SherbyNet - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Line integral convolution</title>

</head>

Line integral convolution

</h1>

</header>

In <a href="Scientific_visualization" title="Scientific visualization">scientific visualization</a>, line integral convolution (LIC) is a method to visualize a <a href="Vector_field" title="Vector field">vector field</a> (such as <a href="Fluid_motion" class="mw-redirect" title="Fluid motion">fluid motion</a>) at high spatial resolutions.<a href="#cite_note-Stalling_1995-1">[1]</a> The LIC technique was first proposed by Brian Cabral and Leith Casey Leedom in 1993.<a href="#cite_note-Cabral_1993-2">[2]</a>

In LIC, discrete numerical <a href="Line_integral" title="Line integral">line integration</a> is performed along the <a href="Field_line" title="Field line">field lines</a> (curves) of the vector field on a <a href="Regular_grid" title="Regular grid">uniform grid</a>. The integral operation is a <a href="Convolution" title="Convolution">convolution</a> of a filter <a href="Kernel_(image_processing)" title="Kernel (image processing)">kernel</a> and an input texture, often <a href="White_noise" title="White noise">white noise</a>.<a href="#cite_note-Stalling_1995-1">[1]</a> In <a href="Signal_processing" title="Signal processing">signal processing</a>, this process is known as a <a href="Convolution#Discrete_convolution" title="Convolution">discrete convolution</a>.<a href="#cite_note-:0-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2></div>

Traditional visualizations of vector fields use small arrows or lines to represent vector direction and magnitude. This method has a low <a href="Spatial_resolution" title="Spatial resolution">spatial resolution</a>, which limits the density of presentable data and risks obscuring characteristic features in the data.<a href="#cite_note-Stalling_1995-1">[1]</a><a href="#cite_note-:0-3">[3]</a> More sophisticated methods, such as <a href="Streamlines%2C_streaklines%2C_and_pathlines" title="Streamlines, streaklines, and pathlines">streamlines</a> and particle tracing techniques, can be more revealing but are highly dependent on proper seed points.<a href="#cite_note-Stalling_1995-1">[1]</a> Texture-based methods, like LIC, avoid these problems since they depict the entire vector field at point-like (pixel) resolution.<a href="#cite_note-Stalling_1995-1">[1]</a>

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.

In user testing, LIC was found to be particularly good for identifying critical points.<a href="#cite_note-Laidlaw_2001-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Informal_description">Informal description</h3></div>

LIC causes output values to be strongly correlated along the field lines, but uncorrelated in orthogonal directions.<a href="#cite_note-Stalling_1995-1">[1]</a> As a result, the field lines contrast each other and stand out visually from the background.

Intuitively, the process can be understood with the following example: the flow of a vector field can be visualized by overlaying a fixed, random pattern of dark and light paint. As the flow passes by the paint, the fluid picks up some of the paint's color, averaging it with the color it has already acquired. The result is a randomly striped, smeared texture where points along the same streamline tend to have a similar color. Other physical examples include:

<ul><li>whorl patterns of paint, oil, or foam on a river</li>

<li>visualisation of <a href="Magnetic_field" title="Magnetic field">magnetic field</a> lines using <a href="https://commons.wikimedia.org/wiki/Category:Magnetised_iron_filings" class="extiw external" title="commons:Category:Magnetised iron filings">randomly distributed iron filings</a></li>

<li>fine sand being blown by strong wind<a href="#cite_note-5">[5]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Formal_mathematical_description">Formal mathematical description</h3></div>

Although the input vector field and the result image are discretized, it pays to look at it from a continuous viewpoint.<a href="#cite_note-Stalling_1995-1">[1]</a> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

</semantics>

</math><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"> be the vector field given in some domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy">. Although the input vector field is typically discretized, we regard the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation>

</semantics>

</math><img src="./35c1866e359fbfd2e0f606c725ba5cc37a5195d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }" loading="lazy"> as defined in every point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy">, i.e. we assume an <a href="Interpolation" title="Interpolation">interpolation</a>. Streamlines, or more generally field lines, are tangent to the vector field in each point. They end either at the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {0} }</annotation>

</semantics>

</math><img src="./d31d67261ea15071d6152c53c93b21388cf22e5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.846ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} =\mathbf {0} }" loading="lazy">. For the sake of simplicity, critical points and boundaries are ignored in the following.

