Method for generating spline animation of three-dimensional model surface flow based on hogde decomposition
By transforming the simulation of 3D model surface animation into vector field analysis through Helmholtz-Hodge decomposition, generating irrotation-free and divergence-free fields for time integration, the problems of low efficiency and insufficient accuracy in existing technologies are solved, and efficient and continuous generation of 3D model surface flow spline animation is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- WUHAN UNIV
- Filing Date
- 2023-03-21
- Publication Date
- 2026-07-07
AI Technical Summary
Existing technologies are inefficient and lack accuracy in generating animations on 3D model surfaces. In particular, physical simulation-based methods consume a lot of computational resources, while deep learning-based methods require a large amount of data preparation and have low accuracy.
The Helmholtz-Hodge decomposition method is used to transform the animation simulation of the 3D model surface into vector field analysis. By performing Helmholtz-Hodge decomposition on the input vector field, irrotation-free field, divergence-free field, and harmonic field are generated. The divergence-free field is then used as the initial vector field for time integration to generate continuous spline animation.
It achieves efficient generation of 3D model surface flow spline animation, preserves the original model features, reduces computational resource consumption, and generates continuous and interconnected animation effects, saving labor costs.
Smart Images
Figure CN116385602B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the fields of computer graphics and vector field processing, and in particular to a method for generating surface flow spline animations of three-dimensional models based on Hodge decomposition. Background Technology
[0002] In the field of computer graphics, animation simulation is an important research area and a significant application scenario. Common methods for generating animations on the surface of 3D models include physically based simulation and deep learning-based generation. Physically based simulation provides more realistic simulation effects but requires a larger amount of computation and resources, while deep learning-based generation requires extensive data preparation in advance and cannot provide high accuracy.
[0003] Vector fields are widely used in scientific computing scenarios, such as diffusion process simulation, electromagnetic field simulation, and fluid flow simulation. Vector field analysis on curved surfaces is also a topic of great interest in computer graphics. Helmholtz-Hodge decomposition is a common and effective method for vector field analysis. It decomposes a vector field into divergence-free (rotational) fields, irrotational (non-rotational) fields, and harmonic components. This decomposition allows us to simplify the analysis of complex vector fields by studying the divergence and rotation-related properties separately.
[0004] Transforming the animation simulation of the 3D model surface into vector field analysis and processing, and then processing different components to achieve the simulation effect will greatly improve efficiency. Summary of the Invention
[0005] The technical problem to be solved by the present invention is to overcome the defects of the prior art, introduce vector field analysis technology into the animation simulation of three-dimensional model surface, and provide a method for generating flow spline animation of three-dimensional model surface more efficiently.
[0006] The technical solution adopted in this invention is a method for generating surface flow spline animations of three-dimensional models based on Hodge decomposition, comprising the following steps:
[0007] Step 1: Generate a vector field from the input 3D mesh model;
[0008] Step 2: Perform Helmholtz-Hodge decomposition on the current vector field v to generate an irrotational field v. comp No end v rot and harmonizing field v harm ;
[0009] Step 3, use a non-scattering v rot As the initial vector field, calculate the corresponding curl;
[0010] Step 4: Use curl as velocity to perform time integration on the current vector field to obtain the vector field at the next moment.
[0011] Step 5: Based on the vector field obtained in Step 4, calculate the corresponding curl, then return to Step 4 to calculate the next moment, until the set moment is reached, and proceed to Step 6.
[0012] Step six: Generate spline animation by visualizing the trajectory, as follows:
[0013] The vector fields generated at each time step are all divergence-free, and the vector fields at each time step are continuous based on the time sequence. By visualizing the trajectory of the vector fields at each time step within a fixed time step, a continuous spline animation is obtained as time progresses.
[0014] Furthermore, in step one, based on the input 3D mesh model, an input vector field is generated and denoted as v: Ω represents the set of vectors in a vector field. Represents a field or set. Let represent a d-dimensional Euclidean vector space.
[0015] Furthermore, in step two, the irrotational field v generated by running Helmholtz–Hodge decomposition on the current vector field v comp The curl is zero, and there is no divergence. rot The divergence is zero, and the harmonic field v harm Both curl and divergence are zero; the scalar potential and vector potential are obtained by solving the Poisson problem, and then the compressive component v of the input vector field is obtained by differentiation. comp and rotational component v rot .
