CN110910492A - Method for point matching between non-rigid three-dimensional models - Google Patents

Method for point matching between non-rigid three-dimensional models Download PDF

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CN110910492A
CN110910492A CN201911204811.2A CN201911204811A CN110910492A CN 110910492 A CN110910492 A CN 110910492A CN 201911204811 A CN201911204811 A CN 201911204811A CN 110910492 A CN110910492 A CN 110910492A
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刘圣军
李钦松
胡玲
刘新儒
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Abstract

The invention discloses a point matching method between non-rigid three-dimensional models, which comprises the following steps: firstly, establishing an anisotropic spectrum manifold wavelet descriptor; and then, the established descriptors are used as descriptor constraints of model points, the thermal kernel relation of each point on the model is used as point-to-point relation constraints, and a target function is established to realize the optimal matching among the model points. The invention firstly establishes an anisotropic spectrum manifold wavelet descriptor in the early stage and then adopts the thermonuclear relation as point pair relation constraint. Compared with the existing method, the anisotropic spectrum manifold wavelet descriptor has the advantages of equidistant deformation invariance, capability of distinguishing the intrinsic symmetry of the model, high resolution capability and positioning capability, high calculation efficiency and compact structure; the thermonuclear relation is used as point-to-point relation constraint, and the calculation efficiency and stability are superior to those of other methods adopting geodesic distance; therefore, the method is ensured to be definite in calculation, robust in result and accurate in matching.

Description

Method for point matching between non-rigid three-dimensional models
Technical Field
The invention belongs to the field of computer graphics and computer vision, and particularly relates to a point matching method between non-rigid three-dimensional models.
Background
In the past decade, with the rapid development of three-dimensional model acquisition equipment and related technologies, digital three-dimensional models have been used to represent very rich objects, and their related applications relate to traditional computer aided design, medical engineering, and emerging industries such as multimedia technology, entertainment industry, etc. Retrieval, identification and classification, semantic segmentation, migration and the like of 3D models are basic and important applications widely existing in the fields, and the important premise for realizing the applications is to establish accurate and efficient point matching (hereinafter referred to as model matching) between three-dimensional models.
The core technology of the mainstream method for establishing model matching exists in the following two steps:
(1) firstly, establishing a point descriptor for accurately describing geometric characteristics of a model;
by encoding the multi-scale shape information and the geometric characteristics of each point of the model, a point descriptor (usually expressed as a high-dimensional vector) capable of describing the overall geometric characteristics at each point of the model is constructed, and the descriptor requires high resolution and high calculation efficiency;
(2) and establishing an optimal descriptor matching method in a descriptor space of the model to realize accurate matching of points on the model.
Due to the complex and various forms of the three-dimensional model which actually exists, various deformations can be experienced, and meanwhile, the limitation of the digital scanning device can cause the situation that the model has a large amount of noise and even partial loss, so that the realization of matching with high performance becomes a very challenging problem in the fields of computer graphics and vision.
At present, a spectrum point descriptor (which can be regarded as a characteristic value and a characteristic vector of a group of filters acting on a laplacian-bellter operator) established by using diffusion geometry becomes a main method for analyzing the shape characteristics of a non-rigid model due to the characteristics of high calculation efficiency, intrinsic (equidistant deformation) invariance and the like. Typical work includes thermonuclear signatures (HKS, Heat Kernel Signature) and Wave Kernel signatures (WKS, Wave Kernel Signature), but both methods fail to realize details or contour information of a comprehensive analysis model, fail to realize accurate positioning between model points, and are prone to generate matching noise. Other methods still have deficiencies in the computational efficiency, resolution, and stability of descriptors. In particular, most of these methods fail to identify the symmetry inherent in the model, which greatly reduces the accuracy of descriptor and model matching.
