Kevin M. Holt

Total Nuclear Variation and Jacobian Extensions of Total Variation for Vector Fields

We explore a class of vectorial total variation (VTV) measures formed as the spatial sum of a pixel-wise matrix norm of the Jacobian of a vector field. We give a theoretical treatment that indicates that, while color smearing and affine-coupling bias (often reported as gray-scale bias) are typically cited as drawbacks for VTV, these are actually fundamental to smoothing vector direction (i.e., smoothing hue and saturation in color images). In addition, we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing Jacobian rank. We thus propose total nuclear variation (TNV), and since nuclear norm is the convex envelope of matrix rank, we argue that TNV is the optimal convex regularizer for enforcing shared directions. We also propose extended Jacobians, which use larger neighborhoods than the conventional finite difference operator, and we discuss efficient VTV optimization algorithms. In simple color image denoising experiments, TNV outperformed other common VTV regularizers, and was further improved by using extended Jacobians. TNV was also competitive with the method of nonlocal means, often outperforming it by 0.25-2 dB when using extended Jacobians.

Angular regularization of vector-valued signals

A new class of pair-wise regularizers is proposed for vector-valued signals, including several new regularizers that penalize the angular difference between neighboring vectors. The pair-wise model is a generalization of the conventional difference model that includes discretized Tikhonov, L1, and Total Variation (TV) as special cases. Whereas Tikhonov and TV regularization have become popular for encouraging signal values to be smooth or piecewise smooth, respectively, angular regularizers are proposed to encourage vector directions to be smooth or piecewise smooth. The regularizers are simple, effective, and general, making them easily applicable to a wide array of problems. Experimental results demonstrate the methods effectiveness in denoising vector fields and color images.

Geometric calibration of detectors with discrete irregularities for computed tomography

For accurate CT reconstruction, it is important to know the geometric position of every detector channel relative to the X-ray source and the rotation axis. Often, such as for truly equally spaced detectors, it may suffice just to accurately know the gross geometry. However, for some detector designs, a detailed description of the fine-scale channel locations may also be necessary. While there are numerous methods to perform fine-scale calibration, such methods generally assume a continuous distortion (typically for image intensifiers) and are thus unsuitable for detectors with discrete distortions such as irregularly placed discrete sensors, tiled flat panels, or multiple flat segments arranged to form a polygonal approximation to an arc. In this paper, a method is proposed to measure both gross and fine geometry from a single simple calibration scan in a way that properly characterizes discrete irregularities. Experimental results show the proposed method to be rather effective on polygonal arrays. While the method is derived and demonstrated for fan beam, a discussion is given on extending it to cone beam CT.

Geometric calibration and distortion calibration for CT from scans of unknown objects using complementary rays

To achieve good image quality for computed tomography, it is important to accurately know the geometrical relationship between the X-ray source, the axis of rotation, and all of the detector channels. This usually involves knowing gross parameters such as iso-ray coordinate, detector pixel pitch, and source-to-detector distance, but for some detector types such as distorted arrays, polygonal or tiled arrays, or arrays of irregularly placed sparse detectors, it is beneficial to measure a more detailed description of the individual channel locations. Typically, geometric calibration and distortion calibration are performed using specialized phantoms, such as a pin, an array of pellets, or a wire grid, but these can have their practical downsides for certain applications. A promising recent alternative is to calibrate geometry in a way that requires no particular phantom or a priori knowledge of the scanned object -- these approaches are particularly helpful for high magnifications, large heavy objects, frequent calibration, and retrospective calibration. However, until now these approaches have only addressed gross geometry. In this paper, a framework is given which allows one to calibrate both gross and fine geometry from unknown objects. Example images demonstrate the success of the proposed methods on both real and simulated data.