M010 · Measurement

Motion Measurement & Numerical Differentiation

Record simulated camera pixels, recover derivatives using finite differences or a local fit, then inspect noise, aliasing, calibration bias and integration drift.

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 / FIT / COMPARE

Let the pixels speak

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\,

A simulated one-dimensional target combines constant acceleration and a sinusoid. An ideal orthographic camera maps it to a 2048-column sensor with exact timestamps and instantaneous exposure. The transparent surface is a virtual image plane, not an optical lens or a 3D data chart. This Lab does not upload or analyze real video.

The record contains only frame number, time, noisy horizontal pixel and tracking status. Noise is deterministic and bounded; optional nearest-pixel rounding adds up to half a pixel. Out-of-frame targets have missing pixels. No clipping or interpolation fabricates a tracked position.

Inference receives only recorded pixels and the assumed calibration. Three-point central differences use three frames; a centered quadratic least-squares fit uses the selected odd window. Both need a full contiguous, uniformly timed window, so endpoints and gaps remain unavailable. The analytic trajectory is a separately labeled reference.

More frames reduce the time step and can amplify pixel noise in derivatives. A wider window suppresses noise but also erases rapid motion. At or below twice the declared sinusoid frequency, aliasing is possible; being above that threshold alone does not guarantee an accurate derivative. No confidence interval is claimed.

Recovered velocity is integrated by trapezoids. Each contiguous estimate segment starts at its first calibrated raw position; gaps are never bridged. The optional added velocity bias is an explicit demonstration of drift. Calibration, fit settings, bias and replay preserve acquired pixels; changes to motion or the camera clear them.

Prescribed teaching motion; explicit interpolation and measurement boundaries.

Start with the essentials

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

Core mathematical model

Declared synthetic trajectory

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

This law generates pixels and an independent comparison reference; it is withheld from the estimator.

Calibrate only the recorded pixel

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

Scale is pixels per metre; an origin error offsets position but cancels in exact differences.

Noise enters derivatives

\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}

A smaller time step can amplify the same position error.

Local quadratic reconstruction

\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}

Only a complete centered window is used; a wider window also suppresses real rapid motion.

An explicit bias accumulates

\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 known added bias illustrates linear drift; it is not inferred camera behavior.

Common difficulties

Model and evidence

Typical misconceptionEnough frames make numerical acceleration exact.

Better mental modelMore frames reduce the time step and can amplify pixel noise in derivatives. A wider window suppresses noise but also erases rapid motion. At or below twice the declared sinusoid frequency, aliasing is possible; being above that threshold alone does not guarantee an accurate derivative. No confidence interval is claimed.

Run the experiment

01
Acquire first
Acquire Noisy oscillation; open the raw table and export pixels.
What to observe: No true derivative is present in the CSV.
02
Trade noise for detail
Compare three-point differences with a wider quadratic fit.
What to observe: Noise can shrink while rapid motion is lost.
03
Find indistinguishable samples
Try Aliased motion and Stroboscopic rest.
What to observe: Sparse frames cannot uniquely determine the continuous motion.
04
Audit calibration and gaps
Change assumed scale, add integration bias, then try Leaves the sensor.
What to observe: Analysis settings preserve raw pixels; missing frames stay missing.