Bart M. ter Haar Romeny

Efficient and reliable schemes for nonlinear diffusion filtering

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.

Geometry-Driven Diffusion in Computer Vision

Preface. Foreword. 1. Linear Scale-Space I: Basic Theory T. Lindeberg, B.M. ter Haar Romeny. 2. Linear Scale-Space II: Early Visual Operations T. Lindeberg, B.M. ter Haar Romeny. 3. Anisotropic Diffusion P. Perona, T. Shiota, J. Malik. 4. Vector-Valued Diffusion R. Whitaker, G. Gerig. 5. Bayesian Rationale for the Variational Formulation D. Mumford. 6. Variational Problems with a Free Discontinuity Set A. Leaci, S. Solimini. 7. Minimization of Energy Functional with Curve-Represented Edges N. Nordstroem. 8. Approximation, Computation, and Distortion in the Variational Formulation T. Richardson, S. Mitter. 9. Coupled Geometry-Driven Diffusion Equations for Low-Level Vision M. Proesmans, E. Pauwels, L. van Gool. 10. Morphological Approach to Multiscale Analysis: From Principles to Equations l. Alvarez, J.-M. Morel. 11. Differential Invariant Signatures and Flows in Computer Vision: a Symmetry Group Approach P. Olver, G. Sapiro, A. Tannenbaum. 12. On Optimal Control Methods in Computer Vision and Image Processing B. Kimia, A. Tannenbaum, S. Zucker. 13. Nonlinear Scale-Space L. Florack, A. Salden, B. ter Haar Romeny, J. Koenderink, M. Viergever. 14. A Differential Geometric Approach to Anisotropic Diffusion D. Eberly. 15. Numerical Analysis of Geometry-Driven Diffusion Equations W. Niessen, B.M. ter Haar Romeny, M. Viergever. Bibliography. Index.

Scale and the differential structure of images

Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, resettling operator) and its linear partial derivatives. They enable local image analysis through the detection of local differential structure in a robust way, while at the same time capturing global features through the extra scale degree of freedom. In this paper we show why the operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, cartesian differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles found in mammalian frontend visual systems.

The Gaussian scale-space paradigm and the multiscale local jet

A representation of local image structure is proposed which takes into account both the images spatial structure at a given location, as well as its “deep structure”, that is, its local behaviour as a function of scale or resolution (scale-space). This is of interest for several low-level image tasks. The proposed basis of scale-space, for example, enables a precise local study of interactions of neighbouring image intensities in the course of the blurring process. It also provides an extrapolation scheme for local image data, obtained at a given spatial location and resolution, to a finite scale-space neighbourhood. This is especially useful for the determination of sampling rates and for interpolation algorithms in a multilocal context. Another, particularly straightforward application is image enhancement or deblurring, which is an instance of data extrapolation in the high-resolution direction.A potentially interesting feature of the proposed local image parametrisation is that it captures a trade-off between spatial and scale extrapolations from a given interior point that do not exceed a given tolerance. This (rade-off suggests the possibility of a fairly coarse scale sampling at the expense of a dense spatial sampling large relative spatial overlap of scale-space kernels).The central concept developed in this paper is an equivalence class called the multiscale local jet, which is a hierarchical, local characterisation of the image in a full scale-space neighbourhood. For this local jet, a basis of fundamental polynomials is constructed that captures the scale-space paradigm at the local level up to any given order.

Scale-Space: Its Natural Operators and Differential Invariants

Why and how one should study a scale-space is prescribed by the universal physical law of scale invariance, expressed by the so-called Pi-theorem. The fact that any image is a physical observable with an inner and outer scale bound, necessarily gives rise to a ‘scale-space representation’, in which a given image is represented by a one-dimensional family of images representing that image on various levels of inner spatial scale. An early vision system is completely ignorant of the geometry of its input. Its primary task is to establish this geometry at any available scale. The absence of geometrical knowledge poses additional constraints on the construction of a scale-space, notably linearity, spatial shift invariance and isotropy, thereby defining a complete hierarchical family of scaled partial differential operators: the Gaussian kernel (the lowest order, rescaling operator) and its linear partial derivatives. They enable local image analysis in a robust way, while at the same time capturing global features through the extra scale degree of freedom. The operations of scaling and differentiation cannot be separated. This framework permits us to construct in a systematic way multiscale, orthogonal differential invariants, i.e. true image descriptors that exhibit manifest invariance with respect to a change of cartesian coordinates. The scale-space operators closely resemble the receptive field profiles in the mammalian front-end visual system.

Higher order differential structure of images

This paper is meant as a tutorial on the basic concepts for vision in the ‘Koenderink’ school. The concept of scale-space is a necessity, if the extraction of structure from measured physical signals (i.e. images) is at stage. The Gaussian derivative kernels for physical signals are then the natural analogues of the mathematical differential operators. This paper discusses some interesting properties of the Gaussian derivative kernels, like their orthogonality and behaviour with noisy input data. Geometrical structure to extract is expressed in terms of differential invariants, in this paper limited to invariants under orthogonal transformations. Three representations are summarized: Cartesian, gauge and manifest invariant notation. Many explicit examples are given. A section is included about the computer implementation of the calculation of higher order invariant structure.

On the Axioms of Scale Space Theory

We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all “reasonable” axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane {(x, s) ∈ ℝd × ℝ | s > 0}. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.

Automatic segmentation of lung fields in chest radiographs

The delineation of important structures in chest radiographs is an essential preprocessing step in order to automatically analyze these images, e.g., for tuberculosis screening support or in computer assisted diagnosis. We present algorithms for the automatic segmentation of lung fields in chest radiographs. We compare several segmentation techniques: a matching approach; pixel classifiers based on several combinations of features; a new rule-based scheme that detects lung contours using a general framework for the detection of oriented edges and ridges in images; and a hybrid scheme. Each approach is discussed and the performance of nine systems is compared with interobserver variability and results available from the literature. The best performance is obtained by the hybrid scheme that combines the rule-based segmentation algorithm with a pixel classification approach. The combinations of two complementary techniques leads to robust performance; the accuracy is above 94% for all 115 images in the test set. The average accuracy of the scheme is 0.969 +/- 0.0080, which is close to the interobserver variability of 0.984 +/- 0.0048. The methods are fast, and implemented on a standard PC platform.

Structural and Resting State Functional Connectivity of the Subthalamic Nucleus: Identification of Motor STN Parts and the Hyperdirect Pathway

Deep brain stimulation (DBS) for Parkinson’s disease often alleviates the motor symptoms, but causes cognitive and emotional side effects in a substantial number of cases. Identification of the motor part of the subthalamic nucleus (STN) as part of the presurgical workup could minimize these adverse effects. In this study, we assessed the STN’s connectivity to motor, associative, and limbic brain areas, based on structural and functional connectivity analysis of volunteer data. For the structural connectivity, we used streamline counts derived from HARDI fiber tracking. The resulting tracks supported the existence of the so-called “hyperdirect” pathway in humans. Furthermore, we determined the connectivity of each STN voxel with the motor cortical areas. Functional connectivity was calculated based on functional MRI, as the correlation of the signal within a given brain voxel with the signal in the STN. Also, the signal per STN voxel was explained in terms of the correlation with motor or limbic brain seed ROI areas. Both right and left STN ROIs appeared to be structurally and functionally connected to brain areas that are part of the motor, associative, and limbic circuit. Furthermore, this study enabled us to assess the level of segregation of the STN motor part, which is relevant for the planning of STN DBS procedures.

Geodesic deformable models for medical image analysis

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.