The Math Behind Erasing Crowds with Long Exposures

A person walking through a scene during a four-minute exposure will be invisible in the final photograph, according to a mathematical analysis published by Fstoppers. The effect hinges on the fraction of time a moving subject occupies any single camera pixel.

Putting a 35mm lens on a full-frame camera about 20 meters from a plaza means each pixel covers roughly 3.4 mm of ground. A person walking at a normal pace clears their shoulder width in about a third of a second. Over a 240-second exposure, they cover any single pixel for only about 0.13 percent of the total time. This tiny fraction is the core mechanism for making crowds disappear.

Dwell Time Determines Visibility

A camera sensor pixel simply counts photons during an exposure. It has no memory of when they arrived. If an object covers a pixel for time t out of a total exposure T, it contributes roughly t/T of that pixel's final value. The outcome depends solely on two factors: the fraction of the exposure a spot is covered and how far the covering object sits from the background for luminance and color.

Luminance is the obvious factor, meaning a dark coat on bright pavement leaves the strongest signal. Color is often forgotten. A red coat and gray stone can have identical luminance but still leave a colored stain because the camera samples red, green, and blue channels separately.

Running the numbers for the example plaza, a person roughly 0.45 meters across occupies about 130 pixels of width. Walking at 1.4 m/s, they need 0.32 seconds to clear their own width. During a 240-second exposure, that's 0.13 percent of the total. A completely black silhouette blocking a pixel for that sliver removes about 0.13 percent of the light that pixel would have collected. In a well-exposed midtone, the pixel's own shot noise is several times larger than that change, burying the walker's contribution. One walker does not come out faint; in an ordinary print or screen view, they do not come out at all.

However, the source cautions against pushing this too far. A walker leaves a faint, smeared band along their path, not a body-shaped silhouette. This trail is structured, not random noise. Aggressive tone curves, heavy local contrast, or denoising software can potentially recover it. The defensible claim is that a single pass is normally imperceptible under ordinary viewing, not that the file contains no record.

Now consider if the same person stops to check a map for 90 seconds. That's 37.5 percent of the exposure, removing roughly 37 percent of the light from those pixels. The result is a translucent, recognizable human figure in the otherwise empty plaza. This is one of two failure modes. The first is people who stop, like tourists posing, someone eating lunch, or a vendor working from one spot.

The second failure mode involves occupancy accumulation across a crowd. What governs a pixel is the total time during which anybody is covering it. Individual crossings add only when they do not overlap. Two people passing the same spot together count once, not twice. A tight group therefore costs less than the same number of people spread out.

Treating occupancy alone as the answer is exact only for the black-silhouette case and is a good approximation when a crowd is generally darker than the ground. With fifty separate, non-overlapping crossings, a spot is occupied for about 16 seconds, or 6.7 percent of a four-minute exposure, making it comfortably visible. At the one percent threshold, roughly seven or eight clean crossings of the same pixel is the budget. This is easy on a quiet morning but can be exceeded in a couple of minutes on a busy walkway.

Practical Thresholds and Traffic Patterns

The practical threshold for visibility sits around one percent of the total exposure time. This is not the sensor's noise floor but closer to the point where a difference stops mattering to a viewer, aligning roughly with the limits of human luminance discrimination. The background also matters significantly. Flat pavement, still water, or clear sky reveal a ghost at a fraction of a percent, while textured surfaces like cobblestone or foliage can swallow a two or three percent change.

Consequently, the source states that honest advice is about traffic patterns rather than just shutter speeds. A steady, diffuse flow spreads occupancy thinly. If the total occupancy of any one spot stays under about one percent, it averages to nothing. However, a sustained, scattered crowd can still lay down a faint veil. Concentrated traffic at a queue, doorway, crosswalk, or popular photo spot will haunt the frame even if no one stops walking.

Method: One Very Long Exposure

The classic technique is a single, very long exposure, often requiring strong neutral density (ND) filters in bright light. Each stop of filtration halves the light, compounding the exposure time dramatically.

These numbers lead to very different outcomes. At f/11 and ISO 100 on a sunny day (metering at about 1/200s), a 10-stop filter yields only 5.1 seconds, which is "nowhere near enough to erase anyone." A 15-stop filter yields about 164 seconds, bringing the exposure into useful territory. A 10-stop filter only achieves a four-minute exposure when the ambient light is already low, around 1/4 second, which corresponds to deep shade or blue hour. Heavy overcast, at about 1/30s, becomes only 30 seconds with a 10-stop filter.

The source cites Lee's filter range as a common reference: the Little Stopper (6 stops), Big Stopper (10 stops), and Super Stopper (15 stops). For screw-on 10-stop filters, it lists the Breakthrough Photography X4 ND 3.0, B+W Master 810 ND 3.0, NiSi ND1000, and Haida NanoPro ND1000. Any ND filter labeled with a 3.0 density or 1000x rating is a 10-stop filter.

Finally, the analysis notes a key practical problem for digital photography: heat. Unlike film, digital sensors do not suffer from reciprocity failure and accumulate charge linearly. However, dark current rises with sensor temperature and exposure length, meaning multi-minute frames on a warm day can produce hot pixels.