A Three-Dimensional Mandible Reconstruction Method Based on Principal Geodesy Analysis

Abstract

This invention relates to a three-dimensional mandible reconstruction method based on principal geodesic analysis, comprising: Step 1, constructing a shape space by removing Lie group effects of translation, scaling, and rotation from the original three-dimensional skull and projecting it into the shape space; Step 2, a training phase by projecting training skull samples into the shape space, followed by principal geodesic analysis at the intrinsic mean on all training samples, and finally obtaining the relationship between the shape of the maxilla and the shape of the mandible from the training samples for use in the reconstruction phase; Step 3, a reconstruction phase by projecting the input maxilla model into the shape space, constructing the mandibular shape space based on the prior knowledge obtained in the training phase, and finally restoring it to Euclidean space. The advantage of this invention is that the three-dimensional mandible reconstruction method based on principal geodesic analysis described herein can effectively reconstruct a mandible that closely approximates the true value.

Description

Technical Field

[0001] This invention relates to the field of three-dimensional mandibular reconstruction, and more specifically to a three-dimensional mandibular reconstruction method based on principal geodesic analysis. Background Technology

[0002] With the advent of CT and DICOM technologies, a multitude of skull reconstruction techniques have emerged, gradually replacing traditional manual reconstruction techniques with computer-aided design and manufacturing technologies, such as mirror-based reconstruction methods, surface interpolation methods, and data-driven methods. Since the human skull is roughly symmetrical, mirror-based methods offer significant convenience in the virtual planning process of skull reconstruction. However, this method requires the defect to be strictly limited to one side of the skull, making it unsuitable for bilateral defects or defects crossing the facial midline. Surface interpolation methods are useful for defects crossing the midline, ensuring a smooth approximation of the skull shape to the defect area. However, for large reconstruction areas, this method lacks constraints, resulting in overly flat surfaces. Another type of data-driven method is a general tool for handling large or bilateral defects. This approach utilizes deformation models to extract corresponding templates from a skull database to guide the reconstruction of regions of interest. For example, Marreirors et al. proposed a semi-automated geometric morphology reconstruction method that uses thin-plate splines to generate templates from a database to guide the reconstruction of large curved surfaces; Fuessinger et al. designed a statistical shape model based on the intact skull, using captured shape variations as prior knowledge to estimate missing portions. Data-driven methods overcome the prerequisites of other skull reconstruction methods, and are not constrained by the symmetry of the skull or the closed boundaries of defects / holes. They can combine templates and coefficients to reconstruct skull shapes with individual characteristics.

[0003] Most existing data-driven methods directly calculate skull templates in Euclidean space, ignoring the inherent characteristics of the shape itself. The Cartesian coordinate representation in Euclidean space does not fully reflect the information about the skull shape itself, as it also includes other information unrelated to the skull shape. In contrast, the shape space proposed by Kendall et al. is an intrinsic representation that focuses solely on the shape itself, allowing for a more precise depiction of the details and local variations of the skull shape.

[0004] In existing published patent literature, for example, Chinese patent application publication number CN111563953A discloses a method, device, terminal, and medium for jawbone defect reconstruction based on machine learning. This includes: collecting jawbone CT data from multiple sampled individuals within a preset population range; establishing multiple jawbone feature points characteristic of the maxilla and mandible surface for each sampled individual based on the jawbone CT data; obtaining the correlation between these jawbone feature points through a machine learning algorithm, and then reconstructing the jawbone defect based on this correlation. This invention utilizes machine learning to restore jawbone feature points, providing precise and personalized solutions in complex jawbone reconstruction processes, solving the clinical problem of relying solely on experience and lacking reference points for reconstruction of large-scale jawbone defects across the midline. Furthermore, based on jawbone CT data within a preset population range, this invention provides targeted jawbone reconstruction strategies for specific groups, ensuring bone segment blood supply and preserving implant sites while more closely approximating the facial features of that specific group.

