JP4325469B2 - Object surface flattening apparatus and program - Google Patents

Description

  The present invention relates to an object surface flattening apparatus, an object surface flattening method, and a program.

  Computer graphic (hereinafter, CG) technology is widely used as a method for generating a realistic image on a computer. In the CG technology, there is a method called surface model or polygon graphics. These approximate the surface shape of an object as an aggregate of polygons. For example, the following application examples can be considered by developing the object surface expressed by a polygonal aggregate.

  As an application example of the distortion correction method of the image scanner, for example, the technique described in Non-Patent Document 1 (Conventional Example 1) can be cited. The method described in Non-Patent Document 1 is to convert the surface shape of a photographed curved object into a planar shape on a computer. The present invention can be applied to a case where an image must be input while being curved due to restrictions at the time of image scan input even though it is originally a flat surface like the surface shape of a book.

  For example, a method described in Non-Patent Document 2 (conventional example 2) has been proposed as a method for performing texture mapping with as little distortion as possible on a complex-shaped surface. As shown in Non-Patent Document 2, texture mapping with as little distortion as possible can be performed by developing a surface that cannot be developed on a plane. Here, if the object surface in the three-dimensional space can be converted to a two-dimensional plane without changing the distance between any two points, the object surface is defined as expandable and converted to a two-dimensional plane. If it cannot, define it as undeployable.

  In general, it is known that a curved surface having a Gaussian curvature of 0 over the entire surface can be developed on a plane, and a curved surface having a non-Gaussian curvature of 0 cannot be developed on a plane. For example, the side surface of a cylinder can be developed on a plane, but a spherical surface cannot be developed on a plane because the Gaussian curvature is not zero.

  In the method described in Non-Patent Document 2, first, the distance between the points extracted from the surface of the three-dimensional shape (the length of the geodesic line) is measured, and the surface is flattened so that the magnitude of the change in the distance is as small as possible. expand. Next, the texture is mapped to the developed plane. Finally, the development from the curved surface to the plane is performed in reverse to convert the plane texture map into the original three-dimensional shape. As described above, the method of developing a three-dimensional shape on a two-dimensional plane can be used as a texture mapping method with a small apparent distortion.

  Next, as another example of a method of developing a non-developable curved surface into a plane, a method of performing development using an object surface (developable body surface) that can be developed into a plane as a medium (Patent Document 1) is shown. The method described in Patent Document 1 is a method for efficiently performing texture mapping by developing a texture surface. The method described in Patent Document 1 relates to a method for displaying an image of an arbitrary viewpoint using three-dimensional shape restoration processing from photographed two-dimensional data.

  Further, the method described in Patent Document 1 relates to a technique for reducing the amount of texture image information data in this display method. In this display method, the texture of each minute plane obtained from the two-dimensional data is not held for each individual minute plane, but the three-dimensional shape is developed on the plane, and the texture of the developed view with as few gaps as possible is obtained. The object is to hold image data efficiently by holding.

The method described in Patent Document 1 projects the feature points of a three-dimensional object onto a projection surface of a known shape surface (cylinder, column, cone, etc.) that is a developable surface and develops it on a plane. The texture mapping is performed on the development plane for the polygon. As described above, by projecting the object surface that can be developed once, a complicated three-dimensional shape can be developed on a plane. From the above three conventional examples, the following three application examples have been shown.
(1) After an image is scanned in by an image scanner, an image distorted due to scanning conditions is corrected. For example, the surface of a book is flattened.
(2) By converting a complicated three-dimensional curved surface to a two-dimensional plane, texture mapping can be easily performed on the two-dimensional plane.
(3) By converting a complex three-dimensional curved surface into a two-dimensional plane, the amount of texture information to be held can be reduced.

"Curved Document Shooting with an Eye Scanner" (The IEICE Transactions D-II, Vol. J86-D-II, No. 3, March 2003, pp.409-417) "Texture Mapping Using Surface Flattening via Multidimensional Scaling" (IEEE Trans. Visualization and Computer Graphics, vol. 8, no. 2, pp. 198-206, Apr.-June, 2002) JP 2000-20735 A

  However, (1) The method described in Non-Patent Document 1 has a problem that the object surface to be applied is limited to the surface of a book and cannot be applied to a general object surface that cannot be developed.

  (2) The method described in Non-Patent Document 2 has a problem that the calculation amount for obtaining a geodesic line from a three-dimensional shape is very large.

  (3) On the other hand, the method described in Patent Document 1 uses a method in which development is performed by once projecting the surface of an object in a three-dimensional space onto a developable surface without obtaining the distance of the geodesic line strictly. Therefore, the calculation amount is small. However, in the method described in Patent Document 1, if the three-dimensional shape of the original object cannot be developed on a plane, even if a developable surface that is as close as possible to the surface of the original object is selected, the result is eventually projected onto the developable surface. There was a problem that distortion occurred at the stage.

  Therefore, in view of the problems of the conventional example described above, the present invention has many cases where the object surface cannot be expanded when the object surface in the three-dimensional space is represented as a polygonal aggregate. An object of the present invention is to provide an object surface flattening apparatus, an object surface flattening method, and a program capable of arranging squares so that distortion is minimized on a plane.

In order to solve the above-described problems, the present invention provides an object surface flattening that develops on a plane an object surface in a three-dimensional space expressed as a polygonal aggregate on a computer. A device for extracting polygon shape information from input data; a means for extracting shape information of a peripheral polygon sharing a side with the polygon; and shape information of the polygon and the peripheral polygon From the shape information, the function for the distance between the sides shared between the polygon and the surrounding polygon on the plane is obtained, and the condition that the distance between the shared sides is minimized in the function. By applying the coefficient acquisition means for creating a linear equation for determining the position of each polygon for each of the polygons, and solving the linear equation for determining the polygons, For each polygon Characterized by comprising a polygonal position calculating means for calculating a location, and means for arranging the polygon the position is calculated.

