Sine and cosine transforms

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Sine_and_cosine_transforms

In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function. The modern Fourier transform concisely contains both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely correspond to Joseph Fourier's original transform equations and are still preferred in some signal processing and statistics applications and may be better suited as an introduction to Fourier analysis.

Definition

Fourier transforms relate a time-domain function (red) to a frequency-domain function (blue). Sine or cosine waves that make up the original function will appear as peaks in the frequency domain functions produced by the sine or cosine transform, respectively.

The Fourier sine transform of ${\displaystyle f(t)}$ is:

Fourier sine transform

${\displaystyle {\hat{f}}^s(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\sin (2\pi \xi t)dt.}$

If ${\displaystyle t}$ means time, then ${\displaystyle \xi }$ is frequency in cycles per unit time, but in the abstract, they can be any dual pair of variables (e.g. position and spatial frequency).

The sine transform is necessarily an odd function of frequency, i.e. for all ${\displaystyle \xi }$:

$${\displaystyle {\hat{f}}^s(-\xi )=-{\hat{f}}^s(\xi ).}$$

The cosine transform of a simple rectangular function (of height ${\displaystyle \frac{1}{a}}$ and width ${\displaystyle a}$) is the normalized sinc${\displaystyle (a\xi )}$ plotted above.

The Fourier cosine transform of ${\displaystyle f(t)}$ is:

Fourier cosine transform

${\displaystyle {\hat{f}}^c(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\cos (2\pi \xi t)dt.}$

The cosine transform is necessarily an even function of frequency, i.e. for all ${\displaystyle \xi }$:

$${\displaystyle {\hat{f}}^c(\xi )={\hat{f}}^c(-\xi ).}$$

Odd and even simplification

Like all even functions, the left half of a Gaussian function is a mirror image of its right half and its sine transform is entirely 0. Gaussians have the form ${\displaystyle e^{-\alpha t^2}}$ and their cosine transform:
$${\displaystyle {\hat{f}}^c(\xi )=\sqrt{\frac{\pi }{\alpha }}{e}^{-\frac{(\pi \xi {)}^2}{\alpha }}}$$
also is a Gaussian. The plotted Gaussian uses α=π and is its own cosine transform.

The multiplication rules for even and odd functions shown in the overbraces in the following equations dramatically simplify the integrands when transforming even and odd functions. Some authors even only define the cosine transform for even functions ${\displaystyle f_{\text{even}}(t)}$. Since cosine is an even function and because the integral of an even function from ${\displaystyle -\mathrm{\infty }}$ to ${\displaystyle \mathrm{\infty }}$ is twice its integral from ${\displaystyle 0}$ to ${\displaystyle \mathrm{\infty }}$, the cosine transform of any even function can be simplified to avoid negative ${\displaystyle t}$:

$${\displaystyle {\hat{f}}^c(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\stackrel{\text{even·even=even}}{\overbrace{f_{\text{even}}(t)\cdot \cos (2\pi \xi t)}}dt=2{\int }_0^{\mathrm{\infty }}{f}_{\text{even}}(t)\cos (2\pi \xi t)dt.}$$

And because the integral from ${\displaystyle -\mathrm{\infty }}$ to ${\displaystyle \mathrm{\infty }}$ of any odd function from is zero, the cosine transform of any odd function is simply zero:

$${\displaystyle {\hat{f}}^c(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\stackrel{\text{odd·even=odd}}{\overbrace{f_{\text{odd}}(t)\cdot \cos (2\pi \xi t)}}dt=0.}$$

Odd functions are unchanged if rotated 180 degrees about the origin. Their cosine transform is entirely zero. The above odd function contains two half-sized time-shifted Dirac delta functions. Its sine transform is simply ${\displaystyle \sin (a\xi ).}$ Likewise, the sine transform of ${\displaystyle \sin (a\xi )}$ is the above plot. Thus, the sine wave function and the time-shifted Dirac delta function form a transform pair.

Similarly, because sin is odd, the sine transform of any odd function ${\displaystyle f_{\text{odd}}(t)}$ also simplifies to avoid negative ${\displaystyle t}$:

$${\displaystyle {\hat{f}}^s(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\stackrel{\text{odd·odd=even}}{\overbrace{f_{\text{odd}}(t)\cdot \sin (2\pi \xi t)}}dt=2{\int }_0^{\mathrm{\infty }}{f}_{\text{odd}}(t)\sin (2\pi \xi t)dt}$$

and the sine transform of any even function is simply zero:

$${\displaystyle {\hat{f}}^s(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\stackrel{\text{even·odd=odd}}{\overbrace{f_{\text{even}}(t)\cdot \sin (2\pi \xi t)}}dt=0.}$$

The sine transform represents the odd part of a function, while the cosine transform represents the even part of a function.

Other conventions

Just like the Fourier transform takes the form of different equations with different constant factors (see Fourier transform § Unitarity and definition for square integrable functions for discussion), other authors also define the cosine transform as $${\displaystyle {\hat{f}}^c(\xi )=\sqrt{\frac{2}{\pi }}{\int }_0^{\mathrm{\infty }}f(t)\cos (2\pi \xi t)dt}$$ and the sine transform as $${\displaystyle {\hat{f}}^s(\xi )=\sqrt{\frac{2}{\pi }}{\int }_0^{\mathrm{\infty }}f(t)\sin (2\pi \xi t)dt.}$$Another convention defines the cosine transform as $${\displaystyle F_c(\alpha )=\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}f(x)\cos (\alpha x)dx}$$ and the sine transform as $${\displaystyle F_s(\alpha )=\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}f(x)\sin (\alpha x)dx}$$using ${\displaystyle \alpha }$ as the transformation variable. And while ${\displaystyle t}$ is typically used to represent the time domain, ${\displaystyle x}$ is often instead used to represent a spatial domain when transforming to spatial frequencies.

