JPH11328442A - Method for forming surface of three-dimensional object - Google Patents

Abstract

PROBLEM TO BE SOLVED: To form a smooth interpolation face by a simple processing based on an outline plotted on the cross-section of a three-dimensional object even when topology is different for each cross-section of the three-dimensional object. SOLUTION: Outline expression data for specifying the outline of a three- dimensional object for each cross-section are inputted in a first step 50. Second- dimensional density distribution is generated for each cross-section in a second step 51. A three-dimensional field is formed based on the generated second- dimensional density distribution in a third step 52. An interpolation face is formed based on the formed three-dimensional field by using a threshold processing.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to three-dimensional modeling, and more particularly, to a method for forming a surface of a three-dimensional object based on a contour appearing in a cross section of the three-dimensional object.

[0002]

2. Description of the Related Art Contour interpolation processing is important in various fields such as medical tomography (CT), magnetic resonance imaging (MRI), visualization of terrain information, and three-dimensional modeling. It is becoming. By using such interpolation processing, for example, as shown in FIG. 1, the surface of the three-dimensional object 10 can be reconstructed based on a plurality of contour lines 11a to 11c appearing in a two-dimensional cross section. In general, to obtain such a contour, a three-dimensional object is sliced by a limited number of cutting planes parallel to each other.

[0003] Each section of a three-dimensional object that has been sliced has
For each section, one or more closed contour lines appearing indicating the boundary between the three-dimensional object and the outside world. For example, in FIG. 1, one closed contour line 11b appears for one cross section, and two closed contour lines 11c appear for other cross sections. By using interpolation between contours, the surface of the three-dimensional object can be generated based on the contours appearing in the two-dimensional cross section of the three-dimensional object. The surface of the three-dimensional object generated in this way is called an “interpolated plane”. An interpolation plane passing through two contour lines appearing in two cross sections connects the two contour lines to each other. For example,
It can be said that the partial surface 12a connects the contour lines 11a and 11b. It can be said that the partial surface 12b connects the contour lines 11b and 11c. If the three-dimensional object is branched between two cross sections, a single contour line 11b appearing on one cross section will be connected to two contour lines 11c appearing on the other cross section.

[0004] It is desired that such interpolation between contour lines generates a smooth interpolated surface. From a mathematical perspective,
Smoothness relates to the differentiability of the interpolated surface. If differentiation is possible at each point on the interpolation plane, it can be said that the interpolation plane is smooth. Since such differentiation cannot be applied to discrete data such as a binary bitmap image used in an embodiment of the present invention described later, smoothness is defined as visual smoothness. Can be It is only necessary that the visualized interpolation plane look natural. In general, a three-dimensional object obtained by interpolation between contour lines should not be angular unless the original three-dimensional object lacks smoothness. Even if the original 3D object lacks smoothness,
It is desirable that an interpolation plane as smooth as possible is formed. When the three-dimensional object is branched, that is, when the two cross sections have different topologies, it is particularly difficult to form a smooth interpolation surface.

In the interpolation between contour lines, it is desired that the connection relation between the contour lines is controlled. Even when cross sections parallel to each other are relatively displaced, it is desired that the interpolation plane describe which contour line should be connected between the two cross sections. Such a demand increases especially when the three-dimensional object is branched.

Furthermore, it is desirable that the interpolation between the contour lines is simple and is realized with as few calculation processes as possible.

[0007]

Heretofore, various methods have been proposed for forming a surface from a plurality of contours. For example, H. Fuchs et al., "Optimal Surfac
e Reconstruction from Planar Contours "(1977
Year; Communications of the ACM, 20 (10); pp693
According to the approach proposed in 702), the interpolation problem is replaced by a problem relating to the minimum cost cycles of the oriented toroidal graph. However, this technique can only be used to generate surfaces that connect contour lines one-to-one.
In other words, this method cannot be applied to the case where the topology of each cross section is different, such as when the three-dimensional object involves branching, that is, when the number of contour lines differs for each cross section.

In addition, Shinagawa et al., The Homotopy Model:
A Generalized Model for Smooth Surface Generation
from Cross Sectional Data "(1991; The Visu
al Computer 7; pp. 72-86), a discrete toroidal graph can be generalized to a continuous toroidal graph. In this technique,
Points corresponding to the two contours are connected based on homotopy. According to such a method, the problem of a triangle caused when a discrete toroidal graph is used is solved. A smooth surface can be obtained based on homotopy. However, this method can also be used only when connecting contour lines one-to-one.

Further, D. Meyers et al., Surfac
es from Contours "(1992; ACM Transactions o
According to the method proposed in n Graphics 11 (3); pp. 228 to 258), branching of a three-dimensional object described by a contour line can be handled. However, this approach can be complicated. In particular, when the shape of the contour is complicated, the complexity increases.

Furthermore, M.I. Kass et al., "Snake
s: Active Contour Models "(1988; Internati
onal Journal Computer Vision 1 (4);
The movable contour model (active contour m) proposed in 31)
According to odels), the energy function of the contour is minimized. On the other hand, L. D. Cohen says, "On Active Contou
r Models and Baloons "(1991; CVGIP: Image Un
derstanding 53 (2); pp. 211 to 218) further developed this movable contour model. However, it is difficult for these movable contour models to determine a weighting function included in the energy function.

Furthermore, Komatsu, "Reconstruction of
Three-Dimensional Object Based on Shrinking Deform
ation of its Boundary "(1995; Ph.D. Thesi
s, The Graduate School of The University of Toky
In the method proposed in o), a physical simulation method is used to automatically generate a smooth interpolation surface. However, this technique can only be applied to contour lines that gradually shrink.

Furthermore, S.I. P. "Sh
ape-Based Interpolation of Multidimensional Object
s "(1992; IEEE Transactions on Medical Ima
ging9 (1); T. Herman
"Shape-Based Interpolation"(1992;
IEEE Computer Graphics and Applications 12 (3); p
According to the method proposed in pp. 69-79), a two-dimensional gray scale map is applied to the contour line for each section. In this approach, interpolation is applied to the grayscale map, resulting in an interpolated surface. The gray scale map is created using the minimum distance from each contour line. Linear interpolation or spline function interpolation is used for the interpolation. This method can also be used only when connecting contour lines one-to-one.

In addition, for example, in J. F. "A
Generalization of Algebraic Surface Drawing "(1
982; ACM Transactions 2; pp 235-256)
The basic three-dimensional computer graphic tool is provided by the metaball method described in US Pat. According to this metaball method, a reference point (sour
ce points) are set to any space. In the space, a "field" is formed for each reference point. The reference point can be compared to a charge, where the field can be compared to an electric field (electric field). The "field" calculated at any point in space is defined by the "field" generated at all reference points. In visualizing the “place” of space,
A surface that includes the three-dimensional region is formed based on the three-dimensional field. Points within the three-dimensional area are given a "field" value greater than the threshold, and points outside the three-dimensional area are given a "field" value less than the threshold. In this specification, such an approach is referred to as thresholding-based surface formation. Generally, such techniques produce a smooth surface. Two or more metaballs that have a positive “field” and are close together interact with each other and are visualized using appropriate thresholds to form a three-dimensional region that encompasses and connects those metaballs Create a “place” that forms In the metaball method, the function of the “field” associated with each reference point is not limited. Any function may be applied. If the function of "field" is smooth,
A smooth "field" or surface can be formed.

