JP2006139367A - Image transformation method, image transformation apparatus, and image transformation program - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an image transformation method for performing affine transformation whereby a certain triangle is transformed into a certain triangle by expansion and contraction along X-axis and Y-axis, rotational movement about the original point, and combination of coordinate planes capable of parallel translation. <P>SOLUTION: An image transformation apparatus performs (1) a step for moving the apex A of a triangle ABC to the original point on a plane M1 according to the apex coordinates of the original triangle ABC and the desired triangle A'B'C', and superposing the bisector of the angle A on X-axis; (2) a step for expanding the triangle ABC along Y-axis on a plane M2, and making the angle A orthogonal; (3) a step for expanding the sides AB and AC respectively at first and second predetermined magnifications on a plane M3; (4) and a step for expanding the angle A along Y-axis at a third predetermined magnification on a plane M4 so that the angle A equals the angle A', and causing rotational movement and parallel translation so that the coordinate positions of the triangle ABC coincide with the coordinate positions of the triangle A'B'C'. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to an image conversion method, an image conversion apparatus, and an image conversion program for displaying an image after geometric conversion.

  2. Description of the Related Art Conventionally, an image processing apparatus that generates a three-dimensional image by a computer and displays the generated three-dimensional image is known (see, for example, Patent Document 1). A conventional image processing apparatus can generate an image in which a target is viewed from an arbitrary angle by dividing the target by triangular patches and geometrically converting each triangular patch. There is also known a method of deforming an image therein when a three-dimensional patch is deformed (see, for example, Non-Patent Document 1).

  By the way, affine transformation, which is one of geometric transformations, is expressed by equation (1). In equation (1), the coordinates P (x, y) are converted into coordinates P ′ (x ′, y ′) by the matrix characterized by the parameters a to f. By these parameters a to f, conversion such as translation, rotation, enlargement, reduction, and inclination can be expressed.

Here, an example of affine transformation will be described. 2A and 2B are diagrams for explaining how an image is converted. FIG. 2A shows an image before conversion, and FIG. 2B shows an image after conversion. In the frame shown in FIG. 2, the upper left point is the origin O (0, 0), the upper side is the X axis (right direction is positive), and the left side is the Y axis (lower direction is positive). As shown in FIG. 2A, the letter “F” as an image is arranged in the inner area of the triangle ABC. Consider a case where the image area of the triangle ABC is converted into an image area of a triangle A′B′C ′ as shown in FIG. Respective vertices of the triangle ABC before conversion are represented by A (x ₀ , y ₀ ), B (x ₁ , y ₁ ), C (x ₂ , y ₂ ), and the target converted triangle A′B′C ′. Let A ′ (x ₀ ′, y ₀ ′), B ′ (x ₁ ′, y ₁ ′), and C ′ (x ₂ ′, y ₂ ′). The conversion at this time can be expressed as shown in Equation (2).

  In order to obtain the parameters a to f of the equation (2), the inverse matrix of the second matrix on the right side of the equation (2) is multiplied from both sides of the equation (2) from the right to obtain the equation (3). It is done. If the three points A ′, B ′, and C ′ are not on the same straight line, the second matrix on the right side of Equation (2) is a regular matrix, and therefore there is an inverse matrix. By calculating the right side of the equation (3), specific values of the parameters a to f can be obtained. By substituting the obtained parameters a to f into the equation (1) and using affine transformation, the respective coordinate points after the pixels in the image inside the triangle ABC are transformed into the inside of the triangle A′B′C ′ are calculated. be able to.

  As described above, when individual pixels can be set, the image can be converted (deformed) by performing the conversion by the affine transformation shown in Expression (1). For example, in a page description language such as PostScript language, an affine parameter for deformation can be specified.

By the way, Flash (registered trademark) of Macromedia Co., Ltd. manages drawing objects on a coordinate plane called MovieClip (hereinafter abbreviated as MC). The features of this MC will be described with reference to FIGS. 3A and 3B are diagrams for explaining geometric transformation in the coordinate plane (MC), where FIG. 3A shows enlargement / reduction, FIG. 3B shows rotational movement, and FIG. 3C shows parallel movement. FIG. 4 is a perspective view showing the arrangement of coordinate planes.
In MC, as shown in FIG. 3A, an arbitrary image area can be enlarged (or reduced) in the X-axis and Y-axis directions. The enlargement ratio (reduction ratio) in the X-axis and Y-axis directions is specified by a parameter xscale and a parameter yscale, respectively. Further, in MC, as shown in FIG. 3B, an arbitrary image region can be rotated around the origin. The rotation angle at this time is specified by the parameter rotation. In FIGS. 3A and 3B, the direction of the illustrated arrow is positive. Further, in MC, as shown in FIG. 3C, an arbitrary image region can be translated by designating a difference x in the X direction and a difference y in the Y direction. Moreover, MC can take a hierarchical structure as shown in FIG. In the example shown in FIG. 4, the coordinate plane M1 is arranged in the lowest layer in the four-layer structure, and the coordinate plane M2, the coordinate plane M3, and the coordinate plane M4 are sequentially arranged in the z direction.
JP 2002-222430 A (paragraphs 0036-0039, FIG. 6) "Digital Image Warping" by George Wolberg, IEEE Computer Society Press Monograph, 1990, full page

  However, this MC has a problem that affine parameters cannot be directly specified.

  Therefore, the present invention solves the above-described problem and combines an arbitrary triangle with an arbitrary triangle by combining coordinate planes that can be expanded / reduced in the X-axis and Y-axis directions, rotated around the origin, and translated. An object of the present invention is to provide an image conversion method, an image conversion apparatus, and an image conversion program capable of performing affine transformation to be converted into an image.

  In order to solve the above-described problem, the image conversion method according to claim 1, in the images on the first to fourth coordinate planes sequentially stacked, enlargement or reduction in the X-axis direction and the Y-axis direction, and setting the origin The rotational movement around the center and the parallel movement to an arbitrary coordinate point are made possible in this order, and the image area of the triangle ABC on the first coordinate plane as the lowermost layer is changed to the fourth coordinate as the uppermost layer. In the image conversion method in the image conversion apparatus for converting into the image area of the triangle A′B′C ′ on the plane, the image conversion apparatus includes a first input step, a second input step, a first conversion step, The second conversion step, the third conversion step, the fourth conversion step, and the fifth conversion step are executed.

  According to such a procedure, the image conversion apparatus inputs the vertex coordinates of the triangle ABC in the first input step, and inputs the vertex coordinates of the triangle A′B′C ′ in the second input step. Then, in the first conversion step, the image conversion apparatus moves the vertex A of the triangle ABC input to the first coordinate plane to the origin and superimposes the bisector of the angle A on the X axis or the Y axis. In the second conversion step, the triangle ABC is enlarged or reduced in the direction orthogonal to the bisector of the angle A so that the angle A is a right angle on the second coordinate plane. As a result, a first triangle in which the conversion source triangle is deformed is generated. Since the first triangle is a right triangle, the lengths of the side AB and the side AC can be freely changed by rotating the side AB and the side AC so as to be on the X axis and the Y axis, respectively.

  Therefore, in the third conversion step, the image conversion apparatus enlarges or reduces the side AB and the side AC of the right triangle ABC at the first and second predetermined magnifications on the third coordinate plane, respectively. Here, the first and second predetermined magnifications are such that the side AB and the side AC are finally subjected to all the conversion steps, and finally the side triangle A′B ′ and the side A of the target triangle A′B′C ′. It is determined by back calculation so as to be equal to 'C'. Then, in the fourth conversion step, the image conversion apparatus converts the triangle ABC to the angle A so that the angle A is equal to the angle A ′ of the triangle A′B′C ′ on the fourth coordinate plane. Enlarge or reduce in a direction orthogonal to the bisector at a third predetermined magnification. As a result, the generated triangle ABC becomes congruent with the triangle A′B′C ′. Further, in the fifth conversion step, the image conversion apparatus causes the triangle ABC to coincide with the coordinate position of the triangle A′B′C ′ by rotating and translating the triangle ABC on the fourth coordinate plane. As a result, the image area of the original triangle ABC is converted into the image area of the triangle A′B′C ′.

  The image conversion method according to claim 2 is the image conversion method according to claim 1, wherein the first predetermined magnification is the triangle ABC generated in the second conversion step as a bisector of the angle A. The side AB after being expanded or reduced in the orthogonal direction at the third predetermined magnification is equal to the magnification when the side AB ′ of the triangle A′B′C ′ is enlarged or reduced, and the second predetermined magnification is The side AC after the triangle ABC generated in the second conversion step is enlarged or reduced at the third predetermined magnification in the direction orthogonal to the bisector of the angle A is represented by the triangle A′B′C ′. It is characterized by being equal to the magnification when enlarging or reducing to the side A′C ′.

  In this way, the triangle ABC generated by expanding or reducing the triangle ABC generated in the second conversion step at the third predetermined magnification in the direction orthogonal to the bisector of the angle A is The angle A becomes equal to the angle A ′ of the triangle A′B′C ′. Therefore, if the length of the two sides AB and AC having the angle A as an included angle is expanded to fit the sides A′B ′ and A′C ′, the triangle ABC is divided into the triangle A′B′C ′. And become congruent. However, since the enlargement direction is limited to the X-axis and Y-axis directions, it is difficult to enlarge both sides AB and AC with the included angle A fixed. Therefore, the enlargement factor calculated as if such conversion is possible is set as the first and second predetermined magnifications in the third conversion step, and an enlargement operation is performed before fixing the included angle. Conversion can be possible.

  An image conversion method according to a third aspect is the image conversion method according to the first or second aspect, wherein the image conversion device is configured to perform the first conversion when the vertex coordinates of the triangle ABC are input in the first input step. The first and second conversion steps are executed, and when the vertex coordinates of the triangle A′B′C ′ are input by the second input step, the third to fifth conversion steps are executed. .

  According to such a procedure, the calculation when the vertex coordinates of the conversion source triangle ABC are changed or the vertex coordinates of the converted triangle A′B′C ′ are changed can be executed independently. Time can be shortened. For example, if the vertex coordinates of the triangle ABC are captured in the first input step in advance and the first and second conversion steps are executed, then the vertex coordinates of the triangle A′B′C ′ are input. Accordingly, various converted images can be displayed in a short time.

  An image conversion program according to a fourth aspect causes a computer to execute the image conversion method according to any one of the first to third aspects. With this configuration, a computer installed with this program can realize each function based on the program.

  The image conversion apparatus according to claim 5, with respect to the images on the first to fourth coordinate planes sequentially stacked, enlargement or reduction in the X-axis direction and the Y-axis direction, rotational movement about the origin, and The image area of the triangle ABC on the first coordinate plane that is the lowermost layer is represented as a triangle A′B′C ′ on the fourth coordinate plane that is the uppermost layer. In the image conversion apparatus for converting to the image area, the image processing apparatus includes an input unit, a first coordinate plane processing unit, a second coordinate plane processing unit, a third coordinate plane processing unit, and a fourth coordinate plane processing unit. did.

  According to this configuration, the image conversion apparatus inputs the vertex coordinates of the triangle ABC and the triangle A′B′C ′ by the input unit. Then, the image conversion apparatus moves the vertex A of the triangle ABC input to the first coordinate plane to the origin by the first coordinate plane processing unit and superimposes the bisector of the angle A on the X axis or the Y axis. The triangle ABC is enlarged or reduced in the direction orthogonal to the bisector of the angle A on the second coordinate plane by the second coordinate plane processing unit, and the first triangle ABC with the angle A being a right angle. Convert to Then, the image conversion apparatus enlarges or reduces the side AB and the side AC of the first triangle ABC at the first and second predetermined magnifications on the third coordinate plane by the third coordinate plane processing unit. To the second triangle ABC. Further, the image conversion apparatus is configured such that the fourth coordinate plane processing unit causes the fourth coordinate plane so that the angle A of the second triangle ABC becomes equal to the angle A ′ of the triangle A′B′C ′ on the fourth coordinate plane. The second triangle ABC is enlarged or reduced at a third predetermined magnification in a direction perpendicular to the bisector of the angle A to be transformed into the third triangle ABC, and the third triangle ABC is rotated and translated. By doing so, the coordinate position of the triangle A′B′C ′ is made coincident.

  The image conversion apparatus according to claim 6 is the image conversion apparatus according to claim 5, wherein the first predetermined magnification is the third triangle ABC in the direction orthogonal to the bisector of the angle A. The side AB of the fourth triangle ABC generated by enlarging or reducing at a predetermined magnification is equal to the magnification at the time of enlarging or reducing to the side A′B ′ of the triangle A′B′C ′. The magnification is obtained by expanding the side AC of the fourth triangle ABC generated by enlarging or reducing the first right triangle ABC at a third predetermined magnification in the direction perpendicular to the bisector of the angle A, and the triangle A ′. It is equal to the magnification when enlarging or reducing to the side A′C ′ of B′C ′.

  According to this configuration, the fourth triangle ABC has the angle A equal to the angle A ′ of the triangle A′B′C ′. Accordingly, the fourth triangle ABC can be obtained by expanding the length of the two sides AB and AC having the angle A as an included angle so as to fit the sides A′B ′ and A′C ′. It becomes congruent with C '. However, since the enlargement direction is limited to the X-axis and Y-axis directions, it is difficult to enlarge both sides AB and AC with the included angle A fixed. Therefore, any affine transformation can be performed by performing an enlargement operation using the third coordinate plane processing unit as the first and second predetermined magnifications with the enlargement factor calculated as if such conversion is possible. Can be.

According to the present invention, an image area of an arbitrary triangle ABC can be obtained by using four layers of coordinate planes that can be enlarged / reduced in the X-axis and Y-axis directions, rotated and translated around the origin in this order. Can be converted into an image area of an arbitrary triangle A′B′C ′.
Moreover, since it is possible to separate into two independent conversion operations, a conversion operation according to the input of the vertex coordinates of the triangle before conversion and a conversion operation according to the input of the vertex coordinates of the triangle after conversion, depending on the input Thus, only the necessary part can be calculated, and the calculation time can be reduced.

  In the following, first, a method for image conversion in the image conversion device will be described, and an image conversion device that performs image conversion based on the method will be described sequentially.

[Prerequisites for image conversion method]
The premise of the image conversion method of the present invention will be described with reference to FIGS. According to the image conversion method of the present invention, an image area of a triangle ABC having a predetermined shape existing at a predetermined position is converted into a triangle A′B′C ′ (vertices A ′, B ′, C ′ having different shapes existing at another position. Are converted so as to be image regions of vertices A, B, and C). As shown in FIG. 2A, the vertices of the triangle ABC before conversion are represented by A (x ₀ , y ₀ ), B (x ₁ , y ₁ ), C (x ₂ , y ₂ ), and after conversion. The vertices of the triangle A′B′C ′ are A ′ (x ₀ ′, y ₀ ′), B ′ (x ₁ ′, y ₁ ′), and C ′ (x ₂ ′, y ₂ ′). .

  In the coordinate plane assumed in the present invention, conversion (conversion parameters xscale, yscale) in the X-axis direction and the Y-axis direction shown in FIG. 3A (conversion parameters xscale, yscale) is shown in FIG. It is assumed that conversion by rotation movement around the origin (conversion parameter rotation) and conversion by parallel movement to an arbitrary coordinate point shown in FIG. 3C are possible. Further, as shown in FIG. 4, the coordinate planes are arranged in order from the bottom layer: coordinate plane M1 (first coordinate plane), coordinate plane M2 (second coordinate plane), coordinate plane M3 (third coordinate plane). ) And a coordinate plane M4 (fourth coordinate plane).

  In each coordinate plane, three types of conversion (enlargement or reduction, rotational movement, translation) are possible, but in one coordinate plane, (1) enlargement or reduction, (2) rotational movement, (3) translation The conversion operation is defined in the order of. Therefore, the image conversion method of the present invention is executed by appropriately combining in consideration of the order of the three types of conversion operations that can be performed on each of the four coordinate planes. As an example of the coordinate plane assumed in the present invention, there can be mentioned Flash (registered trademark) MovieClip of Macromedia.

[One Embodiment of Image Conversion Method]
Next, an embodiment of the image conversion method of the present invention will be described with reference to FIGS. FIG. 5 is a flowchart showing an embodiment of the image conversion method of the present invention. FIG. 6 is a diagram for explaining a triangle to be converted. (A) shows a triangle before conversion, and (b) shows a triangle after conversion. FIG. 7 is a diagram for explaining the image conversion method of the present embodiment, and (a) to (i) show triangles before conversion (broken line) and after conversion (solid line) in the conversion process, respectively. Yes.

(First input step)
As a preparation stage of the image conversion method of the present embodiment, first, vertex coordinates of the triangle ABC are input to the lowest coordinate plane M1. Here, as shown in FIG. 6A, the coordinates of the vertices of the triangle ABC are A (x ₀ , y ₀ ), B (x ₁ , y ₁ ), and C (x ₂ , y ₂ ). To do. Further, as shown in FIG. 6A, the angle formed by the side AB and the X axis is r1, and the angle formed by the side AC and the X axis is r2. The angle is positive in the clockwise direction. At this time, the following expressions (4) to (7) are satisfied. Here, atan represents an arctangent function arctan.
r1 = atan ((y ₁ −y ₀ ) / (x ₁ −x ₀ )) (when x ₁ ≠ x ₀ ) (4)
r1 = −90 (when x ₁ = x ₀ ) (5)
r2 = atan ((y ₂ −y ₀ ) / (x ₂ −x ₀ )) (when x ₂ ≠ x ₀ ) (6)
r2 = 90 (when x ₂ = x ₀ ) (7)

(Second input step)
Similarly, as a preparation stage of the image conversion method of the present embodiment, the vertex coordinates of the triangle A′B′C ′ are input to the uppermost coordinate plane M4. Here, as shown in FIG. 6B, the coordinates of the vertices of the triangle A′B′C ′ are represented by A ′ (x ₀ ′, y ₀ ′), B ′ (x ₁ ′, y ₁ ′). ), C ′ (x ₂ ′, y ₂ ′). Further, as shown in FIG. 6B, the angle formed by the side A′B ′ and the X axis is r1 ′, and the angle formed by the side A′C ′ and the X axis is r2 ′. At this time, the following expressions (8) to (11) are satisfied.
_{r1 '= atan ((y 1} ' -y 0 ') / (x 1' -x 0 ')) (x 1' when _{≠ x 0 ') ... (8} )
_{r1 '= - 90 (x 1} ' when _{= x 0 ') ... (9} )
_{r2 '= atan ((y 2} ' -y 0 ') / (x 2' -x 0 ')) (x 2' when _{≠ x 0 ') ... (10} )
r2 '= 90 (x _2' when _{= x 0 ') ... (11} )

  Next, the conversion process of the triangle ABC will be described in detail with reference to FIG. 5 (refer to FIGS. 7 and 6 as appropriate). In the frame shown in FIG. 7, the upper left point is the origin O (0, 0), the upper side is the X axis (right direction is positive), and the left side is the Y axis (down direction is positive). Moreover, the display of M1-M4 represents the coordinate plane which sets a parameter at the stage, a dotted line represents the shape before conversion, and a continuous line represents the shape after conversion.

(First conversion step)
First, as shown in FIG. 7A, on the coordinate plane M1, the triangle ABC is rotationally moved so that the bisector of the corner A of the triangle ABC and the X axis are parallel to each other (step S1). . The rotation angle at this time includes an angle r1 (see Formula (4) or Formula (5)) formed by the side AB and the X axis, and an angle r2 (see Formula (6) or Formula (7)) formed by the side AC and the X axis. ), And the rotation direction is a negative direction (counterclockwise). Also, the conversion operation at this time is described as Expression (12) because the coordinate plane is M1, the conversion type is “rotation operation”, and the rotation angle (parameter) is −rm. Similarly, in the following expression representing the conversion operation, “(coordinate plane). (Type of conversion) = (parameter value)” is described.
M1. rotation = −rm (12)
However, rm = (r1 + r2) / 2 (13)
The triangle ABC converted by this conversion operation (step S1) is defined as a triangle A ₁ B ₁ C ₁ (congruent with the triangle ABC). At this time, a conversion equation for converting the vertex A (x ₀ −y ₀ ) into the vertex A ₁ (X, Y) is expressed by Equation (14).

Next, as shown in FIG. 7B, the triangle A ₁ B ₁ C ₁ is translated on the coordinate plane M1 so that the vertex A ₁ overlaps the origin (step S2). The conversion operation at this time is described as Expression (15) and Expression (16) because the coordinate plane is M1 and the conversion type is “parallel movement”. Incidentally, the right side of the equation (15) and (16) are obtained as the difference between the origin of coordinates, the coordinates (X, Y) of the vertex A ₁ represented by formula (14) and.
M1. x = −cos (rm) · x ₀ −sin (rm) · y ₀ (15)
M1. y = sin (rm) · x ₀ −cos (rm) · y ₀ (16)

The triangle A ₁ B ₁ C ₁ converted by this conversion operation (step S2) is defined as a triangle A ₂ B ₂ C ₂ (congruent with the triangle ABC). As shown in FIG. 7B, the bisector of the corner A ₂ is on the X axis. Here, an angle rd (a bisector of the angle A ₂ ) formed by the side A ₂ B ₂ and the X axis is expressed by Expression (17).
rd = (r2-r1) / 2 (17)
Therefore, the coordinates of the point B ₂ are (| AB | cos (rd), − | AB | sin (rd)) (| AB | is the length of the side A ₂ B ₂ = the length of the side AB).

(Second conversion step)
Next, as shown in FIG. 7C, on the upper coordinate plane M2, the angle A ₂ (corresponding to the angle A) of the triangle A ₂ B ₂ C ₂ is 90 degrees in the Y-axis direction. Enlarge (or reduce) (step S3). The triangle A ₂ B ₂ C ₂ converted by this conversion operation (step S3) is defined as a triangle A ₃ B ₃ C ₃ (a first triangle having a different shape from the original triangle). In other words, the conversion operation is to enlarge (or reduce) the Y-axis direction so that the bisector of the angle A ₃ becomes 45 degrees. At this time, the absolute value of the y coordinate of the vertex B ₃ is equal to the value of the x coordinate, that is, equal to | AB | cos (rd). Thereby, the enlargement ratio in the Y-axis direction by this conversion can be obtained as a quotient (1 / tan (rd)) between the absolute value of the y coordinate of the vertex B ₃ and the absolute value of the y coordinate of the vertex B _2. . Further, the conversion operation at this time is described as Expression (18) because the coordinate plane is M2 and the type of conversion is “enlargement / reduction”.

Next, as shown in FIG. 7D, on the coordinate plane M2, the side A ₃ B ₃ (corresponding to the side AB) is on the X axis, and the side A ₃ C ₃ (corresponding to the side AC) is on the Y axis. The triangles A ₃ B ₃ C ₃ are rotationally moved so as to overlap each other (step S4). That is, it moves 45 degrees in the forward direction. The conversion operation (step S4) at this time is described as in equation (19).
M2. rotation = 45 (19)
The triangle A ₃ B ₃ C ₃ converted by this conversion operation (step S4) is defined as a triangle A ₄ B ₄ C ₄ (congruent with the first triangle). As shown in FIG. 7D, the triangle A ₄ B ₄ C ₄ is arranged such that the vertex A ₄ is arranged at the origin, the angle A ₄ is a right angle, the vertex B ₄ is the X axis, and the vertex C ₄ is the Y axis. Is riding.

(Third conversion step)
Next, as shown in FIG. 7E, on the upper coordinate plane M3, the side A ₄ B ₄ (corresponding to the side AB) and the side A ₄ C ₄ (corresponding to the side AC) are set in the X-axis direction. Are reduced (or enlarged) by predetermined magnifications α and β described later in the Y-axis direction (step S5). The conversion operation (step S5) at this time is described as in equations (20) and (21). The triangle A ₄ B ₄ C ₄ converted by this conversion operation (step S5) is defined as a triangle A ₅ B ₅ C ₅ (second triangle having a different shape from the original triangle).
M3. xscale = α (20)
M3. yscale = β (21)

Next, as shown in (f) of FIG. 7, on the coordinate plane M3, the triangle A ₅ B _{5 is} such that the bisector of the angle A ₅ (corresponding to the angle A) and the X axis are parallel to each other. 45 ° rotational movement in the negative direction C ₅ (step S6). The conversion operation at this time is described as in Expression (22). The triangle A ₅ B ₅ C ₅ converted by this conversion operation (step S6) is defined as a triangle A ₆ B ₆ C ₆ (congruent with the second triangle).
M3. rotation = −45 (22)

(Fourth conversion step)
Next, as shown in FIG. 7 (g), on the coordinate plane M4, the triangle A ₆ B ₆ C ₆ has an angle A ₆ (corresponding to the angle A) at an angle A ′ of the triangle A′B′C ′. Enlarge (or reduce) in the Y-axis direction so as to be equal (step S7). The triangle A ₆ B ₆ C ₆ converted by this conversion operation (step S7) is defined as a triangle A ₇ B ₇ C ₇ (a third triangle having a different shape from the original triangle).
Here, the bisector angle rd ′ of the angle A ′ of the triangle A′B′C ′ is expressed by the equation (23) similarly to the equation (17).
rd '= (r2'-r1') / 2 (23)
In step S7, since the same idea as in step S3 can be applied in reverse, tan (rd '), which is the reciprocal of the enlargement ratio in step S3, may be used as the enlargement ratio (reduction ratio). Therefore, the conversion operation at this time is described as in Expression (24).
M4. yscale = tan (rd ′) (24)

(Fifth conversion step)
Next, as shown in FIG. 7H, on the coordinate plane M4, the side A ₇ B ₇ (corresponding to the side AB) and the side A ₇ C ₇ (corresponding to the side AC) of the triangle A ₇ B ₇ C ₇ ) Are rotated so as to be parallel to the sides A'B 'and A'C' of the triangle A'B'C '(step S8). Here, the angle formed by the side A ₇ B ₇ of the triangle A ₇ B ₇ C ₇ and the X axis is rd ′ (see equation (23)), and the sides A′C ′ and X of the triangle A′B′C ′ The angle formed by the axes is r2 '(see formula (10) or formula (11)). Therefore, this conversion operation corresponds to the reverse rotation of the triangle A ₇ B ₇ C ₇ by (rd′−r2 ′). The conversion operation at this time is described as in Expression (25). The triangle A ₇ B ₇ C ₇ converted by this conversion operation (step S8) is defined as a triangle A ₈ B ₈ C ₈ (congruent with the third triangle).
M4. rotation = r2′−rd ′ (25)

Next, as shown in (i) in FIG. 7, on the coordinate plane M4, the apex A ₈ of the triangle A ₈ B ₈ C ₈ (corresponding to the corner A) is in the 'apex A' of the triangle A'B'C The triangle A ₈ B ₈ C ₈ is translated so as to overlap (step S9). The conversion operation at this time is described as in Expression (26) and Expression (27). The triangle A ₈ B ₈ C ₈ converted by this conversion operation (step S9) is defined as a triangle A ₉ B ₉ C ₉ (congruent with the third triangle).
M4. x = x ₀ ′ (26)
M4. y = y ₀ ′ (27)

By step S9, finally, on the coordinate plane M4, the length of the side A ₉ B ₉ of the triangle A ₉ B ₉ C ₉ (the length of A ₈ B ₈ = the length of A ₇ B ₇ ) and the side The length of A ₉ C ₉ (the length of A ₈ C ₈ = the length of A ₇ C ₇ ) is the length of the side A′B ′ of the triangle A′B′C ′ and the side A′C, respectively. It must be equal to the length of ′. In order to do this, in this image conversion method, the values of the predetermined magnifications α and β in step S5 described above are determined. Hereinafter, the predetermined magnifications α and β will be described with reference to FIG. FIG. 8 is a diagram for explaining a predetermined magnification in the image conversion method of the present embodiment, and FIGS. 8A to 8D show a state of deformation of a triangle in the process of deriving the predetermined magnification.

(A) to (c) of FIG. 8 are the conversions of steps S6 to S8 in the shape (first triangle) as it is without expanding (reducing) the triangle A ₄ B ₄ C ₄ in step S5. A change in the triangle ABC when the operation is performed is shown, and FIG. 8D corresponds to the conversion operation in step S5. In FIG. 8, an alternate long and short dash line indicates a frame in which two sides (extension lines of AB and AC) are on the X axis and the Y axis on the coordinate plane M3.

The conversion operation at this time will be described. As shown in FIG. 8A, after rotating the first triangle ABC (triangle A ₄ B ₄ C ₄ : see step S4 and FIG. 7D) on the coordinate plane M3 by −45 degrees, As shown in FIG. 8B, on the coordinate plane M4, the angle A is equal to the angle A ′ of the target triangle A′B′C ′ (see FIG. 2B). The first triangle ABC is enlarged (or reduced) in the Y-axis direction. A figure formed by this conversion operation is a fourth triangle ABC. Then, as shown in FIG. 8C, on the coordinate plane M4, the side AB and the side AC of the fourth triangle are respectively the side A′B ′ and the side A of the triangle A′B′C ′. Rotate and move in parallel with 'C'. Then, as shown in FIG. 8D, in the direction AB, the length of the side AB of the third triangle is equal to the length of the side A′B ′ of the triangle A′B′C ′. Similarly, the side AC is enlarged (or reduced) at a predetermined magnification β in the direction of AC.

  Here, the predetermined magnifications α and β used in the process of generating the fourth triangle ABC are the predetermined magnifications used in step S5. This fourth triangle ABC is congruent with the triangle A′B′C ′. Finally, when the step S9 is executed and the fourth triangle is translated and aligned, a series of conversions is completed. However, what is important here is that the enlargement operation shown in FIG. 8D must be performed on the coordinate plane 1 because the enlargement operation cannot be performed after the rotational movement on the coordinate plane M4. is there. Therefore, the conversion operation shown in FIG. 8 is performed on the coordinate plane M4 as if it was executed, and the predetermined magnifications α and β are calculated backward, and the obtained α and β are used to perform step S5 (FIG. 7 (e)). The conversion operation was performed.

The process of deriving the predetermined magnifications α and β is as follows.
Here, the length d1 of the side AB and the length d2 of the side AC of the original triangle ABC are represented by Expression (28) and Expression (29), respectively.

  Further, the length d1 ′ of the side A′B ′ and the length d2 ′ of the side A′C ′ of the triangle A′B′C ′ after the final conversion are expressed by the equations (30) and (31), respectively. Indicated.

The side length | A ₃ B ₃ | (= | A ₄ B ₄ |) of the triangle A ₃ B ₃ C ₃ (first triangle) generated in the step S3 is d1 × cos (rd) × √ 2 (see FIG. 7C). However, d1 is the length of the side AB of the original triangle ABC. That is, the conversion operation in step S3 is an operation of enlarging the length of the side A ₂ B ₂ (the length of AB) to cos (rd) × √2. Further, this step S3 is an operation for converting the angle formed by the side A ₂ B ₂ (side AB) of the triangle ABC and the X axis from rd to 45 degrees for conversion.

On the other hand, the conversion operation shown in FIG. 8B is an operation for converting the angle formed by the side AB (side A ₄ B ₄ ) of the first triangle ABC and the X axis from 45 degrees to rd ′. It has become. That is, this conversion operation is the reverse of the conversion operation in step S3. Therefore, the conversion operation shown in FIG. 8B is an operation for expanding the length of the side AB (side A ₄ B ₄ ) to a reciprocal multiple of (cos (rd ′) × √2). . As a result, the length | AB | of the side AB of the fourth triangle ABC generated by the conversion operation shown in FIG. 8B is expressed by Expression (32). However, d1 is the length of the side AB of the original triangle ABC.
| AB | = d1 · cos (rd) / cos (rd ′) (32)
Similarly, the length | AC | of the side AC of the fourth triangle ABC is expressed by Expression (33).
| AC | = d 2 · cos (rd) / cos (rd ′) (33)
Since the equations (32) and (33) are equal to d1 ′ (see equation (30)) and d2 ′ (see equation (31)), respectively, the predetermined magnifications α and β are expressed by the following equations (34) ) And formula (35).

In addition, Formula (20) and Formula (21) which show conversion operation in step S5 are rewritten like Formula (36) and Formula (37).
M3. xscale = (d1 ′ · cos (rd ′)) / (d1 · cos (rd)) (36)
M3. yscale = (d2 ′ · cos (rd ′)) / (d2 · cos (rd)) (37)

  As described above, in the coordinate planes M1 to M4, the expressions (12), (15), (16), (18), (19), (36), (37), ( 22) The image area of the original triangle ABC is converted into the image area of the triangle A′B′C ′ (see FIG. 2B) by sequentially calculating the expressions (24) to (27). Can do.

In addition, the expansion direction of the triangle in step S3 and / or step S6 is not limited to the Y axis, and may be the X axis direction. In this case, on the coordinate plane M1, the triangle ABC is rotated (90-rm) degrees so that the bisector of the angle A of the triangle ABC and the Y axis are parallel (step S1 ′: FIG. corresponding to 7 (a)), on the coordinate plane M2, enlarged in the X-axis direction as the corner a ₂ of the triangle a ₂ B ₂ C ₂ is 90 degrees (or reduced) (step S3 ': 7 (Corresponding to (c) of FIG. 7), the triangle A ₃ B ₃ C ₃ is rotated 45 degrees in the negative direction on the coordinate plane M2 (step S4 ′: corresponding to FIG. 7D). Further, on the coordinate plane M3, the triangle A ₅ B ₅ C ₅ is rotated 45 degrees in the positive direction so that the bisector of the angle A ₅ is parallel to the Y axis (step S6 ′: FIG. 7). On the coordinate plane M4, the triangle A ₆ B ₆ C ₆ is expanded (or reduced) in the X-axis direction so that the angle A ₆ is equal to the angle A ′ of the triangle A′B′C ′. (Step S7 ': corresponding to (g) of FIG. 7).

[Configuration of Image Conversion Device]
Next, an image conversion apparatus that executes the image conversion method of the present invention will be described in detail. FIG. 1 is a configuration diagram of an image conversion apparatus according to the present embodiment. The image conversion apparatus 1 uses an image area of an arbitrary triangle ABC (see FIG. 2A) developed on a memory as an image of another triangle A′B′C ′ (see FIG. 2B). The image is converted into an area and the converted image is displayed. The image conversion apparatus 1 includes an input unit 10, a network I / F unit 12, a storage unit 14, an output unit 16, and a CPU 20.

The input means 10 is composed of, for example, a keyboard or a mouse, and inputs the vertex coordinates of the triangle ABC before conversion and the vertex coordinates of the triangle A′B′C ′ after conversion.
The network I / F unit 12 inputs a predetermined image from the network 2.
The storage unit 14 stores an image before conversion and an image after conversion, and is a general storage device such as a hard disk.
The output means 16 is composed of a display, for example, and displays a predetermined image or the like.

The CPU 20 performs various processes to be described later by executing a predetermined program, and includes a first coordinate plane processing unit 21, a second coordinate plane processing unit 22, a third coordinate plane processing unit 23, A fourth coordinate plane processing unit 24, a storage unit 25, and an image processing unit 26 are provided.
The first coordinate plane processing unit 21 executes a conversion process related to the coordinate plane M1. The first coordinate plane processing unit 21 rotates and moves the triangle ABC with respect to the triangle ABC developed in the storage unit 25 so that the bisector of the corner A and the X axis are parallel to each other (step S1: FIG. 5), the triangle ABC is translated so that the vertex A overlaps the origin (step S2: see FIG. 5).
The second coordinate plane processing unit 22 executes a conversion process related to the coordinate plane M2. The second coordinate plane processing unit 22 enlarges or reduces the triangle ABC in the Y-axis direction so that the angle A is 90 degrees with respect to the triangle ABC developed in the storage unit 25 (step S3: see FIG. 5). The triangle ABC is rotated 45 degrees so that the side AB overlaps the X axis and the side AC overlaps the Y axis (step S4: see FIG. 5).

The third coordinate plane processing unit 23 executes a conversion process related to the coordinate plane M3. The third coordinate plane processing unit 23 enlarges or reduces the side AB of the triangle ABC developed in the storage unit 25 by α times in the X axis direction and β times in the Y axis direction (step S5: see FIG. 5). The triangle ABC is rotated 45 degrees so that the bisector of the angle A and the X axis are parallel to each other (step S6: see FIG. 5).
The fourth coordinate plane processing unit 24 executes a conversion process related to the coordinate plane M4. The fourth coordinate plane processing unit 24 converts the triangle ABC in the Y-axis direction so that the triangle A developed in the storage unit 25 is equal to the angle A ′ of the target triangle A′B′C ′. (Step S7: see FIG. 5), and the sides AB and AC are rotated so that the sides A'B 'and A'C' are parallel to each other (step S8: see FIG. 5). ) The triangle ABC is translated so that the vertex A overlaps the vertex A ′ (step S9: see FIG. 5).

The storage unit 25 stores values of parameters for conversion operations, coordinates of pixels in a triangular image area in the conversion process, and the like, and includes a semiconductor memory or a magnetic memory.
The image processing unit 26 stores the input image in the storage unit 14, and appropriately switches the input image and the converted image in accordance with a command input from the input unit 10 and outputs the input image to the output unit 16.

  Note that the image conversion apparatus 1 can also be realized by executing an image conversion program that causes each of the above-described units to function in a general computer. This image conversion program can be distributed via a communication line, or can be distributed by writing in a recording medium such as a CD-ROM.

