In this note we study the local behavior of singularities occurring in scale space under Gaussian blurring. Based on ideas from singularity theory for vector fields this is done by considering deformations or unfoldings. To deal with the special nature of the problem the concept of Gaussian deformation is introduced. Using singularity theory the stability of these deformations is considered. New concepts of one-sided stability and one-sided equivalence are introduced. This way a classification of stable singularities is obtained which agrees with those known in literature.
Blurring an image with a Gaussian of width σ and considering σ as an extra dimension, extends the image to an Gaussian scale space ( $\mathcal{GSS}$ ) image. In this $\mathcal{GSS}$ -image the iso-intensity manifolds behave in an nicely pre-determined manner. As a result of that, the $\mathcal{GSS}$ -image directly generates a hierarchy in the form of a binary ordered rooted tree, that can be used for segmentation, indexing, recognition and retrieval. Understanding the geometry of the manifolds allows fast methods to derive the hierarchy. In this paper we discuss the relevant geometric properties of $\mathcal{GSS}$ images, as well as their implications for algorithms used for the tree extraction. Examples show the applicability and increased speed of the proposed method compared to traditional ones.
The Symmetry Set (\(\mathcal{SS}\)) and its representation in parameter space, the pre-Symmetry Set, can be used to describe a shape with a linear data structure containing strings. As shape descriptor one specific string can be chosen. This string represents not only the major axis of the shape, but it also contains information of the complete shape. The string is augmented with information about the special points along the (pre-) Symmetry Set that it resembles. Changes in this simple line structure are directly related to so-called transitions (topological changes) of the \(\mathcal{SS}\) and the Pre- \(\mathcal{SS}\). It also carries information about the skeleton, or Medial Axis.
Multi-Scale Singularity Trees(MSSTs) [10] are multi-scale image descriptors aimed at representing the deep structures of images. Changes in images are directly translated to changes in the deep structures; therefore transitions in MSSTs. Because MSSTs can be used to represent the deep structure of images efficiently, it is important to investigate and understand their transitions and impacts. We present four kinds of MSST transitions and discuss the potential advantages of Saddle-Based MSSTs over Extrema-Based MSSTs. The study of MSST transitions presented in this paper is an important step towards the development of the image matching and indexing algorithms based on MSSTs.
We show that the pre-symmetry set of a smooth surface in 3-space has the structure of the graph of a function from ℝ2 to ℝ2 in many cases of interest, generalising known results for the pre-symmetry set of a curve in the plane. We explain how this function is obtained, and illustrate with examples both on and off the diagonal. There are other cases where the pre-symmetry set is singular; we mention some of these cases but leave their investigation to another occasion.
We present a methodology for estimating the probability of multi-object anatomic complexes that reflects both the individual objects' variability and the variability of the inter-relationships between objects. The method is based on m-reps and the idea of augmenting medial atoms from one object's m-rep to the set of atoms of an object being described. We describe the training of these probabilities, and we present an example of calculating the statistics of the bladder, prostate, rectum complex in the male pelvis. Via examples from the real world and from Monte-Carlo simulation, we show that this means of representing multi-object statistics yields samples that are nearly geometrically proper and means and principal modes of variations that are intuitively reasonable.
The geometry of "empty" scale space is investigated. By virtue of the proposed geometric axioms the generating PDE, the linear isotropic heat equation, can be presented in covariant, or geometrical form. The postulate of a metric for scale space cannot be upheld, as it is incompatible with the generating equation. Two familiar instances of scale spaces consistent with the geometric axioms are considered by way of example, viz. classical, homogeneous scale space, and foveal scale space.
