JP4333452B2 - Coordinate transformation apparatus and program - Google Patents

Description

  The present invention relates to a coordinate conversion device, a coordinate conversion method, and a program.

  Conventionally, a computer graphic (hereinafter referred to as CG) technique has been widely used as a method of generating a realistic image on a computer. In CG technology, texture mapping is one method for expressing a more realistic object surface. Furthermore, a projective transformation method is known as a method for mapping a two-dimensional texture image to an object in a projection view obtained by perspective projection of a three-dimensional space. For example, the contents of the projective transformation method are introduced in Non-Patent Document 1.

  The projective transformation method of Non-Patent Document 1 will be described with reference to FIG. FIG. 1 is a perspective explanatory view of a planar texture image. In FIG. 1A, it is assumed that there is a quadrilateral ABCD on a two-dimensional plane. Further, it is assumed that a texture exists in the rectangle ABCD. FIG. 1B is a perspective view obtained by perspectively projecting a three-dimensional space. Assume that the quadrangle ABCD in FIG. 1A is perspective-projected, resulting in the quadrangle abcd in FIG. Here, it is assumed that the point abcd is on the same plane. The texture in the rectangle ABCD is mapped in the rectangle abcd.

  The method introduced in Non-Patent Document 1 is a general method (projection conversion method) that is usually performed to map the texture in the rectangle ABCD into the rectangle abcd. Let the coordinates of the points in the input rectangle ABCD be (u, v). Assuming that the texture on the point (u, v) appears in the coordinates (S, L) in the output quadrangle abcd, the relationship between (u, v) and (S, L) is as follows: 2).

上記（１）（２）式内の８つの係数ｐ₀、ｐ_１、ｐ₂、ｑ₀、ｑ_１、ｑ₂、ｒ₁、ｒ₂の値を求めることができれば、四角形ＡＢＣＤ内のテクスチャを四角形ａｂｃｄにマッピングすることができる。ところで、今、頂点Ａは、頂点ａと対応している。頂点Ａの座標を（ｕＡ,ｖＡ）、頂点ａの座標を（Ｓａ,Ｌａ）とすると、以下の関係式（３）、（４）を得ることができる。

If the values of the eight coefficients p ₀ , p ₁ , p ₂ , q ₀ , q ₁ , q ₂ , r ₁ , r _{2 in} the above equations (1) and (2) can be obtained, the texture in the rectangle ABCD can be obtained. It can be mapped to a rectangle abcd. Now, the vertex A corresponds to the vertex a. When the coordinates of the vertex A are (uA, vA) and the coordinates of the vertex a are (Sa, La), the following relational expressions (3) and (4) can be obtained.

同様に、頂点Ｂと頂点ｂの対応、頂点Ｃと頂点ｃの対応、頂点Ｄと頂点ｄの対応から、それぞれ２つの関係式を得ることができる。全てあわせると８つの関係式(一次式)を得る。よって、上記８つの未知数ｐ₀、ｐ₁、ｐ₂、ｑ₀、ｑ₁、ｑ₂、ｒ₁、ｒ₂は、これらの８つの連立一次方程式を解くことにより、決定することができる。以下、これら８つの変数ｐ₀、ｐ₁、ｐ₂、ｑ₀、ｑ₁、ｑ₂、ｒ₁、r₂を射影変換パラメタと呼ぶことにする。

Similarly, two relational expressions can be obtained from the correspondence between the vertex B and the vertex b, the correspondence between the vertex C and the vertex c, and the correspondence between the vertex D and the vertex d. When all are combined, eight relational expressions (primary expressions) are obtained. Therefore, the eight unknowns p ₀ , p ₁ , p ₂ , q ₀ , q ₁ , q ₂ , r ₁ , r ₂ can be determined by solving these eight simultaneous linear equations. Hereinafter, these eight variables p ₀ , p ₁ , p ₂ , q ₀ , q ₁ , q ₂ , r ₁ , r ₂ will be referred to as projective transformation parameters.

  Now, an example in which texture mapping is performed using this projective transformation will be described. In the method described in Patent Document 1, an example in which an input texture image is divided into a plurality of quadrangles is described. FIG. 2 is a diagram for explaining the method described in Patent Document 1. In FIG. Consider a case where an input texture image (rectangle ABCD) is divided into nine rectangles as shown in FIG. In the method described in Patent Document 1, the correspondence between vertices is not performed for all (9) quadrangular vertices, but only for four representative vertices. For example, it is performed only for the four points of vertices A, B, C, and D and vertices a, b, c, and d. Other vertex positions are calculated. In the method described in Patent Document 1, an example of a method of adding a random error to the vertex position when calculating the vertex position is described in order to perform more realistic texture mapping.

  Next, Fumiko Ito et al., “Texture Simulation System Considering Convenience and Reality,” Information Processing Society 39th (late 1989) National Convention, introduced as Known Example 1 in Patent Document 1, will be described (conventional) Technology 3). In the prior art 3, the positions of (A and a), (B and b), (C and c), (D and d), etc. in FIG. 1 are designated by using a mouse. Projective transformation parameters can be calculated by substituting coordinate information designated by the mouse into the equations (1) and (2).

  As described above, in the method described in Patent Document 1 and Conventional Example 3, a texture on a two-dimensional plane is mapped to a projection view that is perspective-projected on the plane. This perspective projection method has been described in Non-Patent Document 1.

"Texture Mapping 3D Models of Real-World Scenes" ACM Computing Surveys, Vol. 29, No. 4, December 1997 pp. 325-365 JP-A-5-120439

  However, when associating vertices as in the conventional example, the method described in Patent Document 1 has a problem that an error is artificially given to the coordinate position of the vertex to be mapped. In Conventional Example 3, the vertex position to be mapped is given manually. In this case, there is a problem that an error occurs when the position is designated. Further, in the method described in Patent Document 1 and Conventional Example 3, the object to which the two-dimensional texture image is mapped is limited to a plane.

  However, the target surface on which texture mapping is performed is not necessarily a flat surface. When texture mapping is to be performed on a surface other than a plane, as shown in FIG. 2, the texture image is divided into small squares (hereinafter referred to as small squares) as shown in FIG. A method is conceivable in which mapping is performed by taking projection correspondence for each quadrangle by taking correspondence between vertices every time. However, depending on the shape of the target surface, it may be impossible to arrange the small squares on the target surface without gaps or overlap. In general, the curved surface cannot be developed except for a curved surface having a Gaussian curvature of 0 over the entire surface, and the small squares cannot be arranged on the target surface without gaps or overlaps.

The three examples listed above are
(1) When an error occurs in the vertex position of the rectangle, or
(2) The existence when a plurality of small squares cannot be arranged on the object surface in three dimensions without gaps or overlaps is shown. In these cases, the projective transformation with respect to the vertex positions is successful, but there is a problem that the texture becomes discontinuous at the edges between the vertices.

  Hereinafter, this problem will be described with reference to FIG. FIG. 3 is an explanatory diagram relating to texture discontinuities. FIG. 3A is a planar texture image. It is assumed that the planar texture image is divided into two quadrangular ABCD and quadrangular DCEF. It is assumed that the rectangle ABCD and the rectangle DCEF share the side CD. Assume that the quadrangle ABCD and the quadrangle DCEF in FIG. 3A are projected to become a quadrangle abcd and a quadrangle dcef in FIG. Here, vertex A is projected onto a. Similarly, B is projected on b, C is projected on c, D is projected on d, E is projected on e, and F is projected on f.

  Actually, when there is no error or when it is projected from the plane to the plane, the shape as shown in FIG. 3B is not obtained. However, as described above, there is an error, When projecting from a plane onto a non-deployable surface, this shape can occur.

  Now consider points G, H, and I in FIG. Here, as an example, G is the midpoint of the line segment AB, H is the midpoint of the line segment CD, and I is the midpoint of the line segment EF. When projective transformation is performed from the rectangle ABCD to the rectangle abcd, the point A appears in a, the point B in b, the point C in c, and the point D in d. The point G appears on the point g in the line segment ab.

  At this time, G is the midpoint of the line segment AB, but g is not always the midpoint of the line segment ab as a result of the projective transformation. As shown in FIG. 3, when the rectangle ABCD is reflected in the trapezoid abcd and ad <bc, g is closer to a than b. Similarly, H also appears at a point h1 on the line segment cd. h1 is not a midpoint of the line segment cd but a point closer to d than c. Similarly, a projection conversion from the square DCEF to the square dcef is considered. A point H on the side of the quadrangle DCEF appears in a point h2 on the side of the quadrangle dcef. Similarly, at this time, the position of h2 is not necessarily the midpoint of the line segment cd.

  As a result, as shown in FIG. 3A, the straight line GI that should have been a straight line becomes discontinuous at the point H as shown in FIG. The problem of end up occurs. That is, since the positions of the points to be shared between the adjacent quadrangles differ depending on the quadrilateral, there arises a problem that the texture becomes discontinuous on the sides of the quadrilateral.

