---
kind: concept
slug: time-of-flight-depth-sensing
title: Time-of-flight depth sensing
summary: >-
  Time-of-flight (ToF) sensors measure distance from the round trip of light, either by timing a pulse or by reading the phase of a modulated signal. Pulsed sensors suit long outdoor range; continuous-wave cameras suit short indoor range and need an unwrapping rule. Calibration turns raw readings into 3-D points.
tags: [time-of-flight, depth-sensing, lidar, camera-calibration]
papers: ["2012.06772"]
related: [calibration, stereo-matching]
status: draft
updated_at: 2026-11-06
---

## Overview

Every ToF variant turns a round trip of light into a distance. With travel
time τ and speed of light c, depth is d = c τ / 2 [arxiv:2012.06772 §4.3].
Pulsed sensors measure τ directly. Continuous-wave (CW) sensors measure a
phase φ = 2π f τ for a modulation frequency f [arxiv:2012.06772 §1]
[arxiv:2012.06772 §4.1].

The shared constraint is timing precision. A centimetre of depth needs timing
better than a nanosecond, because light covers about 30 cm per nanosecond
[arxiv:2012.06772 §2.1]. CW sensors trade direct timing for phase, which
repeats every full cycle, so depth is ambiguous beyond one unambiguous range
[arxiv:2012.06772 §4.3].

Only one card (the survey by [Horaud et al.](/papers/2012.06772)) covers this
concept. The paper has no experiments of its own, so the figures below are
formulas and device specifications the paper reports. They are not measured
performance.

## Variants

### Pulsed-light range scanners

A laser or laser diode emits pulses, and single-photon avalanche diodes
(SPADs) with per-pixel time-to-digital converters record arrival times
[arxiv:2012.06772 §2] [arxiv:2012.06772 §2.2]. The scanner uses one laser on a
rotating mirror, or a head of 16, 32 or 64 laser-detector pairs at fixed
vertical angles [arxiv:2012.06772 §2]. The Velodyne HDL-64E, HDL-32E and VLP-16
are the models in the paper's table, at 905 nm with 2–120 m or 2–100 m range
and 2 cm accuracy [arxiv:2012.06772 tab. 1]. Card: [Horaud et al. survey](/papers/2012.06772).

### Pulsed-light flash cameras

One diffused laser pulse floods the scene, and a 2-D SPAD array records depth
in one shot [arxiv:2012.06772 §2] [arxiv:2012.06772 §3.3]. The paper's table
lists TigerEye and DragonEye units at 128×128 pixels with ranges up to 1100 m
and 1500 m, all outdoor-capable [arxiv:2012.06772 tab. 2]. The MIT Lincoln
prototype used a 1 ns pulse at 532 nm [arxiv:2012.06772 §3.3]. Card: [Horaud et al. survey](/papers/2012.06772).

### Continuous-wave cameras

The emitter sends a sinusoidally modulated signal, and each pixel
demodulates the return [arxiv:2012.06772 §4.1]. A real pixel takes four
samples, each through a shutter open for a quarter of the modulation period,
and sums them over many periods, up to hundreds of thousands, to raise the
signal-to-noise ratio [arxiv:2012.06772 §4.2]. The paper's table lists
SR4000, SR4500, DS311, DS325, E70, E40 and Kinect v2 at 160×120 to 512×424
pixels and 30 to 60 FPS [arxiv:2012.06772 tab. 3]. Most are indoor-only;
E70 and E40 are marked outdoor-capable there. Card: [Horaud et al. survey](/papers/2012.06772).

