Method and system for improving aerial image simulation speeds
A method and system for improving aerial image simulation speeds. The method includes receiving a mask; generating a matrix of node values based on the mask, wherein each node value corresponds to a node of a plurality of nodes in a lattice; performing a one-dimensional (1-D) approximation of a plurality of first approximation values at corresponding first approximation points between pairs of nodes of the plurality of nodes; performing a two-dimensional (2-D) approximation of second approximation values at corresponding second approximation points between pairs of first approximation points, wherein Chebyshev polynomials are used to approximate the first approximation values and the second approximation values. According to the method and system disclosed herein, approximating values using Chebyshev polynomials results in high-resolution aerial images that are generated at faster speeds.
1. A method for generating an aerial image, the method comprising:
receiving a mask;
generating a matrix of node values based on the mask, wherein each node value corresponds to a node of a plurality of nodes in a lattice;
performing a one-dimensional (1-D) approximation of a plurality of first approximation values at corresponding first approximation points between pairs of nodes of the plurality of nodes;
performing a two-dimensional (2-d) approximation of second approximation values at corresponding second approximation points between pairs of first approximation points, wherein chebyshev polynomials are used to approximate the first approximation values and the second approximation values; and
wherein a Chebyshev polynomial is defined as T n (cos((2i+1)*/(2n)))=0, wherein n is a degree of the polynomial and i=0 . . . n−1 is applied to the approximation process.
2. The method of claim 1 wherein the first approximation values and the second approximation values are approximated without calculating first derivatives of the node values.
3. The method of claim 1 wherein the node values, the first approximation values, and the second approximation values are intensity values of a function f(x,y).
4. The method of claim 3 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each point represents at least one of a node, a first approximation point, and a second approximation point.
5. The method of claim 3 wherein each value is approximated according to the following expression:
φ*=c 0 *T 0 (0)+c 1 *T 1 (0)+ . . . +c N−1 *T N−1 (0), wherein φ* represents the approximated value of the function, wherein c 0 , c 1 , . . . c N−1 denote coefficients, wherein T 0 (0), T 1 (0), . . . T N−1 (0) are Chebyshev polynomials, and N is a number of the first approximation points.
6. The method of claim 3 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each of the node values, first approximation values, and second approximation values are approximated according to the following expression:
φ*≅f((x 0 +x N−1 )/2), wherein φ* represents the approximated value of the function.
7. The method of claim 1 wherein the lattice comprises a plurality of squares having top sides, bottom sides, left sides, right sides, and corners, wherein a node is positioned at each corner, wherein the first approximation values are approximated between pairs of nodes at either the top and bottom sides and/or the left and right sides, and wherein the second approximation values are approximated between pairs of first approximation values at either the top and bottom sides and/or the left and right sides.
8. The method of claim 1 wherein the lattice is an equal-step lattice.
9. The method of claim 1 wherein after the 2-D approximation a resulting lattice has a border, wherein the border comprises one or more non-equal-step lattices.
10. A simulator system for generating an aerial image, comprising:
a processor; and
a simulator application executed on the processor for approximating values by,
receiving a mask;
generating a matrix of node values based on the mask, wherein each node value corresponds to a node of a plurality of nodes in a lattice;
performing a one-dimensional (1-D) approximation of a plurality of first approximation values at corresponding first approximation points between pairs of nodes of the plurality of nodes; and
performing a two-dimensional (2-d) approximation of second approximation values at corresponding second approximation points between pairs of first approximation points, wherein chebyshev polynomials are used to approximate the first approximation values and the second approximation values,
wherein the node values, the first approximation values, and the second approximation values are intensity values of a function f(x,y), and
wherein each value is approximated according to the following expression:
φ*=c 0 *T 0 (0)+c 1 *T 1 (0)+ . . . +c N−1 *T N−1 (0), wherein φ* represents the approximated value of the function, wherein c 0 , c 1 , . . . c N−1 denote coefficients, wherein T 0 (0), T 1 (0), . . . T N−1 (0) are Chebyshev polynomials, and N is a number of the first approximation points.
