Yen-Hsi Richard Tsai

Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations

We derive a Godunov-type numerical flux for the class of strictly convex, homogeneous Hamiltonians that includes

Fast Sweeping Methods for Static Hamilton--Jacobi Equations

H(p,q)=\sqrt{ap^{2}+bq^{2}-2cpq},

Redistancing by flow of time dependent eikonal equation

c^{2}<ab.

Estimation of 3D Surface Shape and Smooth Radiance from 2D Images: A Level Set Approach

We combine our Godunov numerical fluxes with simple Gauss--Seidel-type iterations for solving the corresponding Hamilton--Jacobi (HJ) equations. The resulting algorithm is fast since it does not require a sorting strategy as found, e.g., in the fast marching method. In addition, it providesa way to compute solutions to a class of HJ equations for which the conventional fast marching method is not applicable. Our experiments indicate convergence after a few iterations, even in rather difficult cases.

Automated Polyp Detection in Colon Capsule Endoscopy

We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions.

Total variation and level set methods in image science

The heterogeneous multiscale methods (HMM) is a general framework for the numerical approximation of multiscale problems. It is here developed for ordinary differential equations containing differe ...

A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations

Construction of signed distance to a given interface is a topic of special interest to level set methods. There are currently, however, few algorithms that can efficiently produce highly accurate solutions. We introduce an algorithm for constructing an approximate signed distance function through manipulation of values calculated from flow of time dependent eikonal equations. We provide operation counts and experimental results to show that this algorithm can efficiently generate solutions with a high order of accuracy. Comparison with the standard level set reinitialization algorithm shows ours is superior in terms of predictability and local construction, which, for example, are critical in local level set methods. We further apply the same ideas to extension of values off interfaces. Together, our proposed approaches can be used to advance the level set method for fast and accurate computations of the latest scientific problems.

Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem

We cast the problem of shape reconstruction of a scene as the global region segmentation of a collection of calibrated images. We assume that the scene is composed of a number of smooth surfaces and a background, both of which support smooth Lambertian radiance functions. We formulate the problem in a variational framework, where the solution (both the shape and radiance of the scene) is a minimizer of a global cost functional which combines a geometric prior on shape, a smoothness prior on radiance and a data fitness score. We estimate the shape and radiance via an alternating minimization: The radiance is computed as the solutions of partial differential equations defined on the surface and the background. The shape is estimated using a gradient descent flow, which is implemented using the level set method. Our algorithm works for scenes with smooth radiances as well as fine homogeneous textures, which are known challenges to traditional stereo algorithms based on local correspondence.

Robotic Path Planning and Visibility with Limited Sensor Data

Colorectal polyps are important precursors to colon cancer, a major health problem. Colon capsule endoscopy is a safe and minimally invasive examination procedure, in which the images of the intestine are obtained via digital cameras on board of a small capsule ingested by a patient. The video sequence is then analyzed for the presence of polyps. We propose an algorithm that relieves the labor of a human operator analyzing the frames in the video sequence. The algorithm acts as a binary classifier, which labels the frame as either containing polyps or not, based on the geometrical analysis and the texture content of the frame.We assume that the polyps are characterized as protrusions that are mostly round in shape. Thus, a best fit ball radius is used as a decision parameter of the classifier. We present a statistical performance evaluation of our approach on a data set containing over 18 900 frames from the endoscopic video sequences of five adult patients. The algorithm achieves 47% sensitivity per frame and 81% sensitivity per polyp at a specificity level of 90%. On average, with a video sequence length of 3747 frames, only 367 false positive frames need to be inspected by an operator.