  In view of the above-described problems of the conventional example, the present invention provides a coordinate conversion apparatus, a coordinate conversion method, and a program that can perform coordinate conversion so that coordinate values are continuous between adjacent quadrangles. And

  In order to solve the above-described problem, the present invention provides a coordinate conversion device for converting coordinates in a plurality of adjacent rectangles into a plurality of adjacent rectangles, as described in claim 1, wherein the coordinates are adjacent to input coordinates. A weighting factor calculation unit that calculates a weighting factor using the distance between the rectangles to be coordinated, coordinates obtained using projective transformation corresponding to the rectangle to which the input coordinates belong, and a rectangle adjacent to the rectangle to which the input coordinates belong And a coordinate calculation unit that calculates the output coordinates by weighting the coordinates obtained by using the projective transformation with the weighting factor and adding the coordinates.

  According to the first aspect of the present invention, the coordinates obtained using the projective transformation corresponding to the quadrangle to which the input coordinates belong, and the coordinates obtained using the projective transformation corresponding to the quadrangle adjacent to the quadrangle to which the input coordinates belong are obtained. Since the output coordinates are calculated by weighting and adding, the coordinates in the other rectangle can be calculated from the coordinates in one rectangle so that the coordinate values are continuous between adjacent rectangles. it can. For example, even when a plurality of adjacent rectangles are perspective-projected, texture mapping can be performed so that the texture does not become discontinuous on the side shared between the rectangles.

  According to a second aspect of the present invention, in the coordinate transformation device according to the first aspect, the weighting factor calculation unit calculates the weighting factor so that the vertex position of the quadrangular projection destination does not change. It is characterized by. According to the second aspect of the present invention, it is possible to eliminate an error in the vertex position of the quadrangle by restraining the vertex of the quadrangle so that the vertex of the quadrangle of the projection destination is not changed.

  According to a third aspect of the present invention, in the coordinate transformation device according to the first or second aspect, the weighting factor calculation unit is configured so that the two of the two sides are shared by two squares. The weighting factor corresponding to the quadrangle is made equal. According to the third aspect of the invention, discontinuity on the shared side can be eliminated by making the weighting coefficients corresponding to the two quadrilaterals equal on the side shared by the two quadrangles.

  According to a fourth aspect of the present invention, in the coordinate transformation device according to any one of the first to third aspects, the quadrangle to which the input coordinates belong is further divided into four regions. An area determination unit, and the weighting factor calculation unit calculates a weighting factor corresponding to four adjacent quadrilaterals including the quadrilateral to which the input coordinates belong in each of the divided regions, and calculates the weighting factor of the other quadrangle. It is characterized by zero. According to the invention described in claim 4, the quadrangle to which the input coordinates belong is divided into four regions, and in each of the divided regions, weighting corresponding to four adjacent quadrilaterals including the quadrangle to which the input coordinates belong is performed, Since the weighting coefficient of the quadrangle is 0, natural coordinate conversion can be performed by decreasing the weighting coefficient for a farther quadrangle.

  According to a fifth aspect of the present invention, in the coordinate conversion device according to any one of the first to fourth aspects, the input coordinate is further changed to an adjacent quadrangle existing in the vertical direction. And a distance calculation unit that calculates a distance from the input coordinates to an adjacent quadrangle that exists in the horizontal direction, and the weighting factor calculation unit calculates a ratio of the distance to the quadrangle that exists in the vertical direction and the horizontal The weighting factor is calculated by multiplying the ratio of the distance to the square existing in the direction. According to the fifth aspect of the invention, the weighting factor is calculated by multiplying the ratio of the distance in the vertical direction and the ratio of the distance in the horizontal direction. More natural coordinate transformation.

  Further, according to a sixth aspect of the present invention, in the coordinate transformation device according to the fourth or fifth aspect, the distance calculation unit includes the input coordinates and each side of the region to which the input coordinates belong. The distance between each side is calculated as the distance between the side and the adjacent rectangle. According to the sixth aspect of the invention, it is possible to solve the problem that the calculation increases when the distance between each side is set as the distance between the side and the adjacent rectangle.

  Further, according to the present invention, as described in claim 7, in the coordinate transformation device according to claim 5, the weighting factor calculation unit newly calculates a value obtained by substituting the distance ratio into a monotonically increasing function. It is used as a ratio of various distances. According to the seventh aspect of the invention, since the value obtained by substituting the distance ratio into the monotonically increasing function is used as the new distance ratio, natural coordinate conversion according to the distance ratio can be performed. .

  Further, according to the present invention, as described in claim 8, in the coordinate conversion device according to claim 6, the distance calculation unit sets a region to which the input coordinates belong to a rectangle, and each vertex of the rectangle is arbitrarily parallel. The input coordinates are mapped into the parallelogram using projective transformation projected onto each vertex of the quadrilateral, and the distance to the side of each region is calculated using the coordinates of the mapping destination. According to the eighth aspect of the present invention, the input coordinates are mapped into the parallelogram by projective transformation that maps each vertex of the rectangle to which the input coordinates belong to each vertex of an arbitrary parallelogram, and the coordinates of the mapping destination Is used to measure the distance to the side of each region, so that a more natural coordinate transformation can be performed by using straight lines that move continuously in the parallelogram.

  Further, according to a ninth aspect of the present invention, in the coordinate transformation device according to any one of the first to eighth aspects, the weight coefficient calculation unit is configured such that the weight coefficient calculation unit has a vertex on each vertex of the quadrangle. The weighting factor corresponding to the quadrangle that shares is reduced to a quarter. According to the ninth aspect of the present invention, it is possible to exert the same effect from the four quadrangles sharing the vertex.

  Further, according to a tenth aspect of the present invention, in the coordinate transformation device according to any one of the first to ninth aspects, the weighting factor calculation unit is configured such that the weighting factor calculating unit is arranged on each side of the quadrangle. The weight coefficient corresponding to the quadrangle that shares is set to at least one point that is ½.

  Further, according to the present invention, as described in claim 11, in the coordinate transformation device according to any one of claims 1 to 10, the weighting factor calculation unit includes a weighting factor of the rectangle within the rectangle. 1 or more, and one or more points where the weighting factor of other squares is 0 are set.

  Further, according to the present invention, as described in claim 12, in the coordinate transformation device according to any one of claims 4 to 11, the region determination unit is a straight line passing through a midpoint of a square side. Alternatively, the region is divided by a curve.

  Further, according to the present invention, as described in claim 13, in the coordinate transformation device according to any one of claims 4 to 11, the region determination unit is configured such that the area determination units are arranged at midpoints of opposite sides of a quadrangle. The region is divided by a straight line passing through a straight line connecting the two.

  Further, according to the present invention, as described in claim 14, in the coordinate transformation device according to any one of claims 1 to 13, the weighting factor calculation unit sets the sum of the weighting factors of each square to 1. The weighting factor is a real number of 0 or more and 1 or less.

  Further, according to the present invention, as described in claim 15, in the coordinate transformation device according to any one of claims 4 to 14, the weight coefficient calculation unit is configured to have a function other than a square side of each region. On the boundary line, the weighting coefficient corresponding to a rectangle other than the rectangle closest to the rectangle where the input coordinates exist is set to 0. According to the fifteenth aspect of the present invention, the weighting factor corresponding to a farther quadrangle can be made smaller.

According to a sixteenth aspect of the present invention, there is provided a program for calculating a distance between an input coordinate and an adjacent rectangle in order to convert coordinates in a plurality of adjacent rectangles into a plurality of adjacent rectangles. Using a weighting factor calculating step to calculate a weighting factor, coordinates obtained using a projective transformation corresponding to the quadrangle to which the input coordinate belongs, and a projective transformation corresponding to a quadrangle adjacent to the quadrangle to which the input coordinate belongs. A coordinate calculation step of calculating an output coordinate by weighting and adding the obtained coordinates with the weighting coefficient.

According to the invention described in claim 16 , the coordinates obtained by using the projective transformation corresponding to the quadrangle to which the input coordinates belong, and the coordinates obtained by using the projective transformation corresponding to the quadrangle adjacent to the quadrangle to which the input coordinates belong are obtained. Since the output coordinates are calculated by weighting and adding, the coordinates in the other rectangle can be calculated from the coordinates in one rectangle so that the coordinate values are continuous between adjacent rectangles. it can. For example, even when a plurality of adjacent rectangles are perspective-projected, texture mapping can be performed so that the texture does not become discontinuous on the side shared between the rectangles.

  According to the present invention, it is possible to provide a coordinate conversion device, a coordinate conversion method, and a program capable of performing coordinate conversion so that coordinate values are continuous between adjacent quadrangles.

  Hereinafter, the best mode for carrying out the present invention will be described. First, the concept of the present invention will be described. 4A and 4B are diagrams for explaining the concept of the invention. FIG. 4A is a diagram showing an image before projection, and FIG. 4B is a diagram showing an image after projection. Assume that nine adjacent squares (0 to 9) as shown in FIG. 4A are perspective-projected into nine adjacent squares (0 to 9) as shown in FIG. 4B. In addition, in FIG. 4A and FIG. 4B, it is assumed that squares assigned with the same number correspond to each other.

  Here, an example is shown in which the texture in the square 0 of the pre-projection image in FIG. 4A is mapped into the square 0 of the post-projection image in FIG. 4B. To map the texture in the post-projection image shown in FIG. 4 (a) to the post-projection image shown in FIG. 4 (b), input the coordinates (u, v) in the post-projection image and correspond to the coordinates. It is only necessary that the coordinates in the pre-projection image to be obtained can be obtained. That is, if the vertices of the square 0 of the pre-projection image are A, B, C, and D, and the vertices of the square 0 of the post-projection image are a, b, c, and d, the coordinates of the points in the quadrangle abcd are input, It is only necessary that the coordinates can be obtained by mapping to a point in the rectangle ABCD. Here, the output coordinates in the pre-projection image corresponding to the input coordinates (u, v) in the post-projection image are X (u, v). X (u, v) is a two-dimensional vector.

