From analogy to symmetry detection

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From analogy to symmetry detection

• Ron Ferguson
• CS/ISYE/PSYC 7790 Cognitive Modeling
• Fall 2003

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The Story So Far

• Analogy has a number of interesting
  characteristics
• Sensitivity to higher-order structure
• Very fast
• Ability to generate candidate inferences
• Today
• Can we extend this model to cover symmetry
  perception?

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Stages of analogical reasoning
target description
base description
ACCESS
MAPPING
possible inferences
Casebase of descriptions
Revised inferences
TWEAKING
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Characteristics of symmetry studied in previous
work

• Visual
• Exact
• Structurally simple
• Guiding metaphors
• Symmetry as perception
• Symmetry as a mathematical symmetry

5
Classic problems in symmetry detection

• Goodness of multiple symmetries (Garner,
  Attneave)
• Preference for vertical symmetry (Mach,
  Corballis, Palmer Hemenway, many others)
• Symmetry interactions with object-centered
  reference frames (Palmer)

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Expanding the range of our understanding of
symmetry

• Inexact or approximate symmetry
• Structurally complex symmetry
• Non-visual symmetry
• Guiding metaphor
• Symmetry as analogy

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Approximate symmetry

• How approximate can symmetry be?
• Are there different kinds of asymmetry?

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Complex symmetry

• Existing models of symmetry detection assume a
  limited number of visual element and relation
  types
• Dot patterns
• Dash or iron filings patterns
• Boundaries
• Polygons

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How other models simplify visual structure types

• Transformational invariance (Weyl, 1952 many
  others)
• Point-to-point measurements (Jenkins, 1983
  Labonte, 1995 Wagemans, 1993)
• Brushfire methods (Blum, 1978 Brady, 1983
  Burbeck Pizer, 1995 many others)
• Measures of asymmetry (Zabrodsky, 1992 1995)

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Non-visual symmetry

• Conceptual symmetry
• Physical laws
• Narratives
• Visual symmetry is often used to support
  conceptual symmetry

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The Symmetry as Self-Similarity Hypothesis

• Instead of looking for transformational
  invariance, find relational self-similarity
• Key idea Symmetry detection utilizes same
  cognitive process as similarity comparison
• Use Structure Mapping Theory (Gentner, 1983) as
  basis

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Starting Point The Structure Mapping Engine (SME)
Gentner (1983, 1989), Falkenhainer, Forbus
Gentner (1989), Forbus, Ferguson Gentner (1994)

• Compares twodescriptions
• Constraints
• Identicality
• One-to-one correspondence
• Parallel connectivity
• Systematicity
• Generates candidate inferences

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Symmetry as Structure Mapping
Mapping a description of O. Henrys The Gift of
the MAGI.

• Synopsis
• A wife sells her hair to purchase a chain for her
  husbands watch.
• A husband sells his watch to purchase combs for
  his wifes hair.

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How MAGI extends Structure Mapping

• Self-similarity mapping
• Instead of base and target, description maps to
  self.
• Two additional mapping constraints

• Limited self-matching An expression or entity
  cannot map to itself, except as the argument of
  two non-identical expressions
• For commutative expressions arguments must be
  permuted.

• Maximal differentiation
• maximize intraconnectivity of mapped expressions
• minimize interconnectivity of mapped expressions
• Backward compatible with SMT and SME.

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Visual symmetry detection using MAGI
INPUT Line drawing
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GeoRep A Qualitative Spatial Representation
Engine
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Spatial relations used by MAGI
GeoRep
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Sample spatial relations from GeoRep
(mid-connect ltL2gt ltL13gt)(hort-interval-equal
ltL7gt ltL6gt ltreference-frame
1gt)(parallel-above ltL10gt ltL3gt
ltreference-frame 1gt)(above ltL13gt ltL11gt
ltreference-frame 1gt)(horizontal ltL11gt
ltreference-frame 1gt)(h-aligned (corner ltL10gt
ltL11gt) (corner ltL11gt ltL12gt)
ltreference-frame 1gt)(during ltL1gt ltL2gt)(before
ltL1gt ltL5gt)