[Operation of image converter]
Next, the operation of the image conversion apparatus 1 will be described. In the image conversion apparatus 1, the following operation is performed in advance. That is, the image conversion apparatus 1 inputs a predetermined image by the network I / F unit 12, stores the input image in the storage unit 14 by the image processing unit 26, and outputs it to the output unit 16. Then, the operator of the image conversion apparatus 1 determines the vertex coordinates of the triangle ABC assuming a triangle patch including a part or the whole of the predetermined image displayed on the output unit 16 of the image conversion apparatus 1, and further, a desired The vertex coordinates of the triangle A′B′C ′ converted into a shape are determined in advance. Then, according to the operation of the operator, the image conversion apparatus 1 uses the input means 10 as the apex coordinates of the triangle ABC on the lowest coordinate plane M1 as A (x ₀ , y ₀ ), B (x ₁ , y ₁ ) and C (x ₂ , y ₂ ) are input. Furthermore, the image conversion apparatus 1 uses the input means 10 as the vertex coordinates of the triangle A′B′C ′ on the uppermost coordinate plane M4 according to the operation of the operator as A ′ (x ₀ ′, y ₀ ′), B ′ (x ₁ ′, y ₁ ′), C ′ (x ₂ ′, y ₂ ′). The input vertex coordinates of the two triangles are stored in the storage unit 25 by the image processing unit 26 of the image conversion apparatus 1.

Based on the data input or the operator's operation, the image conversion apparatus 1 causes the first coordinate plane processing unit 21 to each pixel (including the vertex) of the image area of the triangle ABC based on the equation (12). , And the coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₁ B ₁ C ₁ (see FIG. 7A). Then, the image conversion apparatus 1 uses the first coordinate plane processing unit 21 to convert each pixel (including the vertex) of the image area of the triangle A ₁ B ₁ C _{1 in} the X-axis direction based on the equation (15). Further, based on the above equation (16), it is translated in the Y-axis direction, and the coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₂ B ₂ C ₂ (see FIG. 7B).

Subsequently, the image conversion apparatus 1 uses the second coordinate plane processing unit 22 to convert each pixel (including the vertex) of the image area of the triangle A ₂ B ₂ C _{2 in} the Y-axis direction based on the equation (18). The coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set as a triangle A ₃ B ₃ C ₃ (see FIG. 7C). Then, the image conversion apparatus 1 uses the second coordinate plane processing unit 22 to convert each pixel (including the vertex) of the image area of the triangle A ₃ B ₃ C ₃ to 45 in the positive direction based on the above equation (19). The coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₄ B ₄ C ₄ (see FIG. 7D).

Subsequently, the image conversion apparatus 1 uses the third coordinate plane processing unit 23 to convert each pixel (including the vertex) of the image area of the triangle A ₄ B ₄ C _{4 in} the X-axis direction based on the equation (36). Is reduced (or enlarged) and reduced (or enlarged) in the Y-axis direction based on the equation (37), and the coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₅ B ₅ C ₅ (see FIG. 7E). Then, the image conversion apparatus 1 causes the third coordinate plane processing unit 23 to set each pixel (including the vertex) of the image area of the triangle A ₅ B ₅ C _{5 in} the negative direction based on the equation (22). It rotates 45 degrees and stores the coordinate point of the movement destination in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₆ B ₆ C ₆ (see (f) in FIG. 7).

Subsequently, the image conversion apparatus 1 uses the fourth coordinate plane processing unit 24 to convert each pixel (including the apex) of the image area of the triangle A ₆ B ₆ C _{6 in} the Y-axis direction based on the above equation (24). The coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₇ B ₇ C ₇ (see (g) of FIG. 7). Then, the image conversion apparatus 1 uses the fourth coordinate plane processing unit 24 to set each pixel (including the vertex) of the image area of the triangle A ₇ B ₇ C _{7 in} the positive direction based on the equation (25) ( r2′−rd ′) degrees, and the coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set as a triangle A ₈ B ₈ C ₈ (see (h) in FIG. 7). Furthermore, the image conversion apparatus 1 uses the fourth coordinate plane processing unit 24 to convert each pixel (including the vertex) of the image area of the triangle A ₈ B ₈ C _{8 in} the X-axis direction based on the above equation (26). Further, based on the equation (27), the translation is performed, and the coordinate point of the movement destination is stored in the storage unit 25. At this time, the destination triangle ABC is set to triangle A ₉ B ₉ C ₉ (see (i) in FIG. 7). The vertex coordinates of the triangle A ₉ B ₉ C ₉ coincide with the vertex coordinates of the triangle A′B′C ′ on the uppermost coordinate plane M 4 input by the input means 10.

Finally, the image conversion apparatus 1 reads out the coordinate point of each pixel stored in the storage unit 25 by the image processing unit 26, outputs it to the output unit 16, and converts it into the target image area by the output unit 16. Displayed images.
According to the image conversion apparatus of the present embodiment, a desired affine transformation can be performed and displayed on a triangular image region having an arbitrary shape.

  As mentioned above, although embodiment of this invention was described, this invention is not limited to this, It can implement variously in the range which does not change the meaning. For example, in the present embodiment, the image conversion apparatus 1 inputs the vertex coordinates of the triangle patch to be converted and the vertex coordinates of the triangle A′B′C ′ converted into a desired shape, and then the above steps S1 to S1. Although it has been described that the conversion operation of step S9 is executed, only the vertex coordinates of the conversion source triangle patch are input to execute the conversion operation of steps S1 to S4 in advance, and then a triangle having a desired shape is obtained. The vertex coordinates of A′B′C ′ may be input, and the processing may be separated and executed so as to execute the conversion operation in steps S5 to S9. In this case, since the calculation when the coordinates of the triangle patch of the conversion source are changed or the coordinates of the converted triangle A′B′C ′ are changed can be executed independently, the calculation time can be shortened. Can do. For example, if a large amount of data is acquired in advance as a conversion source triangle patch and the conversion operation in steps S1 to S4 is executed, various converted images can be displayed in a short time thereafter. become.

In addition to the configuration of the present embodiment, the configuration of the image conversion apparatus may be provided with, for example, a triangle patch generation unit, or may receive a triangle patch from a network. Further, the image conversion apparatus of the present embodiment may be applied to, for example, a 3D model display apparatus. An example of this case will be described with reference to FIG. FIG. 9 shows an example in which the present invention is applied to synthesis of a face image, where (a) shows an image before conversion and (b) shows an image after conversion. For example, with the function of the image conversion device, as shown in FIG. 9A, a triangular patch generation means is provided in a region where a face photograph viewed from the front of a certain person is taken and converted as a predetermined image. To generate a plurality of triangular patches. On the other hand, a frame (deformed triangle) obtained by rotationally moving a triangle patch (parallel projection conversion) by the function of the 3D model display device is generated. By inputting the coordinates of this frame to the image conversion apparatus, the desired region of the face photograph is affine transformed, and as shown in FIG. 9B, the image (conversion) when the desired region of the face photograph is rotated. Image) can be displayed. According to this, texture mapping can be performed easily.
The image conversion apparatus of the present invention can also be applied to, for example, transforming an image by affine transformation and displaying it in a pseudo manner on the result of projective transformation of a triangular patch.

