Dictionary

the integral curve of the vector field = field line of vector field = streamline of steady ( time independent) flow
In mathematics, convolution is a special type of binary operation on two functions. Here discrete convolution is used

elements

input:

texture, usually stationary white noise ( random texture)
vector field: a stationary vector field defined by a map $v:R^{2}\to R^{2}$

kernel:^[2]

one dimensional matrix ( array) of weights
with an odd kernel size to ensure there is a valid integer (x, y)-coordinate at the center of the image
normalized. Normalization is defined as the division of each element in the kernel by the sum of all kernel elements, so that the sum of the elements of a normalized kernel is unity. This will ensure the average pixel in the modified image is as bright as the average pixel in the original image.
often simple box filter
often symmetric (isotropic) = the convolution kernel is symmetric across its zero point.

Convolution formula ( convolution is not matrix multiplication, when kernel matrix is used )

output image = Final LIC image of input vector field. It is a smeared version of the input texture , where the direction of smearing is determined by the vector field.

vector field

Source

synthetic data
experimental data

Method

array of values
functions defining vector field

algorithm

General steps for LIC image:

For each pixel of output 2D array compute it's color
save complete output array as 2D static image file

Substeps performed for each pixel of LIC image:

choose pixel $p_{0}(x,y)$ from output array
go to the Vector array
- choose value at pixel $p_{0}(x,y)$
- find segment of local field line

dimensions

all images ( textures, domains) have the same dimensions =

segment of field line

The field lines are calculated on basis of a vector field
a segment of locations $l(s)=(i,j)+sV(i,j)$ where $s\in \left\{-L,-L+1,...,0,...,L-1,L\right\}$
a segment of pixels intensities $I(l(s))$ of length 2L + 1

  $I(i,j)\to O(i,j)$

tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

Properties The general rule for the identity of two $n$ -tuples is

(a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})

if and only if

a_{1}=b_{1},{\text{ }}a_{2}=b_{2},{\text{ }}\ldots ,{\text{ }}a_{n}=b_{n}

.

Thus a tuple has properties that distinguish it from a set:

A tuple may contain multiple instances of the same element, so
tuple $(1,2,2,3)\neq (1,2,3)$ ; but set $\{1,2,2,3\}=\{1,2,3\}$ .
Tuple elements are ordered: tuple $(1,2,3)\neq (3,2,1)$ , but set $\{1,2,3\}=\{3,2,1\}$ .
A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements.