Line integral convolution: Difference between revisions

Revision as of 06:39, 28 October 2022

Revision as of 03:55, 9 November 2022

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[[File:Rotation of the Large Magellanic Cloud ESA393163.png|thumb|right|The Large Magellanic Cloud (LMC), one of the nearest galaxies to our Milky Way. This image was created with LIC]]

In [[scientific visualization]], '''line integral convolution''' ('''LIC''') is a method to visualize a [[vector field]], such as [[fluid motion]].

==Features==

* global method

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===Convolution===

In [[signal processing]] this process is known as [[Convolution#Discrete_convolutionDiscrete convolution|discrete convolution]].<ref>[https://web.cs.wpi.edu/~matt/courses/cs563/talks/flowvis/lic.html Line Integral Convolution for Flow Visualization by Dr. Matthew O. Ward]</ref>

=== Integration-based method ===

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===Global method===

Compared to other [[Integral|integration]]-based techniques that compute [[field line]]s 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, itr shows the topology of the vector field.

In user testing, LIC was found to be particularly good for identifying critical points.<ref name="Laidlaw 2001">{{cite conference | first1 = David H. | last1 = Laidlaw | first2 = Robert M. | last2 = Kirby | first3 = J. Scott | last3 = Davidson | first4 = Timothy S. | last4 = Miller | first5 = Marco | last5 = da Silva | first6 = William H. | last6 = Warren | first7 = Michael J. | last7 = Tarr | title = Quantitative Comparative Evaluation of 2D Vector Field Visualization Methods | book-title = IEEE Visualization 2001, VIS '01. Proceedings | date = October 21–26, 2001 | place = San Diego, CA, USA | pages = 143–150}}</ref>

===other===

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* visualisation of [[magnetic field]] lines using [[:commons:Category:Magnetised iron filings|randomly distributed iron filings]]

* "emulates what happens when a rectangular area of massless fine sand is blown by strong wind"<ref>[http://www.zhanpingliu.org/research/flowvis/LIC/LIC.htm LIC by Zhanping Liu]</ref>

Intuitively, the flow of a [[vector field]] in some ___domain is visualized by adding a static random pattern of dark and light paint sources. As the flow passes by the sources, each parcel of fluid picks up some of the source color, averaging it with the color it has already acquired in a manner similar to throwing paint in a river. The result is a random striped texture where points along the same streamline tend to have similar color.

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where <math>k(s)</math> is the convolution 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, [[Field line|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 output image will normally be colored in some way. Typically some scalar field in <math>\Omega</math> is used, like the vector length, to determine the hue, while the gray-scale LIC image determines the brightness of the color.

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====Problem====

Visualise field lines and singularities of a 2D stationary vector field (stream lines of a 2D steady flow)

====Input====

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====Steps====

====constraints ====

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Since the computation of a LIC image is expensive but inherently parallel, it has also been parallelized<ref name="Zoeckler 1996">{{cite journal | last1 = Zöckler | first1 = Malte | last2 = Stalling | first2 = Detlev | last3 = Hege | first3 = Hans-Christian | title = Parallel Line Integral Convolution | journal = Parallel Computing | volume = 23 | issue = 7 | pages = 975–989 | publisher = North Holland | ___location = Amsterdam | url = http://www.zib.de/visual-publications/sources/src-1996/parLIC.pdf | year = 1997 | issn = 0167-8191 | doi = 10.1016/S0167-8191(97)00039-2 }}</ref> and, with availability of GPU-based implementations, it has become interactive on PCs. Also for UFLIC an interactive GPU-based implementation has been presented.<ref name="Ding 2015">{{cite conference | first = Zi'ang | last = Ding | first2 = Zhanping | last2 = Liu | first3 = Yang | last3 = Yu | first4 = Wei | last4 = Chen | title = Parallel unsteady flow line integral convolution for high-performance dense visualization | book-title = 2015 IEEE Pacific Visualization Symposium, PacificVis 2015 | place = Hangzhou, China | pages = 25–30 | year = 2015}}</ref>

===Multidimensional===

Note that ___domain <math>\Omega</math> does not have to be a 2D ___domain: the method is applicable to higher dimensional domains using multidimensional noise fields. However, the visualization of the higher-dimensional LIC texture is problematic; one way is to use interactive exploration with 2D slices that are manually positioned and rotated. The ___domain <math>\Omega</math> does not have to be flat either; the LIC texture can be computed also for arbitrarily shaped 2D surfaces in 3D space.<ref name="Battke 1997">{{cite book | first1 = Henrik | last1 = Battke | first2 = Detlev | last2 = Stalling | first3 = Hans-Christian | last3 = Hege | chapter = Fast Line Integral Convolution for Arbitrary Surfaces in 3D | editor1-first = Hans-Christian | editor1-last = Hege | editor2-first = Konrad | editor2-last = Polthier | title = Visualization and Mathematics: Experiments, Simulations, and Environments | url = https://archive.org/details/visualizationmat00fran | url-access = limited | publisher = [[Springer Science+Business Media|Springer]] | ___location = Berlin, New York | pages = [https://archive.org/details/visualizationmat00fran/page/n191 181]–195 | year = 1997 | citeseerx = 10.1.1.71.7228 | doi = 10.1007/978-3-642-59195-2_12| isbn = 3-540-61269-6 }}</ref>

==Applications==

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Applications:

* representing vector fields by aproximationapproximation a [[field line]]s of the input vector field

** flow-visualization method for steady ( time independent) flows.<ref>[https://daac.hpc.mil/gettingStarted/Line%20Integral%20Convolution.html DAAC: Line Integral Convolution]</ref> Here field lines are called [[Streamlines, streaklines, and pathlines|stremlines]]

** Visual exploration of 2D autonomous dynamical systems<ref>[https://iopscience.iop.org/article/10.1088/0143-0807/36/3/035007 Visual exploration of 2D autonomous dynamical systems Thomas Müller2,1 and Filip Sadlo1 Published 26 February 2015 • © 2015 IOP Publishing Ltd European Journal of Physics, Volume 36, Number 3]</ref>

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** Automatic generation of hair texture<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>

* Terrain generalization: creating generalized [[Terrain_cartographyTerrain cartography#Shaded_reliefShaded relief|shaded relief]]<ref>[https://www.tandfonline.com/doi/full/10.1080/15230406.2020.1833762 Bernhard Jenny (2021) Terrain generalization with line integral convolution, Cartography and Geographic Information Science, 48:1, 78-92, DOI: 10.1080/15230406.2020.1833762]</ref>

==Implementations==

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* [https://github.com/andresbejarano/2DFlowVisualization A 2D flow visualization tool based on LIC and RK4. Developed using C++ and VTK. by Andres Bejarano]

* [https://reference.wolfram.com/language/ref/LineIntegralConvolutionPlot.html Wolfram Research (2008), LineIntegralConvolutionPlot, Wolfram Language function, https://reference.wolfram.com/language/ref/LineIntegralConvolutionPlot.html (updated 2014).]

==See also==

* [[Weighted arithmetic mean|Weighted arithmetic mean or Weighted average]]

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{{Commons category|Line integral convolution}}

{{Wikibooks|Fractals/Mathematics/LIC}}

[[Category:Numerical function drawing]]

[[Category:Vector calculus]]