After establishing accurate model point descriptors, matching between models can be achieved by reducing distortion as much as possible. There are two main strategies in the current method. A method for finding matching points by using a nearest neighbor method in a descriptor space is simple and easy to implement, but the matching performance depends heavily on the used descriptor, and the method fails to consider the geometric association characteristic between the points and the points on a model, so that the discontinuity of matching is easily caused. To this end, another approach imposes constraints on the geometric relationship between the descriptors while taking into account the point descriptors as little distortion as possible. Compared with the method simply using the nearest neighbor method, the method greatly improves the matching accuracy. But the performance of the descriptors used earlier and the correct establishment of the geometric association between the points still seriously affect the accuracy of the final matching.
Disclosure of Invention
The invention aims to provide a method capable of improving the accuracy of point matching between non-rigid three-dimensional models aiming at the defects in the prior art.
The invention provides a method for point matching between non-rigid three-dimensional models, which comprises the following steps:
step one, establishing an anisotropic spectrum manifold wavelet descriptor;
and step two, taking the descriptor established in the step one as descriptor constraint of the model point, adopting the thermonuclear relation of each point on the model as point pair relation constraint, establishing a target function, and realizing optimal matching between the model points.
In one embodiment, the step of establishing the anisotropic spectral manifold wavelet descriptor in the step one comprises the following steps:
step (1), calculating an anisotropic Laplace-Belleville matrix of the model and carrying out generalized characteristic decomposition,
step (2), defining anisotropic spectrum manifold wavelet transform,
and (3) establishing an anisotropic spectrum manifold wavelet descriptor.
Further, in the step (1), an anisotropic Laplace-Bell Meter matrix Lαθ＝-A-1Bαθ，
Wherein A ═ diag (a)1,…,aN) For a diagonal matrix, the elements on the diagonal represent the neighborhood area of the corresponding vertex,
matrix BETAαθ＝(bij) Is a weight matrix, wherein
Figure RE-GDA0002369069950000031
Generalized characteristic decomposition, solving equation Bαθφαθ,k＝λαθ,kAφαθ,k，
Obtaining the characteristic value
Figure RE-GDA0002369069950000032
And feature vector set
Figure RE-GDA0002369069950000033
Accordingly, in step (2), the wavelet generating kernel function is:
Figure RE-GDA0002369069950000034
for1 is less than or equal to x is less than or equal to 2, and the corresponding scale function generating kernel function is
Figure RE-GDA0002369069950000035
Where γ is set equal to the maximum value of g (x).
Further, in step (3), the descriptor at vertex i of the mesh is denoted as ASMWD (i),
Figure RE-GDA0002369069950000036
wherein the content of the first and second substances,
Figure RE-GDA0002369069950000037
Figure RE-GDA0002369069950000038
Figure RE-GDA0002369069950000039
Figure RE-GDA00023690699500000310
in the specific implementation, in the second step,
given a grid model X and a grid model Y, calculating an anisotropic spectral manifold wavelet descriptor of each point X and each point Y to obtain a matrix FX,
Figure RE-GDA00023690699500000311
Selecting time t according to formula
Figure RE-GDA00023690699500000312
Computing a point-to-point relationship matrix K determined by a thermokernelX,
Figure RE-GDA00023690699500000313
Establishing an optimization objective function for point descriptors and point pair relationships:
Figure RE-GDA0002369069950000041
and (6) solving.
The invention firstly establishes an anisotropic spectrum manifold wavelet descriptor in the early stage and then adopts the thermonuclear relation as point pair relation constraint. Compared with the existing method, the anisotropic spectrum manifold wavelet descriptor has the advantages of equidistant deformation invariance, capability of distinguishing the intrinsic symmetry of the model, high resolution capability and positioning capability, high calculation efficiency and compact structure; the thermonuclear relation is used as point-to-point relation constraint, and the calculation efficiency and stability are superior to those of other methods adopting geodesic distance; therefore, the method is ensured to be definite in calculation, robust in result and accurate in matching.
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FIG. 1 is a schematic flow diagram of a preferred embodiment of the present invention.
Fig. 2 is an anisotropic spectral manifold wavelet descriptor property display diagram (equidistant deformation invariance).
Fig. 3 is an anisotropic spectral manifold wavelet descriptor property display diagram (implicit symmetry differentiation).
FIGS. 4(a) -4 (f) are schematic diagrams illustrating the performance comparison result between the descriptor in this embodiment and other descriptors in the prior art.
