High-dimensional Shape Space Research Articles

Machine learning approaches for the optimization of packing densities in granular matter.

The fundamental question of how densely granular matter can pack and how this density depends on the shape of the constituent particles has been a longstanding scientific problem. Previous work has mainly focused on empirical approaches based on simulations or mean-field theory to investigate the effect of shape variation on the resulting packing densities, focusing on a small set of pre-defined shapes like dimers, ellipsoids, and spherocylinders. Here we discuss how machine learning methods can support the search for optimally dense packing shapes in a high-dimensional shape space. We apply dimensional reduction and regression techniques based on random forests and neural networks to find novel dense packing shapes by numerical optimization. Moreover, an investigation of the regression function in the dimensionally reduced shape representation allows us to identify directions in the packing density landscape that lead to a strongly non-monotonic variation of the packing density. The predictions obtained by machine learning are compared with packing simulations. Our approach can be more widely applied to optimize the properties of granular matter by varying the shape of its constituent particles.

Adaptive Shape Kernel-Based Mean Shift Tracker in Robot Vision System

This paper proposes an adaptive shape kernel-based mean shift tracker using a single static camera for the robot vision system. The question that we address in this paper is how to construct such a kernel shape that is adaptive to the object shape. We perform nonlinear manifold learning technique to obtain the low-dimensional shape space which is trained by training data with the same view as the tracking video. The proposed kernel searches the shape in the low-dimensional shape space obtained by nonlinear manifold learning technique and constructs the adaptive kernel shape in the high-dimensional shape space. It can improve mean shift tracker performance to track object position and object contour and avoid the background clutter. In the experimental part, we take the walking human as example to validate that our method is accurate and robust to track human position and describe human contour.

Real-Time Nonlinear Shape Interpolation

We introduce a scheme for real-time nonlinear interpolation of a set of shapes. The scheme exploits the structure of the shape interpolation problem, in particular the fact that the set of all possible interpolated shapes is a low-dimensional object in a high-dimensional shape space. The interpolated shapes are defined as the minimizers of a nonlinear objective functional on the shape space. Our approach is to construct a reduced optimization problem that approximates its unreduced counterpart and can be solved in milliseconds. To achieve this, we restrict the optimization to a low-dimensional subspace that is specifically designed for the shape interpolation problem. The construction of the subspace is based on two components: a formula for the calculation of derivatives of the interpolated shapes and a Krylov-type sequence that combines the derivatives and the Hessian of the objective functional. To make the computational cost for solving the reduced optimization problem independent of the resolution of the example shapes, we combine the dimensional reduction with schemes for the efficient approximation of the reduced nonlinear objective functional and its gradient. In our experiments, we obtain rates of 20--100 interpolated shapes per second, even for the largest examples which have 500k vertices per example shape.

Analysis of principal nested spheres

A general framework for a novel non-geodesic decomposition of high-dimensional spheres or high-dimensional shape spaces for planar landmarks is discussed. The decomposition, principal nested spheres, leads to a sequence of submanifolds with decreasing intrinsic dimensions, which can be interpreted as an analogue of principal component analysis. In a number of real datasets, an apparent one-dimensional mode of variation curving through more than one geodesic component is captured in the one-dimensional component of principal nested spheres. While analysis of principal nested spheres provides an intuitive and flexible decomposition of the high-dimensional sphere, an interesting special case of the analysis results in finding principal geodesics, similar to those from previous approaches to manifold principal component analysis. An adaptation of our method to Kendall's shape space is discussed, and a computational algorithm for fitting principal nested spheres is proposed. The result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation, as exemplified by several datasets.

