Anuj Srivastava

• 
  Statistical Models
  Karthik Bharath, Anuj Srivastava
  • Statistical modeling of functional data: Karcher mean, multiple registration, fPCA in quotient space.
  • Curves on manifolds: smoothing splines.
  • Trajectories on Manifolds: issues; transported SRVFs, vector bundle representations, tangent bundle representation.
• 
  Elastic Shape Analysis of Surfaces
  Anuj Srivastava
  • Goals and motivation; past approaches; Iterative Closest Point (ICP) algorithm.
  • Representations of surfaces: coordinate functions, gradient field, surface normal, 1nd fundamental form, 2nd fundamental form.
  • Registration problem; elastic Riemannian metric; square-root normal fields (SRNFs); inversion problem; shape geodesics.
  • Statistics of shapes: sample mean, shape PCA, shape models, clustering and classification; symmetric shapes.
• 
  Fundamental Mathematical Formulations, Recent Progress and Open Problems
  Eric Klassen, Anuj Srivastava
  • scalar FDA
  • characterization of orbits
  • existence of optimal matching for piecewise-linear curves
  • C1 functions
• 
  Elastic Shape Analysis of Curves
  Anuj Srivastava
  • Goals and motivation; Past approaches in shape analysis; shape-preserving group actions; Kendall's shape analysis; active shape models; registration problem; non-elastic framework.
  • Elastic Riemannian metric; SRVF; open curves and closed curves; shape geodesics.
  • Statistics of shapes: sample mean, shape PCA, shape models, clustering and classification; symmetric shapes.
• 
  Elastic Functional Data Analysis
  Anuj Srivastava
  • Background from geometry and algebra: manifolds, Riemannian metric, geodesics, exponential and inverse exponential map, Karcher/Frechet means; functional spaces of interest: set of probability density functions (pdfs), set of warping functions, group, group actions, quotient spaces.
  • Phase variability and the registration problem; L2 metric and pinching problem; dynamic programming algorithm; penalized L2 metric.
  • Invariance, square-root velocity function (SRVF); Fisher-Rao Riemannian metric; change of variables; registration using SRVFs; exact solution and approximation; quotient space metric, amplitude and phase distances.
• 
  Introduction and Background
  Anuj Srivastava
  • Introduction and motivation.
  • Function Spaces: norms, Hilbert space; L2 metric; complete orthonormal basis; sample mean, sample covariance; functional Principal Component Analysis (fPCA); functional regression models; generative models for functional data; least squares curve fitting.
• 
  Generalizations of the Square Root Velocity Framework to Trajectories in Manifolds
  Eric Klassen, Anuj Srivastava

  In previous talks, SRVF method has been discussed as a powerful and efficient way to analyze collections of curves in a Euclidean space. However, in many applications the data to be analyzed consists of trajectories in a manifold, rather than in a Euclidean space. Examples include hurricane paths on the surface of the earth, paths of covariance matrices that arise in the study of brain connectivity, and paths of images that lie in a shape space. The comparison of curves in a manifold is more subtle than in a Euclidean space, because one cannot subtract velocity vectors that lie in two different tangent spaces! In this talk, we discuss several methods of adapting the SRVF method to this situation, and comment on the advantages and disadvantages of each method.

• 
  A Formal Definition of Phase and Amplitude in Functional Data
  Anuj Srivastava
  A Formal Definition of Phase and Amplitude in Functional Data
• 
  Metric-Based Approaches for shape analysis
  Anuj Srivastava

  Shape analysis of objects intrinsically involves registration of points across objects. While registration is historically treated as a pre-processing step, a more recent trend is to incorporate it as an integral step in shape comparison. There are two distinct approaches for shape registration -- model based and metric based.

  
  

  (1) In the first case, one assumes a model upfront that is generally of the type:
  observation = deformed template & noise.

  
  

  The goal then is to estimate the template and individual deformations given multiple observations.

  
  

  (2) The second approach does not start with a model but uses a metric that forms an objective function for both: registration of points across objects and quantification of difference in shapes. This metric facilitates a consistent framework where registration and ensuing shape analysis (summarization, modes of variations, etc) are all performed under the same metric (and not as either pre- and/or post-processing).

  
  

  It is the second approach that I will elaborate upon in this talk. The key property of valid metrics for registration is that the re-parameterization group acts by isometries under this metric. While such metrics are beginning to emerge for shape analysis of curves, surfaces, and even images, they are often too difficult to allow efficient solutions. In some cases a simple change of variable converts these invariant metrics into standard Euclidean metrics, the classical computational solutions apply.

  
  

  Examples include square-root velocity functions for curves and square-root normal fields for surfaces.

  
  

  I will demonstrate these ideas using different types of objects: curves in Euclidean spaces, trajectories on Riemannian manifolds, 2D surfaces, and vector-valued images.