We studied image texture due to the shading of corrugated (3D textured) surfaces, which are Lambertian on the micro scale. Our theory applies to physically canonical cases of isotropic Gaussian random surfaces, under collimated illumination. In this investigation we analyze effects of oblique viewing, extending our theory which applied to normal viewing conditions only [5]. The theory for normal views predicts the structure tensors from either the gradient or the Hessian of the image intensity and allows for inferences of the orientation of irradiation of the surface. Even for surfaces that are not at all Gaussian, the BRDF [10] far from Lambertian, with vignetting and multiple scattering present, such inferences of the orientation of irradiation were accurate up to a few degrees. In this paper we derive predictions for oblique viewing conditions, for which the inferences of the irradiation orientation will deviate from the veridical value in a systematic manner, depending on the viewing and illumination directions. Theoretical predictions are compared with empirical data, for rendered and for real rough surfaces, and found to be in good agreement. We discuss issues of scale selection and robustness.
We present a novel approach to statistically characterize histograms of model-relative image regions. A multiscale model is used as an aperture to define image regions at multiple scales. We use this image description to define an appearance model for deformable model segmentation. Appearance models measure the likelihood of an object given a target image. To determine this likelihood we compute pixel intensity histograms of local model-relative image regions from a 3D image volume near the object boundary. We use a Gaussian model to statistically characterize the variation of non-parametric histograms mapped to Euclidean space using the Earth Mover’s distance. The new method is illustrated and evaluated in a deformable model segmentation study on CT images of the human bladder, prostate, and rectum. Results show improvement over a previous profile based appearance model, out-performance of statistically modeled histograms over simple histogram measurements, and advantages of regional histograms at a fixed local scale over a fixed global scale.
A maximum likelihood local scale estimation principle is presented. An actual implementation of the estimation principle uses second order moments of multiple measurements at a fixed location in the image. These measurements consist of Gaussian derivatives possibly taken at several scales and/or having different derivative orders.Although the principle is applicable to a wide variety of image models, the main focus here is on the Brownian model and its use for scale selection in natural images. Furthermore, in the examples provided, the simplifying assumption is made that the behavior of the measurements is completely characterized by all moments up to second order.
We discuss the topic of correlation in a scale space setting. Correlation involves two distinct scales. The “outer scale” is the scale of the region over which the correlation will be calculated. Classically this is the whole space of interest, but in many cases one desires the correlation over some region of interest. The “inner scale” is the scale at which the signals to be correlated are represented. Classically this means infinite precision. For our purposes we define “correlation” as the point–wise product of two signals, “blurred correlation” as the integration of this correlation over the region of interest, and “correlation blur” as this point–wise correlation applied to the signals represented at the inner scale. For generic purposes we are interested in “blurred correlation blur”. We discuss a well known (and practically important) example of blurred correlation for essentially zero inner scale. Such a situation leads to apparently paradoxical results. We then discuss correlation blur, which can be understood as a form of “regularized” correlation, leading to intuitively acceptable results even for the case of point sets (e.g., temporal events or point sets in space). We develop the formal structure and present a number of examples.
In this paper we discuss the implementation of methods to derive 3D Symmetry Sets, given a parameterized shape, as well as an unorganized point cloud. It presents a geometric method to derive the Symmetry Set, that is an extension of the one given in [6]. Although the mathematics is a simple extension of the 2D case, the visualization, numerical computations and their stability are much more complicated. An example is given by means of an ellipsoid. In this example the Symmetry Set can be computed exactly and results can be compared to the ground truth.
In this paper, we present a novel framework to carry out computations on tensors, i.e. symmetric positive definite matrices. We endow the space of tensors with an affine-invariant Riemannian metric, which leads to strong theoretical properties: The space of positive definite symmetric matrices is replaced by a regular and geodesically complete manifold without boundaries. Thus, tensors with non-positive eigenvalues are at an infinite distance of any positive definite matrix. Moreover, the tools of differential geometry apply and we generalize to tensors numerous algorithms that were reserved to vector spaces. The application of this framework to the processing of diffusion tensor images shows very promising results, We apply this framework to the processing of structure tensor images and show that it could help to extract low-level features thanks to the affine-invariance of our metric. However, the same affine-invariance causes the whole framework to be noise sensitive and we believe that the choice of a more adapted metric could significantly improve the robustness of the result.
