Brownian images) will be discussed in relation to linear scale-space theory, and it will be shown empirically that the second order statistics of natural images mapped into jet space may, within some scale interval, be modeled by the Brownian image model. This is consistent with the 1/f2 power spectrum law that apparently governs natural images. Furthermore, the distribution of Brownian images mapped into jet space is Gaussian and an analytical expression can be derived for the covariance matrix of Brownian images in jet space. This matrix is also a good approximation of the covariance matrix of natural images in jet space. The consequence of these results is that the Brownian image model can be used as a least committed model of the covariance structure of the distribution of natural images. [ABSTRACT FROM AUTHOR]

Estimation of the mode of a distribution over ℝn from discrete samples is introduced and three methods for its solution are developed and evaluated. The first solution is based on Fréchet's definition of central tendencies. We show that algorithms based on this approach have only limited success due to the non-differentiability of the Fréchet measures. The second solution is based on tracking maxima through a Scale Space built from the samples. We show that this is more accurate than the Fréchet approach, but that tracking to very fine scales is unwarranted and undesirable. For our third method we analyze the reliability of the information across scale using an exact bootstrap analysis. This leads to a modified version of the Scale Space approach where unreliable information is downgraded (pessimistically) so that tracking into such regions does not occur. This modification improves the accuracy of mode estimation. We conclude with demonstrations on high-dimensional real and synthetic data, which confirm the technique's accuracy and utility. [ABSTRACT FROM AUTHOR]

Gaussian scale space permits one to compute image derivatives. The limitation to some finite order is not inherent in the paradigm itself (Gaussian blurred functions are always smooth), but is caused by the interplay of (at least) two external factors. One is the ratio of the Gaussian scale parameter versus the atomic scale that limits physically or perceptually meaningful sizes (e.g. pixel size, or in general any scale at which the image is "locally orderless"). The second factor involved is the fiducial level of tolerance. Together these factors conspire to determine a maximal order beyond which differential structure becomes meaningless. Thus they give rise to the notion of regularity classes for images akin to the conceptual Ck-classification pertaining to mathematical functions. We study the relationship between the maximal differential order k, the ratio of inner scale to atomic scale, and the prescribed tolerance level, and draw several conclusions that are of practical interest when considering image derivatives. [ABSTRACT FROM AUTHOR]

Linear scale space methodology uses Gaussian probes at scale s to observe the differential structure. In observing the differential image structure through the Gaussian derivative probes at scale s we implicitly construct the Taylor series expansion of the smoothed image. The Gaussian facet model, as a generalization of the classic Haralick facet model, constructs a polynomial approximation of the unsmoothed image. The measured differential structure therefore is closer to the ‘real' structure then the differential structure measured using Gaussian derivatives. At the points in an image where the differential structure changes abruptly (because of discontinuities in the imaging conditions, e.g. a material change, or a depth discontinuity) both the Gaussian derivatives and the Gaussian facet model diffuse the information from both sides of the discontinuity (smoothing across the edge). Robust estimators that are classically meant to deal with statistical outliers can also be used to deal with these ‘mixed model distributions'. In this paper we introduce the robust estimators of local image structure. Starting with the Gaussian facet model model where we replace the quadratic error norm with a robust (Gaussian) error norm leads to a robust Gaussian facet model. We will show examples of using the robust differential structure estimators for luminance and color images, for zero and higher order differential structure. Furthermore we look at a ‘robustified' structure tensor that forms the basis of robust orientation estimation. [ABSTRACT FROM AUTHOR]

