Distance-Based Shape Statistics

International audienceThis article deals with statistics on sets of shapes. The approach is based on the Hausdorff distance between shapes. The choice of the Hausdorff distance between shapes is itself not fundamental since the same framework could be applied with another distance. We first define a smooth approximation of the Hausdorff distance and build non-supervised warpings between shapes by a gradient descent of the approximation. Local minima can be avoided by changing the scalar product in the tangent space of the shape being warped.When non-supervised warping fails, we present a way to guide the evolution with a small number of landmarks. Thanks to the warping fields, we can define the mean of a set of shapes and express statistics on them. Finally, we come back to the initial distance between shapes and use it to represent a set of shapes by a graph, which with the technic of graph Laplacian leads to a way of projecting shapes onto a low dimensional space

Manifold Learning in MR spectroscopy using nonlinear dimensionality reduction and unsupervised clustering

Purpose To investigate whether nonlinear dimensionality reduction improves unsupervised classification of 1H MRS brain tumor data compared with a linear method. Methods In vivo single-voxel 1H magnetic resonance spectroscopy (55 patients) and 1H magnetic resonance spectroscopy imaging (MRSI) (29 patients) data were acquired from histopathologically diagnosed gliomas. Data reduction using Laplacian eigenmaps (LE) or independent component analysis (ICA) was followed by k-means clustering or agglomerative hierarchical clustering (AHC) for unsupervised learning to assess tumor grade and for tissue type segmentation of MRSI data. Results An accuracy of 93% in classification of glioma grade II and grade IV, with 100% accuracy in distinguishing tumor and normal spectra, was obtained by LE with unsupervised clustering, but not with the combination of k-means and ICA. With 1H MRSI data, LE provided a more linear distribution of data for cluster analysis and better cluster stability than ICA. LE combined with k-means or AHC provided 91% accuracy for classifying tumor grade and 100% accuracy for identifying normal tissue voxels. Color-coded visualization of normal brain, tumor core, and infiltration regions was achieved with LE combined with AHC. Conclusion Purpose To investigate whether nonlinear dimensionality reduction improves unsupervised classification of 1H MRS brain tumor data compared with a linear method. Methods In vivo single-voxel 1H magnetic resonance spectroscopy (55 patients) and 1H magnetic resonance spectroscopy imaging (MRSI) (29 patients) data were acquired from histopathologically diagnosed gliomas. Data reduction using Laplacian eigenmaps (LE) or independent component analysis (ICA) was followed by k-means clustering or agglomerative hierarchical clustering (AHC) for unsupervised learning to assess tumor grade and for tissue type segmentation of MRSI data. Results An accuracy of 93% in classification of glioma grade II and grade IV, with 100% accuracy in distinguishing tumor and normal spectra, was obtained by LE with unsupervised clustering, but not with the combination of k-means and ICA. With 1H MRSI data, LE provided a more linear distribution of data for cluster analysis and better cluster stability than ICA. LE combined with k-means or AHC provided 91% accuracy for classifying tumor grade and 100% accuracy for identifying normal tissue voxels. Color-coded visualization of normal brain, tumor core, and infiltration regions was achieved with LE combined with AHC. Conclusion The LE method is promising for unsupervised clustering to separate brain and tumor tissue with automated color-coding for visualization of 1H MRSI data after cluster analysis

Modified Large Margin Nearest Neighbor Metric Learning for Regression

The main objective of this letter is to formulate a new approach of learning a Mahalanobis distance metric for nearest neighbor regression from a training sample set. We propose a modified version of the large margin nearest neighbor metric learning method to deal with regression problems. As an application, the prediction of post-operative trunk 3-D shapes in scoliosis surgery using nearest neighbor regression is described. Accuracy of the proposed method is quantitatively evaluated through experiments on real medical data.IRSC / CIH

Statistical model based 3D shape prediction of postoperative trunks for non-invasive scoliosis surgery planning

One of the major concerns of scoliosis patients undergoing surgical treatment is the aesthetic aspect of the surgery outcome. It would be useful to predict the postoperative appearance of the patient trunk in the course of a surgery planning process in order to take into account the expectations of the patient. In this paper, we propose to use least squares support vector regression for the prediction of the postoperative trunk 3D shape after spine surgery for adolescent idiopathic scoliosis. Five dimensionality reduction techniques used in conjunction with the support vector machine are compared. The methods are evaluated in terms of their accuracy, based on the leave-one-out cross-validation performed on a database of 141 cases. The results indicate that the 3D shape predictions using a dimensionality reduction obtained by simultaneous decomposition of the predictors and response variables have the best accuracy.CIHR / IRS

A physically based trunk soft tissue modeling for scoliosis surgery planning systems

One of the major concerns of scoliotic patients undergoing spinal correction surgery is the trunk's external appearance after the surgery. This paper presents a novel incremental approach for simulating postoperative trunk shape in scoliosis surgery. Preoperative and postoperative trunk shapes data were obtained using three-dimensional medical imaging techniques for seven patients with adolescent idiopathic scoliosis. Results of qualitative and quantitative evaluations, based on the comparison of the simulated and actual postoperative trunk surfaces, showed an adequate accuracy of the method. Our approach provides a candidate simulation tool to be used in a clinical environment for the surgery planning process.IRSC / CIH

Spatio-Temporal Video Segmentation with Shape Growth or Shrinkage Constraint

We propose a new method for joint segmentation of monotonously growing or shrinking shapes in a time sequence of noisy images. The task of segmenting the image time series is expressed as an optimization problem using the spatio-temporal graph of pixels, in which we are able to impose the constraint of shape growth or of shrinkage by introducing monodirectional infinite links connecting pixels at the same spatial locations in successive image frames. The globally optimal solution is computed with a graph cut. The performance of the proposed method is validated on three applications: segmentation of melting sea ice floes and of growing burned areas from time series of 2D satellite images, and segmentation of a growing brain tumor from sequences of 3D medical scans. In the latter application, we impose an additional intersequences inclusion constraint by adding directed infinite links between pixels of dependent image structures

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound