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# Webber's law, Fechner's law and Stevens' law

### Weber's Law

Weber, in the early 19th century, discovered that for a person lifting weights in an experiment, the just distinguishable additional weight depended proportionally on the weight of the original weight. In other words, if the original weight is $$I$$ and the just distinguishable additional weight is $$\Delta I$$, then $$\frac{\Delta l}{I}$$ is a constant.

### Fechner's Law

Fechner wished to derive a transformation that would transform the physical intensity scale to a "perceptual scale" where units on the perceputal scale corresponded to perceptual changes of the same magnitude.

Fechner started with two assumptions:

• Weber's law was valid.
• A just noticeable difference can be considered a unit of perception.

### Stevens' law

Stevens showed that many relationships between just noticable differences and physical stimulus changes actually follow a power law. Steven's work finds its way into some fundamental aspects of color measurement.

### More details

Below, the details on the front side are expanded upon.

### Weber's law

Weber's law holds approximately for many perceptural stimuli. It can be used to explain why stars cannot be seen during the daytime. At night, the stars represnt a certain increment in intensity $$\delta I$$ that is relatively large in comparison to the background intensity. Thus, $$\frac{\delta I}{I}$$ is large enough to be perceptable. During the day, $$I$$ is much larger and $$\frac{\delta I}{I}$$ is smaller than $$\frac{\Delta I}{I}$$.

### Fechner's law

With Weber's law, on the physical scale, just noticeable differences are proportional to the current value. Having a scale that acts as the exponent will make just noticeable differences be constant in this scale. Thus, transforming the physical scale by taking the logarithm is what we need.

Fechner's law states that the perceived intensity of a stimulus is proportional to the logarithm of the physical stimulus intensity.

Fechner's law is why color is often manipulated in "perceptual linear" aka manipulating the exponent (the logorathim of the physical color value).

#### Source

Color Appearance Models, Fairchild