Visualizing Flow Conditions Using a Stream Digital Twin Jerry Tessendorf October, 2022 South Carolina Water Resources Conference 2022 presentation 

Normal Maps for Rendering Vast Ocean Scenes Jerry Tessendorf, Liang Gao, Colin Reinhardt 2023 Technical Report 

Whitecap Phenomenology for Ocean Surface Simulation Jerry Tessendorf, Colin Reinhardt, Liang Gao 2022 Technical Report 

Visualization of Simulated Ocean Scenes Employing Whitecap Fraction Phenomenology Jerry Tessendorf, Liang Gao, Colin Reinhardt March, 2022 Ocean Sciences 2022 Conference presentation 

A Note on Computation of a Ray Bending Path Jerry Tessendorf August, 2017 Technical Report 

Gilligan: A Prototype Framework for Simulating and Rendering Maritime Environments Jerry Tessendorf February, 2017 Technical Report 

A Note on Representing Multiple Scatter as Multiple Internal Lights Jerry Tessendorf April, 2016 Technical Report 

Beam spread functions calculated using Feynman path integrals Paul Kilgo and Jerry Tessendorf March, 2016 J. Quant. Spectr. Rad. Transf. A method of solving the radiative transfer equation using Feynman path integrals (FPIs) is discussed. The FPI approach is a mathematical framework for computing multiple scattering in participating media. Its numerical behavior is not well known, and techniques are being developed to solve the FPI approach numerically. A missing numerical technique is detailed and used to calculate beam spread functions (BSFs), a commonly studied experimental property of many types of media. The calculations are com pared against measured BSFs of sea ice. Analysis shows differentlyshaped BSFs, and suggests the width parameter of the calculated BSF's Gaussian fit approaches a value in the limit of the number of path segments. A projection is attempted, but suggests a larger number of path segments would not increase the width of the calculated BSF. The trial suggests the approach is numerically stable, but requires further testing to ensure scientific accuracy 

Towards Validation of a Monte Carlo Rendering Technique Paul Kilgo and Jerry Tessendorf August, 2015 Poster, Siggraph 2015 

Accelerated Path Generation and Visualization for Numerical Integration of Feynman Path Integrals for Radiative Transfer Paul Kilgo and Jerry Tessendorf April, 2015 ANS MC2015  Joint International Conference on Mathematics and Computation (M&C), Supercomputing in Nuclear Applications (SNA) and the Monte Carlo (MC) Method Nashville, Tennessee 

Advection Solver Performance with Long Time Steps, and Strategies for Fast and Accurate Numerical Implementation Jerry Tessendorf February, 2015 Technical Note A detailed look at solving advection equations by using a Characteristic Map. Three items in this note: (1) Using the CM as the tool for advection, substeps can be generated by efficiently, in fact logarithmically fast; (2) The exact solution for the CM is given, leading to an accurate numerical algorithm for using it. Because the solution involves exponentiated 3X3 matrices, the algorithm is relatively slow although very accurate. In fact, for rigid rotations the algorithm is exact. (3) Several standard tests of advection accuracy are evaluated and the error in several popular advection schemes are quantified by comparing them with the exact solution. 

eWave: Using an Exponential Solver on the iWave Problem Jerry Tessendorf March 2014 Technical Note The iWave approach to simulating surface waves is fast and efficient, but suffers stability and artifact problems. By rephrasing the algorithm to employ two first order diffential equations for the displacement and velocity potential, and using an exponential solver, much more accurate simulations result. Importantly, the solution is free of the stability and artifact issues of the previous approach, and maintains high efficiency and flexibility. 

The Characteristic Map for Fast and Efficient VFX Fluid Simulations Jerry Tessendorf and Brandon Pelfrey June, 2011 Computer Graphics International Workshop on VFX, Computer Animation, and Stereo Movies The Method of Characteristics is examined as a tool for making fluid simulation more efficient and effective in VFX production. A mathematical frame work for a Characteristic Map is shown to be a general ization of the previous methods called Gridless Advec tion and SemiLagrangian Mapping. We demonstrate that the Characteristic Map can be used to modify a fluid flow postsimulation, including injecting higher resolution motion, and precise flow control from blend ing Characteristic Maps. 

