Injective 3D deformations based on 2D mesh deformations

1 . A method comprising:

defining at least one two-dimensional (2D) mesh deformation based on a designated position of an object represented by an input neural radiance field;

applying the at least one 2D mesh deformation to a three-dimensional (3D) piecewise-linear map that operates over a plane and preserves a normal direction to produce a plurality of prismatic maps defining plurality of layers;

composing a 3D deformation for the object from the plurality of layers using the plurality of prismatic maps;

parameterizing the 3D piecewise-linear map using the 3D deformation and parameters of the plurality of the prismatic maps; and

storing, using the 3D piecewise-linear map, a deformed neural radiance field injectively representing the object in the designated position.

2 . The method of claim 1 , further comprising rendering the object, as deformed, on a presentation device.

3 . The method of claim 2 , further comprising:

determining a desired plane of deformation; and

rotating a coordinate system to apply the 3D deformation in the desired plane.

4 . The method of claim 1 , further comprising:

parameterizing a 2D injective mesh deformation space using a Tutte embedding to provide deformed vertices; and

using the deformed vertices to define the at least one 2D mesh deformation.

5 . The method of claim 1 , further comprising regularizing the plurality of layers to reduce Jacobian distortion of the 3D piecewise-linear map.

6 . The method of claim 1 , further comprising parameterizing a boundary of the at least one 2D mesh deformation using positive angle increments so that the boundary is defined by a convex polygon.

7 . The method of claim 1 , further comprising:

selectively triangulating each unit square of a disk-topology mesh using same-size isosceles triangles to produce at least one 2D triangular mesh; and

using the at least one 2D triangular mesh to define the at least one 2D mesh deformation.

8 . A system comprising:

a memory component including a designated position of an object and an input neural radiance field representing the object; and

a processing device coupled to the memory component to perform operations comprising:

defining at least one two-dimensional (2D) mesh deformation based on the designated position and the input neural radiance field;

applying the at least one 2D mesh deformation to a three-dimensional (3D) piecewise-linear map that operates over a plane and preserves a normal direction to produce a plurality of prismatic maps defining a plurality of layers;

composing a 3D deformation for the object from the plurality of layers using the plurality of prismatic maps;

parameterizing the 3D piecewise-linear map using the 3D deformation and parameters of the plurality of the prismatic maps; and

storing, using the 3D piecewise-linear map, a deformed neural radiance field injectively representing the object in the designated position.

9 . The system of claim 8 , wherein the processing device additionally performs an operation of rendering the object, as deformed, on a presentation device.

10 . The system of claim 9 , wherein the processing device additionally performs operations comprising:

determining a desired plane of deformation; and

rotating a coordinate system to apply the 3D deformation in the desired plane.

11 . The system of claim 8 , wherein the processing device additionally performs operations comprising:

parameterizing a 2D injective mesh deformation space using a Tutte embedding to provide deformed vertices; and

using the deformed vertices to define the at least one 2D mesh deformation.

12 . The system of claim 8 , wherein the processing device additionally performs an operation of regularizing the plurality of layers to reduce Jacobian distortion of the 3D piecewise-linear map.

13 . The system of claim 8 , wherein the processing device additionally performs an operation of parameterizing a boundary of the at least one 2D mesh deformation using positive angle increments so that the boundary is defined by a convex polygon.

14 . The system of claim 8 , wherein the processing device additionally performs operations comprising:

selectively triangulating each unit square of a disk-topology mesh using same-size isosceles triangles to produce at least one 2D triangular mesh; and

using the at least one 2D triangular mesh to define the at least one 2D mesh deformation.

15 . A non-transitory computer-readable medium storing executable instructions, which when executed by a processing device, cause the processing device to perform operations comprising:

a step for producing a plurality of prismatic maps from an input neural radiance field representing an object;

a step for providing a three-dimensional (3D) piecewise-linear map based on the plurality of prismatic maps;

producing, using the 3D piecewise-linear map, a deformed neural radiance field representing the object in a designated position; and

storing, using the deformed neural radiance field, an injective representation of the object.

16 . The non-transitory computer-readable medium of claim 15 , wherein the instructions cause the processing device to define at least one two-dimensional (2D) mesh deformation based on a designated position of an object represented by the input neural radiance field.

17 . The non-transitory computer-readable medium of claim 15 , wherein the instructions cause the processing device to render the injective representation on a presentation device.

18 . The non-transitory computer-readable medium of claim 17 , wherein the instructions cause the processing device to perform operations comprising:

determining a desired plane of deformation; and

rotating a coordinate system to the desired plane of deformation.

19 . The non-transitory computer-readable medium of claim 15 , wherein the instructions cause the processing device to perform operations comprising:

parameterizing a 2D injective mesh deformation space using a Tutte embedding to provide deformed vertices; and

using the deformed vertices to define a 2D mesh deformation.

20 . The non-transitory computer-readable medium of claim 19 , wherein the instructions cause the processing device to perform operations comprising:

selectively triangulating each unit square of a disk-topology mesh using same-size isosceles triangles to produce at least one 2D triangular mesh; and

using the at least one 2D triangular mesh to define the 2D mesh deformation.