  According to the first aspect of the present invention, the position of the input polygon can be determined by solving an expression relating to the position of the polygon, and the input polygon can be arranged on a plane. As a result, when the object surface in the three-dimensional space is represented as a collection of polygons, even if the object surface cannot be expanded, the distortion of the polygon on the plane is minimized. Can be lined up. In addition, since it is not necessary to obtain the geodesic distance required to create a development with a small distortion, the amount of calculation for obtaining the geodesic distance can be reduced.

  Further, according to the present invention, as described in claim 2, the object surface flattening device according to claim 1, further comprising means for correcting the shapes of the arranged polygons. . According to the second aspect of the present invention, when there is an overlap or a gap between the arranged polygons, it is possible to correct the shape of the polygon to create a plane without the overlap or the gap.

  Further, according to the present invention, as described in claim 3, in the object surface flattening device according to claim 1 or 2, there is further a vertex of another polygon on a side other than the vertex of the polygon. And a polygon dividing means that draws a line segment or a half line from the vertex on the side of the polygon and sets the intersection of the vertex and the side as a new vertex. . According to the invention of claim 3, since the vertex on the side of the polygon can be eliminated, even when there is a vertex on the side of the input polygon, it is the same as when there is no vertex on the side of the input polygon The object surface can be developed on a plane.

  Further, according to the present invention, as described in claim 4, in the object surface flattening device according to any one of claims 1 to 3, the input data includes an object surface in a three-dimensional space. Including data representing the shape of the polygon that constitutes the data, and the data representing the shape of the polygon is a two-dimensional vector from a common starting point to each vertex of the polygon, and the vector to the common starting point is a polygonal shape. It is a vector representing a position. According to the invention described in claim 4, by expressing the data representing the shape of the polygon by a vector representing a common starting point and a two-dimensional vector from the common starting point to the vertex, the position and shape of the polygon Can be operated separately.

  Further, according to the present invention, as described in claim 5, in the object surface flattening apparatus according to any one of claims 1 to 4, the input data includes an object surface in a three-dimensional space. Including data representing the relative positions of the polygons constituting the polygon, and the data representing the relative positions is determined by which polygons share each side with respect to each side of the polygon constituting the object surface in the three-dimensional space. Information indicating whether or not and / or information indicating which side exists in the polygon that forms the object surface in the three-dimensional space, and data of vertex indices at both ends of each side And According to the fifth aspect of the present invention, the surrounding polygon can be extracted efficiently.

Further, according to the present invention, as described in claim 6, in the object surface flattening device according to any one of claims 1 to 5, the coefficient acquisition unit includes a center polygon, Create a linear equation that indicates the relative positions of the center polygon and the surrounding polygons so that the sum of the distances between the sides to be shared is minimized between the center polygon and the surrounding polygons that share the sides. It is characterized by that. According to the sixth aspect of the invention, by obtaining the coefficient of the expression indicating the relative position between the center polygon and the surrounding polygon so that the sum of the distances between the sides to be shared is minimized, Can be arranged without gaps and without duplication.

  Further, according to the present invention, in the object surface flattening device according to claim 6, the distance between the sides is the sum of squares of the distances between the vertices to be shared. It is characterized by being multiplied by. According to the seventh aspect of the invention, the sum of squares of the distances between the vertices that should be at the same position can be weighted by the length of the side to be shared.

  Further, according to the present invention, as described in claim 8, in the object surface flattening device according to any one of claims 1 to 7, the polygon position calculation means is simultaneous by a least square method. The position of each polygon is calculated by solving a linear equation. According to the invention described in claim 8, the position of each polygon is calculated by solving simultaneous linear equations using the least square method. Thereby, the position of the polygon can be calculated using a known method of solving simultaneous linear equations by the least square method.

  Further, according to the present invention, as described in claim 9, in the object surface flattening device according to any one of claims 1 to 3, the polygon included in the input data is a two-dimensional matrix. A polygon arranged so as to be mapped to the polygon, and the means for extracting the shape of the polygon is a polygon that forms the object surface in the three-dimensional space from the polygon index corresponding to the arrangement location of the rectangle. Data representing a relative position is determined, and the coefficient acquisition unit determines a coefficient of an expression relating to the position of each polygon from the relative position of the polygon. According to the ninth aspect of the present invention, when the input polygon is a quadrangle arranged so as to be mapped to a two-dimensional matrix, the amount of input data and the amount of calculation for expansion processing are reduced. be able to.

  According to a tenth aspect of the present invention, in the object surface flattening apparatus according to the fourth aspect, the common starting point is one of polygonal vertices. According to the tenth aspect of the present invention, one piece of two-dimensional vector data relating to this vertex can be omitted from the data representing the polygonal shape.

Further, according to the present invention, the polygon shape information is input from the input data to the computer in order to develop the object surface in the three-dimensional space expressed as a collection of polygons on a plane. Extracting the shape information of the surrounding polygon that shares an edge with the polygon, the shape information of the polygon and the shape information of the surrounding polygon, and the polygon on the plane. By obtaining a function for the distance between sides shared with the surrounding polygon, and applying the condition that the distance between the shared sides is minimized to the function, and creating an equation for determining the position of each polygon by solving linear equations to determine the position of each polygon, and calculating the position of each polygon, the Characterized by a step of arranging a polygon location is calculated.

The program of the present invention, as described in claim 11 , extracts a polygon shape information from input data to a computer in order to develop an object surface in a three-dimensional space on a plane, A step of extracting shape information of a neighboring polygon sharing a side with the polygon, and an expression relating to a position of each polygon for each of the polygons from the shape information of the polygon and the shape information of the surrounding polygon The step of obtaining the coefficient, the step of calculating the position of each polygon by solving the formula relating to the position of each polygon, and the step of arranging the polygons for which the positions have been calculated are executed.

