Camera Field of View and Pixels per Foot

Why this matters

The call every video installer dreads: "Someone stole a package off the porch, the camera got it, and you can't tell who it is." The camera worked. It recorded a person, in color, at the right time. It was never able to show a face, because the face covered a few dozen pixels in a frame built to show the whole yard. That was decided the day the camera went up, by the lens and the distance, and it can be worked out on paper before a single hole is drilled. The unit that does the working is pixels per foot.

The one formula

Pixels per foot (ppf) = horizontal pixels in the recorded image / width of the scene in feet, at the distance where the target will be.

Both halves matter. The horizontal pixels are those of the stream the recorder actually keeps. A camera sold as 2560 pixels wide that is set to record a lower-resolution substream records fewer, and the ppf drops with it. The scene width is not a property of the camera; it depends on how far away you are measuring, because the field of view spreads out from the lens as a wedge.

From above, a camera's horizontal field of view is a triangle with the lens at the point. Here is a camera whose recorded image is 2560 pixels across, with a lens that shows a scene 16 ft wide at 20 ft from the camera:

             +-------------------------------+
               \   40 ft out: 32 ft wide   /
                 \        80 ppf         /
                   \ 20 ft: 16 ft wide /
                     +---------------+
                       \  160 ppf  /
                         \       /
                           \   /
                          CAMERA
              2560 px across, about 44 degrees

The scene doubles in width when the distance doubles: 16 ft at 20 ft, 32 ft at 40 ft. The 2560 pixels stay the same, so the ppf halves: 2560 divided by 16 is 160 ppf at 20 ft, and 2560 divided by 32 is 80 ppf at 40 ft.

Getting the scene width without a chart. Width = 2 x distance x tan(half the horizontal angle). For this lens, 16 ft wide at 20 ft means tan(half angle) = 8 / 20 = 0.4, which is a half angle of about 21.8 degrees and a full horizontal angle of about 44 degrees. Run it forward as a check: at 40 ft, 2 x 40 x 0.4 = 32 ft, which matches the diagram. Most camera makers publish the horizontal angle for each lens; some publish a calculator that does this for you. Use the horizontal angle, not the diagonal one, and use the figure for the resolution and aspect ratio you will record at, since some cameras crop the sensor at other settings.

How many pixels per foot you need

How much detail is enough depends on the job the image has to do. The most widely used framework is the one in IEC 62676-4, the international standard for video surveillance system application, which grades image detail by task in pixels per meter. Converted to feet (dividing by 3.28):

Treat these as the floor under good conditions, not a target. They assume a well-lit, in-focus, uncompressed-looking image of a target facing the camera. Real conditions take detail away:

• Night. Under infrared the image goes monochrome and noisy, and the encoder throws away fine detail first. The same ppf that identifies by day may only recognize at night.
• Compression. A recorder set to a low bitrate smears faces before it smears a wall. More on that in Recorder Storage Math: Bitrate, Days and Cameras.
• Motion. A person walking past at an angle with a slow shutter blurs.
• Angle. A face seen from well above, or from the side, presents fewer useful pixels than the same face head-on. A camera mounted high and tilted steeply down sees the tops of heads.

So a shop default that holds up is to design for the identify figure plus a margin wherever identification is the point, and to put the camera where people face it.

The trade a wider lens makes

A wider lens covers more ground with the same pixels, so every foot of scene gets fewer of them. Swap the 44 degree lens in the diagram for a wide lens with about a 100 degree horizontal view on the same 2560 pixel camera:

• At 20 ft: width = 2 x 20 x tan(50 degrees) = 2 x 20 x 1.19 = about 47.7 ft. 2560 / 47.7 = about 54 ppf. Recognize, not identify.
• At 40 ft: width = about 95.3 ft. 2560 / 95.3 = about 27 ppf. Observe, not recognize.

The wide lens sees the whole front yard and both neighbors' driveways. It will show that someone came up the walk and what they carried. It will not identify them at the door, let alone at the curb. Neither lens is wrong. They do different jobs, and a homeowner usually needs both: one camera for the overview and one for the place a face will be.