Angular Smoothing and Spatial Diffusion from the Feynman Path Integral Representation of Radiative Transfer Jerry Tessendorf October, 2010 Journal of Quantitative Spectroscopy and Radiative Transfer The propagation kernel for time dependent radiative transfer is represented by a Feynman Path Integral (FPI). The FPI is approximately evaluated in the spatialFourier domain. Spatial diffusion is exhibited in the kernel when the approximations lead to a gaussian dependence on the Fourier domain wave vector. The approximations provide an explicit expression for the diffusion matrix. They also provide an asymptotic criterion for the selfconsistency of the diffusion approximation. The criterion is weakly violated in the limit of large numbers of scattering lengths. Additional expansion of higherorder terms may resolve whether this weak violation is significant. 

I Love It When A Cloud Comes Together Sho Hasegawa, Jason Iversen, Hideki Okano, and Jerry Tessendorf July, 2010 Siggraph, Los Angeles One of the action sequences in the film The ATeam takes place within a growing system of storm clouds. The story plot required the creation of a fully 3D environment of clouds covering tens of kilometers of evolving storm, modeled and simulated at high reso lution because the camera and story elements are embedded within it. The cloud system covered approximately 20 kilometers, at a resolution as small at 0.1 meters. A variety of clouds types were modeled, corresponding to the different cloud taxonomies in differ ent regions of a storm supercell. Individual cloud structure was controlled and directed down to arbitrarily fine spatial detail. This talk discusses the software tools developed to model, simulate, and render this complex, high resolution cloud system. 

Resolution Independent Volumes Jerry Tessendorf and Michael Kowalski July, 2011 Siggraph, Los Angeles Course notes for the 2011 Siggraph course "Production Volume Rendering 2". Emphasises scriptbased manipulation of volumetric data, illustrated with a scripting language called Felt. 

Monocoupled 3D and 2D River Simulations Sanjit Patel, Jerry Tessendorf, and Jeroen Molemaker August, 2009 Symposium on Computer Animation, New Orleans This is a poster presented at SCA in New Orleans. To achieve a high resolution simulation of a river, we employed two different simulation methods. A full NavierStokes 3D simulation at a medium resolution to get the overall motion, followed by a high resolution simulation of surface displacement waves. The dynamics of the surface displacements was driven in part by the motion of the 3D simulation. abstract poster 

Numerical Integration of the Feynman Path Integral for Radiative Transport Jerry Tessendorf May, 2009 International Conference on Mathematics, Computational Methods and Reactor Physics, Saratoga Springs, New York, May 37, 2009, American Nuclear Society. The radiative transport problem is cast in integral form using a transport kernel. The transport kernel has an explicit representation in terms of a Feynman Path Integral over all paths between selected points in a volume. This representation is setup in detail. Numerical evaluation of this Path Integral is formulated with a FrenetSerret based procedure for generating valid random paths, and with a numerical evaluation of the weight for each valid path. Very early sanity checks of a numerical implementation are reported. Approaches to optimization are identified. 

Simulation of Interactive Surface Waves Jerry Tessendorf October, 2008 This is a set of notes for a very short course on water surface wave simulation via the iWave algorithm. I presented the course over two class sessions at the Clemson University, and at University of Pennsylvania. The notes are in two files: one that covers background and enough detail to execute the algorithm in code, and in the second pdf file there are technical details of the meaning of the square root of a derivative operator. Many thanks to Robert Geist, Don House, and Mike Westall for inviting me to Clemson and taking care of me while there, and to Norm Badler, Alla Safanova, Stephen Lane, and Jessica Marcus for inviting me to UPenn and taking care of me. course notes pdf square root of a derivative pdf 

Production Volume Rendering Jerry Tessendorf November, 2008 This is a set of notes for a very short course on volume rendering. I presented the course over two class sessions at the University of Pennsylvania just before Thanksgiving. Despite the brevity of the class time, I was very pleased to hear afterward from several students in the class who had successfully written volume renderers from the notes (and from some rendering code they had previously built in the class). Some of the students have put up websites (http://www.alinenormoyle.com/projects/clouds/index.html, www.sunilkamat.com/files/ReadMe.pdf, and http://www.seas.upenn.edu/~tgorkin/VolumeRenderer/volumeRenderer.html). Many thanks to Norm Badler, Alla Safanova, Stephen Lane, and Jessica Marcus for inviting me to UPenn and taking care of me during my stay. 