• (polygon ltpolygon-1gt)(polygon-member ltpolygon-2gt
  ltL7gt)(number-of-sides ltpolygon-1gt 6)
• (obtuse (corner ltL11gt ltL12gt)
  ltpolygon-1gt)(corner ltL9gt ltL10gt)(acute (corner
  ltL9gt ltL14gt) ltpolygon-1gt)(concave
  (corner ltL9gt ltL10gt) ltpolygon-1gt)(conve
  x (corner ltL9gt ltL14gt)
  ltpolygon-1gt)(perpendicular-in (corner ltL9gt
  ltL10gt) ltpolygon-1gt)(indentation ltpolygon-1gt
  (set (corner ltL9gt ltL10gt)))(protrusion
  ltpolygon-2gt (set (corner ltL7gt ltL8gt)
• (corner ltL3gt ltL8gt) (corner ltL3gt
  ltL4gt)
• (corner ltL4gt ltL5gt)
• (corner ltL5gt ltL6gt)))

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Visual symmetry detection using MAGI

• Finds regularity (symmetry or repetition) using
  structure mapping
• Based on Structure-Mapping Theory (Gentner, 1983)
• Runs mostly in parallel
• Uses SMEs constraints, plus 2 additional
  constraints

INPUT Line drawing
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Axis detection in MAGI
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Expanding the range of our understanding of
symmetry

• Inexact or approximate symmetry
• Structurally complex symmetry
• Non-visual symmetry

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21 entities 160 relations SES 14.361 9.9 sec.
13 entities 113 relations SES 4.924 4.2 sec.
14 entities 112 relations SES 12.245 10.3 sec.
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Effects of structure on symmetry detection
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Are there two kinds of asymmetry?

• (Ferguson, Aminoff Gentner, in preparation)

• If symmetry detection involves aligning
  qualitative relations
• Qualitatively asymmetric figures should be
  easier to detect than quantitatively asymmetric
  figures
• Should be true even when accounting for other
  quantitative factors, such as differences in
  perimeter or radial length between the sides.

Easier
Harder
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Results of Experiment

• Experiment 1
• Symmetry judgment task using symmetric and
  asymmetric 16-gons.
• 50 ms. display time with backward and forward
  masking.
• Subjects more accurate for asymmetric figures
  containing concavity differences (F(1,13)24.3,
  plt.0005).
• Experiment 2
• Same task using 12-gons.
• Fast condition As before, with better display
  conditions and no masking.
• Slow condition As before, but with no display
  time limit.
• Results similar to Experiment 1, but
  statistically weaker.
• In fast condition, more accurate with concavity
  (F(1,53)14.7, plt.001) or orientation differences
  (F(1,53)9.71, plt.005).
• In slow condition, subjects were faster to
  classify figures with concavity differences
  (F(1,53)3.75,plt.06) but not orientation
  differences.
• Experimental results simulated using MAGI

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Expanding the range of our understanding of
symmetry

• Inexact or approximate symmetry
• Structurally complex symmetry
• Non-visual symmetry

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Summary of new phenomena

• Approximate symmetry
• MAGI makes a testable distinction between two
  types of asymmetry
• Structurally complex symmetry
• MAGI can detect it
• Predicts that some structure types are better
  than other types
• Non-visual symmetry
• MAGI can detect non-visual symmetry
• Predicts an interaction between visual and
  functional symmetry

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Mapping Perceptual and Conceptual Regularity
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Classic problems in symmetry detection

• Goodness of multiple symmetries (Garner,
  Attneave)
• Preference for vertical symmetry (Mach,
  Corballis, Palmer Hemenway, many others)
• Symmetry interactions with object-centered
  reference frames (Palmer)

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The Preference for Vertical Symmetry
Harder

• Geometric symmetry doesnt depend on figure
  orientation
• Human symmetry detection does
• Vertical symmetry is detected most easily (Mach,
  1896 many others)
• Explanations
• Evolutionary explanations
• Retinocentric explanations
• But...