[0005] For example, Chinese Patent Publication No. CN112785691A discloses a mandibular bone reconstruction method, device, electronic device, and storage medium. The mandibular bone defect reconstruction method acquires CT data of the sampled individual and performs three-dimensional reconstruction. It selects relevant mandibular feature points according to a preset scheme and saves their coordinates to establish a normal mandibular bone database. A virtual osteotomy is performed to obtain a mandibular bone defect model. The mandibular bone defect model is standardized based on the relevant mandibular feature points, and the most similar mandibular bone is retrieved and matched in the normal mandibular bone database using a preset mandibular bone database retrieval and matching algorithm. The most similar mandibular bone obtained from the retrieval and matching is then registered with the defect mandibular bone model. This provides a highly repeatable reference, in addition to physician experience, for the digital design of mandibular bone defects across the midline, secondary mandibular defects, and bilateral facial asymmetry, effectively addressing the shortcomings of mirror technology in clinical application.

[0006] For example, Chinese patent application publication number CN113397702A discloses a mandibular prosthesis and its manufacturing method. The mandible includes a mandibular body with a notch. The mandibular prosthesis includes a prosthesis body and a connector. The prosthesis body is used to fill the notch in the mandibular body. The prosthesis body includes an outer mesh body and an inner mesh body. The inner mesh body is fitted inside the outer mesh body and connected to the inner wall of the outer mesh body. The prosthesis body is connected to the mandibular body through the connector. The manufacturing method of the mandibular prosthesis includes S1, modeling the mandibular body to obtain the notch structure data of the mandibular body; S2, setting the body model according to the notch structure data, the body model including an outer mesh body model and an inner mesh body model, fitting the inner mesh body model inside the outer mesh body model and connecting it to the outer mesh body model; S3, connecting the body model to the mandibular body through the connector model.

[0007] None of the aforementioned patent applications address the need for more precise characterization of skull shape details and local variations within shape space, nor do they address a three-dimensional mandibular reconstruction method employing principal geodesic analysis (PGA). Therefore, this invention combines a data-driven approach with the shape space proposed by Kendall et al., proposing a three-dimensional mandibular reconstruction method based on Principal Geodesic Analysis (PGA). Summary of the Invention

[0008] Based on the aforementioned technical problems existing in the prior art, this invention proposes a three-dimensional mandibular reconstruction method based on principal geodesic analysis, comprising the following steps:

[0009] Step 1: Construct the shape space. Based on the shape definition proposed by Kendall, remove the Lie group effects of translation, scaling and rotation from the original 3D skull and project it into the shape space.

[0010] Step 2, training phase: After projecting the skull training samples into shape space, the intrinsic mean in shape space is found using the method proposed by Fréchet. Then, principal geodesic analysis is performed on all training samples at the intrinsic mean. Finally, the relationship between the shape of the maxilla and the shape of the mandible is obtained from the training samples as prior knowledge for the reconstruction phase.

[0011] Step 3, the reconstruction stage, projects the input upper skull model into the shape space, constructs the mandibular shape space based on the prior knowledge obtained in the training stage, and finally restores it to the Euclidean space.

[0012] Furthermore, in step 1, a shape space is constructed. Based on the shape definition proposed by Kendall, the original 3D skull is removed from the Lie group effects of translation, scaling, and rotation, and projected into the shape space, including the following steps:

[0013] Step 1.1, Representation of the 3D Skull Shape: A centralized 3D skull point cloud consisting of m points in 3D space is denoted as... x here h (h = 1, 2, ..., m) are the three-dimensional coordinates of the h-th point in the three-dimensional skull point cloud. The point cloud data is transformed into a column vector, i.e., an element in the configuration space, using vectorization operators:

[0014]

[0015] Step 1.2: Project onto shape space, use the Helmert matrix or centering matrix in Kendall shape space theory to remove the translation effect on the configuration elements, and then use the centroid size to remove the scaling effect to obtain the pre-shape x. Then, perform Protodyakonov analysis on the pre-shape to remove the effect of the rotation transformation group and obtain the element [x] in shape space. The element [x] in shape space is the set of elements of the pre-shape x under the rotation.