## When to use which

| Situation | Use | Why |
|---|---|---|
| Outdoor, long range (vehicles, robot navigation) | Pulsed-light scanner or flash camera | The paper says CW systems do not work under adverse outdoor light [arxiv:2012.06772 §8]. Scanners list 2–120 m [arxiv:2012.06772 tab. 1]; flash cameras list up to 1100 m, outdoor-capable [arxiv:2012.06772 tab. 2]. |
| Indoor, short range (about 0–10 m) | Continuous-wave camera | SR4000 covers 0–5 or 0–10 m and is indoor-only [arxiv:2012.06772 tab. 3]. Each pixel reads four samples to get a phase, not a pulse arrival time [arxiv:2012.06772 §4.1]. |
| Highest frame rate | CW camera | DS311 and DS325 list 60 FPS [arxiv:2012.06772 tab. 3]. Scanners list 5–20 Hz [arxiv:2012.06772 tab. 1]. |
| Several cameras in one scene | CW camera with a distinct frequency per unit | SR4000 units support six frequencies [arxiv:2012.06772 §5.1]. Many devices do not let the user set the frequency [arxiv:2012.06772 §7]. |
| Need depth on untextured surfaces with colour alignment | ToF plus colour camera | ToF gives depth regardless of texture [arxiv:2012.06772 §7]. Kinect v2 and DS311/DS325 have a colour camera built in [arxiv:2012.06772 §7]. |
| Need a measured accuracy figure | Not decidable from this paper | Most devices have no accuracy figure in the paper, because the paper says it depends on surface, lighting and frame rate [arxiv:2012.06772 tab. 2] [arxiv:2012.06772 tab. 3]. Measure your own setup. |

## Implementation sketch

The common core is: get a per-pixel quantity (phase or arrival time), convert
it to a radial distance, then back-project through a calibrated camera model.
Calibration covers intrinsics, distortion and a depth-wiggling correction
[arxiv:2012.06772 §6].

The code below follows the paper's CW formulas. The paper's LaTeX source
contains no code link.

```pseudocode
# Four-bucket demodulation of one CW pixel  [arxiv:2012.06772 §4.1] [§4.2]
# Input:  c0..c3 = samples at 0, pi/2, pi, 3pi/2, each summed over many
#         periods  [arxiv:2012.06772 §4.2]
#         a1, a2 = emitted offset and amplitude (source not stated)  [§4.1]
# Output: phase (rad), amplitude, offset
def demodulate_pixel(c0, c1, c2, c3, a1, a2):
    # Phase from the arctan; atan2 picks the quadrant (implementer's choice)
    phase = atan2(c3 - c1, c0 - c2)
    # Map into one period; the paper does not say how to wrap  [§4.1]
    phase = phase % (2.0 * pi)
    # Amplitude, scaled by the emitted amplitude a2  [§4.1]
    amplitude = sqrt((c3 - c1)**2 + (c0 - c2)**2) / a2
    # Offset, scaled by the emitted offset a1  [§4.1]
    offset = (c0 + c1 + c2 + c3) / (4.0 * a1)
    return phase, amplitude, offset


# Depth from phase  [arxiv:2012.06772 §4.3]
# Input:  phase (rad) in [0, 2*pi), modulation frequency f (Hz)
# Output: depth (m) inside the unambiguous range [0, d_max)
def depth_from_phase(phase, f):
    # Unambiguous range: depth at phase 2*pi  [§4.3]
    d_max = C_LIGHT / (2.0 * f)
    # d = c * phase / (4*pi*f), equal to d_max * phase / (2*pi)  [§4.3]
    return d_max * phase / (2.0 * pi)


# Pixel to camera-frame point, pinhole model  [arxiv:2012.06772 §6.1]
# Input:  (u, v) pixel after undistortion  [§6.3], radial distance d (m),
#         A 3x3 intrinsic matrix (no skew term in the paper)
# Output: point (X, Y, Z) in camera coordinates
def pixel_to_point(u, v, d, A):
    # Ray direction from the pixel  [§6.1]
    ray = inverse(A) @ [u, v, 1]
    # Scale the ray so its length equals the radial distance d  [§6.1]
    return d / norm(ray) * ray
```

Notes for implementers, by variant:

- **Pulsed sensors** skip the demodulation step. They use arrival time τ
  directly, with d = c τ / 2 [arxiv:2012.06772 §4.3].
- **CW sensors** must unwrap phase. The paper cites several methods but
  describes none. Multi-frequency unwrapping is cited without detail
  [arxiv:2012.06772 §4.3].
- **Depth-wiggling correction.** A systematic nonlinear error ("wiggling")
  can be corrected by a lookup table or a cubic B-spline correction whose
  coefficients come from least squares on known depths [arxiv:2012.06772 §6.4].
  The number of coefficients and calibration depths is not stated, so
  calibrate per device.
- **Colour alignment.** Projective alignment solved by SVD, refined by
  bundle adjustment, is one option. The paper states the method without
  equations [arxiv:2012.06772 §7].
- **Practical calibration.** The amplitude-plus-offset image of a CW camera
  can be calibrated with standard colour-camera tools. Because ToF images
  are low resolution, the paper uses a corner detector that fits two pencils
  of lines to the pattern [arxiv:2012.06772 §6.5].

Pitfalls the card records:

- **Radial distance, not z-depth.** The pinhole model in
  [arxiv:2012.06772 §6.1] assumes the sensor reports radial distance from the
  optical centre. The paper's footnote says the sensor measures to the image
  plane and assumes a small offset [arxiv:2012.06772 §6.1]. Many SDKs return
  z-depth, so check the device before using `pixel_to_point`.
- **No skew term.** The intrinsic matrix has none, and the paper does not
  justify dropping it [arxiv:2012.06772 §6.1].
- **Scale constants.** The amplitude and offset formulas need a1 and a2, and
  the paper does not say how to get them. Using the sensor's own
  amplitude-plus-offset output is the practical route [arxiv:2012.06772 §6.5],
  but the paper does not give that as a formula.
- **Motion blur.** Long integration means moving objects blur, and fast
  shutter speeds are not possible [arxiv:2012.06772 §4.2] [arxiv:2012.06772 §4.3].
- **Error sources.** The paper lists temperature, motion blur, distance,
  background light, phase wrapping, scattering and multi-path errors
  [arxiv:2012.06772 §4.3]. It gives magnitudes for few of them.

## Contradictions

- **Phase formula, within the paper.** The four-sample phase is
  arctan((C3 − C1) / (C0 − C2)) [arxiv:2012.06772 §4.1]. The depth formula
  writes arctan((C3 − C1) / (C0 − C1)) [arxiv:2012.06772 §4.3]. The paper
  does not reconcile them, and this page does not pick one. Check the
  derivation before implementing.
- **SR4500 range.** The paper states a 9 m maximum at 13.5 MHz
  [arxiv:2012.06772 §5.1] [arxiv:2012.06772 tab. 3]. Its own relation
  c / (2f) gives about 11.1 m at that frequency [arxiv:2012.06772 §4.3].
  Other SR4000 values match the relation: 5.17 m at 29 MHz
  [arxiv:2012.06772 §5.1].
- **HDL-64E pulse width.** The text gives 5 ns [arxiv:2012.06772 §3.1].
  Table 1 gives 10 ns [arxiv:2012.06772 tab. 1].
- **HDL-64E minimum range.** Figure 2 gives 1 m [arxiv:2012.06772 fig. 2].
  Table 1 gives 2 m [arxiv:2012.06772 tab. 1].
- **SPAD timing jitter.** The introduction says about 10 ps
  [arxiv:2012.06772 §1]. Section 2.2 says a few tens of picoseconds
  [arxiv:2012.06772 §2.2].
- **DragonEye frame rate.** The text gives 10 FPS [arxiv:2012.06772 §3.3].
  Table 2 gives 10–30 FPS [arxiv:2012.06772 tab. 2].
- **SR4000 frame rate.** Table 3 gives 30 FPS maximum [arxiv:2012.06772 tab. 3].
  The text says 10–15 FPS is the most accurate setting [arxiv:2012.06772 §5.1].
- **CW outdoor use.** The paper says CW systems do not work under adverse
  outdoor light [arxiv:2012.06772 §8]. Table 3 marks E70 and E40 as
  outdoor-capable [arxiv:2012.06772 tab. 3]. The paper does not explain the
  difference, and "adverse" is not defined.

## Papers

- [Horaud et al. survey](/papers/2012.06772): survey of pulsed-light and CW ToF sensing, with depth-from-phase formulas, device specifications and a pinhole-plus-distortion calibration model for ToF cameras. No new experiments.