11. The system of claim 10 wherein a Chebyshev polynomial property T n (cos((2i+1)*/(2n)))=0, wherein n is a degree of the polynomial and i=0 . . . n−1 is applied to the approximation process.
12. The system of claim 10 wherein the first approximation values and the second approximation values are approximated without calculating first derivatives of the node values.
13. The system of claim 10 wherein the node values, the first approximation values, and the second approximation values are intensity values of a function f(x,y).
14. The system of claim 13 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each point represents at least one of a node, a first approximation point, and a second approximation point.
15. The system of claim 13 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each of the node values, first approximation values, and second approximation values are approximated according to the following expression:
φ*≅f((x 0 +x N−1 )/2), wherein φ* represents the approximated value of the function.
16. The system of claim 10 wherein the lattice comprises a plurality of squares having top sides, bottom sides, left sides, right sides, and corners, wherein a node is positioned at each corner, wherein the first approximation values are approximated between pairs of nodes at either the top and bottom sides and/or the left and right sides, and wherein the second approximation values are approximated between pairs of first approximation values at either the top and bottom sides and/or the left and right sides.
17. The system of claim 10 wherein the lattice is an equal-step lattice.
18. The system of claim 10 wherein after the 2-D approximation a resulting lattice has a border, wherein the border comprises one or more non-equal-step lattices.
19. A computer-readable medium containing program instructions for generating an aerial image, the program instructions which when executed by a computer system cause the computer system to execute a method comprising:
receiving a mask;
generating a matrix of node values based on the mask, wherein each node value corresponds to a node of a plurality of nodes in a lattice;
performing a one-dimensional (1-D) approximation of a plurality of first approximation values at corresponding first approximation points between pairs of nodes of the plurality of nodes;
performing a two-dimensional (2-d) approximation of second approximation values at corresponding second approximation points between pairs of first approximation points, wherein chebyshev polynomials are used to approximate the first approximation values and the second approximation values; and
wherein a Chebyshev polynomial is defined as T n (cos((2i+1)*/(2n)))=0, wherein n is a degree of the polynomial and i=0 . . . n−1 is applied to the approximation process.
20. The computer-readable medium of claim 19 wherein the first approximation values and the second approximation values are approximated without calculating first derivatives of the node values.
21. The computer-readable medium of claim 19 wherein the node values, the first approximation values, and the second approximation values are intensity values of a function f(x,y).
22. The computer-readable medium of claim 21 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each point represents at least one of a node, a first approximation point, and a second approximation point.
23. The computer-readable medium of claim 21 wherein each value is approximated according to the following expression:
φ*=c 0 *T 0 (0)+c 1 *T 1 (0)+ . . . +c N−1 *T N−1 (0), wherein φ* represents the approximated value of the function, wherein c 0 , c 1 , . . . c N−1 denote coefficients, wherein T 0 (0), T 1 (0), . . . T N−1 (0) are Chebyshev polynomials, and N is a number of the first approximation points.
24. The computer-readable medium of claim 21 wherein the function f(x,y) is defined as a quantity of points x 0 , x 1 , . . . xN−1, and wherein each of the node values, first approximation values, and second approximation values are approximated according to the following expression:
φ*≅f((x 0 +x N−1 )/2), wherein φ* represents the approximated value of the function.
25. The computer-readable medium of claim 19 wherein the lattice comprises a plurality of squares having top sides, bottom sides, left sides, right sides, and corners, wherein a node is positioned at each corner, wherein the first approximation values are approximated between pairs of nodes at either the top and bottom sides and/or the left and right sides, and wherein the second approximation values are approximated between pairs of first approximation values at either the top and bottom sides and/or the left and right sides.
26. The computer-readable medium of claim 19 wherein the lattice is an equal-step lattice.
27. The computer-readable medium of claim 19 wherein after the 2-D approximation a resulting lattice has a border, wherein the border comprises one or more non-equal-step lattices.