[0016] Furthermore, in step 2, after projecting the skull training samples onto the shape space, the intrinsic mean in the shape space is found using the method proposed by Fréchet. Then, principal geodesic analysis is performed on all training samples at the intrinsic mean. Finally, the relationship between the shape of the maxilla and the shape of the mandible is obtained from the training samples as prior knowledge for the reconstruction stage, including the following steps:

[0017] Step 2.1, Logarithmic Mapping of Shape Space: By analyzing the shape in shape space, and considering that shape space is differentially homeomorphic to Euclidean space in the neighborhood of a point on it, the local shape space is mapped to Euclidean space. The analysis process is completed in Euclidean space, and the results are mapped back to shape space. Since the distribution of 3D skull shapes in shape space is very concentrated, they can be regarded as being in the neighborhood of a point. Projecting the 3D skull shapes onto the tangent space of that point allows for convenient analysis of the essential features of the shapes. Let p and q be two points on the pre-shaped sphere, and [p] and [q] be the shape representations of these two pre-shaped shapes, respectively. Using Protodyakonov's analysis method, the representative elements p and q of [p] and [q] in the pre-shaped space are calculated. ro , here q ro Let q represent the element after rotation and alignment from p. Then the logarithmic mapping in shape space is expressed as follows (2):

[0018]

[0019] In equation (2) above, Γ * It is the optimal rotation matrix aligned from q to p calculated by Protodyakonov analysis. At this time, the logarithmic mapping is represented by equal intervals, and the shortest geodesic distance between the corresponding shapes of p and q can be expressed as the following equation (3):

[0020] d([p],[q])=||Log p (q ro )||……(3);

[0021] Step 2.2, calculate the intrinsic mean of the three-dimensional skull model. First, construct the pre-shape space of the skull data. Then, Protodyakonov analysis was used to eliminate the rotation effect, removing all pre-shaped x... i To the first pre-shape x 1Alignment yields representative elements of the skull shape in the pre-shape space, denoted as x. 1,ro ,x 2,ro ,…,x N,ro The shape space composed of all skull shapes is represented as Based on Fréchet's intrinsic mean algorithm and the geodesic distance representation in shape space in formula (3), the definition of intrinsic mean in the three-dimensional skull shape space is given as follows (4):

[0022]

[0023] In equation (4) above, μ * Let x represent the point in shape space that minimizes the sum of squared geodesic distances to all shapes. i,ro As the initial value of the intrinsic mean, the gradient descent method is used to solve equation (4) above. The gradient of the three-dimensional skull mean point is given by equation (5) below:

[0024]

[0025] The intrinsic mean of the three-dimensional skull shape obtained from each iteration update is expressed as follows (6):

[0026]

[0027] Step 2.3 involves performing principal geodesic analysis on the 3D skull model. The principal geodesic in shape space is a generalization of the principal directions in linear space, preserving the most significant changes in shape within shape space. Utilizing the property that the local differential of shape space is homeomorphic to Euclidean space, the 3D skull shape in shape space is projected onto the tangent space of the intrinsic mean point. The main directions of change of shape in the tangent space can then be calculated. These directions are then projected back onto shape space to obtain the principal geodesic of shape space. The manifold formed by the principal geodesic is a geodesic submanifold of the 3D skull shape space. Intrinsic mean μ * The variance of the three-dimensional skull shape is calculated as follows (7):

[0028]

[0029] The covariance matrix of the three-dimensional skull sample shape is calculated according to the above equation (7), as shown in equations (8) and (9):

[0030]

[0031] in:

[0032] The eigenvectors in the tangent space at the intrinsic mean are further obtained from the covariance matrix. and eigenvalues ​​λ1,…,λ3m The number of principal components k of the geodesic submanifold is determined by the cumulative variance contribution rate through eigenvectors and eigenvalues, in order to construct the projection matrix. The original shapes are projected onto the geodesic submanifolds using a projection matrix, resulting in their dimension-reduced representations in the low-dimensional tangent space (10):

[0033]

[0034] The above describes the process of shape analysis of a three-dimensional skull model using the principal geodesic analysis method. The principal geodesic analysis method used is universal for three-dimensional point cloud models.

