mach.wavefront.spherical

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mach.wavefront.spherical#

mach.wavefront.spherical(
origin_m: Real[Array, 'xyz=3'],
points_m: Real[Array, '*points xyz=3'],
focus_m: Real[Array, 'xyz=3'],
) → Real[Array, '*points']#

Spherical-wave transmit distance (also known as focused or diverging waves).

Spherical waves propagate like a collapsing sphere focusing onto a point, or an expanding sphere diverging from a point.

Parameters:

  • origin_m – xyz-position of the transmitting element/sender in meters. distance=0 at the origin.

  • points_m – xyz-positions of the points to compute the transmit-arrival time for, in meters.

  • focus_m – xyz-position of the focal point where spherical waves converge, in meters. sometimes called the source or apex. for a focused wave, the focus is in front of the origin. for a diverging wave, the focus is behind the origin. Note: ‘focus’ refers to the convergence point, while ‘origin’ refers to the physical transducer element that transmits the wave.

Returns:

The transmit-wavefront-arrival distance for each point in meters, calculated as the difference between the origin-to-focus distance and the focus-to-point distance. To convert to time, divide by sound speed: distance_m / sound_speed_m_s

Notes

Similar to Equation 5 / Figure 2 from:

https://www.biomecardio.com/publis/ultrasonics21.pdf So you think you can DAS? A viewpoint on delay-and-sum beamforming Perrot, Polichetti, Varray, Garcia 2021

Modifications: assume L=0 (use wave-time through origin), and extend to 3D.

The sign convention accounts for the direction of wave propagation. For typical ultrasound imaging where z increases with depth, negative values indicate the wavefront arrives before the reference time, positive values after.

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