According to the eleventh aspect of the present invention, the position of the input polygon can be determined by solving a formula relating to the position of the polygon, and the input polygon can be arranged on a plane. As a result, when the object surface in the three-dimensional space is represented as a collection of polygons, even if the object surface cannot be expanded, the distortion of the polygon on the plane is minimized. Can be lined up. In addition, since it is not necessary to obtain the geodesic distance required to create a development with a small distortion, the amount of calculation for obtaining the geodesic distance can be reduced.

Further, according to a twelfth aspect of the present invention, in the program according to the eleventh aspect , the computer further causes the computer to execute a step of correcting the shapes of the arranged polygons. According to the twelfth aspect of the present invention, when there is an overlap or a gap between the arranged polygons, it is possible to correct the shape of the polygon to create a plane without the overlap or the gap. This program can be provided to a computer using a communication line or a recording medium.

According to a thirteenth aspect of the present invention, there is provided a computer-readable recording medium on which the program is recorded.

  According to the present invention, when the object surface in the three-dimensional space is represented as an aggregate of polygons, even if the object surface cannot be expanded, the polygon is reduced as much as possible on the plane. It is possible to provide an object surface flattening apparatus, an object surface flattening method, and a program that can be arranged as follows.

  Hereinafter, the best mode for carrying out the present invention will be described. First, the principle of the present invention will be described with reference to FIGS. FIG. 1 is a diagram for explaining side sharing of a polygon. FIG. 2 is a diagram for explaining a case where a vertex of a polygon is divided into two points. In the invention, it is assumed that the surface of an object in a three-dimensional space is formed by a set of polyhedrons, and the shape of a polygon constituting the polyhedron is known in advance. When arranging the polygons on a two-dimensional plane, there are cases where it is better to rotate the polygons so that there is less overlap or gap between the polygons.

  However, in the present invention, this polygon rotation is not optimized, and only the parallel movement of each polygon is optimized. As shown in FIG. 1, polygons adjacent on a three-dimensional curved surface share sides. Here, it is assumed that one side is shared only by a maximum of two polygons. For example, in FIG. 1, the side 511 is shared by the polygon 512 and the polygon 513. Even when these polygons are developed and arranged on a plane, the side 511 is preferably shared between the adjacent polygons 512 and 513.

  However, in the case of a non-developable surface, the relative positional relationship between the polygons changes at the stage of two-dimensional development, making it impossible to arrange the polygons without gaps and without overlapping. In this case, as shown in FIG. 2, the position of the side to be shared is divided into two sides 5110 and 5111.

  The present invention intends to create a developed image with small distortion by arranging polygons so that the distance between the two divided sides is as small as possible. As described above, in the present invention, a planar texture image can be generated directly from an output polygon without obtaining a geodesic line or projecting a vertex onto another developable surface. Embodiments of the present invention will be described below with reference to the drawings.

  FIG. 3 is a block diagram of the object surface flattening apparatus 100 according to the first embodiment. As shown in FIG. 3, the object surface flattening device 100 includes a polygon scanning unit 102, a surrounding polygon extracting unit 103, a linear equation creating unit 104, a polygon position calculating unit 105, an alignment unit 106, a polygon shape A correction unit 107 is provided.

  The object surface flattening apparatus 100 develops an object surface in a three-dimensional space expressed on a computer on a plane. The object surface flattening device 100 receives input data 101 representing a polygon of an object surface in a three-dimensional space from an external device such as a three-dimensional shape modeling device (not shown).

  The polygon scanning unit 102 extracts shape information of the center polygon from the input data 101. The surrounding polygon extraction unit 103 extracts the polygon information around the center polygon extracted by the polygon scanning unit 102, that is, the shape information of the surrounding polygon sharing the side with the center polygon. Based on the polygon shape information extracted by the polygon scan unit 102 and the peripheral polygon shape information extracted by the surrounding polygon extraction unit 103, the linear equation creation unit 104 performs each polygon for each polygon. Create a linear equation that includes a coefficient for the position of the square. The linear equation creation unit 104 corresponds to a coefficient acquisition unit in the claims.

  The polygon position calculation unit 105 calculates the positions of all polygons by solving this linear equation. The alignment unit 106 arranges the polygons based on the polygon positions calculated by the polygon position calculation unit 105. The polygon shape correction unit 107 corrects the shapes of the arranged polygons so that there are no gaps or overlaps, and outputs final output data 108.

  Next, the operation of each component will be described in detail. First, the input data 101 will be described. The input data 101 is data representing the shape of a polygon forming the object surface in the three-dimensional space and the relative position information of the polygon. Here, an index indicating each polygon constituting the object surface is i. When there are a total of P polygons, for example, numbers from 1 to P can be assigned to the polygons as indexes. In addition, an index indicating each side of the polygon on the object surface is set as h. When there are a total of Q sides, for example, numbers from 1 to Q can be assigned to each side as an index. Normally, each side is shared from a plurality of polygons.

  Here, in the first embodiment, it is assumed that the polygons on the object surface have the following conditions. FIG. 7A shows a state in which vertices of other polygons do not exist on sides other than the vertices of a certain polygon. FIG. 7B shows a state in which vertices of other polygons exist on sides other than the vertices of a certain polygon. In the present embodiment, a condition is imposed that the existence of vertices of other polygons is not allowed on sides other than the vertices of all polygons. This condition is equivalent to the condition that one side is shared only by a maximum of two polygons.

  In this embodiment, first, the arranging unit 106 translates the polygons while keeping the shape of each polygon, and arranges the polygons in a two-dimensional plane. Next, the polygon shape correction unit 107 corrects the polygon shape. That is, the input polygonal shape is stored before the polygonal shape correcting unit 107. For this reason, it is desirable to hold data by dividing it into a part indicating the polygonal shape and a part indicating the polygonal position. Therefore, it is assumed that the polygon i has N vertices k1, k2,..., KN.

Furthermore, in the present embodiment, the shape of the polygon i is represented using N two-dimensional vectors a _ik (k = k1, k2,..., KN) from the common starting point to each vertex of the polygon. The position of the polygon i is expressed using a two-dimensional vector v _i . The common starting point vector corresponds to a vector representing the position of the polygon.

