CN106952339A - A kind of Points Sample method theoretical based on optimal transmission - Google Patents
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Abstract
The present invention relates to a kind of Points Sample method theoretical based on optimal transmission, belong to 3-D view processing technology field.This method comprises the following steps:S1:Input cloud data to be sampled, mass distribution is carried out to the cloud data of input;S2:Random down-sampling is carried out to cloud data, sampled point set is obtained, i.e., random that n point is chosen from original point cloud data as sampled point, wherein n represents the quantity of sampled point;S3:Utilize the optimal transmission plan between optimal transmission principle solving cloud data and sampled point;S4:Transmission plan calculates transmission cost according to required by S3, and adjusts the position of sampled point;S5:Step S3 and S4 are repeated to the sampled point after adjustment, until transmission cost convergence, final sampled result is obtained.This method has preferable robustness and adaptability, and sampled result possesses good making an uproar property of indigo plant, can efficiently solve insensitive for noise present in the current point cloud method of sampling, poor for applicability, the low problem of efficiency.
Description
Technical Field
The invention belongs to the technical field of three-dimensional image processing, and relates to a point cloud sampling method based on an optimal transmission theory.
Background
Since the advent of three-dimensional scanning equipment, three-dimensional scanning technology has rapidly developed, and current technology has enabled rapid digitization of three-dimensional information in the real world. The surface shape information of an object in the real world can be quickly converted and stored as point cloud data. The point cloud is a discrete point set scattered in a three-dimensional space, and the obtained point cloud data may contain some additional information (such as color of points, normal vector, etc.) in addition to three-dimensional coordinate position information according to different scanning devices. For a point cloud, which is a discrete data set, researchers have developed various researches, such as denoising, rendering, surface reconstruction, and the like. The point cloud data is the biggest difference from the mesh data in that it does not have connection information of edges and faces. Therefore, the point cloud data greatly simplifies the data expression and provides more flexible space for a plurality of applications in the field of computer graphics. However, due to the influence of the sampling environment, the point cloud data often has a large amount of noise or data loss, and these problems will bring challenges to relevant research and application.
Sampling is an important research direction in computer graphics, and has wide application in the aspects of dot rendering, halftone, visualization and the like. Currently, the techniques used for sampling can be generally divided into three categories: the first type is the poisson disc sampling method, which randomly generates poisson discs within a sampling domain according to a given sampling radius, and if a currently generated poisson disc collides with a previously generated poisson disc, it is rejected, otherwise it is accepted, and the process is repeated until a continuous rejection event is observed to occur. The second type is a tiling-based sampling method whose core is to compute one or more tiles in advance and then place them next to each other to form a set of arbitrarily-sized points. The third category is relaxation-based sampling methods, which typically include two steps, generating an initial set of sample points, and using iterative optimization of point locations until convergence.
The sampling method has the advantages and the disadvantages, the principle of the Poisson disc sampling method is simple and easy to realize, but the efficiency is low, and the calculation time cannot be accurately estimated; the paving-based sampling method can generate a large sampling point set in real time, but the sampling quality is sacrificed; relaxation-based sampling methods can generate high-quality sets of sample points.
Disclosure of Invention
In view of the above, the present invention provides a point cloud sampling method based on an optimal transmission theory, which can solve the problems of the existing point cloud sampling method, such as insensitivity to noise, poor applicability, and low efficiency. The method is based on the optimal transmission theory, has strong robustness and adaptability, not only expands the application range of the traditional loose sampling method, but also reduces the calculation complexity through entropy constraint, enlarges the range of quality transmission, and can effectively solve the problems of the existing point cloud sampling method.
In order to achieve the purpose, the invention provides the following technical scheme:
a point cloud sampling method based on an optimal transmission theory mainly comprises the following steps: s1: inputting point cloud data to be sampled, and performing quality distribution on the input point cloud data; s2: randomly down-sampling the point cloud data to obtain a sampling point set, namely randomly selecting n points from the original point cloud data as sampling points, wherein n represents the number of the sampling points; s3: solving an optimal transmission plan between the point cloud data and the sampling points by utilizing an optimal transmission principle; s4: calculating the transmission cost according to the transmission plan obtained in the step S3, and adjusting the positions of the sampling points; s5: and repeating the steps S3 and S4 on the adjusted sampling points until the transmission cost is converged to obtain a final sampling result.