Easier
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Explaining the preference for vertical symmetry

• Another possible explanation
• Reference frames affect shape perception
  (Goldmeier, 1978 Rock, 1963, 1983).
• Conjecture by Rock (1983) Preference depends on
  phenomenological characteristics of shapes
• MAGIs explanation
• Symmetry as self-similarity hypothesis
• A structure-mapping model of symmetry
• Key variable stimulus relational structure
• Claim preference may result from how reference
  frames affect visual relational structure

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Overview of the MAGI model

• Finds regularity (symmetry or repetition) using
  structure mapping
• Based on Structure-Mapping Theory (Gentner, 1983)
• Runs mostly in parallel
• Uses SMEs constraints, plus 2 additional
  constraints

INPUT Line drawing
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MAGIs simulation of orientation effects in
symmetry detection

• Depends on visual relations
• - above-below relations are directed and
  salient - left-right relations are
  commutative
• Relations that depend on the reference frame
  change with object orientation

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Handling figures with good intrinsic axes
In some figures, the available visual relations
are sufficient for symmetry detection at any
orientation.
Figure from Wiser (1981).
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Figures with an almost-intrinsic axis
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Summary of MAGIs explanation for the preference
for vertical symmetry

• The preference for vertical symmetry depends on
  visual relations that are reference-frame
  dependent
• Figures with intrinsic axes attenuate the
  preference for vertical symmetry
• Visual structure in these figures are sufficient
  for symmetry detection, even without
  reference-frame dependent visual relations
• Figures may even contain almost-intrinsic axes
• Visual structure in the figure may generate a
  partial symmetry, even without reference-frame
  dependent visual relations.
• This partial symmetry may allow the viewer to
  reset the frame of reference, and capture the
  full symmetry.

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Simulation of Orientation Effects

• Stimuli from Palmer Hemenway (1978)
• Set of 30 16-gons
• Divided evenly between 4x, 2x, single, near and
  rotational symmetry conditions
• Shown at 4 orientations tilted left, vertical,
  tilted right, horizontal
• Subjects had to decide if figures were
  mirror-symmetric
• Same figures given to MAGI. Systematicity rating
  (SES) used to estimate the strength of the
  symmetry in each figure

Humans (Palmer and Hemenway, Exp 1).
MAGI.
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Orientation effects in symmetry detection

• MAGI provides a novel explanation for a
  century-old puzzle in symmetry detection
• Solution is based on how relational structure
  changes with the frame of reference
• Explains
• The preference for vertical symmetry
• Why the preference for vertical symmetry is not
  retinocentric
• Which figures have good intrinsic axes
• Can simulate results from Palmer Hemenway study

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Summary Why Symmetry is Like Analogy

• Symmetry detection is broader than perfect,
  visual symmetry
• To handle this, we must restructure symmetry as
  a similarity recognition process
• This restructuring has psychologically-testable
  results
• New phenomena (different types of asymmetry)
• New explanations for old phenomena (preference
  for vertical symmetry)

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Future work

• Examine effects of visual structure on symmetry
  detection
• Diagrammatic reasoning system that utilizes
  repetition and symmetry
• Aligned differences are key
• Provide a better definition for objects with
  intrinsic axes (Wiser)

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Coda Is symmetry detection amodal?

• What does symmetry detection tell us about the
  relationship between vision, other modalities,
  and cognition?
• The MAGI model appears to remove symmetry
  detection from perception per se
• Removes a conundrum is non-visual symmetry
  merely analogous to visual symmetry?
• Adds a problem If a general facility, why is 99
  of symmetry visual?
• Have we made a bad bargain?

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Evidence of a good bargain

• Good Commonality of process (analogy,
  similarity, and symmetry) constrains our model
• Yet, the results are additional predictions that
  can be made (vertical symmetry, multiple
  symmetries, intrinsic axes, qualitative symmetry,
  symmetry-based inferences, symmetry without a
  straight axis)
• Better Symmetry provides an account of how
  similarity and analogy itself might have arisen
  from the development of visual and perceptual
  capabilities

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Vision uniquely provides a rich information flow
suitable for symmetry detection

• Among modalities, vision is unique
• Simultaneous presentation of highly structured
  percepts
• Requires little or no memory store
• Requires some relational abstraction
• Symmetry detection (in terms of MAGI) geared
  toward this kind of information flow

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Vision -gt Symmetry -gt Analogy

• In evolutionary terms, symmetry detection may be
  early
• Driven by visual processing alone
• Arising before similarity
• Not general capability at first
• Symmetry detection can become similarity
  comparison
• In MAGI, moving a single wire is enough to turn
  it into a model of similarity
• Thus, visually-driven symmetry detection may have
  provided the evolutionary basis for a broader
  ability to perform similarity comparisons