FIG. 4 is an explanatory diagram of polygonal shapes and positions. By using these two types of vectors, as shown in FIG. 4, the coordinates of each vertex of the polygon i when the polygon i is arranged on a two-dimensional plane are
v _i + a _ik (k = k1, k2,..., kN)
It shall be expressed as

These vertices k are shared by a plurality of polygons in the three-dimensional space. However, on the two-dimensional plane, the position of the vertex k of the polygon i is the same vertex k.
v _i + a _ik
And the position of the vertex k of the polygon j is
v _j + a _jk
And will be in a different location. Similarly, each side having each vertex as an end point is not shared on the two-dimensional plane but is another side even if shared in the three-dimensional space.

Now, by designating the coordinates of each vertex of the polygon as described above, when it is desired to translate only the position of the polygon while preserving the shape of the polygon, a vector a indicating the shape of the polygon i. _It is only necessary to change the vector v _i indicating the position of the polygon i without changing _ik . It is assumed that the input data 101 has data regarding the polygonal shape thus determined, that is, a _ik . Further, the input data 101 includes data representing the relative position information of polygons constituting the object surface in the three-dimensional space. The data regarding the relative position information is used in the surrounding polygon extraction unit 103. The relative position information is (Information 1), (Information 2), and (Information 3) shown below.

(Information 1) Polygon index data sharing sides indicating which polygons share each side with respect to each side of the polygon forming the object surface in the three-dimensional space. For example, when the index indicating each side is h, it is a polygon set Fh sharing the side h.
(Information 2) Data of vertex indexes at both ends of each side. For example, when the index indicating each side is h, it is a set Gh of vertex indexes at both ends of the side h.
(Information 3) Polygon side index data indicating which side exists in each polygon constituting the object surface in the three-dimensional space. For example, when the index indicating each polygon is i, it is a set Hi of the indices of the sides of the polygon i.

  Since (Information 1) and (Information 3) can be converted to each other, only one of them needs to be held. However, when the amount of calculation of the above mutual conversion is large, it is of course possible to hold both (information 1) and (information 3). Alternatively, when polygons are regularly arranged and the relative position information of the polygon, that is, (Information 1), (Information 2), and (Information 3) described above can be easily obtained from the polygon index. The relative position information need not be included in the input data 101.

Further, as shown in FIG. 4, the position of each vertex is obtained by extending each vector a _ik from the end point of the vector v _i . Since the positional relationship between the end point of the vector v _i and the polygon i is arbitrary, by fixing the position of the polygon i and the end point of the vector v _i in advance, a _ik (k = k1, k2, .., KN), it is possible to omit having one vector as input data 101. For example, it is assumed that the common starting point of the vector a _ik indicating the position of the polygon i is the vertex k1 of the polygon i. In other words, a _ik1 = 0 is always set. In this way, it is not necessary to hold a _ik1 as input data. Therefore, one piece of two-dimensional vector data related to the vertex can be omitted. This is one example, and as another example, for example, it may of course be provided that the centroid of the polygon i is the end point of the vector v _i . This is the end of the description of the input data 101.

  Next, the polygon scanning unit 102 and the surrounding polygon extracting unit 103 will be described. The polygon scanning unit 102 extracts all the input polygons one by one. Here, it is assumed that the polygon i is extracted by the polygon scanning unit 102. The surrounding polygon extraction unit 103 extracts all polygons that share sides with the polygon i. This concludes the description of the polygon scanning unit 102 and the surrounding polygon extracting unit 103.

Next, the linear equation creation unit 104 will be described. When the polygons are arranged on a two-dimensional plane, the linear equation creating unit 104 determines the distance between the sides to be shared between a certain center polygon and the neighboring polygons sharing the side with the center polygon. A coefficient of a linear equation indicating the relative position between the central polygon and the peripheral polygon that minimizes the sum is created. Here, the following distance function is defined, and the polygon position is determined by obtaining a polygon position vector v _i that minimizes the distance function.

  First, an index indicating each vertex is k, an index indicating each side is h, and an index indicating each polygon is i. Furthermore, as shown in (Information 1), (Information 2), and (Information 3), Fh is a set of polygonal indexes sharing the side h (polygonal index set), and vertices at both ends of the side h A set of indices (vertex index set) is Gh, and a set of indexes on the sides of the polygon i (side index set) is Hi.

Here, a set Fh of polygonal indexes sharing the side h will be described with reference to FIG. FIG. 5 is an explanatory diagram of the polygonal index set Fh. For example, assume that polygon 1 and polygon 2 share side h on the surface of an object in a three-dimensional space, as shown in FIG. At this time, the polygon index set is
Fh = {1, 2}
It is. Also, as shown in FIG. 5, when the indices of the vertices at both ends of the side h are 4 and 5, the vertex index set is
Gh = {4,5}
It is.

Next, a set of indexes on the side of the polygon i will be described with reference to FIG. FIG. 6 is an explanatory diagram of the edge index set Hi. As shown in FIG. 6, it is assumed that the polygon i has sides 1, 2, 3, 4, 5, and 6 and has no other sides or vertices. At this time, the edge index set is
Hi = {1, 2, 3, 4, 5, 6}
It is.

Here, a distance function between the side h1 and the side h2 when a side h is divided into the side h1 and the side h2 on the two-dimensional plane will be described with reference to FIG. FIG. 8 is a diagram for explaining the distance function. In FIG. 8, polygon 1 and polygon 2 are adjacent in the three-dimensional space, and vertices k11 and k21 at that time exist at the same position. In addition, it is assumed that the vertex k12 and the vertex k22 exist at the same position. On the two-dimensional plane, the side h1 and the side h2 may not match. In this case, the distance between the side h1 and the side h2 is defined by the following formula (1).
(Square of distance between vertex k11 and vertex k21 + square of distance between vertex k12 and vertex k22) × (length of side h1 + length of side h2) / 2 (1)
The distance between the sides is obtained by multiplying the square sum of the distances between the vertices to be shared by the length of the side, and the square sum of the distances between the vertices that should be at the same position is the length of the side to be shared. It can be said that the distance is weighted by the distance. This is because the influence when the vertices k11 and k21 and the vertices k12 and k22 are separated is considered to be proportional to the lengths of the side h1 and the side h2.