[0035] Step 2.4: Relation training based on principal geodesic analysis. Principal geodesic analysis is used to analyze the training samples of the maxilla and mandible respectively, obtaining the principal components Φ of the tangent space of the maxilla shape model at the intrinsic mean. X and its dimensionality reduction representation D X And the principal component Φ of the tangent space at the mean of the mandibular shape model within its shape space. Y and its dimensionality reduction representation D Y Given that the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention reconstructs the complete mandible from the maxilla model, it is necessary to find the transformation relationship between the maxilla and the mandible. The transformation relationship between the two is expressed as the following formula (11):

[0036] D Y =D X U * ……(11),

[0037] In equation (11) above, U * The optimal transformation matrix obtained by the least squares method is given. It should be noted that during dimensionality reduction, the number of principal directions of the maxilla and mandible must be kept consistent to establish a one-to-one linear relationship, thereby obtaining the relationship matrix U. * This gives us all the prior knowledge needed to reconstruct the mandible, the relation matrix U. * As shown in equation (12),

[0038]

[0039] Furthermore, in step 3, the input maxillary bone model x′ is projected into shape space, the mandibular shape space is constructed based on the prior knowledge obtained during the training phase, and finally it is restored to Euclidean space, including the following steps:

[0040] Step 3.1: For an input upper skull shape model x′, first remove translation and scaling, project it into a pre-shape space, then use Protodyakonov analysis to remove rotation effects and align it with the training sample shape; utilizing prior knowledge from the training phase, project x′ onto the upper skull mean point μ. X * In the tangent space at the intrinsic mean, and by performing principal geodesic analysis, we obtain the dimensionality-reduced representation D of the input skull in the tangent space at the intrinsic mean. [x′] ;

[0041] Step 3.2 Use the supercranial model x′ obtained during the training phase to tangent the principal component Φ of the shape space at the intrinsic mean of the shape space. X Project it into this tangent space, and then use the spatial transformation matrix U. * Transform it to the tangent space at the intrinsic mean of the mandibular shape space, and then use... Projecting it into the shape space of the mandible, we obtain the dimensionality-reduced representation of the predicted mandible in the subspace, as shown in Equation (13):

[0042] D [y′] =D [x′] U * ……(13),

[0043] Using the exponential mapping, which is the inverse of the logarithmic mapping, the predicted mandible is projected onto the shape space as shown in equation (14):

[0044]

[0045] By restoring the position, size, and rotation of the mandible in shape space and converting it back to Euclidean space, we obtain a three-dimensional mandible y′ that is adapted to the input maxilla and has personalized features.

[0046] Step 3.3: In order to restore the shape of the mandible to the Euclidean space that matches the input maxilla, it is necessary to preserve the absolute position, size, and rotation matrix used for alignment in shape space of the input maxilla.

[0047] Compared with existing technologies, the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention has the following superior technical effects:

[0048] 1. Using the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention, a mandibular bone that closely approximates the true value can be effectively reconstructed;

[0049] 2. Using the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention, the predicted mandible has a very small error compared with the true value, and is close to the true value in terms of shape details. Attached Figure Description

[0050] Figure 1 This is a flowchart of the three-dimensional mandibular reconstruction method based on principal geodesic analysis as described in this invention. Detailed Implementation

[0051] The following describes in detail, with reference to the accompanying drawings, the specific implementation of the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention.

