Brain point cloud registration method based on Clifford algebra and manifold features

Abstract

The invention discloses a brain point cloud registration method based on Clifford algebra and manifold features. The brain point cloud registration method comprises the following steps: calculating a brain curvature field based on manifold features; extracting three-dimensional ridge line features; three-dimensional feature main direction construction based on Clifford algebra is carried out; constructing a Clifford geometric invariant feature descriptor; bidirectional consistency matching based on ratio checking; carrying out space attitude calculation based on a high-order spinor mapping kernel; carrying out dense feedback alignment based on manifold constraint; according to the method, manifold differential geometry is introduced to extract high-robustness brain ridge features, a geometric invariant descriptor with holographic characteristics is constructed in combination with Clifford algebra, and meanwhile, singular-point-free rapid attitude calculation is realized by using a high-order spinor analysis method, so that the calculation efficiency is improved on the premise that the calculation efficiency is ensured. And the precision and robustness of multi-modal point cloud registration in neurosurgical navigation are remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of computer vision and medical image processing technology, and in particular relates to a brain point cloud registration method based on Clifford algebra and manifold features. Background Technology

[0002] In minimally invasive neurosurgery, precise surgical path planning and navigation require accurate matching (i.e., registration) between preoperative high-resolution images (such as MRI and CT) and the patient's actual anatomical location during surgery. Due to the complexity of the brain's structure and the presence of numerous irregular surfaces such as sulci and gyri, image data typically needs to be converted into a 3D point cloud model for processing. Currently, point cloud registration techniques are mainly divided into point-based registration (such as the ICP algorithm) and feature-based registration. Among these, feature-based registration has become a research hotspot in this field because it does not require a good initial location and can handle partially overlapping data.

[0003] Currently, the closest existing technology typically employs a scheme combining coarse registration based on Euclidean geometric features with fine registration using Iterative Closest Point (ICP). The specific implementation process is as follows: First, by calculating the normal vector and curvature of each point in the point cloud, salient feature points (such as ISS feature points) are extracted. Second, feature descriptors (such as FPFH or SHOT descriptors) are constructed using geometric relationships in Euclidean space (such as the distance and angle between a point and its neighbors). Corresponding point pairs are found by matching Euclidean distances between descriptors, and the initial transformation matrix is ​​solved using SVD decomposition. Finally, the initial transformation is applied to the floating point cloud, and the standard ICP algorithm is used to iteratively optimize the transformation matrix by minimizing the point-to-point distance error, thus completing the registration.

[0004] The aforementioned existing technologies suffer from the following major drawbacks when processing complex brain point cloud data: Insufficient robustness of feature descriptors: Existing descriptors such as FPFH are constructed based on Euclidean scalars, resulting in a single information dimension and a lack of joint description capability for spatial "direction" and "area" (poor holographicity). When faced with numerous flat regions or repetitive textures (gyri) with similar geometric structures in the brain, they lack strict rotation and translation invariance, easily leading to mismatches. Sensitivity to noise and outliers: Brain MRI reconstructed point clouds are often accompanied by reconstruction noise. Existing technologies typically perform feature calculations directly based on the original point cloud or simple PCA principal directions, resulting in weak noise resistance and unstable extracted principal directions, severely affecting the accuracy of initial registration values. Singularity and efficiency bottlenecks in attitude calculation: Traditional methods often use Euler angles or quaternions for iterative optimization when solving rotation matrices. Euler angle representation suffers from "gimbal lock" problems, while iterative calculations based on SVD or quaternions are prone to getting trapped in local minima in nonlinear optimization, and the computational convergence speed is slow, making it difficult to meet the real-time requirements of surgical navigation. Summary of the Invention

[0005] Purpose of the invention: The purpose of this invention is to provide a brain point cloud registration method based on Clifford algebra and manifold features that can improve registration accuracy and robustness.

[0006] Technical solution: The present invention provides a brain point cloud registration method based on Clifford algebra and manifold features, comprising:

[0007] (1) Input the pre-acquired brain reference point cloud and floating point cloud, and construct a multi-scale neighborhood of each point using spatial index; construct a local weighted covariance matrix based on the multi-scale neighborhood and perform eigenvalue decomposition, and use the decomposed eigenvalues ​​to calculate the surface change index of each point to obtain the manifold curvature parameter;

[0008] (2) Based on the manifold curvature parameter, set the curvature threshold to filter ridge candidate points; for each candidate point, use the non-maximum suppression algorithm along the principal curvature gradient to eliminate non-local maxima along the cross-sectional direction and retain the ridge center point; then use Euclidean distance growth clustering to connect all ridge center points into a line and eliminate debris noise to output a three-dimensional ridge set.

