Information processing apparatus

• US 6,970,178 B2
• Filed: 06/20/2002
• Issued: 11/29/2005
• Est. Priority Date: 06/21/2001
• Status: Expired due to Term

• Current Assignee: HI Corporation (Micware Co., Ltd.)
• Original Assignee: HI Corporation (Micware Co., Ltd.)
• Inventors: Kawabata, Kazuo, Tsutsumi, Junya, Oda, Kazutoshi, Tanioka, Yasuhisa
• Primary Examiner(s): Zimmerman, Mark
• Assistant Examiner(s): ELKASSABGI, AHMED Y
• Application Number:
  US10/296,811
• Publication Number:
  US 20030184543A1
• Time in Patent Office: 1,258 Days
• Field of Search: 345/648, 345/419, 345/643, 345/949, 345/950, 345/951, 345/952, 345/953, 345/954, 345/955, 345/956, 345/957, 345/958, 345/959, 345/960
• US Class Current: 345/643
• CPC Class Codes: G06T 3/02 Affine transformations for image registration G06T3/147; for image mosaicing G06T3/4038

Abstract

In an information processing apparatus for conducting an affine transformation representative of $\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)=A\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right),$
a matrix $A^{\prime }=\left(\begin{array}{ccc}{a}_{11}^{\prime }& a_{12}^{\prime }& a_{13}^{\prime }\\ a_{21}^{\prime }& a_{22}^{\prime }& a_{23}^{\prime }\\ a_{31}^{\prime }& a_{32}^{\prime }& a_{33}^{\prime }\end{array}\right)=\lambda A=\left(\begin{array}{ccc}\lambda a_{11}& \lambda a_{12}& \lambda a_{13}\\ \lambda a_{21}& \lambda a_{22}& \lambda a_{23}\\ \lambda a_{31}& \lambda a_{32}& \lambda a_{33}\end{array}\right)$
obtained by multiplying a matrix A for the affine transformation by λ(≠0) is stored in a memory section in advance. In case of conducting calculation processing for transforming coordinates (x,y,z)^t into coordinates (x′,y′,z′)^t by means of the affine transformation, A′ and a matrix (t₁,t₂,t₃)^t are read from the memory section,
x′=(a′₁₁*x+a′₁₂*y+a′₁₃*z)/λ+t₁;
y′=(a′₂₁*x+a′₂₂*y+a′₂₃*z)/λ+t₂;
z′=(a′₃₁*x+a′₃₂*y+a′₃₃*z)/λ+t₃;
are calculated, and the coordinates (x′,y′,z′)^t are obtained.

Claims

1. An information processing apparatus for conducting imaging of a three-dimensional image, in which coordinate values are limited to integers, and transformation processing is applied to coordinates (x,y,z)^t to create coordinates (x′

,y′

,z′

)^t by means of an affine transformation representative of $\left(\begin{array}{c}{x}^{\mathrm{\prime <br><br>}}\\ y^{\mathrm{\prime <br><br>}}\\ z^{\mathrm{\prime <br><br>}}\end{array}\right)=A<br><br>\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right),$

characterized in that the apparatus has;

a memory in which a matrix $A^{\mathrm{\prime <br><br>}}=(\text{\hspace{1em}<br><br>}<br><br>\begin{array}{ccc}{a}_{11}^{\mathrm{\prime <br><br>}}& a_{12}^{\mathrm{\prime <br><br>}}& a_{13}^{\mathrm{\prime <br><br>}}\\ a_{21}^{\mathrm{\prime <br><br>}}& a_{22}^{\mathrm{\prime <br><br>}}& a_{23}^{\mathrm{\prime <br><br>}}\\ a_{31}^{\mathrm{\prime <br><br>}}& a_{32}^{\mathrm{\prime <br><br>}}& a_{33}^{\mathrm{\prime <br><br>}}\end{array}<br><br>\text{\hspace{1em}<br><br>})=\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>A=(\text{\hspace{1em}<br><br>}<br><br>\begin{array}{ccc}\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{11}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{12}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{13}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{21}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{22}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{23}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{31}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{32}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{33}\end{array}<br><br>\text{\hspace{1em}<br><br>})$

that is obtained by multiplying an affine transformation matrix $A=(\text{\hspace{1em}<br><br>}<br><br>\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}<br><br>\text{\hspace{1em}<br><br>})$

by λ

(≠

0),

a matrix (t₁,t₂,t₃)^t, and a shape data are stored; and

calculation means for, in transforming the coordinates (x,y,z)^t of said shape data into the coordinates (x′

,y′

,z′

)^t by means of an affine transformation of the matrix A and the matrix (t₁,t₂,t₃)^t reading the matrix A′ and

the matrix (t₁,t₂,t₃)^t from said memory, and calculating

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

and calculating the coordinates (x′

,y′

,z′

)^t.

2. An information processing apparatus recited in claim 1, characterized in that said λ

is limited to 2ⁿ(n≧

1), and

said calculation means is means for calculating divisional calculation of

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

by conducting right-shifting by n digits.

3. An information processing apparatus recited in claim 2, characterized in that said calculation means is means for conducting the calculation by conducting right-shifting by n digits after adding a constant λ

/2 to each number to be divided.

4. An information processing apparatus recited in any of claim 1 to claim 3, characterized in that the apparatus has synthesis means for synthesizing two or more parameters that are multiplied by λ

(≠

0) in advance.

5. An information processing apparatus recited in any of claim 1 to claim 3, characterized in that the apparatus further has:

a memory in which a constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with a quantity of a change of y coordinates; and

calculation means for, in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to the y coordinates,

reading μ

corresponding to the quantity of the change of said y coordinates from said memory, and calculating

Δ

x′

=μ

*(x₂−

x₁);

by means of multiplication, and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

21. An information processing apparatus recited in claim 4, characterized in that the apparatus further has:

a memory in which a constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with a quantity of a change of y coordinates; and

calculation means for, in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to the y coordinates,

reading μ

corresponding to the quantity of the change of said y coordinates from said memory, and calculating

Δ

x′

=μ

*(x₂−

x₁);

by means of multiplication, and

calculating Δ

x by right-shifting the calculated Δ

x by n digits.

6. An imaging processing method of a three-dimensional image, in which coordinate values are limited to integers, and transformation processing is applied to coordinates (x,y,z)^t to create coordinates (x′

,y′

,z′

)^t by means of an affine transformation representative of $\left(\begin{array}{c}{x}^{\mathrm{\prime <br><br>}}\\ y^{\mathrm{\prime <br><br>}}\\ z^{\mathrm{\prime <br><br>}}\end{array}\right)=A<br><br>\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right),$

characterized in that the method has steps of;

in transforming the coordinates (x,y,z)^t of a shape data into the coordinates (x′

,y′

,z′

)^t by means of an affine transformation of a matrix A and a matrix (t₁,t₂,t₃)^t, reading a parameter A′

$A^{\mathrm{\prime <br><br>}}=(\text{\hspace{1em}<br><br>}<br><br>\begin{array}{ccc}{a}_{11}^{\mathrm{\prime <br><br>}}& a_{12}^{\mathrm{\prime <br><br>}}& a_{13}^{\mathrm{\prime <br><br>}}\\ a_{21}^{\mathrm{\prime <br><br>}}& a_{22}^{\mathrm{\prime <br><br>}}& a_{23}^{\mathrm{\prime <br><br>}}\\ a_{31}^{\mathrm{\prime <br><br>}}& a_{32}^{\mathrm{\prime <br><br>}}& a_{33}^{\mathrm{\prime <br><br>}}\end{array}<br><br>\text{\hspace{1em}<br><br>})=\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>A=(\text{\hspace{1em}<br><br>}<br><br>\begin{array}{ccc}\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{11}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{12}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{13}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{21}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{22}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{23}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{31}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{32}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{33}\end{array}<br><br>\text{\hspace{1em}<br><br>})$

that is obtained by multiplying the matrix A by λ

(≠

0), and the matrix (t₁,t₂,t₃)^t, which are stored in a memory; and

based on said read matrix A′ and

matrix (t₁,t₂,t₃)^t, calculating

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

and calculating the coordinates (x′

,y′

,z′

)^t.