Further, in step S1, the input point cloud data to be sampled and the sampling points obtained by random down-sampling have initial mass distributions that respectively satisfy the following conditions:
wherein m represents the number of points in the point cloud,representing the ith point x in a point cloudiN denotes the number of sample points, p (y)j) Representing the jth point y in the set of sample pointsjThe initial mass of (a).
Further, each point x in the point cloudiInitial mass ofSolving an approximate value through a covariance matrix, wherein the approximate value measures the change of a local curved surface and reflects the characteristic information at the point, and the method comprises the following steps:
first, x is constructed as followsiCovariance matrix C of (a):
wherein,represents a distance xiThe k-th point of the nearest distance,to representk is the centroid of the nearest neighbor;
then, three eigenvalues λ of the covariance matrix C which is the minimum are calculated0≤λ1≤λ2:
C·vl=λl·vl,l∈{0,1,2}
Finally, calculating the approximate value of the mass distribution according to the obtained characteristic valueThen will beNormalization:
further, in step S3, the method for solving the optimal transmission plan between the point cloud and the sampling points using the optimal transmission theory mainly includes the following steps:
firstly, measuring the transmission cost between the point cloud and the sampling point by using the p-Wasserstein distance, and when the transmission cost is less than the threshold value, obtaining the point cloudThen, for the probability measures μ and v, the p-Wasserstein distance between them is:
where d (x, y) denotes the cost of the transfer from x to y, and pi denotes a transfer plan, all sets of transfer plans ii (μ, v) must satisfy the following condition: pi (μ, v) ═ { pi | pi (·, Ω) ═ μ, pi (Ω,) v }
Then, solving an optimal transmission plan between the point cloud and the sampling point set by using the Wasserstein gravity center, wherein the Wasserstein gravity center is defined as:
wherein X ═ { X ═ X1,x2,…,xm};
Finally, since the point cloud data and the sampling points are discrete measurements, the above formula is discretized:
wherein the transmission plan is WijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe mass of (c); | xi-yjAnd | l represents the distance measurement between the ith point in the point cloud and the jth point in the sampling point set.
Further, the Wasserstein distance is regularized by an entropy regularization term:
wherein, epsilon is a positive regularization parameter, H (pi) is the entropy of pi; then, after adding the entropy regulation term, an entropy regulation term smooth objective function is formed as follows:
further, the entropy regular term smooth objective function is solved iteratively through the following formula:
wherein,m is a distance matrix between the point cloud and the sampling points,a mass distribution matrix representing the point cloud, p (y) a mass distribution matrix representing the sample points, u and v are iteration parameters, and after the iteration is finished, the optimal transmission planning matrix is W ═ diag (u) kdiag (v).
Further, according to the calculated optimal transmission matrix W, the minimum transmission cost between the point cloud and the sampling point and the coordinate position of the movement of the sampling point are calculated, and the method specifically comprises the following steps:
firstly, calculating the minimum transmission cost between the point cloud and the sampling point according to the following formula for judging whether iteration converges or not:
wherein E represents a transmission cost; wijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe mass of (c); mijRepresenting the ith point x from the point cloudiTo the jth point y in the sample point setjThe distance of (d);
then, the coordinate position of the movement of the sampling point is calculated according to the following formula:
Y=θY|(1 θ)XWdiag(p(y)-1)
wherein X is (X)1,x2,…,xm) Indicating the coordinate position of the point cloud, Y ═ Y1,y2,…,ynDenotes the coordinate position of the sample point, and θ is a constant between 0 and 1 for controlling the moving speed of the sample point.
Further, taking the newly generated coordinate position Y of the sampling point as a starting point, and repeating the steps S3 and S4 until the transmission cost converges, so as to obtain a final sampling result.
