Weber–Fechner law
The Weber–Fechner laws are two related scientific laws in the field of psychophysics, known as Weber's law and Fechner's law. Both relate to human perception, more specifically the relation between the actual change in a physical stimulus and the perceived change. This includes stimuli to all senses: vision, hearing, taste, touch, and smell.
Ernst Heinrich Weber states that "the minimum increase of stimulus which will produce a perceptible increase of sensation is proportional to the pre-existent stimulus," while Gustav Fechner's law is an inference from Weber's law which states that the intensity of our sensation increases as the logarithm of an increase in energy rather than as rapidly as the increase.
History and formulation of the laws
Both Weber's law and Fechner's law were formulated by Gustav Theodor Fechner. They were first published in 1860 in the work Elemente der Psychophysik. This publication was the first work ever in this field, and where Fechner coined the term psychophysics to describe the interdisciplinary study of how humans perceive physical magnitudes. He made the claim that "...psycho-physics is an exact doctrine of the relation of function or dependence between body and soul."Weber's law
was one of the first persons to approach the study of the human response to a physical stimulus in a quantitative fashion. Fechner was a student of Weber and named his first law in honor of his mentor, since it was Weber who had conducted the experiments needed to formulate the law.In sensation and contrast, change is defined as "contrast over time". The sense detect the change, specifically in Weber's law, it detects the relative change - not absolute change. In Weber's law, to notice a change in stimulus, the change must be constant proportion of the original stimulus. Weber's law states that just noticeable difference is proportional to the magnitude of the initial stimulus. The brain is a percentage of change detector. The differential threshold is the smallest difference needed to differentiate two stimuli for each sense has been studied by using similarly methods to signal detection. For instance, holding an object that weighs 1,2,10, and 11lbs. If one were to hold the lightest object, and then swap out that object for the 2lb object, one could easily tell that the second object is heavier than the first. There is a 100 % weight increase from the first object to the second. However, it becomes more difficult when the difference is a smaller percentage of the initial object's weight. For example, if one were to hold the 10lb object, and then swap it out and hold the 11lb object, it would be difficult for one to tell which of the two is heavier, because there is only a 10 % increase from the first to the second object. This demonstrates that bigger stimuli require a larger difference in order to be noticed. This is Weber's Law.
Fechner formulated several versions of the law, all communicating the same idea. One formulation states:
What this means is that the perceived change in stimuli is inversely proportional to the initial stimuli.
Weber's law also incorporates the just-noticeable difference. Let be some reference stimulus, and the smallest change in this stimulus that can be perceived. This means that for any the stimulus is indistinguishable from. Weber's law states that and are proportional to one another,,
where is some constant of proportionality.
Weber's law always fails at low intensities, near and below the absolute detection threshold, and often also at high intensities, but may be approximately true across a wide middle range of intensities.
Weber contrast
Although Weber's law includes a statement of the proportionality of a perceived change to initial stimuli, Weber only refers to this as a rule of thumb regarding human perception. It was Fechner who formulated this statement as a mathematical expression referred to as Weber contrast.where is how much the perception changes when the stimulus,, changes by an amount.
is another proportionality constant. Plugging in the JND,, we see the proportionality constant in Weber's law is related to the new constant and the smallest perceptual change,. If then Weber's law states that. In Weber contrast this is not the case, so, though the mathematical relationships look similar, they differ in content.
Weber contrast, when integrated, explains Fechner's law.
Starting at some base stimulus,, and changing it to, the total change in perception is
Fechner's law
Fechner noticed in his own studies that different individuals have different sensitivity to certain stimuli. For example, the ability to perceive differences in light intensity could be related to how good that individual's vision is. He also noted that how the human sensitivity to stimuli changes depends on which sense is affected. He used this to formulate another version of Weber's law that he named die Maßformel, the "measurement formula". Fechner's law states that the subjective sensation is proportional to the logarithm of the stimulus intensity. According to this law, human perceptions of sight and sound work as follows: Perceived loudness/brightness is proportional to logarithm of the actual intensity measured with an accurate nonhuman instrument,The relationship between stimulus and perception is logarithmic. A logarithmic relationship means that if a stimulus varies as a geometric progression, the corresponding perception is altered in an arithmetic progression. For example, if a stimulus is tripled in strength, the corresponding perception may be two times as strong as its original value. If the stimulus is again tripled in strength, the corresponding perception will be three times as strong as its original value. Hence, for multiplications in stimulus strength, the strength of perception only adds. The mathematical derivations of the torques on a simple beam balance produce a description that is strictly compatible with Weber's law.
Since Weber's law fails at low intensity, so does Fechner's law.
An early reference to "Fechner's... law" was in 1875 by Ludimar Hermann in Elements of Human Physiology.
Types of perception
Weber and Fechner conducted research on differences in light intensity and the perceived difference in weight. Other sense modalities provide only mixed support for either Weber's law or Fechner's law.Weight perception
Weber found that the just noticeable difference between two weights was approximately proportional to the weights. Thus, if the weight of 105 g can be distinguished from that of 100 g, the JND is 5 g. If the mass is doubled, the differential threshold also doubles to 10 g, so that 210 g can be distinguished from 200 g. In this example, a weight seems to have to increase by 5% for someone to be able to reliably detect the increase, and this minimum required fractional increase is referred to as the "Weber fraction" for detecting changes in weight. Other discrimination tasks, such as detecting changes in brightness, or in tone height, or in the length of a line shown on a screen, may have different Weber fractions, but they all obey Weber's law in that observed values need to change by at least some small but constant proportion of the current value to ensure human observers will reliably be able to detect that change.Fechner did not conduct any experiments on how perceived heaviness increased with the mass of the stimulus. Instead, he assumed that all JNDs are subjectively equal, and argued mathematically that this would produce a logarithmic relation between the stimulus intensity and the sensation. These assumptions have both been questioned.
Following the work of S. S. Stevens, many researchers came to believe in the 1960s that the Stevens's power law was a more general psychophysical principle than Fechner's logarithmic law.
Sound
Weber's law does not quite hold for loudness. It is a fair approximation for higher intensities, but not for lower amplitudes.Limitation of Weber's law in the auditory system
Weber's law does not hold at perception of higher intensities. Intensity discrimination improves at higher intensities. The first demonstration of the phenomena was presented by Riesz in 1928, in Physical Review. This deviation of the Weber's law is known as the "near miss" of the Weber's law. This term was coined by McGill and Goldberg in their paper of 1968 in Perception & Psychophysics. Their study consisted of intensity discrimination in pure tones. Further studies have shown that the near miss is observed in noise stimuli as well. Jesteadt et al. demonstrated that the near miss holds across all the frequencies, and that the intensity discrimination is not a function of frequency, and that the change in discrimination with level can be represented by a single function across all frequencies:.Vision
The eye senses brightness approximately logarithmically over a moderate range and stellar magnitude is measured on a logarithmic scale.This magnitude scale was invented by the ancient Greek astronomer Hipparchus in about 150 B.C. He ranked the stars he could see in terms of their brightness, with 1 representing the brightest down to 6 representing the faintest, though now the scale has been extended beyond these limits; an increase in 5 magnitudes corresponds to a decrease in brightness by a factor of 100.
Modern researchers have attempted to incorporate such perceptual effects into mathematical models of vision.