A field line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation>

</semantics>

</math><img src="./e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}" loading="lazy">, parametrized by arc length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy">, is defined as

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d{\boldsymbol {\sigma }}(s)}{ds}}={\frac {\mathbf {v} ({\boldsymbol {\sigma }}(s))}{|\mathbf {v} ({\boldsymbol {\sigma }}(s))|}}.}</annotation>

</semantics>

</math>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}_{\mathbf {r} }(s)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}_{\mathbf {r} }(s)}</annotation>

</semantics>

</math><img src="./6f9036c59a0f288319deb37b3a323e260f6c73ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.506ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\sigma }}_{\mathbf {r} }(s)}" loading="lazy"> be the field line that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</math><img src="./eca0f46511c4c986c48b254073732c0bd98ae0c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=0}</annotation>

</semantics>

</math><img src="./7903b8069a44c70f6f96511675bdd9a4ff200ed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=0}" loading="lazy">. Then the image gray value at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation>

</semantics>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(\mathbf {r} )=\int _{-L/2}^{L/2}k(s)N({\boldsymbol {\sigma }}_{\mathbf {r} }(s))ds}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k(s)}</annotation>

</semantics>

</math><img src="./ac3a0f3d91f1d43e2c27409fd7eb41b887590563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.111ex; height:2.843ex;" alt="{\displaystyle k(s)}" loading="lazy"> is the convolution <a href="Kernel_(image_processing)" title="Kernel (image processing)">kernel</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\mathbf {r} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./64f23430688609bf1cbb00423b25543c7c8ede84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.975ex; height:2.843ex;" alt="{\displaystyle N(\mathbf {r} )}" loading="lazy"> is the noise image, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is the length of field <a href="Line_segment" title="Line segment">line segment</a> that is followed.

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D(\mathbf {r} )}</annotation>

</semantics>

</math><img src="./907edb8a67e55bb812185b922494ca3f78ddc085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.836ex; height:2.843ex;" alt="{\displaystyle D(\mathbf {r} )}" loading="lazy"> 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 <a href="Numerical_methods_for_ordinary_differential_equations" title="Numerical methods for ordinary differential equations">numerical method for solving ordinary differential equations</a>, like a <a href="Runge%E2%80%93Kutta_methods" title="Runge–Kutta methods">Runge–Kutta method</a>, 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 <a href="Scalar_field" title="Scalar field">scalar field</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

Different choices of convolution kernels and random noise produce different textures; for example, <a href="Pink_noise" title="Pink noise">pink noise</a> 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.<a href="#cite_note-Weiskopf_2009-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Programming_description">Programming description</h3></div>

Algorithmically, LIC takes a vector field and noise texture as input, and outputs a texture. The process starts by generating in the domain of the vector field a random <a href="Grayscale" title="Grayscale">gray level image</a> at the desired output resolution. Then, for every pixel in this image, the forward and backward streamline of a fixed <a href="Arc_length" title="Arc length">arc length</a> is calculated. The value assigned to the current pixel is computed by a <a href="Convolution" title="Convolution">convolution</a> of a suitable convolution kernel with the gray levels of all the noise pixels lying on a segment of this streamline. This creates a gray level LIC image.

<div class="mw-heading mw-heading2"><h2 id="Versions">Versions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Basic">Basic</h3></div>

Basic LIC images are grayscale images, without color and animation. While such LIC images convey the direction of the field vectors, they do not indicate orientation; for stationary fields, this can be remedied by animation. Basic LIC images do not show the length of the vectors (or the strength of the field).

<div class="mw-heading mw-heading3"><h3 id="Color">Color</h3></div>

The length of the vectors (or the strength of the field) is usually coded in color; alternatively, animation can be used.<a href="#cite_note-Cabral_1993-2">[2]</a><a href="#cite_note-Stalling_1995-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="Animation">Animation</h3></div>

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 <a href="Periodic_function" title="Periodic function">periodic function</a> multiplied by a <a href="Hann_function" title="Hann function">Hann function</a> acting as a window (in order to prevent artifacts) is used. The periodic function is then shifted along the period to create an animation.

