Inverse Hyperbolic Sine

This article describes how inverse hyperbolic functions are used as activators in the digital replication of ganglion and bipolar retinal cells.

Inverse hyperbolic sine is often used in quantization and audio signal processing. It works very well to compress the high-frequency imaging signal or highlight a bend in cinematography.

Humans see the relative change in brightness, while camera image sensors are developed with a linear response to the strength of a light source. To compress and map the linear image signal from the image sensor to the perceptual domain in imaging, often, gamma functions defined by logarithms are used.

However, when compressing a high-frequency signal that is zero-centered, the logarithms are not suitable due to their behavior near zero. We need a function whose derivative would behave like y=x near zero, behave similarly to log, and satisfy y(-x)=-y(x), and inverse hyperbolic sine is very good for it.

It is always eye-opening to see the behavior of this function of a complex argument

It is good to see the derivation process to remember the function behavior.