A Simple Improvement of Gas Simulation Quality Victor Grant, Charles Anderson, Nathan Ortiz, Jerry Tessendorf January, 2008 With some very simple additional processing, lowresolution fluid simulations can be rendered with fine detail. This is a sketch that was submitted to siggraph 2008, but unfortunately not accepted. video 

Golden Compass Auroras Nathan Ortiz, Eric Horton, Michael Kowalski, Jerry Tessendorf January, 2008 Talk, Siggraph 2008 Describes the many layers of volumetric elements constructed and animated for the Auroras at the end of Golden Compass. 

Golden Compass Daemon Deaths Scott Townsend, Eric Horton, Sanjit Patel, Jerry Tessendorf January, 2008 Talk, Siggraph 2008 Describes the many layers of volumetric elements and fluid simulations for the daemon deaths in Golden Compass. 

MultipleForwardScattering in Volume Rendering of Participating Media Jerry Tessendorf January, 2006 Natural volumetric media have phase functions which typically are sharply peaked in the forward scattering direction, with backscatter accounting for only a few percent of the total angular redistribution from a single scattering event. This property has been exploited in the past in the smallangle approximation for radiative transfer, successfully for many engineering and science applications. The smallangle approximation also robustly describes the multipleforwardscattered behavior of the light field many scattering lengths into the participating medium, including the asymptotic regime, in agreement with experimental measurements and computationally intensive simulations. Physically, the important missing ingredient not found in the smallangle approximation is occasional large angle scatters that reverse the propagation direction of some of the light. This paper introduces a quantitative model of multiple scattering which contains both the multipleforwardscatter character and a few largeangle scattering events. The model is derived directly from the Green's function representation of radiative transfer, and path integrals are used to construct the appropriate form of the small angle approximation. The model is suitable for media that have internal structure. 

Interactive Water Surfaces Jerry Tessendorf 2004 Game Programming Gems 4, Charles River Media Chapter from Game Programming Gems 4 on the basic iWave algorithm for simulating water surface interaction with obstructions in the water. 

Motion Blur Algorithm for Clipped Triangle Rendering Jerry Tessendorf December, 2004 Notes on a general algorithm from tracking the motion of vertices on an image plane as the 3D positions of the camera can vertices move. 

Efficient Rendering of Atmospheric Phenomena Kirk Riley, David S. Ebert, Martin Kraus, Jerry Tessendorf, and Charles Hansen September, 2004 Eurographics Symposium on Rendering, H. W. Jensen and A. Keller (eds), 2004 Rendering of atmospheric bodies involves modeling the complex interaction of light throughout the highly scattering medium of water and air particles. Scattering by these particles creates many wellknown atmospheric optical phenomena including rainbows, halos, the corona, and the glory. Unfortunately, most radiative transport approximations in computer graphics are illsuited to render complex angularly dependent effects in the presence of multiple scattering at reasonable frame rates. Therefore, this paper introduces a multiplemodel lighting system that efficiently captures these essential atmospheric effects. We have solved the rendering of fine angularly dependent effects in the presence of multiple scattering by designing a lighting approximation based upon multiple scattering phase functions. This model captures gradual blurring of chromatic atmospheric optical phenomena by handling the gradual angular spreading of the sunlight as it experiences multiple scattering events with anisotropic scattering particles. It has been designed to take advantage of modern graphics hardware; thus, it is capable of rendering these effects at near interactive frame rates. 

Practical Rendering of Multiple Scattering Effects in Participating Media Simon Premoze, Michael Ashikhmin, Jerry Tessendorf, Ravi Ramamoorthi, and Shree Nayar September, 2004 Eurographics Symposium on Rendering, H. W. Jensen and A. Keller (eds), 2004 Volumetric light transport effects are significant for many materials like skin, smoke, clouds, snow or water. In particular, one must consider the multiple scattering of light within the volume. While it is possible to simulate such media using volumetric Monte Carlo or finite element techniques, those methods are very computationally expensive. On the other hand, simple analytic models have so far been limited to homogeneous and/or optically dense media and cannot be easily extended to include strongly directional effects and visibility in spatially varying volumes. We present a practical method for rendering volumetric effects that include multiple scattering. We show an expression for the point spread function that captures blurring of radiance due to multiple scattering. We develop a general framework for incorporating this point spread function, while considering inhomogeneous media¿this framework could also be used with other analytic multiple scattering models. 

