Left-invariant evolutions of wavelet transforms on the Similitude Group

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

Automatic plant pest detection and recognition using k-means clustering algorithm and correspondence filters

Plant pest recognition and detection is vital for food security, quality of life and a stable agricultural economy. This research demonstrates the combination of the k-means clustering algorithm and the correspondence filter to achieve pest detection and recognition. The detection of the dataset is achieved by partitioning the data space into Voronoi cells, which tends to find clusters of comparable spatial extents, thereby separating the objects (pests) from the background (pest habitat). The detection is established by extracting the variant distinctive attributes between the pest and its habitat (leaf, stem) and using the correspondence filter to identify the plant pests to obtain correlation peak values for different datasets. This work further establishes that the recognition probability from the pest image is directly proportional to the height of the output signal and inversely proportional to the viewing angles, which further confirmed that the recognition of plant pests is a function of their position and viewing angle. It is encouraging to note that the correspondence filter can achieve rotational invariance of pests up to angles of 360 degrees, which proves the effectiveness of the algorithm for the detection and recognition of plant pests

Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well-defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations $U:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R}$ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group $SE(d)$, $d=2,3$. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in $SE(d)$. These curve fits minimize first or second order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on $SE(d)$. We include these gauge frames in differential invariants and crossing preserving PDE-flows acting on extended data representation $U$ and we show their advantage compared to the standard left-invariant frame on $SE(d)$. Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores

Generalized hardi invariants by method of tensor contraction

pre-printWe propose a 3D object recognition technique to construct rotation invariant feature vectors for high angular resolution diffusion imaging (HARDI). This method uses the spherical harmonics (SH) expansion and is based on generating rank-1 contravariant tensors using the SH coefficients, and contracting them with covariant tensors to obtain invariants. The proposed technique enables the systematic construction of invariants for SH expansions of any order using simple mathematical operations. In addition, it allows construction of a large set of invariants, even for low order expansions, thus providing rich feature vectors for image analysis tasks such as classification and segmentation. In this paper, we use this technique to construct feature vectors for eighth-order fiber orientation distributions (FODs) reconstructed using constrained spherical deconvolution (CSD). Using simulated and in vivo brain data, we show that these invariants are robust to noise, enable voxel-wise classification, and capture meaningful information on the underlying white matter structure

Finsler Active Contours

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TPAMI.2007.70713In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor that is chosen depends only upon position and is in this sense isotropic. Although directional information has been studied previously for other segmentation frameworks, here, we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming-based schemes. Finally, we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery

Feature based estimation of myocardial motion from tagged MR images

In the past few years we witnessed an increase in mortality due to cancer relative to mortality due to cardiovascular diseases. In 2008, the Netherlands Statistics Agency reports that 33.900 people died of cancer against 33.100 deaths due to cardiovascular diseases, making cancer the number one cause of death in the Netherlands [33]. Even if the rate of people affected by heart diseases is continually rising, they "simply don’t die of it", according to the research director Prof. Mat Daemen of research institute CARIM of the University of Maastricht [50]. The reason for this is the early diagnosis, and the treatment of people with identified risk factors for diseases like ischemic heart disease, hypertrophic cardiomyopathy, thoracic aortic disease, pericardial (sac around the heart) disease, cardiac tumors, pulmonary artery disease, valvular disease, and congenital heart disease before and after surgical repair. Cardiac imaging plays a crucial role in the early diagnosis, since it allows the accurate investigation of a large amount of imaging data in a small amount of time. Moreover, cardiac imaging reduces costs of inpatient care, as has been shown in recent studies [77]. With this in mind, in this work we have provided several tools with the aim to help the investigation of the cardiac motion. In chapters 2 and 3 we have explored a novel variational optic flow methodology based on multi-scale feature points to extract cardiac motion from tagged MR images. Compared to constant brightness methods, this new approach exhibits several advantages. Although the intensity of critical points is also influenced by fading, critical points do retain their characteristic even in the presence of intensity changes, such as in MR imaging. In an experiment in section 5.4 we have applied this optic flow approach directly on tagged MR images. A visual inspection confirmed that the extracted motion fields realistically depicted the cardiac wall motion. The method exploits also the advantages from the multiscale framework. Because sparse velocity formulas 2.9, 3.7, 6.21, and 7.5 provide a number of equations equal to the number of unknowns, the method does not suffer from the aperture problem in retrieving velocities associated to the critical points. In chapters 2 and 3 we have moreover introduced a smoothness component of the optic flow equation described by means of covariant derivatives. This is a novelty in the optic flow literature. Many variational optic flow methods present a smoothness component that penalizes for changes from global assumptions such as isotropic or anisotropic smoothness. In the smoothness term proposed deviations from a predefined motion model are penalized. Moreover, the proposed optic flow equation has been decomposed in rotation-free and divergence-free components. This decomposition allows independent tuning of the two components during the vector field reconstruction. The experiments and the Table of errors provided in 3.8 showed that the combination of the smoothness term, influenced by a predefined motion model, and the Helmholtz decomposition in the optic flow equation reduces the average angular error substantially (20%-25%) with respect to a similar technique that employs only standard derivatives in the smoothness term. In section 5.3 we extracted the motion field of a phantom of which we know the ground truth of and compared the performance of this optic flow method with the performance of other optic flow methods well known in the literature, such as the Horn and Schunck [76] approach, the Lucas and Kanade [111] technique and the tuple image multi-scale optic flow constraint equation of Van Assen et al. [163]. Tests showed that the proposed optic flow methodology provides the smallest average angular error (AAE = 3.84 degrees) and L2 norm = 0.1. In this work we employed the Helmholtz decomposition also to study the cardiac behavior, since the vector field decomposition allows to investigate cardiac contraction and cardiac rotation independently. In chapter 4 we carried out an analysis of cardiac motion of ten volunteers and one patient where we estimated the kinetic energy for the different components. This decomposition is useful since it allows to visualize and quantify the contributions of each single vector field component to the heart beat. Local measurements of the kinetic energy have also been used to detect areas of the cardiac walls with little movement. Experiments on a patient and a comparison between a late enhancement cardiac image and an illustration of the cardiac kinetic energy on a bull’s eye plot illustrated that a correspondence between an infarcted area and an area with very small kinetic energy exists. With the aim to extend in the future the proposed optic flow equation to a 3D approach, in chapter 6 we investigated the 3D winding number approach as a tool to locate critical points in volume images. We simplified the mathematics involved with respect to a previous work [150] and we provided several examples and applications such as cardiac motion estimation from 3-dimensional tagged images, follicle and neuronal cell counting. Finally in chapter 7 we continued our investigation on volume tagged MR images, by retrieving the cardiac motion field using a 3-dimensional and simple version of the proposed optic flow equation based on standard derivatives. We showed that the retrieved motion fields display the contracting and rotating behavior of the cardiac muscle. We moreover extracted the through-plane component, which provides a realistic illustration of the vector field and is missed by 2-dimensional approaches

Zoom invariant vision of figural shape: The mathematics of cores

Believing that figural zoom invariance and the cross-figural boundary linking implied by medial loci are important aspects of object shape, we present the mathematics of and algorithms for the extraction of medial loci directly from image intensities. The medial loci called cores are defined as generalized maxima in scale space of a form of medial information that is invariant to translation, rotation, and in particular, zoom. These loci are very insensitive to image disturbances, in strong contrast to previously available medial loci, as demonstrated in a companion paper. Core-related geometric properties and image object representations are laid out which, together with the aforementioned insensitivities, allow the core to be used effectively for a variety of image analysis objectives.

Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences