A method for open surface reconstruction of a three-dimensional skull point cloud
By designing and optimizing the energy functional minimization problem based on variational level set technology and Euler's Elastica regularization, the problem of coarse open surface reconstruction results of 3D skull point cloud in the prior art is solved, and accurate and robust open surface reconstruction is achieved, which is suitable for cultural heritage protection and human medical research.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- QINGDAO UNIV
- Filing Date
- 2022-10-09
- Publication Date
- 2026-06-26
AI Technical Summary
Existing level set methods suffer from coarse and inaccurate reconstruction results when reconstructing open surfaces of 3D skull point clouds with complex topological relationships in the medical field. This is especially true when dealing with topology-insensitive implicit representations, where it is difficult to obtain accurate open surface reconstruction results.
A method based on variational level set technology and Euler's Elastica regularization is adopted. By designing an energy functional minimization problem, an open surface is fitted using two level set functions ψ and φ. The augmented Lagrangian method is then used for optimization until the energy functional converges. The surface with ψ = 0 and φ ≥ 0 is selected to represent the open surface corresponding to the target skull point cloud.
It achieves accurate and robust reconstruction of open surfaces from discrete point cloud data, with smooth and clear reconstruction results, suitable for cultural heritage protection and human medical research.
Smart Images
Figure CN115457239B_ABST
Abstract
Description
Technical fields:
[0001] This invention belongs to the field of computer open surface reconstruction, specifically relating to an open surface reconstruction method for three-dimensional skull point clouds based on variational level set technology. It can digitize and protect cultural heritage, and can also be widely used in medical research and human body structure processing technology. Background technology:
[0002] In existing technologies, surface reconstruction typically falls under the domains of computer graphics and computer vision, with broad application prospects in fields such as digital preservation of cultural heritage and medicine. Recently, implicit representation has proven to be a promising method for surface representation. This involves defining a level set function on a rectangular mesh using a known dataset, and representing the implicit surface with the zero level set of this function. Implicit representation is topology-insensitive and can handle arbitrary and dynamically changing surface topologies well. Therefore, it is a suitable choice for reconstructing skull surfaces from topologically complex 3D skull point clouds in the medical field. The level set method, a method of solving implicit partial differential equations using the Euler method, was first proposed by Osher et al. and subsequently introduced into computer vision and image processing. However, the level set method is used to represent closed surfaces that begin and end at the boundaries of the computational domain. Reconstructing accurate, smooth, and bounded open surfaces from skull point cloud data remains very challenging.
[0003] A review of existing technical literature revealed relatively little research on open surface reconstruction based on the level set method. Matthias Desamer et al. proposed a method for automatically trimming the boundary region of 3D open surface mesh models (patent number: CN113168731A), but this method uses an open surface mesh to trim the open surface boundary. Early open surface reconstruction was accomplished by tracing the curve representing the open surface boundary on an implicitly defined surface and using convex hull partitioning to perform piecewise linear approximation of the initial surface boundary. While this method is suitable for fitting open surfaces to data points, the results obtained for data with complex topologies are rather coarse. In recent years, with the introduction of constraints and optimization algorithms such as normal, curvature, and signed distance functions, surface reconstruction based on the level set method has become faster and more accurate. Research using neural implicit functions to learn 3D shapes has also made remarkable progress, overcoming the previous obstacle of low resolution caused by the diversity of topologies. Therefore, most existing techniques for reconstructing open surfaces are limited to closed surfaces. Obtaining accurate and robust surface reconstruction results from discrete points remains an open problem. Seeking a completely new technique for reconstructing open surfaces has significant theoretical and practical value. Summary of the Invention:
[0004] The purpose of this invention is to overcome and solve the above-mentioned problems in the prior art, and to design a new method for open surface reconstruction of three-dimensional skull point clouds based on variational level set technology and Euler's Elastica regularization terms, so as to effectively use it for the protection of cultural heritage and human medical research.
[0005] To achieve the above objectives, this invention designs an energy functional for minimizing two level set functions ψ and φ to fit an open surface, and selects the surface when ψ = 0 and φ ≥ 0 to represent the open surface corresponding to the target skull point cloud. The energy functional minimization process is optimized using the augmented Lagrangian method until the energy functional converges. The final optimization result is selected when ψ and φ reach the minimum value of the energy functional.
[0006] The method for reconstructing the open surface of a three-dimensional skull point cloud involved in this invention includes the following specific steps:
[0007] Step (1): First, obtain the initialized level set functions ψ and φ from the three-dimensional skull point cloud data S to be reconstructed;
[0008] Step (2): Using two level set functions ψ and φ, design an energy functional that includes an open surface area term, an open surface boundary length term, and an Euler's Elastica regularization term to obtain an open surface reconstruction model;
[0009] Step (3): Iteratively optimize the energy functional using the augmented Lagrange method until convergence is achieved, and obtain the optimized ψ and φ;
[0010] Step (4): Select the surface representing the open surface corresponding to the point cloud when ψ = 0 and φ ≥ 0. The specific formula is as follows:
[0011] Γ:ψ(x)=0∩H(φ(x))=1
[0012] In the formula, H(·) is the Heaviside function.
[0013] In step (1) of this invention, the Fast Sweeping algorithm is used to obtain the initialized level set functions ψ and φ, wherein the level set functions are initialized as the corresponding symbolic distance functions.
[0014] The specific process method for obtaining the open surface reconstruction model in step (2) of this invention is as follows:
[0015] Step (2.1): Fit the surface Γ to the target skull point cloud data S, that is, use two level set functions ψ and φ to design the open surface area term, as shown in the following formula:
[0016]
[0017] In the formula, d(x) is the distance function from a point on Γ to S;
[0018] Step (2.2): Determine the open surface boundary, i.e., use two level set functions ψ and φ to design the open surface boundary length term, as shown in the following formula:
[0019]
[0020] In the formula, It is a projection matrix;
[0021] Step (2.3): Introduce Euler's Elastica regularization term as a constraint, as shown in the following formula:
[0022]
[0023] Step (2.4): Design an energy functional minimization process to obtain the open surface reconstruction model, the specific formula of which is as follows:
[0024]
[0025] In the formula, δ(·) is the derivative of the Heaviside function.
[0026] The specific method for solving the energy functional minimization process using the augmented Lagrange method in step (3) of this invention is as follows:
[0027] Step (3.1): Introduce auxiliary variables to transform the energy functional minimization problem into a constrained minimization problem, as shown in the following formula:
[0028]
[0029]
[0030] Step (3.2): Introducing Lagrange multipliers and penalty parameters, the minimization process is transformed into an augmented Lagrange functional, the specific formula of which is as follows:
[0031]
[0032] st|w1|=1,|w2|=1,|n|≤1
[0033] Step (3.3): Decompose the augmented Lagrange functional into several subproblems, iteratively optimize them to achieve convergence, and obtain the optimized ψ and φ.
[0034] The specific process method for decomposing the augmented Lagrange functional into several sub-problems and solving them in step (3.3) of this invention is as follows:
[0035] Step (3.3.1): Obtain the corresponding gradient descent equation from the sub-optimization problem of ψ, and discretize it to solve it through finite difference;
[0036] Step (3.3.2): Obtain the Euler-Lagrange equation from the sub-optimization problem of w1, and solve it by discretization; the constraint |w1|=1 corresponding to the sub-problem can be calculated using the projection formula as follows:
[0037]
[0038] Step (3.3.3): Obtain the corresponding gradient descent equation from the sub-optimization problem of φ, and discretize it to solve it through finite difference;
[0039] Step (3.3.4): The Euler-Lagrange equation is obtained from the sub-optimization problem of w2, and then discretized and solved; the constraint |w2|=1 corresponding to the sub-problem can be calculated using the projection formula as follows:
[0040]
[0041] Step (3.3.5): The Euler-Lagrange equation is obtained from the sub-optimization problem of n, and discretized and solved using FFT; the constraint |n|≤1 corresponding to the sub-problem can be calculated using the projection formula as follows:
[0042]
[0043] Step (3.3.6): From the sub-optimization problem of q, the solution in analytical form is obtained directly through analytical formulas;
[0044] Step (3.3.7): The Lagrange multipliers can be updated according to the corresponding auxiliary variables, as shown in the following formula:
[0045]
[0046] Step (3.3.8): Repeat the above steps until the energy functional converges.
[0047] The method for obtaining the open surface corresponding to the point cloud of the target skull in step (4) of this invention is as follows:
[0048] Step (4.1): Extract the points V and surfaces F on the zero level set of the level set function ψ;
[0049] Step (4.2): Determine whether point V is inside or outside the zero level set of the level set function φ;
[0050] Step (4.3): Retain the triangular face F with all three vertices on the outside, which is represented as the surface when ψ=0 and φ≥0, which is the open surface corresponding to the target skull point cloud.
[0051] Compared with existing technologies, this invention has the following advantages: By designing an energy functional minimization problem to fit an open surface on three-dimensional skull point cloud data S using two level set functions ψ and φ, and selecting the surface where ψ = 0 and φ ≥ 0 to represent the open surface corresponding to the target skull point cloud; further experimental results on three-dimensional skull point cloud data and face point cloud data show that the open surface reconstructed using this method is accurate and robust, not only able to reconstruct open surfaces from discrete point cloud data, but also able to obtain smooth and accurate reconstruction results; its designed process method is scientifically sound and reasonable, the processing and calculation operations are simple and flexible, the open surface reconstruction is clear, its usability is strong, its development prospects are broad, and its application environment and effects are friendly. Attached image description:
[0052] Figure 1 This is a schematic flowchart of the reconstruction method involved in the present invention.
[0053] Figure 2 This is a schematic diagram of the cranial surface with ψ=0 in the reconstruction method described in this invention.
[0054] Figure 3 This is a schematic diagram of the open surface corresponding to the point cloud of the target skull determined by the reconstruction method of the present invention. In the diagram: points inside are points where ψ = 0 and φ < 0, and points outside are points where ψ = 0 and φ ≥ 0.
[0055] Figure 4 This is a schematic diagram of the open surface result reconstructed from the skull point cloud in the reconstruction method described in this invention. Detailed implementation method:
[0056] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0057] This embodiment relates to a specific method for reconstructing open surfaces from three-dimensional cranial point clouds, as shown in the flowchart below. Figure 1 As shown, the specific process includes the following steps:
[0058] Step S100: Obtain the initialized level set functions ψ and φ;
[0059] From the three-dimensional skull point cloud data S to be reconstructed, the Fast Sweeping algorithm is used to obtain the initialized level set functions ψ and φ, where the level set functions are initialized to the corresponding signed distance functions;
[0060] Step S200: Design an energy functional minimization problem to obtain an open surface reconstruction model. The energy functional includes an open surface area term that fits the surface Γ to the target skull point cloud data S, an open surface boundary length term that determines the open surface boundary, and an Euler's Elastica regularization term as a constraint, i.e.:
[0061]
[0062] In the formula, d(x) is the distance function from a point on Γ to S. H is the projection matrix, H(·) is the Heaviside function, δ(·) is the derivative of the Heaviside function, and a, b > 0, γ and η are parameters.
[0063] Step S300: Calculate the energy functional minimization problem using the augmented Lagrange method;
[0064] First, by introducing auxiliary variables w1, w2, n, and q, the energy functional minimization problem is transformed into a constrained minimization problem, as shown in the following formula:
[0065]
[0066]
[0067] Then, by introducing Lagrange multipliers λ1, λ2, λ3, λ4 and penalty parameters μ1, μ2, μ3, μ4, the constrained minimization problem is transformed into an augmented Lagrange functional, as shown in the following formula:
[0068]
[0069] st|w1|=1,|w2|=1,|n|≤1
[0070] Considering that |n|≤1, a more relaxed constraint |w1|-n·w1=0 and |n|≤1 can be used to constrain n. Additionally, when updating the level set function, attention needs to be paid to the reinitialization of the level set function; therefore, the constraints |w1|=1 and |w2|=1 are introduced.