Simulating Ocean Surface Jerry Tessendorf January, 2004 Siggraph course notes, 19992004 Notes and slides from a course given at Siggraph from 1999 to 2004. Notes 2004 Slides 2004 Notes 2002 Slides 2002 Slides 2001 

Tetrad Volume and Particle Rendering in X2 Bill La Barge, Jerry Tessendorf, and Vijoy Gaddipati August, 2003 Siggraph Sketch In the movie X2 XMen United, Cerebro is a large spherical cavity that extends mutant mental capabilities. To depict its large, cavernous, dynamic nature, and the connnection between the machine and the characters, the atmospheric element of the scene was built based on a fully 3D volumetric rendering technology developed at Cinesite. In addition, this volumetric technique was also used to connect floating vinnettes in the space with the land masses on the borders of Cerebro. The teleportation effect of the character Nightcrawler is accompanied by a dynamic smokey filament effect using turbulent particle dynamics and particle rendering. 

Concepts for Volume Rendering Jerry Tessendorf August, 2003 Notes on the path integral mathematics for doing volume rendering. Written to be succinct, not as a tutorial or explanatory. Suggests a full algorithm for doing multiple scattering. 

Efficiently Rendering Gobs and Gobs of Particles Jerry Tessendorf May, 2002 Software has been developed and deployed which is able to render unlimited numbers of particles in very low amounts of RAM. This approach also generates volumetric lighting and opacity, including selfshadowing, making the particle renderer an efficient volume renderer. The algorithms combine several channels of alpha and depth data with techniques to transform the hiding and shadow map problems to a compositing operation. 

Deforming Geometric Volumes: Kinematics, Dynamics, Constraints, and Collisions Jerry Tessendorf February, 2002 These notes are intended as an outline of some geometric techniques that may be applicable to modeling and dynamics problems that include volumetric objects. Of particular interest are objects consisting of an enclosed volume, with the volume undergoing deformations while subjected to external forces, collisions, and boundary constraints that combine to reshape the volume dynamically. 

Algorithm for Capturing a 3D Model from Multiple Camera Views Jerry Tessendorf March, 2000 Neat and clean theory for taking rotoscope outlines from multiple images and reconstructing the 3D object in the view. Underwent very limited testing at the time, with success. 

Fast Wake Algorithm Derivation Jerry Tessendorf March, 2000 One pager on creating boat wakes in the FFT water surface method. 

The Map of a Sphere to and from the Image Plane Jerry Tessendorf November, 1999 A brief note laying the map of projecting a sphere to the image, and back. 

Implementation of Curved Strands I: FrenetSerret Framework J Tessendorf and D D Weston January, 1999 Implements a representation of curved hair strands in terms of a FrenetSerret geometric framework. This paper lays out the framework and implements a numerical scheme. 

Implementation of Curved Strands II: Dynamics J Tessendorf and D D Weston December, 1998 Sets up the dynamics of strands under the FrenetSerret framework. 

Renormalized Rendering of Unresolved Objects Jerry Tessendorf February, 1998 This note describes the theory and practice behind a statistical renormalization procedure, which gives a few unresolved objects the appearance of more opacity than actually rendered. 

Impact of Multiple Scattering on Simulated Infrared Cloud Scene Images Jerry Tessendorf and David Wasson April, 1994 Characterization and Propagation of Sources and Backgrounds, SPIE Proceedings, vol 2223, 462473, (1994) The threedimensional volumetric character of clouds is a critically important factor in determining cloud structure as seen in infrared imagery. Using a longwave cloud scene simulator which images a threedimensional cloud volume, the 3D structure has been shown to be particularly important when viewing at low grazing angles. In order to conduct analyses of cloud scene structure in MW and visible bands as well, the longwave simulator has been significantly upgraded to perform imaging of clouds with multiple scattering included. The multiple scattering algorithm is based on a WKB approximation method for the exact radiative transfer problem, and comprehends the spatial variations in optical properties within the cloud volume. As a first analysis, we have generated a cloud scene which is backlit by the sun, and systematically assess the contributions of the thermal, solar, and multiple scattering mechanisms within the imagery. As might be expected, multiple scattering has its greatest impact at the cloud edges in the MW band, where the "silver lining" is formed. In the MW band, scattering can also play a role at cloud edges and create additional clutter by scattering earthshine into the field of view of the low grazing angle camera. In principle, this simulator is capable of operating throughout the visible and infrared bands, for realistically size clouds. 

