Method and system for generating polygon meshes approximating surfaces using iteration for mesh vertex positions
Generating polygon meshes that approximate surfaces using iteration for mesh vertex positions. In some implementations, a method includes receiving input data that represents a surface distinguishing a volume, where a voxel grid includes the surface. Particular voxels of the voxel grid are identified, which the surface intersects. A surface-approximating mesh is generated including polygons defined by vertices in the particular voxels. Generating the mesh includes determining approximate positions of a subset of the vertices in a subset of the particular voxels, based on interpolation of locations in the voxel subset where the surface intersects the voxel subset. Errors between approximate voxel values (based on the approximate positions) and assigned voxel values of the particular voxels (based on the input data) are determined, and the approximate position of at least one vertex of the subset of the vertices is adjusted using a successive over-relaxation technique to reduce the errors.
1. A computer-implemented method comprising:
receiving, by one or more processors, input data that represents a surface that distinguishes an inside and an outside of an object volume;
determining, by the one or more processors, a voxel grid that includes the surface, wherein the voxel grid includes a plurality of voxels;
identifying, by the one or more processors, particular voxels of the plurality of voxels which the surface intersects; and
generating, by the one or more processors, a mesh that approximates the surface, the mesh including a plurality of polygons that are defined by vertices of the mesh in the particular voxels, wherein generating the mesh includes:
determining approximate positions of a subset of the vertices of the polygons of the mesh in a subset of the particular voxels;
determining an approximate volume value for each voxel of the subset of the particular voxels, the approximate volume value representing an approximate volume of the mesh occupying the voxel based on the approximate positions of the subset of the vertices;
determining errors between the approximate volume values and corresponding voxel values of the particular voxels, wherein corresponding voxel values are determined from the input data; and
adjusting the approximate position of at least one vertex of the subset of the vertices using an iterative technique to reduce the errors.
2. The computer-implemented method of claim 1 , wherein adjusting the approximate position of the at least one vertex includes iterating, using the one or more processors, the determining the errors and the adjusting until the errors meet predetermined criteria.
3. The computer-implemented method of claim 2 , wherein the errors meet predetermined criteria in response to a predetermined number of iterations being performed or in response to the errors meeting one or more predetermined thresholds.
4. The computer-implemented method of claim 1 , wherein the approximate positions of the subset of the vertices of the polygons are determined based on interpolation of locations on edges of the subset of the particular voxels where the surface intersects the edges.
5. The computer-implemented method of claim 1 , further comprising, mapping, by the one or more processors, each voxel of the plurality of voxels to a respective intersection case from a set of stored intersection cases based on the input data, wherein each mapped intersection case indicates an intersection path of the surface through the voxel, wherein identifying the particular voxels includes identifying edges or other line segments of the particular voxels which the surface intersects based on the mapped intersection cases.
6. The computer-implemented method of claim 1 , wherein determining the approximate positions of the subset of the vertices of the polygons of the mesh includes determining initial approximate positions of the vertices using a marching cubes technique or a surface nets technique.
7. The computer-implemented method of claim 1 , wherein adjusting the approximate position of the at least one vertex of the subset of the vertices includes determining a plurality of approximate positions of the subset of vertices of the polygons, and wherein the subset of the particular voxels includes a plurality of adjacent voxels of the mesh.
8. The computer-implemented method of claim 1 , wherein the subset of the particular voxels includes a plurality of adjacent voxels of the voxel grid, and further comprising determining approximate positions of a second subset of vertices of the polygons of the mesh in a second subset of adjacent voxels of the particular voxels.
9. The computer-implemented method of claim 1 , wherein determining the approximate volume value representing the approximate volume of the mesh occupying each voxel of the subset of the particular voxels includes determining a 3-dimensional integral of the mesh within the voxel.
10. The computer-implemented method of claim 1 , further comprising generating a feature surface based on the mesh, the feature surface displayable by a display device.
11. The computer-implemented method of claim 1 , wherein the input data includes scalar values that indicate a density or an occupancy of voxels by the object volume.
12. A system comprising:
at least one processor; and
a memory coupled to the at least one processor, with instructions stored thereon that, when executed by the at least one processor, cause the at least one processor to perform operations comprising:
receiving input data that represents a surface that distinguishes an inside and an outside of an object volume;
determining a voxel grid that includes the surface, wherein the voxel grid includes a plurality of voxels;
identifying particular voxels of the plurality of voxels which the surface intersects; and
generating a mesh that approximates the surface, the mesh including a plurality of polygons that are defined by vertices of the mesh in the particular voxels, wherein generating the mesh includes:
determining approximate positions of a subset of the vertices of the polygons of the mesh in a subset of the particular voxels;
determining an approximate volume value for each voxel of the subset of the particular voxels, the approximate volume value representing an approximate volume of the mesh occupying the voxel based on the approximate positions of the subset of the vertices;
determining errors between the approximate volume values and corresponding voxel values of the particular voxels, wherein the corresponding voxel values are determined from the input data; and
adjusting the approximate position of at least one vertex of the subset of the vertices using an iterative technique to reduce the errors.
13. The system of claim 12 , wherein the operation of adjusting the approximate position of the at least one vertex includes iterating the determining the errors and the adjusting until the errors meet predetermined criteria.
14. The system of claim 12 , wherein the approximate positions of the subset of the vertices of the polygons are determined based on interpolation of locations on edges of the subset of the particular voxels where the surface intersects the edges.
15. The system of claim 12 , wherein the subset of the particular voxels includes a plurality of adjacent voxels of the voxel grid, and the at least one processor performs further operations comprising determining approximate positions of a second subset of vertices of the polygons of the mesh in a second subset of adjacent voxels of the particular voxels.
16. The system of claim 12 , wherein the operation of determining the approximate volume value representing the approximate volume of the mesh occupying each voxel of the subset of the particular voxels includes determining a 3-dimensional integral of the mesh within the voxel.
17. The system of claim 12 , wherein the at least one processor performs further operations comprising generating a feature surface based on the mesh, the feature surface displayable by a display device.
18. The system of claim 12 , wherein the input data includes scalar values that indicate a density or an occupancy of voxels by the object volume.
19. A non-transitory computer readable medium has stored thereon software instructions that, when executed by a processor of a device, cause the processor to perform operations comprising:
receiving input data that represents a surface that distinguishes an inside and an outside of an object volume;
determining a voxel grid that includes the surface, wherein the voxel grid includes a plurality of voxels;
identifying particular voxels of the plurality of voxels which the surface intersects; and
generating a mesh that approximates the surface, the mesh including a plurality of polygons that are defined by vertices of the mesh in the particular voxels, wherein generating the mesh includes:
determining approximate positions of a subset of the vertices of the polygons of the mesh in a subset of the particular voxels, based on interpolation of locations in the subset of the particular voxels where the surface intersects the subset of the particular voxels;
determining an approximate volume value for each voxel of the subset of the particular voxels, the approximate volume value representing an approximate volume of the mesh occupying the voxel based on the approximate positions of the subset of the vertices;
determining errors between the approximate volume values and corresponding voxel values of the particular voxels, wherein the corresponding voxel values are determined from the input data; and
adjusting the approximate position of at least one vertex of the subset of the vertices using a successive over-relaxation technique to reduce the errors.
20. The non-transitory computer readable medium of claim 19 , wherein the input data includes scalar values that indicate a density or an occupancy of voxels by the object volume, and wherein the operation of determining the approximate volume value representing the approximate volume of the mesh occupying each voxel of the subset of the particular voxels includes determining a 3-dimensional integral of the mesh within the voxel.