[0052] like Figure 1 As shown, the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention includes the following steps:

[0053] Step 1: Construct the shape space. Based on Kendall's shape definition, remove the Lie group effects of translation, scaling, and rotation from the original 3D skull and project it into the shape. This includes the following steps:

[0054] Step 1.1, Representation of the 3D Skull Shape: A centralized 3D skull point cloud consisting of m points in 3D space is denoted as... x here h (h = 1, 2, ..., m) are the three-dimensional coordinates of the h-th point in the three-dimensional skull point cloud. The point cloud data is transformed into a column vector, i.e., an element in the configuration space, using vectorization operators:

[0055]

[0056] Step 1.2: Project onto shape space, use the Helmert matrix or centering matrix in Kendall shape space theory to remove the translation effect for the configuration elements, and then use the centroid size to remove the scaling effect to obtain the pre-shape x. Then, perform Protodyakonov analysis on the pre-shape to remove the effect of the rotation transformation group and obtain the element [x] in shape space. The element [x] in shape space is the set of elements of the pre-shape x under the rotation.

[0057] Step 2, the training phase, involves projecting the skull training samples onto shape space and then using Fréchet's method to find the intrinsic mean in shape space. Principal geodesy analysis is then performed on all training samples at the intrinsic mean. Finally, the relationship between the shape of the maxilla and the shape of the mandible is obtained from the training samples and used as prior knowledge in the reconstruction phase. This includes the following steps:

[0058] Step 2.1, Logarithmic mapping of shape space: By analyzing the shape in shape space, and given that shape space is differentially homeomorphic to Euclidean space in the neighborhood of a point on it, the local shape space is mapped to Euclidean space. The analysis process is completed in Euclidean space, and the results are mapped back to shape space. Since the distribution of the three-dimensional skull shape in shape space is very concentrated, they can be regarded as being in the neighborhood of a point. By projecting the three-dimensional skull shape onto the tangent space of that point, the essential features of the shape can be easily analyzed. Let p and q be two points on the pre-shaped sphere, and [p] and [q] be the shape representations of these two pre-shaped shapes, respectively. Using the Protodyakonov analysis method, the representative elements p and q of [p] and [q] in the pre-shaped space are calculated. ro , here q ro Let q represent the element after rotation and alignment from p. Then the logarithmic mapping in shape space can be expressed as equation (2):

[0059]

[0060] In equation (2) above, Γ * It is the optimal rotation matrix aligned from q to p calculated by Protodyakonov analysis. At this time, the logarithmic mapping is represented by equal intervals, and the shortest geodesic distance between the corresponding shapes of p and q can be expressed as the following equation (3):

[0061] d([p],[q])=||Log p (q ro )||……(3),

[0062] Step 2.2, calculate the intrinsic mean of the three-dimensional skull model. First, construct the pre-shape space of the skull data. Then, Protodyakonov analysis was used to eliminate the rotation effect, removing all pre-shaped x... i To the first pre-shape x 1 Alignment yields representative elements of the skull shape in the pre-shape space, denoted as x. 1,ro ,x 2,ro ,…,x N,ro The shape space composed of all skull shapes is represented as Based on Fréchet's intrinsic mean algorithm and the geodesic distance representation of shape space in formula (3), the definition of intrinsic mean in the three-dimensional skull shape space can be given as follows (4):

[0063]

[0064] In equation (4) above, μ * Let x represent the point in shape space that minimizes the sum of squared geodesic distances to all shapes. i,roAs the initial value of the intrinsic mean, the gradient descent method is used to solve equation (4) above. The gradient of the three-dimensional skull mean point is given by equation (5) below:

[0065]

[0066] The intrinsic mean of the three-dimensional skull shape obtained from each iteration update is expressed as follows (6):

[0067]

[0068] Step 2.3 involves performing principal geodesic analysis on the 3D skull model. The principal geodesic in shape space is a generalization of the principal directions in linear space, preserving the most significant changes in shape within shape space. Utilizing the property that the local differential of shape space is homeomorphic to Euclidean space, the 3D skull shape in shape space is projected onto the tangent space of the intrinsic mean point. The main directions of change of shape in the tangent space can then be calculated. These directions are then projected back onto shape space to obtain the principal geodesic of shape space. The manifold formed by the principal geodesic is a geodesic submanifold of the 3D skull shape space. Intrinsic mean μ * The variance of the three-dimensional skull shape is calculated as follows (7):