[0009] (3) Select feature corner points on the ridges in the three-dimensional ridge set; model in the Clifford algebra space corresponding to the three-dimensional Euclidean space, and construct an energy function that minimizes the sum of the squares of the outer product of all neighborhood points of the feature corner point and the principal direction vector; solve the matrix eigenvalues ​​corresponding to the energy function by the Lagrange multiplier method, and take the eigenvector corresponding to the smallest eigenvalue of the matrix eigenvalues ​​as the optimal geometric principal direction of the feature corner point;

[0010] (4) Based on the feature corner points and the optimal principal direction unit vector of the feature corner points, the geometric product formula of Clifford algebra is used to operate on the neighborhood points, and the resulting geometric integral solution is divided into a scalar part and a double vector part. The scalar part and the double vector part are combined to construct the feature descriptor vector.

[0011] (5) Define the Euclidean distance between feature descriptor vectors as similarity; perform forward matching on the feature points of the reference point cloud and the floating point cloud and calculate the similarity to find the best match and the second best match. Use the ratio of the best match to the second best match for filtering. Combined with the reverse verification strategy, retain the matching pairs that are the nearest neighbors of the forward match and the reverse verification as high confidence matching pairs.

[0012] (6) Using the Clifford algebra even subalgebra to define the spinor transformation kernel, construct the maximum likelihood estimation function based on the high confidence matching pair set, transform the maximum likelihood estimation function into a matrix eigenvalue problem, solve the eigenspin corresponding to the largest positive eigenvalue, analyze the optimal space rotation parameters, and achieve coarse registration.

[0013] (7) Apply the transformation obtained from coarse registration to the floating point cloud; use the improved point-to-surface iterative nearest point algorithm to construct the error objective function that minimizes the vertical distance between the source point and the target point tangent plane; solve the incremental transformation matrix iteratively until the convergence condition is met, and output the final registration result.

[0014] Further, step (1) includes:

[0015] (11) Determine the input data:

[0016] Let the pre-acquired set of floating point clouds of the brain be... The reference point cloud set is ;

[0017] (12) Construction of multi-scale neighborhood and local weighted covariance matrix:

[0018] For reference point cloud With floating point clouds For any target point in the dataset, use a kd-tree for spatial indexing to search for the set of nearest neighbors of that target point. ; Increase the number of neighborhood searches The value is set as an adaptive value; then a locally weighted covariance matrix is ​​constructed based on the voxel values ​​of the neighborhood. And perform eigenvalue decomposition to obtain and the eigenvectors corresponding to the eigenvalues and ;

[0019] (13) Extraction of manifold curvature parameters:

[0020] Calculate the manifold curvature parameter of the target point in step (12). The formula is as follows:

[0021]

[0022] Among them, features Used to describe the amount of change in the surface normal direction, characteristic The direction corresponding to the greatest curvature is the direction that crosses the ridge line.

[0023] Further, step (2) includes:

[0024] (21) Initial screening of candidate regions:

[0025] Select manifold curvature parameter Points exceeding a preset threshold are selected as ridge candidate points; the threshold Set to 1.5 to 2.0 times the overall average curvature of the point cloud;

[0026] (22) Non-maximum suppression along the principal curvature gradient: for candidate points Extract the center point of the ridge line;

[0027] (23) Euclidean distance growing clustering: Connect the center points of the ridges refined in step (22) into a line, remove fragment noise, and output a three-dimensional ridge set.

[0028] Further, step (22) includes:

[0029] (221) Obtain candidate points The eigenvector corresponding to the largest eigenvalue , The direction is perpendicular to the ridgeline and points in the direction of the steepest descent.

[0030] (222) in along Search for neighboring points within a small neighborhood along a direction; the range of the small neighborhood is the radius. Within the range of multiples of the average point distance;

[0031] (223) If point manifold curvature parameter If the manifold curvature parameter is less than that of any neighboring point in that direction, then it is determined that... Slopes located on the ridgeline, rather than the summit, should be excluded; only when When the value is the maximum value within the local linear neighborhood, it is retained to obtain the center point of the ridge.

[0032] Further, step (23) includes:

[0033] (231) Create an empty list of clusters. Mark all feature points as unvisited;

[0034] (232) Randomly select an unvisited point as the seed point. Establish a new cluster ;

[0035] (233) Search preset radius Add all feature points within And push it onto the search stack;

[0036] (234) Recursively execute steps (231) to (233) until the stack is empty;

[0037] (235) If a certain cluster The number of points included is less than the preset threshold. This is considered noise and discarded; the final output is a set of three-dimensional ridge lines. .

[0038] Further, step (3) includes:

[0039] (31) Definition of characteristic corner point:

[0040] Any ridge in the three-dimensional ridge set Calculate the geodesic curvature of points on the line, and select local maxima as characteristic corner points. ;

[0041] (32) Clifford algebraic space modeling:

[0042] Clifford algebra in three-dimensional Euclidean space In the diagram, let the basis vectors be... , and ,correspond , and Axis; define characteristic corner points With the origin as the origin, the attitude vector to be determined is a unit vector. Select the neighborhood point set distributed along the ridge line before and after the feature corner point; , The value ranges from 5 to 8. For the first The neighborhood points are represented as vectors in Clifford algebra;

[0043] (33) Construct the outer product optimization objective function:

[0044] Constructing the energy function , indicating all Minimize the sum of the squares of the magnitudes of the outer products of the neighboring points and the principal direction vector:

[0045] ;

[0046] In Clifford algebra, Represented by vector form and The area of ​​the parallelogram formed;

[0047] (34) Derivation of matrix solution:

[0048] Expand the outer product terms:

[0049] ;

[0050] in, , and The unknowns are the coordinates of the main direction;

[0051] Introducing Lagrange multipliers constraint ,right Taking the partial derivatives and setting them to 0, we derive the following system of linear equations:

[0052] ;

[0053] Where the matrix The specific form is as follows:

[0054] ;

[0055] (35) Results Acquisition:

[0056] For matrix Perform eigenvalue decomposition, and the eigenvector corresponding to the smallest eigenvalue after decomposition is the optimal geometric principal direction of the feature corner point. .

[0057] Further, step (4) includes:

[0058] (41) Input data:

[0059] For the current feature corner point as well as The optimal principal direction unit vector Then there is Surrounding a set of neighborhood points ;

[0060] (42) Calculate the Clifford geometric product:

[0061] For each neighboring point Using Clifford's algebra's geometric product formula and principal direction Perform the calculation:

[0062] ;

[0063] This formula breaks down the geometric relationship into two parts:

[0064] scalar part, ; Vertical features are used to represent neighborhood points. The projected length on the principal direction axis;

[0065] Double vector part: , used to represent vector form and The directed area of ​​the enclosed parallelogram is taken as... ,get It is a lateral feature used to represent the vertical distance of a neighboring point from the principal direction axis;

[0066] (43) Generate feature vectors:

[0067] Vertical features of all neighboring points and lateral features Combine to construct the feature descriptor vector of the feature corner points. :

[0068] .

[0069] Further, step (5) includes:

[0070] (51) Definition of distance metric function:

[0071] Define two feature descriptor vectors and The Euclidean distance between two feature corner points is used to measure the geometric similarity between them, and the formula is as follows:

[0072] ;

[0073] in, The number of descriptors, Euclidean distance The smaller the value, the more similar the local geometry of the two feature corner points;

[0074] (52) Positive matching and ratio test:

[0075] For each feature corner point in the reference point cloud :

[0076] (a) Calculate the characteristic corner points of the floating point cloud and... distance ;

[0077] (b) Find the feature corner point of the nearest floating point cloud. The corresponding distance is denoted as ;

[0078] (c) Find the feature corner point of the second closest floating point cloud. The corresponding distance is denoted as ;

[0079] (d) Calculate the ratio Preset threshold ,like Then keep ,like Then discard ;

[0080] (53) Reverse verification:

[0081] Perform reverse verification on the retained feature corner point pairs, and select the retained ones. Re-search for its nearest neighbor in the reference point cloud, when The nearest neighbor in the reference point cloud is the initial one. At that time, and The matched pairs were confirmed as high-confidence matched pairs.

[0082] Further, step (6) includes:

[0083] (61) Definition of spinor transformation nucleus:

[0084] Define spinor transformation kernel For spinors in hypercomplex space:

[0085] ;

[0086] in, , and Let be the basis vectors of the Clifford algebra. , , and For parameters;

[0087] (62) Full-space fitting optimization:

[0088] Constructing a set of high-confidence matching pairs Maximum likelihood estimation function:

[0089] ;

[0090] in, for conjugate, This serves as an index for iterating through all high-confidence matching pairs. Is assigned to the first The weight values ​​of each matching pair are determined by the set of matching pairs. of After normalization, we get The distance between the feature corner point of the reference point cloud and the feature corner point of the nearest floating point cloud; and Representing the first For the reference point cloud vector and the floating point cloud vector in the matching points;

[0091] (63) Solving for the characteristic system:

[0092] The problem of the maximum likelihood estimation function is linearized as follows: Constructing a matrix The intrinsic problem, calculation The feature spinor corresponding to the largest positive eigenvalue, which corresponds to the optimal spatial rotation, is used to achieve coarse registration of the point cloud.

[0093] Further, step (7) includes:

[0094] (71) Initial value loading:

[0095] spinor transformation kernel and the translation of the center of mass Applied to floating point clouds, resulting in pre-aligned point clouds. ;

[0096] (72) Minimize the normal vector projection error:

[0097] Reference point cloud China Each point Find the nearest neighbor and obtain Surface normal vector at the location Construct the error objective function , used to minimize arrive Perpendicular distance of the tangent plane:

[0098]

[0099] in, and This is the incremental transformation matrix;

[0100] (73) Iterative convergence control:

[0101] Solving using linear least squares and Update the floating point cloud position;

[0102] (74) In the iteration steps (72) to (73), when the preset number of iterations is reached, or when the change in root mean square error between two iterations is less than the preset threshold, it is determined to be converged and the final registration result is output.