[0016] Furthermore, the method for calculating the curl of the corresponding vector field is as follows:
[0017] For any three-dimensional vector field The formula for calculating curl is:
[0018]
[0019] in, Let x, y, and z be the components of the vector field in three dimensions, respectively. This represents a partial differential operator.
[0020] Furthermore, the vector field v at time t+1 is calculated based on the forward Euler method. t+1 :
[0021] v t+1 =v t +Δt×curl t ;
[0022] Among them, curl t v represents the curl of the vector field at time t. t Let Δt be the vector field at time t, and Δt represent the time step.
[0023] The recursion then has
[0024] v t+1 =v t +Δt×curl t =v t-1 +Δt×curl t-1 +Δt×curl t =…
[0025] =v0+Δt×curl0+Δt×curl1+…Δt×curl t
[0026] but
[0027]
[0028] Among them, v0…v t-1 Let curl0…curl represent the vector field from time 0 to time t-1. t-1 The curl of the vector field from time 0 to t-1 is represented.
[0029] The obtained vector field v t+1 If it is also a divergence-free field, then the vector field at each time step is a divergence-free field.
[0030] Moreover, it is used to simulate bone growth in animal or human models.
[0031] Alternatively, it can be used to simulate the plant growth process.
[0032] Alternatively, it can be used to simulate fluid diffusion.
[0033] On the other hand, the present invention provides a three-dimensional model surface flow spline animation generation system based on Hodge decomposition, including a processor and a memory. The memory is used to store program instructions, and the processor is used to call the stored instructions in the memory to execute a three-dimensional model surface flow spline animation generation method based on Hodge decomposition as described above.
[0034] On the other hand, the present invention provides a three-dimensional model surface flow spline animation generation device based on Hodge decomposition, including a readable storage medium storing a computer program, wherein when the computer program is executed, it implements a three-dimensional model surface flow spline animation generation method based on Hodge decomposition as described above.
[0035] The beneficial effects of this invention are as follows: Based on Helmholtz-Hodge decomposition, the rotational component of the original vector field can be quickly generated as the initial divergence-free field. This component retains some features of the original vector field, i.e., the features of the original input 3D mesh model, while the vectors in the divergence-free field do not intersect. Based on the property of curl, the divergence of curl is zero. By performing forward Euler time integration based on the curl of the divergence-free field, it is ensured that the vector field at the next time step retains some features of the previous vector field while having the divergence-free property. Based on the time integration method, it is ensured that the vector fields are continuously related and do not intersect in each consecutive time step, ensuring the continuous flow pattern of the generated visual trajectory. It can automatically and efficiently simulate the animation effect of flowing lines, saving a lot of manpower and having significant market value. Attached Figure Description
[0036] Figure 1 This is a flowchart of a method for generating animations of flowing surface splines in a three-dimensional model based on Helmholtz–Hodge decomposition, as proposed in an embodiment of the present invention. Detailed Implementation
[0037] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.
[0038] Example 1
[0039] See Figure 1 To address the problem of generating 3D model surface spline animations as proposed in this invention, this embodiment provides a method for generating 3D model surface flow spline animations based on Hodge decomposition, comprising the following steps:
[0040] Step S1: Based on the input 3D mesh model, generate the input vector field and denote it as v: Ω represents the set of vectors in a vector field. Represents a field or set. Let represent a d-dimensional Euclidean vector space.
[0041] In the example, an input three-dimensional vector field v is generated:
[0042] Where Ω represents the set of vectors in the vector field, Represents a field or set. This represents a 3-dimensional Euclidean vector space.
[0043] Step S2: Perform Helmholtz-Hodge decomposition on the current vector field v to generate an irrotational field v. comp (The curl of this vector field is zero) No end v rot (The divergence of this vector field is zero) and harmonizing field v harm(The curl and divergence of this vector field are both zero), that is, v = v comp +v rot +v harm ;
[0044] in, Represents the gradient operator. Represents the curl operator, This represents the divergence operator.
[0045] Since the divergence of the divergence of the divergence-free field and the harmonic field is zero, it can be deduced that... Since the curl of irrotational and harmonic fields is zero, it can be deduced that... Based on the property of irrotational fields, i.e., divergence-free fields, the irrotational field components can be expressed as the gradient of a scalar potential, i.e. The divergence-free component can be derived from the vector potential, i.e. The following Poisson equation can then be derived:
[0046]
[0047]
[0048] Where Δ is the scalar Laplace operator. For vector Laplacian operator but Represents scalar potential gradient, Let A represent the gradient of the vector potential A. The scalar potential is obtained by solving the Poisson equation. The vector potential A can be differentiated to obtain the compression component v of the input vector field. comp and rotational component v rot .