Fig. 5 is a diagram illustrating comparison between the matching method of the present embodiment and the conventional matching method.
Fig. 6(a) -6 (d) are schematic diagrams illustrating visual comparison of matching results of the matching method in the present embodiment and the existing matching method.
Detailed Description
This embodiment discloses a high performance three-dimensional model matching method, which we call ASMWD-PMF (product manifold spatial filtering based on anisotropic spectral manifold wavelet descriptor). The method can accurately realize the matching between the deformation models with approximate equal distances, has high calculation efficiency and performance far superior to that of the existing matching method, and can provide important technical guarantee for the subsequent application of the three-dimensional model.
As shown in fig. 1, in specific implementation, the detailed calculation steps are as follows:
1. an anisotropic spectral manifold wavelet descriptor is established.
The descriptor algorithm can be specifically divided into the following 3 steps:
1.1 calculating the anisotropic Laplace-Bell meter matrix of the model and the characteristic decomposition thereof;
an anisotropic laplacian-belter operator is first calculated, and a three-dimensional model discrete form is generally represented as a triangular grid M (V, E, F) including a vertex set V ═ 1., N }, an edge set E and a patch set F { (i, j, k) }, and a vector
Figure RE-GDA0002369069950000042
Representing a function defined on a grid.
Given the anisotropy parameter α and the rotation angle θ, the principal curvature direction V on each patch is first calculatedM,VmAnd establishing an orthogonal coordinate system on the patch
Figure RE-GDA0002369069950000051
Where n is the unit normal on the patch. Calculated in a coordinate system UijkTensor matrix of lower representation
Figure RE-GDA0002369069950000052
Vector quantity
Figure RE-GDA0002369069950000053
Representing a unit vector pointing from the triangle vertex i to the vertex j, defining an edge ekjAnd ekiH of (A) to (B)θ-Inner product
Figure RE-GDA0002369069950000054
Sparse matrix L with discrete representation of anisotropic Laplace-Belleville operators as NxNαθ＝-A-1BαθWherein A ═ diag (a)1,…,aN) For a diagonal matrix, the elements on the diagonal represent the neighborhood area of the corresponding vertex. Matrix BETAαθ＝(bij) Is a weight matrix, wherein
Figure RE-GDA0002369069950000055
For the above anisotropic Laplace-Bell Mite matrix LαθPerforming generalized eigen decomposition, i.e. solving equations
Bαθφαθ,k＝λαθ,kAφαθ,k(2)
The obtained characteristic value
Figure RE-GDA0002369069950000056
And corresponding feature vectors
Figure RE-GDA0002369069950000057
They are called the frequency and fourier basis functions on the manifold, respectively. Correspondingly, Fourier coefficients of the signal f
Figure RE-GDA0002369069950000058
Can be calculated from the following formula:
Figure RE-GDA0002369069950000059
in particular, the Fourier coefficient of the impulse function at point n is
Figure RE-GDA0002369069950000061
1.2 anisotropic spectral manifold wavelet transform;
an anisotropic spectral manifold wavelet transform on a manifold is first created. The classical wavelet transform essentially filters the fourier coefficients in the frequency domain, so that an anisotropic spectral manifold wavelet transform can be defined by means of the frequency domain defined by the anisotropic laplace-bellter-metric operator signature system.
In practical application, J discrete scales are sampled firstly
Figure RE-GDA0002369069950000062
At point m, has a dimension sjAnisotropic spectral manifold wavelet function of
Figure RE-GDA0002369069950000063
Is defined as
Figure RE-GDA0002369069950000064
Corresponding scale function at the point
Figure RE-GDA0002369069950000065
Is composed of
Figure RE-GDA0002369069950000066
We define the anisotropic spectral manifold wavelet transform of a signal as the inner product of the signal f and the corresponding wavelet function, i.e.
Figure RE-GDA0002369069950000067
Figure RE-GDA0002369069950000068
In the above wavelet function calculation, the wavelet kernel function g (x) and the scale kernel function h (x) correspond to a filter function in the frequency domain, and the cubic B-spline wavelet generation kernel function may be selected
Figure RE-GDA0002369069950000069
Corresponding scale function generating kernel function
Figure RE-GDA00023690699500000610
Where γ is set equal to the maximum value of g (x).