[0103] Beneficial Effects: Compared with the prior art, the present invention has the following significant advantages: 1. The present invention, through manifold curvature field solution and ridge extraction, first extracts a highly robust topological skeleton (ridge) with anatomical significance from scattered point clouds, providing high-quality input with clear directionality for Clifford algebra operations; this combination strategy of "topological screening first, algebraic encoding later" effectively overcomes the deficiency of low discriminability in low curvature regions by single geometric algebra methods, so that the final generated descriptor retains both mathematical rotational invariance and strong anatomical structural saliency; it achieves deep coupling between manifold topological features and algebraic invariants, significantly improving the registration robustness of weakly textured regions; 2. The present invention The invention creatively uses the analytical solution of high-order spinor as the initial value and seamlessly integrates it into the dense feedback alignment of manifold constraints; it constructs a closed-loop alignment system that complements "analytical global coarse registration" and "manifold constraint local fine registration"; this combination not only solves the problem of "initial value sensitivity" by utilizing the advantages of the spinor method (Spinor) in being non-iterative and resistant to large-angle misalignment, but also corrects the linearized residual of the spinor method by using the subsequent normal vector projection error minimization strategy; this cross-level combination from "algebraic analysis" to "geometric iteration" achieves a registration effect that can still reach sub-voxel level accuracy under large initial misalignment; 3. This invention combines the optimization of the main direction of the outer product (physical geometric features) with the two-way consistency ratio test (statistical probability features). This combination constitutes a cascaded filtering mechanism: first, it uses outer product fitting to eliminate noise interference, and then it uses ratio testing to remove ambiguous points that are geometrically similar but have unreasonable spatial distributions. This multi-dimensional combination of "physical features" and "statistical features" significantly reduces the mismatch rate, ensures the uniqueness and reliability of feature correspondence, and solves the problem of ambiguous matching caused by repetitive structures in brain images. 4. In the feature extraction stage, this invention abandons the traditional PCA method and instead adopts a Clifford outer product optimization objective function based on ridge neighborhood, combined with multi-scale neighborhood construction. By utilizing the natural geometric filtering characteristics of the double vector structure, it directly suppresses high-frequency noise in MRI reconstruction data at the algebraic level. This allows the invention to directly extract accurate structural principal directions without additional preprocessing smoothing steps when processing low signal-to-noise ratio medical images, demonstrating the synergistic noise resistance advantages of manifold preprocessing and algebraic optimization, and improving the stability of feature extraction in complex noise environments. Attached Figure Description

[0104] Figure 1 This is a flowchart of a method according to an embodiment of the present invention. Detailed Implementation

[0105] To enable those skilled in the art to better understand the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings.

[0106] As attached Figure 1 As shown, the brain point cloud registration method based on Clifford algebra and manifold features in this embodiment includes:

[0107] Step 1: Input the pre-acquired brain reference point cloud and floating point cloud, and construct a multi-scale neighborhood for each point using spatial indexing; construct a local weighted covariance matrix based on the multi-scale neighborhood and perform eigenvalue decomposition; use the decomposed eigenvalues ​​to calculate the surface change index of each point to obtain the manifold curvature parameter.

[0108] Step 2: Based on the manifold curvature parameter, set a curvature threshold to filter candidate ridge points; for each candidate point, use the non-maximum suppression algorithm along the principal curvature gradient to eliminate non-local maxima along the cross-sectional direction and retain the ridge center point; then use Euclidean distance growing clustering to connect all ridge center points into a line, and remove debris noise to output a three-dimensional ridge set.

[0109] Step 3: Select feature corner points on the ridges in the 3D ridge set; model in the Clifford algebra space corresponding to the 3D Euclidean space, and construct an energy function that minimizes the sum of the squares of the outer product of the modulus of all neighborhood points of the feature corner point and the principal direction vector; solve for the matrix eigenvalues ​​corresponding to the energy function using the Lagrange multiplier method, and take the eigenvector corresponding to the smallest eigenvalue of the matrix eigenvalues ​​as the optimal geometric principal direction of the feature corner point;

[0110] Step 4: Based on the feature corner points and the optimal principal direction unit vector of the feature corner points, use the geometric product formula of Clifford algebra to operate on the neighborhood points, and divide the obtained geometric integral solution into a scalar part and a double vector part. Combine the scalar part and the double vector part to construct the feature descriptor vector.

[0111] Step 5: Define the Euclidean distance between feature descriptor vectors as similarity; perform forward matching on the feature points of the reference point cloud and the floating point cloud and calculate the similarity to find the best match and the second best match. Use the ratio of the best match to the second best match for filtering. Combined with the reverse verification strategy, retain the matching pairs where the forward match and the reverse verification are each other's nearest neighbors as high-confidence matching pairs.

[0112] Step 6: Using Clifford's algebraic even subalgebra to define the spinor transformation kernel, construct the maximum likelihood estimation function based on the high confidence matching pair set, transform the maximum likelihood estimation function into a matrix eigenvalue problem, solve for the eigenspin corresponding to the largest positive eigenvalue, analyze the optimal space rotation parameters, and achieve coarse registration;

[0113] Step 7: Apply the transformation obtained from coarse registration to the floating point cloud; use the improved point-to-surface iterative nearest point algorithm to construct an error objective function that minimizes the perpendicular distance between the source point and the tangent plane of the target point; solve the incremental transformation matrix iteratively until the convergence condition is met, and output the final registration result.