Fourier inversion

The original function ${\displaystyle f}$ can be recovered from its sine and cosine transforms under the usual hypotheses using the inversion formula:

Fourier inversion (from the sine and cosine transforms)

${\displaystyle f(t)=\underset{\text{odd component of }f(t)}{\underbrace{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{f}}^s(\xi )\sin (2\pi \xi t)d\xi }}+\underset{\text{even component of }f(t)}{\underbrace{{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{f}}^c(\xi )\cos (2\pi \xi t)d\xi }}.}$

Simplifications

Note that since both integrands are even functions of ${\displaystyle \xi }$, the concept of negative frequency can be avoided by doubling the result of integrating over non-negative frequencies:

$${\displaystyle f(t)=2{\int }_0^{\mathrm{\infty }}{\hat{f}}^s(\xi )\sin (2\pi \xi t)d\xi +2{\int }_0^{\mathrm{\infty }}{\hat{f}}^c(\xi )\cos (2\pi \xi t)d\xi .}$$

Also, if ${\displaystyle f}$ is an odd function, then the cosine transform is zero, so its inversion simplifies to:$${\displaystyle f(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{f}}^s(\xi )\sin (2\pi \xi t)d\xi ,\text{ only if }f(t)\text{ is odd.}}$$

Likewise, if the original function ${\displaystyle f}$ is an even function, then the sine transform is zero, so its inversion also simplifies to:

$${\displaystyle f(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\hat{f}}^c(\xi )\cos (2\pi \xi t)d\xi ,\text{ only if }f(t)\text{ is even.}}$$

Remarkably, these last two simplified inversion formulas look identical to the original sine and cosine transforms, respectively, though with ${\displaystyle t}$ swapped with ${\displaystyle \xi }$ (and with ${\displaystyle f}$ swapped with ${\displaystyle {\hat{f}}^s}$ or ${\displaystyle {\hat{f}}^c}$). A consequence of this symmetry is that their inversion and transform processes still work when the two functions are swapped. Two such functions are called transform pairs.

Overview of inversion proof

Using the addition formula for cosine, the full inversion formula can also be rewritten as Fourier's integral formula: $${\displaystyle f(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\cos (2\pi \xi (x-t))dxd\xi .}$$ This theorem is often stated under different hypotheses, that ${\displaystyle f}$ is integrable, and is of bounded variation on an open interval containing the point ${\displaystyle t}$, in which case $${\displaystyle \frac{1}{2}\underset{h\to 0}{lim}\left(f(t+h)+f(t-h)\right)=2{\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\cos (2\pi \xi (x-t))dxd\xi .}$$

This latter form is a useful intermediate step in proving the inverse formulae for the since and cosine transforms. One method of deriving it, due to Cauchy is to insert a ${\displaystyle e^{-\delta \xi }}$ into the integral, where ${\displaystyle \delta >0}$ is fixed. Then $${\displaystyle 2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{e}^{-\delta \xi }\cos (2\pi \xi (x-t))d\xi f(x)dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\frac{2\delta }{{\delta }^2+4{\pi }^2(x-t{)}^2}dx.}$$ Now when ${\displaystyle \delta \to 0}$, the integrand tends to zero except at ${\displaystyle x=t}$, so that formally the above is $${\displaystyle f(t){\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{2\delta }{{\delta }^2+4{\pi }^2(x-t{)}^2}dx=f(t).}$$

Relation with complex exponentials

The complex exponential form of the Fourier transform used more often today is $${\displaystyle \begin{array}{rl}\hat{f}(\xi )& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)e^{-2\pi i\xi t}dt\end{array}}$$where ${\displaystyle i}$ is the square root of negative one. By applying Euler's formula $(e^{ix}=\cos x+i\sin x),$ it can be shown (for real-valued functions) that the Fourier transform's real component is the cosine transform (representing the even component of the original function) and the Fourier transform's imaginary component is the negative of the sine transform (representing the odd component of the original function):$${\displaystyle \begin{array}{rlrl}\hat{f}(\xi )& ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\left(\cos (2\pi \xi t)-i\sin (2\pi \xi t)\right)dt& & \text{Euler's Formula}\\ & =\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\cos (2\pi \xi t)dt\right)-i\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\sin (2\pi \xi t)dt\right)\\ & ={\hat{f}}^c(\xi )-i{\hat{f}}^s(\xi ).\end{array}}$$Because of this relationship, the cosine transform of functions whose Fourier transform is known (e.g. in Fourier transform § Tables of important Fourier transforms) can be simply found by taking the real part of the Fourier transform:$${\displaystyle {\hat{f}}^c(\xi )=\mathrm{R}\mathrm{e}[\hat{f}(\xi )]}$$while the sine transform is simply the negative of the imaginary part of the Fourier transform:$${\displaystyle {\hat{f}}^s(\xi )=-\mathrm{I}\mathrm{m}[\hat{f}(\xi )].}$$

Pros and cons

Adding a sine wave (red) and a cosine wave (blue) of the same frequency results a phase-shifted sine wave (green) of that same frequency, but whose amplitude and phase depends on the amplitudes of the original sine and cosine wave. Hence, at a particular frequency, the sine transform and the cosine transform together essentially only represent one sine wave that could have any phase shift.

An advantage of the modern Fourier transform is that while the sine and cosine transforms together are required to extract the phase information of a frequency, the modern Fourier transform instead compactly packs both phase and amplitude information inside its complex valued result. But a disadvantage is its requirement on understanding complex numbers, complex exponentials, and negative frequency.

The sine and cosine transforms meanwhile have the advantage that all quantities are real. Since positive frequencies can fully express them, the non-trivial concept of negative frequency needed in the regular Fourier transform can be avoided. They may also be convenient when the original function is already even or odd or can be made even or odd, in which case only the cosine or the sine transform respectively is needed. For instance, even though an input may not be even or odd, a discrete cosine transform may start by assuming an even extension of its input while a discrete sine transform may start by assuming an odd extension of its input, to avoid having to compute the entire discrete Fourier transform.

Numerical evaluation

Using standard methods of numerical evaluation for Fourier integrals, such as Gaussian or tanh-sinh quadrature, is likely to lead to completely incorrect results, as the quadrature sum is (for most integrands of interest) highly ill-conditioned. Special numerical methods which exploit the structure of the oscillation are required, an example of which is Ooura's method for Fourier integrals. This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.

Salience (neuroscience)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Salience_(neuroscience)

Salience (also called saliency, from Latin saliō meaning “leap, spring”) is the property by which some thing stands out. Salient events are an attentional mechanism by which organisms learn and survive; those organisms can focus their limited perceptual and cognitive resources on the pertinent (that is, salient) subset of the sensory data available to them.

Saliency typically arises from contrasts between items and their neighborhood. They might be represented, for example, by a red dot surrounded by white dots, or by a flickering message indicator of an answering machine, or a loud noise in an otherwise quiet environment. Saliency detection is often studied in the context of the visual system, but similar mechanisms operate in other sensory systems. Just what is salient can be influenced by training: for example, for human subjects particular letters can become salient by training. There can be a sequence of necessary events, each of which has to be salient, in turn, in order for successful training in the sequence; the alternative is a failure, as in an illustrated sequence when tying a bowline; in the list of illustrations, even the first illustration is a salient: the rope in the list must cross over, and not under the bitter end of the rope (which can remain fixed, and not free to move); failure to notice that the first salient has not been satisfied means the knot will fail to hold, even when the remaining salient events have been satisfied.

When attention deployment is driven by salient stimuli, it is considered to be bottom-up, memory-free, and reactive. Conversely, attention can also be guided by top-down, memory-dependent, or anticipatory mechanisms, such as when looking ahead of moving objects or sideways before crossing streets. Humans and other animals have difficulty paying attention to more than one item simultaneously, so they are faced with the challenge of continuously integrating and prioritizing different bottom-up and top-down influences.

Neuroanatomy

The brain component named the hippocampus helps with the assessment of salience and context by using past memories to filter new incoming stimuli, and placing those that are most important into long term memory. The entorhinal cortex is the pathway into and out of the hippocampus, and is an important part of the brain's memory network; research shows that it is a brain region that suffers damage early on in Alzheimer's disease, one of the effects of which is altered (diminished) salience.

The pulvinar nuclei (in the thalamus) modulate physical/perceptual salience in attentional selection.

One group of neurons (i.e., D1-type medium spiny neurons) within the nucleus accumbens shell (NAcc shell) assigns appetitive motivational salience ("want" and "desire", which includes a motivational component), aka incentive salience, to rewarding stimuli, while another group of neurons (i.e., D2-type medium spiny neurons) within the NAcc shell assigns aversive motivational salience to aversive stimuli.

The primary visual cortex (V1) generates a bottom-up saliency map from visual inputs to guide reflexive attentional shifts or gaze shifts. According to V1 Saliency Hypothesis, the saliency of a location is higher when V1 neurons give higher responses to that location relative to V1 neurons' responses to other visual locations. For example, a unique red item among green items, or a unique vertical bar among horizontal bars, is salient since it evokes higher V1 responses and attracts attention or gaze. The V1 neural responses are sent to the superior colliculus to guide gaze shifts to the salient locations. A fingerprint of the saliency map in V1 is that attention or gaze can be captured by the location of an eye-of-origin singleton in visual inputs, e.g., a bar uniquely shown to the left eye in a background of many other bars shown to the right eye, even when observers cannot tell the difference between the singleton and the background bars.

In psychology

The term is widely used in the study of perception and cognition to refer to any aspect of a stimulus that, for any of many reasons, stands out from the rest. Salience may be the result of emotional, motivational or cognitive factors and is not necessarily associated with physical factors such as intensity, clarity or size. Although salience is thought to determine attentional selection, salience associated with physical factors does not necessarily influence selection of a stimulus.

Salience Bias Example: attention is drawn to the second image due to the more prominent color (red), as opposed to the less vivid color (light blue) of the first image, biased to the more salient stimulus.

Salience bias

Salience bias (also referred to as perceptual salience) is a cognitive bias that predisposes individuals to focus on or attend to items, information, or stimuli that are more prominent, visible, or emotionally striking. This is as opposed to stimuli that are unremarkable, or less salient, even though this difference is often irrelevant by objective standards. The American Psychological Association (APA) defines the salience hypothesis as a theory regarding perception where “motivationally significant” information is more readily perceived than information with little or less significant motivational importance. Perceptual salience (salience bias) is linked to the vividness effect, whereby a more pronounced response is produced by a more vivid perception of a stimulus than the mere knowledge of the stimulus. Salience bias assumes that more dynamic, conspicuous, or distinctive stimuli engage attention more than less prominent stimuli, disproportionately impacting decision making, it is a bias which favors more salient information.