Imaging brain fibrillar β-amyloid in adults with Down syndrome (DS) using [11C] PiB-PET: acceptability, feasibility and preliminary results

age-related changes in the hippocampal shapes of healthy subjects across lifespan. Methods: We automatically delineate hippocampi from 302 MRI scans of healthy subjects (18-94 years; 114 males and 188 females) and apply LDDMM to find the optimal initial momentum that describes the hippocampal shape, referred as shape descriptor. We then apply LLDME to embed the high-dimensional hippocampal shape descriptor to a low dimensional space whose first few dimensions characterize the most of age-related variations of the hippocampal shape. Line arregression is performed with linear, quadratic, cubic terms of age as main factors and the embedding dimensions of hippocampal shapes as dependent variable. Hippocampi are first segmented from MRI scans by atlas-based segmentation approach, then hippocampal surface sare registered to a common template hippocampal surface through large diffeomorphic metric surface-based mapping LDDMM. Each hippocampal surfaceis then represented by the initial momentum computed from LDDMM, which encodes full information needed to transform the template to the target surface.Dimensionality of the high dimensional shape space is reduced by the locally linear diffeomorphic metric embedding (LLDME) approach that we developed recently in order to map anatomical variations onto low dimensional space for visualization and analysis. After projection, statistical association between low dimensional embedding and age is explored by regression analysis using polynomial model. Aging groups are also classified based on the low dimensional embeddings, and leave-one-out approach is used to estimate the classification error. Results: The shape descriptor of the hippocampus is projected to a low dimensional space whose first dimension is highly correlated with age (r 1⁄4 0.76, p < 0.0001). Figure 1 shows its distribution with respect to age. When it is fitted with age, a third-order polynomial model gives the best fit (R1⁄4 0.66), suggesting age-related processes of the hippocampal shape are non-linear throughout the lifespan with no/slow change before 50s, but accelerated change after 50s. In contrast, such nonlinear age-related processes are not clearly apparent in the hippocampal volume normalized by the total brain volume (Figure 2). The linear (green line) and quadratic fittings (redline) of the normalized hippocampal volume are very close to each other with R 1⁄4 0.014and R 1⁄4 0.03. Conclusions: Nonlinear age-related changes of hippocampal morphology are similar to those of whole brain. Such nonlinear relation is present in terms of the local hippocampal shape but not obvious in its size.

Dynamic Multi‐View Exploration of Shape Spaces

AbstractStatistical shape modeling is a widely used technique for the representation and analysis of the shapes and shape variations present in a population. A statistical shape model models the distribution in a high dimensional shape space, where each shape is represented by a single point.We present a design study on the intuitive exploration and visualization of shape spaces and shape models. Our approach focuses on the dual‐space nature of these spaces. The high‐dimensional shape space represents the population, whereas object space represents the shape of the 3D object associated with a point in shape space.A 3D object view provides local details for a single shape. The high dimensional points in shape space are visualized using a 2D scatter plot projection, the axes of which can be manipulated interactively. This results in a dynamic scatter plot, with the further extension that each point is visualized as a small version of the object shape that it represents. We further enhance the population‐object duality with a new type of view aimed at shape comparison. This new “shape evolution view” visualizes shape variability along a single trajectory in shape space, and serves as a link between the two spaces described above.Our three‐view exploration concept strongly emphasizes linked interaction between all spaces. Moving the cursor over the scatter plot or evolution views, shapes are dynamically interpolated and shown in the object view. Conversely, camera manipulation in the object view affects the object visualizations in the other views. We present a GPU‐accelerated implementation, and show the effectiveness of the three‐view approach using a number of real‐world cases. In these, we demonstrate how this multi‐view approach can be used to visually explore important aspects of a statistical shape model, including specificity, compactness and reconstruction error.

MONTE CARLO SIMULATION OF ISING-LIKE IMMUNOLOGICAL SHAPE SPACE

The high-dimensional shape space for the antibodies of the immune system is simulated with an Ising-like interaction. However, instead of the molecular field being linear in the sum of the neighbor spins, we take it as quadratic and negative. In this way the bell-shaped response curve of biological immune systems is approximated, as a probabilistic generalization of window automata. We find phase transitions only in five and more dimensions, not in two to four, for nearest-neighbor interactions.