Objective

2

Shape editing has been extensively researched for its applications in the fields of computer graphics and computer animation. Shape editing methods based on control units are widely used because they are intuitive

effective

and easy to implement during operation on animation characters. Generally

for an input 3D animation shape

the editing method first establishes the relationship between the shape and control units. It then calculates the weights of the points on the shape affected by each control unit and completes shape editing by applying transformations on the control units and updating the points on the shape by a linear combination of the control units with the pre-computed weights. Usually

the editing result depends seriously on weight calculation. Although numerous methods of weight computation have been introduced

most of them cannot effectively or efficiently handle control skeletons. In addition

the linear combination of rotation is no longer a rotation. Thus

the transformation of 3D skeletons is very complicated

and the transformation of local frame at bone joint points is difficult to determine. In this study

our proposed algorithm is aimed at editing 2D images

from 2D domain to 3D domain. In our editing method

the control units are classified into real units (or user-defined units) and virtual units. The virtual control units are generated by our virtual unit insertion algorithm. A novel insertion algorithm of virtual control unit is developed for the control skeletons in 3D space. An improved transformation method of skeleton joint point frame is also proposed to maintain the volume of the shape around the joint points of the skeletons

resulting in naturally smoother editing results.

Method

2

To achieve a smooth editing result

we utilize the Bezier function with 5° as the basic function for weight computation because of its

$${C^2}$$

weighting properties. The virtual con

trol points based on the

$${C^2}$$

continuous weight calculation function are inserted into the input model based on the pre-input control units. The points of the shape are then controlled by the virtual and real control points

thereby realizing a smooth and efficient shape editing algorithm. The proposed method mainly includes several key steps: 1) calculating the internal distance for region decomposition

2) inserting virtual control units

and 3) calculating control weights. The input 3D shape is voxelized. Next

the internal distances of the points on the shape and the control units are computed in the closed 3D space. The 3D space is decomposed into different parts according to the internal distances to obtain initial control region of each control unit. In our method

the support domain of a control point is also defined to achieve intuitive editing results. No other control point should be in a determined support region. If the regions do not satisfy the constraint of control point support domain

the insertion of virtual control points is required. In this work

the control units are divided into two classes: control points and control skeletons. The virtual control points inserted according to the pre-input control points become the first class of virtual control points

whereas the ones inserted according to the control skeletons are called the second class of virtual control points. When editing a 3D shape

the virtual control points cannot be directly operated by the user. Therefore

the transformation rules of the virtual control points following the real control points are also defined. 1) For the first type of virtual control points

the transformation is general translation and rotation. 2) For the second class of virtual control points

the local frame must be constrained to have an axis along the skeleton direction during transformation. When all support domains of the control units fulfill the constraints

the transformation of the

real and virtual control points is then determined by solving a sparse linear system. Finally

the weights of all control points (including real and virtual control points) on the points of the 3D shape are computed using the

$${C^2}$$

weighting function. During the editing process

users can conduct transformations on the real control units

and the input shape will be updated by setting new coordinates of the points on the shape with the transformation propagations of the control units. In our pipeline

the distance computation

weight computation

and shape updating steps can be computed in parallel. Thus

we incorporate graphics processing unit (GPU) to conduct the computations in these steps

which accelerate the computation significantly.

Result

2

In this study

we compare our method with existing methods: the SR-ARAP(smooth rotation as-rigid-as-possible) method

and Enhanced-ARAP(smooth rotation enhanced as-rigid-as-possible) method. Our experiments mainly compare these algorithms according to four aspects: editing details

adaptability of the algorithm to the non-triangular mesh models

multiple closed-domain overlay models

and computation time of the editing process. Experimental results show that the proposed algorithm never over-deforms and maintains smooth local details of the model

especially the shape around the control skeleton region. For the non-triangular mesh model and multiple closed-domain overlay models

the proposed algorithm is still applicable. However

the other two algorithms are not adaptive. Importantly

our algorithm requires no iteration

and due to GPU parallel computation

the editing efficiency is significantly improved.

Conclusion

2

Our proposed method is easy to understand and implement. The editing effect is natural

intuitive

and smooth

and the model details can be preserved quite well. Thus

the proposed method satisfies the general requirements of 3D shape editing. At the same time

the algorithm efficiency is greatly improved given the GPU parallel computing

which delivers feedback to the user operation faster. It basically achieves the effect of real-time interaction and thus improves user experience.