Perception and Action Modelling

• The lecture discusses modelling perception and action, covering a brief historical overview, mathematical basics, probability theory, probabilistic and dynamic models, and modelling of perceptual performance and action.

History of Perception and Action Modelling

• Gustav Theodor Fechner (1801-1887) introduced the term "psychophysics" to describe the study of relationships between the body and soul, and more broadly, physical and mental phenomena.
• Ernst Heinrich Weber (1834) originally formulated Weber's Law, which suggests that the difference in sensation is constant if the relative difference in stimulus remains the same.
• The Weber-Fechner Law describes a functional formula between stimulus and sensation, such that y = k(log β – log b),
• In the Weber-Fechner Law, k is dependent on units and the logarithmic system, while b denotes the threshold value of stimulus β at which sensation y disappears.
• Scalar variability: the standard deviation of responses increases proportionally with stimulus size. (Gibbon; 1977)
• Stevens' Power Law (1957) states that perceived intensities follow a power law, acting almost as contradiction to the Weber-Fechner law.
• Herrmann Ludwig Ferdinand von Helmholtz (1821-1894) stated that psychical activities of ordinary perception can be described as unconscious inferences, deriving ideas of causes the effects on our senses.
• Sir Isaac Newton (Opticks, 1730) believed that colors in objects are solely dispositions to reflect certain rays more copiously.

Colour Perception

• Vermilion, or any other colour, is not inherently a specific colour, but is distinguished differently depending on the observer's eyes.

Inverse Problem

• Retinal sensation interpretation involves consideration of illumination, reflectance, and transmittance (Purves et al., PNAS, 2011).
• Rods facilitate achromatic night vision, while Cones enable trichromatic color vision. (Purves et al 2001)
• Metamers: Humans cannot distinguish between monochromatic yellow and additive color mixing yellow.

Colour Constancy, Contrast, and Context

• Reflected spectral composition varies based on illumination
• Perceived color is context-dependent
• Spatial color contrast can change color appearance, relative to colour constancy

Perception vs Sensation

• The checkerboard illusion demonstrates that perception is not solely sensation.

Estimation Process

• The estimation process involves interactions between the illumination, retina opponent channel response, and internal models, to determine properties of a viewed object.

Priori Knowledge

• Priori knowledge is existing knowledge or assumptions applied in models to better understand sensory information
• Prior knowledge for bananas will infer them as slightly bluish in order to produce a grey colour

Internal Models for Perception

• Internal models aid in inferring properties of objects within the world using sensory stimuli (Rao, 1999; from O'Reilly, 1996).

Numerical Simulation

• Script-based models use Matlab, various solvers are available, much more flexible
• Simulink based models use simulin graphical interface, easier development, inbuilt visualisation
• du/dt = v(t)-u(t) / T

Bayesian Probability

• Bayesian probability interprets probability as reasonable expectation or personal belief quantification.
• A Bayesian probabilist updates an hypothesis from new evidence, compared to prior.

Probability

• P(A): Probability of event A
• P(A∩B) : Joint probability of events A and B
• P(AIB) : Conditional probability of A given B

Probability Density Distributions

• Function: : P(X,Y) = P(Y|X) * P(X) = P(X|Y) * P(Y)
• P(X) is the probability that measures density of x

Bayes' Theorem

• Bayes' theorem (Bayes, 1763) enhances our knowledge with new evidence.
• P(DIH) is the likelihood of data given the hypothesis.
• P(H) is the prior probability of the hypothesis.
• P(HID) is the posterior probability, given data.
• Formula: P(H | D) = P(DH)·P(H) / P(D)

Bayesian Statistics

• Bayesian Parameter (q) = (1 ∀ q ∈ [0,1] / 0 otherwise

Bayesian Updating

• The updated posterior is the result of the likelihood and the prior

von-Mises distribution

The distributions used for the example on Locating a Signal are von-Mises distributions (normal distribution on a circle).

Empirical Research

• Vierordt's Law (1868): short durations are overestimated, while long durations are underestimated.
• Central Tendency of Judgement (Hollingworth, 1909/1910): short lengths are overestimated and long lengths are underestimated.
• Range Effect (Poulton, 1968): Results depend on the range of stimuli