Application

Cognitive Psychology

Salience bias, like all other cognitive biases, is an applicable concept to various disciplines. For example, cognitive psychology investigates cognitive functions and processes, such as perception, attention, memory, problem solving, and decision making, all of which could be influenced by salience bias. Salience bias acts to combat cognitive overload by focusing attention on prominent stimuli, which affects how individuals perceive the world as other, less vivid stimuli that could add to or change this perception, are ignored. Human attention gravitates towards novel and relevant stimuli and unconsciously filters out less prominent information, demonstrating salience bias, which influences behavior as human behavior is affected by what is attended to. Behavioral economists Tversky and Kahneman also suggest that the retrieval of instances is influenced by their salience, such as how witnessing or experiencing an event first-hand has a greater impact than when it is less salient, like if it were read about, implying that memory is affected by salience.

Language

It is also relevant in language understanding and acquisition. Focusing on more salient phenomena allows people to detect language patterns and dialect variations more easily, making dialect categorization more efficient.

Social Behavior

Furthermore, social behaviors and interactions can also be influenced by perceptual salience. Changes in the perceptual salience of an individual heavily influences their social behavior and subjective experience of their social interactions, confirming a “social salience effect”. Social salience relates to how individuals perceive and respond to other people.

Behavioral Science

The connection between salience bias and other heuristics, like availability and representativeness, links it to the fields of behavioral science and behavioral economics. Salience bias is closely related to the availability heuristic in behavioral economics, based on the influence of information vividness and visibility, such as recency or frequency, on judgements, for example:

Accessibility and salience are closely related to availability, and they are important as well. If you have personally experienced a serious earthquake, you’re more likely to believe that an earthquake is likely than if you read about it in a weekly magazine. Thus, vivid and easily imagined causes of death (for example, tornadoes) often receive inflated estimates of probability, and less-vivid causes (for example, asthma attacks) receive low estimates, even if they occur with a far greater frequency (here, by a factor of twenty). Timing counts too: more recent events have a greater impact on our behavior, and on our fears, than earlier ones.

— Richard H. Thaler, Nudge: Improving Decisions about Health, Wealth, and Happiness (2008-04-08)

Humans have bounded rationality, which refers to their limited ability to be rational in decision making, due to a limited capacity to process information and cognitive ability. Heuristics, such as availability, are employed to reduce the complexity of cognitive and social tasks or judgements, in order to decrease the cognitive load that result from bounded rationality. Despite the effectiveness of heuristics in doing so, they are limited by systematic errors that occur, often the result of influencing biases, such as salience. This can lead to misdirected or misinformed judgements, based on an overemphasis or overweighting of certain, more salient information. For example, the irrational behavior of procrastination occurs because costs in the present, like sacrificing free time, are disproportionately salient to future costs, because at that time they are more vivid. The more prominent information is more readily available than the less salient information, and thus has a larger impact on decision making and behavior, resulting in errors in judgement.

Other fields such as philosophy, economics, finance, and political science have also investigated the effects of salience, such as in relation to taxes, where salience bias is applied to real-world behaviors, affecting systems like the economy. The existence of salience bias in humans can make behavior more predictable and this bias can be leveraged to influence behavior, such as through nudges.

Evaluation

Salience bias is one of many explanations for why humans deviate from rational decision making: by being overly focused on or biased to the most visible data and ignoring other potentially important information that could result in a more reasonable judgment. As a concept it is supported in psychological and economic literature, through its relationship with the availability heuristic outlined by Tversky and Kahneman, and its applicability to behaviors relevant to multiple disciplines, such as economics.

Despite this support, salience bias is limited for various reasons, one example being its difficulty in quantifying, operationalizing, and universally defining. Salience is often confused with other terms in literature, for example, one article states that salience, which is defined as a cognitive bias referring to “visibility and prominence”, is often confused with terms like transparency and complexity in public finance literature. This limits salience bias as the confusion negates its importance as an individual term, and therefore the influence it has on tax related behavior. Likewise, the APA definition of salience refers to motivational importance, which is based on subjective judgement, adding to the difficulty. According to psychologist S. Taylor “some people are more salient than others” and these differences can further bias judgements.

Biased judgements have far-reaching consequences, beyond poor decision making, such as overgeneralizing and stereotyping. Studies into solo status or token integration demonstrate this. The token is an individual in a group different to the other members in that social environment, like a female in an all-male workplace. The token is viewed as symbolic of their social group, whereby judgments made about the solo individual predict judgements of their social group, which can result in inaccurate perceptions of that group and potential stereotyping. The distinctiveness of the individual in that environment “fosters a salience bias” and hence predisposes those generalized judgements, positive or negative.

In interaction design

Salience in design draws from the cognitive aspects of attention, and applies it to the making of 2D and 3D objects. When designing computer and screen interfaces, salience helps draw attention to certain objects like buttons and signify affordance, so designers can utilize this aspect of perception to guide users.

There are several variables used to direct attention:

• Color. Hue, saturation, and value can all be used to call attention to areas or objects within an interface, and de-emphasize others.
• Size. Object size and proportion to surrounding elements creates visual hierarchy, both in interactive elements like buttons, but also within informative elements like text.
• Position. An object's orientation or spatial arrangement in relation to the surrounding objects creates differentiation to invite action.

Accessibility

A consideration for salience in interaction design is accessibility. Many interfaces used today rely on visual salience for guiding user interaction, and people with disabilities like color-blindness may have trouble interacting with interfaces using color or contrast to create salience.

Aberrant salience hypothesis of schizophrenia

Further information: Dopamine hypothesis of schizophrenia

Kapur (2003) proposed that a hyperdopaminergic state, at a "brain" level of description, leads to an aberrant assignment of salience to the elements of one's experience, at a "mind" level. These aberrant salience attributions have been associated with altered activities in the mesolimbic system, including the striatum, the amygdala, the hippocampus, the parahippocampal gyrus., the anterior cingulate cortex and the insula. Dopamine mediates the conversion of the neural representation of an external stimulus from a neutral bit of information into an attractive or aversive entity, i.e. a salient event. Symptoms of schizophrenia may arise out of 'the aberrant assignment of salience to external objects and internal representations', and antipsychotic medications reduce positive symptoms by attenuating aberrant motivational salience via blockade of the dopamine D2 receptors (Kapur, 2003).

