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==Overview==
Traditional visualizations of vector fields use small arrows or lines to represent vector direction and magnitude. This method has a low [[spatial resolution]], which limits the density of presentable data and risks obscuring characteristic features in the data.<ref name="Stalling 1995" /><ref name=":0" /> More sophisticated methods, such as [[Streamlines, streaklines, and pathlines|streamlines]] and particle tracing techniques, can be more revealing but are highly dependent on proper seed points.<ref name="Stalling 1995" /> Texture-based methods, like LIC, avoid these problems since they depict the entire vector field at point-like (pixel) resolution.<ref name="Stalling 1995" />
Compared to other integration-based techniques that compute field lines of the input vector field, LIC has the advantage that all structural features of the vector field are displayed, without the need to adapt the start and end points of field lines to the specific vector field. In other words, it shows the topology of the vector field.{{Citation needed|date=July 2024}}
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Although the input vector field and the result image are discretized, it pays to look at it from a continuous viewpoint.<ref name="Stalling 1995">{{cite conference | first1 = Detlev | last1 = Stalling | first2 = Hans-Christian | last2 = Hege | title = Fast and Resolution Independent Line Integral Convolution | citeseerx = 10.1.1.45.5526 | book-title = Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques | conference = SIGGRAPH '95 | date = August 6–11, 1995 | ___location = Los Angeles, California | pages = [https://archive.org/details/computergraphics00sigg/page/249 249–256] | doi = 10.1145/218380.218448 | isbn = 0-89791-701-4 | url = https://archive.org/details/computergraphics00sigg/page/249 }}</ref> Let <math>\mathbf{v}</math> be the vector field given in some ___domain <math>\Omega</math>. Although the input vector field is typically discretized, we regard the field <math>\mathbf{v}</math> as defined in every point of <math>\Omega</math>, i.e. we assume an [[interpolation]]. Streamlines, or more generally field lines, are tangent to the vector field in each point. They end either at the boundary of <math>\Omega</math> or at critical points where <math>\mathbf{v}=\mathbf{0}</math>. For the sake of simplicity, critical points and boundaries are ignored in the following.
A field line <math>\boldsymbol \sigma</math>, parametrized by arc length <math>s</math>, is defined as
<math display="block">\frac{d\boldsymbol \sigma (s)}{ds} = \frac{\mathbf{v}(\boldsymbol {\sigma}(s))}{|\mathbf{v}(\boldsymbol{\sigma}(s))|}.</math> Let <math>\boldsymbol{\sigma}_\mathbf{r}(s)</math> be the field line that passes through the point <math>\mathbf{r}</math> for <math>s=0</math>. Then the image gray value at <math>\mathbf{r}</math> is set to <math display="block">D(\mathbf{r}) = \int_{-L/2}^{L/2} k(s)N(\boldsymbol{\sigma}_{\mathbf{r}}(s)) ds</math>
where <math>k(s)</math> is the convolution [[Kernel (image processing)|kernel]], <math>N(\mathbf{r})</math> is the noise image, and <math>L</math> is the length of field [[line segment]] that is followed. <math>D(\mathbf{r})</math> has to be computed for each pixel in the LIC image. If carried out naively, this is quite expensive. First, the field lines have to be computed using a [[Numerical methods for ordinary differential equations|numerical method for solving ordinary differential equations]], like a [[Runge–Kutta methods|Runge–Kutta method]], and then for each pixel the convolution along a field line segment has to be calculated.
The final image will normally be colored in some way. Typically, some [[scalar field]] in <math>\Omega</math> (like the vector length) is used to determine the hue, while the grayscale LIC output determines the [[Lightness|brightness]].
Different choices of convolution kernels and random noise produce different textures; for example, [[pink noise]] produces a cloudy pattern where areas of higher flow stand out as smearing, suitable for weather visualization. Further refinements in the convolution can improve the quality of the image.<ref name="Weiskopf 2009">{{cite book |last1=Weiskopf |first1=Daniel |chapter-url=https://scholar.archive.org/
===Programming description===
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[[File:Animated LIC.svg|thumb|Illustration of how LIC is animated.<br/>Top: normal [[box filter]] (average).<br/>Middle: sinusoidal filter at <math>t</math>.<br/>Bottom: sinusoidal filter at <math>t + \delta t</math>.]]
LIC images can be animated by using a kernel that changes over time. Samples at a constant time from the streamline would still be used, but instead of averaging all pixels in a streamline with a static kernel, a ripple-like kernel constructed from a [[periodic function]] multiplied by a [[Hann function]] acting as a window (in order to prevent artifacts) is used. The periodic function is then shifted along the period to create an animation.
=== Fast LIC (FLIC)===
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Artistic effects for image generation and stylization:
* pencil drawing<ref>[https://www.researchgate.net/publication/
* automatic generation of hair textures<ref>[https://ieeexplore.ieee.org/document/859772?arnumber=859772 Xiaoyang Mao, M. Kikukawa, K. Kashio and A. Imamiya, "Automatic generation of hair texture with line integral convolution," 2000 IEEE Conference on Information Visualization. An International Conference on Computer Visualization and Graphics, 2000, pp. 303-308, doi: 10.1109/IV.2000.859772.]</ref>
* creating marbling texture<ref>[https://dl.acm.org/doi/10.1145/604471.604489 Xiaoyang Mao, Toshikazu Suzuki, and Atsumi Imamiya. 2003. AtelierM: a physically based interactive system for creating traditional marbling textures. In Proceedings of the 1st international conference on Computer graphics and interactive techniques in Australasia and South East Asia (GRAPHITE '03). Association for Computing Machinery, New York, NY, USA, 79–ff. https://doi.org/10.1145/604471.604489]</ref>
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