Hereinafter, the calculation method of X (u, v) will be described in order. FIG. 5 is a process flowchart of the coordinate conversion method.
(1) First, projective transformation parameters between corresponding quadrangles are calculated (step S1). As described in the conventional example, the projective transformation parameters can be calculated for each quadrangle using the coordinates of the four vertices. The example described in the conventional example is a mathematical formula for obtaining the coordinates (u, v) from the coordinates (S, L), but in order to obtain the coordinates (S, L) from the coordinates (u, v), What is necessary is just to convert a variable suitably. Hereinafter, the projective transformation corresponding to the rectangle i is denoted by Si.
(2) The output coordinates X0 (u, v) in the pre-projection image shown in FIG. 4 (a) corresponding to the input coordinates (u, v) in the square 0 of the post-projection image shown in FIG. Calculation can be performed using the projective transformation parameter of the square 0 (step S2). X0 (u, v) is a two-dimensional vector.
(3) Similarly, the output coordinates Xi (u, v) in the pre-projection image of FIG. 4 (a) corresponding to the input coordinates (u, v) in the quadrangle 0 of the post-projection image of FIG. Calculation can be performed using the projective transformation parameters of the rectangle i (i = 1, 2,..., 8) (step S3). Xi (u, v) is a two-dimensional vector.

  (4) FIG. 4 (b) The final output coordinates X (u, v) in the pre-projection image in FIG. 4 (a) corresponding to the input coordinates (u, v) in the square 0 of the post-projection image are projected. The output coordinates Xi (u, v) in the previous image are weighted and added with a weighting coefficient Wi (u, v) (i = 0, 1, 2,..., 8) that is a function of the square number i. (Step S4). This is shown in equation (5).

ただし、０≦Ｗi(ｕ,ｖ)≦１、かつ、

However, 0 ≦ Wi (u, v) ≦ 1, and

である。すなわち、各四角形の重み係数の和を１とし、かつ、重み係数は０以上１以下の実数とする。

It is. That is, the sum of the weighting factors of each square is set to 1, and the weighting factor is a real number from 0 to 1.

  The above equation (5) is obtained by converting the coordinate X0 (u, v) obtained using the projective transformation S0 corresponding to the rectangle 0 to which the input coordinate (u, v) belongs and the rectangle 0 to which the input coordinate (u, v) belongs. The final output coordinates X (u, v) are obtained by weighting and adding the coordinates Xi (u, v) obtained by using the projective transformation Si corresponding to the adjacent quadrilateral i (i ≠ 0) with the weighting factor Wi. v) can be calculated.

  Hereinafter, the weighting coefficient of the point x corresponding to the quadrangle number i may be written as Wi (x), and the output coordinate value of the point x corresponding to the quadrangle number i may be written as Xi (x). Alternatively, the weighting coefficient of a point on the line segment xy is also expressed as Wi (xy), the weighting coefficient in the region R is also expressed as Wi (R), and the like.

Next, a method for calculating the weighting coefficient Wi (u, v) will be described. In the present invention, in order to eliminate the discontinuity of the texture, the weighting factor Wi (u, v) is determined so as to satisfy the following condition.
Condition (1) “Point a appears at point A. Similarly, point b appears at point B, point c appears at point C, and point d appears at point D (restriction condition for quadrangular vertices).”
Condition (2) “The result when the point on the side ab is calculated as belonging to the rectangle 0 is equal to the result when the point is calculated as belonging to the rectangle 1. Similarly, it is assumed that the point on the side bc belongs to the rectangle 0. The calculation result is the same as the result when it is calculated that it belongs to the rectangle 3. The result when the point on the side cd is calculated as belonging to the rectangle 0 and the result when it is calculated as belonging to the rectangle 5 are equal. The result when the point on the side da is calculated as belonging to the rectangle 0 is equal to the result when the point is calculated as belonging to the rectangle 7. (Requirements for eliminating the discontinuity on the shared side) . "
Condition (3) “The weighting factor changes as smoothly as possible in response to a smooth change in input coordinates. Also, the weighting factor corresponding to a farther quadrangle becomes smaller. Further, the weight corresponding to a quadrangle at an equal distance. Coefficients are equal (conditions for a more natural texture). Since the definition of distance is not considered here, this condition (3) is qualitative.

Next, an appropriate weighting factor Wi that satisfies the above conditions is determined. Now, taking the vertex a as an example, consider the condition of the weighting coefficient Wi to satisfy the condition (1). The rectangles sharing the vertex a are the rectangles 0, 1, 7, and 8. When the point a is transformed by the projective transformation S0 corresponding to the square 0, it appears in A. That is, X0 (a) = A. Similarly, when the point a is converted by the projective transformation S1 corresponding to the rectangle 1, it is shown in A. That is, X1 (a) = A. When the point a is transformed by the projective transformation S7 corresponding to the rectangle 7, it appears in A. That is, X7 (a) = A. When the point a is transformed by the projective transformation S8 corresponding to the rectangle 8, it appears in A. That is, X8 (a) = A. From these results, at point a, the weighting factor is
wi (a) = 0 (i ≠ 0,1,7,8)
Thus, the equation (5) can be set to X (a) = Σwi × Xi (a) = A.

Therefore, in the present invention, the weighting factor of the rectangle sharing the vertex is
Wi (a) = 0 (i ≠ 0,1,7,8),
Wi (b) = 0 (i ≠ 0,1,2,3),
Wi (c) = 0 (i ≠ 0,3,4,5),
Wi (d) = 0 (i ≠ 0,5,6,7)
And In this way, a weighting coefficient that does not change the vertex position of the quadrangle at the projection destination is used.

In addition, it is natural that each vertex is considered to have the same influence from four rectangles that share the vertex. Alternatively, it is natural that the weighting coefficients at the four vertices of the square 0 are given symmetrically. Therefore, hereinafter, the weighting coefficients at the four vertices of the square 0 are
Wi (a) = 1/4 (i = 0,1,7,8),
Wi (b) = 1/4 (i = 0,1,2,3),
Wi (c) = 1/4 (i = 0,3,4,5),
Wi (d) = 1/4 (i = 0,5,6,7)
And In this way, on each vertex of the quadrangle, by setting the weighting factor corresponding to the quadrangle sharing the vertex to a quarter, it is possible to exert the same influence from the four quadrangle sharing the vertex.

  Next, consideration will be given using FIG. FIG. 6 is an explanatory diagram of the inventive concept. In FIG. 6, for example, consider a curve (possibly a straight line) that is equidistant from the rectangle 3 and the rectangle 7 within the rectangle 0. Let e be the intersection of this curve and side ab, and g be the intersection of side cd. Similarly, consider a curve (possibly a straight line) that is equidistant from the rectangle 1 and the rectangle 5 within the rectangle 0. Let the intersection of this curve and the side bc be f, and the intersection of the side da be h. Let p be the intersection of the curve eg and the curve fh. p can be a point at the farthest distance from the squares 1, 3, 5, 7.

If the weighting factor of the square 0 is W0 = 1 and the weighting factor of the other square is determined as one point where Wi = 0 (i ≠ 0) among the points in the square 0, this point It is natural to choose p. Originally (when weighting is not performed), W0 = 1 and Wi = 0 (i ≠ 0) over the entire surface of the square 0, so that W0 = 1 and Wi = 0 (i It is natural that there is a point where ≠ 0). Therefore, hereinafter, in the present invention, the weighting coefficient of the point p is
W0 (p) = 1, Wi (p) = 0 (i ≠ 0)
And In this way, by setting one or more points in the quadrangle where the weighting factor of the quadrangle is 1 and the weighting factor of the other quadrangle is 0, natural coordinate conversion is possible. Further, this point p is hereinafter referred to as a center point of the square 0.

  Next, consider the weighting factors of points on the curve pe. The point on the curve pe is considered to be farther from the squares 4, 5, 6 than the point p. Moreover, it is natural to think that it is far from the squares 3 and 7. At the point p, the weighting factor is W1 = W2 = W3 = W4 = W5 = W6 = W7 = 0. Therefore, from the condition (3), on the curve pe, it is natural that the weighting coefficient is W3 = W4 = W5 = W6 = W7 = 0.

  Further, from the point on the curve pe, it is natural to think that the square 3 is closer to the square 2. Therefore, it is natural that the weighting factor W3 = 0 is changed to the weighting factor W2 = 0. Similarly, since it is natural that the square 8 is farther than the square 7, it is natural to set the weighting factor W8 = 0. Therefore, in the present invention, the weighting coefficient is set to Wi = 0 (i = 2, 3, 4, 5, 6, 7, 8) on the curve pe.

  Similarly, on the curve pg, the weighting coefficient is Wi = 0 (i = 6,7,8,1,2,3,4), and on the curve pf, the weighting coefficient is Wi = 0 (i = 4,5 , 6, 7, 8, 1, 2), and on the curve ph, the weighting coefficient is Wi = 0 (i = 8, 1, 2, 3, 4, 5, 6).