7. An imaging processing method recited in claim 6, characterized in that said λ

is limited to 2ⁿ(n≧

1), and

said calculating step is a step of calculating division of

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

by conducting right-shifting by n digits to calculate the coordinates (x′

,y′

,z′

)^t.

8. An imaging processing method recited in claim 7, characterized in that the method further has a step of adding a constant λ

/2 to each number to be divided before conducting said right-shifting by n digits.

9. An imaging processing method recited in any of claim 6 to claim 8, characterized in that the method further has a step of synthesizing two or more parameters that are multiplied by λ

(≠

0) in advance.

10. An imaging processing method recited in any of claim 6 to claim 8;

characterized in that the method further has steps of;

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates,

reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

22. An imaging processing method recited in claim 9, characterized in that the method further has steps of:

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates,

reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

11. A program for recorded on a computer readable medium for making an information processing apparatus apply transformation processing to coordinates (x,y,z)^t to create coordinates (x′

,y′

,z′

)^t by means of an affine transformation representative of $\left(\begin{array}{c}{x}^{\mathrm{\prime <br><br>}}\\ y^{\mathrm{\prime <br><br>}}\\ z^{\mathrm{\prime <br><br>}}\end{array}\right)=A<br><br>\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right),$

in which coordinate values are limited to integers, characterized in that the program has steps of;

in transforming the coordinates (x,y,z)^t of a shape data into the coordinates (x′

,y′

,z′

)^t by means of an affine transformation of a matrix A and a matrix (t₁,t_2,t3)^t, reading a parameter A′

$A^{\mathrm{\prime <br><br>}}=\left(\begin{array}{ccc}{a}_{11}^{\mathrm{\prime <br><br>}}& a_{12}^{\mathrm{\prime <br><br>}}& a_{13}^{\mathrm{\prime <br><br>}}\\ a_{21}^{\mathrm{\prime <br><br>}}& a_{22}^{\mathrm{\prime <br><br>}}& a_{23}^{\mathrm{\prime <br><br>}}\\ a_{31}^{\mathrm{\prime <br><br>}}& a_{32}^{\mathrm{\prime <br><br>}}& a_{33}^{\mathrm{\prime <br><br>}}\end{array}\right)=\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>A=\left(\begin{array}{ccc}\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{11}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{12}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{13}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{21}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{22}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{23}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{31}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{32}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{33}\end{array}\right)$

that is obtained by multiplying the matrix A by λ

(≠

0), and the matrix (t₁,t₂t₃₎^t, which are stored in a memory; and

based on said read matrix A′ and

matrix (t₁,t₂,t₃₎^t, calculating

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

and calculating the coordinates (x′

y′

,z′

)^t.

12. A program recited in claim 11, characterized in that, in case that said λ

is limited to 2ⁿ(n≧

1),

said calculating step in said program is a step of, in the information processing apparatus, calculating division of

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

by conducting right-shifting by n digits to calculate the coordinates (x′

,y′

,z′

)^t.

13. A program recited in claim 12, characterized in that, in the information processing apparatus, the program further has a step of adding a constant λ

/2 to each number to be divided before conducting said right-shifting by n digits.

14. A program recited in any of claim 11 to claim 13, characterized in that, in the information processing apparatus, the program further has a step of synthesizing two or more parameters that are multiplied by λ

(≠

0) in advance.

