JP2006039622A - Curved surface reconstruction method for three-dimensional graphic, and curved surface reconstruction program for three-dimensional graphic - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a curved surface reconstruction method for a three-dimensional graphic and a program therefor which can complement the defective part of a graphic from ambient data. <P>SOLUTION: This curved surface reconfiguration method comprises: an input process for inputting data obtained by discretely sampling a distance place with sign with normal information, which is configured of a distance with sign from a predetermined sample point 11 to an object surface 10 expressed with a vector p and a normal to a nearest point 12 in a three-dimensional space; and an approximation process for carrying out the local secondary approximation of the distance place with sign with normal information expressed with the components of a predetermined normal vector n, distance s with sign and symmetric matrix M to a normal vector n<SB>i</SB>in the neighborhood of the i-th sample point deviated only by a vector Δp<SB>i</SB>from the sample point 10 expressed with the vector p and a sample value s<SB>i</SB>of the distance with sign. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a method for reconstructing a curved surface of a three-dimensional figure and a program for reconstructing a curved surface of a three-dimensional figure to complement a missing part of the shape model.

  In recent years, various object shape measuring means have been proposed due to improvements in performance, miniaturization, and cost reduction of computers, devices, and sensors. The optical measurement method includes a passive method using only a light receiving sensor and an active method using light and the like to project the response.

  As a general framework for complementing the so-called 2.5-dimensional data deficiency that can be described as a graph curved surface, there is energy minimization by regularization. By representing the fit and continuity to the data in the form of an energy function and minimizing the energy function so that the compatibility constraints are balanced, a continuous plausible surface can be generated. In computer vision, many methods for complementing the sparse shape obtained by stereo or the like in this framework have been proposed (for example, see Non-Patent Documents 1 to 3 below).

  There has been proposed a method of describing a shape by a signed distance field (SDF) obtained by discretely sampling a signed distance from the object surface as a scalar field. Since a smooth surface can be reconstructed from the signed distance field as an isosurface with a distance of 0, many methods have been proposed for integrating the entire shape model as an average of a plurality of shape measurement data (for example, Non-patent documents 4 to 6 below).

  Further, it has been shown that SDF can be used not only for integration of distance data but also for determination of erroneous measurement points and simultaneous alignment of a plurality of distance data (for example, see Non-Patent Document 7 below). In this method, assuming that the data shape is roughly aligned to some extent in advance by some method, robust shape integration processing with outlier processing and alignment to the integrated shape are repeated alternately. The integrated shape model described by the SDF can be automatically generated while determining subtle erroneous measurement points and data missing portions that cannot be understood unless they are overlapped. From the integrated shape described by SDF, the shape description by polygon can be reconstructed by simple processing.

  As a method for integrating a plurality of distance images, there has been proposed a method of attaching three types of labels, object (inside), surface (surface), and space (outside), to a voxel (see, for example, Non-Patent Document 8 below). In this method, an unmeasured portion can be determined as a portion where the inside and the outside are in contact with each other without sandwiching the surface. In order to apply this method, not only the positional relationship between the object and the sensor but also the optical characteristics need to be known. Since the shape description by voxels is not smooth, surface restoration via SDF is proposed, and the missing part can be simplified by classifying voxels as empty (external), near surface (surface), and unsenen (internal) in the same way. It is shown that it can be filled (for example, refer to the following non-patent document 5).

  In order to reconstruct a smooth shape model from coarse or noisy distance data, a method using a level-set method has been proposed (for example, see Non-Patent Document 9 below). The level-set method describes the propagation in the space of the interface with a differential equation. The field update rule can be created using the term of time differentiation, and the propagation speed depending on the curvature can be set. And smoothing of the field. Applying the level-set method, smooth interpolation of the surface in the signed distance field (for example, see Non-Patent Document 10 below) and resolution of mismatches occurring when integrating multiple distance data (for example, Non-Patent Document below) 11).

  Since the level-set method generally sets a curvature-dependent value as the propagation speed of the level-set, the magnitude of the generated field gradient is not always constant as in the SDF. Further, the curvature of the curved surface can be obtained by using the normal information of the SDF (see, for example, Non-Patent Document 12 below).