The computation can be significantly accelerated by re-using parts of already computed field lines, specializing to a box function as convolution kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k(s)}</annotation>

</semantics>

</math><img src="./ac3a0f3d91f1d43e2c27409fd7eb41b887590563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.111ex; height:2.843ex;" alt="{\displaystyle k(s)}" loading="lazy"> and avoiding redundant computations during convolution.<a href="#cite_note-Stalling_1995-1">[1]</a> The resulting fast LIC method can be generalized to convolution kernels that are arbitrary polynomials.<a href="#cite_note-Hege_1998-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Oriented_Line_Integral_Convolution_(OLIC)">Oriented Line Integral Convolution (OLIC)</h3></div>

Because LIC does not encode flow orientation, it cannot distinguish between streamlines of equal direction but opposite orientation.<a href="#cite_note-:1-8">[8]</a> Oriented Line Integral Convolution (OLIC) solves this issue by using a ramp-like asymmetric kernel and a low-density noise texture.<a href="#cite_note-:1-8">[8]</a> The kernel asymmetrically modulates the intensity along the streamline, producing a trace that encodes orientation; the low-density of the noise texture prevents smeared traces from overlapping, aiding readability.

Fast Rendering of Oriented Line Integral Convolution (FROLIC) is a variation that approximates OLIC by rendering each trace in discrete steps instead of as a continuous smear.<a href="#cite_note-:1-8">[8]</a><a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Unsteady_Flow_LIC_(UFLIC)">Unsteady Flow LIC (UFLIC)</h3></div>

For time-dependent vector fields (unsteady flow), a variant called Unsteady Flow LIC has been designed that maintains the coherence of the flow animation.<a href="#cite_note-Shen_1998-10">[10]</a> An interactive GPU-based implementation of UFLIC has been presented.<a href="#cite_note-Ding_2015-11">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="Parallel">Parallel</h3></div>

Since the computation of an LIC image is expensive but inherently parallel, the process has been <a href="Parallel_computing" title="Parallel computing">parallelized</a><a href="#cite_note-Zoeckler_1996-12">[12]</a> and, with availability of GPU-based implementations, interactive on PCs.

<div class="mw-heading mw-heading3"><h3 id="Multidimensional">Multidimensional</h3></div>

Note that the domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

This technique has been applied to a wide range of problems since it first was published in 1993, both scientific and creative, including:

Representing vector fields:

<ul><li>visualization of steady (time-independent) flows (streamlines)<a href="#cite_note-14">[14]</a></li>

<li>visual exploration of 2D autonomous dynamical systems<a href="#cite_note-15">[15]</a></li>

<li>wind mapping<a href="#cite_note-16">[16]</a></li>

<li>water flow mapping</li></ul>

Artistic effects for image generation and stylization:

<ul><li>pencil drawing<a href="#cite_note-17">[17]</a> (automatic pencil drawing generation technique using LIC pencil filter<a href="#cite_note-18">[18]</a>)</li>

<li>automatic generation of hair textures<a href="#cite_note-19">[19]</a></li>

<li>creating marbling texture<a href="#cite_note-20">[20]</a></li></ul>

Terrain generalization:

<ul><li>creating generalized <a href="Terrain_cartography#Shaded_relief" title="Terrain cartography">shaded relief</a><a href="#cite_note-21">[21]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Implementations">Implementations</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://github.com/RaymondMcGuire/GPU-Based-Image-Processing-Tools">GPU Based Image Processing Tools by Raymond McGuire</a></li>

<li><a rel="nofollow" class="external text" href="https://www.paraview.org/Wiki/ParaView/Line%20Integral%20Convolution">ParaView : Line Integral Convolution</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/andresbejarano/2DFlowVisualization">A 2D flow visualization tool based on LIC and RK4. Developed using C++ and VTK. by Andres Bejarano</a></li>

<li><a rel="nofollow" class="external text" href="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).</a></li></ul>

<ul><li><a href="Weighted_arithmetic_mean" title="Weighted arithmetic mean">Weighted arithmetic mean</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFStallingHege1995" class="citation conference cs1">Stalling, Detlev; Hege, Hans-Christian (August 6–11, 1995). <a rel="nofollow" class="external text" href="https://archive.org/details/computergraphics00sigg/page/249">"Fast and Resolution Independent Line Integral Convolution"</a>. Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques. SIGGRAPH '95. Los Angeles, California. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/computergraphics00sigg/page/249">249–256</a>. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5526">10.1.1.45.5526</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F218380.218448">10.1145/218380.218448</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89791-701-4</bdi>.</cite>