Furthermore, the distance function described above is formulated below. A set of polygons sharing side h is Fh. Let i be the polygon index which is each element of Fh. That is, iεFh. Further, kεGh, where k is the vertex index at both ends of the side h. The position vector indicating the position of the vertex k of the polygon i when expanded on a two-dimensional plane is
v _i + a _ik
It is.

Ideally, the position vector is the same for all i (i∈Fh) and k (k∈Gh), but an error occurs on the surface of an object that cannot be developed. That is, the position vector v _i + a _ik of the vertex k (k∈Gh) of the polygon i (i∈Fh) and the position vector of the vertex k (k∈Gh) of the polygon j (j∈Fh, j ≠ i). v _j + a _jk may be different. Here, a value obtained by weighting the sum of squares of the error regarding the vertex k (kεGh) by the length of the side h is defined as a distance function Eh of the side h. The distance function Eh of the side h can be expressed as the following formula (2).

ただし、

However,

とする。ここで、ｉ＝ｊの時には、（ｖ_i＋ａ_ik）−（ｖ_j＋ａ_jk）=0であるので、（２）式は、

And Here, when i = j, (v _i + a _ik ) − (v _j + a _jk ) = 0, so equation (2) is

とできる。次に、全辺に渡って、Ｅｈを加算したものが、全体の距離関数Ｅである。

And can. Next, the total distance function E is obtained by adding Eh over the entire side.

ここで、距離関数Ｅは、ｖ_iの２次形式であり、２次の項の係数は正であるので、全ての多角形インデクスｉに関して、

Here, the distance function E is a quadratic form of v _i and the coefficient of the quadratic term is positive.

となるｖ_iが、距離関数Ｅを最小化する多角形の位置ベクトルとなる。そこで、ｉ＝ｍの時の最小化条件を求める。

V _i becomes a polygonal position vector that minimizes the distance function E. Therefore, the minimization condition when i = m is obtained.

よって、ｉ＝ｍの時の最小化条件は、

Therefore, the minimization condition when i = m is

とできる。（８）式を変形すると、以下のようになる。

And can. When the equation (8) is modified, it becomes as follows.

（９）式は、ｖ_i(ｉ＝ｍ,ｊ、ただし、ｈ∈Ｈｍ,ｊ∈Ｆｈ,かつ,ｋ∈Ｇｈ)の一次式である。

Expression (9) is a linear expression of v _i (i = m, j, where h∈Hm, j∈Fh, and k∈Gh).

In the primary equation creation unit 104, the polygon m extracted by the polygon scanning unit 102 and the polygon j around the polygon m extracted by the surrounding polygon extraction unit 103 (h∈Hm and j∈Fh). ) Of the linear equation (9) is created from the shape information. The shape information shown above is a _mk (k∈Gh, h∈Hm) and a _jk (j∈Fh, k∈Gh, h∈Hm). Further, although the above equation (9) is a two-dimensional vector equation, an equation can be created in the same manner as a normal scalar value by performing an operation for each element of the two-dimensional vector.

  As described above, a linear equation related to the polygon m could be made. The polygon scanning unit 102 scans all polygons. As a result, the linear equation creation unit 104 can create linear equations for all polygons i.

Here, the unknown information that we are seeking is v _i . Here, if the number of polygons is N, there are N unknowns v ₁ , v ₂ ,..., V _N. (In fact, since v _i is a two-dimensional vector, the unknown scalar value is 2 × N, but here the vector is treated as one number and the unknown is represented as N). In addition, a linear equation represented by equation (9) can be created for one polygon m. Since the number of polygons is N, N linear equations can be created. This simultaneous linear equation can be expressed using a matrix as follows.
Ax = b (10)
A is a coefficient matrix, and can be calculated from the following part of equation (9).

ＡはＮ×Ｎの行列である。
ｘはＮ行１列の未知ベクトルであり、（ｖ₁（ｕ）,ｖ₂（ｕ）,…,ｖ_N（ｕ））^Tである。

A is an N × N matrix.
x is an unknown vector of N rows and 1 column, and is (v ₁ (u), v ₂ (u),..., v _N (u)) ^T.

Here, v _i (u) represents the u-th component of vector v _i (u = 1, 2). Further, b is a known vector of N rows and 1 column shown below. This is shown in equation (12).

ここで、（１０）式のＡは実は特異行列であり、逆行列を持たない。そのためこのままでは（１０）式を解くことができない。これは、（９）式あるいは、（１０）式が多角形の位置ベクトルｖ_i各々の相対値のみを規定しているためである。何らかの方法によって、絶対的な位置を指定する必要がある。たとえば、インデクスが１の多角形の位置ベクトルの値を、ｖ₁＝０とすることによって、（１０）式を解くことが可能となる。このとき、（１０）式の係数行列Ａの第１列の数値は任意となる。そこで、行列Ａから第１列を抜いたＮ行（Ｎ−１）列の行列をＢとする。ｖ₁＝０としたとき、（１０）式は
Ｂｘ＝ｂ （１３）
となる。
以上で、一次方程式作成部１０４の説明を終える。

Here, A in the equation (10) is actually a singular matrix and has no inverse matrix. Therefore, equation (10) cannot be solved as it is. This is because (9) or (10) defines only the relative value of each of the polygonal position vectors v _i . It is necessary to specify an absolute position by some method. For example, by setting the value of the position vector of a polygon with an index of 1 to v ₁ = 0, it is possible to solve equation (10). At this time, the numerical value in the first column of the coefficient matrix A in the equation (10) is arbitrary. Therefore, a matrix of N rows (N−1) columns obtained by removing the first column from the matrix A is defined as B. When v ₁ = 0, the equation (10) is expressed as Bx = b (13)
It becomes.
This is the end of the description of the primary equation creation unit 104.

