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Visual Time to Collision (TTC)

Time to collision (TTC) is the estimated time until an approaching object reaches the camera, assuming its current relative velocity stays constant.

A camera can estimate TTC without knowing the object's real size or distance. The key observation is looming: an object appears larger in the image as it gets closer.

Warning

A small TTC does not by itself prove that a collision will occur. The object's path must also overlap the camera or robot's path.

1. Start with pinhole-camera geometry

For an object facing the camera:

  • \(W\): real object width;
  • \(Z\): distance from the camera;
  • \(f\): camera focal length in pixels;
  • \(w\): object width measured in the image, in pixels.

Similar triangles give:

\[ \frac{w}{f} = \frac{W}{Z} \]

Therefore:

\[ w = \frac{fW}{Z} \]

The real width \(W\) and focal length \(f\) are constant. When distance \(Z\) decreases, image width \(w\) increases.

2. Relate image growth to TTC

Differentiate \(w=fW/Z\) with respect to time:

\[ \dot{w} = -\frac{w\dot{Z}}{Z} \]

For an approaching object, \(\dot{Z}<0\). TTC is distance divided by closing speed:

\[ TTC = \frac{Z}{-\dot{Z}} \]

Substituting the image-growth equation gives:

\[ \boxed{TTC = \frac{w}{\dot{w}}} \]

The unknown object width and camera focal length cancel. Only the object's image size and growth rate are needed.

3. Estimate TTC from two frames

Let:

  • \(w_0\) be the measured width in the first frame;
  • \(w_1\) be the width in the second frame;
  • \(\Delta t\) be the time between frames.

Assuming constant closing velocity, the TTC at the second frame is:

\[ \boxed{TTC_1 = \Delta t\frac{w_0}{w_1-w_0}} \]

This two-frame form comes directly from \(Z=fW/w\), so \(fW\) still cancels.

If \(w_1 \le w_0\), the object is not growing in the image. Do not report a finite approaching TTC from these two measurements.

Simple example

An object's bounding-box width changes from \(80\) pixels to \(100\) pixels in \(0.1\) seconds:

\[ w_0=80, \qquad w_1=100, \qquad \Delta t=0.1\text{ s} \]

Apply the two-frame formula:

\[ TTC_1 = 0.1\frac{80}{100-80} = 0.1\frac{80}{20} = 0.4\text{ s} \]

At the time of the second frame, the object is estimated to be \(0.4\) seconds from the camera plane if the same motion continues.

Interactive simulation

Press Start to watch an object approach the camera at a constant \(2\) m/s. The side view shows the decreasing distance. The camera view shows the same object expanding according to \(w=fW/Z\).

The actual value is \(Z/v\). Every \(0.25\) seconds, the demo measures two projected widths and calculates \(TTC_1=\Delta t\,w_0/(w_1-w_0)\). After the second sample, the estimated and actual TTC should closely match. The dashed rectangle marks the latest sampled image width.

Practical steps

  1. Detect or track the same object in consecutive frames.
  2. Measure the same size quantity in both frames, such as bounding-box width.
  3. Calculate \(\Delta t\) from the frame timestamps, not only the nominal frame rate.
  4. Confirm that the measured size increased.
  5. Calculate \(TTC_1=\Delta t\,w_0/(w_1-w_0)\).
  6. Combine TTC with object position or predicted path before declaring a collision risk.

If object area \(A\) is more stable than width, use the linear scale \(w=\sqrt{A}\) in the same formula.

Assumptions and limitations

The simple geometry assumes:

  • constant relative velocity between the two frames;
  • the same rigid object is tracked in both frames;
  • object orientation and visible shape do not change significantly;
  • image-size growth is caused mainly by motion toward the camera;
  • the camera and timestamps are stable enough to measure the growth.

Bounding-box jitter, rotation, partial occlusion, camera motion, and very small changes in image size can make TTC noisy. Real collision-avoidance systems normally smooth several measurements and combine visual TTC with trajectory or range information.

Reference