[0049] Step S3: Use a divergence-free v rot As the initial vector field v0, calculate its curl.
[0050] For any three-dimensional vector field Its curl calculation formula is:
[0051]
[0052] in, Let x, y, and z be the components of the vector field in three dimensions, respectively. This represents a partial differential operator.
[0053] Step S4: Integrate the current vector field over time using curl as the velocity to obtain the vector field at the next moment.
[0054] The first execution of step S4 involves the initial vector field. Perform time integration. in, Let Δt represent the vector field from the initial time to the next time step; Δt represents the time step; from the initial vector field For no end v rot (From the rotation component) we can see that, Since the divergence of curl is zero, we know
[0055]
[0056] At this point, the vector field can be calculated. divergence:
[0057]
[0058] That is, vector field It also means there is no end to the event;
[0059] In subsequent iterations, step S4 involves integrating the vector field at the current time t based on the results of the previous iteration to obtain the vector field at the next time t+1.
[0060] Step S5: Based on the current time vector field calculated in step S4, calculate its curl, and then return to step S4 to calculate the next time step until the set time is reached before proceeding to step S6.
[0061] Based on the vector field at the current time t calculated in step S4 Calculate its curl
[0062] This invention proposes a time integration method based on the forward Euler method, which can be expressed as follows:
[0063] q k+1 =q k +τf(q k )
[0064] Where τ is the time step, and q k f(q) represents the current value. k ) represents the current velocity, q k+1 This will be the value at the next time.
[0065] Calculation of the vector field at time t+1 based on the forward Euler method
[0066]
[0067] Among them, curl t Let represent the curl of the vector field at time t.
[0068] Referring to the proof in step S4, we can see from the recursive formula that:
[0069]
[0070] Then we have:
[0071]
[0072] in, Let curl0…curl represent the vector field from time 0 to time t-1. t-1 This represents the curl of the vector field from time 0 to time t-1.
[0073] From the above proof, we can see that the vector field It is also divergence-free; that is, the vector field at each time step is divergence-free. Step S6: Generate spline animation by visualizing the trajectory. As proved by steps S4 and S5, the vector field generated at each time step is divergence-free, and the vector field at each time step is continuous based on the time sequence. By visualizing the trajectory of the vector field at each time step within a fixed time step, a continuous spline animation can be obtained as time progresses.
[0074] The visualization of the time-vector field trajectory is implemented as follows:
[0075] Specifically, the vector field position at the most recent n time moments is stored as the trajectory point. Based on the temporal relationship, the trajectory points at these n time moments are connected and drawn according to the direction of the vector field. As time progresses, the vector field position at the most recent n time moments is updated, and the trajectories before the most recent n time moments are removed, thereby realizing the visualization of the vector field trajectory.
[0076] Optionally, n can be set to any integer value. The smaller the value of n, the shorter the visualized trajectory; the larger the value of n, the longer the visualized trajectory.
[0077] Example 2
[0078] Creative animation generation based on continuous spline animation technology: After executing the process as in Example 1, this embodiment generates creative video scenes by continuously generating splines.
[0079] Example 3
[0080] Animal or human model spline growth animation drawing based on continuous spline animation technology: After executing the process as in Example 1, this embodiment creates an animation effect of animal or human model skeleton growth by using spline trajectories based on the continuous growth of the animal or human model outline.
[0081] Example 4
[0082] Plant growth animation based on continuous spline animation technology: After executing the process as in Example 1, this embodiment simulates the animation effect of plant growth by using random continuous splines based on a specific starting point.
[0083] Example 5
[0084] Simulation of diffusion effect based on continuous spline animation technology: After executing the process as in Example 1, this embodiment simulates the animation effect of fluid diffusion through continuous spline animation.
[0085] In specific implementation, the method proposed in the technical solution of this invention can be automatically executed by those skilled in the art using computer software technology. System devices for implementing the method, such as computer-readable storage media storing the corresponding computer program of the technical solution of this invention and computer equipment including the computer program running the corresponding computer program, should also be within the protection scope of this invention.