An object of the present invention is to provide a surface forming method capable of constructing a smooth surface shape based on a plurality of contour lines. Further, the present invention provides a surface which can reliably connect contour lines of different cutting planes on a surface formed by interpolation when a different number of contour lines appear on each cross section, such as when the topologies of the cross sections are different. It is an object to provide a forming method. Another object of the present invention is to provide a surface forming method which has excellent control characteristics and can clearly describe the connection relationship between contour lines. Still another object of the present invention is to provide a surface forming method capable of restoring a real object as precisely as possible. Still another object of the present invention is to provide a surface forming method which is simple and can be realized with a relatively small amount of calculation processing.

[0015]

In order to achieve the above object, according to a first aspect of the present invention, a three-dimensional object is defined based on cross-sectional shape data describing a contour of the three-dimensional object on a cutting plane for specifying a cross section of the three-dimensional object. A step of generating a two-dimensional density distribution for each cutting plane; and a step of performing interpolation between the two-dimensional density distributions and forming a surface of a three-dimensional object between a pair of cutting planes. A method for forming a surface of a three-dimensional object is provided.

According to the second aspect of the present invention, a threshold value is set for each cutting plane along the contour line based on the sectional shape data describing the contour line of the three-dimensional object on the cutting plane that specifies the cross section of the three-dimensional object. Generating a two-dimensional density distribution to be arranged;
Interpolate between the two-dimensional density distributions, at least one
Forming a three-dimensional field in a space between the pair of cutting planes, and forming a surface of the three-dimensional object using threshold processing based on a threshold specified in the three-dimensional field. A method for forming a surface of a three-dimensional object is provided.

[0017]

An embodiment of the present invention will be described below with reference to the accompanying drawings.

As shown in FIG. 2, according to the surface forming method according to the present invention, in a first step 50, contour expression data for specifying a contour of a three-dimensional object is input for each cross section. In the second step 51, a two-dimensional density distribution is generated for each section. In a third step 52, a three-dimensional field is formed based on the generated two-dimensional density distribution. In a fourth step 53, an interpolation plane is formed based on the formed three-dimensional field using threshold processing. The interpolation surface thus formed may be displayed on a display screen in the fifth step 54, or may be stored in a storage device or a recording medium.

The details of the first to fifth steps 50 to 54 will be described later. However, in each of the steps 50 to 54, specific conditions for finally forming a smooth interpolation surface must be applied. Moreover, in each of steps 50 to 54,
Certain conditions must be applied for superimposing each contour line with the finally formed interpolation plane.

The two-dimensional density distribution used when forming a three-dimensional field preferably satisfies the following requirements. First
In addition, it is desirable that the two-dimensional density distribution is smooth. Second, it is desirable to show a higher density inside the three-dimensional object than outside the three-dimensional object. Third, it is desirable that the density on the contour line be fixed to an arbitrary threshold. In addition, it is desirable that the maximum value of the density in the three-dimensional object increases as the contour line increases. One specific example of such a density distribution is shown in FIG. 3, for example. This density distribution 14
According to, the maximum density 14a appearing on the relatively large contour line 13a has a larger value than the maximum density 14b appearing on the contour line 13b smaller than the contour line 13a. However,
Any density distribution may be used as long as the conditions and requirements described above are satisfied. Methods for generating a density distribution satisfying the above conditions and requirements include, for example, two methods such as a physical simulation and a pseudo metaball filter method. Details of these methods will be described later.

In superposing the contour and the interpolated surface formed by the threshold value processing in the fourth step 53, the three-dimensional field formed in the third step 52 may show a specified value on the contour. desirable. Fourth step 5
This is because such a specific value can be used as a threshold value in the threshold value processing of No. 3. In addition, according to such a three-dimensional field, when the interpolation plane is formed in the fourth step 53, it is desirable that the contour lines are reliably connected. For example, if linear interpolation is used, these two conditions are surely satisfied in forming a three-dimensional field. According to this linear interpolation, when forming a field in a space between two cutting planes, interpolation is performed linearly between density distributions defined in each cutting plane. According to such linear interpolation, as long as the cross-sections appearing in each cutting plane when viewed from the vertical direction of the two cutting planes at least partially overlap, the finally obtained interpolation plane is the same as the outline of the cross-section. Can be connected. This is because, in a region where two cross sections overlap, the density value is considered to be higher than the threshold value in any of the cutting planes. As a result, in a space sandwiched by regions where the cross sections overlap, a field value larger than the threshold is obtained according to the linear interpolation. Further, according to the linear interpolation, the processing time can be reduced.