[0069]

[0070] The covariance matrix of the three-dimensional skull sample shape is calculated according to the above equation (7), as shown in equations (8) and (9):

[0071]

[0072] in:

[0073] The eigenvectors in the tangent space at the intrinsic mean are further obtained from the covariance matrix. and eigenvalues ​​λ1,…,λ 3m The number of principal components k of the geodesic submanifold is determined by the cumulative variance contribution rate through eigenvectors and eigenvalues, in order to construct the projection matrix. The original shapes are projected onto the geodesic submanifolds using a projection matrix, resulting in their dimension-reduced representations in the low-dimensional tangent space (10):

[0074]

[0075] The above describes the process of shape analysis of a three-dimensional skull model using the principal geodesic analysis method. The principal geodesic analysis method used is universal for three-dimensional point cloud models.

[0076] Step 2.4: Relation training based on principal geodesic analysis. Principal geodesic analysis is used to analyze the training samples of the maxilla and mandible respectively, obtaining the principal components Φ of the tangent space of the maxilla shape model at the intrinsic mean. X and its dimensionality reduction representation D X And the principal component Φ of the tangent space at the mean of the mandibular shape model within its shape space. Y and its dimensionality reduction representation D Y Given that the three-dimensional mandibular reconstruction method based on principal geodesic analysis described in this invention reconstructs the complete mandible from the maxilla model, it is necessary to find the transformation relationship between the maxilla and the mandible. The transformation relationship between the two is expressed as the following formula (11):

[0077] D Y =D X U * ……(11),

[0078] In equation (11) above, U * The optimal transformation matrix obtained by the least squares method is given. It should be noted that, in the dimensionality reduction representation, the number of principal directions of the maxilla and mandible must be kept consistent to establish a one-to-one linear relationship, thereby obtaining the relationship matrix U. * This gives us all the prior knowledge needed to reconstruct the mandible, the relation matrix U. * As shown in equation (12):

[0079]

[0080] Step 3, the reconstruction stage, involves projecting the input maxillary model into shape space, constructing the mandibular shape space based on prior knowledge obtained during the training stage, and finally restoring it to Euclidean space. This includes the following steps:

[0081] Step 3.1: For an input upper skull shape model x′, first remove translation and scaling, project it into a pre-shape space, then use Protodyakonov analysis to remove rotation effects and align it with the training sample shape; utilizing prior knowledge from the training phase, project x′ onto the upper skull mean point μ. X * In the tangent space at the intrinsic mean, and by performing principal geodesic analysis, we obtain the dimensionality-reduced representation D of the input skull in the tangent space at the intrinsic mean. [x′] ;

[0082] Step 3.2 Use the supercranial model x′ obtained during the training phase to tangent the principal component Φ of the shape space at the intrinsic mean of the shape space. X Project it into this tangent space, and then use the spatial transformation matrix U. * Transform it to the tangent space at the intrinsic mean of the mandibular shape space, and then use... Projecting it into the shape space of the mandible, we obtain the dimensionality-reduced representation of the predicted mandible in the subspace, as shown in Equation (13):

[0083] D [y′] =D [x′] U * ……(13),

[0084] Using the exponential mapping, which is the inverse of the logarithmic mapping, the predicted mandible is projected into the shape space, as shown in equation (14):

[0085]

[0086] By restoring the position, size, and rotation of the mandible in shape space and converting it back to Euclidean space, we obtain a three-dimensional mandible y′ that is adapted to the input maxilla and has personalized features.

[0087] Step 3.3: In order to restore the shape of the mandible to the Euclidean space that matches the input maxilla, it is necessary to preserve the absolute position, size, and rotation matrix used for alignment in shape space of the input maxilla.

[0088] This invention is not limited to the above embodiments. The embodiments and descriptions in the specification are only illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from the concept and scope of the invention, and all such changes and modifications will fall within the protection scope specified in the claims of this invention.