It is a block diagram of the image conversion apparatus of this embodiment. It is a figure explaining a mode that an image is converted, (a) shows an image before conversion and (b) shows an image after conversion. It is a figure explaining the geometric transformation in a coordinate plane, (a) is expansion / contraction, (b) shows rotational movement, (c) shows parallel movement. It is a perspective view which shows arrangement | positioning of a coordinate plane. It is a flowchart showing one Embodiment of the image conversion method of this invention. It is a figure explaining the triangle to convert, Comprising: (a) has shown the triangle before conversion, (b) has shown the triangle after conversion. It is a figure for demonstrating the image conversion method of this embodiment, Comprising: (a)-(i) has each shown the triangle before conversion (broken line) and the post-conversion (solid line) in step S1-step S9. . It is a figure explaining the predetermined magnification in the image conversion method of this embodiment, Comprising: (a)-(d) has shown the mode of the deformation | transformation of the triangle in the process of deriving the predetermined magnification. It is an example which applied this invention to the synthesis | combination of a face image, Comprising: (a) is the image before conversion, (b) has shown the image after conversion.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Image converter 10 Input means 12 Network I / F means 14 Storage means 16 Output means 20 CPU
21 first coordinate plane processing unit 22 second coordinate plane processing unit 23 third coordinate plane processing unit 24 fourth coordinate plane processing unit 25 storage unit 26 image processing unit

Claims (6)

For the sequentially stacked images on the first to fourth coordinate planes, enlargement or reduction in the X-axis direction and Y-axis direction, rotational movement around the origin, and parallel movement to any coordinate point are in this order. The image conversion of the triangle ABC on the first coordinate plane that is the lowest layer and the image area of the triangle A′B′C ′ on the fourth coordinate plane that is the top layer In the image conversion method in the apparatus,
The image conversion device includes:
A first input step of inputting vertex coordinates of the triangle ABC;
A second input step of inputting vertex coordinates of the triangle A′B′C ′;
Moving the vertex A of the triangle ABC input to the first coordinate plane to the origin, and superimposing the bisector of the angle A on the X axis or the Y axis;
The triangle ABC transformed by the first transformation step is enlarged or reduced in the direction orthogonal to the bisector of the angle A so that the angle A is a right angle on the second coordinate plane. A conversion step;
A third conversion step of enlarging or reducing the side AB and the side AC of the triangle ABC converted by the second conversion step at first and second predetermined magnifications on the third coordinate plane;
The triangle ABC transformed by the third transformation step is bisected by the angle A so that the angle A is equal to the angle A ′ of the triangle A′B′C ′ on the fourth coordinate plane. A fourth conversion step for enlarging or reducing at a third predetermined magnification in a direction orthogonal to
Fifth conversion step for matching the coordinate position of the triangle A′B′C ′ by rotating and translating the triangle ABC converted by the fourth conversion step on the fourth coordinate plane. When,
The image conversion method characterized by performing. The first predetermined magnification is obtained by multiplying the triangle ABC generated in the second conversion step by the third predetermined magnification in the direction orthogonal to the bisector of the angle A at the side AB. , Equal to the magnification when enlarging or reducing to the side A′B ′ of the triangle A′B′C ′,
The second predetermined magnification is obtained by enlarging or reducing the side AC of the triangle ABC generated in the second conversion step in the direction orthogonal to the bisector of the angle A at the third predetermined magnification. , Equal to the magnification when enlarging or reducing to the side A′C ′ of the triangle A′B′C ′,
The image conversion method according to claim 1.   The image conversion apparatus executes the first and second conversion steps when the vertex coordinates of the triangle ABC are input by the first input step, and the triangle A ′ by the second input step. 3. The image conversion method according to claim 1, wherein when the vertex coordinates of B′C ′ are input, the third to fifth conversion steps are executed.   An image conversion program for causing a computer to execute the image conversion method according to any one of claims 1 to 3. The images on the first to fourth coordinate planes that are sequentially stacked can be enlarged or reduced in the X-axis direction and Y-axis direction, rotated around the origin, and translated to any coordinate point. In the image conversion apparatus configured to convert the image area of the triangle ABC on the first coordinate plane as the lowermost layer into the image area of the triangle A′B′C ′ on the fourth coordinate plane as the uppermost layer,
Input means for inputting vertex coordinates of the triangle ABC and the triangle A′B′C ′;
A first coordinate plane processing unit that moves the vertex A of the triangle ABC input to the first coordinate plane to the origin and superimposes the bisector of the angle A on the X axis or the Y axis;
On the second coordinate plane, the triangle ABC is enlarged or reduced in a direction perpendicular to the bisector of the angle A, and converted into a first triangle ABC having the angle A being a right angle. A processing unit;
On the third coordinate plane, a third coordinate plane that converts the side AB and the side AC of the first triangle ABC into the second triangle ABC by enlarging or reducing the first and second predetermined magnifications, respectively. A processing unit;
On the fourth coordinate plane, the second triangle ABC is bisected of the angle A so that the angle A of the second triangle ABC is equal to the angle A ′ of the triangle A′B′C ′. The triangle A′B ′ is obtained by enlarging or reducing the third triangle ABC in a direction orthogonal to the segment by a third predetermined magnification, and rotating and translating the third triangle ABC. A fourth coordinate plane processing unit for matching the coordinate position of C ′;
An image conversion apparatus comprising: The first predetermined magnification is obtained by enlarging or reducing the first triangle ABC in the direction orthogonal to the bisector of the angle A at the third predetermined magnification. Equal to the magnification when AB is enlarged or reduced to the side A′B ′ of the triangle A′B′C ′;
The second predetermined magnification is obtained by enlarging or reducing the first right triangle ABC in the direction perpendicular to the bisector of the angle A at the third predetermined magnification. Equal to the magnification when the side AC is enlarged or reduced to the side A′C ′ of the triangle A′B′C ′;
The image conversion apparatus according to claim 5.

Images