Next, the polygon position calculation unit 105 calculates a solution of Expression (13). Polygon position calculation unit 105 solves simultaneous linear equation (13) in which the unknown number is N-1 and the number of equations is N. Equation (13) may be solved using the least square method. For example,
x = (B ^T B) ⁻¹ B ^T b
And so on.

  Thereby, the position of the polygon can be calculated using a known method of solving simultaneous linear equations by the least square method. This is the end of the description of the polygon position calculation unit 105.

Next, the alignment unit 106 will be described. The alignment unit 106 performs polygon alignment based on the polygon position v _i obtained by the polygon position calculation unit 105, the known polygon shape a _ik , and the relative position information of the polygon. Do. Here, it is assumed that (information 2) already described is held as polygon relative position information. That is, for each polygon index i, the vertex index set Gi of the polygon i is known. Under such conditions, the coordinates of each vertex of each polygon are (i = 1, 2,..., N),
v _i + a _ik (i = 1, 2,..., N and kεGi)
Can be obtained as This completes the description of the alignment unit 106.

Next, the operation of the polygon shape correcting unit 107 and the output data 108 will be described. Although a certain vertex k should be shared by a plurality of polygons, there is a case where the coordinates of the vertex k of each polygon are different as a result of arranging the polygons by the aligning unit 106. In this state, since there are gaps and overlaps between the arranged polygons, the polygonal shape is corrected so as to eliminate the gaps and overlaps. Actually, the vertex coordinates are corrected. The following processing is performed for each vertex k. That is, the coordinate vector x _k of the vertex k is determined as the center of gravity of the vertex k belonging to each polygon as follows.

ここで、Ｊｋは、頂点ｋを共有する多角形インデクスの集合であり、｜Ｊｋ｜は、集合Ｊｋの要素数を示す。この頂点座標ｘ_kが出力データ１０８である。あるいは、出力データ１０８は、頂点座標ではなくて、多角形位置ｖ_iと、補正された多角形形状ａ’_ikとする場合も考えられる。この場合、多角形形状補正部１０７では、補正された多角形形状ａ’_ikの計算を以下のように行えば良い。
ａ’_ik＝ｘ_k−ｖ_i
以上で、多角形形状補正部１０７の動作および出力データ１０８の説明を終える。これで、実施例１の動作の説明を終える。

Here, Jk is a set of polygon indexes sharing the vertex k, and | Jk | indicates the number of elements of the set Jk. This vertex coordinate x _k is the output data 108. Alternatively, the output data 108 may be a polygon position v _i and a corrected polygon shape a ′ _ik instead of the vertex coordinates. In this case, the polygon shape correcting unit 107 may calculate the corrected polygon shape a ′ _ik as follows.
a ′ _ik = x _k −v _i
This is the end of the description of the operation of the polygonal shape correcting unit 107 and the output data 108. This completes the description of the operation of the first embodiment.

  According to the first embodiment, the position of the input polygon can be determined by solving an expression related to the position of the polygon, and the input polygon can be arranged on a plane. As a result, when the object surface in the three-dimensional space is represented as a collection of polygons, even if the object surface cannot be expanded, the distortion of the polygon on the plane is minimized. Can be lined up. In addition, since it is not necessary to obtain the geodesic distance required to create a development with a small distortion, the amount of calculation for obtaining the geodesic distance can be reduced.

  Next, Example 2 will be described. In the first embodiment, the case where the object surface is formed of a general polygon has been described. In the second embodiment, the case where the object surface is formed of a convex quadrangle will be described. Further, in the second embodiment, polygons are regularly arranged, and the relative position information of the polygons, that is, the above (Information 1), (Information 2), or (Information 3) is easy from the polygon index. An example in the case where it is required is described.

Further, the second embodiment shows an example in which the positional relationship between the coordinates of the polygon _{i and} the end point of the position vector v _i of the polygon i is clarified. Furthermore, also in the second embodiment, as in the first embodiment, a condition is imposed that the existence of vertices of other polygons is not allowed on sides other than the vertices of all polygons. Hereinafter, the present embodiment will be specifically described.

  FIG. 9 is an explanatory diagram showing a state in which the rectangles on the surface of the object are regularly arranged. In the present embodiment, as shown in FIG. 9, it is assumed that the rectangles forming the object surface are regularly arranged so as to correspond to the rows and columns of the two-dimensional array. The numbers written in the rectangle in FIG. 9 indicate the index of the rectangle (polygon). For example, polygons (squares) 1, 2, 3, 4 are arranged in a first row, polygons (squares) 2, 6, 10, 14 are arranged in a two-dimensional array such that they can be mapped into a second column, etc. ing.

  In this case, if the polygon scanning unit 102 knows the polygon index corresponding to the square arrangement location, the polygon scanning unit 102 can easily obtain the polygon index around the polygon and the vertex index of the polygon. It is possible to determine data representing the relative positions of polygons constituting the object surface in the three-dimensional space as described in the first embodiment, that is, (information 1), (information 2), and (information 3). Therefore, it is not necessary to have such information as input data. Of course, you can have this information.

  FIG. 10 is an explanatory diagram showing a state in which the rectangles on the object surface are not regularly arranged. However, in the positional relationship of the quadrangle as shown in FIG. 10, each quadrangle cannot be mapped to a two-dimensional array, and thus this example is not applicable. That is, in the second embodiment, the conditions of the first embodiment are narrowed, more specifically, and will be described in an easy-to-understand manner.

  Now, also in the present Example 2, a linear equation is created using Formula (9). FIG. 11 is a diagram for explaining the operation of the second embodiment. Here, for simplicity, as shown in FIG. 11, the polygon index extracted by the polygon scan unit 102 is 0, and the polygon index extracted by the surrounding polygon extraction unit 103 is 1 to 4. And Also, as shown in FIG. 11, the edge index is 1 to 4 and the vertex index is 1 to 4. At this time, in equation (9), m = 0, Hm = H0 = {1,2,3,4}, F1 = {0,1}, F2 = {0,2}, F3 = {0,3} , F4 = {0,4}, and G1 = {1,3}, G2 = {2,4}, G3 = {1,2}, G4 = {3,4}.