Recently, several different approaches for digital inpainting have been proposed in the literature. We give a review and introduce a novel approach based on the complex Ginzburg-Landau equation. The use of this equation is motivated by some of its remarkable analytical properties. While common inpainting technology is especially designed for restorations of two dimensional image data, the Ginzburg-Landau equation can straight forwardly be applied to restore higher dimensional data, which has applications in frame interpolation, improving sparsely sampled volumetric data and to fill in fragmentary surfaces. The latter application is of importance in architectural heritage preservation. We discuss a stable and efficient scheme for the numerical solution of the Ginzburg-Landau equation and present some numerical experiments. We compare the performance of our algorithm with other well established methods for inpainting. [ABSTRACT FROM AUTHOR]

In this paper we present a method to implement the monogenic scale space on a bounded domain and show some applications. The monogenic scale space is a vector valued scale space based on the Poisson scale space, which establishes a sophisticated alternative to the Gaussian scale space. The features of the monogenic scale space, including local amplitude, local phase, local orientation, local frequency, and phase congruency, are much easier to interpret in terms of image features evolving through scale than in the Gaussian case. Furthermore, applying results from harmonic analysis, relations between the features are obtained which improve the understanding of image analysis. As applications, we present a very simple but still accurate approach to image reconstruction from local amplitude and local phase and a method for extracting the evolution of lines and edges through scale. [ABSTRACT FROM AUTHOR]

We study, in this work, the maximum principle for the Beltrami color flow and the stability of the flow's numerical approximation by finite difference schemes. We discuss, in the continuous case, the theoretical properties of this system and prove the maximum principle in the strong and the weak formulations. In the discrete case, all the second order explicit schemes, that are currently used, violate, in general, the maximum principle. For these schemes we give a theoretical stability proof, accompanied by several numerical examples. [ABSTRACT FROM AUTHOR]

We present a new approach to model edges in monochrome images. The method is divided in two parts: the localization of possible edge points and their valuation. The first part is based on the theory of minimal paths, where the selection of an energy and a set of sources determines a partition of the domain. Then, the valuation is obtained by the creation of a contrast driven hierarchy of partitions. The method uses only the original image and supplies a set of closed contours that preserve semantically important characteristics of edges. [ABSTRACT FROM AUTHOR]

Previously, we presented a method for contour registration using minimal surfaces. This method involves embedding each of two unregistered two-dimensional contours into two parallel planes separated in three-dimensional space. The minimal surface is then computed between the two contours via mean curvature flow. We then evolve the rigid registration of one of the two contours which in turn changes the minimal surface. Mean curvature flow of the surface and evolution of the curve registration both support a consistent energy functional, i.e., area of the connecting surface. We review the implementation details and show an example registration. In this paper we concentrate on developing this method as a registration scale space. The separation of the two contour planes serves as a scale space parameter, larger separations producing coarser registrations. At the finest scale, which occurs as the separation distance approaches zero, this registration method is identical to minimizing the set-symmetric difference between the interiors of the contours. Thus, this method can be viewed as a geometric generalization of set-symmetric difference registration. We explain the scale space properties of this registration method theoretically and experimentally. Through examples we show how at increasingly coarser scales, our method overcomes increasingly coarser local minima apparent in set-symmetric difference registration. In addition, we present sufficient conditions for existence of the minimal surface connecting two contours. This condition yields an upper bound for the separation distance between two contours and gives an estimate for the coarsest registration scale. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of a higher dimensional discrete scale-space (DSS) formulation. The continuous linear scale-space theory provides a unique framework for visual front-end processes. In practice, a higher dimensional DSS formulation is necessary since higher dimensional discrete signals must be dealt with. In this paper, first we examine the approximation fidelity of the commonly used sampled Gaussian. Second, we propose a generalized DSS formulation for 2-D and 3-D signals. The DSS theory has been presented at first by Lindeberg. While his 1-D DSS formulation is complete, the formulation as related to the extension to higher dimensions has not been fully derived. Furthermore, we investigate the properties of our derived DSS kernels and present the results of a validation study with respect to both smoothing and differentiation performance. [ABSTRACT FROM AUTHOR]

