Refining Cellular Automatic Introduction Updating Rules to Model - PowerPoint PPT Presentation

Mean Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Refining Cellular Automatic Introduction Updating Rules to Model Mean Normal Vectors Curvature Flow Amount of Life to Remove Current William B. Byrd, Walter L. Gerych, Kateri A. Sternberg Evolution Mentor: Jeremy S. LeCrone Real World Time Future Kansas State University Research Questions July 21, 2015

Mean Introduction Curvature Flow W. Byrd W. Gerych • Mean curvature flow is a type of surface flow where the K.Sternberg J. LeCrone velocity of the surface along it’s normal direction is Introduction proportional to the mean curvature of the surface. Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions • We are trying to determine efficient rules to make a cellular automatic system mimic the behavior of a surface that undergoes mean curvature flow.

Mean Introduction Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research A Cellular Automata is a collection of grid cells each with a Questions finite number of states that evolves with each generation. The state of a cell in a certain generation is determined by the states of the neighboring cells in the previous generation and a set of evolution rules.

Mean Previous Research Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction • Proved that area calculations converge to the curvature. Normal Vectors • Approximated curvature at a cell with area calculations. Amount of Life to Remove • Implemented burning algorithm to more accurately calculate Current curvature Evolution • Added/Removed cells based on curvature calculations. Real World Time • Allowed for obstacles to affect surface flow. Future Research • Cells that had fractional quantities of life Questions

Mean What We’ve Done Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone • Determined an effective volumetric method of calculating Introduction normal vectors to the surface. Normal Vectors • Altered code to evolve the surface in the direction of the Amount of Life to normal vector. Remove • Determined that separating cells by neighbor counts does Current Evolution not improve accuracy Real World Time • Uncovered data that could be implemented to improve the Future accuracy of the evolution Research Questions • Discussed theoretical relationship between real world time and the evolution.

Mean Theoretical Convergence Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction � x � Normal Vectors �� R ( � p − ) dA y Amount of � Life to N = Remove 2 3 δ 3 Current Evolution Real World Time Theorem Future lim δ → 0 + � Research N is the outward unit normal vector at the Questions point p.

Mean Theoretical Convergence Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Mean Proof of Convergence Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone The Linear Convergence Introduction √ δ 2 − x 2 � x � � � � δ � 0 � δ x Normal √ δ 2 − x 2 − dydx dx Vectors 2 ( δ 2 − x 2 ) − 1 − δ − − δ y = − Amount of 2 2 3 δ 3 3 δ 3 Life to Remove Current � 0 2 ( δ 2 − δ 2 ) 3 / 2 + 1 2 ( δ 2 − δ 2 ) 3 / 2 Evolution � � − 1 � Real World ( − δ 3 2 + δ 3 6 ) − ( δ 3 2 − δ 3 6 ) 3 δ 3 2 Time = − = Future 2 2 3 δ 3 3 δ 3 Research Questions � 0 � = 1

Mean The Convergence of Error Curvature Flow W. Byrd x-component W. Gerych K.Sternberg J. LeCrone � δ � | f ( x ) | � δ | x | dydx − δ | xf ( x ) | dx − δ 0 E x ≤ = Introduction 2 2 3 δ 3 3 δ 3 Normal Vectors � δ � δ Amount of − δ M | x 3 | dx 0 Mx 3 dx = 2 M 1 4 δ 4 = 2 = 3 M Life to E x ≤ 4 δ Remove 2 2 2 3 δ 3 3 δ 3 3 δ 3 Current Evolution y-component Real World Time � δ � | f ( x ) | � δ � | f ( x ) | � δ 2 | f ( x ) | 2 dx 1 | y | dydx ydydx Future − δ 0 − δ 0 − δ E y ≤ = = Research 2 2 2 3 δ 3 3 δ 3 3 δ 3 Questions � δ � δ 1 2 M 2 | x 4 | dx 0 M 2 x 4 dx 5 M 2 δ 5 1 = 3 M 2 − δ 10 δ 2 E y ≤ = = 2 2 2 3 δ 3 3 δ 3 3 δ 3

Mean Error in Calculated Normal Vector Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Mean Determining the Life Lost Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction √ Normal r 02 − 2 t Vectors • r ( t ) = Amount of Life to • Analyzed flow of discrete circle of radius 5 from Remove Current time t = 0 to time t = 12, in . 1 increments. Evolution Real World • Calculated difference in life between Time Future generations Research Questions

Mean Separating by Neighbor Count Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions 5 Neighbors 7 Neighbors

Mean Different Neighbor Regressions Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current 4 Neighbors Evolution Real World Time Future Research Questions 7 Neighbors

Mean Evolution Comparison Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Evolution of Circles Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions Neighbor Count Evolution Discrete Circle

Mean Finding a New Regression Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Data For Regular Circles Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Original No Zeros Final Questions

Mean Inverted Circle Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Mean Fixing Inverted Circle Data Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Data For Inverted Circles Vectors Amount of Life to Remove Current Evolution Real World Time With Future Research Zeros No Zeros Final Questions

Mean Regression Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Mean How to Proceed Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal • Isolate the areas where life should be growing and shrink the Vectors dead region in those areas. Amount of Life to Remove • Could be done using the information gathered from the Current shrinking circle. Evolution • Would accomplish one of our goals of ensuring that a Real World Time curve’s evolution is unaffected by whether or not the life is Future Research on the inside or outside. Questions

Mean Current Evolution Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions

Mean Time vs Generations Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction • A ′ = Γ( t ) � � k · � n ds = − 2 π Normal Vectors • Rate of change of area is constant Amount of Life to Remove • Allows us to calculate the time elapsed between Current Evolution generations by calculating the loss of area Real World Time • Different time steps between different Future Research generations Questions

Mean Possible Future Research Curvature Flow W. Byrd W. Gerych K.Sternberg J. LeCrone Introduction Normal • Integrate real world time calculations into Vectors evolution Amount of Life to Remove • Apply our ideas for new evolution rules and Current Evolution compare them to discrete circles for accuracy Real World Time • Redefine the interface to possibly increase Future Research accuracy Questions

Mean Acknowledgements Curvature Flow W. Byrd W. Gerych K.Sternberg This research was carried out as a part of the Summer J. LeCrone Undergraduate Mathematics Research REU at Kansas State Introduction University funded by NSF award #1262877. The authors would Normal like to thank Kansas State University for their hospitality. Vectors Amount of Life to Remove Current Evolution Real World Time Future Research Questions #thejourney