[0114] Specifically, step one: brain curvature field calculation based on manifold features; this step aims to extract the differential geometric properties of the sulci and gyri from the scattered point cloud based on three-dimensional curvature features.

[0115] 1. Input data: Reference point cloud set Floating point cloud collection ;

[0116] 2. Multi-scale neighborhood covariance construction: For the reference point cloud With floating point clouds any target point in Using a kd-tree for spatial indexing, we search for the nearest neighbor set of the target point. Parameter settings: Number of neighborhood searches Set as an adaptive value; the recommended value range is [range to be specified]. To balance local noise suppression with preservation of small structures; for high-resolution MRI reconstruction data, the preferred method is... Subsequently, a locally weighted covariance matrix is ​​constructed based on the voxel values ​​of the neighborhood. And perform eigenvalue decomposition to obtain and its corresponding eigenvectors and .

[0117] 3. Manifold curvature parameter extraction: Calculate the manifold curvature parameter for the target point. (Approximate average curvature), the formula is as follows:

[0118]

[0119] Here It describes the amount of change in the direction of the surface normal. The direction corresponding to the greatest curvature (i.e., the direction that crosses the cerebral gyri).

[0120] In step two, three-dimensional ridge feature extraction is performed.

[0121] 1. Initial screening of candidate regions: retain Points exceeding a certain threshold are selected as ridge candidate points. Parameter setting: Curvature threshold. Set to 1.5 to 2.0 times the overall average curvature of the point cloud.

[0122] 2. Non-maximum suppression along the principal curvature gradient (Directional NMS): targeting candidate points To extract the exact center of the ridge, it is necessary to determine whether it is a local maxima along its "cross-section". Specific steps:

[0123] a) Acquisition Points The corresponding eigenvector with the largest eigenvalue (This direction is perpendicular to the ridgeline and points in the direction of the steepest descent.)

[0124] b) In along Within a small neighborhood of the direction (e.g., radius) Search for neighboring points within the range of multiples of the average point distance.

[0125] c) Judgment logic: If point manifold curvature parameter If the manifold curvature parameter is less than that of any neighboring point in that direction, then it is determined that... The "slope" rather than the "summit" of the ridgeline should be excluded; only when... If the value is the maximum value within the local linear neighborhood, it is retained.

[0126] 3. Euclidean Region Growing (ERG): Connects discrete points refined by NMS into lines. Specific steps:

[0127] a) Create an empty list of clusters. Mark all feature points as "unvisited".

[0128] b) Randomly select an unvisited point as the seed point. Establish a new cluster .

[0129] c) Search radius (It is recommended to set the distance to 3mm-5mm) All feature points within this range should be added if the distance meets the requirement. And push it onto the search stack.

[0130] d) Recursively execute the growth process from a) to c) above until the stack is empty.

[0131] e) Fragment removal: If a cluster The number of points contained is less than (15 points are recommended), these are considered noise and discarded. The final output is a set of 3D ridge lines. .

[0132] In step three: the principal directions of the three-dimensional features are constructed based on Clifford algebra;

[0133] 1. Feature Corner Definition: In each extracted ridge line Calculate the geodesic curvature of points on the line, and select local maxima as characteristic corner points. .

[0134] 2. Clifford Algebra Space Modeling: The Clifford Algebra corresponding to three-dimensional Euclidean space In the given information, the basis vectors are... (Corresponding to the x, y, z axes). Let the characteristic corner point be... With the origin as the origin, the attitude vector to be determined is a unit vector. Select the neighborhood point set distributed along the ridge line before and after the feature corner point. Number of neighboring points Select 5 to 8 points before and after the key point to ensure that the linear features of the local geometry are fully captured. For the first The ...

[0135] 3. Constructing the outer product optimization objective function: To find the axis that best represents the spatial orientation of the ridge line, an energy function is constructed. That is, minimizing the sum of the squares of the magnitudes of the outer products of all neighboring points and the principal direction vector:

[0136]

[0137] Physical meaning: In Clifford algebra, Represented by vector form and The area of ​​the parallelogram formed (i.e., the perpendicular distance from the point to the line). Minimizing this value is equivalent to the overall least squares fit in three-dimensional space.

[0138] 4. Matrix solution derivation (detailed steps): Expanding the outer product terms:

[0139] ;

[0140] in, , and The unknowns are the coordinates of the main direction;

[0141] Introducing Lagrange multipliers constraint ,right Taking the partial derivatives and setting them to 0, we derive the following system of linear equations:

[0142] ;

[0143] Where the matrix ( The specific form of a real symmetric matrix is:

[0144]

[0145] 5. Result Acquisition: For the matrix Eigenvalue decomposition is performed, and the eigenvector corresponding to the smallest eigenvalue is the optimal geometric principal direction of that key point. Advantages: Compared to PCA, the fitting based on Clifford's outer product directly utilizes the double vector structure, making it more robust to noise and naturally adaptable to spatial linear structures.

