In order to do that I first have to introduce the concept of psychophysics.

So what is psychophysics?

Psychophysics is a branch of physics and

psychology that basically studies the relationship

between the physical intensity of a signal and

the perceived sensation from a human being, okay?

Let me give you an example.

Let's say that there is an experiment where the scientist

is turning a knob to change the physical intensity of a given signal.

Let's say, for instance an audio signal that's somebody's listening to and

we have the volume of the signal.

Or say, a light, or the intensity of the light is manipulated,

is controlled by this knob, okay?

On the other side of the experiment, there is a person who's

actually observing this light or listening to the volume of this sound.

And every time the knob is turned, the experimenter

asks the subject how much bigger is the volume, or

the light compared to the previous value, okay?

So now, by doing this kind of experiment, what the experimenter

can do is to find a relationship between how much difference

there is between the physical intensity of the signal that

can be measured by some physical measurements technology,

and the perceived intensity from the human subject.

So this is exactly what Fechner did by

inventing the whole idea of psychophysics.

As I said before, the idea of studying what is the relationship between

physical intensity and perceived intensity of a signal.

So Fechner invented the whole idea of psychophysics

that was in the late 19th century.

And in particular, he ran a number of experiments and

the most important result that he found, or the foundations

of his theory is that the relationship between the physical intensity and

the perceived intensity of a signal always follows a power law.

So what is a power law?

It's basically a mathematical equation or

a model that describes the relationship between these two variables,

physical intensity and perceived intensity, as a power equation, okay?

So in the equation that you see in this image,

we have the physical intensity represented by the I.

And the perceived intensity represented by the S symbol.

And as you can see the mathematical relationship is I to

a given number represents the perceived sensation.

And that's a power law, okay?

You can also see a graph here.

And in this graph, you see examples of curves that have been

experimentally generated through experiments in psychophysics.

And here are some of the signals that have been studied.

So we have for instance, the brightness of a light,

the depth of something, the area of something,

the length of something, the color saturation of something, and so on.

And as you can see these curves, while belonging to the same

categories of mathematical objects, they are all power laws, right?

They are obtained from equations that have different exponents, okay?

In particular what you can see is that with some of these curves,

if for some of these curves we have under-estimation of the signal,

whereas with some other curves with over-estimation of the signal.

What do I mean by under and over-estimation?

It means that when the physically intensity is changed,

the way it is perceived maybe when it's under-estimated

the sensation that is perceived is actually smaller

than the change that has been made on the physical intensity.

And the opposite happens when we have over estimation.

So why is this important for visualization?

Well, because we are studying visual channels,

each visual channel can be expressed In terms of this power loss.

And it's important for us to know whether a given channel

is visually underestimated or overestimated.

In particular, in this graph, you can see for instance that brightness,

if you use brightness of something to represent a quantity,

you can expect under estimation.

Whereas if you use length of something, you can expect over estimation.

And of course this is very important in the use of visual channels for

visual encoding.