Method of converting continuous three-dimensional geometrical representations into discrete three-dimensional voxel-based representations within a three-dimensional voxel-based system
First Claim
1. A method of converting a continuous 3-D straight line segment into a discrete set of voxels connected together in discrete 3-D voxel space, said 3-D straight line segment being defined by two endpoints P1 and P2 each having integer coordinates and specifying extents of x, y and z coordinate directions of said 3-D straight line segment, said discrete 3-D voxel space being characterized by orthogonal x, y, and z coordinate directions, the addresses of said discrete 3-D voxel space being specified by integer x, y and z coordinate values of said voxels, said method comprising the sequence of steps of:
- (a) computing the value of an integer n to determine the number of sample points sampled along said continuous 3-D straight line segment, said sample points and said integer n represents the number of said voxels in said discrete set;
(b) defininginteger voxel-coordinate error variables for said x, y, and z coordinate directions, andfirst and second error variable increments along each said x, y and z coordinate directions;
(c) specifying the initial values of said integer error variables;
(d) placing into said discrete 3-D voxel space, said voxel having x, y and z coordinate values of said first end point P1 of said sampled 3-D straight line segment; and
(e) converting voxels corresponding to said sample points of said discrete set of voxels, in said discrete 3-D voxel space.
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Abstract
Method of converting continuous 3-D geometrical representations into a discrete set of voxels in discrete 3-D voxel space. In one embodiment, a method is provided for converting a continuous 3-D straight line segment into a discrete set of voxels connected together in discrete 3-D voxel space. In another embodiment, a method is provided for converting a continuous 3-D parametric polynomial curve segment into a discrete set of voxels connected together in discrete 3-D voxel space. In an alternative embodiment, a method is provided for converting a continuous 3-D parametric polynomial surface patch into a discrete set of voxels connected together in discrete 3-D voxel space. Yet in another embodiment of the present invention, a method is provided for converting a continuous 3-D parametric polynomial volume element into a discrete set of voxels connected together in discrete 3-D voxel space. The method of the present invention is incremental in nature and uses all integer arithmetic. The method of the present invention is also characterized by symmetrical decisional process loops using substantially the same decisional process logic in each of the x, y and z coordinate directions, and thus is capable of generating discrete sets of voxels having a variety of connectivities.
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Citations
15 Claims
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1. A method of converting a continuous 3-D straight line segment into a discrete set of voxels connected together in discrete 3-D voxel space, said 3-D straight line segment being defined by two endpoints P1 and P2 each having integer coordinates and specifying extents of x, y and z coordinate directions of said 3-D straight line segment, said discrete 3-D voxel space being characterized by orthogonal x, y, and z coordinate directions, the addresses of said discrete 3-D voxel space being specified by integer x, y and z coordinate values of said voxels, said method comprising the sequence of steps of:
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(a) computing the value of an integer n to determine the number of sample points sampled along said continuous 3-D straight line segment, said sample points and said integer n represents the number of said voxels in said discrete set; (b) defining integer voxel-coordinate error variables for said x, y, and z coordinate directions, and first and second error variable increments along each said x, y and z coordinate directions; (c) specifying the initial values of said integer error variables; (d) placing into said discrete 3-D voxel space, said voxel having x, y and z coordinate values of said first end point P1 of said sampled 3-D straight line segment; and (e) converting voxels corresponding to said sample points of said discrete set of voxels, in said discrete 3-D voxel space. - View Dependent Claims (2, 3, 4, 5, 6)
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7. A method of converting a continuous 3-D parametric polynomial curve segment into a discrete set of voxels connected together in discrete,3-D voxel space, said 3-D parametric curve segment having first and second endpoints and being defined by a kth order polynomial vector T, a geometric basis matrix M, a geometric control point vector G, parameter t, and an integer step size along said parameter t, said discrete 3-D voxel space being characterized by orthogonal x, y, z coordinate directions, the addresses of said discrete 3-D voxel space being specified by integer x, y and z coordinate values of said voxels, said method comprising the sequence of steps of:
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(a) computing the value of integer n corresponding to the number of sample to be sampled along said parameter t of said continuous 3-D parametric polynomial curve segment, said integer n being determined so that consecutive voxels of said discrete set of voxels are connected; (b) defining a Kth order integer finite forward difference matrix for said 3-D parametric polynomial curve segment, for which said parameter t takes on integer values from o to n; (c) determining an initial Kth order finite forward difference vector for said 3-D parametric polynomial curve segment, on the basis of said Kth order finite forward difference matrix, said geometric basis matrix M, and said geometric control point vector G; (d) defining integer decision variables for said x, y and z coordinate directions, first and second decision thresholds based on n, and a decision variable increment based on n; (e) specifying an initial value for each said integer decision variable of step (d); (f) placing into said discrete 3-D voxel space, said voxel having x, y, and z coordinate values corresponding to the first endpoint of said 3-D parametric polynomial curve segment; and (g) converting said continuous 3-D parametric polynomial curve segment into said discrete set of voxels, said conversion being based on said integer decision variables, said first and second thresholds, said decision variable increment, and said first endpoint. - View Dependent Claims (8)
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9. A method of converting a continuous 3-D parametric polynomial surface patch into a discrete set of voxels connected together in discrete 3-D voxel space, said 3-D parametric polynomial surface patch being defined by bi-Kth order polynomial vectors T and U, a geometric basis M, a geometric control point matrix G, and parameters t and u each with an integer step size, said 3-D parametric polynomial surface patch being formed by a plurality of 3-D parametric polynomial curve segments each having first and second endpoints, said discrete 3-D voxel space being characterized by orthogonal x, y, and z coordinate directions, the addresses of said discrete 3-D voxel space being specified by integer x, y, and z coordinate values of said voxels, said method comprising the sequence of steps of:
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(a) computing the values of integers n and m corresponding to the number of sample points to be sampled along said parameters t and u, respectively, of said continuous 3-D parametric polynomial surface patch, said integers n and m being determined so that a resulting set of voxels lack tunnels; (b) defining first and second bi-Kth order integer finite forward difference matrices for said 3-D parametric polynomial surface patch, said first bi-Kth order finite forward difference matrix corresponding to said parameter t which takes on integer values from o to n, and said second bi-Kth order finite forward difference matrix corresponding to said parameter u which takes on integer values from o to m; (c) determining an initial bi-Kth order finite forward difference matrix for said 3-D parametric polynomial surface patch, on the basis of said first and second bi-Kth order finite forward difference matrices, said geometric basis matrix M, and said geometric control point matrix G; (d) defining surface integer decision variables for said x, y and z coordinate directions, and first and second decision thresholds based on n and m, and a decision variable increment based on n and m; (e) specifying an initial value for each said surface integer decision variable of step (d); and (f) converting said continuous 3-D parametric polynomial surface patch into said discrete set of voxels, said conversion being based on said integer decision variables, said first and second thresholds, said decision variable increment, and said first endpoint. - View Dependent Claims (10, 11)
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12. A method of converting a continuous 3-D parametric polynomial volume element into a discrete set of voxels connected together in discrete 3-D voxel space, said 3-D parametrical polynomial volume element being defined by cubic order polynomial vectors T, U and V, a geometric basis M, a geometrical control point tensor G, and parameters t, u, and v each with an integer step size, said 3-D parametric polynomial volume element being formed by a plurality of 3-D parametric polynomial surface patches each of which is formed by a plurality of 3-D parametric polynomial curve segments each having first and second endpoints, said discrete 3-D voxel space being characterized by orthogonal x, y and z coordinate directions, the addresses of said discrete 3-D voxel space being specified by integer x, y and z coordinate values of said voxels said method comprising the sequence of steps of:
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(a) computing the values of integers n, m and 1 corresponding to the number of sample points to be sampled along said parameters t, u and v, respectively, of said 3-D parametric polynomial volume element, said integers n, m and 1 being determined so that a resulting set of voxels lack cavities; (b) defining first, second and third integer finite forward difference matrices for said 3-D parametric polynomial voxel element, said first finite forward difference matrix corresponding to said parameter t which takes on integer values from o to n, said second finite forward difference matrix corresponding to said parameter u which take on integer values from o to m, and said third finite forward difference matrix corresponding to said parameter v which takes on integer values from o to 1; (c) determining an initial order finite forward difference tensor for said 3-D parametric polynomial volume element, on the basis of said first, second and third order finite forward difference matrices, said geometric basis matrix M, and said geometric control point tensor G; (d) defining volume integer decision variables for said x, y and z coordinate directions, first and second decision thresholds based on n, m and 1, and a decision variable increment based on n, m, and 1; (e) specifying an initial value for each said integer decision variable of step (d); and (f) converting said continuous 3-D parametric polynomial volume element into said discrete set of voxels, said conversion being based on said integer decision variables, said first and second decision thresholds, said decision variable increment, and said first endpoint. - View Dependent Claims (13, 14, 15)
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Specification