The invention has the beneficial effects that: the invention provides a point cloud sampling method based on an optimal transmission theory, which is characterized in that on the basis of the optimal transmission theory, the optimal transmission plan between a point cloud and a sampling point is obtained by iteratively calculating the center of gravity of Wasserstein, and the application range of the sampling method based on relaxation is expanded. Meanwhile, a rule item is added, the calculation complexity is reduced through entropy constraint, the quality transmission range is expanded from the recent transmission to the whole domain, and a more flexible and relaxed transmission scheme is provided. The point cloud sampling method provided by the invention can effectively solve the problems of the existing point cloud sampling method.
Drawings
In order to make the object, technical scheme and beneficial effect of the invention more clear, the invention provides the following drawings for explanation:
FIG. 1 is a flow chart of a point cloud sampling method based on an optimal transmission theory according to the present invention;
FIG. 2(a) is input point cloud data to be sampled;
FIG. 2(b) shows the original point cloud quality distribution result;
FIG. 2(c) shows the final sample point distribution;
FIG. 2(d) shows the reconstruction effect of the sampling point set;
fig. 3(a) is inputting point cloud data to be sampled;
FIG. 3(b) shows the original point cloud and the reconstruction effect of the sampling point set;
fig. 3(c) shows the reconstruction result of the sampling point set and the locally enlarged sampling grid structure.
Detailed Description
Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings.
The invention provides a point cloud sampling method based on optimal quality transmission theory, as shown in figure 1, the method specifically comprises the following steps:
s1: inputting point cloud data to be sampled, and performing quality distribution on the input point cloud data;
s2: randomly downsampling the point cloud data to obtain a sampling point set;
s3: solving an optimal transmission plan between the point cloud data and the sampling points by utilizing an optimal transmission principle;
s4: calculating the transmission cost according to the transmission plan obtained in the step S3, and adjusting the positions of the sampling points;
s5: and repeating the steps S3 and S4 on the adjusted sampling points until the transmission cost is converged to obtain a final sampling result.
Specifically, the method comprises the following steps:
step 1: in the present embodiment, the point cloud data X to be sampled is input as { X ═ X1,x2,…,xmAnd in order to facilitate calculation, the sum of the initial masses of all points in the point cloud meets the following condition:
wherein m represents the number of points in the point cloud,representing the ith point x in a point cloudiThe initial mass of (a). In order to improve the robustness and applicability of the sampling method, in the embodiment, each point in the point cloud is advanced according to the position of the point in the point cloudLine quality assignment, comprising the steps of:
step 101: calculating point xiCovariance matrix C of (a):
wherein,represents a distance xiThe k-th point of the nearest distance,represents a distance xiThe centroid of the nearest k points. Where k is 25, satisfactory results are usually obtained.
Step 102: calculating three eigenvalues of the least covariance matrix C and lambda0≤λ1≤λ2:
C·vl=λl·vl,l∈{0,1,2}
Step 103: calculating an approximation of the mass distribution from the determined characteristic valuesFor convenience of calculation, willNormalization:
step 2: randomly selecting n points Y ═ Y from cloud data1,y2,…,ynAs sample points, where n represents the number of sample points. The sum of the initial masses of each point in the sampling point set meets the following condition:
in order to ensure that the sampling result has good blue noise characteristics, the quality of each sampling point is equal, so the quality distribution of the sampling points is
And step 3: the optimal transmission plan between the point cloud and the sampling points is solved by using an optimal transmission theory, and the method mainly comprises the following steps:
step 301: firstly, measuring the transmission cost between the point cloud and the sampling point by using the p-Wasserstein distance. And then solving an optimal transmission plan between the point cloud and the sampling point set by utilizing the Wasserstein gravity center. Finally, because the point cloud data and the sampling points are discrete measures, the formula is discretized, and the sampling problem is converted into the most value problem for solving the transmission cost:
wherein the transmission plan is WijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe mass of (c); | xi-yjAnd | l represents the distance measurement between the ith point in the point cloud and the jth point in the sampling point set.
Step 302: regularization of Wasserstein distance by an entropy regularization term:
where e is a positive regularization parameter and H (π) is π entropy. The entropy regularization term extends the transmission range in the conventional relaxation method from the most recent transmission to the entire domain, providing a transmission scheme that is flexible and relaxed.