W. E. L. Grimsson, “An implementation of a computational theory of visual surface information”, “Computer Vision, Graphics, and Image Processing”, 22, 1983, p. 39-69 D. Terzopoulos, “The computation of visible-surface representations”, IEEE Trans. PAMI, 1988, Vol. 10, No. 4, p. 417-438 R. Szeliski, “Fast surface interpolation using hierarchical basis functions”, IEEE Trans. PAMI, 1990, Vol. 12, No. 6, p. 513-528 A. Hilton, A .; J. et al. Staudart, J .; Illingworth and T.W. Windeatt, “Reliable surface restructuring from multiple range images”, Proc. ECCV 96, 1996, p. 117-126 B. Curless and M.M. Levoy, “A volumetric method for building complex models from range images”, Proc. SIGGRAPH 96, 1996, p. 303-312 M.M. D. Wheeler, Y .; Sato and K.M. Ikeuchi, “Consensus surfaces for modeling 3D objects from multiple range images”, Proc. ICCV97, 1997, p. 917-924 Takeshi Masuda, “Generating Shape Models from Multiple Distance Images by Matching Signed Distance Fields”, IPSJ Transactions on Computer Vision and Image Media, 2003, p. 30-40 Y. Sakaguchi, H .; Kato, K .; Sato and S. Inokuchi, “Acquisition of endurance surface data based on fusion of range data”, Trans. IEICE, E-74, No. 10, 1991, p. 3417-3422 R. T.A. Whitaker, “A level-set approach to 3d reconstitution from range data”, “International Journal of Computer Vision”, Vol. 29, No. 3, 1998, p. 203-231 J. et al. Davis, S.M. R. Marschner, M.M. Garr and M.M. Levoy, “Filling holes in complex surfaces using volumetric diffusion”, Proc. 3DPVT 2002, 2002, p. 428-437 R. Sagawa and K.A. Ikeuchi, “Taking consensus of signed distance field for compiling unobservable surface”, Proc. 3DIM 2003, 2003, p. 410-417 T.A. Masuda, “Surface curvature estimation from the signed distance field”, Proc. 3DIM 2003, 2003, p. 361-368

  A sensor has a limited range that can be measured from the principle of measurement and geometrical arrangement, and it is generally difficult to measure the entire surface of an object at once. For example, when the surface is specularly reflected, is a black portion, or the optical path is blocked by the front portion, the reflection of the projected light does not return, and therefore cannot be measured by the active method.

  A uniform surface with no texture cannot be measured by passive methods. The measurable spatial range is limited by the focal length of the imaging sensor and the interval between the sensors. The sensor can be installed only in a free space, and the viewpoint is limited not only on the ground, platform, wall surface, and other objects existing around the object, but also on the object itself. Also, only the front side of the object can be photographed at one time.

  When the position of the viewpoint is limited in this way, a very large number of viewpoints may be required to measure the entire surface even with a simple shape. In practice, it is necessary to supplement missing portions that are impossible to measure, require a lot of labor to measure, or remain unmeasured.

  The present invention provides a method for reconstructing a curved surface of a three-dimensional figure and a program for reconstructing a curved surface of a three-dimensional figure, which can complement a missing part of the figure from surrounding data in order to solve the above-described problems caused by the prior art. For the purpose.

In order to solve the above-described problems and achieve the object, a method for reconstructing a curved surface of a three-dimensional figure according to the present invention is a shaped surface that is a directional curved surface from a predetermined sample point represented by a vector p in a three-dimensional space. The data obtained by discretely sampling the signed distance field with normal information composed of the signed distance to and the normal to the nearest point is input, and the vector Δp from the sample point represented by the vector p _i just shifted i th sample point of the normal vector n _i and the signed distance of the sample values s _i in the vicinity of, the law represented by components in a predetermined normal vector n, the signed distance s and symmetric matrix M A local quadratic approximation of a signed distance field with line information is performed.

  According to the present invention, it is possible to supplement a region left unmeasured in shape data obtained by integrating distance data measured from a plurality of directions as post-processing. Using a signed distance field with normal information as a shape representation, it is possible to estimate missing parts by iteratively performing a local quadratic function approximation. Can be obtained.

  A three-dimensional figure curved surface reconstruction program according to the present invention causes a computer to execute the above-described three-dimensional figure curved surface reconstruction method.

