Digital images need to account for a simple fact: human eyes don't perceive light in a straightforward, linear way. We are much more sensitive to changes in dark tones than we are to changes in bright tones. If a camera stored brightness values exactly as it measured them, the resulting image would look unnatural to us, with the dark areas appearing crushed and lacking detail.

Gamma correction is the process of adjusting the brightness and contrast of an image to match our non-linear perception of light. It's a crucial step that ensures the colors and tones you see on your screen look the way they're supposed to.

At its core, gamma correction is a power-law function. It takes an input brightness value and raises it to the power of a specific exponent, called gamma (represented by the Greek letter $\gamma$).

In this formula:

• $V_{in}$ is the original luminance (brightness) value of a pixel, typically on a scale from 0 (black) to 1 (white).
• $V_{out}$ is the new, gamma-corrected luminance value.
• $\gamma$ is the gamma value. This number determines how the brightness is adjusted.

When $\gamma$ is greater than 1 (a common value is 2.2), the output values are lower than the input values. This makes the image appear darker overall. It compresses the brighter tones and expands the darker ones, adding more detail in the shadows.

When $\gamma$ is less than 1, the opposite happens. The output values are higher than the input values, making the image brighter. This expands the highlights and compresses the shadows.

The main goal of gamma correction is to make digital images look right to the human eye. Because our eyes are more attuned to subtle differences in shadows, it makes sense to dedicate more of an image's data to representing these darker tones.

This process is often a two-step dance. First, a camera performs gamma encoding. It applies a gamma value greater than 1 (like 2.2) to the raw image data. This redistributes the tonal levels, assigning more bits to the darker areas where our eyes are most sensitive. This creates a more perceptually uniform image that is ideal for storage and transmission.

Then, when you view the image, your monitor or display performs gamma decoding. It applies an inverse gamma value (like $1/2.2$, which is approximately 0.45) to brighten the image back up. This decoding step cancels out the initial encoding, ensuring the final image displayed on the screen has a linear and accurate relationship to the original scene's brightness, all while looking natural to our eyes.

Without this encoding and decoding process, a linear-encoded image would appear washed out, with bright areas that are too bright and dark areas that lack sufficient detail. Gamma correction is the invisible transformation that bridges the gap between how a camera captures light and how we actually see it.