</li>

<li id="cite_note-Cabral_1993-2">^ <a href="#cite_ref-Cabral_1993_2-0">a</a> <a href="#cite_ref-Cabral_1993_2-1">b</a> <cite id="CITEREFCabralLeedom1993" class="citation conference cs1">Cabral, Brian; Leedom, Leith Casey (August 2–6, 1993). "Imaging Vector Fields Using Line Integral Convolution". Proceedings of the 20th annual conference on Computer graphics and interactive techniques. SIGGRAPH '93. Anaheim, California. pp. 263–270. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.115.1636">10.1.1.115.1636</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F166117.166151">10.1145/166117.166151</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-89791-601-8</bdi>.</cite>

</li>

<li id="cite_note-:0-3">^ <a href="#cite_ref-:0_3-0">a</a> <a href="#cite_ref-:0_3-1">b</a> <cite id="CITEREFWard1996" class="citation web cs1">Ward, Matthew O. (February 5, 1996). <a rel="nofollow" class="external text" href="https://web.cs.wpi.edu/~matt/courses/cs563/talks/flowvis/lic.html">"Line Integral Convolution for Flow Visualization"</a>. Worcester Polytechnic Institute. Retrieved 2024-07-21.</cite>

</li>

<li id="cite_note-Laidlaw_2001-4"><a href="#cite_ref-Laidlaw_2001_4-0">^</a> <cite id="CITEREFLaidlawKirbyDavidsonMiller2001" class="citation conference cs1">Laidlaw, David H.; Kirby, Robert M.; Davidson, J. Scott; Miller, Timothy S.; da Silva, Marco; Warren, William H.; Tarr, Michael J. (October 21–26, 2001). "Quantitative Comparative Evaluation of 2D Vector Field Visualization Methods". IEEE Visualization 2001, VIS '01. Proceedings. San Diego, CA, USA. pp. 143–150.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFLiu2007" class="citation web cs1">Liu, Zhanping (Aug 21, 2007). <a rel="nofollow" class="external text" href="http://www.zhanpingliu.org/research/flowvis/LIC/LIC.htm">"LIC (Line Integral Convolution)"</a>. Retrieved 2024-07-21.</cite>

</li>

<li id="cite_note-Weiskopf_2009-6"><a href="#cite_ref-Weiskopf_2009_6-0">^</a> <cite id="CITEREFWeiskopf2009" class="citation book cs1">Weiskopf, Daniel (2009). "Iterative Twofold Line Integral Convolution for Texture-Based Vector Field Visualization". In Möller, Torsten; Hamann, Bernd; Russell, Robert D. (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalfoun00mlle">Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration</a>. Mathematics and Visualization. Berlin, New York: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalfoun00mlle/page/n195">191</a>–211. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.66.3013">10.1.1.66.3013</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fb106657_10">10.1007/b106657_10</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-25076-0</bdi>.</cite>

</li>

<li id="cite_note-Hege_1998-7"><a href="#cite_ref-Hege_1998_7-0">^</a> <cite id="CITEREFHegeStalling1998" class="citation cs2">Hege, Hans-Christian; Stalling, Detlev (1998), "Fast LIC with Piecewise Polynomial Filter Kernels", in Hege, Hans-Christian; Polthier, Konrad (eds.), Mathematical Visualization, Berlin, Heidelberg: Springer-Verlag, pp. 295–314, <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.31.504">10.1.1.31.504</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-03567-2_22">10.1007/978-3-662-03567-2_22</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-08373-0</bdi></cite>

</li>

<li id="cite_note-:1-8">^ <a href="#cite_ref-:1_8-0">a</a> <a href="#cite_ref-:1_8-1">b</a> <a href="#cite_ref-:1_8-2">c</a> <cite id="CITEREFWegenkittlGröller1997" class="citation book cs1">Wegenkittl, Rainer; Gröller, Eduard (24 October 1997). <a rel="nofollow" class="external text" href="https://www.cg.tuwien.ac.at/research/vis/dynsys/frolic/frolic_crc.pdf">"Fast oriented line integral convolution for vector field visualization via the Internet"</a> (PDF). Proceedings. Visualization '97 (Cat. No. 97CB36155). pp. 309–316. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FVISUAL.1997.663897">10.1109/VISUAL.1997.663897</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8186-8262-0</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a rel="nofollow" class="external text" href="https://www.cg.tuwien.ac.at/research/vis/dynsys/frolic/">Java Exploration Tool for Dynamical Systems by R. Wegenkittl and E. Gröller.</a>

