Three dimensional trajectory model and system

A computing device comprising:

• an image capture device communicatively coupled to a programmable processor;
• a non-transient machine-readable medium storing instructions that, when executed by the programmable processor, cause the programmable processor to perform operations comprising:
• capturing, using the image capture device, one or more two-dimensional videos of a three-dimensional projectile object;
• calculating a data group based on the one or more two-dimensional videos;
• computing, using the programmable processor, a solution that is based on the data group, the solution corresponding to at least one characteristic of the three-dimensional projectile object;
• and
• displaying a visual representation of the three-dimensional projectile object and at least one trajectory for the projectile object based on the solution,
• wherein the projectile trajectory model is implemented according to a function (⊖,t) that returns three-dimensional coordinates of the projectile object at time t governed by the projectile trajectory model with an initial state ⊖,
• wherein the initial state ⊖ is a tuple (x, y, z, vx, vy, vy, wx, wy, wz) which denotes the coordinates, velocity and spin of the projectile object in a three-dimensional environment, and
• wherein a loss function ƒ of the projectile trajectory model is defined by the function (⊖,t) as:
• min
• θ
• ,
• τ
• ∑
• i
• ∑
• j
• d
• ⁡
• (
• P
• i
• ·
• f
• ⁡
• (
• θ
• ,
• t
• ij
• +
• τ
• i
• )
• ,
• x
• ij
• t
• )
• wherein xijt denotes a sample trajectory object J observed by at least one of the one or more image capture devices i at time t and Pi is the projection matrix for the capture device i and ti represents the time lag of the capture device i, and the function d measures distances between two homogeneous points on the capture device iframe.

A computer program product comprising a non-transitory, machine-readable medium storing instructions which, when executed by at least one programmable processor, cause the at least one programmable processor to perform operations comprising:

• capturing one or more videos of a projectile object trajectory for a projectile object, using one or more image capture devices coupled to at least a first computing device, the first computing device communicatively connected to a second computing device over a communications network;
• transmitting the one or more videos to the second computing device over the communications network, wherein the one or more videos from the one or more image capture devices are sorted into one or more data groups, at least one solution for the one or more data groups being calculated using a projectile trajectory model;
• and
• receiving the solution computed for the one or more data groups,
• wherein the projectile trajectory model is implemented according to a function (⊖,t) that returns three-dimensional coordinates of the projectile object at time t governed by the projectile trajectory model with an initial state ⊖,
• wherein the initial state ⊖ is a tuple (x, y, z, vx, vy, vy, wx, wy, wz) which denotes the coordinates, velocity and spin of the projectile object in a three-dimensional environment, and
• wherein a loss function ƒ of the projectile trajectory model is defined by the function (⊖,t) as:
• min
• θ
• ,
• τ
• ∑
• i
• ∑
• j
• d
• ⁡
• (
• P
• i
• ·
• f
• ⁡
• (
• θ
• ,
• t
• ij
• +
• τ
• i
• )
• ,
• x
• ij
• t
• )
• wherein xijt denotes a sample trajectory object J observed by at least one of the one or more image capture devices i at time t and Pi is the projection matrix for the capture device i and τi represents the time lag of the capture device i, and the function d measures distances between two homogeneous points on the capture device i frame.

A computer-implemented method of reconstructing a three-dimensional projectile object trajectory, the method comprising:

• receiving, at a remote computing device, one or more videos of a projectile object trajectory for a projectile object, the one or more videos being captured using one or more image capture devices coupled to a local computing device, the local computing device and the remote computing device being communicatively connected over a communications network;
• sorting the one or more videos from the one or more image capture devices into one or more data groups;
• computing at least one solution for the one or more data groups using a projectile trajectory model;
• and
• transmitting the solution computed for the one or more data groups to the local computing device over the communications network,
• wherein the projectile trajectory model is implemented according to a function (⊖,t) that returns three-dimensional coordinates of the projectile object at time t governed by the projectile trajectory model with an initial state ⊖,
• wherein the initial state ⊖ is a tuple (x, y, z, vx, vy, vy, wx, wy, wz) which denotes the coordinates, velocity and spin of the projectile object in a three-dimensional environment, and
• wherein a loss function ƒ of the projectile trajectory model is defined by the function (⊖,t) as:
• min
• θ
• ,
• τ
• ∑
• i
• ∑
• j
• d
• ⁡
• (
• P
• i
• ·
• f
• ⁡
• (
• θ
• ,
• t
• ij
• +
• τ
• i
• )
• ,
• x
• ij
• t
• )
• wherein xijt denotes a sample trajectory object J observed by at least one of the one or more image capture devices i at time t and Pi is the projection matrix for the capture device i and τi represents the time lag of the capture device i, and the function d measures distances between two homogeneous points on the capture device i frame.