Jeff Calder

Optimal real-time Q-ball imaging using regularized Kalman filtering with incremental orientation sets.

Diffusion MRI has become an established research tool for the investigation of tissue structure and orientation. Since its inception, Diffusion MRI has expanded considerably to include a number of variations such as diffusion tensor imaging (DTI), diffusion spectrum imaging (DSI) and Q-ball imaging (QBI). The acquisition and analysis of such data is very challenging due to its complexity. Recently, an exciting new Kalman filtering framework has been proposed for DTI and QBI reconstructions in real-time during the repetition time (TR) of the acquisition sequence. In this article, we first revisit and thoroughly analyze this approach and show it is actually sub-optimal and not recursively minimizing the intended criterion due to the Laplace-Beltrami regularization term. Then, we propose a new approach that implements the QBI reconstruction algorithm in real-time using a fast and robust Laplace-Beltrami regularization without sacrificing the optimality of the Kalman filter. We demonstrate that our method solves the correct minimization problem at each iteration and recursively provides the optimal QBI solution. We validate with real QBI data that our proposed real-time method is equivalent in terms of QBI estimation accuracy to the standard offline processing techniques and outperforms the existing solution. Last, we propose a fast algorithm to recursively compute gradient orientation sets whose partial subsets are almost uniform and show that it can also be applied to the problem of efficiently ordering an existing point-set of any size. This work enables a clinician to start an acquisition with just the minimum number of gradient directions and an initial estimate of the orientation distribution functions (ODF) and then the next gradient directions and ODF estimates can be recursively and optimally determined, allowing the acquisition to be stopped as soon as desired or at any iteration with the optimal ODF estimates. This opens new and interesting opportunities for real-time feedback for clinicians during an acquisition and also for researchers investigating into optimal diffusion orientation sets and real-time fiber tracking and connectivity mapping.

A variational approach to bone segmentation in CT images

We present a variational approach for segmenting bone structures in Computed Tomography (CT) images. We introduce a novel functional on the space of image segmentations, and subsequently minimize this functional through a gradient descent partial differential equation. The functional we propose provides a measure of similarity of the intensity characteristics of the bone and tissue regions through a comparison of their cumulative distribution functions; minimizing this similarity measure therefore yields the maximal separation between the two regions. We perform the minimization of our proposed functional using level set partial differential equations; in addition to numerical stability, this yields topology independence, which is especially useful in the context of CT bone segmentation where a bone region may consist of several disjoint pieces. Finally, we present an extensive validation of our method against expert manual segmentation on CT images of the wrist, ankle, foot, and pelvis.

Image Sharpening via Sobolev Gradient Flows

Motivated by some recent work in active contour applications, we study the use of Sobolev gradients for PDE-based image diffusion and sharpening. We begin by studying, for the case of isotropic diffusion, the gradient descent/ascent equation obtained by modifying the usual metric on the space of images, which is the

A Hamilton--Jacobi Equation for the Continuum Limit of Nondominated Sorting

L^2

A PDE-based approach to nondominated sorting

metric, to a Sobolev metric. We present existence and uniqueness results for the Sobolev isotropic diffusion, derive a number of maximum principles, and show that the differential equations are stable and well-posed both in the forward and backward directions. This allows us to apply the Sobolev flow in the backward direction for sharpening. Favorable comparisons to the well-known shock filter for sharpening are demonstrated. Finally, we continue to exploit this same well-posed behavior both forward and backward in order to formulate new constrained gradient flows on higher order energy functionals which preserve the first order energy of the original image for interesting combined smoothing and sharpening effects.

Directed Last Passage Percolation with Discontinuous Weights

We show that nondominated sorting of a sequence

Pareto-Depth for Multiple-Query Image Retrieval

X_1,\dots,X_n

Multicriteria Similarity-Based Anomaly Detection Using Pareto Depth Analysis

of independent and identically distributed random variables in

New Possibilities in Image Diffusion and Sharpening via High-Order Sobolev Gradient Flows

\mathbb{R}^d

On the circular area signature for graphs

has a continuum limit that corresponds to solving a Hamilton--Jacobi equation involving the probability density function