Then, after adding the entropy regulation term, an entropy regulation term smooth objective function is formed as follows:
step 303: to solve equation (1), the entropy regularization term smooth objective function is iteratively solved by the following Sinkhorn iterative method:
wherein,m is a distance matrix between the point cloud and the sampling points,represents the mass distribution matrix of the point cloud, and p (y) represents the mass distribution matrix of the sample points. u and v are iteration parameters, and after the iteration is finished, the optimal transmission planning matrix is W ═ diag (u) kdiag (v).
And 4, step 4: according to the calculated optimal transmission matrix W, calculating the minimum transmission cost between the point cloud and the sampling point and the coordinate position of the movement of the sampling point, and specifically comprising the following steps:
calculating the minimum transmission cost between the point cloud and the sampling point according to the following formula:
wherein E represents a transmission cost; wijRepresenting the ith point x from the point cloudiTransmitted to a central sampling pointJ point yjThe mass of (c); mijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe distance of (c).
Then, the coordinate position of the movement of the sampling point is calculated according to the following formula:
Y=θ·Y+(1-θ)·XWdiag(p(y)-1)-1)
wherein X ═ { X ═ X1,x2,…,xmDenotes the point cloud coordinate position, Y ═ Y1,y2,…,ynDenotes the coordinate position of the sample point, and θ is a constant between 0 and 1 for controlling the moving speed of the sample point.
And 5: and taking the newly generated coordinate position of the sampling point as a starting point, and calculating the optimal transmission plan and transmission cost of the new starting point until the transmission cost is converged, thereby obtaining a final sampling result.
In this embodiment, the distance measure between the point cloud and the sampling point is measured by the transmission cost of the optimal transmission theory, so that the sampling problem becomes the optimization problem. The point cloud sampling method provided by the embodiment can effectively solve the problems of insensitive noise, poor applicability, low efficiency and the like in the current point cloud sampling method. FIG. 2(a) is raw point cloud data; FIG. 2(b) shows the mass distribution calculated from the point cloud distribution; FIG. 2(c) is a diagram illustrating a point cloud sampling result obtained by utilizing the most efficient transmission theory; fig. 2(d) sample results reconstruct the effect. Fig. 3 shows a sampling result and a reconstruction effect of the point cloud sampling method provided by the present invention under a high noise condition.
Finally, it is noted that the above-mentioned preferred embodiments illustrate rather than limit the invention, and that, although the invention has been described in detail with reference to the above-mentioned preferred embodiments, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the scope of the invention as defined by the appended claims.
Claims (8)
1. A point cloud sampling method based on an optimal transmission theory is characterized in that: the method mainly comprises the following steps: s1: inputting point cloud data to be sampled, and performing quality distribution on the input point cloud data; s2: randomly down-sampling the point cloud data to obtain a sampling point set, namely randomly selecting n points from the original point cloud data as sampling points, wherein n represents the number of the sampling points; s3: solving an optimal transmission plan between the point cloud data and the sampling points by utilizing an optimal transmission principle; s4: calculating the transmission cost according to the transmission plan obtained in the step S3, and adjusting the positions of the sampling points; s5: and repeating the steps S3 and S4 on the adjusted sampling points until the transmission cost is converged to obtain a final sampling result.
2. The point cloud sampling method based on the optimal transmission theory as claimed in claim 1, wherein: in step S1, the initial mass distributions of the input point cloud data to be sampled and the sampling points obtained by random down-sampling satisfy the following conditions:
wherein m represents the number of points in the point cloud,representing the ith point x in a point cloudiN denotes the number of sample points, p (y)j) Representing the jth point y in the set of sample pointsjThe initial mass of (a).
