Content deleted Content added
m Fixed broken citation |
Link suggestions feature: 3 links added. Tags: Visual edit Mobile edit Mobile web edit Newcomer task Suggested: add links |
||
Line 7:
==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}}
Line 35:
<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 |url=https://archive.org/details/mathematicalfoun00mlle |title=Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration |publisher=[[Springer Science+Business Media|Springer]] |year=2009 |isbn=978-3-540-25076-0 |editor1-last=Möller |editor1-first=Torsten |series=Mathematics and Visualization |___location=Berlin, New York |pages=[https://archive.org/details/mathematicalfoun00mlle/page/n195 191]–211 |chapter=Iterative Twofold Line Integral Convolution for Texture-Based Vector Field Visualization |citeseerx=10.1.1.66.3013 |doi=10.1007/b106657_10 |editor2-last=Hamann |editor2-first=Bernd |editor3-last=Russell |editor3-first=Robert D. |url-access=limited }}{{dead link|date=May 2024}}</ref>
| |||