34.6 Doppler / velocity imaging🔗⧉

There are two ways to know how fast something in a scene is moving. The usual way is to infer it: take two frames, find where each bit of the scene went, and divide displacement by time, optical flow (Chapter 12.8, Optical flow). The other way is to measure it, over a measurement interval, by exploiting a fact about waves: when a surface moves toward or away from you, it shifts the frequency of any wave bouncing off it. A coherent source, such as a laser or a radar transmitter, sends out a wave of known frequency; the surface's motion adds a small Doppler shift to the return; and that shift, read per pixel, is the surface's velocity along the line of sight. This is Doppler imaging: a sensor that returns radial velocity at every pixel.

34.6.1 The Doppler shift, as a velocity sensor🔗⧉

The effect is the one everyone has heard, the siren that drops in pitch as the ambulance passes. A source approaching you crowds its wavefronts together, raising the frequency you receive; a receding source stretches them, lowering it. For light or radar bouncing off a moving surface (a round trip, so the effect counts twice), for a colocated transmitter and receiver and speeds much smaller than the wave speed, the fractional frequency shift is approximately

where $v_r$ is the radial velocity, the component of the surface's motion along the line of sight, positive toward the sensor under this sign convention, and $c$ is the wave's speed. Measure $\Delta f$ and you have $v_r$ directly. Note what the formula does not contain: brightness, texture, or a second frame. The sign of the shift tells you toward (blue, higher $f$) versus away (red, lower $f$); the magnitude tells you how fast. The catch, equally visible in the formula, is that it sees only the radial component: motion purely across the field of view produces no shift, exactly the blind spot that optical flow handles best, which is why the two are complementary rather than competing.

34.6.2 The instruments: vibrometry, Doppler LiDAR, radar🔗⧉

The same principle is realized at three wavelengths, for three jobs.

Laser Doppler vibrometry (LDV) points a laser at a surface and reads the Doppler shift of the reflected beam to recover the surface's velocity and, when integrated, its displacement, without touching it. It is sensitive enough to register vibrations of nanometers, which makes it the standard non-contact tool for measuring how an engine panel, a loudspeaker cone, or a bridge resonates. It is also the literal, hardware version of an idea this book meets elsewhere as a computational trick: the visual microphone, which recovers sound from the minute vibrations a video camera can see on a bag of chips or a plant leaf (Davis et al. 2014; Part 38, Revealing the invisible). LDV does optically and directly what the visual microphone does by inference from ordinary frames: measure a surface's micro-motion and read the sound from it, and the comparison is the cleanest illustration of "measure, don't infer" in the book.

Doppler LiDAR adds velocity to the range LiDAR already gives. A frequency-modulated continuous-wave (FMCW) LiDAR sweeps its laser frequency and beats the return against the outgoing beam; the beat frequency carries both the time-of-flight (hence range) and the Doppler shift (hence range-rate) in the beat signal. One constant-slope chirp generally mixes range and velocity; multiple chirp slopes or repeated chirps provide the independent information needed to separate them. So an FMCW LiDAR returns a point cloud where every point knows both where it is and how fast it is approaching, a decisive advantage for self-driving cars, which can tell a stationary lamppost from an oncoming car within a chirp sequence, without matching objects across camera frames. Ego-motion must still be accounted for when deciding whether an object is stationary in the world.

Doppler radar does the same at radio wavelengths, the version most readers have already seen on television: weather radar colors clouds by the radial velocity of rain (toward/away), revealing rotation in a storm, and automotive radar reports the closing speed of the car ahead. Radar is often more robust than optical sensing to fog and poor visibility, although rain attenuation, clutter, multipath, and target reflectivity still matter, at the cost of coarse angular resolution, which is why a car fuses radar's reliable range-rate with a camera's fine angular detail.

34.6.3 Where it fits🔗⧉

Doppler imaging belongs to the same family as the rest of this part: a sensor engineered to measure, per pixel, a quantity an ordinary camera throws away, here radial velocity, rather than Chapter 34.5, Single-photon sensors (SPAD, avalanche, photon counting) that gave range. Its natural partner is time-of-flight: range tells you where a surface is, range-rate tells you where it is going, and a sensor that returns both at once (FMCW LiDAR) hands perception a fuller picture than either alone. Its natural foil is optical flow, which infers full two-dimensional motion from image change but needs texture, light, and two frames; Doppler measures only the radial component but can do so in darkness or on an untextured surface if the coherent return is strong enough. Frequency resolution requires observation time; there is no instantaneous velocity readout. Like Doppler ultrasound and weather radar, it shows that when you can illuminate a scene coherently, velocity is something you can read directly off the returning wave, not something you must reconstruct after the fact.