Alternative areas of investigation include supplementary motor areas, frontal eye fields and parietal eye fields. These areas of the brain are involved with calculating predictions and visual salience. Changing expectations on where to look restructures these areas of the brain. This cognitive repatterning can result in some of the symptoms found in such disorders.

Visual saliency modeling

In the domain of psychology, efforts have been made in modeling the mechanism of human attention, including the learning of prioritizing the different bottom-up and top-down influences.

In the domain of computer vision, efforts have been made in modeling the mechanism of human attention, especially the bottom-up attentional mechanism, including both spatial and temporal attention. Such a process is also called visual saliency detection.

Generally speaking, there are two kinds of models to mimic the bottom-up saliency mechanism. One way is based on the spatial contrast analysis: for example, a center-surround mechanism is used to define saliency across scales, which is inspired by the putative neural mechanism. The other way is based on the frequency domain analysis. While they used the amplitude spectrum to assign saliency to rarely occurring magnitudes, Guo et al. use the phase spectrum instead. Recently, Li et al. introduced a system that uses both the amplitude and the phase information.

A key limitation in many such approaches is their computational complexity leading to less than real-time performance, even on modern computer hardware. Some recent work attempts to overcome these issues at the expense of saliency detection quality under some conditions. Other work suggests that saliency and associated speed-accuracy phenomena may be a fundamental mechanisms determined during recognition through gradient descent, needing not be spatial in nature.

Change blindness

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Change_blindness 

Example of images that can be used in a change blindness task. Although similar, the two images have a number of differences.

Change blindness is a perceptual phenomenon that occurs when a change in a visual stimulus is introduced and the observer does not notice it. For example, observers often fail to notice major differences introduced into an image while it flickers off and on again. People's poor ability to detect changes has been argued to reflect fundamental limitations of human attention. Change blindness has become a highly researched topic and some have argued that it may have important practical implications in areas such as eyewitness testimony and distractions while driving.

History

Early anecdotal observations

Outside of the domain of psychology, phenomena related to change blindness have been discussed since the 19th century. When film editing was introduced in movies, editors began to notice that changes to the background were not noticed by those watching the film. Going back earlier, William James (1842–1910) was the first to mention the lack of ability to detect change in his book Principles of Psychology (1890).

Earliest experimental reports

Research on change blindness developed from investigation in other phenomena such as eye movements and working memory. Although individuals have a very good memory as to whether or not they have seen an image, they are generally poor at recalling the smaller details in that image. When we are visually stimulated with a complex picture, it is more likely that individuals retain only a gist of an image and not the image in its entirety.

The laboratory study of change blindness began in the 1970s within the context of eye movement research. George McConkie conducted the first studies on change blindness involving changes in words and texts; in these studies, the changes were introduced while the observer performed a saccadic eye movement. Observers often failed to notice these changes.

In the late 1980s, the first clear experimental demonstration was published showing very poor change detection in complex displays over brief intervals without eye movements being involved. Pashler (1988) showed that observers were poor at detecting changes introduced into arrays of letters while the display was flickered off and on, even if the offset was as brief as 67 milliseconds (although offsets briefer than that produced much more effective change detection). Pashler concluded by noting how odd it was that people generally report having a "clear sense of apprehending the identities and locations of large numbers of objects in a scene" (p. 377), and that given this introspective sense, it seemed quite surprising how poor is their detection of changes.

Research in the 1990s and 2000s

Saccadic eye movements have been known to induce change blindness

With the rise of the ability to present complex, real-world images on a computer screen, McConkie, in the early 1990s, as part of new research at the new Beckman Institute for Advanced Science and Technology, renewed investigations of why the world looks stable and continuous despite the shifting retinal input signal that accompanied each saccade. This research began when John Grimes and Dr. George McConkie (1996) began to use actual photographs to study visual stability. This development in change blindness research was able to show the effects of change blindness in more realistic settings. Additionally, further research stated that rather large changes will not be detected when they occur during saccadic movements of the eye. In the first experiment of this kind, in 1995, Blackmore et al. forced saccades by moving the image and making a change in the scene at the same time. Observers' ability to detect the changes fell to chance. The effect was stronger using this method than when using brief grey flashes between images, although subsequent research has mostly used grey flashes or masking stimuli. Another finding based on similar studies stated that a change was easily picked up on by participants when the eye was fixated on the point of change. Therefore, the eye must be directly fixated on the area of change for it to be noticed. This was called the saccade target theory of transsaccadic memory of visual stability. However, other research in the mid-1990s has indicated that individuals still have difficulty detecting change even when they are directly fixated on a particular scene. Rensink, O'Regan, and Clarke presented a picture, followed by a blank, masking screen, followed by the initial picture with a change. The masking screen acts like a saccadic eye movement. This was a critical contribution to change blindness research because it demonstrated that a change can remain unnoticed with the smallest disruptions.

Research on change blindness proceeded one step further into practical applications of this phenomenon. For example, there does not have to be a masking stimulus in order for individuals to miss a change in a scene. Individuals often take significantly longer to notice certain changes if there are a few small, high contrast shapes that are temporarily splattered over a picture. This method for testing change blindness is called “mudsplashes”. This method is particularly relevant to individuals driving in a car when there is a visual obstruction on the windshield. This obstruction may impair an individual's ability to detect a change in their environment which could result in severe negative consequences while driving.

