Compact Representations for Fast Nonrigid Registration of Medical Images

We develop efficient techniques for the non-rigid registration of medical images by using representations that adapt to the anatomy found in such images. Images of anatomical structures typically have uniform intensity interiors and smooth boundaries. We create methods to represent such regions compactly using tetrahedra. Unlike voxel-based representations, tetrahedra can accurately describe the expected smooth surfaces of medical objects. Furthermore, the interior of such objects can be represented using a small number of tetrahedra. Rather than describing a medical object using tens of thousands of voxels, our representations generally contain only a few thousand elements. Tetrahedra facilitate the creation of efficient non-rigid registration algorithms based on finite element methods (FEM). We create a fast, FEM-based method to non-rigidly register segmented anatomical structures from two subjects. Using our compact tetrahedral representations, this method generally requires less than one minute of processing time on a desktop PC. We also create a novel method for the non-rigid registration of gray scale images. To facilitate a fast method, we create a tetrahedral representation of a displacement field that automatically adapts to both the anatomy in an image and to the displacement field. The resulting algorithm has a computational cost that is dominated by the number of nodes in the mesh (about 10,000), rather than the number of voxels in an image (nearly 10,000,000). For many non-rigid registration problems, we can find a transformation from one image to another in five minutes. This speed is important as it allows use of the algorithm during surgery. We apply our algorithms to find correlations between the shape of anatomical structures and the presence of schizophrenia. We show that a study based on our representations outperforms studies based on other representations. We also use the results of our non-rigid registration algorithm as the basis of a segmentation algorithm. That algorithm also outperforms other methods in our tests, producing smoother segmentations and more accurately reproducing manual segmentations.

Virtual Model Control of a Biped Walking Robot

The transformation from high level task specification to low level motion control is a fundamental issue in sensorimotor control in animals and robots. This thesis develops a control scheme called virtual model control which addresses this issue. Virtual model control is a motion control language which uses simulations of imagined mechanical components to create forces, which are applied through joint torques, thereby creating the illusion that the components are connected to the robot. Due to the intuitive nature of this technique, designing a virtual model controller requires the same skills as designing the mechanism itself. A high level control system can be cascaded with the low level virtual model controller to modulate the parameters of the virtual mechanisms. Discrete commands from the high level controller would then result in fluid motion. An extension of Gardner's Partitioned Actuator Set Control method is developed. This method allows for the specification of constraints on the generalized forces which each serial path of a parallel mechanism can apply. Virtual model control has been applied to a bipedal walking robot. A simple algorithm utilizing a simple set of virtual components has successfully compelled the robot to walk eight consecutive steps.

Virtual Model Control of a Hexapod Walking Robot

Since robots are typically designed with an individual actuator at each joint, the control of these systems is often difficult and non-intuitive. This thesis explains a more intuitive control scheme called Virtual Model Control. This thesis also demonstrates the simplicity and ease of this control method by using it to control a simulated walking hexapod. Virtual Model Control uses imagined mechanical components to create virtual forces, which are applied through the joint torques of real actuators. This method produces a straightforward means of controlling joint torques to produce a desired robot behavior. Due to the intuitive nature of this control scheme, the design of a virtual model controller is similar to the design of a controller with basic mechanical components. The ease of this control scheme facilitates the use of a high level control system which can be used above the low level virtual model controllers to modulate the parameters of the imaginary mechanical components. In order to apply Virtual Model Control to parallel mechanisms, a solution to the force distribution problem is required. This thesis uses an extension of Gardner`s Partitioned Force Control method which allows for the specification of constrained degrees of freedom. This virtual model control technique was applied to a simulated hexapod robot. Although the hexapod is a highly non-linear, parallel mechanism, the virtual models allowed text-book control solutions to be used while the robot was walking. Using a simple linear control law, the robot walked while simultaneously balancing a pendulum and tracking an object.

Image-Based View Synthesis

We present a new method for rendering novel images of flexible 3D objects from a small number of example images in correspondence. The strength of the method is the ability to synthesize images whose viewing position is significantly far away from the viewing cone of the example images ("view extrapolation"), yet without ever modeling the 3D structure of the scene. The method relies on synthesizing a chain of "trilinear tensors" that governs the warping function from the example images to the novel image, together with a multi-dimensional interpolation function that synthesizes the non-rigid motions of the viewed object from the virtual camera position. We show that two closely spaced example images alone are sufficient in practice to synthesize a significant viewing cone, thus demonstrating the ability of representing an object by a relatively small number of model images --- for the purpose of cheap and fast viewers that can run on standard hardware.

An Immersed Interface Method for the Incompressible Navier-Stokes Equations in Irregular Domains

We present an immersed interface method for the incompressible Navier Stokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the no-slip condition on the boundary is satisfied, singular forces are applied on the fluid at the immersed boundary. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of singular forces is determined by solving a small system of equations at each time step. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity.

FastAero – A Precorrected FFT – Fast Multipole Tree Steady and Unsteady Potential Flow Solver

In this paper a precorrected FFT-Fast Multipole Tree (pFFT-FMT) method for solving the potential flow around arbitrary three dimensional bodies is presented. The method takes advantage of the efficiency of the pFFT and FMT algorithms to facilitate more demanding computations such as automatic wake generation and hands-off steady and unsteady aerodynamic simulations. The velocity potential on the body surfaces and in the domain is determined using a pFFT Boundary Element Method (BEM) approach based on the Green’s Theorem Boundary Integral Equation. The vorticity trailing all lifting surfaces in the domain is represented using a Fast Multipole Tree, time advected, vortex participle method. Some simple steady state flow solutions are performed to demonstrate the basic capabilities of the solver. Although this paper focuses primarily on steady state solutions, it should be noted that this approach is designed to be a robust and efficient unsteady potential flow simulation tool, useful for rapid computational prototyping.