  As shown in FIG. 6, the region surrounded by ae, ep, ph, and ha is Ra, the region surrounded by eb, bf, fp, and pe is Rb, and is surrounded by pf, fc, cg, and gp. The region surrounded by hp, pg, gd, and dh is Rd. Thus, on the boundary lines (curves pe, pf, pg, ph) other than the sides of the rectangle of each region (Ra, Rb, Rc, Rd), the weights corresponding to the rectangle other than the rectangle closest to the rectangle where the input coordinates exist. By setting the coefficient to 0, the weight coefficient corresponding to a farther square can be made smaller.

  Next, the weighting factors in these areas will be described.

  Since the points in the region Ra are farther from the squares 3, 4, and 5 than p, the weighting factor is W3 = W4 = W5 = 0 and far from the squares 2, 3, and 4 than pe, so the weighting factor is W2 = W3 = Since W4 = 0, and further away from the squares 4, 5, and 6 than ph, the weight coefficient is set to W4 = W5 = W6. Therefore, it is natural that the weighting coefficient is Wi = 0 (i = 2, 3, 4, 5, 6) in the region Ra.

  Similarly, since the point in the region Rb is farther from the squares 5, 6, and 7 than p, the weighting factor is set to W5 = W6 = W7 = 0, and since the point far from the squares 6, 7, and 8 is less than pe, the weighting factor is set to W6. = W7 = W8 = 0, and further, the weight coefficient is W4 = W5 = W6 because it is farther from the squares 4, 5, and 6 than pf. Therefore, it is natural that the weighting coefficient is Wi = 0 (i = 4, 5, 6, 7, 8) in the region Ra.

  Since the points in the region Rc are farther from the squares 1, 7, and 8 than p, the weighting coefficient is W1 = W7 = W8 = 0, and the points that are farther from the squares 6, 7, and 8 than pg are the weighting coefficients W6 = W7 = Since W8 = 0 and further from the squares 1, 2, and 8 than pf, the weighting coefficient is set to W1 = W2 = W8. Therefore, in the region Rc, the weighting coefficient is set to Wi = 0 (i = 6, 7, 8, 1, 2).

  Since the points in the region Rd are far from the squares 1, 2, and 3 from p, the weighting factor is W1 = W2 = W3 = 0, and the points from the squares 2, 3, and 4 are far from pg, so the weighting factor is W2 = W3 = Since W4 = 0 and further far from the squares 1, 2, and 8 from ph, the weighting coefficient is set to W1 = W2 = W8. Therefore, in the region Rd, the weight coefficient is set to Wi = 0 (i = 8, 1, 2, 3, 4).

  In this way, the quadrilateral 0 to which the input coordinates belong is divided into four areas (Ra, Rb, Rc, Rd), and the weights corresponding to the four adjacent quadrilaterals including the quadrilateral 0 to which the input coordinates belong in each divided area. And the weighting factors for other rectangles are set to 0, so that the weighting factors for the farther rectangles can be reduced.

  Next, the limitation on the positions of the points e, f, g, and h will be considered. On the side ab, the point where the weighting coefficient is W3 = W7 is e. In addition, it is natural that the point where the weight coefficient is W2 = W8 is also the point e. That is, the positions of points (points corresponding to e, f, g, and h) given to divide each quadrangle on each side are the same on the shared side. Further, the point e needs to be determined in consideration of the situation of the square 0 and the square 1 together. In such a case, a reasonable and natural position is to make point e the midpoint of side ab.

  For example, the point e may be the midpoint of the side ab, the point f may be the midpoint of the side bc, the point g may be the midpoint of the side cd, and the point h may be the midpoint of the side da. So far, the positions eg and fh that are equidistant from the two quadrangles have been curved, but in the following, the equidistant curve is assumed to be a straight line.

  Next, a point on the line segment be will be considered. From the condition (2), the point on the line segment be only needs to be the same in the case where the point exists in the square 0 and in the case where the point exists in the square 1. That is, on the side shared by the two quadrangles, the discontinuity of the coordinate transformation on the shared side can be eliminated by making the weighting coefficients corresponding to the two quadrangles equal. For example, if the weighting coefficient on the line be is symmetric, the condition (2) is satisfied. That is, the weighting coefficients may be W0 (be) = W1 (be) and W3 (be) = W2 (be). Similarly, the weighting factor can be set to W0 (bf) = W3 (bf), W1 (bf) = W2 (bf), and the like.

The above considerations are summarized. For example, in the region Rb (rectangle Pebf), the weighting factor is
Point p condition: W0 = 1, Wi = 0 (i ≠ 0)
Point e condition: W0 = W1 = 1/2, Wi = 0 (i ≠ 0,1)
Point f condition: W0 = W3 = 1/2, Wi = 0 (i ≠ 0,3)
Point b condition: W0 = W1 = W2 = W3 = 1/4, Wi = 0 (i ≠ 0,1,2,3)
Line pe condition: Wi = 0 (i ≠ 0,1)
Line segment pf condition: Wi = 0 (i ≠ 0,3)
Line segment be condition: W0 = W1, W3 = W2, Wi = 0 (i ≠ 4,5,6,7,8)
Line segment bf condition: W0 = W3, W1 = W2, Wi = 0 (i ≠ 4,5,6,7,8)
However, if the line segment pe condition and the line segment pf condition are satisfied, the point p condition is also satisfied. If the line segment pe condition and the line segment be condition are satisfied, the point e condition is satisfied. Similarly, if the line segment pf condition and the line segment bf condition are satisfied, the point f condition is also satisfied. If the line segment be condition and the line segment bf condition are satisfied, the point b condition is also satisfied. Therefore, there are the following four boundary conditions.

<Boundary conditions>
Line pe condition: Wi = 0 (i ≠ 0,1)
Line segment pf condition: Wi = 0 (i ≠ 0,3)
Line segment be condition: W0 = W1, W3 = W2, Wi = 0 (i ≠ 4,5,6,7,8)
Line segment bf condition: W0 = W3, W1 = W2, Wi = 0 (i ≠ 4,5,6,7,8)
A method for determining a weighting factor that satisfies the above boundary conditions will be described below.

  Hereinafter, the case where the input coordinates are in the region Rb is taken as an example, and the weighting coefficient setting method in Rb will be described. FIG. 7 illustrates a method for setting a weighting factor in Rb, taking as an example the case where the input coordinates are in the region Rb. In FIG. 7, it is assumed that the input coordinate Q = (u, v) is in the region Rb. At this time, first, the weighting factor can be W4 = W5 = W6 = W7 = W8 = 0.

  Hereinafter, the distance from the point Q to the squares 0, 1, and 3 will be divided into two directions, the horizontal direction (the direction in which the squares 0 and 1 are arranged) and the vertical direction (the direction in which the squares 0 and 3 are arranged). Using the following distances α, β, γ, and δ, the weighting factor is determined so as to reduce the weighting factor corresponding to the quadrangle having the center point in the distance. The horizontal distance from point Q to square 0 is α, the horizontal distance from point Q to square 1 is β, the vertical distance from point Q to square 0 is γ, and the vertical distance from point Q to square 3 is Let δ.

  In the present invention, the weighting coefficient Wi (u, v) is set as in the following formulas (9) to (12). First, λ and μ are determined as shown in equations (7) and (8).

とする。

And

ただし、ここで、Ｆ()はＦ(０)＝０、Ｆ(１／２)＝１／２となる単調増加関数である。

Here, F () is a monotonically increasing function that satisfies F (0) = 0 and F (1/2) = 1/2.

  The above formulas (9) to (12) are obtained from the distance from the input coordinate Q to the adjacent quadrangle 3 existing in the vertical direction and the input coordinate Q when the adjacent squares are regularly arranged in the vertical and horizontal directions. The distance to the adjacent quadrangle 1 existing in the horizontal direction is calculated, and the weighting coefficient is obtained by multiplying the ratio of the distance in the vertical direction and the ratio of the distance in the horizontal direction. The above formulas (9) to (12) are as follows when, for example, F (x) = x.

上記式（１３）〜（１６）は、距離の比λ、μを単調増加関数Ｆ()に代入して得られた新たな値を距離の比として用いて、この距離の比を乗算することによって重み係数を求めることを示している。

The above equations (13) to (16) are obtained by multiplying the distance ratio by using the new values obtained by substituting the distance ratios λ and μ into the monotonically increasing function F () as the distance ratio. Indicates that the weighting coefficient is obtained.

  Here, if true, it is ideal to calculate the weighting coefficient using the distances β and δ to the neighboring squares. However, it is very difficult to set a distance that always satisfies the above <boundary conditions> (1) to (4). In addition, when the shape of a neighboring quadrangle is taken into account, there arises a problem that the calculation is increased. Therefore, in the present invention, β and δ are approximated considering only the shape in the rectangle 0.

Specifically, first, α is a distance from the point Q to the line segment pf. γ is a distance from the point Q to the line segment pe. Further, ε is the distance from the point Q to the line segment be, and ζ is the distance from the point Q to the line segment bf. here,
β = α + 2ε, δ = γ + 2ζ (17)
It is considered. In this way, the above equations (7) and (8) can be obtained as follows.