3. The point cloud sampling method based on the optimal transmission theory as claimed in claim 2, wherein: each point x in the point cloudiInitial mass ofSolving an approximate value through a covariance matrix, wherein the approximate value measures the change of a local curved surface and reflects the characteristic information at the point, and the method comprises the following steps:
first, x is constructed as followsiCovariance matrix C of (a):
wherein,represents a distance xiThe k-th point of the nearest distance,representing the centroid of the k neighbor;
then, three eigenvalues λ of the covariance matrix C which is the minimum are calculated0≤λ1≤λ2:
c·vl=λl·vl,l∈{0,1,2}
Finally, calculating the approximate value of the mass distribution according to the obtained characteristic valueThen will beNormalization:
4. the point cloud sampling method based on the optimal transmission theory as claimed in claim 1, wherein: in step S3, the optimal transmission plan between the point cloud and the sampling points is solved using the optimal transmission theory, which mainly includes the following steps:
firstly, measuring the transmission cost between the point cloud and the sampling point by using p-Wasserstein distance, and when p belongs to [1, ∞ ], the p-Wasserstein distance between the probability measures mu and v is as follows:
where d (x, y) denotes the cost of the transfer from x to y, and pi denotes a transfer plan, all sets of which pi (μ, v) must satisfy the following condition: pi (μ, v) { pi | pi (·, Ω) ═ μ, pi (Ω,) v }
Then, solving an optimal transmission plan between the point cloud and the sampling point set by using the Wasserstein gravity center, wherein the Wasserstein gravity center is defined as:
wherein X ═ { X ═ X1,x2,…,xm};
Finally, since the point cloud data and the sampling points are discrete measurements, the above formula is discretized:
wherein the transmission plan is WijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe mass of (c); | xi-yiAnd | l represents the distance measurement between the ith point in the point cloud and the jth point in the sampling point set.
5. The point cloud sampling method based on the optimal transmission theory as claimed in claim 4, wherein: regularization of Wasserstein distance by an entropy regularization term:
wherein, is a positive regularization parameter, and H (π) is π entropy; then, after adding the entropy regulation term, an entropy regulation term smooth objective function is formed as follows:
6. the point cloud sampling method based on the optimal transmission theory as claimed in claim 5, wherein: iteratively solving the entropy regular term smooth objective function by the following formula:
wherein,m is a distance matrix between the point cloud and the sampling points,a mass distribution matrix representing the point cloud, p (y) a mass distribution matrix representing the sample points, u and v are iteration parameters, and after the iteration is finished, the optimal transmission planning matrix is W ═ diag (u) kdiag (v).
7. The point cloud sampling method based on the optimal transmission theory as claimed in claim 6, wherein: according to the calculated optimal transmission matrix W, calculating the minimum transmission cost between the point cloud and the sampling point and the moving coordinate position of the sampling point, and the specific steps comprise:
firstly, calculating the minimum transmission cost between the point cloud and the sampling point according to the following formula for judging whether iteration converges or not:
wherein E represents a transmission cost; wijRepresenting the ith point x from the point cloudiTo the jth point y in the set of sample pointsjThe mass of (c); mijRepresenting the ith point x from the point cloudiTo the jth point y in the sample point setjThe distance of (d);
then, the coordinate position of the movement of the sampling point is calculated according to the following formula:
Y=θY+(1-θ)XWdiag(p(y)-1)
wherein X ═ { X ═ X1,x2,…,xmDenotes the point cloud coordinate position, Y ═ Y1,y2,…,ynDenotes the coordinate position of the sample point, and θ is a constant between 0 and 1 for controlling the moving speed of the sample point.
8. The point cloud sampling method based on the optimal transmission theory as claimed in claim 7, wherein: and (5) taking the newly generated coordinate position Y of the sampling point as a starting point, and repeating the steps S3 and S4 until the transmission cost is converged to obtain a final sampling result.
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CN113762154A (en) * | 2021-09-07 | 2021-12-07 | 西安理工大学 | Part feature identification method based on point cloud data set |
CN113762154B (en) * | 2021-09-07 | 2024-04-09 | 西安理工大学 | Part feature recognition method based on point cloud data set |
CN113936070A (en) * | 2021-10-14 | 2022-01-14 | 厦门大学 | Two-dimensional point cloud shape regional parallel reconstruction method based on optimal transmission theory |
CN113936070B (en) * | 2021-10-14 | 2024-06-04 | 厦门大学 | Two-dimensional point cloud shape and region parallel reconstruction method based on optimal transmission theory |
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