Various known techniques can be used to form an interpolation plane using threshold processing. Such techniques include, for example, W.S. E. FIG. Lorensen and H.C. E. FIG. Marching Cubes: A High R by Cine
esolution 3D Surface Reconstruction Algorithm ''
(1987; Computer Graphics 21 (4); pp163)
To 169). According to the marching cube method, an algorithm is provided that executes processing at high speed and easily in generating a triangular polygon for a surface. The algorithm defines that points with a density greater than the threshold are located inside the three-dimensional object to be constructed. In this marching cube method, the space is divided into small cubes. Based on the vertices of the cube cut at the surface of the three-dimensional object, an optimal triangular polygon is finally generated.

FIG. 4 shows a specific example of the physical simulation method. In this physical simulation method,
The elastic grid 21 is superimposed on the outline 22 appearing in the cross section. It is desirable that the elastic grid 21 draw rectangular coordinates. However, it is not limited to rectangular coordinates. Here, it is assumed that charges are set at each lattice point. Positive unit charges are placed on lattice points 23 existing inside the three-dimensional object. On the other hand, a negative unit charge is placed on the lattice point 24 existing outside the three-dimensional object. At this time, the lattice points 25 (star marks) existing within a predetermined distance from the contour line 22 are fixed to the cross section. Other lattice points 23 and 24 can move in a direction orthogonal to the cross section. The grid lines 26 connecting adjacent grid points can extend from the unloaded length established when located on the cross-section. The elastic modulus of each grid line 26 is made uniform throughout the elastic grid 21. The smaller the size of each grid, the more accurate a contour, that is, a smooth contour can be reproduced by discrete grid points.

The elastic grid 21 on which electric charges are set as described above
When an electric field is applied from a direction perpendicular to the cross section, the lattice points 23 and 24 that are not fixed on the cross section move in a direction perpendicular to the cross section until the force acting from the electric field and the elastic force of the grid lines 26 are balanced. Go to When a balance is established between the force of the electric field and the elastic force of the grid lines 26, the shape of the elastic grid 21 will represent a two-dimensional density distribution.

By using such a physical model of the elastic lattice, the shape of the elastic lattice 21 can be simulated. For example, it is assumed that the distance between an arbitrary grid point (i, j) and a cross section is represented by x _ij . Grid point (i, j)
Four grid points (i-1, j) (i + 1, j) adjacent to
Assuming that the distance between (i, j-1) (i, j + 1) and the cross section is specified similarly, the force acting on the grid point (i, j) is

(Equation 1) Is represented by Here, E indicates a force acting from an electric field. k indicates a constant proportional to the elastic modulus of the lattice line.
l indicates the length of the grid line when no load is applied. The shape of the elastic lattice 21 can be simulated by using such accurate physical equations. Here, the following equations approximating such physical equations are used.

[0026]

(Equation 2) According to this simplification, it can be seen that the force acting on one grid point on another grid point is proportional to the distance specified between the two grid points in the direction orthogonal to the cross section. The solution of the above simultaneous equations can be derived by a direct solution such as LU decomposition. Generally, such solutions require enormous storage capacity and enormous computational processing. For example, to simulate the shape of an elastic lattice having a lattice number of 200 × 200, 4000
A solution of 40000 simultaneous equations must be derived using a 0x40000 matrix. Here, in simulating the shape of the elastic lattice, an iterative calculation process for deriving an approximate solution of the simultaneous equations is used.

