Fast rendering of quadrics and marking of silhouettes thereof

1. Apparatus, comprising:

a display, comprising a screen; and

a processor, configured:

to define a bounding region on the screen,

to render a quadric, which is defined in a parameter space, over a three-dimensional electroanatomical map of a surface of a heart that is displayed on the screen, by:

for each pixel in the bounding region, transforming, to the parameter space, a virtual ray that passes through the pixel, and ascertaining whether a point of intersection between the transformed virtual ray and the quadric exists in the parameter space, and

for each pixel in the bounding region for which the point of intersection exists, rendering the pixel on the screen, based on properties of the point of intersection; and

to mark a silhouette of the quadric on the screen, by, for each pixel in the bounding region for which the point of intersection does not exist:

estimating a shortest distance between the transformed virtual ray and the quadric, and

provided that the estimated shortest distance is less than a threshold, marking the pixel as belonging to the silhouette.

2. The apparatus according to claim 1 , wherein the processor is configured to mark the silhouette of the quadric in response to a user selecting the quadric on the screen.

3. The apparatus according to claim 1 , wherein, for each pixel in the bounding region for which the point of intersection does not exist, the processor is configured to:

ascertain that the point of intersection does not exist, by:

substituting an equation of the transformed virtual ray into an equation of the quadric, such as to yield a quadratic equation at 2 +bt+c=0, and

ascertaining that a discriminant b 2 -4ac of the quadratic equation is negative, and

in response to ascertaining that the point of intersection does not exist, estimate the shortest distance between the transformed virtual ray and the quadric as Ic-b 2 /4a1.

4. The apparatus according to claim 1 , wherein the processor is further configured to define the quadric, in the parameter space, such that the quadric is bounded by a cube having eight corners, two of which are at (−1,−1,−1) and (1,1,1), respectively.

5. The apparatus according to claim 4 , wherein the processor is configured to define the bounding region by:

transforming the corners of the cube to a screen space, which is defined in terms of a coordinate system of the screen, and

defining the bounding region such that the bounding region is a minimum bounding rectangle of the transformed corners.

6. The apparatus according to claim 1 , wherein the processor is further configured to define the quadric such that the quadric is representable by a 4×4 diagonal matrix Q.

7. The apparatus according to claim 6 ,

wherein the virtual ray has a ray origin O and a ray-direction vector D,

wherein the processor is configured to transform the virtual ray by computing O′, which is the ray origin O transformed to the parameter space, and D′, which is the ray-direction vector D transformed to the parameter space, and

wherein the processor is configured to ascertain whether the point of intersection exists by:

computing a first coefficient a=D′ T QD′, where D′ T is a transpose of D′, a second coefficient b=2D′ T QO′, and a third coefficient c=O′ T QO′, and

subsequently, ascertaining whether a quadratic equation at 2 +bt+c=0 has any real roots.

8. The apparatus according to claim 7 , wherein the processor is further configured to represent Q as a four-element vector Q D , and wherein the processor is configured to compute each of the first coefficient a, the second coefficient b, and the third coefficient c by performing an element-wise multiplication of Q D .

9. The apparatus according to claim 1 , wherein the processor is further configured to receive a signal that indicates a location of a distal end of an intrabody catheter, and wherein the processor is configured to render the quadric over a portion of the three-dimensional electroanatomical map that corresponds to the indicated location.

10. The apparatus according to claim 9 , wherein the processor is configured to render the quadric in response to an ablating signal being passed into the surface of the heart, by the distal end of the intrabody catheter, at the indicated location.

11. The apparatus according to claim 1 , wherein the processor is configured to render the pixel on the screen by:

computing a normal vector to the quadric at the point of intersection, and

rendering the pixel, based on a coloring of the quadric at the point of intersection, and the normal vector.

12. A method, comprising:

using a processor, defining a bounding region on a screen;

rendering a quadric, which is defined in a parameter space, over a three-dimensional electroanatomical map of a surface of a heart that is displayed on the screen, by:

for each pixel in the bounding region, transforming, to the parameter space, a virtual ray that passes through the pixel, and ascertaining whether a point of intersection between the transformed virtual ray and the quadric exists in the parameter space, and

for each pixel in the bounding region for which the point of intersection exists, rendering the pixel on the screen, based on properties of the point of intersection; and

marking a silhouette of the quadric on the screen, by, for each pixel in the bounding region for which the point of intersection does not exist:

estimating a shortest distance between the transformed virtual ray and the quadric, and

provided that the estimated shortest distance is less than a threshold, marking the pixel as belonging to the silhouette.