Current research (2010–present)

Change detection

Research indicates that detecting changes in a change blindness task is easier when items are holistically processed, such as faces. Individuals notice a change faster when required to detect changes in facial features than when required to detect changes in images of houses. However, individuals are better at identifying the nature of the change in houses.

Other researchers have discovered that mental processing in change blindness begins even before the change is presented. More specifically, there is increased brain activity in the parietal-occipital and occipital regions prior to the emergence of a change in a change blindness task.

Researchers have also indicated there is a difference in brain activity between detecting a change and identifying change in an image. Detecting a change is associated with a higher ERP (Event-related potential) whereas identifying change is associated with an increased ERP before and after the change was presented.

Additional research using fluctuations in ERPs has observed that changes in pictures (change blindness) are represented in the brain, even without the perceiver's conscious awareness of the change.

Change blindness can be effectively used in the process of visualizing actual changes detected in 3D scenes. With appropriate techniques it is possible to enhance the perception of the portion of a 3D scene that is changed while hiding non significant, but otherwise still visible, changes.

Lucid dreaming

Lucid dreaming occurs when one realizes that the events experienced within a dream are bizarre or would not occur in one's waking life. As such, the inability to notice the bizarre nature of the dream has been coined as an example of change blindness, also known as individuals who are non-lucid dreamers. However, a recent study found that lucid dreamers did not perform better on a change blindness task than non-lucid dreamers. Therefore, the relation between lucid dreamers and change blindness has been discredited to some degree.

In teams

Another interesting area of research is the decreased susceptibility to change blindness when individuals are placed in teams. Although change blindness is still observed within teams, research has indicated that changes between images are noticed more when individuals work in teams as opposed to individually. Both teamwork and communication assist teams in correctly identifying changes between images.

Expertise

Another recent study looked at the relation between expertise and change blindness. Physics experts were more likely to notice a change between two physics problems than novices. It is hypothesized that experts are better at analyzing problems on a deeper level whereas novices employ a surface-level analysis. This research suggests that observing the phenomenon of change blindness may be conditional upon the context of the task.

Choice blindness

Main article: Introspection illusion § Choice blindness

Cognitive psychologists expanded the study of change blindness into decision-making. In one study, they showed participants ten pairs of faces and asked them to choose which face was more attractive. For some pairs, the experimenter used sleight of hand to show participants a face they had not chosen. Only 26% of subjects noticed the mismatch between their choice of face and the different face they were shown instead. The experimenters tested pairs of faces that were either high in similarity or low in similarity, but the detection rate was no different between those conditions. Subjects were also asked to give reasons why they had chosen a face (although due to the sleight of hand they actually hadn't chosen it). Despite the mismatch, subjects gave responses that were comparable in emotionality, specificity, and certainty for faces they had or had not actually chosen. Further research has shown that the failure to detect mismatches between intention and outcome exists in consumer product choices and in political attitudes.

Counteraction

Prior research in the early part of the decade had shown that change blindness can be counteracted by a number of methods. Shifting attention with a visual cue can help lower the negative effects of change blindness. Stimulation of the superior colliculus improves performance and reaction time in the same way. However, recent research has also been done on countering tactile change blindness. A 2016 study by Riggs et al. shows that three successful methods for limiting tactile change blindness in distinguishing changes in vibration patterns are attention guidance, signal gradation and direct comparison. All three methods seek to bring attention to the area of change. Attention guidance works proactively by increasing the frequency of a cue. The second and third methods are reactive and based on error-feedback. Signal gradation further increases the intensity of the vibration after the change has been missed. Direct comparison pairs the pre-change and post-change vibration intensities without a gap in between after a change has been missed to support the use of relative judgment rather than absolute. While all significantly improve performance, the second and third countermeasures are most effective. Concentration and attention are also a major factors in avoiding change blindness.

Non-humans

Though comparatively little research has been done on change blindness in other animals, a few species of animals exhibited the same effects of change blindness as humans. Using the same motion detection paradigm for monkeys as humans, researchers found the results were the same in showing change blindness in motion. Pigeons not only demonstrate change blindness, but also are influenced by the salience and timing of the change in scenery like humans. Chimpanzees similarly have difficulty with detecting change in flicker-type visual search after a blank display was shown. Positional switches of a stimulus are the most difficult for chimpanzees to detect. The results show that the same levels of attention is demanded for chimpanzees as humans in these tasks.

Change detection methods

Saccade forcing paradigm

This method was used in the first, 1995, experiment. A change is made in an image at the same time as the image is moved in an unpredictable direction, forcing a saccade. This method mimics eye movements and can detect change blindness without introducing blank screens, masking stimuli or mudsplashes. However, it is unclear if small additions to an image will predict if people will be unable to notice larger changes in an image to the same position to their eye.

Flicker paradigm

In this paradigm, an image and an altered image are switched back and forth with a blank screen in the middle. This procedure is performed at a very high rate and observers are instructed to click a button as soon as they see the difference between the two images. This method of studying change blindness has helped researchers discover two very important findings. The first finding is that it usually takes a while for individuals to notice a change even though they are being instructed to search for a change. In some cases, it can even take individuals over one minute of constant flickers to determine the location of the change. The second important finding is that changes to more important areas of a photograph are noticed at a faster rate than changes to areas of less interest. Although the flicker paradigm was first used in the late 1990s, it is still commonly used in current research on change blindness and has contributed to current knowledge on change blindness.