このとき、線分ｂｅ上ではε＝０なので、λ＝１／２となる。このとき、重み係数はW0=W1={1-F(μ)}/2、W3=W2=F(μ)/2となり、線分ｂｅ条件が成り立つ。同様に、線分ｂｆ上ではζ＝０なので、μ＝１／２であり、重み係数はW0=W3={1-F(λ)}/2、W1=W2=F(λ)/2となり、線分ｂｆ条件が成り立つ。線分ｐｅ上では、γ＝０なので、μ＝０となる。よって重み係数はW3=W2=0となり、線分ｐｅ条件が成り立つ。同様に線分ｐｆ上では、α＝０なので、λ＝０となる。よって重み係数はW1=W2=0となり、線分ｐｆ条件が成り立つ。以上より、（１７）式を導入したとき、境界条件（１）から（４）が成り立つことを示した。

At this time, since ε = 0 on the line segment be, λ = 1/2. At this time, the weighting coefficients are W0 = W1 = {1-F (μ)} / 2 and W3 = W2 = F (μ) / 2, and the line segment be condition is satisfied. Similarly, since ζ = 0 on the line segment bf, μ = 1/2, and the weighting coefficients are W0 = W3 = {1-F (λ)} / 2 and W1 = W2 = F (λ) / 2. The line segment bf condition is satisfied. Since γ = 0 on the line segment pe, μ = 0. Therefore, the weight coefficient is W3 = W2 = 0, and the line segment pe condition is satisfied. Similarly, since α = 0 on the line segment pf, λ = 0. Therefore, the weighting coefficient becomes W1 = W2 = 0, and the line segment pf condition is satisfied. From the above, it was shown that the boundary conditions (1) to (4) are satisfied when the equation (17) is introduced.

  Thus, the amount of calculation is reduced by measuring the distance between the input coordinate and each side of the region to which the input coordinate belongs, and setting the distance ratio between each side as the distance between the side and the adjacent rectangle. be able to.

  Next, an example of the content of the monotonically increasing function F () is shown. FIG. 8 is a diagram showing an example of the content of the monotonically increasing function F. FIG. 8 shows various examples of the monotonically increasing function F () and the shape of the weighting factor W0 corresponding to the monotonically increasing function F (). FIG. 8 is an example when the input coordinates are in the region Rb. 8A to 8D, the horizontal axis indicates the distance x from the point p, the vertical axis indicates the monotonically increasing function F (), and (b) indicates the weighting coefficient WO of each point. Show.

(1) When the monotonically increasing function F (x) = x, as shown in FIG. 8A, the weight coefficient WO is 1 at the point p, 1/2 at the points e and f, and 1 / at the point b. 4.
(2) When the monotonically increasing function F (x) has a shape as shown in FIG. 8 (a) (when there is a portion where F (x) = 0 in the vicinity of x = 0), FIG. As shown in FIG. 5, a region (shaded portion) where the weighting coefficient around the center point p is W0 = 1 can be formed.
(3) When the monotonically increasing function F (x) has a shape as shown in FIG. 8C (when there is a portion where F (x) = 1/2 in the vicinity of x = 1/2) As shown in FIG. 8C, a region (shaded portion) where the weight coefficient W0 = 1/4 around the point b can be formed.
(4) This is a case where the monotonically increasing function F (x) is nonlinear. In the linear function as described above, it is considered that the discontinuity may be visually observed at the portion where the derivative is discontinuous. In that case, the monotonically increasing function F (x) may be a non-linear function as shown in FIG. The joints with the surrounding area can be shown smoothly. In (b) of FIGS. 8A to 8C, an example is set in which at least one point at which a weighting factor corresponding to a quadrangle sharing the side is divided by half is set on each side of the quadrangle. Show.

  Next, a method for measuring the distance from the point Q to the line segments pf, pe, be, bf is determined. FIG. 9 is a diagram for explaining a method of measuring the distance from the point Q to the line segments pf, pe, be, bf. As shown in FIG. 9, p1 is a center point of the rectangle 1 and f1 is an intersection of a straight line equidistant from the left and right rectangles of the rectangle 1 and the shared side of the rectangle 1 and the rectangle 2. The distance that satisfies the above condition (2) is defined.

  For example, when the point Q on the line segment be is in the square 0, the distance from that point to the line segment bf is ζ0, and the distance to the line segment pe is γ0. Similarly, when the point Q on the line segment be is in the square 1, the distance from that point to the line segment b-f1 is ζ1, and the distance to the line segment p1-e is γ1. At this time, μ 0 when it is assumed that the point Q exists in the square 0

と、点Ｑが四角形１内に存在していると仮定した時のμ１

And μ1 when it is assumed that the point Q exists in the rectangle 1

が等しくなければならない。μ0＝μ1とするための、最も簡単な方法は、γ０=γ１、かつ、ζ０＝ζ１とすることである。γ0＝γ1、かつ、ζ0＝ζ1とするために、本発明では、γ0＝γ1＝線分Ｑｅの長さ、ζ0＝ζ1＝線分Ｑｂの長さとして定義する。

Must be equal. The simplest method for setting μ0 = μ1 is to set γ0 = γ1 and ζ0 = ζ1. In order to satisfy γ0 = γ1 and ζ0 = ζ1, in the present invention, γ0 = γ1 = the length of the line segment Qe and ζ0 = ζ1 = the length of the line segment Qb are defined.

  Next, consider a case where the point Q is on the line segment pf. There are two cases: the case where the point Q is in the region Rb and the case where the point Q is in the region Rc. Regardless of the region where the point Q exists, the distance must be defined so that the weighting factor is the same. In this case as well, the distance from the point Q to the line segment pe = the distance from the point Q to the line segment pg = the length of the line segment Qp, the distance from the point Q to the line segment fb = from the point Q to the line segment fc The distance may be defined as the length of the line segment Qf.

  Although the distance in the horizontal direction has been described above, the distance in the vertical direction can be similarly defined. Thus, the distance when the point Q is on the side of the square pebf can be defined.

  Next, consider a case where the point Q is inside the square pebf. FIG. 10 is a diagram for explaining a case where the point Q is inside the square pebf. It is necessary to define the boundary condition (if it is on the edge) and the distance when it is inside so as to be as continuous as possible. For example, it is assumed that the square pebf has a shape as shown in FIG. As discussed so far, on each side, the distance may be measured along the direction of that side.

  FIG. 10A shows the distance when the point Q is on the line segment be or on the line segment pf. It is assumed that the point Q moves from the line be to the line pf. At this time, the straight line for measuring the distance from the point Q to the line segment bf and the distance on the line segment pe moves from the line segment be to the line segment pf. This movement is desired to be as continuous as possible as shown in FIG. In order to obtain such a continuously moving straight line, the following method is used in this embodiment.

FIG. 11 is a diagram for explaining a method for obtaining a straight line that moves continuously. Assume that there is a square pebf as shown in FIG. A rectangular PEBF as shown in FIG. 11B is prepared. The PEBF may be a square or a parallelogram.
(1) Find a projective transformation S that maps the vertices p, e, b, and f of the quadrangle pebf to which the input coordinate q belongs to the vertices P, E, B, and F of the rectangular PEBF that is an arbitrary parallelogram.
(2) Similarly, a projective transformation S ′ (inverse transformation of S) is obtained in which the vertices P, E, B, and F of the rectangular PEBF are copied to the vertices p, e, b, and f of the quadrilateral pebf.
(3) The point q in the quadrangle pebf is copied to the point Q in the quadrangle PEBF using the projective transformation S. Hereinafter, the distance to the side of each region is calculated using the coordinates Q of the mapping destination.
(4) A straight line passing through the point Q and parallel to the sides PF and BE is drawn. The intersection between the straight line and PE is J, and the intersection with BF is K.
(5) Draw a straight line passing through the point Q and parallel to the sides PE and BF. Let L be the intersection of the straight line and PF, and M be the intersection of BE.
(6) Copy J, K, L, and M onto the sides of the square pebf using the projective transformation S ′. Let j, k, l, and m be the captured points, respectively.
(7) Distance from q to line segment pe = length of line segment qj, distance from q to line segment bf = length of line segment qk, distance from q to line segment pf = length of line segment ql , Q to the line segment be = the length of the line segment qm.

この変換の未知数は、射影変換パラメタの未知数と同じ8つである。射影変換パラメタを求めたときと同様に、四角形と四角形の対応関係を基に、8つの一次式を得ることができる。この連立一次方程式を解くことによって、バイリニア変換係数p₀,p₁,p₂,p₃, q₀,q₁,q₂,q₃を求めることができる。
（１）図７における点Qの座標を上記バイリニア変換で変換する。変換後の点をQ’とする。
（２）点Q’と線分ｐ’ｆ’とのユークリッド距離をαとする(四角形ｐ’ｅ’ｂ’ｆ’が長方形の場合はユークリッド距離、四角形ｐ’ｅ’ｂ’ｆ’が平行四辺形の場合は点Q’から線分ｂ’ｆ’に平行に引いた線上で距離を計測する。以下同様)。点Ｑ’と線分ｅ’ｂ’とのユークリッド距離をεとする。点Ｑ’と線分ｐ’ｅ’とのユークリッド距離をγとする。点Ｑ’と線分ｂ’ｆ’とのユークリッド距離をζとする。 Although the method for measuring the distances α, ε, γ, and ζ has been described above, other methods can be adopted. Hereinafter, other examples of measuring the distances α, ε, γ, and ζ will be described.
(1) Prepare a square p'e'b'f '. This square may be a parallelogram. Any size is acceptable.
(2) A bilinear transformation coefficient between the square pebf and the square p'e'b'f 'is obtained. Bilinear conversion can be defined by the following arithmetic expressions (22) and (23).

