There have been a variety of flow visualization methods, among which texture-based ones [1] are gaining significant attention for the dense representation of a flow. Traditional geometry-based approaches such as arrow plots [2], streamlines [3], [4], [5], [6], [7], [8], pathlines [9], [10], [11], [12], timelines [12], streaklines [12], particle tracing [12], [13], [14], [15], surface particles [16], stream ribbons [8], stream polygons [17], stream surfaces [18], [19], [20], stream arrows [21], [22], stream tubes [8], stream balls [23], flow volumes [24], [25], and topological analysis [26], [27], [28], [29] tend to result in an incomplete view or a cluttered image of the flow unless an effective seeding strategy is adopted. Van Wijk presented a texture synthesis technique called Spot Noise [30], [31], [32] that visualizes flow data by stretching in the flow direction a collection of oval texture splats placed within the field. Later Cabral and Leedom [33] proposed LIC (Line Integral Convolution) that convolves an input noise texture using a low-pass filter along pixel-centered symmetrically bi-directional streamlines to exploit spatial correlation in the flow direction. LIC synthesizes an image that provides a global dense representation of the flow, analogous to the resulting pattern of wind-blown sand. There have been many optimizations of and extensions to LIC such as fast LIC [34], parallel LIC [35], LIC on curvilinear grids [36], LIC on triangulated surfaces [37], multi-frequency LIC [38], oriented LIC [39], enhanced LIC [40], dyed LIC [41], volume LIC [42], [43], and HyperLIC [44]. Heidrich et al. [45] incorporated indirect pixel texture addressing and additive/subtractive texture blending to accelerate streamline integration and texture convolution in LIC. Preuber and Rumpf [46] adopted another texture-based method, anisotropic nonlinear diffusion, to low-pass filter a noise texture along streamlines, but enhance edges in the orthogonal flow direction. Later Burkle et al. [47] adapted this approach with texture transport for unsteady flows. Max and Becker [48] applied texture mapping in either forward mesh warping or backward texture coordinates advection for time-dependent flow visualization. Forssell and Cohen [36] used pathlines as convolution paths in LIC to visualize unsteady flow fields. Verma et al. [49] presented PLIC (Pseudo LIC) in which pre-synthesized template textures are mapped on sparsely placed pathline ribbons to emulate a dense representation of time-dependent flows. Some methods take advantage of hardware capabilities to achieve high performance visualization. Jobard et al. [50], [51] designed an efficient rendering pipeline based on indirect pixel texture addressing [45] for fast texture/dye advection and feature extraction. Weiskopf et al. [52] exploited the pixel texture unit to visualize time-varying 3D vector fields. Van Wijk proposed IBFV (Image Based Flow Visualization) [53] in which a sequence of temporally-spatially low-pass filtered noise textures are advected via forward texture mapping on warped meshes in combination with the blending of successive frames. This easy-to-implement, efficient, versatile method can emulate a wide range of techniques such as particles, arrow plots, streamlines, timelines, spot noise, LIC, and topological analysis at high frame rates using basic hardware features. IBFV was then used to visualize flow on 3D curved surfaces and enhance surface shape cueing by means of flow-aligned textures [54]. IBFV was even extended to 3D flows by decomposing the 3D advection to planar and longitudinal advections [55]. Jobard et al. presented LEA [56] to visualize unsteady flow fields also at high frame rates despite its independence of hardware acceleration [57]. To compute each pixel value of a frame, backward integration is used to retrieve at the last time step the contributing particle's footprint into which the iteratively advected texture, with noise injected at in-flow areas, is indexed for the contributed texture value. Successive textures are blended to create temporal coherence along pathlines in the flow evolution and short-length directional low-pass filtering is performed to improve spatial coherence along streamlines in an image. Visually pleasing effects can be obtained when arbitrarily shaped field domains are addressed using contextual masking. Recently Laramee et al. [58] presented a novel method based on IBFV and LEA for a direct dense representation of unsteady flow on arbitrary triangular surfaces to address large, unstructured, dynamic meshes by projecting the surface geometry to image space to which backward mesh advection and successive texture blending are applied. Weiskopf et al. proposed UFAC [59], which, based on a generic spacetime-coherent framework, establishes temporal coherence by property advection along pathlines while building spatial correlation by texture convolution along instantaneous streamlines. A well-known hardware-independent algorithm is UFLIC (Unsteady Flow LIC) proposed earlier by Shen and Kao [60]. UFLIC uses a time-accurate value scattering scheme and a successive