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.

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