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Sums of Power01:22

Sums of Power

In definite integration, Riemann sums approximate the area under a curve by dividing it into subintervals and summing the areas of rectangles. When these approximations follow predictable numerical patterns, such as arithmetic or polynomial sequences, sum formulas offer a more efficient and accurate way to compute the result. In particular, the sum of consecutive integers, squares, and cubes plays an essential role in simplifying these calculations, especially when dealing with uniform...
Summation Notation01:25

Summation Notation

Sigma notation, also known as summation notation, provides a concise method for representing the sum of a sequence of terms that follow a regular pattern. It utilizes the uppercase Greek letter sigma (∑), A typical expression is:In this form, k the index of summation is 1, the starting value, and n the ending value. The term ak​ represents the general term of the sequence.For example, the increasing sequence 5, 7, 9, ..., 23 over 10 terms can be expressed as:This simplifies the representation...
Improper Integrals: Infinite Intervals01:29

Improper Integrals: Infinite Intervals

An integral is classified as improper due to an infinite interval when at least one of its limits of integration extends to positive or negative infinity. In such cases, the region under the curve is unbounded, and standard techniques for evaluating definite integrals are not directly applicable. Instead, the improper integral is defined through a limiting process that allows one to determine whether the accumulated area remains finite despite the infinite domain.Application to Exponential...
Geometric Sequences01:30

Geometric Sequences

In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred...
Sampling Theorem01:15

Sampling Theorem

In signal processing, the analysis of continuous-time signals, denoted as x(t), often involves sampling techniques to convert these signals into discrete-time signals. This process is essential for digital representation and manipulation. A critical component in sampling is the train of impulses, characterized by the sampling interval and the sampling frequency. The relationship between these parameters and the original signal's properties dictates the success of the sampling process.
Probability in Statistics01:14

Probability in Statistics

Probability is the likelihood of an event occurring. The term event is defined as a collection of results of a procedure. An event is a simple event when an outcome cannot be divided into simpler parts.
An example of a simple event is a coin toss. The result of a coin toss is either a head or a tail. Here, head and tail are two simple events. These two simple events make up the sample space. Further, the probability of an event occurring falls within the range of 0 to 1. The probability of an...

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Related Experiment Video

Updated: May 13, 2026

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments
08:12

A Psychophysics Paradigm for the Collection and Analysis of Similarity Judgments

Published on: March 1, 2022

Probability summation--a critique.

Donald Laming1

  • 1University of Cambridge, Department of Experimental Psychology, Cambridge, UK. drjl@hermes.cam.ac.uk

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|March 5, 2013
PubMed
Summary

Probability summation, a visual science theory, struggles with sub-threshold stimuli. Signal-detection theory with a nonlinear transform offers a more accurate explanation for contrast detection and summation phenomena.

Area of Science:

  • Visual science
  • Perception science
  • Neuroscience

Background:

  • Probability summation, particularly the high-threshold assumption, has been used to explain visual detection.
  • This model can account for contrast detectability, grating component summation, and temporal summation.
  • However, it fails to adequately explain phenomena involving stimuli below threshold.

Purpose of the Study:

  • To challenge the utility of probability summation as an explanatory model in visual science.
  • To propose signal-detection theory with a nonlinear transform as a superior alternative.
  • To demonstrate the derivation of this transform from basic sensory neuron properties.

Main Methods:

  • Critically evaluating the probability summation model, focusing on its high-threshold assumption.

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  • Applying signal-detection theory with an accelerated nonlinear transform to explain visual phenomena.
  • Illustrating the transform's behavior, which is power-law for low contrasts and linear for high contrasts.
  • Deriving the proposed transform from fundamental properties of sensory neurons.

    • Main Results:

    • Probability summation, despite explaining some phenomena, falters with sub-threshold stimuli.
    • Signal-detection theory combined with a nonlinear transform accurately models contrast detection, summation effects, and sub-threshold responses.
    • The proposed transform, a fourth power at low contrasts, aligns with neuronal properties.

    Conclusions:

    • The high-threshold assumption in probability summation is an outdated concept hindering progress in visual science.
    • Signal-detection theory with an accelerated nonlinear transform provides a more comprehensive and accurate framework for understanding visual perception.
    • This approach offers a biologically plausible explanation for near-threshold visual processing.