Affine Registration of label maps in Label Space

Two key aspects of coupled multi-object shape\ud analysis and atlas generation are the choice of representation\ud and subsequent registration methods used to align the sample\ud set. For example, a typical brain image can be labeled into\ud three structures: grey matter, white matter and cerebrospinal\ud fluid. Many manipulations such as interpolation, transformation,\ud smoothing, or registration need to be performed on these images\ud before they can be used in further analysis. Current techniques\ud for such analysis tend to trade off performance between the two\ud tasks, performing well for one task but developing problems when\ud used for the other.\ud This article proposes to use a representation that is both\ud flexible and well suited for both tasks. We propose to map object\ud labels to vertices of a regular simplex, e.g. the unit interval for\ud two labels, a triangle for three labels, a tetrahedron for four\ud labels, etc. This representation, which is routinely used in fuzzy\ud classification, is ideally suited for representing and registering\ud multiple shapes. On closer examination, this representation\ud reveals several desirable properties: algebraic operations may\ud be done directly, label uncertainty is expressed as a weighted\ud mixture of labels (probabilistic interpretation), interpolation is\ud unbiased toward any label or the background, and registration\ud may be performed directly.\ud We demonstrate these properties by using label space in a gradient\ud descent based registration scheme to obtain a probabilistic\ud atlas. While straightforward, this iterative method is very slow,\ud could get stuck in local minima, and depends heavily on the initial\ud conditions. To address these issues, two fast methods are proposed\ud which serve as coarse registration schemes following which the\ud iterative descent method can be used to refine the results. Further,\ud we derive an analytical formulation for direct computation of the\ud "group mean" from the parameters of pairwise registration of all\ud the images in the sample set. We show results on richly labeled\ud 2D and 3D data sets

Nonrigid reconstruction of 3D breast surfaces with a low-cost RGBD camera for surgical planning and aesthetic evaluation

Accounting for 26% of all new cancer cases worldwide, breast cancer remains the most common form of cancer in women. Although early breast cancer has a favourable long-term prognosis, roughly a third of patients suffer from a suboptimal aesthetic outcome despite breast conserving cancer treatment. Clinical-quality 3D modelling of the breast surface therefore assumes an increasingly important role in advancing treatment planning, prediction and evaluation of breast cosmesis. Yet, existing 3D torso scanners are expensive and either infrastructure-heavy or subject to motion artefacts. In this paper we employ a single consumer-grade RGBD camera with an ICP-based registration approach to jointly align all points from a sequence of depth images non-rigidly. Subtle body deformation due to postural sway and respiration is successfully mitigated leading to a higher geometric accuracy through regularised locally affine transformations. We present results from 6 clinical cases where our method compares well with the gold standard and outperforms a previous approach. We show that our method produces better reconstructions qualitatively by visual assessment and quantitatively by consistently obtaining lower landmark error scores and yielding more accurate breast volume estimates

Stochastic surface mesh reconstruction

This research was funded by TUBITAK – The Scientific and Technological Research Council of Turkey (Project ID: 115Y239) and by the Scientific Research Projects of Bülent Ecevit University (Project ID: 2015-47912266-01)A generic and practical methodology is presented for 3D surface mesh reconstruction from the terrestrial laser scanner (TLS) derived point clouds. It has two main steps. The first step deals with developing an anisotropic point error model, which is capable of computing the theoretical precisions of 3D coordinates of each individual point in the point cloud. The magnitude and direction of the errors are represented in the form of error ellipsoids. The following second step is focused on the stochastic surface mesh reconstruction. It exploits the previously determined error ellipsoids by computing a point-wise quality measure, which takes into account the semi-diagonal axis length of the error ellipsoid. The points only with the least errors are used in the surface triangulation. The remaining ones are automatically discarded.Publisher's Versio

Enhancing 3D Autonomous Navigation Through Obstacle Fields: Homogeneous Localisation and Mapping, with Obstacle-Aware Trajectory Optimisation

Small flying robots have numerous potential applications, from quadrotors for search and rescue, infrastructure inspection and package delivery to free-flying satellites for assistance activities inside a space station. To enable these applications, a key challenge is autonomous navigation in 3D, near obstacles on a power, mass and computation constrained platform. This challenge requires a robot to perform localisation, mapping, dynamics-aware trajectory planning and control. The current state-of-the-art uses separate algorithms for each component. Here, the aim is for a more homogeneous approach in the search for improved efficiencies and capabilities. First, an algorithm is described to perform Simultaneous Localisation And Mapping (SLAM) with physical, 3D map representation that can also be used to represent obstacles for trajectory planning: Non-Uniform Rational B-Spline (NURBS) surfaces. Termed NURBSLAM, this algorithm is shown to combine the typically separate tasks of localisation and obstacle mapping. Second, a trajectory optimisation algorithm is presented that produces dynamically-optimal trajectories with direct consideration of obstacles, providing a middle ground between path planners and trajectory smoothers. Called the Admissible Subspace TRajectory Optimiser (ASTRO), the algorithm can produce trajectories that are easier to track than the state-of-the-art for flight near obstacles, as shown in flight tests with quadrotors. For quadrotors to track trajectories, a critical component is the differential flatness transformation that links position and attitude controllers. Existing singularities in this transformation are analysed, solutions are proposed and are then demonstrated in flight tests. Finally, a combined system of NURBSLAM and ASTRO are brought together and tested against the state-of-the-art in a novel simulation environment to prove the concept that a single 3D representation can be used for localisation, mapping, and planning

Smooth Distance Approximation

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in $\mathbb{R}^d$ from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute $\varepsilon$-approximation to this query in time $O(\log (1/\varepsilon))$ using $O(1/\varepsilon^{d/2})$ storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian.Comment: To appear in the European Symposium on Algorithms (ESA) 202

Smooth Distance Approximation

Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously. In many real-world applications of geometric data structures, it is assumed that query results are continuous, free of jump discontinuities. This is at odds with many modern data structures in computational geometry, which employ approximations to achieve efficiency, but these approximations often suffer from discontinuities. In this paper, we present a general method for transforming an approximate but discontinuous data structure into one that produces a smooth approximation, while matching the asymptotic space efficiencies of the original. We achieve this by adapting an approach called the partition-of-unity method, which smoothly blends multiple local approximations into a single smooth global approximation. We illustrate the use of this technique in a specific application of approximating the distance to the boundary of a convex polytope in ?^d from any point in its interior. We begin by developing a novel data structure that efficiently computes an absolute ?-approximation to this query in time O(log (1/?)) using O(1/?^{d/2}) storage space. Then, we proceed to apply the proposed partition-of-unity blending to guarantee the smoothness of the approximate distance field, establishing optimal asymptotic bounds on the norms of its gradient and Hessian