According to the iterative calculation process according to this embodiment,
Only the sign of F _ij is considered. For example, if F _ij is positive, the coordinate value x _ij of the grid point (i, j) is increased by one unit in the next calculation process. On the other hand, if F _ij is negative, the coordinate value x _ij of the grid point (i, j) is reduced by one unit in the next calculation process. Assuming that the elastic lattice initially shows a flat shape, a smooth distribution of x-coordinate values can be obtained as shown in FIG. 5 when the calculation processing is repeated a specified number of times. The repetition of the calculation process may be terminated when the solution converges, for example, the change caused by the iterative calculation process falls below a predetermined value. In addition, the repetition of the calculation process may be terminated when the repetition of the specific number of times is completed.

According to such a method, it is not necessary to solve the simultaneous equations. In addition, the coordinates of the calculated or recorded grid points are in integer form. These features will dramatically improve processing speed. However, other techniques may be used to simulate the shape of the elastic lattice 21 as long as it is obvious to those skilled in the art. Such other techniques often involve undesirable conditions in computer processing compared to iterative calculations.

The x-coordinate value obtained by the simulation is
It extends from the negative minimum value to the positive maximum value around the threshold “0”. Such an x-coordinate value can be normalized to a positive region extending from “0” to “1” around the threshold “0.5”, for example. In order to achieve such normalization, the minimum value and “0” and the maximum value are each set to “0”.
A monotonous and smooth mapping process such as a spline function corresponding to “0.5” and “1” may be performed.

FIG. 6 shows a specific example of the pseudo metaball filter method. In this pseudo metaball filter method,
The density given to a point on the cross section is distinguished inside and outside the three-dimensional object and depends on the distance from the contour. As is clear from FIG. 6, the cross section can be represented by, for example, a binary bitmap image. "0" is set for the pixel 32 located outside the contour line 31 of the three-dimensional object.
Is given a value. On the other hand, the pixel 33 located inside the outline 31 is given a value of “1”. Here, the following N × N (N is an odd number) transformation matrix 34 is defined. Each element of the transformation matrix 34 is set to a value corresponding to the distance from the center element. The value of the element decreases as the distance increases. It is desirable that the elements located at the same distance from the central element be given the same value. The pixel size of the conversion matrix 34 is set to be the same as that of the bitmap image.

In calculating the density distribution at the pixel P,
The conversion matrix 34 is superimposed on the bitmap image so that the central element matches the pixel P. Subsequently, a total value is calculated. When the pixel P is located inside the contour of the three-dimensional object, that is, when the pixel P is given a value of “1”, the transformation matrix 3 located inside the contour is
The total value of each element of No. 4 is calculated. That is, the total value of each element of the transformation matrix 34 superimposed on the “1” value of the bitmap image is calculated. On the other hand, when the pixel P is located outside the contour of the three-dimensional object, that is, when the pixel P is given a value of “0”, each of the conversion matrices 34 located outside the contour is The sum of the elements is calculated. That is, the total value of each element of the transformation matrix 34 superimposed on the “0” value of the bitmap image is calculated. In addition, the sign of the total value is inverted. As a result of such processing, the larger the pixel position is inside the contour of the three-dimensional object, the larger the positive total value is given, and the further the pixel position is outside the contour line, the larger the negative total value is. Become. At this time, if the threshold value is fixed to “0”, the normalized total value can be used as the density of each pixel P.

However, as will be apparent to those skilled in the art, the conversion matrix is a continuous conversion profile (continuous con-
version profile), where the sum is an approximation of the integral of the entire section. Therefore, such a continuous conversion profile or integration method may be used.

Also in this case, the total value is, for example, a threshold value “0.5”.
Can be normalized to a positive region extending from “0” to “1” around Since the maximum value is equal to the sum of the elements included in the transformation matrix and the minimum value is equal to the negative sum, normalization can be simply realized by a linear transformation matrix. However, a monotonous and smooth mapping process such as a spline function may be performed.

