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I have been trying to understand how a better spectrum of color can be represented in an image and to avoid banding it sounds like the image would need to have more bits per pixel. When the resolution of an image increases, do the bits per pixel also increase?

Image resolution (amount of pixels) and bit depth (bits per pixel) can be changed independently. Any combination of high/low resolution and more/less bits per pixel is possible.

Sometimes, however, higher resolution can make up for lower bit depth. Individual pixels become invisible and we start perceiving patterns of pixels as shades, as if the color depth were increased.

This is also the case of film photography, where individual "pixels" can only be black or white, and yet we can clearly see the shades of gray, because the individual image elements are so tiny. (ignoring color film photography for simplicity)

Not in common digital imaging terminology. Instead, we call bits-per-pixel bit depth.

Once unpacked from whatever compression format they were stored in, digital images are usually represented as triplets of red, green, and blue ­— each of these is a "channel". (There may also be a channel representing transparency, although that's more of an image editing or graphics design thing than of interest to photography.) If you have more of these triplets (like, 6000×4000 instead of 3000×2000), we say you have more resolution. If you have more bits per channel (say, 16 instead of 8) we say you have more bit depth.

It gets more complicated than this when you consider non-linear encoding, color spaces, and so on, but that's the basics. For your purpose, it's true that greater bit depth can help reduce banding. Be aware, though, that most devices can only show 8 bits per channel and many consumer devices are much worse. And common file formats online are only 8 bit. So, you probably want to reduce to 8 bits on your own system even if you work in higher bit depth, so you can control how the banding looks.

One approach is to introduce a dithering pattern. In fact, having higher resolution can help here — you basically trade some of the spacial detail for colors.

The resolution is a measure of the pixels per unit of physical length. So to increase the resolution of the image you mostly have to increase the pixel count.

The number of bits per pixels (a.k.a pixel depth) doesn't change the resolution. However, any global color change induces color loss (several different values are mapped to the same output value), and banding appears when this produces large uniform areas next to each other. With the usualy 8-bit/channel, color loss happens quickly, losing 10 to 20% of the color values is not uncommon. Increasing bit depth decreases the relative color loss, so banding is a lot less likely to happen.

This is one purpose of "shooting raw", the output of the sensor is 12 to 14 bits per channel (depending on cameral model), somewhat better than the 8-bits in the camera JPEG, so you have more material to work on (though in practice in your factor in sensor noise and demosaicing you are closer to 10-12 bits/channel).

But high-bit-depth is mostly for storage and processing, most images formats are 8-bit/Channel and many displays are barely good enough for 8-bit depths.