[0071] Finally, the augmented Lagrange functional is decomposed into several subproblems, and iterative optimization is performed to achieve convergence, yielding the optimized ψ and φ. The cranial surface with ψ = 0 is shown below. Figure 2 As shown, the augmented Lagrange functional is transformed into the following six subproblems:
[0072] Subproblem 1:
[0073]
[0074] Subproblem 2:
[0075]
[0076] Sub-problem 3:
[0077]
[0078] Sub-problem 4:
[0079]
[0080] Sub-problem 5:
[0081]
[0082] Sub-problem 6:
[0083]
[0084] Step S400: Select the open surface corresponding to the point cloud when ψ = 0 and φ ≥ 0, such as... Figure 3 As shown;
[0085] The 3D skull point cloud data S is optimized through an energy functional minimization problem to obtain the level set functions ψ and φ that minimize the energy functional, where the level set functions are represented by the signed distance function. A surface is selected where ψ = 0 and φ ≥ 0 to represent the open surface corresponding to the skull point cloud, and the calculation formula is as follows:
[0086] Γ:ψ(x)=0∩H(φ(x))=1
[0087] Clearly, ψ(x) = 0 represents the closed surface reconstructed from the three-dimensional skull point cloud S, and H(φ(x)) = 1 represents the point outside the isosurface corresponding to φ(x) = 0.
[0088] In step S200 of this embodiment, the method for designing an energy functional minimization problem to obtain an open surface reconstruction model includes the following steps:
[0089] S210: Fit the surface Γ to the target skull point cloud data S, that is, use two level set functions ψ and φ to design the open surface area term;
[0090] The term attempts to find the zero isosurface Γ that minimizes the above energy functional, similar to an elastic membrane approximating a given point cloud;
[0091] S220: Determine the boundary of the open surface, that is, use two level set functions ψ and φ to design the length term of the open surface boundary to find the outermost isodesic line;
[0092] S230: Introduce Euler's Elastica regularization term as a constraint;
[0093] S240: Combining the above steps, design the energy functional minimization problem to obtain the open surface reconstruction model.
[0094] In step S300 of this embodiment, the method for calculating the energy functional minimization problem includes the following steps:
[0095] S310: Introduce auxiliary variables to transform the energy functional minimization problem into a constraint minimization problem;
[0096] S320: Introducing Lagrange multipliers and penalty parameters transforms the constrained minimization problem into an augmented Lagrange functional;
[0097] S330: Decompose the augmented Lagrange functional into several subproblems, iteratively optimize them to achieve convergence, and obtain the optimized ψ and φ.
[0098] In step S330 of this embodiment, the method of decomposing the augmented Lagrange functional into several sub-problems and solving them includes the following steps:
[0099] S331: The following gradient descent equation is obtained from the sub-optimization problem of ψ, and then discretized and solved by finite difference:
[0100]
[0101] S332: The following Euler-Lagrange equation is obtained from the sub-optimization problem of w1, and then discretized and solved:
[0102]
[0103] The constraint |w1|=1 corresponding to the subproblem can be calculated using the projection formula as follows:
[0104]
[0105] S333: The following gradient descent equation is obtained from the sub-optimization problem of φ, and then discretized and solved by finite difference:
[0106]
[0107] S334: The following Euler-Lagrange equation is obtained from the sub-optimization problem of w2, and then discretized and solved:
[0108]
[0109] The constraint |w2|=1 corresponding to the subproblem can be calculated using the projection formula as follows:
[0110]
[0111] S335: The following Euler-Lagrange equation is obtained from the sub-optimization problem of n, and then discretized and solved using FFT:
[0112]
[0113] The constraint |n|≤1 corresponding to the subproblem can be calculated using the projection formula as follows:
[0114]
[0115] S336: The solution to the sub-optimization problem of q can be obtained directly in analytical form using the following analytical formula:
[0116]
[0117] S337: The Lagrange multipliers can be updated based on the corresponding auxiliary variables, as shown in the following formula:
[0118]
[0119] S338: Repeat the above steps until convergence.
[0120] In step S400 of this embodiment, the method for selecting the open surface corresponding to the point cloud when ψ=0 and φ≥0 includes the following steps:
[0121] S410: Extract the points V and surfaces F on the zero level set of the level set function ψ;
[0122] S420: Determine whether point V is inside or outside the zero level set of the level set function φ;
[0123] S430: Retain the triangular face F with all three vertices on the outside, represented as the surface where ψ = 0 and φ ≥ 0, which is the open surface corresponding to the target skull point cloud, such as... Figure 4 As shown.