According to such a pseudo metaball filter method, calculation processing can be realized for each pixel. As described above, when constructing a three-dimensional field from contours drawn on two cross sections using interpolation, the regions of the cross sections that overlap when viewed from the direction orthogonal to the cross sections are always connected to each other. That is, it exists inside the plane formed by interpolation. On the other hand, regions that do not belong to any cross section when viewed from a direction orthogonal to the cross section are not connected to each other. Therefore, it is not necessary to calculate the density distribution and the field in these areas when forming the interpolation plane. The density of the maximum value may be set for the overlapping area, and the density of the minimum value may be set for the area that does not belong to any cross section. In other words, it is sufficient that the calculation of the field is performed in a region included in one cross section and not included in the other cross section. In such processing, exclusive logic (XOR) may be used for a binary bitmap image that defines a cross section. In FIG. 7, a region calculated by exclusive logic based on the two contour lines 41 and 42 is indicated by oblique lines. In many cases, the two contours are similar to each other, so it is preferable that the calculation is performed only in the region calculated by the exclusive logic.

Next, a method of operating the connection relation of the contour lines will be described in detail with reference to FIG. As described above, when the cross sections are at least partially overlapped when viewed from the direction perpendicular to the cutting plane, the two contour lines are connected by the finally obtained interpolation plane. On the other hand, if the cross sections do not overlap at all, the contour lines are not connected. For example, as shown in FIG. 8A, the contour C1 of the plane P1 and the contour C2 of the plane P2 are not connected. This is because the projected image C1p of the contour C1 does not overlap with the contour C2 on the plane P2. Therefore, in the linear interpolation method, the three-dimensional field between the contours C1 and C2 is smaller than the threshold, and the contours C1 and C2 are not connected. To connect the contour lines C1 and C2 to each other, the planes P1 and P2 are relatively moved, for example, as shown in FIG.
As shown in (b), the contour line C is at least partially
1, C2 should just overlap. That is, the contour lines C1 and C2 may be moved within the planes P1 and P2.

[0036]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Actually, computer software for realizing the above-mentioned surface forming method was created, and the result of executing the computer software will be described in detail below. Computer with pentium 133MHz IB
An MPC / AT compatible personal computer was used. as a result,
As shown in FIGS. 9 to 14, (c) an interpolation plane could be drawn based on (a) the upper cross section and (b) the lower cross section.

9 to 12, a physical simulation method was used. The number of repetitions was set to 1000.
The grid used was a 200 × 200 grid. 9 and 10 show interpolation with simple branching. The density distribution could be calculated in a processing time of about one minute. FIG. 11 shows an interpolation based on one cross section on the upper side and one cross section on the lower side. However, two outlines appear on the upper section, and one outline appears on the lower section. The number of contour lines is different between the upper section and the lower section. Heretofore, such interpolation could not be easily realized. In FIG. 12, the connection relation of the contour lines is operated. here,
The contour 101 and the contour 103 are connected by the interpolation plane 106, and the contour 102 and the contours 104 and 105 are connected by the interpolation plane 107. The outline 101 and the outlines 104 and 105 are not connected, and the outline 102 and the outline 103 are not connected. The cross section defined by the contour 101 overlaps the cross section of the contour 103 while the contour 104,
Since they do not overlap the cross-section of 105, such a specific connection relationship can be realized. Similarly, contour line 1
Although the section of 02 overlaps the sections of the contour lines 104 and 105, it does not overlap the section of the contour line 103. That is,
It has been confirmed that the connection relationship between the contour lines can be easily operated by controlling the overlap between the cross sections.

