System and method for simulated linearization of curved surface
View Patent ↗In accordance with embodiments of the present disclosure, a method may include mapping an ellipse with its center at the origin of an xy-coordinate plane to a circle with its center at the origin of the xy-coordinate plane. The method may also include transforming a three-dimensional subject image including a curved profile in the xy-coordinate plane to an intermediate image based at least on the mapping of the ellipse to a base circle. The method may further include transforming the intermediate image to a transformed image having a linear profile by distorting each of one or more points of interest of the intermediate image based at least on a dimension of the circle and a coordinate of such point.
1. A method comprising:
mapping an ellipse with its center at the origin of an xy-coordinate plane to a circle with its center at the origin of the xy-coordinate plane;
transforming a three-dimensional subject image including a curved profile in the xy-coordinate plane to an intermediate image based at least on the mapping of the ellipse to the circle; and
transforming the intermediate image to a transformed image having a linear profile by distorting each of one or more points of interest of the intermediate image based at least on a dimension of the circle and a coordinate of such point.
2. A method according to claim 1 , wherein a radius of the circle is equal to one of a major radius of the ellipse or a minor radius of the ellipse.
3. A method according to claim 1 , wherein transforming the subject image to the intermediate image is in accordance with the following equations:
P x ′=P x ×b/a
P y ′=P y
P z ′=P z
where P x , P y , and P z are respectively the x, y, and z coordinates of each point P i in the subject image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point P i ′ in the intermediate image, a is a minor radius of the ellipse, and b is a major radius of the ellipse.
4. A method according to claim 1 , wherein transforming the intermediate image to the transformed image is in accordance with the following equations:
P x ″=b ×arctan( P x ′/P y ′)
P y ″=√( P x ′ 2 +P y ′ 2 )
P z ″=P z ′
where P x ″, P y ″, and P z ″ are respectively the x, y, and z coordinates of each point of interest in the transformed image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point of interest in the intermediate image, and b is the radius of the circle.
5. A method according to claim 1 , wherein each of the subject image and the transformed image is a segmented surface image.
6. A method according to claim 1 , in which the subject image is a graphical representation of one of a mandible and a maxilla.
7. An article of manufacture comprising:
a non-transitory computer-readable medium; and
computer-executable instructions carried on the computer-readable medium, the instructions executable by one or more processors and configured to cause the one or more processors to:
map an ellipse with its center at the origin of an xy-coordinate plane to a circle with its center at the origin of the xy-coordinate plane;
transform a three-dimensional subject image including a curved profile in the xy-coordinate plane to an intermediate image based at least on the mapping of the ellipse to the circle; and
transform the intermediate image to a transformed image having a linear profile by distorting each of one or more points of interest of the intermediate image based at least on a dimension of the circle and a coordinate of such point.
8. An article of manufacture according to claim 7 , wherein a radius of the circle is equal to one of a major radius of the ellipse or a minor radius of the ellipse.
9. An article of manufacture according to claim 7 , wherein transforming the subject image to the intermediate image is in accordance with the following equations:
P x ′=P x ×b/a
P y ′=P y
P z ′=P z
where P x , P y , and P z are respectively the x, y, and z coordinates of each point P i in the subject image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point P i ′ in the intermediate image, a is a minor radius of the ellipse, and b is a major radius of the ellipse.
10. An article of manufacture according to claim 7 , wherein transforming the intermediate image to the transformed image is in accordance with the following equations:
P x ″=b ×arctan( P x ′/P y ′)
P y ″=√( P x ′ 2 +P y ′ 2 )
P z ″=P z ′
where P x ″, P y ″, and P z ″ are respectively the x, y, and z coordinates of each point of interest in the transformed image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point of interest in the intermediate image, and b is the radius of the circle.
11. An article of manufacture according to claim 7 , wherein each of the subject image and the transformed image is a segmented surface image.
12. An article of manufacture according to claim 7 , in which the subject image is a graphical representation of one of a mandible and a maxilla.
13. A computing system, comprising:
a processor; and
a memory communicatively coupled to the processor and having stored thereon a program of instructions configured to, when executed by the processor:
map an ellipse with its center at the origin of an xy-coordinate plane to a circle with its center at the origin of the xy-coordinate plane;
transform a three-dimensional subject image including a curved profile in the xy-coordinate plane to an intermediate image based at least on the mapping of the ellipse to the circle; and
transform the intermediate image to a transformed image having a linear profile by distorting each of one or more points of interest of the intermediate image based at least on a dimension of the circle and a coordinate of such point.
14. A computing system according to claim 13 , wherein a radius of the circle is equal to one of a major radius of the ellipse or a minor radius of the ellipse.
15. A computing system according to claim 13 , wherein transforming the subject image to the intermediate image is in accordance with the following equations:
P x ′=P x ×b/a
P y ′=P y
P z ′=P z
where P x , P y , and P z are respectively the x, y, and z coordinates of each point P i in the subject image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point P i ′ in the intermediate image, a is a minor radius of the ellipse, and b is a major radius of the ellipse.
16. A computing system according to claim 13 , wherein transforming the intermediate image to the transformed image is in accordance with the following equations:
P x ″=b ×arctan( P x ′/P y ′)
P y ″=√( P x ′ 2 +P y ′ 2 )
P z ″=P z ′
where P x ″, P y ″, and P z ″ are respectively the x, y, and z coordinates of each point of interest in the transformed image, P x ′, P y ′, and P z ′ are respectively the x, y, and z coordinates of each corresponding point of interest in the intermediate image, and b is the radius of the circle.
17. A computing system according to claim 13 , wherein each of the subject image and the transformed image is a segmented surface image.
18. A computing system according to claim 13 , in which the subject image is a graphical representation of one of a mandible and a maxilla.