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)Lic by 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 kernel:Kernel_(image_processing) in wikipedia 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 from output array go to the Vector array choose value at pixel 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 where a segment of pixels intensities of length 2L + 1 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 if and only if . Thus a tuple has properties that distinguish it from a set: A tuple may contain multiple instances of the same element, so tuple ; but set . Tuple elements are ordered: tuple , but set . 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 , corresponding non-negative weights is 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)Fast Oriented Line Integral Convolution for Vector Field Visualization via the Internet by Rainer Wegenkittl and Eduard Gr¨ollerJava Exploration Tool for Dynamical Systems by R. Wegenkittl and E. Gröller. Implementation GPU-Based-Image-Processing-Tools by RaymondMcGuire in python LIC (Line Integral Convolution) principle and Python implementation github LicPy References