</li>

<li id="cite_note-Shen_1998-10"><a href="#cite_ref-Shen_1998_10-0">^</a> <cite id="CITEREFShenKam1998" class="citation journal cs1">Shen, Han-Wei; Kam, David L. (1998). <a rel="nofollow" class="external text" href="http://web.cse.ohio-state.edu/~shen.94/Research/Gravity/Project_2_files/Shen1998a.pdf">"A New Line Integral Convolution Algorithm for Visualizing Time-Varying Flow Fields"</a> (PDF). IEEE Trans Vis Comput Graph. 4 (2). Los Alamitos: IEEE: 98–108. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F2945.694952">10.1109/2945.694952</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1077-2626">1077-2626</a>.</cite>

</li>

<li id="cite_note-Ding_2015-11"><a href="#cite_ref-Ding_2015_11-0">^</a> <cite id="CITEREFDingLiuYuChen2015" class="citation conference cs1">Ding, Zi'ang; Liu, Zhanping; Yu, Yang; Chen, Wei (2015). "Parallel unsteady flow line integral convolution for high-performance dense visualization". 2015 IEEE Pacific Visualization Symposium, PacificVis 2015. Hangzhou, China. pp. 25–30.</cite>

</li>

<li id="cite_note-Zoeckler_1996-12"><a href="#cite_ref-Zoeckler_1996_12-0">^</a> <cite id="CITEREFZöcklerStallingHege1997" class="citation journal cs1">Zöckler, Malte; Stalling, Detlev; Hege, Hans-Christian (1997). <a rel="nofollow" class="external text" href="http://www.zib.de/visual-publications/sources/src-1996/parLIC.pdf">"Parallel Line Integral Convolution"</a> (PDF). Parallel Computing. 23 (7). Amsterdam: North Holland: 975–989. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0167-8191%2897%2900039-2">10.1016/S0167-8191(97)00039-2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0167-8191">0167-8191</a>.</cite>

</li>

<li id="cite_note-Battke_1997-13"><a href="#cite_ref-Battke_1997_13-0">^</a> <cite id="CITEREFBattkeStallingHege1997" class="citation book cs1">Battke, Henrik; Stalling, Detlev; Hege, Hans-Christian (1997). "Fast Line Integral Convolution for Arbitrary Surfaces in 3D". In Hege, Hans-Christian; Polthier, Konrad (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/visualizationmat00fran">Visualization and Mathematics: Experiments, Simulations, and Environments</a>. Berlin, New York: <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/visualizationmat00fran/page/n191">181</a>–195. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.7228">10.1.1.71.7228</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-59195-2_12">10.1007/978-3-642-59195-2_12</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-61269-6</bdi>.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a rel="nofollow" class="external text" href="https://daac.hpc.mil/gettingStarted/Line%20Integral%20Convolution.html">DAAC: Line Integral Convolution</a>

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a rel="nofollow" class="external text" href="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</a>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a rel="nofollow" class="external text" href="http://hint.fm/wind/">A real-time map of the wind in the U.S. by Fernanda Viégas and Martin Wattenberg.</a>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/228663813_Efficient_Region-Based_Pencil_Drawing">researchgate publication: Sun, Shuo & Huang, Dongwei. (2022). Efficient Region-Based Pencil Drawing.</a>

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/1323994?arnumber=1323994">S. Yamamoto, Xiaoyang Mo and A. Imamiya, "Enhanced LIC pencil filter," Proceedings. International Conference on Computer Graphics, Imaging and Visualization, 2004. CGIV 2004., 2004, pp. 251-256, doi: 10.1109/CGIV.2004.1323994.</a>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a rel="nofollow" class="external text" href="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.</a>

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> <a rel="nofollow" class="external text" href="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</a>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <a rel="nofollow" class="external text" href="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</a>

</li>

</ol></div></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .infobox .side-box{font-size:100%}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1237033735">

/* start https://en.wikipedia.org/ */

@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}

/* end https://en.wikipedia.org/ */

</style><div class="side-box side-box-right sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">

/* start https://en.wikipedia.org/ */

.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}

/* end https://en.wikipedia.org/ */

</style>

<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Line_integral_convolution" class="extiw external" title="commons:Category:Line integral convolution">Line integral convolution</a>.</div></div>

</div>

<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Fractals/Mathematics/LIC" class="extiw external" title="wikibooks:Fractals/Mathematics/LIC">Fractals/Mathematics/LIC</a></div></div>

</div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-26" href="https://en.wikipedia.org/wiki/?title=Line_integral_convolution&oldid=1302620119">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>