In FIGS. 13 and 14, the pseudo metaball filter method was used. An 81 × 81 transformation matrix was used. The density function of the transformation matrix is the distance d from the center element
And specified in (40-d). FIG. 13 shows a contour similar to that of FIG. 9 described above. The same result as in FIG. 9 was obtained. The density distribution could be calculated in a processing time of about 5 seconds. That is, it was confirmed that the pseudo metaball filter method achieves a significantly higher speed than the physical simulation method described above. However, undesired results may occur depending on the size of the transformation matrix and the type of the density function. In particular, when the cross-sectional shapes of the objects are significantly different, such undesirable results tend to be obtained.
There was no difference between the two in the operability regarding the connection relationship of the contour lines.

The above surface forming method can be realized by computer processing based on computer software. Such computer software can be stored on a recording medium such as a magnetic disk or an optical disk or other storage device. However, the surface forming method according to the present invention may be realized by hardware processing based on an electronic circuit. The interpolated surface formed based on the surface forming method according to the present invention may be displayed on a screen of a computer device or another display device, or may be printed out on paper. Further, such an interpolation plane may be stored in a storage device prior to display, or may be displayed, printed out, and stored after being transferred to another computer device via a network.

It should be noted that the surface forming method according to the present invention can be used not only for connecting the contour lines appearing on the cutting planes parallel to each other as described above, but also for the contour appearing on the cutting planes inclined to each other. It may be used when connecting lines. Further, the surface forming method according to the present invention,
As described above, it can be used not only when connecting contour lines appearing on two cutting planes, but also when connecting contour lines appearing on three or more cutting planes. In this case, a three-dimensional field may be formed for each pair of cutting planes using nonlinear interpolation such as linear interpolation or spline function interpolation (third step).
In generating a two-dimensional density distribution or performing threshold processing on each cutting plane, the above-described second step or fourth step may be used.

[0041]

As described above, according to the present invention, a smooth interpolation surface can be formed by simple processing based on the contour drawn on a cross section even if the topology differs for each cross section of the three-dimensional object. Can be.

[Brief description of the drawings]

FIG. 1A is a front view of a three-dimensional object, and FIG. 1B is a plan view showing each cross section.

FIG. 2 is a flowchart schematically showing a surface forming method according to the present invention.

FIG. 3 is a diagram showing a specific example of a density distribution generated for one cutting plane.

FIG. 4 is a schematic diagram illustrating an example of a physical simulation for generating a density distribution.

FIG. 5 is a diagram showing a specific example of a density distribution generated using a physical simulation.

FIG. 6 is a schematic diagram illustrating an example of a pseudo metaball filter method for generating a density distribution.

FIG. 7 is a schematic view showing an overlap between cross sections.

FIG. 8 is a schematic diagram showing a connection relationship between contour lines.

FIG. 9 is a diagram showing a specific example of an interpolation plane generated according to the present invention.

FIG. 10 is a diagram showing another specific example of an interpolation plane generated according to the present invention.

FIG. 11 is a diagram showing still another specific example of an interpolation plane generated according to the present invention.

FIG. 12 is a diagram showing still another specific example of an interpolation plane generated according to the present invention.

FIG. 13 is a diagram showing still another specific example of an interpolation plane generated according to the present invention.

FIG. 14 is a diagram showing still another specific example of an interpolation plane generated according to the present invention.

Claims (2)

[Claims] A step of generating a two-dimensional density distribution for each cutting plane based on cross-sectional shape data describing a contour of the three-dimensional object on a cutting plane for specifying a cross section of the three-dimensional object;
Interpolating between the two-dimensional density distributions and forming a surface of the three-dimensional object between a pair of cutting planes. 2. A two-dimensional density distribution for arranging a threshold value along a contour line for each cutting plane based on cross-sectional shape data describing a contour line of the three-dimensional object by a cutting plane specifying a cross section of the three-dimensional object. Generating, interpolating between the two-dimensional density distributions, forming a three-dimensional field in a space sandwiched by at least one pair of cutting planes, and thresholding based on a threshold specified by the three-dimensional field Forming a surface of the three-dimensional object by using the method.