From DXOMark: More pixels offsets noise3

Started May 30, 2009 | Discussions thread
ejmartin Veteran Member • Posts: 6,274
Units and scalings

natureman wrote:

First a little preface about dimensions. One of the first things that one tries to inculcate in science students is an appreciation that physical quantities have dimensions -- mass, length, time, and combinations thereof. There are commonly accepted systems of units in which those quantities are measured -- grams, centimeters, seconds, etc. (There are also goofier ones, based on the size of some medieval king's extremities, etc.)

For instance, it is always a good check on one's calculation that the units on the two sides of an equation are the same; if they are not, a mistake has been made.

There are also scalings. Scalings are useful for comparing different units of measurement -- eg there are a 10,000 microns/centimeter. Both microns and centimeters have dimensions of length, so the scale factor 10,000 is dimensionless.

Should magnification be based on sensor size, or megapixels?

Let's look at the dimensionalities of these quantities.

Magnification is a scaling -- a ratio of sizes. Therefore it is dimension-less, a pure number in any system of units.

Sensor size has units of area (length^2=height x width).

The number of megapixels is a pure number, dimensionless. However, it is not a ratio of sizes, as magnification is; it is not a scaling.

None of these three quantities is directly related to any other -- one is a scaling, another is dimensionful with dimensions of length^2, and the third is dimensionless but not a scaling.

If sensor size is what matters when it comes to magnification,

As above: These two quantities have different meanings and dimensions, they are not the same.

why is
it that the camera manufacturers state that image size depends on the
amount of pixels?

Image size and pixel count are also not directly related, as they have different dimensions.

Image editing and viewing programs have settings for percentages of
magnification. Two images from different cameras with different sized
sensors (with the same aspect ratio) but with the same amount of
pixels would be the same size if magnified by the same percentage in
a computer program.

Two images from different sized sensors, say APS-C with 5µ pixels and FF with 8µ pixels (so that both have the same pixel count), are magnified by different amounts when both are displayed pixel-for pixel on a monitor. Typical monitors have a dot pitch of .025cm, so the APS-C image has been magnified by a scaling factor of

.025cm/5µ x 10000 µ cm = 50

The FF image has been scaled by a magnification factor of

.025cm/8µ x 10000µ cm = 31

Pixel-for-pixel is often referred to as 100% view, or 100% crop. Other percentages represent further rescalings of the image; for instance 50% view is a rescaling of the image size by .5 from the above scalings, so for instance the image taken by the APS-C camera is now magnified by 25 from what the sensor recorded on the image plane of the camera.

So yes, the two images are displayed the same size, but they have been magnified by different amounts from their source. This is why digicam images are so much noisier -- they have been magnified as much as 6x more than images from a FF DSLR, when both cameras have the same pixel count and are viewed the same size on a monitor.

In other words, two cameras with 12mp each (and the same aspect
ratio), with one having an APS-C sensor and the other a FF sensor,
would display at the same size if both were magnified to the same
percentage of magnification in a computer program.

But realize that the images have undergone different magnifications to achieve that 100% view -- by the ratio of the dot pitch of the monitor to the pixel pitch of the sensor.

I don't recall ever seeing a camera company, a website, or a computer
program basing image sizes from a camera on sensor size. It's always
based on the amount of pixels.

Once again you are equating two quantities of different physical dimensions -- the dimensionless pixel count and the dimensionful image size. One only relates a pixel count to an image size when one specifies the size of the pixel being used to reproduce the image. Implicit then is a magnification of the image by the ratio of the output device pixel size (or dot pitch) to the pixel pitch of the sensor that took the image.

Camera companies, websites, and computer programs don't discuss the amount of magnification that the image undergoes from capture on the sensor to display/printing because it's to some extent extraneous to the purpose of image editing and displaying. One is manipulating the sampled image data at that point, and the dot pitch of the printer/monitor and the pixel pitch of the sensor that recorded the image are somewhat extraneous to the task at hand, which is manipulating the pixel data for a desired outcome.

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