15. A program recited in any of claim 11 to claim 13, characterized in that, in the information processing apparatus, the program further has steps of:

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates,

reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

23. A program recited in claim 14, characterized in that, in the information processing apparatus, the program further has steps of:

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates

reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

16. A record medium in which a program for making an information processing apparatus apply transformation processing to coordinates (x,y,z)^tto create coordinates (x′

,y′

z′

)^t by means of an affine transformation representative of $\left(\begin{array}{c}{x}^{\mathrm{\prime <br><br>}}\\ y^{\mathrm{\prime <br><br>}}\\ z^{\mathrm{\prime <br><br>}}\end{array}\right)=A<br><br>\left(\begin{array}{c}x\\ y\\ z\end{array}\right)+\left(\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right)$

is stored, wherein coordinate values are limited to integers, characterized in that said program has steps of;

in transforming the coordinates (x,y,z)^t of a shape data into the coordinates (x′

,y′

,z′

)^t by means of an affins transformation of a matrix A and a matrix (t₁,t₂,t_3,)^t, reading a parameter A′

$A^{\mathrm{\prime <br><br>}}=\left(\begin{array}{ccc}{a}_{11}^{\mathrm{\prime <br><br>}}& a_{12}^{\mathrm{\prime <br><br>}}& a_{13}^{\mathrm{\prime <br><br>}}\\ a_{21}^{\mathrm{\prime <br><br>}}& a_{22}^{\mathrm{\prime <br><br>}}& a_{23}^{\mathrm{\prime <br><br>}}\\ a_{31}^{\mathrm{\prime <br><br>}}& a_{32}^{\mathrm{\prime <br><br>}}& a_{33}^{\mathrm{\prime <br><br>}}\end{array}\right)=\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>A=\left(\begin{array}{ccc}\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{11}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{12}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{13}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{21}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{22}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{23}\\ \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{31}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{32}& \mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>a_{33}\end{array}\right)$

that is obtained by multiplying the matrix A by λ

(≠

0), and the matrix (t₁,t₂,t₃)^t, which are stored in a memory; and

based on said read matrix A′ and

matrix (t₁,t₂,t₃)^t, calculating

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

and calculating the coordinates (x′

,y′

,z′

)^t.

17. A record medium in which the program is stored recited in claim 16, characterized in that, in case that said λ

is limited to 2ⁿ(n≧

1),

said calculating step in said program is a step of, in the information processing apparatus, calculating division of

x′

=(a′

₁₁*x+a′

₁₂*y+a′

₁₃*z)/λ

+t₁;

y′

=(a′

₂₁*x+a′

₂₂*y+a′

₂₃*z)/λ

+t₂;

z′

=(a′

₃₁*x+a′

₃₂*y+a′

₃₃*z)/λ

+t₃;

by conducting right-shifting by n digits to calculate the coordinates (x′

,y′

,z′

)^t.

18. A record medium in which the program is stored recited in claim 17, characterized in that, in the information processing apparatus, said program further has a step of adding a constant λ

/2 to each number to be divided before conducting said right-shifting by n digits.

19. A record medium in which the program is stored recited in any of claim 16 to claim 18, characterized in that, in the information processing apparatus said program further has a step of synthesizing two or more parameters that are multiplied by λ

(≠

0) in advance.

20. A record medium in which the program is stored recited in any of claim 16 to claim 18, characterized in that, in the information processing apparatus, said program further has steps of:

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates,

reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.

24. A record medium in which the program is stored recited in claim 19, characterized in that, in the information processing apparatus, said program further has steps of:

in calculating a quantity of a change of x coordinates $\mathrm{\Delta <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>x=\frac{x_2-x_1}{y_2-y_1}$

to y coordinates, reading a constant μ

corresponding to a quantity of a change to be calculated of said y coordinates from a memory in which the constant μ

representative of $\mathrm{\mu <br><br>}=\frac{\mathrm{\lambda <br><br>}}{y_2-y_1}<br><br>\left(\mathrm{\lambda <br><br>}<br><br>\text{\hspace{1em}<br><br>}<br><br>\mathrm{is}<br><br>\text{\hspace{1em}<br><br>}<br><br>2^n<br><br>\left(n\ge <br><br>1\right)\right)$

is stored in association with the quantity of the change of the y coordinates;

calculating

Δ

x′

=μ

*(x₂−

x₁);

based on the read constant μ

; and

calculating Δ

x by right-shifting the calculated Δ

x′

by n digits.