  According to the present invention, the above-described method for reconstructing a curved surface of a three-dimensional figure can be executed using a computer.

  According to the method of reconstructing a curved surface of a three-dimensional figure and the program for reconstructing a curved surface of a three-dimensional figure according to the present invention, it is possible to complement a missing part of a shape model by applying a local quadratic function of SDF sequentially. To clarify the relationship between the fitted quadratic function model and the curvature of the curved surface, and to complement not only the signed distance but also the normal field with a smooth curvature. There is an effect that can be done.

  Exemplary embodiments of a 3D graphic curved surface reconstruction method and a 3D graphic curved surface reconstruction program according to the present invention will be described below in detail with reference to the accompanying drawings. First, the outline of the SDF will be explained in the explanation of (signed distance field), the relationship between the SDF and the curvature will be explained in the explanation of (signed distance field and curved surface curvature), and a method for complementing the unmeasured part will be derived. Experimental results are shown in the explanation of (Experiment).

(Embodiment)
(Signed distance field)
First, the definition will be described. FIG. 1 is a diagram showing an outline of parameters describing a signed distance field (SDF). When the object surface 10 is regarded as an oriented curved surface S, the closest point 12 which is a point on the curved surface closest to the sample point 11 represented by an arbitrary vector p in a three-dimensional space at a general position is uniquely determined. be able to. An exception is a point that is equidistant from a plurality of points on a curved surface, such as the center of a sphere, the axis of a cylinder, or an equidistant surface from two planes, generally on a Voronoi boundary of the surface. The closest point 12 closest to the object surface 10 from the sample point 11 is expressed by the following equation (1).

If the curved surface is differentiable, the direction of the normal SN [p, S] at the nearest point 12 and CP [p, S] is p (vector value. The character represented by p hereinafter is not a scalar value but a vector value. Is in the same direction as The length of the normal is expressed by the following formula (2):

If the object is directed from the inside to the outside, the signed distance of the point p becomes a scalar field represented by the following equation (3):

And as the following equation (4),

The gradient is a vector field formed by normals. The SDF sample at the point p is expressed by the following equation (5).

The signed distance at a point x near the sample point 11 (vector value. Characters expressed as jx without meaning in the following coordinates are not scalar values but vector values) is (6) It is linearly approximated as

Hereinafter, CP [p, S], SN [p, S], and SD [p, S] are respectively expressed as c, n, and s (both are vector values) unless one object shape is described and there is no fear of confusion. Abbreviated).

  Next, the distance between SDFs will be described. When the SDFs of two data shapes A and B are given in the vicinity of the sample point 11, the distance between them is defined by the integral of the square of the difference of the SDF linearly approximated in the vicinity of the point p (for example, See Reference 7 above.) Using the fact that the SDF is linearly approximated by the equation (6) within the vicinity of the sample point 11, the distance between the SDFs is expressed as the following equation (7).

Where w _n is determined by the shape and size of the neighborhood, but if the length of one side in the vicinity is a cube [delta], becomes w n ₌ δ ^2/12.

  Next, discretization will be described. SDF is conceptually a continuous quantity, but it must be discretized in order to perform calculations. A cube having a length W of one side that can include the target curved surface is taken as a sample region, and its end point is expressed by the following equation (8).

The sample points 11 are set on lattice points at equal intervals in the sample region, and the sample interval δ of the level 1 is set so as to satisfy the relationship of W = 2 ^l δ. The following equation (9) expresses the integer coordinates,

The coordinates of the sample point 11 designated by this equation (9) are taken by equation (10).

  FIG. 2A is a two-dimensional schematic diagram illustrating a process of sampling the SDF of the object surface 10 at the sample points 11 taken on the lattice. FIG. 2-2 is a diagram showing an outline of a vector from the sample point 11 to the nearest point 12. If the nearest point 12 is at the end of the data shape, the sample is invalid.

  Since the SDF sample of the entire space requires a huge storage capacity, only those whose absolute value distance from the data shape is within a threshold value (T = 2δ in implementation) are stored. In addition, when the nearest point 12 is at the end of the data measurement region, the normal line n does not coincide with the direction to the sample point 11, so that it is not adopted as an effective sample.