We assess the feasibility of approximating non-linear diffusion processes with simple local Gaussian filters. The purpose of doing this is twofold. Firstly, the theoretical implications are by themselves interesting. Secondly, a successful method would reduce the need for computationally expensive implementations of non-linear diffusion schemes. We evaluate using isotropic and affine Gaussian filters for the task of approximating the local diffusion for a number of non-linear diffusion schemes. The approximations are firstly explored using an information theoretical approach and secondly evaluated based on their performance on a multi-scale segmentation application. The results show that while the approximations do not perform quite as well as the original non-linear scheme, the decrease in performance is acceptable for the evaluated task. Furthermore, the affine approximations perform significantly better than the isotropic. [ABSTRACT FROM AUTHOR]

It has been stressed that regularisation methods and diffusion processes approximate each other. In this paper we identify a situation where both processes are even identical: the space-discrete 1-D case of total variation (TV) denoising. This equivalence is proved by deriving identical analytical solutions for both processes. The temporal evolution confirms that space-discrete TV methods implement a region merging strategy with finite extinction time. Between two merging events, only extremal segments move. Their speed is inversely proportional to their size. Our results stress the distinguished nature of TV denoising. Furthermore, they enable a mutual transfer of all theoretical and algorithmic achievements between both techniques. [ABSTRACT FROM AUTHOR]

We present a complete system of functionally independent invariants for Abelian Lie transformation groups acting on an image. The invariants are based on measurements, given by inner product of predesigned functions and the image. We build on steerable filters and adopt a Lie theoretical approach that is applicable to any dimensionality. A complete characterization of Lie measurement invariants of a general irreducible component of the group, termed block invariants, is provided. We show that invariants for the entire group can be taken as the union of the invariants of its components. The system is completed by deriving invariants between components of the group, termed cross invariants. [ABSTRACT FROM AUTHOR]

Current models of primary visual cortex (V1) include a linear filtering stage followed by a gain control mechanism that explains some of the nonlinear behavior of neurons. The nonlinear stage has been modeled as a divisive normalization in which each input linear response is squared and then divided by a weighted sum of squared linear responses in a certain neighborhood. In this communication, we show that such a scheme permits an efficient coding of natural image features. In our case, the linear stage is implemented as a four-level Daubechies decomposition, and the nonlinear normalization parameters are determined from the statistics of natural images under the hypothesis that sensory systems are adapted to signals to which they are exposed. In particular, we fix the weights of the divisive normalization to the mutual information of the corresponding pair of linear coefficients. This nonlinear process extracts significant, statistically independent, visual events in the image. [ABSTRACT FROM AUTHOR]

A feasibility study for a new method for content based image retrieval is presented. First, an image representation using multiscale top points is introduced. This representation is validated using a minimal variance reconstruction algorithm. The image retrieval problem can now be translated into comparing distances between point sets. For this purpose the proportional transportation distance (PTD) is used. A method is proposed using multiscale top points and their reconstruction coefficients in the PTD to define these distances between images. We present some experiments with promising results on a database with face images. [ABSTRACT FROM AUTHOR]

In an ordinary 2D image the critical points and the isophotes through the saddle points provide sufficient information for classifying the image into distinct regions belonging to the extrema (i.e. a collection of bright and dark blobs), together with their nesting due to the saddle isophotes. For scale space images, obtained by convolution of the image with a Gaussian filter at a continuous range of widths for the Gaussian, things are more complicated. Here only scale space saddle points occur. They are related to spatial saddle points and spatial extrema and can thus provide a scale space based segmentation and hierarchy. However, a spatial extremum can be related to multiple scale space saddles. The key to solve this ambiguity is the investigation of both the scale space saddles and the iso-intensity manifolds (the extension of isophotes in scale space) through them. I will describe the different situations that one can encounter in this investigation, which scale space saddles are relevant, give examples and show the difference between selecting the relevant and the non-relevant ("void") scale space saddles. [ABSTRACT FROM AUTHOR]