Resolution buys back some of it, not all

The tempting fix for a wide lens is more pixels. Put the same 100 degree lens on a camera that records 3840 pixels across: at 40 ft, 3840 / 95.3 = about 40 ppf, which clears the 38 ppf recognize line by a hair and is still about half of the 76 ppf identify line. Recognition at the sidewalk, identification nowhere past the porch.

Two things limit the trade. First, ppf rises only in proportion to pixels: going from 2560 to 3840 is 1.5 times the pixels and 1.5 times the ppf, while halving the scene width (a narrower lens) doubles it. Second, more pixels per frame need more bitrate to keep the same quality, so a higher-resolution camera on the same recorder settings either eats storage faster or gets compressed harder, and the extra detail is the first thing compression removes. A narrower lens aimed at the place the face will be is usually the cheaper detail.

Night range is a separate limit. A camera's built-in infrared illuminator has a rated range per the maker. A target past that range is underlit however many pixels land on it, so check that the identify distance sits inside the IR range, or that the scene has its own lighting.

Worked example: identifying a visitor at the front door

The customer wants two things: to see who walks up the 40 ft front walk from the sidewalk, and to identify anyone who reaches the door. The only practical mount is the corner of the garage, 40 ft from the sidewalk end of the walk. A visitor standing at the door is 12 ft from that mount, seen at a modest angle. The proposed camera records 2560 pixels across.

Job 1, the walk. The customer wants to recognize regular visitors along the walk. The far end of the walk is 40 ft away. To recognize there, the design needs at least 38 ppf at 40 ft (this article's table). Scene width allowed = 2560 / 38 = about 67 ft. A 44 degree lens gives 32 ft at 40 ft, which is 80 ppf, more than enough; a 100 degree lens gives about 95.3 ft and about 27 ppf, not enough. Anything up to about an 80 degree lens would pass at 40 ft (2 x 40 x tan(40 degrees) = 2 x 40 x 0.84 = about 67 ft). The 44 degree lens covers the walk with margin, but at 32 ft wide at the sidewalk it does not show the street beyond.

Job 2, the door. From the garage corner the door and the whole walk lie inside a narrow wedge, so one aim can hold both, but the tech confirms it on site: at 12 ft, the 44 degree lens shows only 2 x 12 x 0.4 = 9.6 ft of scene, so the door must sit within 4.8 ft of the center line there. 2560 / 9.6 = about 267 ppf, well past the identify figure of 76. Even accounting for the angle and nighttime, faces at the door are covered.

Check the mounting height. The garage corner fixture is at 9 ft. At the door 12 ft away, the camera looks down at a face 5 ft off the ground through a drop of 4 ft over 12 ft, and the arctangent of 4 / 12 is about 18 degrees, which presents the face well. Mounted at the soffit, 14 ft up, the drop to a 5 ft face is 9 ft over 12 ft, about 37 degrees, and a visitor in a cap shows mostly brim. The tech mounts at 9 ft.

The design. One 44 degree camera on the garage corner at 9 ft covers both jobs: 80 ppf at the sidewalk, about 267 ppf at the door. The customer also wanted the street, so a second, wide-lens camera goes on the other corner as an overview, with no expectation that it identifies anyone. That expectation goes into the handover in writing. See related: How to Plan Camera Locations on a House.

Verify on site, not on paper. After mounting, the tech has a helper stand at the door and at the sidewalk, day and again after dark, and checks the recorded footage on the recorder, not the live view on a phone. The live view can be a higher-quality stream than the one being kept.

Where the math misleads

• Varifocal and motorized lenses change the angle. Record the final angle setting in the job notes; the next tech who "zooms out a bit" for a customer changes every number above.
• Fisheye and panoramic cameras spread their pixels across a circle or a very wide band, and the image is de-warped in software. The ppf at the edge is far lower than a simple width calculation suggests.
• The target is not at one distance. A driveway runs from near to far, so ppf varies along it. Design for the farthest point where the task still has to succeed.

References

• IEC 62676-4, Video surveillance systems for use in security applications, Part 4: Application guidelines (detection, observation, recognition and identification levels)
• Camera maker's lens specifications: horizontal angle at the recorded resolution
• See related: How to Plan Camera Locations on a House
• See related: Recorder Storage Math: Bitrate, Days and Cameras