texture feed-forward strategy to achieve very high temporal and spatial coherence. At each time step a value scattering process occurs for which a seed is placed in each pixel. For this seed a pathline is integrated along which the seed's texture value is scattered to the downstream pixels over several time steps. The values received by each pixel at a time step are accumulated and convolved to synthesize the corresponding frame. To address the low performance of UFLIC, Liu and Moorhead presented AUFLIC (Accelerated UFLIC) [61] based on their earilier work [62]. AUFLIC is an order-of-magnitude faster than UFLIC and achieves interactive unsteady flow visualization by reusing pathlines in the time-consuming value scattering process. AUFLIC can be easily extended to time-varying volume flows --- VAUFLIC [63], [64]. As a geometry-based method, streamlines remain one of the most important approaches to flow visualization for the straightforward direction cueing and the low computational expense. Furthermore, a placement of evenly spaced streamlines [65], [66], [67] may provide an aesthetic and informative pattern, without either incompleteness or cluttering, to facilitate mental reconstruction of the flow. The sparseness makes evenly spaced streamlines, particularly coupled with curve illumination, a promising solution for visualizing planar flows, curved surface flows, and volume flows. |

[ 1] Robert S. Laramee, Helwig
Hauser, Helmut Doleisch, Benjamin Vrolijk, Frits H. Post, and Daniel Weiskopf,
"The State of the Art in Flow Visualization: Dense and Texture-based
Techniques," [ 2] Victor R. Klassen and
Steven J. Harrington, "Shadowed Hedgehogs: A Technique for Visualizing
2D Slices of 3D Vector Fields," [ 3] David N. Kenwright and
Gordon D. Mallinson, "A 3-D Srteamline Tracking Algorithm Using Dual
Stream Functions," [ 4] Greg Turk and David Banks,
"Image-Guided Streamline Placement," [ 5] Bruno Jobard and Wilfrid,
"Creating Evenly-Spaced Streamlines of Arbitrary Density," [ 6] Detlev Stalling, Malte
Zockler, and Hans-Christian Hege, "Fast Display of Illuminated Field
Lines," [ 7] Vivek Verma, David Kao,
and Alex Pang, "A Flow-Guided Streamline Seeding Strategy,"
[ 8] Shyh-Kuang Ueng, Christopher
Sikorski, and Kuan-Liu Ma, "Efficient Streamline, Streamribbon, and
Streamtube Constructions on Unstructured Grids," [ 9] Pieter G. Buning, "Numerical
Algorithms in CFD Post-Processing," [10] David A. Lane, "Visualization
of Time-Dependent Flow Fields," [11] David A. Lane, "Scientific
Visualization of Large-Scale Unsteady Fluid Flows," [12] David A. Lane, "UFAT
- A Particle Tracer for Time-Dependent Flow Fields," [13] Andrea J. S. Hin and Frits
H. Post, "Visualization of Turbulent Flow with Particles," [14] David N. Kenwright and
David A, "Lane. Interactive Time-Dependent Particle Tracing Using
Tetrahedral Decomposition," [15] Ralph Bruckschen, Falko
Kuester, Bernd Hamann, and Kenneth I. Joy, "Real-time Out-Of-Core
Visualization of Particle Traces," [16] Jarke J. Van Wijk, "Rendering
Surface-particles," [17] W. J. Schroeder, C. R.
Volpe, and W. E. Lorensen, "The Stream Polygon: A Technique for 3D
Vector Field Visualization," [18] J. P. M. Hultquist, "Constructing
Stream Surfaces in Steady 3D Vector Fields," [19] Jarke J. Van Wijk, "Implicit
Stream Surfaces," [20] Gerik Scheuermann, Tom
Bobach, Hans Hagen, Karim Mahrous, Bernd Hamann, Kennenth I. Joy, and
Wolfgang Kollmann, "A Tetrahedra-Based Stream Surface Algorithm,"
[21] Helwig Loffelmann, Lukas
Mroz, Eduard Groller, and Werner Purgathofer, "Stream Arrows: Enhancing
The Use of Stream Surfaces for The Visualization of Dynamical Systems,"
[22] Helwig Loffelmann, Lukas
Mroz, and Eduard Groller, "Hierarchical Streamarrows for the Visualization
of Dynamical Systems," [23] Manfred Brill, Hans Hagen,
Wladimir Djatschin, and Stanislav V. Klimenko, "Streamball Techniques
for Flow Visualization," [24] Nelson Max, Barry Becker,
and Roger Crawfis, "Flow Volumes for Interactive Vector Field Visualization,"
[25] Barry G. Becker, David
A. Lane, and Nelson L. Max, "Unsteady Flow Volumes," [26] James L. Helman and Lambertus
Hesselink, "Representation and Display of Vector Field Topology in
Fluid Flow Data Sets," [27] James. L. Helman and Lambertus
Hesselink, "Surface Representation of Two- and Three-Dimensional
Fluid Flow Topology," [28] A. Globus, C. Levit, and
T. Lasinski, "A Tool for Visualizing the Topology of Three-Dimensional
Vector Fields," [29] James L. Helman and Lambertus
Hesselink, "Visualizing vector field topology in fluid flows,"
[30] Jarke J. van Wijk, "Spot
Noise: Texture Synthesis for Data Visualization," [31] Willem C. deLeeuw and
Jarke J. J. van Wijk, "Enhanced Spot Noise for Vector Field Visualization,"
[32] Willem C. deLeeuw and
Robert Van Liere, "Comparing LIC and Spot Noise," [33] Brian Cabral and Leith
(Casey) Leedom, "Imaging Vector Fields Using Line Integral Convolution,"
[34] Detlev Stalling and Hans-Christian
Hege, "Fast and Resolution Independent Line Integral Convolution," [35] Detlev Stalling, M. Zockler,
and Hans-Christian Hege, "Parallel Line Integral Convolution," [36] Lisa K. Forssell and S.
