We develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. Based on Riemann surface structure, we can then canonically partition the surface into patches. Each of these patches can be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. To illustrate the technique, we computed conformal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles. We found that the resulting parameterizations were consistent across

Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. Several studies have demonstrated the malfunctioning of postural balance, proprioception, and equilibrium control in patients with AIS. Since these functions are closely related to structures in and around the brainstem, the morphometry of the brainstem surface is of utmost importance. In this paper, we propose an effective method to accurately compute the registration between brainstem surfaces. Four consistent features, which describe the global geometry of the brainstem, are automatically extracted to guide the surface registration. Using the discrete Ricci flow method, brainstem surfaces are parameterized conformally onto the quadrilaterally-faced hexahedron, through which the brainstem registration can be obtained. Our registration algorithm can guarantee the exact landmark correspondence between brainstem surfaces. With the obtained registration, a shape energy can be defined to measure the local shape difference between different brainstem surfaces. We have tested our algorithms on 30 real brainstem surfaces extracted from MRIs of 15 normal subjects and 15 AIS patients. Experimental results show the efficacy of the proposed algorithm to register brainstem surfaces, which matches landmark features consistently. The computed registration can be used for the morphometry of brainstems.

We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.

This paper describes a high-resolution, real-time, three-dimensional shape measurement system using the modified two-plus-one phase-shifting algorithm. The data acquisition speed is as high as 60 frames/s with an image resolution of 640480 pixels per frame. Experiments demonstrated that the system was able to acquire the dynamic changing objects such as facial geometric shape changes when the subject is speaking, and the modified two-plus-one phase-shifting algorithm can further alleviate the error due to motion. Applications of this system include manufacturing, online inspection, medical imaging, compute vision, and computer graphics.

This paper describes a Graphics Processing Unit (GPU)-assisted real-time three-dimensional shape measurement system. Our experiments demonstrated that the absolute coordinates calculation and rendering speed of a GPU is more than four times faster than that of a dual CPU workstation with the same graphics card. By implementing the GPU into our system, we realized simultaneous absolute coordinate acquisition, reconstruction and display at 30 frames per second with a resolution of approximately 266K points per frame. Moreover, a 2+1 phase-shifting algorithm was employed to alleviate the measurement error caused by motion. Applications of the system include medical imaging, manufacturing, entertainment, and security.