Scattering in the 3D Cloud Scene Simulator Jerry Tessendorf and David Wasson February, 1994 Arete Associates Technical Report One of two documents describing early approaches to multiple scattering in a rendering system. 

3D Cloud Scene Simulator V2 Algorithm for Scattering Jerry Tessendorf and David Wasson February, 1994 Arete Associates Technical Report One of two documents describing early approaches to multiple scattering in a rendering system. 

Measured and Simulated Power Spectral Density and Temporal Coherence of Infrared Cirrus and Stratus Cloud Images Morton Farber, Jerry Tessendorf, and Albert Soong April, 1993 Characterization, Propagation, and Simulation of Sources and Backgrounds III, SPIE Publication, vol 1967, 123141, (1993) Sets of image data from the MUSIC (Multi Spectral Infrared Camera) have been analyzed to obtain the spatial and temporal structure of cirrus and stratus cloud scenes at small spatial and temporal scales. The MUSIC data were collected in 1989 in a sidelooking airborne geometry. The large clutter to noise ratios found in thsi analysis for both cirrus and stratus clouds represent a potentially large noise source for IRST. The powerlaw rolloff and anisotropy of the power spectral density of images is compared to the output of a 3D cloud scene simulator and found to be in reasonable agreement; and in very poor agreement with simpler notions which invoke perspective distortion of twodimensional clouds and predict much larger anisotropy. 

Measures of Temporal Pulse Stretching Jerry Tessendorf July, 1992 Ocean Optics XI, SPIE Publication, vol 1750, 407418, (1992) Temporal pulse stretching is a consequence of the multiple scatter by ocean water of a laser pulse. Although the physical process behind pulse stretching is intuitively clear, there is no widely held quantitative definition of it. Here temporal pulse stretching is defined in terms of temporal moments of the radiance at a fixed position and orientation with respect to the initial pulse axis. This definition has been chosen because it is directly measurable from the waveform output of a radiometer. The first temporal moment is a measure of the apparent delay of the pulse, and the variance from the second moment describes the increasing width. Using a WKB approach, an expression is obtained for the first two temporal moments for waveforms measured at positions along the initial pulse axis. Quantitative predictions of the temporal delay and width are made for a pulse with is initially a collimated point. To within an error of no more than 12%, the delay and width are proportional. Stretching effects on waveforms are shown graphically in plots at various distances from the source. 

Structure and Spatial Spectra of Simulated Cloud Scenes at Infrared Wavelengths Jerry Tessendorf, Daniel Weston, and Lisa Taylor April, 1992 Characterization, Propagation, and Simulation of Sources and Backgrounds II, SPIE Publication, vol 1687, 499508, (1992) Longwave infrared imagery of cloud fields are examined in terms of their power spectral density (PSD). In order to systematically investigate the dependence of the PSD on viewing conditions, a cloud scene simulator is employed to generate images of a simulated cloud field. The cloud field is fully three dimensional and is described by its fluctuating temperature and liquid/ice water content fields. The image process accurately calculates the spatially varying attentuation of blackbody emission. Several views of a single cloud field are examined to study the effect of viewing angle on the image PSD. Zenith views produce isotropic PSDs, while nearly horizontal views contain a large amount of foreshortening and a correspondingly anisotropic PSD. One possible component of the foreshortening is simply geometric and can be estimated and compared to simulation output. We find that geometrically induced foreshortening does not describe the PSD effects observed in the simulation for the relatively thin cirruslike cloud simulated here. Possibly this indicates that the threedimensional cloud structure is more important in some views than in others when there are large fluctuations in the cloud optical properties. We are pursuing a more quantitative description of this behavior. 

The Underwater Solar Light Field: Analytical Model from a WKB Evaluation Jerry Tessendorf July, 1991 Underwater Imaging, Photography, and Visibility, SPIE Publication, vol 1537, 1020, (1991) An analytical expression for the underwater radiance distribution due to a purely "delta function" sun is discussed. 