FIG. 12 is a diagram for explaining another operation example of the second embodiment. Further, as shown in FIG. 12, the lengths of the sides of the polygons 1 to 5 are measured when mapping to a two-dimensional plane. As shown in FIG. 12, the length of the side 1 of the polygon 0 is L1, and the length of the side 1 of the polygon 1 is L1 ′. Similarly, the length of side 2 of polygon 0 is L2, the length of side 2 of polygon 2 is L2 ', the length of side 3 of polygon 0 is L3, and the length of side 3 of polygon 3 is L3. 'The length of the side 4 of the polygon 0 is L4, and the length of the side 4 of the polygon 4 is L4'. Here, the values of Li and Li ′ may be different values. Under the above conditions, the length of each side is
L (1) = (L1 + L1 ') / 2
L (2) = (L2 + L2 ') / 2,
L (3) = (L3 + L3 ') / 2,
L (4) = (L4 + L4 ') / 2
It becomes. At this time, the equation (9) becomes as shown in the following equation (15).
2 × {L (1) + L (2) + L (3) + L (4)} × v ₁
-2 × L (1) × v ₁
-2 × L (2) × v ₂
-2 × L (3) × v ₃
-2 × L (4) × v ₄
=
-{L (1) + L (3)} × a _01- {L (3) + L (2)} × a _02- {L (4) + L (1)} × a _03- {L (2 ) + L (4)} × a ₀₄
+ L (1) × (a ₁₁ + a ₁₃ ) + L (2) × (a ₂₂ + a ₂₄ ) + L (3) × (a ₃₁ + a ₃₂ ) + L (4) × (a ₄₄ + a ₄₃ ) (15)
It becomes.

The coefficient matrix A has a coefficient 2 × {L (1) + L (2) + L (3) + L (4)} or −2 × L at a predetermined column (position corresponding to each vector v _i ) in each row. It can be made by putting (h) and putting 0 in other positions. In the second embodiment, the coefficient 2 × {L (1) + L (2) + L (3) + L (4)} or −2 × L (h) is the same regardless of the polygon. For this reason (which vector coefficient is different), it is not necessary to calculate each time because it can be selected immediately if the relative positions of the center polygon and the surrounding polygon are known. For this reason, the polygon position calculation part 105 can determine the coefficient of the linear equation regarding the position of each polygon from the relative position of a polygon.

As described in the first embodiment, since the positional relationship between the end point of the vector v _i and each polygon is arbitrary, the coordinates of the lower left vertex of each quadrilateral i may be matched with the end point of the vector v _i . In this case, since it can be fixed as a ₀₁ = a ₂₂ = a ₄₃ = 0, the data amount of the input data 101 can be omitted. At this time, Expression (15) is as shown in Expression (16) below.
2 × {L (1) + L (2) + L (3) + L (4)} × v1
-2 × L (1) × v ₁
-2 × L (2) × v ₂
-2 × L (3) × v ₃
-2 × L (4) × v ₄
=
-{L (3) + L (2)} × a _02- {L (4) + L (1)} × a _03- {L (2) + L (4)} × a ₀₄
+ L (1) × (a ₁₁ + a ₁₃ ) + L (2) × (a ₂₄ ) + L (3) × (a ₃₁ + a ₃₂ ) + L (4) × (a ₄₄ ) (16)
It becomes.

  Further, consider a case where the input rectangle is limited to a parallelogram. In this case, since the quadrangular shape can be represented by two vectors, the data amount of the input data 101 can be further omitted. The other parts operate in the same manner as in the first embodiment. This is the end of the description of the second embodiment.

  According to the second embodiment, when the input polygon is a quadrangle arranged so as to be mapped to a two-dimensional matrix, the information amount of input data and the calculation amount for the expansion process can be reduced. .

  Next, Example 3 will be described. In Example 1 and Example 2, the condition that the existence of vertices of other polygons is not allowed on the sides other than the vertices of all polygons is imposed. The present invention can also be applied to the case where vertices of other polygons exist on the sides other than the removed vertices. Since the third embodiment can be realized with the same configuration as the object surface flattening device 100 of the first embodiment, it will be described with reference to FIG. First, the case of a general polygon will be described.

  FIG. 13 is a diagram for explaining the operation of the third embodiment. As shown in FIG. 13, consider a case in which vertices V of other polygons 2 and 3 exist on side ST of polygon 1. In this case, the method shown in the first embodiment cannot be applied. In the present embodiment, the technique of the first embodiment is applied by dividing the polygon 1 into two polygons having the position of the vertex V as a new vertex. The polygon is divided by the polygon scanning unit 102 based on the shape information of the polygon and the shape information of the surrounding polygon. Therefore, the polygon scanning unit 102 corresponds to the polygon dividing means in the claims.

There are the following two methods of application. FIG. 14 is a diagram for explaining the polygon division method. FIG. 15 is a diagram for explaining another example of the polygon division method.
(1) By drawing a line from the vertex V to one vertex other than the vertices S and T when the vertices V of the other polygons 2 and 3 exist on the sides other than the vertex of the polygon 1 Divide polygon 1. FIG. 14 shows how this method is divided.
(2) When the vertices V of the other polygons 2 and 3 exist on the sides other than the vertex of the polygon 1, a half line is drawn from the vertex V in the direction inside the polygon 1.

  By drawing a straight line in this way, the vertex V can be made not a vertex on the side of the polygon. Furthermore, in the case of (2) above, by making the direction of the half straight line the same as the line segment between the polygon 2 and the polygon 3 in FIG.