[0124] This embodiment can be applied to the processing of fitting open surfaces to unorganized point clouds, and can be used to represent 3D shapes with arbitrary topological structures.
Claims
1. A method for reconstructing the open surface of a three-dimensional skull point cloud, characterized in that, Includes the following steps: Step 1, from the 3D skull point cloud data to be reconstructed To obtain the initialized level set function and ; Step 2, using two level set functions and We designed an energy functional that includes the open surface area term, the open surface boundary length term, and the Euler's Elastica regularization term to obtain the open surface reconstruction model. Step 3: The energy functional is iteratively optimized using the augmented Lagrangian method until convergence is achieved, yielding the optimized functional. and ; Step 4, select and The surface representation of the point cloud corresponds to the open surface, and the formula is as follows: In the formula, For the Heaviside function; The method for obtaining the open surface reconstruction model in step 2 is as follows: Step 2.1, the curved surface Fitting to target skull point cloud data That is, using two level set functions and The formula for the area term of the open surface is as follows: Step 2.2, determine the boundary of the open surface, i.e., use two level set functions. and The formula for the boundary length term of the open surface is as follows: Step 2.3 introduces Euler's Elastica regularization term as a constraint, as shown in the following formula: Step 2.4: Design an energy functional minimization problem to obtain the open surface reconstruction model, as shown in the following formula: In the formula, It is the derivative of the Dirac function, i.e., the Heaviside function; , , and It is a positive parameter; for The point on The distance function, For gradient operators; It is a projection matrix.
2. The method for reconstructing the open surface of a three-dimensional skull point cloud according to claim 1, characterized in that, Step 1 uses the Fast Sweeping algorithm to obtain the initialized level set function. and The level set function is initialized to the corresponding symbolic distance function.
3. The method for reconstructing the open surface of a three-dimensional skull point cloud according to claim 1, characterized in that, The iterative optimization method using the augmented Lagrange method described in step 3 is as follows: Step 3.1: Introduce auxiliary variables to transform the energy functional minimization problem into a constrained minimization problem, as shown in the following formula: Step 3.2: Introducing Lagrange multipliers and penalty parameters, the constrained minimization problem is transformed into an augmented Lagrange functional, as shown in the following formula: Step 3.3: Decompose the augmented Lagrange functional into several subproblems, iteratively optimize them until convergence, and obtain the optimized result. and .
4. The method for reconstructing the open surface of a three-dimensional skull point cloud according to claim 3, characterized in that, Step 3.3, which involves decomposing the augmented Lagrange functional into several subproblems and solving them, is as follows: Step 3.3.1, by The sub-optimization problem yields the corresponding gradient descent equation, which is then discretized and solved using finite difference calculus. Step 3.3.2, by The sub-optimization problem yields the Euler-Lagrange equation, which is then discretized and solved; the constraints corresponding to the sub-problem... The following can be calculated using the projection formula: Step 3.3.3, by The sub-optimization problem yields the corresponding gradient descent equation, which is then discretized and solved using finite difference calculus. Step 3.3.4, by The sub-optimization problem yields the Euler-Lagrange equation, which is then discretized and solved; the constraints corresponding to the sub-problem... The following can be calculated using the projection formula: Step 3.3.5, by The sub-optimization problem yields the Euler-Lagrange equation, which is discretized and solved using Fast Fourier Transform (FFT); the constraints corresponding to the sub-problem... The following can be calculated using the projection formula: Step 3.3.6, by The sub-optimization problem can be solved directly in analytical form using analytical formulas; Step 3.3.7, the Lagrange multipliers can be updated according to the corresponding auxiliary variables, as shown in the following formula: Step 3.3.8: Repeat the above steps until convergence.
5. The method for reconstructing the open surface of a three-dimensional skull point cloud according to claim 1, characterized in that, The method for obtaining the open surface corresponding to the point cloud of the target skull in step 4 is as follows: Step 4.1, extract the level set function. Points on the zero level set Kneading noodles ; Step 4.2, Determine the point In the level set function Inside / outside the zero level set; Step 4.3: Retain the triangle face where all three vertices are on the outside. , represented as and The surface at that time is the open surface corresponding to the point cloud of the target skull.