Radiative Transfer on Curved Surfaces J. Tessendorf April, 1990 Journal of Mathematical Physics, vol 31, no. 4, 10101019, (1990) After a review of appropriate concepts in local surface geometry, a formally exact solution of the radiative transfer equation is constructed, for transfer from one surface of arbitrary shape to another. 

Downwelling Irradiance Fluctuations in the SmallAngle Approximation Jerry Tessendorf April, 1990 Ocean Optics X, SPIE Publication, vol 1302, 454463, (1990) Mean and rms fluctuations of downwelling irradiance below a rough ocean surface have been modelled using the smallangle approximation for the inwater radiance distribution. 

TimeDependent Radiative Transfer and Pulse Evolution J. Tessendorf February, 1989 Journal of the Optical Society of America A, vol 6, no 2, 280297, (1989) The time dependent radiative transfer equation in an absorbing and scattering medium is recast as an evolution equation that is similar to the global formulation of Preisendorfer. 

Approximate Parametric Receiver Operating Characteristics for Poisson Distributed Noise Jerry Tessendorf January, 1989 Applied Optics, vol 28, 214216, (1989) ROC curves for poisson distributed noise. 

Comparison Between Data and SmallAngle Approximations for the InWater Solar Radiance Distribution J. Tessendorf September, 1988 Journal of the Optical Society of America A, vol 5, no 9, 14101418, (1988) Qualitative and quantitative properties of the inwater distribution of solar radiance, as predicted by the radiative transfer equation, are examined. 

FiniteDifference Evolution of a Scattered Laser Pulse in Ocean Water J. Tessendorf, C. Piotrowski, R.L. Kelly July, 1988 Ocean Optics IX, SPIE Publication, vol 925, 2235, (1988) The propagation of a finitesized laser pulse through ocean water is simulated. 

Radiative Transfer as a Sum over Paths J. Tessendorf January, 1987 Physical Review A, vol 35, no. 2, 872878 (1987) The radiativetransfer equation describes the collection of path taken by an element of radiation as it travels from one location to another. When backscatter can be ignored, the exact solution is constructed as a formal sum (path integral) over all such paths. In the appropriate limit the usual (diffusive) smallangle solution and the multiple scattering solution can be obtained. Another smallangle solution has also been found which includes some of the nonlinear and largeangle behavior not present in the diffusive solution. After several attenuation lengths, length scales are characterized by a parameter constructed our of the absorption and scattering coefficients, and the rms scattering angle per scattering event. The two solutions are compared i n the case of a point beam. 

Green's Functions at Zero Viscosity H. M. Fried and J. Tessendorf April, 1984 Journal of Mathematical Physics, vol 25, 11441154 (1984) Fradkintype propagator representations are written for solutions to NavierStokes and related equations, for arbitrary dimension D and arbitrary source geometry. In the limit of very small viscosity, velocity/vorticity solutions are given in terms of Cauchy position coordinates q of a particle advected by the velocity flow v, using a set of coupled equations for q and v. For localized point vortices in two dimensions, the vectors q become the timedependent position coordinates of interacting vortices, and our equations reduce to those of the familiar, coupled vortex problem. The formalism is, however, able to discuss threedimensional vortex motion, discrete or continuous, including the effects of vortex stretching. The mathematical structure of vortex stretching in a Ddimensional fluid without boundaries is conveniently described in terms of an SU(D) representation of these equations. Several simple examples are given in two dimensions, to anchor the method in the context of previoiusly known, exact solutions. In three dimensions, vortex stretching effects are approximated using a previous "strong coupling" technique of particle physics, enabling one to build a crude model of the intermittent growth of enstrophy, which may signal the onset of turbulence. For isotropic turbulence, the possibility of a singularity in the inviscid enstrophy af a finite time is related to the behavior of a single function characterizing the intermittency. 

MeanField Burgers' Model of Turbulence J. Tessendorf October, 1983 Brown University Dept of Physics Report BROWNHET516 A meanfield expansion of the Martin, Siggia, and Rose functional formalism for turbulence is proposed as a "strongcoupling" approximation. To carry out the expansion, the volume of the wavenumber space must be truncated to a large but finite amount. Burgers' model with homogeneous average flow is examined in detail to several orders of approximation. The two point cumulant is calculated. The meanfield expansion is described as a perturbation expansion, with its small parameter inversely proportional to the truncated volume of wavenumber space. Renormalization and some features of the expansion are explored. 