# A Vector Field Texture Generation Method without Convolution Calculation

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Mesh Unit Filling Preprocessing Method

_{i}is the coordinate of current point i and p

_{i}

_{+1}is the coordinate of the next point, which is the step length for streamline tracing; v(p) is the vector value of p point, and c

_{1}to c

_{4}are four process parameters.

#### 3.1. Texture Pixel

_{i}, y

_{i}) are the coordinates of the four vertices in the counterclockwise direction of the texture pixel starting from the lower left corner, and v

_{i}is the physical field value of these four vertices.

_{min}and y

_{min}are the horizontal and vertical coordinates of the bottom left corner of the smallest rectangle that can cover the physical field, respectively; lx and ly are the horizontal and vertical resolutions of the final regular grid, respectively; and a is the side length of one texture pixel. Below we discuss how to generate a regular grid made of texture pixels.

#### 3.2. Mesh Unit Filling Preprocessing Method

## 4. Random Increment Streamline Method

#### 4.1. Basic Idea

_{i}⌋) is the input primitive value at point P

_{i}passed by the streamline tracking, h

_{i}is the weight point at the time of convolution, L is the number of streamline tracking steps in the forward direction, and L′ is the number of streamline tracking steps in the reverse direction.

_{0}) is the output texture gray value of the sampling point x

_{0}, I(x

_{i}) is the input gray value of the point x

_{i}, L is the tracing length of one side, and k(x

_{i}) is the convolution kernel function.

_{m}, the convolution calculation of x

_{m}

_{+1}no longer contains the factor x

_{m−L}, but rather it contains an extra factor x

_{m}

_{−1−L}. Based on this idea, the FLIC [6] algorithm uses faster calculation to improve the algorithm speed, as follows:

_{m}is a sampling point on the streamline whose gray value has been calculated, and x

_{m}

_{+1}is the next adjacent sampling point on the streamline. Please note that the letters used in Equation (7) and the literature [6] are slightly different in order to ensure consistency in the discussion in this article.

_{i}) − U[0, 255]. Second, in various LIC methods, the convolution kernel function k(x

_{i}) has different forms, but in the FLIC method, only the box kernel function was used. Our RIS method also uses the box kernel function only.

_{LIC}is the original texture obtained by the RIS method, T

_{mag}is the contour graph drawn in line with the vector field modulus value, and t is the mixed parameter. The larger the t, the higher influence the contour graph is. The smaller the t, the more obvious the original texture is.

#### 4.2. Periodic Circulating Animation

## 5. Simplified RIS Algorithm

## 6. Results and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Gray RIS textures with different contrasts, as generated by Equations (10) and (8). (

**a**) The texture that is drawn using Equations (8) and (9) (

**b**) The texture that is drawn using Equations (11) and (9), and the parameter r is set to 30 (

**c**) The texture that is drawn using Equations (11) and (9), and the parameter r is set to 50 (

**d**) The texture that is drawn using Equations (11) and (9), and the parameter r is set to 90.

**Figure 11.**A set of textures for concentric circles in vector field data. (

**a**) is drawn using the LIC algorithm, L = 10 (

**b**) is drawn using the FLIC algorithm, L = 10 (

**c**) is drawn using the RIS algorithm, L = 10 (

**d**) is drawn using the RIS algorithm, L = 5 (

**e**) is drawn using the RIS algorithm, L = 20 (

**f**) is the color texture which be drawn using the RIS algorithm, L = 10.

**Figure 12.**A set of textures for saddle-shaped vector field data. (

**a**) is drawn using the LIC algorithm, L = 10 (

**b**) is drawn using the FLIC algorithm, L = 10 (

**c**) is drawn using the RIS algorithm, L = 10 (

**d**) is drawn using the RIS algorithm, L = 5 (

**e**) is drawn using the RIS algorithm, L = 20 (

**f**) is the color texture which be drawn using the RIS algorithm, L = 10.

**Figure 13.**A set of textures for the first set of wind field data. (

**a**) the gray RIS texture (

**b**) the gray SRIS texture (

**c**) the color SRIS texture.

**Figure 14.**A set of textures for the second set of wind field data. (

**a**) the gray RIS texture (

**b**) the gray SRIS texture (

**c**) the color SRIS texture.

**Figure 15.**A set of textures for the third set of wind field data. (

**a**) the gray RIS texture (

**b**) the gray SRIS texture (

**c**) the color SRIS texture.

Algorithms | Speed | Was the Cyclic Animation Provided? | Two Dimensions/Three Dimensions | With or Without Enhanced Contrast Function |
---|---|---|---|---|