Claims

1. A three-dimensional mandibular reconstruction method based on principal geodesic analysis, characterized in that, Includes the following steps: Step 1. Construct the shape space. Based on the shape definition proposed by Kendall, remove the Lie group effects of translation, scaling and rotation from the original 3D skull and project it into the shape space. Step 2. Training phase: After projecting the skull training samples into shape space, the intrinsic mean in shape space is found using the method proposed by Fréchet. Then, principal geodesic analysis is performed on all training samples at the intrinsic mean. Finally, the relationship between the shape of the maxilla and the shape of the mandible is obtained from the training samples as prior knowledge for the reconstruction phase. Step 3. Reconstruction stage: Project the input maxillary model into shape space, construct the mandibular shape space based on prior knowledge obtained during training, and finally restore it to Euclidean space. This includes the following steps: Step 3.1 For an input upper skull shape model First, translation and scaling are removed, and the image is projected into a pre-shape space. Then, Protodyakonov analysis is used to remove rotation effects and align the image with the training sample shape. Utilizing prior knowledge from the training phase, Projected onto the mean point of the upper skull In the tangent space at the intrinsic mean, and by performing principal geodesic analysis, a dimension-reduced representation of the input skull in the tangent space at the intrinsic mean is obtained. ; Step 3.2 Use the upper skull model obtained during the training phase The principal component of the tangent space at the intrinsic mean in the shape space. Project it into this tangent space, and then use the space transformation matrix. Transform it to the tangent space at the intrinsic mean of the mandibular shape space, and then use... Projecting it into the shape space of the mandible, we obtain the dimensionality-reduced representation of the predicted mandible in the subspace, as shown in Equation (13): ……(13)， Using the exponential mapping, which is the inverse of the logarithmic mapping, the predicted mandible is projected onto the shape space, as shown in equation (14): ……(14)， By restoring the position, size, and rotation of the mandible in shape space and converting it back to Euclidean space, we obtain a three-dimensional mandible with personalized features that matches the input maxilla. ; Step 3.3 In order to restore the shape of the mandible to the Euclidean space that matches the input maxilla, it is necessary to preserve the absolute position, size, and rotation matrix used for alignment in shape space of the input maxilla.

2. The three-dimensional mandible reconstruction method based on principal geodesic analysis according to claim 1, characterized in that, Step 1: Constructing the shape space. Based on Kendall's definition of shape, the original 3D skull is removed from translation, scaling, and rotation Lie group effects and projected into the shape space. This includes the following steps: Step 1.

1. Representation of the three-dimensional skull shape: a shape formed by... The centralized three-dimensional skull point cloud data composed of points is denoted as . , here It is the first in the three-dimensional skull point cloud The three-dimensional coordinates of the points are used to transform the point cloud data into a column vector using vectorization operators, which is an element in the configuration space shown in equation (1): ……（1）， Step 1.

2. Project onto shape space, use the Helmert matrix or centering matrix from Kendall's shape space theory to remove the translation effect on the forming elements, and then use the centroid size to remove the scaling effect to obtain the pre-shape. Next, a Protodyakonov analysis is performed on the pre-shaped form to remove the effect of the rotational transformation group, thus obtaining the elements in the shape space. Elements in shape space It is a pre-shaped A set of elements under rotation.