13. The method according to claim 12 , wherein marking the silhouette of the quadric comprises marking the silhouette of the quadric in response to a user selecting the quadric on the screen.

14. The method according to claim 12 , wherein estimating the shortest distance between the transformed virtual ray and the quadric comprises estimating the shortest distance between the transformed virtual ray and the quadric by, for each pixel in the bounding region for which the point of intersection does not exist:

ascertaining that the point of intersection does not exist, by:

substituting an equation of the transformed virtual ray into an equation of the quadric, such as to yield a quadratic equation at 2 +bt+c=0, and

ascertaining that a discriminant b 2 -4ac of the quadratic equation is negative, and

in response to ascertaining that the point of intersection does not exist, estimating the shortest distance between the transformed virtual ray and the quadric as |c-b 2 /4a|.

15. The method according to claim 12 , further comprising defining the quadric, in the parameter space, such that the quadric is bounded by a cube having eight corners, two of which are at (−1,−1,−1) and (1,1,1), respectively.

16. The method according to claim 15 , wherein defining the bounding region comprises:

transforming the corners of the cube to a screen space, which is defined in terms of a coordinate system of the screen, and

defining the bounding region such that the bounding region is a minimum bounding rectangle of the transformed corners.

17. The method according to claim 12 , further comprising defining the quadric such that the quadric is representable by a 4×4 diagonal matrix Q.

18. The method according to claim 17 ,

wherein the virtual ray has a ray origin O and a ray-direction vector D,

wherein transforming the virtual ray comprises transforming the virtual ray by computing O′, which is the ray origin O transformed to the parameter space, and D′, which is the ray-direction vector D transformed to the parameter space, and

wherein ascertaining whether the point of intersection exists comprises:

computing a first coefficient a=D′ T QD′, where D′ T is a transpose of D′, a second coefficient b=2D′ T QO′, and a third coefficient c=O′ T QO′, and

subsequently, ascertaining whether a quadratic equation at 2 +bt+c=0 has any real roots.

19. The method according to claim 18 , further comprising representing Q as a four-element vector Q D , wherein computing the first coefficient a, the second coefficient b, and the third coefficient c comprises computing each of the first coefficient a, the second coefficient b, and the third coefficient c by performing an element-wise multiplication of Q D .

20. The method according to claim 12 , further comprising receiving a signal that indicates a location of a distal end of an intrabody catheter, wherein rendering the quadric comprises rendering the quadric over a portion of the three-dimensional electroanatomical map that corresponds to the indicated location.

21. The method according to claim 20 , wherein rendering the quadric comprises rendering the quadric in response to an ablating signal being passed into the surface of the heart, by the distal end of the intrabody catheter, at the indicated location.

22. The method according to claim 12 , wherein rendering the pixel on the screen comprises:

computing a normal vector to the quadric at the point of intersection, and

rendering the pixel, based on a coloring of the quadric at the point of intersection, and the normal vector.

23. A computer software product comprising a tangible non-transitory computer-readable medium in which program instructions are stored, which instructions, when read by a processor, cause the processor:

to define a bounding region on a screen,

to render a quadric, which is defined in a parameter space, over a three-dimensional electroanatomical map of a surface of a heart that is displayed on the screen, by:

for each pixel in the bounding region, transforming, to the parameter space, a virtual ray that passes through the pixel, and ascertaining whether a point of intersection between the transformed virtual ray and the quadric exists in the parameter space, and

for each pixel in the bounding region for which the point of intersection exists, rendering the pixel on the screen, based on properties of the point of intersection; and to mark a silhouette of the quadric on the screen, by, for each pixel in the bounding region for which the point of intersection does not exist:

estimating a shortest distance between the transformed virtual ray and the quadric, and

provided that the estimated shortest distance is less than a threshold, marking the pixel as belonging to the silhouette.

24. The computer software product according to claim 23 , wherein the instructions further cause the processor:

to compute a normal vector to the quadric at the point of intersection, and

to render the pixel, based on a coloring of the quadric at the point of intersection, and the normal vector.