[0146] Step 4: Construction of Clifford geometric invariant feature descriptors; the purpose of this step is to transform the geometry around each feature corner point into a string of numbers that a computer can understand. Unlike traditional methods that only calculate point-to-point distances, Clifford algebra can simultaneously calculate both "projected length" and "perpendicular area," thus possessing holographic properties.

[0147] 1. Input data: Current feature corner points The optimal principal direction unit vector of the feature corner point calculated in step three. (Clifford vector); the area around the feature corner. a set of neighborhood points (Coordinates have been converted to) (Relative coordinates of the origin).

[0148] 2. Calculate the Clifford geometric product: for each neighborhood point Using Clifford's algebra's geometric product formula and principal direction Perform the calculation: This formula breaks down the geometric relationship into two parts:

[0149] a) Scalar part ( ): Geometric meaning: Represents a neighboring point. Projected length on the main direction axis (longitudinal feature).

[0150] b) Double vector part ( ): Geometric meaning: Represents a vector form and The directed area of ​​the enclosed parallelogram. We take its modulus. This actually describes the vertical distance (lateral feature) of a neighboring point from the principal direction axis.

[0151] 3. Generate feature vectors: Generate vertical features of all neighboring points. and lateral features Combine them to construct the final feature descriptor vector of the feature corner point. :

[0152]

[0153] because (scalar product) and The (external product modulus) remains unchanged under rigid body rotation, therefore this descriptor has strict rotation and translation invariance.

[0154] Step 5: Bidirectional consistency matching based on ratio test; This step uses the above descriptors to find corresponding point pairs between the reference point cloud and the floating point cloud.

[0155] 1. Distance metric function Definition: In the formula This refers to two feature descriptor vectors. and The Euclidean distance between two feature corner points measures the similarity of their geometric shapes. The formula is:

[0156]

[0157] The smaller the value, the more similar the local geometry of the two points.

[0158] 2. Forward matching and ratio test: For each feature corner point in the reference point cloud :

[0159] a) Among all the feature corner points of the floating point cloud, calculate and... distance .

[0160] b) Find the nearest feature corner point (best match) Distance is denoted as .

[0161] c) Find the second closest feature corner point (second-best match) Distance is denoted as .

[0162] d) Calculate the ratio :like ( If the threshold is set to 0.75, then it is considered... A match that is significantly superior to other points and is considered "unique" will be retained. If... This indicates that the best and second-best matches are too close and ambiguous (e.g., there are many similar flat areas in the brain). This point is considered an unreliable match and is discarded directly.

[0163] 3. Cross-Check: To further eliminate false matches, cross-check is performed on the remaining point pairs: the floating points found in the forward matching are taken. Then, re-search for its nearest neighbor in the reference point cloud. Only if its nearest neighbor in the reference point cloud is exactly the original... Only then is the pair of points confirmed as the final high-confidence match.

[0164] Step 6: Solve the spatial attitude based on the higher-order spinor mapping kernel; use the even subalgebra in Clifford to directly solve the spatial transformation analytically.

[0165] 1. Definition of higher-order spinor kernel: Define the space transformation kernel For spinors in hypercomplex space:

[0166]

[0167] in, , and Let be the basis vectors of the Clifford algebra. , , and The parameter is ; this kernel function encompasses all information about the rotation axis and rotation angle.

[0168] 2. Full-space fitting optimization: Constructing a set of high-confidence matching pairs Maximum likelihood estimation function:

[0169]

[0170] in for conjugate, This serves as an index for iterating through all high-confidence matching pairs. Is assigned to the first The weight values ​​of each matching pair are determined by the set of matching pairs. of After normalization, we get The distance between the feature corner point of the reference point cloud and the feature corner point of the nearest floating point cloud; and Representing the first For the reference point cloud vector and the floating point cloud vector in the matching points.

[0171] 3. Solving the characteristic system: Linearize the nonlinear optimization problem of the maximum likelihood estimation function as follows: Constructing a matrix The intrinsic problem. Calculation. The eigenspiral corresponding to the largest positive eigenvalue. This eigenspiral directly corresponds to the optimal spatial rotation, requiring no trigonometric function calculations or iterations, thus avoiding gimbal lock-up and achieving coarse registration of point clouds.

[0172] Step 7: Dense feedback alignment based on manifold constraints; this step is the fine registration stage, which adopts an improved point-to-surface iterative nearest point algorithm.

[0173] 1. Initial value loading: Apply the spinor transformation kernel calculated in step five. and the translation of the center of mass Applied to floating point clouds, resulting in pre-aligned point clouds. .

[0174] 2. Minimize the normal vector projection error: in the reference point cloud China Each point Find the nearest neighbor and obtain Surface normal vector at the location Construct the error objective function. The goal is to minimize the perpendicular distance between the source point and the tangent plane from the target point:

[0175]

[0176] in and This is the incremental transformation matrix.

[0177] 3. Iterative convergence control: Solving using the linear least squares method and . use and Update the floating point cloud position.

[0178] 4. Parameter settings: Set the maximum number of iterations. Or when the root mean square error changes between two iterations When convergence is achieved, the final registration result is output.

