Psychophysics: Thresholds, Magnitude Estimation, and Perceptual Warping
Psychophysics: Foundations and Core Concepts
Psychophysics = the study of the relationship between physical energy in the environment and our ability to sense or detect basic attributes of that energy.
It focuses on detection and discrimination of basic stimuli, not full object perception.
Key distinction: related to perception but does not explain how we perceive whole objects or complex scenes.
Core idea: perception is grounded in how physical energy is transformed into psychological impressions through sensory systems.
Sensory Adaptation and the Stability of Perception
Sensory adaptation: sensory receptors respond to the presence or absence of physical energy and tend to adapt (fatigue) with constant, unchanging stimulation.
If stimulation is constant, receptors initially fire but eventually fatigue and stop firing, reducing detectability of the stimulus.
Detection relies on multiple neurons responding to changing stimulation, not a single receptor.
Stabilized image as a test of adaptation in vision:
A contact lens with a tiny projector creates a stabilized image on the retina (image does not move with eye movements).
Eye movements normally shift the image across different receptors, helping detect changes.
A stabilized image causes the same set of receptor cells to be continuously stimulated, leading to fading of the image due to adaptation and increased thresholds.
Receptor thresholds adjust with lack of stimulation changes, leading to reduced firing over time if the stimulus is unchanging.
Practical implication: perception relies on changes in stimulation across neurons; our visual system is tuned to detect change rather than absolute, unvarying energy.
Absolute Thresholds vs Arbitrary Thresholds
Absolute threshold (as traditionally taught): a hypothetical fixed energy level at which a stimulus is detected with 50% probability, often depicted as a step-function (0% detection below threshold, 100% above).
Real data show this is not a perfect step; detection likelihood is a curve that increases with stimulus strength, not a perfect 0-to-1 jump.
Textbook note: the term "absolute threshold" an arbitrary threshold derived from data, typically defined at the energy level detected 50% of the time.
Demonstration with sugar in water (moats as units):
X-axis: amount of sugar (e.g., 0–12 moats).
Y-axis: probability of reporting taste of sugar (0 to 1).
An idealized absolute threshold would be a step function: no detection up to a cutoff (e.g., 6 moats), then 100% detection above it.
Actual detection curves are curved (S-shaped or partial slope), indicating a probabilistic relationship rather than a perfect threshold.
Arbitrary threshold (the 50% point): the energy level at which the detection probability is 0.5; used to quantify threshold when the true pattern is not a step function.
Method to estimate the threshold: draw a vertical line from the 50% detection probability to the observed curve and read off the corresponding energy on the x-axis.
Takeaway: perception does not linearly map physical energy to detection; there is perceptual warping, and thresholds must be estimated rather than assumed absolute.
Above-threshold discrimination (just noticeable differences) requires considering how energy changes are perceived when a stimulus is already detectable.
Just Noticeable Difference (JND) and Weber's Law
JND: the smallest amount of change in a stimulus that a person can detect 50% of the time.
Demonstration with weights:
Reference weight 100 g; add 5 g to detect a difference 50% of the time (JND = 5 g in this example).
With a 200 g reference weight, a larger change (e.g., 10 g) is needed to detect a difference 50% of the time.
JND depends on the baseline magnitude—differences are not treated as a constant absolute amount across the scale.
Weber's Law (Weber fraction):
Mathematical form: rac{ riangle I}{I} = k
I = baseline intensity; ΔI = change in intensity needed for detection; k = Weber fraction (constant for a given sensory dimension).
Interpretation: smaller k => greater sensitivity (smaller change needed for detection).
Example: for weight, if ΔI = 5 g at I = 100 g, then k = rac{5}{100} = 0.05. If I = 200 g, ΔI = 10 g gives the same k = 0.05.
Implication: JND scales with baseline magnitude; absolute changes are not equally detectable across the range of intensities.
Weber fraction is not universal across all dimensions; it varies by sense (weight, brightness, loudness, etc.).
Fechner's Law: Linking Physical Energy to Sensation Magnitude
Fechner sought to relate physical energy to subjective magnitude via a mathematical function.
Core idea: larger baseline intensities require disproportionately larger changes to be noticed; the relationship is nonlinear and grows in a way that approaches a logarithmic form.
Fechner's law (in one formulation):
ext{Sensation} = ext{constant} imes
abla igl( ext{log}(I)igr)
In the transcript, the formulation is presented as:
ext{psi} = k \, ext{log}(I) where psi is sensation units, I is stimulus intensity, and k is a constant.
Graphically, Fechner’s law yields a downward-curving (logarithmic) relationship: early increases in I produce large changes in sensation, but later increases yield smaller perceived changes.
Fechner showed that Weber’s law holds in some ranges, but not across the entire spectrum of intensities; the logarithmic relation captures the diminishing returns of perception as energy increases.
Stevens' Power Law and Magnitude Estimation
Stevens criticized Fechner’s limitation and proposed a more general power-law framework to map physical energy to perceived magnitude:
ext{S} = k \, I^{b}
S = perceived magnitude; I = stimulus intensity; k = scale factor; b = exponent that depends on the sensory dimension.