  Next, curved surface reconstruction will be described. The Marching Cube method is often used as a method for converting the integrated shape generated as SDF into shape representation using a general-purpose triangular patch. In this method, the normal information given to SDF is also effective. Used to reconstruct curved surface. FIGS. 3-1 to 3-4 are diagrams illustrating the process of curved surface reconstruction. The sample point 11 is taken by the following equation (11).

  Therefore, the center point of the cubic region having the sample point 11 as a vertex is at ci = o + iδ. For each cubic region, the signed distance at the center point can be obtained by linearly extrapolating the SDF of the sample point 11 at the vertex to the center point. The sign of the average value can determine whether the cubic region should be approximated to the inside or outside of the object.

Although the boundary between the inner and outer cube regions is a polyhedron of a set of squares (see FIG. 3A), the coordinate value of each vertex is set to the nearest point 12 (CP _p ) corresponding to the sample point 11 while maintaining the structure. By replacing with, a smooth surface can be obtained (see FIGS. 3-2 and 3-3). Furthermore, when normal information SN _p of the sample point 11 is also added, it is possible to draw more smoothly (see FIG. 3-4). This method also effectively uses normal information contained in the SDF, and does not cause problems of vertex coordinate interpolation and connection indefiniteness as in the ordinary Marching Cube method (for example, Non-Patent Documents 4 to 6 above). reference.).

(Signed distance field and curved surface curvature)
First, the curved surface curvature will be described. By using the normal information of SDF, the curvature of the curved surface can be obtained (for example, see Non-Patent Document 12). FIG. 4 is a diagram showing an outline of parameters used for calculating the curvature from the SDF in the vicinity of the sample point 11. Let c and n (both are vector values) be the nearest point 12 and the normal line at the sample point 11. Assume that the nearest point 12 and the normal change to c + Δc and n + Δn when the sample point 11 changes to p + Δp. If the projection matrix in the direction of the normal n is the following equation (12),

The tangential plane component of the displacement is given by the following equation (13).

The relationship between the tangential plane components of the displacement is expressed by the following equation (14) using a symmetric matrix K.

Since the matrix K is symmetric, six independent components _{k = (k xx, k yy} , k zz, k yz, k zx, k xy) has, in the following for each neighborhood sample of the actual data to the equation of (15) An equation can be established.

  Since there are more equations than variables, the least squares solution of the components of the symmetric matrix K can be obtained using singular value decomposition. Since the result estimated by solving the equation (15) is real symmetric, it can be diagonalized as the following equation (16).

Here, Δ is a diagonal matrix of real eigenvalues κ ₁ , κ ₂ , and κ ₃ , and the following equation (17) is an orthogonal matrix.

Since transition delta _n c _i and delta _n n _i is a tangent plane components, there are eigenvectors satisfying the following equation (18).

Since the corresponding eigenvalue κ ₃ can take an indefinite value, κ ₃ = 0 can be set, and the matrix K is represented by a symmetric matrix of rank 2. The matrix K gives the correspondence between the surface elements on the curved surface and the surface elements on the Gaussian surface, and the eigenvalues κ ₁ and κ ₂ are the main curvatures, and w ₁ and w ₂ are the main directions. Since the above calculation is equivalent to multiplying by the projection matrix P _n , the normalized estimated value of the matrix K representing the curvature is obtained by Expression (19).

And the relationship of following Formula (20) is formed.

Here, || · || _F is Frobenius norm. When the main direction is unnecessary, the main curvature can be obtained directly from the normalized estimated value component of K using this relational expression. The above calculation method is applicable as long as there is a set of Δ _n c _i and Δ _n n _i .

Next, the parallel curved surface will be described. FIG. 5 is a diagram showing an outline of parameters in a plane element on a parallel curved surface. A surface element on the curved surface in the vicinity of the nearest point 12 is expressed by the following equation (21). Here, _{w 1} and _{w 2} is the main direction.

The change of the normal line accompanying the displacement of the point on the surface element is expressed by the following equation (22) using the main curvatures κ ₁ and κ ₂ .

A change that occurs with respect to a point c _t = c + tn that is a distance t away from the nearest point 12 in the normal direction n with the displacement on the curved surface is expressed by the following equation (23).

Accordingly, the area element of the parallel curved surface that is separated from the curved surface by the distance t is given by the following equation (24).