Several image processing algorithms imitate the lateral interaction of neurons in the visual striate cortex V1 to account for the correlations along contours and lines. Here we focus on two methodologies: tensor voting by Guy and Medioni, and stochastic completion fields by Mumford, Williams and Jacobs. The objective of this article is to compare these two methods and to place them into a common mathematical framework. As a consequence we obtain a sound stochastic foundation of tensor voting, a new tensor voting field, and an analytic approximation of the stochastic completion kernel.
We first describe two stochastic algorithms which build trees in high dimensional Euclidean spaces with some adaptation to the geometry of a chosen target subset. The second one produces search trees and is used to approximately identify in real time the pose of a polyhedron from its external contour. A search tree is first grown in a space of shapes of plane curves which are a set of precomputed polygonal outlines of the polyhedron. The tree is then used to find in real time a best match to the outline of the polyhedron in the current pose. Analyzing the deformation of the curves along the tree thus built, shows progressive differentiation from a simple convex root shape to the various possible external contours, and the tree organizes the complex set of shapes into a more comprehensible object.
This paper considers scale invariance of statistical image models. We study statistical scale invariance of the covariance structure of jet space under scale space blurring and derive the necessary structure and conditions of the jet covariance matrix in order for it to be scale invariant. As part of the derivation, we introduce a blurring operator A t that acts on jet space contrary to doing spatial filtering and a scaling operator S s . The stochastic Brownian image model is an example of a class of functions which are scale invariant with respect to the operators A t and S s . This paper also includes empirical results where we estimate the scale invariant jet covariance of natural images and show that it resembles that of Brownian images.
We compare the topology and deep structure of alternative scale space representations, so called α-scale spaces, 1/2 ≤ α ≤ 1, which are subject to a first order pseudo partial differential equation on the upper half plane {(x,s)∈ℝd ×ℝ|s0}. In particular, the cases α = 1 and α = 1/2, which correspond to respectively Poisson scale space and Gaussian scale space, are considered. Poisson scale space is equivalent to harmonic extension to the upper half plane, inducing potential physics, whereas Gaussian scale space is generated by the diffusion equation on the upper half plane, inducing heat physics. Despite the continuous connection (by parameter 1/2 ≤ α ≤ 1) between these scale spaces and the similarity between their convolution convolution kernels, we show both theoretically and experimentally that there is a strong difference between the topology in the deep structure of these scale spaces.
We face the question of how to produce a scale space of image intensities relative to a scale space of objects or other characteristic image regions filling up the image space, when both images and objects are understood to come from a population. We argue for a schema combining a multi-scale image representation with a multi-scale representation of objects or regions. The objects or regions at one scale level are produced using soft-edged apertures, which are subdivided into sub-regions. The intensities in the regions are represented using histograms. Relevant probabilities of region shape and inter-relations between region geometry and of histograms are described, and the means is given of inter-relating the intensity probabilities and geometric probabilities by producing the probabilities of intensities conditioned on geometry.
In this paper we propose a novel type of scales-spaces which is emerging from the family of inhomogeneous pseudodifferential equations $(I - \tau\Delta)^{\frac{t}{2}}u$ with τ ≥ 0 and scale parameter t ≥ 0. Since they are connected to the convolution semi-group of Bessel potentials we call the associated operators { R $^{n}_{t,{ \tau}}$ | 0≤ τ , t } either Bessel scale-space ( τ =1), R $^{n}_{t}$ for short, or scaled Bessel scale-space ( τ ≠1). This is the first concrete example of a family of scale-spaces that is not originating from a PDE of parabolic type and where the Fourier transforms $\mathcal{F}(R^n_{t,\tau})$ do not have exponential form. These properties make them different from other scale-spaces considered so far in the literature in this field. In contrast to the α -scale-spaces the integral kernels for R $^{n}_{t,{\tau}}$ can be given in explicit form for any t , τ ≥ 0 involving the modified Bessel functions of third kind K ν . In theoretical investigations and numerical experiments on 1D and 2D data we compare this new scale-space with the classical Gaussian one.