3. The three-dimensional mandible reconstruction method based on principal geodesic analysis according to claim 1, characterized in that, Step 2 describes projecting the skull training samples into shape space, then using Fréchet's method to find the intrinsic mean in shape space. Next, principal geodesic analysis is performed on all training samples at the intrinsic mean. Finally, the relationship between the shape of the maxilla and the shape of the mandible is obtained from the training samples and used as prior knowledge in the reconstruction stage. This includes the following steps: Step 2.1 Logarithmic mapping of shape space: By analyzing the shape in shape space, and given that shape space is differentially homeomorphic to Euclidean space in the neighborhood of a point on it, the local shape space is mapped to Euclidean space. The analysis process is completed in Euclidean space, and the results are mapped back to shape space. Since the distribution of 3D skull shapes in shape space is concentrated, they can be regarded as being in the neighborhood of a point. By projecting the 3D skull shapes onto the tangent space of that point, the essential features of the shape can be analyzed, allowing... and These are two points on the pre-shaped sphere. These are the shape representations of the two pre-shaped features, which are then calculated using the Protodyakonov analysis method. Representative elements in the pre-shape space , here represent Towards After rotation and alignment, the logarithmic mapping in shape space is represented by the following equation (2): ……（2）， In the above formula (2), It was calculated by Protodyakonov analysis. Towards The optimal rotation matrix for alignment, where the logarithmic mapping is equidistant. and The shortest geodesic distance between corresponding shapes is expressed by the following formula (3): ……(3)； Step 2.2 Calculate the intrinsic mean of the 3D skull model. First, construct the pre-shape space of the skull data. Then, Protodyakonov analysis was used to eliminate the rotational effect, removing all pre-shaped elements. To the first pre-shape Alignment yields representative elements of the skull shape in the pre-shape space, denoted as follows: The shape space composed of all skull shapes is represented as Based on Fréchet's intrinsic mean algorithm and the geodesic distance representation in shape space in formula (3), the definition of intrinsic mean in the three-dimensional skull shape space is given as follows (4): ……(4)， In the above formula (4), This represents the point in shape space that minimizes the sum of squared geodesic distances to all shapes. It represents any skull shape in shape space. As the initial value of the intrinsic mean, the gradient descent method is used to solve equation (4) above. The gradient of the three-dimensional skull mean point is given by equation (5) below: ……(5)， The intrinsic mean of the three-dimensional skull shape obtained from each iteration update is expressed as follows (6): ……(6)； Step 2.3 Perform principal geodesic analysis on the 3D skull model. The principal geodesic in shape space is a generalization of the principal directions in linear space. It preserves the most significant changes in shape within shape space. Utilizing the property that the local differential of shape space is homeomorphic to Euclidean space, the 3D skull shape in shape space is projected onto the tangent space of the intrinsic mean point. The main directions of change of shape in the tangent space are calculated. These directions are then projected back onto shape space to obtain the principal geodesic of shape space. The manifold formed by the principal geodesic is a geodesic submanifold of the 3D skull shape space. Intrinsic mean The variance of the three-dimensional skull shape is calculated as follows (7): ……(7)， The covariance matrix of the three-dimensional skull sample shape is calculated according to the above equation (7), as shown in equations (8) and (9): ……(8)， in: ...(9); The eigenvectors in the tangent space at the intrinsic mean are further obtained from the covariance matrix. and eigenvalues The number of principal components of a geodesic submanifold is determined by the cumulative variance contribution rate using eigenvectors and eigenvalues. To construct the projection matrix By using a projection matrix to project the original shapes onto the geodesic submanifolds, we obtain their dimension-reduced representations in the low-dimensional tangent space (10): ……(10)， The above describes the process of shape analysis of a three-dimensional skull model using the principal geodesic analysis method. The principal geodesic analysis method is universal for three-dimensional point cloud models. Step 2.4 Relational training based on principal geodesic analysis: The principal geodesic analysis method is used to analyze the training samples of the maxilla and mandible respectively to obtain the principal components of the tangent space of the maxilla shape model at the intrinsic mean. and its dimensionality reduction representation And the principal components of the tangent space at the mean of the mandibular shape model within its shape space. and its dimensionality reduction representation The three-dimensional mandibular reconstruction method based on principal geodesic analysis reconstructs the complete mandible from the maxilla model. To find the transformation relationship between the maxilla and mandible, the transformation relationship between the two is expressed as the following formula (11): ……(11)， In the above formula (11), The optimal transformation matrix is ​​obtained by the least squares method. However, in the dimensionality reduction representation, the number of principal directions of the maxilla and mandible needs to be kept consistent in order to establish a one-to-one linear relationship and thus obtain the relationship matrix. Thus, all the prior knowledge required for mandibular reconstruction was obtained, including the relation matrix. As shown in equation (12), ……(12)。

Images