Since the change of the normal line is common, the curvature on the parallel curved surface is obtained by the following equation (25) from the ratio of the coefficients in equations (22) and (23).

The volume of thickness ± T sandwiching the unit surface elements on the curved surface is given by the following equation (26) when the radius of curvature is T or more.

This calculation result shows that the surface density of the nearest point 12 on the curved surface depends on the Gaussian curvature when the sample points 11 are distributed at a uniform density in a space within a certain width from the curved surface. ing. When the thickness is T = 2δ and K = ± 1 / (2δ) ² , which is the most bent state on the assumption of this calculation, the number of sample points 11 per unit area changes by ± 1/3. Degree.

Next, supplementation of the signed distance field by quadratic approximation will be described. Up to this point, it is shown that the SDF can be linearly interpolated at the point x in the vicinity of the sample point 11, and the relationship between the nearest point CP [x, S] and the normal SN [x, S] in the vicinity of the sample point 11 is linearly approximated. It was shown that the curvature of the curved surface can be obtained. When complementing missing data, the SDF value is not given at the sample point 11 and it is necessary to estimate from the surrounding SDF. Signed distance field is to be approximated by a quadratic expression represented by the following formula for the displacement Delta] p _i of locally sample point 11 (27).

As represented by Expression (3), since the normal is the gradient of the signed distance, the change in the normal can be approximated as the following Expression (28).

  Since the normal field is a gradient of the scalar field, there is no vortex and M must be a symmetric matrix. The secondary approximation parameters M, n, and s for the sample point 11 are collectively described as QSDF (p). The quadratic approximation coefficient QSDF (p) is applied to the actual data so as to minimize the following equation (29).

^{Here, d} 2 (·) is the distance between SDF given in equation (7), samples in _{w i} is a weighting factor for each sample, SDF _[p i, S] is _{p i} on the curved surface S The value, QSDF (p), is the SDF value in the vicinity of the sample point 11 estimated by the local quadratic approximation model obtained for p.

Similar to equation (15), the equation of equation (30) can be established for six independent components, normals, and signed distances of the symmetric matrix K. Here, w _ni = w _i w _n . Find the least squares solution by singular value decomposition of the left-hand side matrix. Since it is in the form of simultaneous equations, it is not necessary to classify the patterns as in Non-Patent Document 2, even when the entire neighborhood is not filled. Since there are 10 variables, at least 10 sample points 11 are required to solve.

About the relationship of the following formula | equation (31) obtained from Formula (1),

When both sides are partially differentiated by p, equation (32) is obtained.

The following equation (33) is obtained from equation (5),

From the equation (17), the following equation (34) is obtained,

Moreover, since it is as the following formula (35),

It can be expressed as the following formula (36).

Using the intrinsic expansion of equation (7) and the curvature on the parallel curved surface of equation (14), the following equation (37) is obtained.

In other words, M is a symmetric matrix of rank 2, and its eigenvalue is the curvature on the parallel curved surface given by equation (14), and the curvatures κ ₁ and κ ₂ of the curved surface can be obtained using the signed distance s. The weighting coefficient is determined by the following equation (38) using the curved surface curvature thus obtained. However, w (Δp _i ) is a weight determined by the way of neighborhood.

A coefficient obtained by solving Expression (30) is expressed as Expression (39).

Henceforth, it uses together with the notation of M (^), n (^), and s (^) for convenience, respectively. These coefficients are normalized so as to satisfy the above conditions. First, the size of the normal is normalized as in the following formula (40).

Next, projection is performed so that the eigenvectors of the matrix M in the quadratic form coincide with the normal line, and the equation (41) is obtained.

Finally, the signed distance is adjusted using the normalized normal line and the quadratic form as shown in Equation (42).

  According to the above calculation, even if there is no SDF sample at the central sample point 11, QSDF (p) can be estimated from the neighboring SDF, so that the missing value of SDF can be complemented. Since the second-order approximation is attempted locally at each position in the space, a third derivative, that is, a field where the change in curvature is small is estimated.