The number of unknowns in this conversion is the same as the number of unknowns in the projective transformation parameter. As in the case of obtaining the projective transformation parameter, eight linear expressions can be obtained based on the correspondence between the rectangles. By solving the simultaneous linear equations, bilinear transformation coefficients p ₀ , p ₁ , p ₂ , p ₃ , q ₀ , q ₁ , q ₂ , q ₃ can be obtained.
(1) The coordinates of the point Q in FIG. 7 are converted by the bilinear conversion. Let Q 'be the converted point.
(2) The Euclidean distance between the point Q ′ and the line segment p′f ′ is α (when the square p′e′b′f ′ is a rectangle, the Euclidean distance is parallel and the square p′e′b′f ′ is parallel). In the case of a quadrilateral, the distance is measured on a line drawn parallel to the line segment b'f 'from the point Q'. The Euclidean distance between the point Q ′ and the line segment e′b ′ is ε. The Euclidean distance between the point Q ′ and the line segment p′e ′ is γ. The Euclidean distance between the point Q ′ and the line segment b′f ′ is ζ.

  Bilinear transformation does not change the distance ratio between a point on the side of the quadrangle and the two end points of the side. Therefore, when the point Q is on the side, the distance to the two end points on the side is the same regardless of whether the point Q is in the square 0 or the square 1. Therefore, the continuity of the texture on the side can be maintained.

  As described above, the method for measuring the distance from the point Q to pf, pe, be, bf can be shown. For the coordinates in the other regions Ra, Rc, Rd, the weight coefficient can be obtained in the same manner. In this way, the region to which the input coordinate q belongs is a square pebf, and the projective transformation S that maps each vertex p, e, b, f of this square to each vertex P, E, B, F of an arbitrary parallelogram PEBF. Then, the input coordinate q is copied into the parallelogram PEBF, and the distance to the side of each region is calculated using the coordinate Q of the mapping destination. For this reason, since the straight line which moves continuously in a parallelogram can be used, more natural coordinate conversion can be performed.

  Next, the following two types of embodiments will be described. Heretofore, an example has been described in which a texture on a two-dimensional plane is mapped and projected onto an object surface in a three-dimensional space. However, since the texture mapping method described here is a method for projective transformation of a quadrangle to a quadrangle, it depends on which quadrangle is on the two-dimensional plane and which plane is on the projection view. Not. Therefore, it is also possible to use the object surface in the projection view obtained by perspective projection of the three-dimensional space for mapping on the two-dimensional texture image. Alternatively, one image does not have to be on a two-dimensional plane. It suffices if the squares of both images are associated by projective transformation.

  In the above description, the texture mapping is performed by obtaining the coordinates in the pre-projection image from the coordinates in the post-projection image. However, in order to perform texture mapping, a method of obtaining coordinates in the post-projection image from coordinates in the pre-projection image may be used. That is, in any case, as a premise, it is only necessary to have a plurality of adjacent quadrangles and a plurality of quadrangles in which the coordinates of the vertices can be individually associated with the quadrangles. An object of the present invention is to calculate the coordinates in the other rectangle from the coordinates in one rectangle so that the coordinate values are continuous between adjacent rectangles. Hereinafter, examples will be described.

  First, Example 1 will be described. The first embodiment is an example in which coordinates in an area divided by parallelograms (or rectangles or squares) having the same shape are input. The precondition of Example 1 is demonstrated using FIG. FIG. 12A is a diagram illustrating an input projection plane according to the first embodiment, and FIG. 12B is a diagram illustrating an output projection plane. As shown in FIG. 12A, a projection plane divided by squares having the same shape is input. Each square vertex dividing the input projection plane corresponds to each vertex on the output projection plane. For example, in FIG. 12, the vertex a of the input projection plane corresponds to the vertex A of the output projection plane. Similarly, vertex b corresponds to vertex B.

  In this embodiment, it is assumed that the correspondence between these vertices and the coordinates of each vertex have already been obtained. That is, in FIG. 12, it is assumed that (a) the coordinates of the 25 vertices of the input projection plane, (b) the coordinates of the 25 vertices of the output projection plane, and the association of the 25 vertices. In the output projection plane shown in FIG. 12 (b), the shape of the input projection plane changes, and a square shape at the time of input is a convex quadrangle. The input projection plane shown here can correspond to the post-projection image described above, and the output projection plane can correspond to the pre-projection image described above. An object of this embodiment is to obtain the coordinates in the pre-projection image using the coordinates in the post-projection image as input. Here, the coordinates of the input projection plane are input, and the coordinates of the output projection plane are output.

  Next, the operation of the first embodiment will be described with reference to FIG. FIG. 13 is a block diagram of the coordinate conversion apparatus according to the first embodiment. As illustrated in FIG. 13, the coordinate conversion apparatus 100 includes a projective conversion parameter calculation unit 102, a region determination unit 104, an adjacent quadrangle distance calculation unit 105, a weight coefficient calculation unit 106, and an output coordinate calculation unit 107.

  The coordinate conversion apparatus 100 converts coordinates in a plurality of adjacent rectangles into a plurality of adjacent rectangles. The input projection plane shown in FIG. 12A and the vertex coordinates 101 of the output projection plane shown in FIG. 12B are input to the projective transformation parameter calculation unit 102. The projective transformation parameter calculation unit 102 calculates a projective transformation parameter based on the input input projection plane and the vertex coordinates 101 of the output projection plane. The input coordinates 103 of the input projection plane shown in FIG. 12A are input to the region determination unit 104 to the region determination unit 104. The area determination unit 104 performs processing such as determining which area the input coordinates 103 belong to based on the input coordinates 103 of the input projection plane. The distance between adjacent rectangles calculation unit 105 calculates the distance between the input coordinates 103 and the adjacent rectangles. The weighting factor calculation unit 106 calculates the weighting factor of the adjacent rectangle using the distance between adjacent rectangles.

  The output coordinate calculation unit 107 calculates the weight coefficient by using the coordinates obtained using the projective transformation corresponding to the rectangle to which the input coordinates belong and the coordinates obtained using the projective transformation corresponding to the rectangle adjacent to the rectangle to which the input coordinates belong. The output coordinates are calculated by weighting with the weighting coefficient calculated by the calculation unit 106 and adding them.

  Each operation will be described in detail below. First, the operation of the projective transformation parameter calculation unit 102 will be described. First, the projective transformation parameter calculation unit 102 extracts the corresponding quadrangles in the input projection plane and the output projection plane one by one based on the vertex coordinates 101. Assume that the quadrilateral I in the output projection plane corresponds to the quadrilateral i in the input projection plane. In addition, it is assumed that the coordinates of the four vertices of the rectangle i are associated with the coordinates of the four vertices of the rectangle I. At this time, in the conventional example, as described using Expressions (1) to (4), it is possible to calculate eight projective transformation parameters for converting the square i into the square I. Hereinafter, the projective transformation parameter corresponding to the rectangle i is assumed to be Si.

  Here, for example, an example in which the coordinate value of the pre-projection image is obtained in reverse from the coordinate value of the post-projection image is shown, so that the projective transformation parameter is defined opposite to equations (1) and (2). This is shown in equations (24) and (25). The input coordinate is (u, v), and the output coordinate value after projective transformation of (u, v) is (S, L).

（ｕ,ｖ）と（Ｓ,Ｌ）に上記の関係があるとき、p₀、p₁、p₂、q₀、q₁、q₂、r₁、r₂を射影変換パラメタとして定義する。図１３において、射影変換パラメタ算出は、重み係数算出と独立に行われているが、必ずしも独立に行う必要は無く、出力座標算出に必要な射影変換パラメタを都度算出するようにしても良い。

When (u, v) and (S, L) have the above relationship, p ₀ , p ₁ , p ₂ , q ₀ , q ₁ , q ₂ , r ₁ , r ₂ are defined as projective transformation parameters. In FIG. 13, the projective transformation parameter calculation is performed independently of the weighting factor calculation. However, it is not always necessary to perform the projective transformation parameter, and the projective transformation parameter necessary for the output coordinate calculation may be calculated each time.

  Next, the operation of the area determination unit 104 will be described. The region determination unit 104 checks in which square in the input projection plane the input coordinate 103 exists. Here, the region determination unit 104 sets the quadrangle where the input coordinates 103 exist as the quadrangle 0. Next, the region determination unit 104 checks in which region (Ra, Rb, Rc, or Rd) the input coordinate 103 exists within the square 0. Here, as an example, the coordinates are input and the rectangle and the area where the coordinates exist are obtained, but actually, the rectangle and the area are first specified, and the coordinates existing therein are determined. Of course, you can do it. FIG. 14 shows a configuration diagram in this case.

  FIG. 14 is a diagram illustrating a modification of the coordinate conversion apparatus according to the first embodiment. As shown in FIG. 14, the coordinate conversion apparatus 200 includes a projective conversion parameter calculation unit 102, a coordinate output unit 110, a distance calculation unit 105 between adjacent rectangles, a weighting factor calculation unit 106, and an output coordinate calculation unit 107. The coordinate output unit 110 determines the coordinates existing in the rectangle based on the input area 109. The same parts as those in FIG. 13 are denoted by the same reference numerals, and the description thereof is omitted.

