M010 · Measurement

Motion Measurement & Numerical Differentiation

Do more frames always give better acceleration? No. Differencing closely spaced frames magnifies the position noise, and acceleration needs a second difference, which magnifies it again. Widely spaced frames skip over changes in the motion. A local fit across many calibrated frames usually beats both.

Record the pixel positions of a moving object with a simulated camera. Estimate velocity and acceleration with finite differences or a local fit, then see how noise, aliasing, calibration bias and integration drift affect the result.

Selected time $\text{Ready}$

Reference position $\text{Ready}$

Reference signed velocity $\text{Ready}$

Reference signed acceleration $\text{Ready}$

Sampling status $\text{Ready}$

Acquired frames $\text{Ready}$

Tracked frames $\text{Ready}$

Valid derivative estimates $\text{Ready}$

Raw position RMS error $\text{Ready}$

Velocity RMS error $\text{Ready}$

Acceleration RMS error $\text{Ready}$

Last segment integration minus raw position $\text{Ready}$

Assumed camera scale $\text{Ready}$

Assumed origin column $\text{Ready}$

Record the frames, then fit and compare

Measure the motion from the pixels

Recording duration

T

4.00\,s

Trajectory offset

x_b

0.00\,m

Trajectory drift velocity

b

0.15\,\frac{\mathrm m}{\mathrm s}

Trajectory constant acceleration

c

0.00\,\frac{\mathrm m}{\mathrm s^2}

Oscillation amplitude

A

1.50\,m

Oscillation frequency

f

0.50\,Hz

Initial oscillation phase

\phi

0.00\,{}^\circ

Frame rate

f_s

30.00\,Hz

True camera scale

s

120.00\,\frac{\mathrm{px}}{\mathrm m}

Pixel noise bound before rounding

\eta_u

0.50\,\mathrm{px}

Integer pixel rounding

Q

1.00\,

Deterministic noise realization

r

17.00\,

Assumed scale error

\epsilon_s

0.00\,\%

Assumed origin pixel error

\delta_u

0.00\,\mathrm{px}

Derivative: 0 three points; 1 quadratic fit

d

1.00\,

Quadratic fit window in frames

n_w

9.00\,

Added velocity bias for integration

\beta_v

0.00\,\frac{\mathrm m}{\mathrm s}

Replay fraction

\tau

0.40\,

The simulated target moves along one line with constant acceleration plus a sinusoidal oscillation. An ideal orthographic camera images it onto a sensor 2048 columns wide, with exact timestamps and instantaneous exposure. The transparent surface is the camera's virtual image plane; it is not a lens or a 3D chart of the data. The Lab does not upload or analyze real video.

Each recorded frame holds only the frame number, the time, the noisy horizontal pixel and whether tracking succeeded. The noise is deterministic and bounded; optional rounding to the nearest pixel adds up to half a pixel more. When the target is outside the frame, its pixel is missing, and the Lab never clips or interpolates to make up a tracked position.

The estimates use only the recorded pixels and the calibration you assume. A three-point central difference uses three frames; a centered quadratic least-squares fit uses the odd-sized window you choose. Both need a complete window of consecutive, evenly timed frames, so there are no estimates at the ends of the record or across gaps. The analytic trajectory is shown separately and labeled as a reference.

More frames mean a shorter time step, which can amplify pixel noise in the derivatives. A wider window smooths out noise but also wipes out fast motion. When the frame rate is at or below twice the sinusoid frequency, aliasing can occur; a frame rate above that does not by itself guarantee an accurate derivative. The Lab gives no confidence interval.

The recovered velocity is integrated with the trapezoid rule. Each unbroken stretch of estimates starts from its own first calibrated raw position, and the integration never jumps across a gap. The optional velocity bias is added on purpose to demonstrate drift. Changing the calibration, fit settings or bias, or replaying, keeps the acquired pixels. Changing the motion or the camera acquires a new simulated record automatically.

The prescribed motion combines drift, constant acceleration and a sinusoidal oscillation. An ideal orthographic camera samples horizontal pixels at exact timestamps; pixel perturbations, rounding and missing targets affect the recovered derivatives.

Start with the essentials

Focus question: Do more frames always give better acceleration?
One-sentence intuition: You can recover only what the acquired samples and the calibration actually support.

Core mathematical model

The synthetic trajectory

\begin{aligned}x(t)&=x_b+bt+\tfrac12ct^2\\&\quad+A\sin(2\pi ft+\phi)\end{aligned}

This law generates the pixels and a separate reference curve for comparison. The estimator never sees it.

Turning recorded pixels into positions

\widehat x_i=\frac{u_i^{\mathrm{obs}}-\widehat u_0}{\widehat s}

The scale is in pixels per metre. An error in the origin shifts every position by the same amount, so it cancels exactly in differences.

Noise in finite differences

\begin{aligned}\widehat v_i&=\frac{\widehat x_{i+1}-\widehat x_{i-1}}{2h}\\\widehat a_i&=\frac{\widehat x_{i+1}-2\widehat x_i+\widehat x_{i-1}}{h^2}\end{aligned}

With a smaller time step, the same position error can become a larger derivative error.

Local quadratic fit

\begin{aligned}p(s)&=b_0+b_1s+b_2s^2\\e_j&=\widehat x_{i+j}-p(jh)\\\min_{\boldsymbol b}\;&\sum_{j=-r}^r e_j^2\\\widehat v_i&=b_1,\quad\widehat a_i=2b_2\end{aligned}

The fit uses only complete windows centered on each sample. A wider window also suppresses real rapid motion.

A known bias builds up as drift

\begin{aligned}\bar v_i&=\tfrac12(\widehat v_i+\widehat v_{i+1})\\\widehat x_{i+1}^{\mathrm{int}}&=\widehat x_i^{\mathrm{int}}+(\bar v_i+\beta_v)h\end{aligned}

The bias is added on purpose and known in advance; it shows how drift grows linearly and is not inferred from the camera.

Common difficulties

More frames are not always better

Typical misconceptionEnough frames make numerical acceleration exact.

Better mental modelMore frames mean a smaller time step, which can amplify pixel noise in the derivatives. A wider window suppresses the noise but also erases rapid motion. At a frame rate at or below twice the stated sinusoid frequency, aliasing is possible; a rate above that does not by itself guarantee an accurate derivative. The Lab gives no confidence interval.

Run the experiment

01
Acquire first
Acquire Noisy oscillation; open the raw table and export pixels.
What to observe: The CSV holds no true derivatives.
02
Trade noise for detail
Compare three-point differences with a wider quadratic fit.
What to observe: The noise can shrink, but rapid motion is lost with it.
03
Find indistinguishable samples
Try Aliased motion and Stroboscopic rest.
What to observe: Sparse frames cannot uniquely determine the continuous motion.
04
Check calibration and gaps
Change assumed scale, add integration bias, then try Leaves the sensor.
What to observe: Changing analysis settings leaves the raw pixels untouched, and missing frames stay missing.

Motion Measurement & Numerical Differentiation