Line Integral Convolution (LIC) [5] | Very slow | Yes | Two dimensions | Without |

Fast Line Integral Convolution (FLIC) [6] | Much faster | Yes | Two dimensions | Without |

Motion Map [7] | Slow | Yes | Two dimensions | Without |

Enhanced Line Integral Convolution (ELIC) [8] | Slow | No | Two dimensions | With |

Unsteady Flow Line Integral Convolution (UFLIC) [19] | Slow | Yes | Two dimensions | Without |

Volume Line Integral Convolution [20] | Slow | No | Three dimensions | Without |

Enhanced 3D Line Integral Convolution [21] | Slow | Yes | Three dimensions | With |

Grid Unit Number | Texture Pixel Number | Walk-Through | Seed Filling Preprocessing (SFP) | Mesh Unit Filling Preprocessing (MUFP) |
---|---|---|---|---|

11,056 | 1,705,600 | 6374.198 | 15.249 | 14.122 |

11,056 | 426,400 | 326.71 | 3.772 | 3.57 |

11,056 | 106,400 | 155.639 | 0.928 | 0.878 |

11,056 | 26,600 | 124.338 | 0.234 | 0.229 |

2740 | 1,705,600 | 322.923 | 16.743 | 14.108 |

2740 | 426,400 | 15.426 | 3.869 | 3.52 |

2740 | 106,400 | 7.271 | 0.952 | 0.872 |

2740 | 26,600 | 6.055 | 0.237 | 0.226 |

720 | 1,705,600 | 150.001 | 17.216 | 14.102 |

720 | 426,400 | 7.223 | 3.944 | 3.46 |

720 | 106,400 | 3.436 | 1.083 | 0.869 |

720 | 26,600 | 2.837 | 0.368 | 0.217 |

176 | 1,705,600 | 68.843 | 18.386 | 14.074 |

176 | 426,400 | 3.428 | 4.075 | 3.43 |

176 | 106,400 | 1.622 | 1.126 | 0.864 |

176 | 26,600 | 1.345 | 0.394 | 0.208 |

Figure 9 | Figure 11 | Figure 12 | Figure 13 | Figure 14 | Figure 15 | |
---|---|---|---|---|---|---|

Grid units | 46 | 5000 | 5000 | 7440 | 45,552 | 7938 |

Texture Pixels | 310,400 | 160,000 | 160,000 | 154,800 | 149,600 | 160,000 |

Mesh Unit Filling Preprocessing | 2.516 | 1.201 | 1.202 | 1.292 | 1.259 | 1.318 |

Line Integral Convolution | 4.915 | 3.526 | 3.527 | 3.523 | 3.474 | 3.528 |

Fast Line Integral Convolution | 0.552 | 0.312 | 0.313 | 0.316 | 0.311 | 0.318 |

Random Increment Streamline | 0.351 | 0.192 | 0.192 | 0.191 | 0.187 | 0.193 |

Simplified Random Increment Streamline | 0.344 | 0.186 | 0.187 | 0.187 | 0.183 | 0.188 |

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**MDPI and ACS Style**

Du, X.; Liu, H.; Tseng, H.-W.; Meen, T.-H.
A Vector Field Texture Generation Method without Convolution Calculation. *Symmetry* **2020**, *12*, 724.
https://doi.org/10.3390/sym12050724

**AMA Style**

Du X, Liu H, Tseng H-W, Meen T-H.
A Vector Field Texture Generation Method without Convolution Calculation. *Symmetry*. 2020; 12(5):724.
https://doi.org/10.3390/sym12050724

**Chicago/Turabian Style**

Du, Xiaofu, Huilin Liu, Hsien-Wei Tseng, and Teen-Hang Meen.
2020. "A Vector Field Texture Generation Method without Convolution Calculation" *Symmetry* 12, no. 5: 724.
https://doi.org/10.3390/sym12050724