Forced choice detection paradigm

Individuals who are tested under the forced choice paradigm are only allowed to view the two pictures once before they make a choice. Both images are also shown for the same amount of time. The flicker paradigm and the forced choice detection paradigm are known as intentional change detection tasks, which means that the participants know they are trying to detect change. These studies have shown that even while participants are focusing their attention and searching for a change, the change may remain unnoticed.

Mudsplashes

Mudsplashes are small, high contrast shapes that are scattered over an image, but do not cover the area of the picture in which the change occurs. This mudsplash effect prevents individuals from noticing the change between the two pictures. A practical application of this paradigm is that dangerous stimuli in a scene may not be noticed if there are slight obstructions in an individual's visual field. Previously, it has been stated that humans hold a very good internal representation of visual stimuli. Studies involving mudsplashes have shown that change blindness may occur because our internal representations of visual stimuli may be much worse than previous studies have shown. Mudsplashes have not been used as frequently as the flicker or forced choice detection paradigms in change blindness research, but have yielded many significant and groundbreaking results.

Foreground-background segregation

The foreground-background segregation method for studying change blindness uses photographs of scenery with a distinct foreground and background. Researchers using this paradigm have found that individuals are usually able to recognize relatively small changes in the foreground of an image. In addition, large changes to the colour of the background take significantly longer to detect. This paradigm is critical to change blindness research because many previous studies have not examined the location of changes in the visual field.

Neuroanatomy

Neuroimaging

MRI image

Various studies have used MRIs (magnetic resonance imaging) to measure brain activity when individuals detect (or fail to detect) a change in the environment. When individuals detect a change, the neural networks of the parietal and right dorsolateral prefrontal lobe regions are strongly activated. If individuals were instructed to detect changes in faces, the fusiform face area was also significantly activated. In addition, other structures such as the pulvinar, cerebellum, and inferior temporal gyrus also showed an increase in activation when individuals reported a change. It has been proposed that the parietal and frontal cortex along with the cerebellum and pulvinar might be used to direct an organism's attention to a change in the environment. A decrease of activation in these brain areas was observed if a change was not detected by the organism. Furthermore, the neurological activation of these highlighted brain areas was correlated with an individual's conscious awareness of change and not the physical change itself.

Other studies using fMRI (functional magnetic resonance imaging) scanners have shown that when change is not consciously detected, there was a significant decrease in the dorsolateral prefrontal and parietal lobe regions. These results further the importance of the dorsolateral prefrontal and parietal cortex in the detection of visual change. In addition to fMRI studies, recent research has used transcranial magnetic stimulation (TMS) in order to inhibit areas of the brain while participants were instructed to try to detect the change between two images. The results show that when the posterior parietal cortex (PPC) is inhibited, individuals are significantly slower at detecting change. The PPC is critical for encoding and maintaining visual images in short term working memory, which demonstrates the importance of the PPC in terms of detecting changes between images. For a change to be detected, the information of the first picture needs to be held in working memory and compared to the second picture. If the PPC is inhibited, the area of the brain responsible for encoding visual images will not function properly. The information will not be encoded and will not be held in working memory and compared to the second picture, thus inducing change blindness.

Role of attention

The role of attention is critical for an organism's ability to detect change. In order for an organism to detect change, visual stimulation must enter through eye and proceed through the visual stream in the brain. A study in 2004 demonstrated that if the superior colliculus (responsible for eye movements) of a monkey's brain is electrically stimulated, there would be a significant decrease in reaction time to detect the change. Therefore, it is critical for organisms to attend to the change in order for it to be detected. Organisms are only able to detect this change once the visual stimulation comes through the eye (its movements are controlled by the superior colliculus) and is subsequently processed through the visual stream.

Influencing factors

Age

Older individuals have been known to have more difficulty detecting changes.

Age has been implicated as one of the factors which modulates the severity of change blindness. In a study conducted by Veiel et al. it was found that older individuals were slower to detect the changes in a change blindness experiment than were younger individuals. This trend was also noticed by Caird et al., who found that drivers aged 65 and older were more prone to making incorrect decisions after a change blindness paradigm was used at an intersection, than were participants aged 18–64. Age differences in change detection become most pronounced when the task is easier. While the actual shift in ability does not occur until at least age 65, people's confidence in their ability to detect change drops significantly at middle-age.

Children from 6–13 years old looked at colored pictures of real world scenes that were manipulated by color, location of objects, or the removal of objects, in the central or peripheral focus of the image. Adults are more accurate when noticing the changes that occur in the picture. Children can accurately detect central changes, but aren't as good at detecting peripheral changes, and their accuracy depends on the type of manipulation.

Younger drivers (average of 22 years old) were compared with older drivers (average of 69 years old). Images were presented on a screen showing various driving situations that included an original image and a modified image, and participants had to identify where a change had occurred in the modified version, if any. Older drivers expressed reduced accuracy, higher reaction times, and more false positive responses compared to younger drivers.

Attention

Attention is another factor that has been implicated in change blindness. Increasing shifts in attention decrease the severity of change blindness and changes in the foreground are detected more readily than changes made to the background of an image, an effect of the intentional bias for foreground elements.

Community volunteers had to focus on a screen and accurately identify if there was a change between series of dots after being fixated on a point in the center of the screen. Distraction of attention by visual disruptions and the observers' ability to focus on potential change were found to have an effect on attention with change blindness.

Object presentation

Object presentation is the way in which objects appear and is a factor that determines the occurrence of change blindness. Change blindness can occur even without a delay between the original image and the altered image, but only if the change in the image forces the viewer to redefine the objects in the image. Additionally, the appearance of a new object is more resistant to change blindness than a looming object, and both the appearance of a new object and the looming of an object are more resistant to change blindness than the receding of an object. Furthermore, the appearance or onset of an object is more resistant to the occurrence of change blindness than the disappearance or offset of an object.

