A photograph holds a little time: sources & model

Photograph a moving toy, shorten the exposure, and unroll the time behind each row. Compare motion blur, rolling shutter, missed focus and low-light noise using your own saved captures.

What supports the explanation?

What this model assumes

1. Authored image-plane model and original 3D rendering; no personal camera measurements or uploaded photographs.
2. Toy motion is rigid horizontal translation with fixed viewpoint and wheel orientation. PBR source shading is an authored signal field, not calibrated irradiance.
3. The focus bench is a separate stationary flat target; it is not a post-blur of a rolling capture.
4. Main pictures show expected signal. The noise loupe is a separate stationary-patch observation, not a noisy full-camera simulation.
5. No diffraction, aberrations, demosaicking, dark current, fixed-pattern noise, full-well capacity, ADC quantization or multiframe processing.
6. Software tests and inspected source material do not constitute independent subject review or learner validation.
7. Define the coordinate system: The reference detector has 640 columns and 360 rows, numbered from zero. Coordinates describe the final upright image. Relative motion is measured in image pixels per second, not a measured toy speed in meters per second.
8. Row timing convention: For row r, c(r) = (r/359 − 1/2)R; reversing scan changes its sign. Start is c − T/2 and end is c + T/2. R is explicitly the first-to-last center delay, not an unqualified manufacturer readout-time specification.
9. Three different quantities: Per-row motion extent is |v|T. The signed full-height center displacement is vR, with scan direction included. The full capture span is R + T, while the adjacent-row delay is R/359. None of these by itself establishes the frame rate.
10. A true image integral: The image operator integrates a continuous row signal over the chosen exposure and a one-pixel detector footprint. Toy source pixels are decoded from sRGB into premultiplied linear RGBA and reconstructed as constant cells. Moving foreground coverage and the stationary background are combined in that space. Encoding for display happens last.
11. Exact within a deliberately narrow model: Translation is constant, horizontal and parallel to a fixed orthographic image plane. Piecewise-polynomial primitives integrate the reconstructed source exactly, including fractional shifts. This does not represent spinning wheels, acceleration, moving perspective, depth-dependent motion or a general camera-shake kernel.
12. A measurable stripe fixture: The calibration bar is eight pixels wide before motion. At four pixels of motion, detector centers at offsets 0, 2, 4 and 6 receive normalized signals 1, 31/32, 1/2 and 1/32. At sixteen pixels of motion the central value is 1/2. The normalized bar area remains eight pixels, with enough image margin.
13. Not every feature spans every row: The full-height calibration bar has the complete vR center displacement. The shorter placard on the toy spans fewer row times. Do not attribute the full-image number to every short edge in the picture.
14. An ideal geometric focus footprint: For the stationary flat target, object distance is 1,000 mm, focal length 8 mm and detector pitch 0.01 mm. Focusing at 500 mm with f/2 produces a geometric footprint diameter of 400/123 ≈ 3.2520 pixels. At f/4 it is half as large. This is a circular footprint diameter, not a Gaussian blur setting.
15. Focus limits: The flat-target inspector convolves a normalized circular footprint, using 64 × 64 area samples per kernel cell. Diffraction, lens aberrations, color filters, lens breathing, autofocus and portrait-mode processing are omitted. The narrow geometric model does not mean that the smallest possible aperture always gives the sharpest photograph.
16. Detected electrons, not incident photons: The separate stationary patch has an authored expected detected-electron rate of 20,000 per second at f/2 and illumination multiplier 1. Expected count μ = 20,000Tq(2/N)² already includes detection efficiency; it is not an incident-photon rate or a measurement of a real phone.
17. Shot noise and read noise: The patch uses P distributed as Poisson(μ), plus independent zero-mean Gaussian read noise with standard deviation 2 electrons. Before clipping, the mean is μ, variance is μ + 4 and input SNR is μ/√(μ + 4). At 10, 100 and 400 expected electrons the SNRs are approximately 2.673, 9.806 and 19.901.
18. Computational photographs: A phone may combine frames and apply denoising, sharpening, stabilization or other processing. Our display gain is not a complete ISO model, and our single-exposure image is not a reconstruction of a particular phone pipeline. A corrected source image is an estimate, not proof that arbitrary lost information can be recovered.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.

About the cover illustration

Original rendered toy and stationary grid, integrated through an explicit exposure model. An authored teaching capture, not a photograph from a real camera.