D. Cohen, "Using Line Integral Convolution for Flow Visualization: Curvilinear
Grids, Variable-Speed Animation, and Unsteady Flows," [37] Christian Teitzel, Robert
Grosso, and Thomas Ertl, "Line Integral Convolution on Triangulated Surfaces,"
[38] Ming-Hoe Kiu and David
C. Banks, "Multi-Frequency Noise for LIC," [39] R. Wegenkittl, E. Groller,
and W. Purgathofer, "Animating Flow Fields: Rendering of Oriented Line
Integral Convolution," [40] A. Okada and D. L. Kao,
"Enhanced Line Integral Convolution with Flow Feature Detection," [41] Han-Wei Shen, C. Johnson,
and Kwan-Liu Ma, "Visualizing Vector Fields using Line Integral Convolution
and Dye Advection," [42] Victoria Interrante and
Chester Grosch, "Strategies for Effectively Visualizing 3D Flow with Volume
LIC," [43] C. Rezk-Salama, P. Hastreiter,
C. Teitzel, and T. Ertl, "Interactive Exploration of Volume Line Integral
Convolution Based on 3D-Texture Mapping," [44] XiaoQiang Zheng and Alex
Pang, "HyperLIC," [45] Wolfgang Heidrich, Rudiger
Westermann, Hans-Peter Seidel, and Thomas Ertl, "Applications of Pixel
Textures in Visualization and Realistic Image Synthesis," [46] T. Preuber and M. Rumpf,
"Anisotropic Nonlinear Diffusion in Flow Visualization," [47] D. Burkle, T. Preuber,
and M. Rumpf, "Transport and Anisotropic Diffusion in Time-Dependent Flow
Visualization," [48] Nelson Max and Barry Becker,
"Flow Visualization Using Moving Textures," [49] Vivek Verma, David Kao,
and Alex Pang, "PLIC: Bridging the Gap Between Streamlines and LIC," [50] Bruno Jobard, Gordon Erlebacher,
and M. Yousuff Hussaini, "Hardware-Assisted Texture Advection for Unsteady
Flow Visualization," [51] Bruno Jobard, Gordon Erlebacher,
and M. Yousuff Hussaini, "Tiled Hardware-Accelerated Texture Advection
for Unsteady Flow Visualization," [52] Daniel Weiskopf, Matthias
Hopf, and Thomas Ertl, "Hardware-Accelerated Visualization of Time-Varying
2D and 3D Vector Fields by Texture Advection via Programmable Per-pixel
Operations," [53] Jarke J. van Wijk, "Image
Based Flow Visualization," [54] Jarke J. van Wijk, "Image
Based Flow Visualization for Curved Surfaces," [55] Alexandru Telea and Jarke
J. van Wijk, "3D IBFV: Hardware-Accelerated 3D Flow Visualization," [56] Bruno Jobard, Gordon Erlebacher,
and M. Yousuff Hussaini, "Lagrangian-Eulerian Advection of Noise and Dye
Textures for Unsteady Flow Visualization," [57] Daniel Weiskopf, Gordon
Erlebacher, Matthias Hopf, and Thomas Ertl, "Hardware-Accelerated Lagrangian-Eulerian
Texture Advection for 2D Flow Visualization," [58] Robert S. Laramee, Bruno
Jobard, and Helwig Hauser, "Image Space Based Visualization of Unsteady
Flow on Surfaces," [59] Daniel Weiskopf, Gordon
Erlebacher, and Thomas Ertl, "A Texture-Based Framework for Spacetime-Coherent
Visualization of Time-Dependent Vector Fields," [60] Han-Wei Shen and David
L. Kao, "A New Line Integral Convolution Algorithm for Visualizing Time-Varying
Flow Fields," [61] Zhanping
Liu and Robert J. Moorhead II, "Accelerated
Unsteady Flow Line Integral Convolution (AUFLIC version 2.0),"
[62] Zhanping
Liu and Robert J. Moorhead II, "AUFLIC:
An Accelerated Algorithm for Unsteady Flow Line Integral Convolution (AUFLIC
version 1.0)," [63] Zhanping
Liu and Robert J. Moorhead II, "Visualizing
Time-Varying Three-Dimensional Flow Fields Using Accelerated UFLIC,"
[64] Zhanping
Liu and Robert J. Moorhead II, "A
Texture-Based Hardware-Independent Technique for Time-Varying Volume Flow
Visualization," J [65] Zhanping
Liu, Robert J. Moorhead II, and Joe Groner, "An
Advanced Evenly Spaced Streamline Placement Algorithm,"
[66] Zhanping
Liu and Robert
J. Moorhead II, "Robust
Loop Detection for Interactively Placing Evenly Spaced Streamlines,"
[67] Zhanping
Liu and Robert J. Moorhead II, "Interactive
View-Driven Evenly Spaced Streamline Placement," |