When the general inverse matrix of the left-side matrix of Equation (30) is obtained, it can be seen that the coefficients of M, n, and s are obtained by applying a three-dimensional linear filter to the data n _i and s _{i on the} right side. That, M of the second derivative with respect to the first-order derivative and _{s i} are coefficients (^) for _{n i} sum, n (^) is the sum of the first derivative of the mean and _{s i} of _{n i,} s (^) is the _{n i} It is obtained as the sum of linear extrapolation and s _i smoothing. If the influence of normalization is not taken into account for the time being, the influence of the error of the data n _i and d _i on the estimated value of the QSDF is the degree of the linear combination.

(About experiment)
Next, an algorithm for complementing unmeasured parts using SDF is proposed. At sites such as stereolithography, interactive 3D shape editing tools are already used, and operations such as filling with a flat surface or leaving the surface with a certain thickness are not difficult if the amount is small. The part that is missing as a curved surface is used.

  According to the erroneous measurement point removal, integration, and alignment performed using SDF, local linear approximation is performed by using not only the signed distance of the scalar field but also the normal that is the vector field of the gradient. In the method of the present embodiment, quadratic approximation is used to perform complementation so as to maintain differential continuity. By obtaining a quadratic approximation, a differential geometric feature such as a curved surface curvature can also be obtained. Inconsistencies that could not be resolved by linear approximation can also be corrected. The algorithm is as follows.

Step 1: Input initial SDF into input SDF: org.
Step 2: Output SDF: ext ← org
Step 3: Extends ext.
Step 4: QSDF is calculated from neighboring org or ext at each sample point of ext, and ext is updated.
Step 5: Return to Step 3 and repeat as many times as necessary.
Step 6: ext is the result.

  org does not change after initialization. In Step 3, a region where ext is expanded with a neighboring mask is set. In Step 4, if there are samples in both org and ext for the same sample point, org is given priority so that ext is not deviated from the initial value. A sample whose distance | d | is equal to or greater than T is calculated by calculating the QSDF, and is suppressed so that the calculation object does not reach the entire space.

  The application result to data will be described. FIG. 6A is a diagram illustrating a result of application of the present embodiment to the ellipsoid in the initial state. FIG. 6B is a diagram illustrating a result of application of the present embodiment to the ellipsoid after 10 repetitions. FIG. 6-3 is a diagram illustrating a result of application of the present embodiment to the ellipsoid after 20 repetitions. FIG. 6-4 is a diagram illustrating a result of application of the present embodiment to the ellipsoid after 50 repetitions. FIG. 6-5 is a diagram showing a change in average curvature when the number of sample points, fitting error, and initial state is 1.

Half of the ellipsoid is given as the initial state, and the state during the repetition is displayed. It is clear that the overall shape is an ellipsoid, but it can be seen that a smooth surface is generated by filling in the missing portion only by repeating the local calculation without using knowledge of the global shape. In this experiment, w (Δp _i ) is a Gaussian distribution with a standard deviation δ, and a neighborhood range (near 33) as shown in the following equation (43) is used.

When plotting the number of sample points and the change in fit error with the number of iterations, it is observed that both initially increase, but tend to decrease as the hole fills in about 40 iterations. The average curvature decreases roughly monotonically.

  FIGS. 7-1 to 7-4 are diagrams showing the results of applying the method of the present embodiment to a model generated from distance data obtained by measuring a horse with a heel. In the generation, the above-described method of Non-Patent Document 7 is used, and extrapolation processing in the method is not performed. Although it was difficult to measure between the legs, it was found that the method was complemented with a continuous surface by repeating the method 20 times. The error and curvature were not as simple as the ellipsoid. The sample point increased from 248k to 282k by the complementary processing, and the calculation time was about 1 hour and a half with a single CPU of 1.7 GHz.

  The missing part of the shape model was complemented by SDF sequential local quadratic function fitting. The relationship between the quadratic function model to be applied and the curvature of the surface was clarified, and it was possible to complement not only the signed distance but also the field of smooth curvature, including the normal field. It is also possible to remove extreme outliers in combination with robust statistics. In addition, this method can be extended to multi-resolution processing, and it can be expected that more stable and high-speed processing can be performed by setting a plurality of locality levels.

  As described above, the method of reconstructing a curved surface of a three-dimensional figure and the program for reconstructing a curved surface of a three-dimensional figure according to the present invention are useful for complementing a missing part of a shape model. It is suitable for complementary processing when a portion that is difficult to measure occurs in a combined three-dimensional shape measurement system or the like.

