I understand the concept of exposure.
I mean that for every camera slower shutter speeds lead to increased noise.
I wonder, are you associating slower shutter speeds with increased noise because you often use 1-second or longer shutter speeds in dark environments and often with higher ISOs? For example, do you use those settings when doing night sky photograhy, nighttime street photograhy, or when traveling and photographing the dark interiors of historic buildings?
How does it compare to amplification via iso setting? Is there a limit causing slow SS noise to overcome amplification. Maybe amplification is always worse than long exposure noise.
Let's go back to the concept of exposure. If you collect all the light passing through the lens to the sensor and spread that evenly across the entire sensor, that's exposure. It's determined by the available light in the scene, the lens aperture and the length of time the sensor is exposured to light.
The more light delivered to a sensor, the greater the exposure - think about it as spreading one or two dollops of peanut butter across a slice of bread. Two dollops is more peanut butter, which means it spreads thicker across the bread. The thicker you can spread light across a sensor, the greater the exposure.
Now, if we're out at night or in a naturally dark environment, there isn't much light in the scene to work with. Even if we use the fastest f-stop available or a really slow shutter speed, there's so little light available in the scene that not much gets to the sensor.
Here's where noise comes into play. The more total light used to make a photo, the less visible shot noise will be. Suppose we spread peanut butter across a slice of bread and a cracker so the peanut butter coats both to the same thickness. If the peanut butter were light, and the bread and cracker were camera sensors, we could say both cameras are working with the same exposure (the peanut butter spread to the same thickness) but that the camera built around the larger sensor is working with more total light. If we scrape all the peanut butter off the bread slice and the cracker, and compare how much PB was spread across both, the slice of bread will have more.
Can you tell I've got a thing for peanut butter?
Anyway, exposure is about the light delivered to the sensor. Noise is about the total light used to make the photo.
ISO has no direct affect either on exposure or the total light used to make the photo. ISO's used to manage the lightness of a photo. If you're making a night sky photo, you might shoot with the lens wide open at f/2.8 and a slow shutter speed of 4-seconds. There may not be much available light in the scene but you're squeezing as much of it as possible onto the sensor.
Even at f/2.8, 4-seconds the test photo may look dark at ISO 200. So, you increase ISO to 800, 1600 or higher to get a lighter photo. But you also see a noisy mess. It wasn't the increase in ISO that introduced or created all the noise. The noise was there at ISO 200. In fact, since you used the same exposure settings (f-stop and shutter speed), the amount of light used to make both photos is the same. Since the amount of light is the same, the amount of noise is the same. The noise in the ISO 800 or 1600 photo is more prominent to the eye because it's a lighter image.
Getting back to your original question about noise visibility in photos made with smaller sensor cameras, it's entirely possible to get clean, noise-free photos with crop sensor cameras given a bright enough scene. The challenge comes when doing photograhy in inherently dark settings, or when doing photograhy that requires very fast shutter speeds or very small lens apertures.
In a dark setting, there just isn't much light to work with. Similarly, a very fast shutter speed (very short exposure time) or a small lens aperture will work to limit how much light gets to the sensor even in bright conditions. These aren't necessarily situations in which it's impossible to make a clean image. But you will need to be conscious of using the other exposure setting (f-stop or shutter speed) to compensate.
Good luck.
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Bill Ferris Photography
Flagstaff, AZ
http://www.billferris.photoshelter.com