[0086] In some possible embodiments, a three-dimensional model surface flow spline animation generation system based on Hodge decomposition is provided, including a processor and a memory. The memory is used to store program instructions, and the processor is used to call the stored instructions in the memory to execute a three-dimensional model surface flow spline animation generation method based on Hodge decomposition as described above.
[0087] In some possible embodiments, a device for generating a three-dimensional model surface flow spline animation based on Hodge decomposition is provided, including a readable storage medium storing a computer program, which, when executed, implements a method for generating a three-dimensional model surface flow spline animation based on Hodge decomposition as described above.
[0088] Although the invention has been described in detail above, those skilled in the art will understand that various changes in form and detail may be made therein without departing from the scope defined by the claims.
Claims
1. A method for generating surface flow spline animations of a 3D model based on Hodge decomposition, characterized in that: Includes the following steps: Step 1: Generate a vector field from the input 3D mesh model; Step 2: Perform Helmholtz-Hodge decomposition on the current vector field v to generate an irrotational field v. comp No end v rot and harmonizing field v harm ; Step 3, use a non-scattering v rot As the initial vector field, calculate the corresponding curl; Step 4: Use curl as velocity to perform time integration on the current vector field to obtain the vector field at the next moment. Step 5: Based on the vector field obtained in Step 4, calculate the corresponding curl, then return to Step 4 to calculate the next moment, until the set moment is reached, and proceed to Step 6. Step six: Generate spline animation by visualizing the trajectory, as follows: The vector fields generated at each time step are all divergence-free, and the vector fields at each time step are continuous based on the time sequence. By visualizing the trajectory of the vector fields at each time step within a fixed time step, a continuous spline animation is obtained as time progresses.
2. The method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition according to claim 1, characterized in that: In step one, based on the input 3D mesh model, an input vector field is generated and denoted as v: Ω represents the set of vectors in a vector field. Represents a field or set. Let represent a d-dimensional Euclidean vector space.
3. The method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition according to claim 1, characterized in that: In step two, the irrotational field v generated by running Helmholtz-Hodge decomposition on the current vector field v is... comp The curl is zero, and there is no divergence. rot The divergence is zero, and the harmonic field v harm Both curl and divergence are zero; By solving the Poisson problem, we obtain the scalar potential and the vector potential. Then, by differentiation, we obtain the compression component v of the input vector field. comp and rotational component v rot .
4. The method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition according to claim 1, characterized in that: The method for calculating the curl of a vector field is as follows: For any three-dimensional vector field The formula for calculating curl is: in, These are the components of the vector field in three dimensions, where x, y, and z are the three dimensions of the vector. This represents a partial differential operator.
5. The method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition according to claim 1, characterized in that: The vector field v at time t+1 is calculated based on the forward Euler method. t+1 : v t+1 =v t +Δt×curl t ; Among them, curl t v represents the curl of the vector field at time t. t Let Δt be the vector field at time t, and Δt represent the time step. The recursion then has v t+1 =v t +Δt×curl t =v t-1 +Δt×curl t-1 +Δt×curl t =… =v0+Δt×curl0+Δt×curl1+…Δt×curl t but Among them, v0…v t-1 Let curl0…curl represent the vector field from time 0 to time t-1. t-1 The curl of the vector field from time 0 to t-1 is represented. The obtained vector field v t+1 If it is also a divergence-free field, then the vector field at each time step is a divergence-free field.
6. A method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition, as described in claim 1, 2, 3, 4, or 5, characterized in that: Used to simulate bone growth in animal or human models.
7. A method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition, as described in claim 1, 2, 3, 4, or 5, characterized in that: Used to simulate the plant growth process.
8. A method for generating surface flow spline animation of a three-dimensional model based on Hodge decomposition, as described in claim 1, 2, 3, 4, or 5, characterized in that: Used to simulate fluid diffusion.
9. A system for generating animations of flow splines on the surface of a 3D model based on Hodge decomposition, characterized in that: It includes a processor and a memory, the memory being used to store program instructions, and the processor being used to call the stored instructions in the memory to execute the method for generating a surface flow spline animation of a three-dimensional model based on Hodge decomposition as described in any one of claims 1-8.
10. A device for generating animation of flow splines on the surface of a three-dimensional model based on Hodge decomposition, characterized in that: The method includes a readable storage medium on which a computer program is stored, and when the computer program is executed, it implements the method for generating a surface flow spline animation of a three-dimensional model based on Hodge decomposition as described in any one of claims 1-8.