Fractal patterns in visual search
Deborah J. Aks


Project Overview



Study the 'internal' process guiding search by examining search patterns over time.


Many questions emerge when looking at complicated eye-movement behavior .


Background information
on fractal concepts & connection to eye-movements
Yale's introductory Fractal Geometry course, sponsored by Michael Frame, Benoit Mandelbrot & Nial Neger,  has excellent demos of many fractal concepts and analyses used to detect fractals.
MIT course: Random Walks and Diffusion by Martin Bazant contains more advanced math background w/ a focus on brownian motion. Here are some additional on-line tutorials related to fractals & modeling.

Mapping eye-mvmts to time-series


What's the connection?

time series animation
animated eye 1 animated eye 1

Eyeballs are modification of  X. Liu's 
if time series animation stops, refresh page to repeat. Credit & thanks to Frame, Mandelbrot & Neger's Yale fractal class


<to top of page>

Key features of experimental design
To evaluate whether there are fractal patterns in eye-movements, experiments are designed to optimize our ability to capture the 'internal' search process by having the following features in our search tasks:
  • We will minimize the stimulus role which is well known in guiding search.
  • Task must be sufficiently challenging to provide enough data to...
    • find temporal (fractal) patterns.
    • elicit wide-ranging and complex eye-movements.


Preliminary experiments & data

These data were collected by Sol Simpson of SR-Research when I was testing out their eye-trackers.

1000HZ sampling rate permits fine-resolution sampling of eye-movement patterns: In just over 3 minutes we collected ~200K data pts !  Importantly, even within such a brief  period, many interesting search patterns emerge. Here's an overview of the 3 trial's data, timing & video samples of the demo search tasks.



Do you notice any patterns in the eye-movements shown below?
time series sample of eye-movements
0-----------------------Time  ------------------------1.6 seconds




Most eye-movement and visual search theory is based on general linear models (GLMs) and assumes that variability across eye-movements and experimental trials is random (or due to random sampling from uniform gaussian or poisson distributions).  The failure to recognize that many processes are not best described by normal distributions may be one of the most overlooked  yet essential property of behavioral phenomena.  Given the ubiquity of fractal patterns being discovered in nature, I expect, through the use of  appropriate statistical tools to uncover search patterns with complex fractal dynamics. Moreover, finding these patterns may guide biologically plausible models such as those that include the compelling notion that visual search can be guided by simple threshold and neuronal interaction rules. Self-organized criticality (SOC) is one such class of models that can produce complex and long-range dynamics. It is described in some of our earlier preliminary work exploring these ideas in the context of human perception. SOC lies within the cellular automata (CA) framework from which this theoretical perspective on eye-movement behavior has been inspired.

Fractal structure will emerge in scan path time series signified by scaling & self-similar properties.  Furthermore, I expect to find 1/f pink noise under some (of the most unstructured) conditions on particular eye-movement parameters. My rationale for this prediction relates to the similarity of processes that can produce 1/f patterns and known properties of sensory/perceptual neural interactions.


Key analyses:
Statistical tests that reveal structure & correlation across eye-movements.
See also
signs of fractal patterns
Probability Distributions (PDFs): --> Skew & Kurtosis
Power spectra (FFT) --> Autocorrelation (Tau)
Rescaled Range Analyses: --> Hurst exponent (H)
Iterated Function Systems (IFS) Detrend Fluctuation Analysis (DFA)


Dependent Variables = Various parameters of eye movement behavior 


  1. Signs of fractal patterns:
    1. noisy, complicated time series w/ frequent-small eye mvmts & rare-large mvmts
    2. Probability Distributions (PDFs) w/ long tails

    3. means and variance change over time
    4. means and variance change  w/ measuring resolution

    5. non-random patterns (despite their random appearance)
    6. statistical self-similarity
    7. what color is the noise: white, brown, pink?
      (wikipedia illustrations of noise color)
      (another illustrations & ways to generate different forms of noise)

      <Note: conventional theory predicts white or brown noise in eye-mvmts>
      <Caution on terminology: fractal brownian motion (fbm) can include both brown & pink forms of noise;
      <......fbm does not necessarily = brown (or brownian) noise


  2. Future Studies will assess..

    <--Fractal networks? -->
    click images for enlargements

    Here are 2 network representions of the html (tag) structure of this 'Fractal patterns' page

    Webs as a Graph software

    Node legend

    this web site as a 2nd fractal


    |D.J. Aks | Eye-tracking research | System noise | Time series & fractal analysis | Visual search | Attention | Satellite Imagery | Tumor detection | Web eval & info search | Illusions