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A new template matching scheme for the detection of objects on the basis of orientations is proposed. The matching scheme is based on correlations in the domain \(\mathbb{R}^2 \rtimes S^1\) of complex valued invertible orientation scores. In invertible orientation scores, a comprehensive overview of how an image is decomposed into local orientations is obtained. The presented approach allows for the efficient detection of orientation patterns in an intuitive and direct way. Furthermore, an energy minimization approach is proposed for the construction of suitable templates. The method is applied to optic nerve head detection in retinal images and extensive testing is done using images from both public and private databases. The method correctly identifies the optic nerve head in 99.7% of 1737 images.

From an image we construct an invertible orientation score, which provides an overview of local orientations in an image. This orientation score is a function on the group SE (2) of both positions and orientations. It allows us to diffuse along multiple local line segments in an image. The transformation from image to orientation score amounts to convolutions with an oriented kernel rotated at multiple angles. Under conditions on the oriented kernel the transform between image and orientation score is unitary. This allows us to relate operators on images to operators on orientation scores in a robust way such that we can deal with crossing lines and orientation uncertainty. To obtain reasonable Euclidean invariant image processing the operator on the orientation score must be both left invariant and non-linear. Therefore we consider non-linear operators on orientation scores which amount to direct products of linear left-invariant scale spaces on SE (2). These linear left-invariant scale spaces correspond to well-known stochastic processes on SE (2) for line completion and line enhancement and are given by group convolution with the corresponding Green's functions. We provide the exact Green's functions and approximations, which we use together with invertible orientation scores for automatic line enhancement and completion.

We propose an iterative approximate reconstruction method where we minimize the difference between reconstructions from subsets of multi scale measurements. To this end we interpret images not as scalar-valued functions but as sections through a fibered space. Information from previous reconstructions, which can be obtained at a coarser scale than the current one, is propagated by means of covariant derivatives on a vector bundle. The gauge field that is used to define the covariant derivatives is defined by the previously reconstructed image. An advantage of using covariant derivatives in the variational formulation of the reconstruction method is that with the number of iterations the accuracy of the approximation increases. The presented reconstruction method allows for a reconstruction at a resolution of choice, which can also be used to speed up the approximation at a finer level. An application of our method to reconstruction from a sparse set of differential features of a scale space representation of an image allows for a weighting of the features based on the sensitivity of those features to noise. To demonstrate the method we apply it to the reconstruction from singular points of a scale space representation of an image.

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study left-invariant diffusion on the 3D Euclidean motion group SE(3), which is useful for processing three-dimensional data. In particular, it is useful for the processing of High Angular Resolution Diffusion Imaging (HARDI) data, since these data can be considered as orientation scores directly, without the need to transform the HARDI data to a different form. In principle, all theory of the 2D case can be mapped to the 3D case. However, one of the complicating factors is that all practical 3D orientation scores are not functions on the entire group SE(3), but rather on a coset space of the group. We will show how we can still conceptually apply processing on the entire group by requiring the operations to preserve the introduced notion of alpha-right-invariance of such functions on SE(3). We introduce left-invariant derivatives and describe how to estimate tangent vectors that locally fit best to the elongated structures in the 3D orientation score. We propose generally applicable techniques for smoothing and enhancing functions on SE(3) using left-invariant diffusion on the group. Finally, we will discuss implementational issues and show a number of results for linear diffusion on artificial HARDI data.

We consider detection of local image symmetry using linear filters. We prove a simple criterion for determining if a filter is sensitive to a group of symmetries. We show that derivative-of-Gaussian (DtG) filters are excellent at detecting local image symmetry. Building on this, we propose a very simple algorithm that, based on the responses of a bank of six DtG filters, classifies each location of an image into one of seven Basic Image Features (BIFs). This effectively and efficiently realizes Marr's proposal for an image primal sketch. We summarize results on the use of BIFs for texture classification, object category detection, and pixel classification.

The dynamic behavior of the cardiac muscle is strongly dependent on heart diseases. Optic flow techniques are essential tools to assess and quantify the contraction of the cardiac walls. Most of the current methods however are restricted to the analysis of 2D MR-tagging image sequences: due to the complex twisting motion combined with longitudinal shortening, a 2D approach will always suffer from through-plane motion. In this paper we investigate a new 3D aperture-problem free optic flow method to study the cardiac motion by tracking stable multi-scale features such as maxima and minima on 3D tagged MR and sine-phase image volumes. We applied harmonic filtering in the Fourier domain to measure the phase. This removes the dependency of intensity changes of the tagging pattern over time due to T1 relaxation. The regular geometry, the size-changing patterns of the MR-tags stretching and compressing along with the tissue, and the phase- and sine-phase plots represent a suitable framework to extract robustly multi-scale landmark features. Experiments were performed on real and phantom data and the results revealed the reliability of the extracted vector field. Our new 3D multi-scale optic flow method is a promising technique for analyzing true 3D cardiac motion at voxel precision, and free of through-plane artifacts present in multiple-2D data sets.

We study three dimensional volumes of higher order tensors, using Finsler geometry. The application considered here is in medical image analysis, specifically High Angular Resolution Diffusion Imaging (HARDI) [1] of the brain. We want to find robust ways to reveal the architecture of the neural fibers in brain white matter. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled with a symmetric positive definite second order tensor, based on the assumption that there exists one dominant direction of fibers restricting the thermal motion of water molecules, leading naturally to a Riemannian framework. HARDI may potentially overcome the shortcomings of DTI by allowing multiple relevant directions, but invalidates the Riemannian approach. Instead Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide the exact criterion to determine whether a field of spherical functions has a Finsler structure. We also show a fiber tracking method in Finsler setting. Our model also incorporates a scale parameter, which is beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as real HARDI data.

Cyclic data arise in various image and signal processing applications such as interferometric synthetic aperture radar, electroencephalogram data analysis, and color image restoration in HSV or LCh spaces. In this paper we introduce a variational inpainting model for cyclic data which utilizes our definition of absolute cyclic second order differences. Based on analytical expressions for the proximal mappings of these differences we propose a cyclic proximal point algorithm (CPPA) for minimizing the corresponding functional. We choose appropriate cycles to implement this algorithm in an efficient way. We further introduce a simple strategy to initialize the unknown inpainting region. Numerical results both for synthetic and real-world data demonstrate the performance of our algorithm.

The aim of this paper is to study the robustness of the pointwise Lipschitz regularity in 2D, which is a measure of the local regularity of the intensity function associated to an image. This regularity can be efficiently computed by an approach based on fine scales. We assess its robustness when the image undergoes various transformations, especially geometric ones. The results we obtain show that the pointwise Lipschitz regularity is a suitable feature for applications in computer vision. ispartof: pages:782-794 ispartof: Scale space and variational methods in computer vision vol:5567 pages:782-794 ispartof: Scale-Space and Variational Methods location:Voss, Norway date:1 Jun - 5 Jun 2009 status: published

We present a novel approach to the reconstruction of depth from light field data. Our method uses dictionary representations and group sparsity constraints to derive a convex formulation. Although our solution results in an increase of the problem dimensionality, we keep numerical complexity at bay by restricting the space of solutions and by exploiting an efficient Primal-Dual formulation. Comparisons with state of the art techniques, on both synthetic and real data, show promising performances.