  Next, the area division performed by the area determination unit 104 will be described. FIG. 15 is a diagram for explaining the region division method. The area determination unit 104 sets the center of each square of the input projection plane, that is, the intersection of the square diagonal lines as the center point P. In addition, the region determination unit 104 divides the square into four regions Ra, Rb, Rc, and Rd using a line segment extending from the center point P to the midpoint of each side.

  In FIG. 15, it is assumed that a square 0 is a square in which the input coordinates 103 exist. As shown in FIG. 15, the vertices of the square 0 are a, b, c, and d. Also, e is the midpoint of the side ab, f is the midpoint of the side bc, G is the midpoint of the side cd, and h is the midpoint of the side da. The center point P of the quadrangle 0 is the intersection of the line segment eg and the line segment fh. The square 0 is divided into four regions Ra, Rb, Rc, and Rd using line segments Pe, Pf, Pg, and Ph. As described above, the region determination unit 104 divides the region by a straight line or a curve passing through the midpoint of the quadrangular side.

  Next, the operation of the distance calculation unit 105 between adjacent squares will be described. FIG. 16 is a diagram for explaining the distance between adjacent parallelograms. The distance between adjacent rectangles calculation unit 105 calculates the distance between adjacent rectangles as follows. As shown in FIG. 16, it is assumed that the square 0 is adjacent to the squares 1 to 8. In addition, it is assumed that the center of the square i (i = 0, 1, 2,..., 8) is Pi. Let the coordinates of Pi be (ui, vi). Further, assume that the input coordinates are Q = (u, v). The distance between adjacent squares is the value of α, β, γ, and δ already described. These four values are calculated as follows.

First, regardless of the region to which the point Q belongs, α = | u-u0 | and γ = | v-v0 |. Also,
(1) When the point Q is in the region Ra, β = | u-u1 |, δ = | v-v7 |
(2) When the point Q is in the region Rb, β = | u-u1 |, δ = | v-v3 |,
(3) When the point Q is in the region Rc, β = | u-u5 |, δ = | v-v3 |,
(4) When the point Q is in the region Rd, β = | u-u5 |, δ = | v-v7 |
It becomes. For example, the lengths of α, β, γ, and δ when the point Q is in the region Rb are illustrated in FIG. In this case, α is the horizontal distance between the points Q and P0, β is the horizontal distance between the points Q and P1, γ is the vertical distance between the points Q and P0, and δ is the vertical distance between the points Q and P3. It is.

  The method described using FIG. 16 is an example of calculating the distance between adjacent rectangles, and the distance calculation unit 105 between adjacent rectangles calculates the distance between adjacent rectangles using the method described in the above concept of the invention. You can also.

  Next, the operation of the weight coefficient calculation unit 106 will be described. The weight coefficient calculation unit 106 calculates the weight coefficient as follows. Here, a weighting coefficient Wi (i = 0, 1,..., 8) corresponding to the input coordinate Q = (u, v) is obtained. First, regardless of the region to which the point Q belongs, the weighting factor W0 is determined by equation (26).

次に、（１）点Ｑが領域Ｒａにあるとき、重み係数W2=W3=W4=W5=W6=0、W1、W7、W8は式（２７）〜（２９）により決定される。

Next, (1) when the point Q is in the region Ra, the weighting factors W2 = W3 = W4 = W5 = W6 = 0, W1, W7, W8 are determined by the equations (27) to (29).

（２）点Ｑが領域Ｒｂにあるとき、重み係数W4=W5=W6=W7=W8=0、W1、W2、W3は式（３０）〜（３２）により決定される。

(2) When the point Q is in the region Rb, the weighting factors W4 = W5 = W6 = W7 = W8 = 0, W1, W2, W3 are determined by the equations (30) to (32).

（３）点Ｑが領域Ｒｃにあるとき、重み係数W1=W2=W6=W7=W8=0、W3、W4、W5は式（３３）〜（３５）により決定される。

(3) When the point Q is in the region Rc, the weighting factors W1 = W2 = W6 = W7 = W8 = 0, W3, W4, W5 are determined by the equations (33) to (35).

（４）点Ｑが領域Ｒｄにあるとき、重み係数W1=W2=W3=W4=W8=0、W5、W6、W7は式（３６）〜（３８）により決定される。

(4) When the point Q is in the region Rd, the weighting factors W1 = W2 = W3 = W4 = W8 = 0, W5, W6, W7 are determined by the equations (36) to (38).

とする。上記では、βやδの距離を計測したが、概念で述べたように、εやζの距離を計算しても、もちろん良い。

And In the above description, the distances β and δ are measured. However, as described in the concept, it is of course possible to calculate the distances ε and ζ.

  Note that the method described here is an example of a weighting factor calculation method, and the weighting factor calculation unit 106 can also calculate the weighting factor using the method described in the above concept of the invention.

Next, the operation of the output coordinate calculation unit 107 will be described. The weighting factor calculation unit 106 has already calculated the weighting factors W0 to W8. In addition, the projective transformation parameter calculation unit 102 has already calculated the projective transformation parameters S0 to S8 of the respective squares 0 to 8. First, the output coordinate calculation unit 107 uses the projective transformation parameters Si (i = 0, 1, 2,..., 8) of the rectangle i and the input coordinates Q = (u, v) to output corresponding to each rectangle. The coordinate Xi is calculated. For example, the projective transformation parameter of the quadrangle 0 is a parameter that projects onto the quadrangle 0 in the input projection plane and the quadrangle 0 in the output projection plane. These parameters are assumed to be p ₀ , p ₁ , p ₂ , q ₀ , q ₁ , q ₂ , r ₁ , r ₂ .

  The output coordinate calculation unit 107 obtains the output coordinate X0 (S0, L0) for the input coordinate Q = (u, v) using the projective transformation parameter S0 for the square 0 and the equations (24) and (25).

  Similarly, the output coordinate calculation unit 107 uses the projective transformation parameters Si corresponding to the other quadrilateral i and the expressions (24) and (25) to output the output coordinates (Si, Li). Here, the output coordinate value calculated using the projective transformation parameter of the square i is assumed to be Xi = (Si, Li). Xi is a two-dimensional vector.

  The output coordinate calculation unit 107 obtains the final output coordinate X (S, L) by weighting and adding the output coordinate Xi with the weighting coefficient Wi as shown in Expression (39).

このとき、iがWi=0の場合の出力座標値Ｘiは特に算出する必要はない。

At this time, it is not necessary to calculate the output coordinate value Xi when i is Wi = 0.

  As described above, the output coordinate calculation unit 107 uses the expression (39) to output the output coordinates X0 (S0, L0) obtained by using the projective transformation S0 corresponding to the rectangle 0 to which the input coordinates Q (u, v) belong. And the coordinates Xi (Si, Li) obtained by using the projective transformation Si corresponding to the quadrilateral i (i ≠ 0) adjacent to the quadrilateral 0 to which the input coordinates Q (u, v) belong are weighted by the weighting factor Wi. It is shown that the final output coordinates X (S, L) can be calculated by adding together.

  Note that the method described here is an example of an output coordinate calculation method, and the output coordinate calculation unit 107 can also calculate output coordinates using the method described in the above concept of the invention.

  Moreover, in the above Example 1, the example in which the input projection figure was divided | segmented by the square of the same shape was shown. The above method can be used in the same manner even when divided by a rectangle having the same shape instead of a square. Furthermore, the above-described method can be used even when divided not by a rectangle but by a parallelogram having the same shape. In the case of a parallelogram, the operation of the distance calculation unit between adjacent squares is slightly different. Of course, since squares and rectangles are also parallelograms, the parallelogram method described below can also be used for squares and rectangles.

  The case of a parallelogram will be described with reference to FIG. In FIG. 16, it is assumed that squares 0 to 8 are parallelograms having the same shape. The square 0 has four vertices a, b, c, and d. The adjacent quadrangle distance calculation unit 105 calculates α, β, γ, and δ as follows. A case where the point Q is in the region Rb will be described as an example. A straight line (straight line 1) parallel to the line segment ad (or line segment bc) is drawn through the point Q. Further, a straight line (straight line 2) parallel to the line segment ab (or line segment cd) is drawn through the point Q.

  The intersection of line 1 and line segment P0-P3 is S30, the intersection of line 1 and line segment P1-P2 is S12, the intersection of line 2 and line segment P2-P3 is S23, and the intersection of line 2 and line segment P0-P1 is S01. α is the length of the line segment Q-S30, β is the length of the line segment Q-S12, γ is the length of the line segment Q-S01, and δ is the length of the line segment Q-S23. Similarly, when the point Q is in another region, the length of the intersection of the parallel line and the center point may be measured by subtracting a parallel line from the point Q to the square side.

  The distance calculation unit 105 between adjacent squares and the output coordinate calculation unit 107 correspond to a distance calculation unit and a coordinate calculation unit in the claims.

  Next, a second embodiment will be described. In the first embodiment, the method of calculating the coordinates in the output projection map from the coordinates of the input projection map divided by the parallelograms having the same shape is shown. The second embodiment shows a method for calculating coordinates in the output projection map from the coordinates of the input projection map divided by the convex quadrangle. The configuration in the second embodiment is the same as that described in the first embodiment with reference to FIG. The differences from the first embodiment are the region dividing method by the region determining unit 104 and the operations of the distance calculation unit 105 between adjacent rectangles and the weighting factor calculating unit 106. Here, it is assumed that the projective transformation parameter has already been obtained. Also, it is assumed that the input coordinate area has been determined.