Discrete wavelet scale is represented by sjFrom LαθThe upper bound of the spectrum is determined. The maximum and minimum dimensions are respectively taken as sJ＝x1/λmax、λmin＝λmaxK, k > 0, the remainder sJ≤sj≤s1Equal logarithmic distribution, with parameter k being selected by the user.
1.3 Anisotropic spectral manifold wavelet descriptor
And selecting the multi-scale anisotropic spectrum wavelet transform coefficient of the pulse function positioned at the point for the vertex i on the grid to encode the multi-scale geometric information of the neighborhood around the point.
Because of the fact that
Figure RE-GDA0002369069950000071
We can get it
Figure RE-GDA0002369069950000072
Figure RE-GDA0002369069950000073
For each direction θ, we first find the descriptor in that direction, i.e.
Figure RE-GDA0002369069950000074
To capture the geometric information in each direction in full, we use L equally distributed rotation angles, i.e. let
Figure RE-GDA0002369069950000075
Finally, we concatenate the descriptors of each sub-direction into a high-dimensional vector of (J +1) xL, which is the descriptor at the vertex i of the mesh, i.e., the descriptor
Figure RE-GDA0002369069950000076
The pseudo code for calculating the ASMWD in this step is shown in the table below:
Figure RE-GDA0002369069950000077
Figure RE-GDA0002369069950000081
2. matching of descriptors.
After calculating the ASMWD at each model vertex, we need to design an optimization algorithm to find the best match between model points and points using the descriptors. Our method will now be described in detail by taking the matching between two models as an example.
Given the mesh models X and Y, their q-dimensional ASMWD at each point is first computed, denoted separately as
Figure RE-GDA0002369069950000082
In the discrete case, the ASMWD at all vertices on the two meshes form a matrix FX,
Figure RE-GDA0002369069950000083
The match between the two models can be viewed as a mapping: [ pi ]: { x1,x2,…,xN}→{y1,y2,…,yNWhere i ═ 1,2 … …, N.xi,yiRespectively, X, Y. The mapping table is shown as a permutation matrix n e {0,1}N×NThe matrix satisfies pi T1 ═ 1, where 1 is the column vector for which all elements are 1. Spatially representing the NxN permutation matrix as PN。
The relationship between pairs of points on each model is represented using a thermal kernel. The thermonuclear is a solution of a thermal diffusion equation on manifold X and can be expressed as an exponential relation of an intrinsic operator, namely the thermonuclear relation of X and y at a time t is
Figure RE-GDA0002369069950000084
Which represents the amount of heat transferred from point x to y at time t. The thermonuclear relations between all points on X, Y are represented as K in discrete casesX,
Figure RE-GDA0002369069950000085
An optimization objective function is now established for the point descriptors and point pair relationships:
Figure RE-GDA0002369069950000086
the solution of the optimization problem can be converted into a corresponding convex optimization problem, and the convex optimization problem can be solved.
The model matching algorithm pseudo-code of this step is shown in the following table
Figure RE-GDA0002369069950000087
Figure RE-GDA0002369069950000091
The deformation experienced by the joints when the human body has different postures is listed.
As shown in figure 2, a three-dimensional model of a human body is established, joints of different parts are selected as selected points on the model, and each point has equidistant deformation invariance.
As shown in fig. 3, the human three-dimensional model has the intrinsic symmetry differentiation, and the use of anisotropy means that the direction information intrinsic in the model is considered when the descriptor is established, so that the descriptor can differentiate the symmetry, the defects of the general spectrum descriptor are overcome, and the matching requirement is more met.
Compared with the performance of the existing descriptors, the method (ASMWD-PMF) has better resolution capability and positioning capability than the existing method, and the matched area is only concentrated in a small neighborhood of the correct matched point.