Claims

1. A brain point cloud registration method based on Clifford algebra and manifold features, characterized in that, include: (1) Input the pre-acquired brain reference point cloud and floating point cloud, and construct the multi-scale neighborhood of each point using spatial index; A local weighted covariance matrix is ​​constructed based on a multi-scale neighborhood and eigenvalue decomposition is performed. The surface change index of each point is calculated using the decomposed eigenvalues ​​to obtain the manifold curvature parameter. (2) Based on the manifold curvature parameter, set a curvature threshold to filter ridge candidate points; For each candidate point, a non-local maximum suppression algorithm along the principal curvature gradient is used to eliminate non-local maximum points along the cross-sectional direction, while retaining the ridge center point. Then, Euclidean distance growing clustering is used to connect all ridge center points into a line, and debris noise is removed to output a three-dimensional ridge set. (3) Select feature corner points on the ridges in the three-dimensional ridge set; Modeling is performed in the Clifford algebra space corresponding to the three-dimensional Euclidean space. An energy function is constructed that minimizes the sum of the squares of the outer product of the moduli of all neighborhood points of the feature corner point and the principal direction vector. The eigenvalues ​​of the matrix corresponding to the energy function are solved by the Lagrange multiplier method. The eigenvector corresponding to the smallest eigenvalue of the matrix is ​​taken as the optimal geometric principal direction of the feature corner point. (4) Based on the feature corner points and the optimal principal direction unit vector of the feature corner points, the geometric product formula of Clifford algebra is used to operate on the neighborhood points, and the resulting geometric integral solution is divided into a scalar part and a double vector part. The scalar part and the double vector part are combined to construct the feature descriptor vector. (5) Define the Euclidean distance between feature descriptor vectors as the similarity; Forward matching is performed on the feature points of the reference point cloud and the floating point cloud, and the similarity is calculated to find the best match and the second best match. The ratio of the best match to the second best match is used for filtering. Combined with the reverse verification strategy, the matching pairs that are the nearest neighbors of the forward match and the reverse verification are retained as high-confidence matching pairs. (6) Using the Clifford algebra even subalgebra to define the spinor transformation kernel, construct the maximum likelihood estimation function based on the high confidence matching pair set, transform the maximum likelihood estimation function into a matrix eigenvalue problem, solve the eigenspin corresponding to the largest positive eigenvalue, analyze the optimal space rotation parameters, and achieve coarse registration. (7) Apply the transformation obtained from coarse registration to the floating point cloud; use the improved point-to-surface iterative nearest point algorithm to construct the error objective function that minimizes the vertical distance between the source point and the target point tangent plane; solve the incremental transformation matrix iteratively until the convergence condition is met, and output the final registration result.

2. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (1) includes: (11) Determine the input data: Let the pre-acquired set of floating point clouds of the brain be... The reference point cloud set is ; (12) Construction of multi-scale neighborhood and local weighted covariance matrix: For reference point cloud With floating point clouds For any target point in the dataset, use a kd-tree for spatial indexing to search for the set of nearest neighbors of that target point. ; Increase the number of neighborhood searches The value is set as an adaptive value; then a locally weighted covariance matrix is ​​constructed based on the voxel values ​​of the neighborhood. And perform eigenvalue decomposition to obtain and the eigenvectors corresponding to the eigenvalues and ; (13) Extraction of manifold curvature parameters: Calculate the manifold curvature parameter of the target point in step (12). The formula is as follows: ； Among them, features Used to describe the amount of change in the surface normal direction, characteristic The direction corresponding to the greatest curvature is the direction that crosses the ridge line.

3. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (2) includes: (21) Initial screening of candidate regions: Select manifold curvature parameter Points exceeding a preset threshold are selected as ridge candidate points; the threshold Set to 1.5 to 2.0 times the overall average curvature of the point cloud; (22) Non-maximum suppression along the principal curvature gradient: for candidate points Extract the center point of the ridge line; (23) Euclidean distance growing clustering: Connect the center points of the ridges refined in step (22) into a line, remove fragment noise, and output a three-dimensional ridge set.

4. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 3, characterized in that, Step (22) includes: (221) Obtain candidate points The eigenvector corresponding to the largest eigenvalue , The direction is perpendicular to the ridgeline and points in the direction of the steepest descent. (222) in along Search for neighboring points within a small neighborhood along a direction; the range of the small neighborhood is the radius. Within the range of multiples of the average point distance; (223) If point manifold curvature parameter If the manifold curvature parameter is less than that of any neighboring point in that direction, then it is determined that... Slopes located on the ridgeline, rather than the summit, should be excluded; only when When the value is the maximum value within the local linear neighborhood, it is retained to obtain the center point of the ridge.

5. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 3, characterized in that, Step (23) includes: (231) Create an empty list of clusters. Mark all feature points as unvisited; (232) Randomly select an unvisited point as the seed point. Establish a new cluster ; (233) Search preset radius Add all feature points within And push it onto the search stack; (234) Recursively execute steps (231) to (233) until the stack is empty; (235) If a certain cluster The number of points included is less than the preset threshold. This is considered noise and discarded; the final output is a set of three-dimensional ridge lines. .

6. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (3) includes: (31) Definition of characteristic corner point: Any ridge in the three-dimensional ridge set Calculate the geodesic curvature of points on the line, and select local maxima as characteristic corner points. ; (32) Clifford algebraic space modeling: Clifford algebra in three-dimensional Euclidean space In the diagram, let the basis vectors be... , and ,correspond , and Axis; define characteristic corner points With the origin as the origin, the attitude vector to be determined is a unit vector. Select the neighborhood point set distributed along the ridge line before and after the feature corner point; , The value ranges from 5 to 8. For the first The neighborhood points are represented as vectors in Clifford algebra; (33) Construct the outer product optimization objective function: Constructing the energy function , indicating all Minimize the sum of the squares of the magnitudes of the outer products of the neighboring points and the principal direction vector: ； In Clifford algebra, Represented by vector form and The area of ​​the parallelogram formed; (34) Derivation of matrix solution: Expand the outer product terms: ； in, , and The unknowns are the coordinates of the main direction; Introducing Lagrange multipliers constraint ,right Taking the partial derivatives and setting them to 0, we derive the following system of linear equations: ； Where the matrix The specific form is as follows: ； (35) Results Acquisition: For matrix Perform eigenvalue decomposition, and the eigenvector corresponding to the smallest eigenvalue after decomposition is the optimal geometric principal direction of the feature corner point. .

7. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (4) includes: (41) Input data: For the current feature corner point as well as The optimal principal direction unit vector Then there is Surrounding a set of neighborhood points ; (42) Calculate the Clifford geometric product: For each neighboring point Using Clifford's algebra's geometric product formula and principal direction Perform the calculation: ； This formula breaks down the geometric relationship into two parts: scalar part, ; Vertical features are used to represent neighborhood points. The projected length on the principal direction axis; Double vector part: , used to represent vector form and The directed area of ​​the enclosed parallelogram is taken as... ,get It is a lateral feature used to represent the vertical distance of a neighboring point from the principal direction axis; (43) Generate feature vectors: Vertical features of all neighboring points and lateral features Combine to construct the feature descriptor vector of the feature corner points. : 。 8. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (5) includes: (51) Definition of distance metric function: Define two feature descriptor vectors and The Euclidean distance between two feature corner points is used to measure the geometric similarity between them, and the formula is as follows: ； in, The number of descriptors, Euclidean distance The smaller the value, the more similar the local geometry of the two feature corner points; (52) Positive matching and ratio test: For each feature corner point in the reference point cloud : (a) Calculate the characteristic corner points of the floating point cloud and... distance ; (b) Find the feature corner point of the nearest floating point cloud. The corresponding distance is denoted as ; (c) Find the feature corner point of the second closest floating point cloud. The corresponding distance is denoted as ; (d) Calculate the ratio Preset threshold ,like Then keep ,like Then discard ; (53) Reverse verification: Perform reverse verification on the retained feature corner point pairs, and select the retained ones. Re-search for its nearest neighbor in the reference point cloud, when The nearest neighbor in the reference point cloud is the initial one. At that time, and The matched pairs were confirmed as high-confidence matched pairs.

9. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (6) includes: (61) Definition of spinor transformation nucleus: Define spinor transformation kernel For spinors in hypercomplex space: ； in, , and Let be the basis vectors of the Clifford algebra. , , and For parameters; (62) Full-space fitting optimization: Constructing a set of high-confidence matching pairs Maximum likelihood estimation function: ； in, for conjugate, This serves as an index for iterating through all high-confidence matching pairs. Is assigned to the first The weight values ​​of each matching pair are determined by the set of matching pairs. of After normalization, we get The distance between the feature corner point of the reference point cloud and the feature corner point of the nearest floating point cloud; and Representing the first For the reference point cloud vector and the floating point cloud vector in the matching points; (63) Solving for the characteristic system: The problem of the maximum likelihood estimation function is linearized as follows: Constructing a matrix The intrinsic problem, calculation The feature spinor corresponding to the largest positive eigenvalue, which corresponds to the optimal spatial rotation, is used to achieve coarse registration of the point cloud.

10. The brain point cloud registration method based on Clifford algebra and manifold features according to claim 1, characterized in that, Step (7) includes: (71) Initial value loading: spinor transformation kernel and centroid translation Applied to floating point clouds, resulting in pre-aligned point clouds. ; (72) Minimize the normal vector projection error: Reference point cloud China Each point Find the nearest neighbor and obtain Surface normal vector at the location Construct the error objective function , used to minimize arrive Perpendicular distance of the tangent plane: ； in, and This is the incremental transformation matrix; (73) Iterative convergence control: Solving using linear least squares and Update the floating point cloud position; (74) In the iteration steps (72) to (73), when the preset number of iterations is reached, or when the change in root mean square error between two iterations is less than the preset threshold, it is determined to be converged and the final registration result is output.

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