Substance use

Substance use has been found to affect the detection biases on change detection tasks. If an individual was presented with two changes simultaneously, those that had a change related to the substance they use regularly reported using the substance more than those detecting the neutral stimuli. This indicates a relationship between substance use and change detection within a change blindness paradigm. This bias for devoting more attention to the drug-relevant stimuli is also observed with problem drinkers. Individuals who have a more severe drinking problem are quicker to detect changes in alcohol-related stimuli than in neutral stimuli.

Alcohol can sometimes improve change blindness. For example, intoxicated participants were quicker at detecting minor changes in large displays of images than sober participants. This could be attributed to more passive viewings of larger images, and the use of alcohol slows down more controlled search processes.

Active viewing involves more saccades than fixations. When viewing an image with a more passive search, more information is processed with each fixation. The alcohol slows down the movement and processing of the brain, therefore causing more fixation points.

In other senses

In addition to change blindness induced by changes in visual images, change blindness also exists for the other senses:

• Change deafness – Change deafness is the concept of change blindness for auditory information. In his experiment, Vitevitch (2003) used a speech shadowing task to demonstrate change deafness. He presented a list of words to participants and had them simultaneously repeat the words they heard. Halfway through the list, either the same or a different speaker presented the second half of the words to participants. At least 40% of participants failed to detect the change in speaker when it occurred. Fenn et al. called participants on the phone and replaced the speaker in the middle of the conversation. Participants rarely noticed change. However, when explicitly monitoring for change, the participants' detection increased. Neuhoff et al. (2015) expanded on the idea of change deafness, and identified a new phenomenon called "slow-change deafness" using a series of four experiments. In the first experiment, he had participants listen to continuous speech that changed three semitones in pitch over time. Fifty percent of participants failed to notice the change. In the second and third experiments, listeners were alerted to the possibility of a change. In these trials, detection rates drastically improved. In the fourth experiment, the magnitude of the change that occurred in the stimulus increased, causing the detection rates to increase. These experiments demonstrated that "slow-change deafness" depends on both the magnitude of a stimulus change and the listeners' expectations.
• Olfactory – Humans are constantly in a state of change blindness due to the poor spatial and temporal resolutions with which scents are detected. Although humans' odor detection thresholds are very low, our olfactory attention is only captured by unusually high odorant concentrations. Olfactory input is made up of a series of sniffs separated in time. The long inter-sniff-interval creates "change anosmia," in which humans have trouble discerning smells that are not highly concentrated. This period of sensory habituation as well as very low concentrations of odorants regularly yield no subjective experience. This behavior is called "experiential nothingness".
• Somatosensory – Somatosensory change blindness for tactile stimuli has been observed, and reveals important information about the distinction from visual change blindness. Auvray et al. (2008) did an experiment on the ability to detect change between two patterns of tactile stimuli presented to fingertips. The experiments presented consecutive patterns which were separated by an empty interval, or by a tactile, visual, or auditory mask. Results showed that performance was impaired when the empty interval was inserted, and even more so when tactile mask was introduced. Changes in tactile displays composed of two or three stimuli with only one distractor in between go unnoticed, while several distractors are needed for visual displays to go unnoticed. These experiments have shown us that our ability to monitor tactile information is affected by more severe limitations than the same ability within the visual modality.

Practical implications

The phenomenon of change blindness has practical implications in the following areas:

Eyewitness testimony

Research in change blindness has uncovered the possibility of inaccuracy in eyewitness testimony. In many cases, witnesses are rarely able to detect a change in the criminal's identity unless first intending to remember the incident in question. This inability to detect a change in identity can lead to inaccuracy in identifying criminals, mistaken eyewitness identification, and wrongful conviction. Therefore, eyewitness testimonies should be handled with caution in court in order to avoid any of these negative consequences.

Driving ability

Traffic collision

Older drivers make more incorrect decisions than younger drivers when faced with a change in the scene at an intersection. This can be attributed to the fact that older individuals notice change at a slower rate compared to younger individuals. In addition, the location and relevance of changes have an effect on what is noticed while driving. The reaction time to changes in the driver's peripherals is much slower than the reaction time to changes that occur towards the center of the driver's visual field. Furthermore, drivers are also able to recognize more relevant changes as opposed to irrelevant ones. Research on the effects of change blindness while driving could provide insight into potential explanations of why car accidents occur.

Military

Military command and control personnel who monitor multiple displays have a delayed time to accurately identify changes due to the necessity of verifying the changes, as well as the effective 'guessing' on some trials. Due to the fact that control personnel have delayed reaction because of change blindness, an interface design of computer work stations may be extremely beneficial to improve the reaction time and accuracy.

Blindness

Change blindness is defined as a misplaced confidence in one's ability to correctly identify visual changes. People are fairly confident in their ability to detect a change, but most people exhibit poor performance on a change blindness task.

Factors

• Perceived Success – A higher perception of success from previous experience inflates the individual's confidence for success in future experiences.
• Search Time – A longer time spent looking for the visual change creates the impression of poor performance on the task. In other words, a shorter time in identifying a visual change creates the impression of good performance and thus the individual will be overconfident in this ability.

Spotlight effect

The spotlight effect is a social phenomenon that is defined as an overestimation of the ability of others to notice us. A seemingly obvious change such as another individual changing a sweater during a memory task is rarely noticed. However, the individuals switching the sweater tend to overestimate the ability of the test writers to notice the change in sweaters. In the spotlight effect, this poor performance is a result of the overestimation of others' ability to notice us whereas in change blindness it is the overestimation of others' ability to notice the sweater change. In other words, it is the distinction between noticing differences on a person and noticing differences between any images.