  First, the region dividing method will be described. FIG. 17 is a diagram for explaining region division by the region determination unit 104. In FIG. 17, only the square 0 is shown, and the adjacent squares 1 to 8 are omitted. As in the first embodiment, the rectangle to which the input coordinate Q = (u, v) belongs is defined as a rectangle 0. Similarly to the first embodiment, as shown in FIG. 17, the center point p, the point e on the side ab, the point f on the side bc, the point g on the side cd, and the point h on the side da are Use the region to be divided. The positions of the points p, e, f, g, and h may be arbitrarily determined.

  However, the point e is also the division point of the side ab in the square 1, the point f is the division point of the side bc in the same way as the square 3, the point g is the division point of the side cd in the same way as the square 5, and the point h is the square 7 Similarly, it must be set to be a dividing point of the side da. For example, the point e is the midpoint of the side ab, the point f is the midpoint of the side bc, the point g is the midpoint of the side cd, the point h is the midpoint of the side da, and the point p is a line segment An example is an intersection of eg and line segment fh. As shown in FIG. 17, a square 0 is divided using line segments pe, pf, pg, and ph to form regions Ra, Rb, Rc, and Rd.

  As described above, the region determination unit 104 divides the region by a straight line that passes through a straight line that connects the midpoints of opposite sides of the quadrangle.

  Next, the operation of the distance calculation unit 105 between adjacent squares will be described. The distance between adjacent squares is the above-described values of α, γ, ε, and ζ. These values can be obtained by the above-described method described as a concept. Next, the operation of the weight coefficient calculation unit 106 will be described. The weighting factor calculation unit 106 obtains a weighting factor using the above values of α, γ, ε, and ζ and the equations (9) to (12). Further, the output coordinate calculation unit 107 obtains output coordinates using the weighting factor obtained above. Since this method is the same as the method described in the first embodiment or the concept portion, the description is omitted here.

  According to each of the embodiments described above, the coordinates obtained using the projective transformation corresponding to the rectangle to which the input coordinates belong and the coordinates obtained using the projective transformation corresponding to the rectangle adjacent to the rectangle to which the input coordinates belong are weighted. Since the output coordinates are calculated by adding the values, the coordinates in the other rectangle can be calculated from the coordinates in one rectangle so that the coordinate values are continuous between adjacent rectangles. As a result, when a plurality of adjacent quadrilaterals are projectively transformed into a plurality of adjacent quadrilaterals and the texture in the quadrangle is mapped, the coordinates are discontinuous on the adjacent sides depending on the shape of the quadrangle. On the other hand, by using this method, the discontinuity of coordinates can be eliminated, which enables higher quality texture mapping.

  In addition, by using the coordinate conversion device or the coordinate conversion method, a figure in which a texture in an image obtained by projecting a surface on one object onto a plane is mapped onto a surface on another object and further projected onto a plane is displayed. It is possible to provide a method and apparatus for creating, and more particularly, a texture mapping apparatus or method for mapping a texture on a two-dimensional plane to a figure obtained by projecting an object in a three-dimensional space onto the two-dimensional plane.

  The coordinate conversion method described in FIG. 5 and the coordinate conversion apparatus described in FIG. 13 or FIG. 14 use, for example, a CPU (Central Processing Unit), a ROM (Read Only Memory), a RAM (Random Access Memory), and the like. FIG. 5 illustrates that the program is installed from a hard disk device, 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. Each step and each function shown in FIG. 13 or FIG. 14 are realized.

  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 perspective projection explanatory drawing of a plane texture image. It is a figure for demonstrating the method of patent document 1. FIG. It is explanatory drawing regarding a texture discontinuity point. It is a figure for demonstrating an invention concept, (a) is a figure which shows a pre-projection image, (b) is a figure which shows a post-projection image. It is a processing flowchart of a coordinate transformation method. It is explanatory drawing of an invention concept. Taking the case where the input coordinates are in the region Rb as an example, a method for setting the weighting factor in Rb will be described. It is a figure which shows the example of the content of the monotone increase function F. It is a figure explaining the method of measuring the distance from the point Q to line segment pf, pe, be, bf. It is a figure explaining the case where the point Q exists in the inside of square pebf. It is a figure explaining the method for calculating | requiring the straight line which moves continuously. (A) is a figure which shows the input projection surface which concerns on Example 1, (b) is a figure which shows an output projection surface. 1 is a block diagram of a coordinate conversion device according to Embodiment 1. FIG. It is a figure which shows the modification of the coordinate transformation apparatus which concerns on Example 1. FIG. It is a figure for demonstrating an area division system. It is a figure for demonstrating the distance between adjacent parallelograms. It is a figure for demonstrating the area | region division by an area | region determination part.

Explanation of symbols

DESCRIPTION OF SYMBOLS 100,200 Coordinate transformation apparatus 102 Projection transformation parameter calculation part 104 Area | region determination part 105 Distance calculation part between adjacent squares 106 Weight coefficient calculation part 107 Output coordinate calculation part 110 Coordinate output part

Claims (16)

A coordinate conversion device that converts coordinates in a plurality of adjacent rectangles into a plurality of adjacent rectangles,
A weighting factor calculating unit that calculates a weighting factor using the distance between the input coordinates and the adjacent rectangle;
The weight obtained by weighting the coordinates obtained using the projective transformation corresponding to the quadrangle to which the input coordinate belongs and the coordinate obtained using the projective transformation corresponding to the quadrangle adjacent to the quadrangle to which the input coordinate belongs A coordinate conversion apparatus comprising: a coordinate calculation unit that calculates an output coordinate by adding. The coordinate conversion apparatus according to claim 1, wherein the weighting factor calculation unit calculates a weighting factor so that a vertex position of a quadrangular projection destination does not change. The coordinate conversion apparatus according to claim 1, wherein the weighting factor calculation unit equalizes the weighting factors corresponding to the two rectangles on an edge shared by the two rectangles. The coordinate conversion device further includes a region determination unit that divides a quadrangle to which the input coordinates belong into four regions,
The weighting factor calculation unit calculates weighting factors corresponding to four adjacent quadrilaterals including the quadrilateral to which the input coordinates belong in each divided area, and sets the weighting factors of the other quadrilaterals to 0. The coordinate transformation device according to any one of claims 1 to 3. The coordinate conversion device further includes a distance calculation unit that calculates a distance from the input coordinates to an adjacent quadrangle that exists in the vertical direction and a distance from the input coordinates to an adjacent quadrangle that exists in the horizontal direction,
The weighting factor calculation unit calculates the weighting factor by multiplying a ratio of a distance to the quadrangle existing in the vertical direction and a ratio of a distance to the quadrangle present in the horizontal direction. The coordinate transformation device according to any one of claims 4 to 5. The distance calculation unit calculates a distance between the input coordinate and each side of the region to which the input coordinate belongs, and sets a distance ratio between the side and a square adjacent to the side as a distance. The coordinate transformation device according to claim 4 or 5. 6. The coordinate transformation apparatus according to claim 5, wherein the weighting factor calculation unit uses a value obtained by substituting the distance ratio into a monotonically increasing function as a new distance ratio. The distance calculation unit takes a region to which the input coordinate belongs as a rectangle and projects the input coordinate into the parallelogram using projective transformation that maps each vertex of the rectangle to each vertex of an arbitrary parallelogram, The coordinate conversion apparatus according to claim 6, wherein the distance to the side of each region is calculated using the coordinates of the mapping destination. 9. The weighting factor calculation unit according to any one of claims 1 to 8, wherein the weighting factor corresponding to the quadrangle sharing the vertex is set to a quarter on each vertex of the quadrangle. Coordinate conversion device. The weighting factor calculation unit sets at least one point or two or more points on each side of the quadrangle that is a half of the weighting factor corresponding to the quadrangle sharing the side. The coordinate transformation device according to any one of the above. The weighting factor calculation unit sets one or more points in the quadrangle where the weighting factor of the quadrangle is set to 1 and the weighting factor of the other quadrangle is set to 0. 11. A coordinate conversion apparatus according to claim 1. The coordinate transformation device according to any one of claims 4 to 11, wherein the region determination unit divides the region by a straight line or a curve passing through a midpoint of a quadrangle side. The coordinate transformation device according to any one of claims 4 to 11, wherein the region determination unit divides the region by a straight line passing through a straight line connecting the midpoints of opposite sides of the quadrangle. The coordinate according to any one of claims 1 to 13, wherein the weighting factor calculation unit sets the sum of the weighting factors of each quadrature to 1, and the weighting factor is a real number of 0 or more and 1 or less. Conversion device. The weighting factor calculation unit sets the weighting factor corresponding to a rectangle other than the rectangle closest to the rectangle in which the input coordinates exist on a boundary line other than the sides of the rectangle of each region to 0. Item 15. The coordinate conversion device according to any one of Items 14 to 14. To convert coordinates in multiple adjacent rectangles into multiple adjacent rectangles,
  A weighting factor calculating step for calculating a weighting factor using the distance between the input coordinates and the adjacent rectangle;
  The coordinates obtained using the projective transformation corresponding to the rectangle to which the input coordinates belong and the coordinates obtained using the projective transformation corresponding to the rectangle adjacent to the rectangle to which the input coordinates belong are added by weighting with the weighting factor. A program for executing a coordinate calculation step of calculating output coordinates by doing.

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