It is a figure which shows the outline | summary of the parameter which describes a signed distance field (SDF). It is a two-dimensional schematic diagram showing the process of sampling the SDF of the data shape S at the sample points taken on the lattice. It is a figure which shows the outline | summary of the vector from a sample point to the nearest point. It is a figure which shows the process of curved surface reconstruction. It is a figure which shows the process of curved surface reconstruction. It is a figure which shows the process of curved surface reconstruction. It is a figure which shows the process of curved surface reconstruction. It is a figure which shows the outline | summary of the parameter used for calculation of the curvature from SDF of the vicinity of a sample point. It is a figure which shows the outline | summary of the parameter in the surface element on a parallel curved surface. It is a figure which shows the application result to the ellipsoid of this Embodiment in an initial state. It is a figure which shows the application result to the ellipsoid of this Embodiment after repeating 10 times. It is a figure which shows the application result to the ellipsoid of this Embodiment after repeating 20 times. It is a figure which shows the application result to the ellipsoid of this Embodiment after 50 repetitions. It is a figure which shows the change of the average curvature when a sample score, fitting error, and an initial state are set to 1. It is a figure which shows the result of applying the method in this Embodiment to the model produced | generated from the distance data which measured the horse of the Minowa. It is a figure which shows the result of applying the method in this Embodiment to the model produced | generated from the distance data which measured the horse of the Minowa. It is a figure which shows the result of applying the method in this Embodiment to the model produced | generated from the distance data which measured the horse of the Minowa. It is a figure which shows the result of applying the method in this Embodiment to the model produced | generated from the distance data which measured the horse of the Minowa.

Explanation of symbols

10 Object surface 11 Sample point 12 Nearest point

Claims (6)

In a three-dimensional space, a signed distance field with normal information is formed by a signed distance from a predetermined sample point represented by a vector p to a shape surface that is a directional curved surface and a normal to the nearest point. An input process for inputting discrete sampled data;
Wherein the sample points represented by the vector p in the vector Delta] p _i shifted by the i-th normal vector n _i and the signed distance of the sample points in the vicinity of the sample values s _i, a predetermined normal vector n, the distance signed an approximation step for performing local quadratic approximation of a signed distance field with normal information represented by components of s and symmetric matrix M;
A method for reconstructing a curved surface of a three-dimensional figure. In the approximating step, local quadratic approximation is performed by obtaining a least squares solution of the following linear simultaneous equations with a weighting factor of a signed distance of the i-th neighborhood as w _i and a weighting factor of a normal component as w _ni. A method for reconstructing a curved surface of a three-dimensional figure.
  The sampled data is graphic data in a three-dimensional space, and the approximation step creates complementary data of graphic data in the three-dimensional space by applying the local quadratic approximation, and the complementary data 3. The method of reconstructing a curved surface of a three-dimensional figure according to claim 1 or 2, wherein the figure is drawn in the three-dimensional space by using. In a three-dimensional space, a signed distance field with normal information is formed by a signed distance from a predetermined sample point represented by a vector p to a shape surface that is a directional curved surface and a normal to the nearest point. An input process for inputting discrete sampled data;
Wherein the sample points represented by the vector p in the vector Delta] p _i shifted by the i-th normal vector n _i and the signed distance of the sample points in the vicinity of the sample values s _i, a predetermined normal vector n, the distance signed an approximation step for performing local quadratic approximation of a signed distance field with normal information represented by components of s and symmetric matrix M;
A program for reconstructing a curved surface of a three-dimensional figure, wherein the computer is executed. In the approximating step, local quadratic approximation is performed by obtaining a least squares solution of the following linear simultaneous equations with a weighting factor of a signed distance of the i-th neighborhood as w _i and a weighting factor of a normal component as w _ni. The three-dimensional figure curved surface reconstruction program according to claim 4.
The sampled data is graphic data in a three-dimensional space, and the approximation step creates complementary data of graphic data in the three-dimensional space by applying the local quadratic approximation, and the complementary data 6. The computer program for reconstructing a curved surface of a three-dimensional figure according to claim 4 or 5, wherein the figure is drawn in the three-dimensional space by using.