Method of magnitude estimation (Stevens’ contribution): get direct numerical reports of perceived magnitude rather than relying on threshold-based thresholds.
Use a modulus (standard) item; participants compare other stimuli to the modulus and assign numbers proportional to perceived magnitude.
Example modulus line assigned 100 units; if a stimulus appears to be half as long, assign 50 units; double—200 units; four times—400 units.
Different sensory dimensions yield different exponents b:
Length of a line: nearly linear with b ≈ 1 (S ≈ k I).
Brightness/brightness changes: sublinear with b ≈ 0.33; this yields a curve similar to a logarithm (since I^0.33 grows slowly at higher I).
Electrical shock: superlinear with b ≈ 3.5; large increases in physical energy lead to large increases in perceived magnitude.
Interpretation: perception is dimension-specific; Stevens’ power law generalizes Fechner but allows for both sublinear and superlinear relationships depending on the modality.
Relationship summary:
Length: linear (b ≈ 1) -> S ∝ I
Brightness: sublinear (b < 1) -> S grows slower than I
Electric shock: superlinear (b > 1) -> S grows faster than I
Practical takeaway: perceptual scales are not universal; different senses transform physical energy into experience in distinct nonlinear ways.
Dimensional Examples and the Warping of Physical Energy
Warping refers to departure from a straight-line mapping between physical energy and perceived magnitude.
Even within the same framework, different senses warp energy differently:
Length estimation (linear): S ∝ I with b ≈ 1.
Brightness (sublinear): S grows quickly at low intensities but levels off at higher intensities (Fechner/Stevens with b ≈ 0.33).
Electric shock (superlinear): Small increases in current at higher levels can lead to rapid increases in perceived intensity (b ≈ 3.5).
The existence of warping demonstrates that perception is not a faithful, exact measure of physical energy; it reflects functional adaptations that may aid discrimination and safety.
Summary insight: perception is not a perfect detector of physical energy; it is a functional system that emphasizes certain distinctions and may facilitate important judgments (e.g., detecting dangerous stimuli sooner).
Practical and Theoretical Implications
Psychophysics provides a toolkit for quantifying how physical energy maps to perception, including thresholds, detection probabilities, and discrimination abilities.
Threshold concepts (arbitrary/threshold estimation) guide experimental design and interpretation of sensory capabilities.
The idea of threshold and JND informs product design (e.g., display brightness, audio levels, tactile feedback) and safety standards.
Dimensional differences in exponent values (b) across modalities explain why some senses are highly sensitive to small changes while others are not.
Perceptual warping has real-world relevance: nonlinearity in perception can be leveraged in design (e.g., compressed dynamic range, perceptual coding) but also poses challenges for accurate measurement of stimulus properties.
Ethical and practical considerations: while the lecture focuses on measurement theory, real-world applications include sensory testing, human factors, and research with animals; ethical guidelines govern such work, though not discussed explicitly in this transcript.
Key Formulas to Remember (LaTeX)
Weber's Law (Just Noticeable Difference relative to baseline):
rac{ riangle I}{I} = k
Arbitrary Threshold (50% detection point):
Threshold is the energy level where detection probability = 0.5, derived from the empirical psychometric function.
For a given curve, locate the 50% point and read off the corresponding energy on the x-axis.
Fechner's Law (Sensation as a function of intensity):
ext{psi} = k \, ext{log}(I)
Stevens' Power Law (Magnitude estimation):
S = k \, I^{b}
Magnitude estimation procedure (example of modulus-based scaling):
If modulus is assigned 100 units, a stimulus appearing as x times the modulus gets a magnitude of $100 imes x^{b}$ (dimension-specific, via the estimated exponent b).
Connections to Prior Principles and Real-World Relevance
Sensory adaptation and the stabilized-image demonstration illustrate the foundational need for change detection across receptors for reliable perception.
Threshold concepts connect to calibration needs in devices and interfaces: detecting presence versus detecting magnitude or change.
The progression from Weber to Fechner to Stevens shows how psychophysics evolved from simple proportional detection to more nuanced, dimension-specific models of sensation.
The idea of perceptual warping clarifies why measurements based on physical energy alone can misrepresent perceptual experience; designers and scientists must account for nonlinear mappings.
Across domains (weight, brightness, line length, electrical shock), the same physical energy can yield very different perceptual scales, highlighting the importance of dimension-specific psychophysical functions when modeling perception or designing stimuli.
Summary Takeaways
Detection depends on adaptation and the collective response of multiple neurons reacting to changing stimulation.
Absolute thresholds as perfect cutoffs do not exist in practice; the meaningful metric is an arbitrary threshold defined by the 50% detection point on a psychometric function.
JNDs increase with baseline magnitude (Weber's law) but the constant k varies by sensory dimension.
Fechner linked physical energy to sensation with a logarithmic relationship, capturing diminishing perceptual returns at higher intensities.
Stevens proposed a general power-law framework that accommodates linear, sublinear, and superlinear relationships across modalities via the exponent b.
Perception is a functional, warped interpretation of physical energy, not a perfect one-to-one mapping, and this warping can be both limiting and advantageous in different contexts.