Determining position/speed of projectile using camera and radar
Disclosed are embodiments for determining the position and speed of a projectile in 3D world coordinates using a calibrated camera and radar. In some embodiments, a method comprises: capturing a sequential set of images of a projectile in motion; determining a radial velocity of the projectile; determining a set of two-dimensional (2D) image points representing respective locations of the projectile in the sequential set of images; determining respective depths of the 2D image points in their respective 2D image planes based on the 2D image points, time intervals between the captured 2D images, the radial velocity and a rotational part of a homographic transformation, the homographic transformation configured to project three-dimensional (3D) world coordinates to the 2D image planes; and determining a set of 3D world points of the projectile in the 3D world coordinates based on the 2D image points, the respective depths thereof and the homographic projection.
1 . A method comprising:
capturing, with a camera, a sequential set of images of a projectile in motion;
determining, with at least one radar, a radial velocity of the projectile;
determining, with at least one processor, a set of two-dimensional (2D) image points representing respective locations of the projectile in the sequential set of images;
determining, with the at least one processor, respective depths of the 2D image points in their respective 2D image planes based on the 2D image points, time intervals between the captured 2D images, the radial velocity and a rotational part of a homographic transformation, wherein the homographic transformation is configured to project three-dimensional (3D) world coordinates to the 2D image planes;
determining, with the at least one processor, a set of 3D world points of the projectile in the 3D world coordinates based on the 2D image points, the respective depths thereof and the homographic projection; and
wherein the sequential set of images includes three images and the set of 2D image points include three 2D image points corresponding to the three images, the method further comprising:
determining a cross-ratio defining a ratio of a first distance between a first 2D image point and a second 2D image point over a second distance between the second 2D image point and a third 2D image point;
determining a second ratio of a first time interval between a first image capture and a second image capture over a second time interval between the second image capture and a third image capture;
determining a first depth corresponding to the first 2D image point as a function of the cross-ratio, the second ratio and a depth corresponding to the second 2D image point;
determining the second depth corresponding to the second 2D image point as a function of the cross-ratio, the second ratio, a time interval between the first and third image captures, the radial velocity as a projection of linear velocity onto a direction from the radar to the projectile, the first, second and third 2D image points, and the rotational part of the homographic transformation; and
determining a third depth corresponding to the third 2D image point as a function of the cross-ratio, the second ratio and the second depth corresponding to the second 2D image point.
2 . The method of claim 1 , further comprising:
determining, with the at least one processor, a speed of the projectile in the 3D world coordinates based on the set of 3D world points and the time intervals.
3 . The method of claim 1 , wherein the time intervals are determined based on a frame rate or flash interval of the camera.
4 . The method of claim 1 , wherein the second depth is determined first, followed by the first depth and then the third depth.
5 . The method of claim 1 , wherein a trajectory of the projectile represented by the 3D world points is substantially linear.
6 . The method of claim 1 , further comprising calibrating the at least one camera without decomposing camera parameters into explicit extrinsic and explicit intrinsic parameters.
7 . The method of claim 6 , wherein the camera is calibrated by modifying one or more implicit extrinsic camera parameters, the modifying comprising adjusting implicit extrinsic camera parameters through an affine correction.
8 . The method of claim 6 , further comprising extracting intrinsic camera parameters implicitly from the homographic transformation.
9 . A system comprising:
a camera;
at least one radar; and
at least one processor configured to:
capture, with the camera, a sequential set of images of a projectile in motion;
determine, with the at least one radar, a radial velocity of the projectile;
determine, with the at least one processor, a set of 2D image points representing respective locations of the projectile in the sequential set of images;
determine, with the at least one processor, respective depths of the 2D image points in their respective 2D image planes based on the 2D image points, time intervals between the captured 2D images, the radial velocity and a rotational part of a homographic transformation, wherein the homographic transformation is configured to project 3D world coordinates to the 2D image planes;
determine, with the at least one processor, a set of 3D world points of the projectile in the 3D world coordinates based on the 2D image points, the respective depths thereof and the homographic projection; and
wherein the sequential set of images includes three images and the set of 2D image points includes three 2D image points corresponding to the three images, the at least one processor is further configured to:
determine a cross-ratio defining a ratio of a first distance between a first 2D image point and a second 2D image point over a second distance between the second 2D image point and a third 2D image point;
determine a second ratio of a first time interval between a first image capture and a second image capture over a second time interval between the second image capture and a third image capture;
determine a first depth corresponding to the first 2D image point as a function of the cross-ratio, the second ratio and a depth corresponding to the second 2D image point;
determine the second depth corresponding to the second 2D image point as a function of the cross-ratio, the second ratio, a time interval between the first and third image captures, the radial velocity as a projection of linear velocity onto a direction from the radar to the projectile, the first, second and third 2D image points, and the rotational part of the homographic transformation; and
determine a third depth corresponding to the third 2D image point as a function of the cross-ratio, the second ratio and the second depth corresponding to the second 2D image point.
10 . The system of claim 9 , wherein the at least one processor is further configured to:
determine, with the at least one processor, a speed of the projectile in the 3D world coordinates based on the set of 3D world points and the time intervals.
11 . The system of claim 9 , wherein the time intervals are determined based on a frame rate or flash interval of the camera.
12 . The system of claim 9 , wherein a trajectory of the projectile represented by the 3D world points is substantially linear.
13 . The system of claim 9 , wherein the at least one radar is a continuous wave (CW) radar.
14 . The system of claim 9 , wherein the at least one camera has been calibrated without decomposing camera parameters into explicit extrinsic and explicit intrinsic parameters.
15 . The system of claim 14 , wherein the at least one processor is further configured to implicitly extract the intrinsic camera parameters from the homographic transformation.
16 . The system of claim 14 , wherein the at least one camera is calibrated by modifying one or more implicit extrinsic camera parameters, the modifying comprising adjusting implicit extrinsic camera parameters through an affine correction.
17 . The system of claim 16 , wherein there are two or more calibrated cameras and capturing of the sequential set of image frames comprises:
controlling, with the at least one processor, a timing of a first calibrated camera and a second calibrated camera physically spaced apart from the first calibrated camera, such that each of the first and second calibrated camera captures the sequential set of images.
18 . The system of claim 17 , wherein the first calibrated camera and the second calibrated camera run according to a common internal clock reference, and the controlling the timing of the first calibrated camera and the second calibrated camera comprises hard synchronizing of the timing of the first calibrated camera and the second calibrated camera.
19 . The system of claim 17 , wherein the controlling the timing of the first calibrated camera and the second calibrated camera comprises soft synchronizing the timing of the first calibrated camera and the second calibrated camera such that each captures an image frame at different time instances, each calibrated camera capturing the sequential set of images of the projectile in motion in a free run mode with each calibrated camera having its own internal synchronization.