As shown in fig. 4a to 4f, compared with the prior art, the method has the advantages of high calculation efficiency and compact structure; the descriptor dimension is lower than the dimension of the similar method, the complex function representing the model characteristic does not need to be calculated in advance, and the solving process does not need to depend on a high-dimensional equation set or the calculation of geodesic distance and the like.
The temporal statistics of the different descriptors for the different vertex models, as shown in the table below,
Figure RE-GDA0002369069950000092
the comparison of the matching results of the method with other methods is shown in fig. 5, and the visual comparison of the matching results of the method with other methods is shown in fig. 6(a) -6 (d), which uses the correspondence obtained by different methods to transfer the color information and connect the corresponding points by straight lines.
The algorithm has the advantages that: by adopting the ASMWD established earlier by the technical scheme, the optimization result is not easy to generate the whole block turning of the model matching region; the thermonuclear relation is adopted as point-to-point relation constraint, and the calculation efficiency and stability are superior to those of other methods adopting geodesic distance; in summary, the present technology provides a new matching method for overcoming the shortcomings of the existing three-dimensional model matching method in terms of accuracy and computational efficiency. The method has the advantages of definite calculation, robust result and accurate matching, and is suitable for processing complex models with equidistant deformation. The technology can greatly promote the development of subsequent application and related industries based on three-dimensional model matching.

Claims (6)

1. A method of point matching between non-rigid three-dimensional models, the method comprising the steps of:
step one, establishing an anisotropic spectrum manifold wavelet descriptor;
and step two, taking the descriptor established in the step one as descriptor constraint of the model point, adopting the thermonuclear relation of each point on the model as point pair relation constraint, establishing a target function, and realizing optimal matching between the model points.
2. The method of point matching between non-rigid three-dimensional models according to claim 1, characterized in that: when the anisotropic spectrum manifold wavelet descriptor is established in the first step, the method comprises the following steps:
step (1), calculating an anisotropic Laplace-Belleville matrix of the model and carrying out generalized characteristic decomposition,
step (2), defining anisotropic spectrum manifold wavelet transform,
and (3) establishing an anisotropic spectrum manifold wavelet descriptor.
3. The method of point matching between non-rigid three-dimensional models according to claim 2, characterized in that: in step (1), the anisotropic Laplace-BellmiterMatrix Lαθ＝-A-1Bαθ，
Wherein A ═ diag (a)1,…,aN) For a diagonal matrix, the elements on the diagonal represent the neighborhood area of the corresponding vertex,
matrix BETAαθ＝(bij) Is a weight matrix, wherein
Figure RE-FDA0002369069940000011
Generalized characteristic decomposition, solving equation Bαθφαθ,k＝λαθ,kAφαθ,k，
Obtaining the characteristic value
Figure RE-FDA0002369069940000012
And feature vector set
Figure RE-FDA0002369069940000013
4. The method of point matching between non-rigid three-dimensional models according to claim 2, characterized in that: in the step (2), the wavelet generating kernel function is as follows:
Figure RE-FDA0002369069940000014
the corresponding scale function generating kernel function is
Figure RE-FDA0002369069940000021
Where γ is set equal to the maximum value of g (x).
5. The method of point matching between non-rigid three-dimensional models according to claim 2, characterized in that: in step (3), the descriptor at the mesh vertex i is denoted as ASMWD (i),
Figure RE-FDA0002369069940000022
wherein the content of the first and second substances,
Figure RE-FDA0002369069940000023
Figure RE-FDA0002369069940000024
Figure RE-FDA0002369069940000025
Figure RE-FDA0002369069940000026
6. the method of point matching between non-rigid three-dimensional models according to claim 1, characterized in that: in the second step, the first step is carried out,
given a grid model X and a grid model Y, calculating an anisotropic spectral manifold wavelet descriptor of each point X and each point Y to obtain a matrix FX,
Figure RE-FDA0002369069940000027
Selecting time t according to formula
Figure RE-FDA0002369069940000028
Computing a point-to-point relationship matrix K determined by a thermokernelX,
Figure RE-FDA0002369069940000029
Establishing an optimization objective function for point descriptors and point pair relationships:
Figure RE-FDA00023690699400000210
and (6) solving.
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