My research is organized into five overlapping project areas.

## Theoretical and Perceptual Bases of Visualization

Carlos Scheidegger and I are developing an algebraic approach to visualization design [VIS-2014] [talks/Geilo-2016a]. It generalizes mathematical principles used in tensor glyph design [VIS-2010] [VisSym-2004] [EV-2016], and it formalizes guidelines I teach in my visualization classes [teach/SciVis] [talks/VIS-2010] [PDV-2016]. Çağatay Demiralp leads a related line of work, with an emphasis on metric spaces [CGnA-2014]. In 2009, I led a discussion at Dagstuhl [talks/DagVis-2009] about mathematical principles of visualization, which introduced and refined some of the ideas that turned into algebraic visualization design.

Describing how visualizations work should ultimately connect back to human perception. Earlier work [VIS-2002] implements one kind of perceptually-guided visualization design.

Having an underlying theory of how to compute, perceive, and use visualizations will benefit the other computational and application-specific projects below.

## Volumetric Image Data Visualization and Analysis

The increasing power and complexity of scientific imaging, and the variety of its applications, demands new ways to visualize and analyze image volumes acquired during scientific and biomedical investigations. My Masters work simplifies creating informative direct volume renderings [CGnA-2001] [MS-1999] [VV-1998] [SAHPC-1996], and subsequent work with Joe Kniss and Chuck Hansen demonstrates GPU-based volume rendering with multidimensional transfer functions [TVCG-2002] [VIS-2001]. Past collaboration with Mario Capecchi, Charles Keller and others developed methods for small-animal imaging [AR-2008] [MI-2005] [CG-2004].

Current collaborators include Patrick La Rivière [SPIE-2010], William Irvine [PNAS-2014], Callum Ross, Victoria Prince, and Kevin White. The Advanced Photon Source at Argonne and the new Zeiss Z-1 light-sheet microscrope on UChicago campus are particularly exciting data sources.

The implementation of image visualization and analysis is often in terms of particle systems, or in the Diderot language.

## Ridge and Valley Features; Sampling Features with Particles

This includes both mathematical and computational aspects. From a mathematical standpoint, ridge and valley features (collectively, "creases") play an interesting role in visualization and image analysis. With Carl-Fredrik Westin and Xavier Tricoche, I show how anisotropy creases capture major white matter structure in DT-MRI of the brain [MIA-2007] [MICCAI-2006]. Lines of degeneracy in tensor fields (of interest to tensor field topologists) are just special cases of crease lines of tensor mode [VIS-2008] [talks/Stanford-2006].

From an computational standpoint, dynamic particle systems provide a way of sampling and depicting information outside the confines of a rectilinear imaging grid. I use particle systems for tensor visualization by glyph packing [VIS-2006], an efficient alternative to reaction-diffusion texture spots [TVCG-2000].

In particular, scale-space particles [VIS-2009] [talks/Geilo-2016c] can sample crease features, and other image features, in continuous four-dimensional scale-space. The refinement and application of scale-space particles continues, much of it in collaboration with Raúl San José Estépar and James Ross for lung CT analysis [MP-2013] [VIS-2013] [AJRCCM-2013] [ISBI-2012b] [ISBI-2012] [MICCAI-2010b]. I am interested in optimizing the computation of particles [talks/KAUST-2010], and using particles to determine feature stability [talks/BIRS-2011].

## Diderot: a Portable Parallel Language for Computing on Tensor Fields

Parallel computing architectures are advancing faster than are the means of efficiently developing new research software that exploits parallelism, so that it is useful on datasets generated by modern scientific imaging and simulation. Perhaps we should let the compiler do some of the hard work of mapping algorithms to parallel hardware.

With John Reppy and students, I'm working on a new portable parallel programming language, called Diderot, which simplifies implementing algorithms for visualizing and analyzing continuous tensor fields. The work is described in papers [VIS-2015] [PLDI-2012], a poster [VIS-2011], and talks [talks/Geilo-2016b] [talks/Bonn-2013] [talks/SCI-2013].

The Diderot Examples page at github provides detailed information about building the Diderot compiler, and concrete starting points for writing new Diderot programs.

## Tensor Glyphs and Fields, Diffusion MRI (dMRI), and other dMRI models

Tensors offer concise descriptions of complex phenomena, I am interested in simplifying how we can see tensors, and structure on tensor fields. Recent work has expanded the class of tensors with readily legible glyphs from second-order symmetric positive-definite to symmetric indefinite [VIS-2010] and assymetric [EV-2016].

Besides anistropy creases [MIA-2007] [MICCAI-2006] and glyph packing [VIS-2006], my post-doctoral work (advised by Carl-Fredrik Westin) applies IGRT to both tensor interpolation via geodesic-loxodromes [MICCAI-2007], and characterizing local changes in fiber structure with Peter Savadjiev [NI-2010] [MICCAI-2009].

Collaboration with Alexandra J. Golby led to clinically-oriented results [ISMRM-2008] [NS-2011], including deterministic tractography that avoids the restrictions of the single-tensor model [MMBIA-2008] [NI-2009] [ISBI-2007].

My PhD [PhD-2004] (advised by Chistopher Johnson) describes

- barycentric shape space of diffusion tensors [VIS-1999b] [TVCG-2000] [MRM-2000],
- superquadric tensor glyphs [VisSym-2004],
- orthogonal systems of invariants [MRM-2006] [MnV-2006b] [talks/DagTen-2004], and
- a framework I term "IGRT" for decomposing tensor field gradients into changes of tensor shape ("invariant gradents", IG) and orientation ("rotation tangents", RT) [APR-2009] [TMI-2007].

Diffusion MRI visualization and analysis research continues in collaboration with Thomas Schultz [MnV-2014] [MICCAI-2010] [EuroVis-2010] and Daniel Ennis [MIA-2014] [MICCAI-2012]. The tension between model selection and model-specific methods is an interesting research direction [talks/DagVis-2011].