Defining Perceptual Metrics in Shape Space

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Defining Perceptual Metrics in Shape Space

• L.L. Walker1, J. Malik1,2
• UC Berkeley, Berkeley, CA
• Vision Science1, Computer Science2

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MOTIVATION

• Shape similarity has traditionally been studied
  in a qualitative setting (e.g. Goldmeier).
• Our goal is to define a quantitative measure of
  the differences between similar (but not
  identical) shapes.
• We validated two measures from the mathematics
  and statistics literature for this purpose
  Kendalls distance and Procrustes distance.

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PROCRUSTES DISTANCE

• Procrustes distance measures the sum of squares
  of Euclidean distances between corresponding
  points, after the two sets have been aligned with
  the best similarity transformation.

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KENDALLS DISTANCE

• In 1984, David Kendall proposed a new,
  non-Euclidean measure which has the advantage of
  being a Riemannian metric in shape space.

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VALIDATING THE PERCEPTUAL METRIC

• Discrimination thresholds measured in the
  appropriate space will not change for a given
  subject when the shapes are transformed by both
  random and systematic changes.

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METHOD

• Target shapes with random change were created by
  jittering the shape vertices.
• Target shapes with systematic change were
  created by an affine transformation.

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Which of the two bottom shapes is more similar to
the top shape?
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RESULTS

• Thresholds measured in Kdist and Pdist for each
  transformation

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CONCLUSIONS

• A statistical analysis of this data indicates
  that the thresholds measured by both metrics are
  consistent when shapes are altered by random or
  systematic changes. Both metrics are therefore
  candidates for measuring shape similarity,
  however, Kdist appears to be the more robust
  measure.

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USING THE PERCEPTUAL METRIC

• How good are we at measuring shape similarity
  when changes in scale or orientation are present?

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SCALE CHANGE
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SCALE CHANGE
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ORIENTATION CHANGE
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ORIENTATION CHANGE
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REFERENCES

• Goldmeier, E. (1972). Similarity in Visually
  Perceived Forms. Psychological Issues, Monograph
  29. 8(1).
• Attneave, F. and Arnoult, M.D. (1956). The
  Quantitative Study of Shape and Pattern
  Perception. Psychological Bulletin. 53(6)
  452-471.
• Kendall, D.G. (1984). Shape Manifolds,
  Procrustean Metrics, and Complex Projective
  Spaces. Bull. London Math. Soc. 16, 81-121.
• Kendall, D.G. (1989). A Survey of the Statistical
  Theory of Shape. Statistical Science. 4(2), 87-99.

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CONTACT INFORMATION

• Laura L. Walker
• UC Berkeley School of Optometry
• 200 Minor Hall 2020
• Berkeley, CA 94720-2020
• lwalk_at_cs.berkeley.edu