Motion — Question 5

Question 5

A moving image is not always a moving object

A camera views a straight track with fixed landmarks A and B, separated by $4.00\ \mathrm{m}$. P is the tracked object. The image may shift horizontally between exposures, but the initial model excludes rescaling and distortion. All three points lie in the same calibrated plane.

Pixel locations increase to the right and are exact in this model: The pixel coordinates $(A,P,B)$ are $(100,260,500)$ in image I, $(140,300,540)$ in image II, and $(120,360,520)$ in image III. The fraction of the A-to-B span occupied by P is $f=(p_P-p_A)/(p_B-p_A)$.

1. Compute $f$ in each image. Does P change its recorded location relative to the track between I and II? Between II and III?

2. Find P’s distance from A in each image. Explain why the raw pixel change of P alone would give a misleading account of I to II.

3. In a fourth image, A and B are 450 pixels apart. Is the original pure-shift model still adequate? Does this observation alone establish that the physical landmarks moved?

4. Do equal track-relative locations in I and II prove P stayed at rest throughout the intervening interval? State the limit of the conclusion.

Question 5 - Solution

A moving image is not always a moving object

(a) Remove the common image offset with landmarks.

$$f=\frac{p_P-p_A}{p_B-p_A}:\quad f_I=\frac{160}{400}=0.4,\quad f_{II}=\frac{160}{400}=0.4,\quad f_{III}=\frac{240}{400}=0.6.$$

The landmark separation is 400 px in each of the first three images. Thus $$f_I=\frac{260-100}{400}=\boxed{0.40},\quad f_{II}=\frac{300-140}{400}=\boxed{0.40},\quad f_{III}=\frac{360-120}{400}=\boxed{0.60}.$$ P has the same recorded track-relative location in I and II. In III it is farther toward B.

(b) Use the physical landmark span.

$$x_{P/A}=4f\ \mathrm{m}:\quad(1.6,1.6,2.4)\ \mathrm{m};\qquad\Delta x_{II\to III}=+0.8\ \mathrm{m}.$$

Multiplying by 4.00 m gives distances from A of $\boxed{1.60,\ 1.60,\ 2.40\ \mathrm{m}}$. The sampled change from II to III is 0.80 m toward B.

Between I and II, A, P, and B all shift by 40 px. That common image shift changes P’s raw pixel coordinate without changing its location relative to the landmarks. Treating the image origin as fixed to the track would misidentify camera/image motion as object motion.

(c) Test the image model before interpreting a new record.

$$p'_B-p'_A=(p_B+c)-(p_A+c)=p_B-p_A=400\ne450.$$

A pure horizontal shift preserves the A–B pixel separation. A change from 400 to 450 px therefore contradicts that model. Rescaling, changed viewing geometry, or another image-model failure could explain it while the landmarks remain physically fixed. Physical landmark motion is not established by the pixel span alone.

(d) Registration does not fill gaps in time.

The equal fractions establish equal track-relative locations at two exposure instants. P could move and return between them. Landmarks resolve the image-reference issue; they do not turn separate snapshots into continuous monitoring of P.