  According to Example 3, when a vertex of another polygon exists on a side other than the vertex of the polygon, the line segment or the half line is drawn from the vertex on the side of the polygon. By making the intersection of the vertex and the side a new vertex, it is possible to eliminate the vertex on the side of the polygon, so that the object surface can be developed on a plane as in the case where there is no vertex on the side of the input polygon.

  Note that the object surface flattening apparatus or the object surface flattening method described in FIG. 3 is realized using, for example, a CPU (Central Processing Unit), a ROM (Read Only Memory), a RAM (Random Access Memory), and the like. The program is installed from a hard disk drive, a portable storage medium such as a CD-ROM, a DVD, or a flexible disk, or downloaded from a communication circuit, and the CPU executes the program, so that each function shown in FIG. Realize.

  Although the preferred embodiments of the present invention have been described in detail above, the present invention is not limited to the specific embodiments, and various modifications, within the scope of the gist of the present invention described in the claims, It can be changed.

It is a figure explaining edge sharing of a polygon. It is a figure explaining the case where the vertex of a polygon is divided | segmented into 2 points | pieces. 1 is a block diagram of an object surface flattening apparatus 100 according to an embodiment. It is explanatory drawing of a polygon shape and a position. It is explanatory drawing of the polygon index set Fh. It is explanatory drawing of edge index set Hi. (a) is a diagram showing a state in which there are no vertices of other polygons on the sides other than the vertices of a certain polygon, and (b) is a diagram showing other polygons on the sides other than the vertices of a certain polygon. It is a figure which shows a mode that the vertex of a square exists. It is a figure for demonstrating a distance function. It is explanatory drawing of a mode that the rectangle of the surface of an object is located regularly. It is explanatory drawing of a mode that the rectangle of the object surface is not regularly arranged. FIG. 6 is an operation explanatory diagram of Embodiment 2. FIG. 10 is a diagram illustrating another operation example of the second embodiment. FIG. 9 is an operation explanatory diagram of Embodiment 3. It is a figure explaining a polygon division | segmentation system. It is a figure explaining the other example of a polygon division system.

Explanation of symbols

DESCRIPTION OF SYMBOLS 100 Object surface flattening apparatus 102 Polygon scanning part 103 Surrounding polygon extraction part 104 Primary equation creation part 105 Polygon position calculating part 106 Arrangement part 107 Polygon shape correction part

Claims (13)

An object surface flattening device that develops on a plane an object surface in a three-dimensional space represented as a collection of polygons on a computer,
Means for extracting polygon shape information from input data;
Means for extracting shape information of a peripheral polygon sharing a side with the polygon;
From the shape information of the polygon and the shape information of the surrounding polygon , a function is calculated for a distance between sides shared between the polygon and the surrounding polygon on the plane, and the function is shared with the function Applying the condition that the distance between sides is minimized, for each polygon, coefficient acquisition means for creating a linear equation for determining the position of each polygon;
Polygon position calculating means for calculating the position of each polygon by solving a linear equation for determining each polygon;
An object surface flattening apparatus comprising: means for arranging the polygons whose positions are calculated. 2. The object surface flattening device according to claim 1, further comprising means for correcting the shapes of the arranged polygons. The object surface flattening device further draws a line segment or a half line from a vertex on the side of the polygon when a vertex of another polygon exists on a side other than the vertex of the polygon. The object surface flattening apparatus according to claim 1 or 2, further comprising polygon dividing means having a new vertex at the intersection of the vertex and the side. The input data includes data representing the shape of a polygon constituting the object surface in a three-dimensional space;
The data representing the shape of the polygon is a two-dimensional vector from a common starting point to each vertex of the polygon,
The object surface flattening device according to any one of claims 1 to 3, wherein the vector to the common starting point is a vector representing a polygonal position. The input data includes data representing the relative positions of polygons constituting the object surface in a three-dimensional space;
The data representing the relative position is
Information indicating which polygon shares each side of the polygon constituting the object surface in the three-dimensional space and / or the polygon constituting the object surface in the three-dimensional space Contains information about which side is present,
5. The object surface flattening device according to claim 1, wherein the data is vertex index data at both ends of each side. 6. The coefficient acquisition means includes a center polygon and a surrounding area that minimize a sum of distances between sides to be shared between the center polygon and a surrounding polygon sharing the side with the center polygon. 6. The object surface flattening device according to claim 1, wherein a linear equation indicating a relative position with respect to the polygon is created . 7. The object surface flattening apparatus according to claim 6, wherein the distance between the sides is obtained by multiplying the square sum of the distances between vertices to be shared by the length of the side. The object surface according to any one of claims 1 to 7, wherein the polygon position calculation means calculates the position of each polygon by solving simultaneous linear equations by a least square method. Flattening equipment. The polygon included in the input data is a quadrangle arranged so as to be mapped to a two-dimensional matrix,
The means for extracting the shape of the polygon determines data representing a relative position of the polygon constituting the object surface in the three-dimensional space from the polygon index corresponding to the location of the quadrangle,
The object surface flatness according to any one of claims 1 to 3, wherein the coefficient acquisition unit determines a coefficient of an expression related to a position of each polygon from a relative position of the polygon. Device. 5. The object surface flattening apparatus according to claim 4, wherein the common starting point is one of polygonal vertices. In order to expand the surface of an object in a three-dimensional space represented as a collection of polygons on a plane,
Extracting polygon shape information from input data;
Extracting shape information of a surrounding polygon that shares a side with the polygon; and
From the shape information of the polygon and the shape information of the surrounding polygon , a function is calculated for a distance between sides shared between the polygon and the surrounding polygon on the plane, and the function is shared with the function Creating a linear equation for determining the position of each polygon for each polygon by applying the condition that the distance between sides is minimized ;
Calculating the position of each polygon by solving a linear equation for determining the position of each polygon;
A program for executing the step of arranging the polygons whose positions are calculated. 12. The program according to claim 11, further causing the computer to execute a step of correcting the shapes of the arranged polygons. A computer-readable recording medium on which the program according to claim 11 or 12 is recorded.

Images