Fractals/Mathematics/LIC

line Integral Convolution (LIC):

• visualize dense flow fields by imaging its integral curves
• "emulates what happens when an area of fine sand is blown by strong wind" ( Zhanping Liu)

Method

• vover domain with a input texture
• blur (convolve) the input texture along the path lines using a specified filter kernel
• smear out the texture along stream lines

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:

• 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:

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

weighted mean

Color of output pixel is computed using weighted mean

The weighted mean of a non-empty finite tuple of:

• data $\left(x_{1},x_{2},\dots ,x_{n}\right)$,
• corresponding non-negative weights $\left(w_{1},w_{2},\dots ,w_{n}\right)$

is

${\bar {x}}={\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}}{\sum \limits _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}$

Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight.

The weights cannot be negative. Some may be zero, but not all of them (since division by zero is not allowed).

versions

line Integral Convolution (LIC)

• Oriented Line Integral Convolution (OLIC)
  • Fast Rendering of Oriented Line Integral Convolution ( FROLIC)

Implementation

• GPU-Based-Image-Processing-Tools by RaymondMcGuire in python
• LIC (Line Integral Convolution) principle and Python implementation
• github
• LicPy