Size Matters: Measurement

1
Size Matters Measurement Modeling in infinite
dimensions

• Kathryn Leonard
• CSUCI Mathematics
• 31 January 2007

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Organization

• Metric spaces
• Measurement in linear spaces
• Modeling

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1. Metric Spaces
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Metric spaces

• A set X with a map is a
  metric space when, for every x, y, z in X

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Metric spaces

• A set X with a map is a
  metric space when, for every x, y, z in X

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Examples

• Finite dimensional
• Linear Nonlinear
• Infinite dimensional
• Linear spaces of sequences, functions
• Nonlinear spaces of curves

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Finite dimensional linear

• Example
• Standard Euclidean metric

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Finite dimensional linear

• Example
• Standard Euclidean metric
• Taxicab metric
• Others

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Finite dimensional linear

• Theorem
• For a finite dimensional linear metric space, any
  two metrics are equivalent.
• ?
• Standard is all there is.
• that respect the linear structure

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Finite dimensional nonlinear

• Geometry-dependent interaction between the
  geometry of the nonlinearity and metrics.

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Infinite dimensional linear

• Sequence spaces (think of ).

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Infinite dimensional linear

• Sequence spaces (think of ).

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Infinite dimensional linear

• Sequence spaces (think of ).
• Question
• Does every real-valued sequence live in this
  metric space?

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Infinite dimensional linear

• Metric determines the space

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Infinite dimensional linear

• Sequence spaces

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Infinite dimensional linear

• Sequence spaces

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Infinite dimensional linear

• Sequence spaces

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Infinite dimensional linear

• One more

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional linear

• Function spaces

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Infinite-dimensional nonlinear

• Spaces of closed curves
• Hausdorff distance (L8-type)
• L1-type distance

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2. Measurement in Linear Spaces
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Measuring size

• Case Study__
• Given a basis , we can write
• If, however, our basis is orthonormal, then
• where

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Norm Equivalence

• Function Spaces
• continuous operation discrete operation
• A norm equivalence happens when

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Norm Equivalence

• Question
• What would guarantee that we have a norm
  equivalence?

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Norm Equivalence

• Question
• When are we guaranteed to have a norm
  equivalence?
• Answer
• If we have an orthonormal basis.

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Norm Equivalence

• Question
• What do we need in a space to have an orthonormal
  basis?

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Norm Equivalence

• Question
• What do we need in a space to have an orthonormal
  basis?
• Answer
• A notion of orthogonality.

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Norm Equivalence

• Who is an inner product space?

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Norm Equivalence

• Who is an inner product space?

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Fourier Basis
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Fourier Basis
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Wavelet Basis
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Wavelet Basis
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3. Modeling Approximation
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What is a model?

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What is a model?
An approximation to the real thing.

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What is a model?
An approximation to the real thing.

What is a good model?
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What is a model?
An approximation to the real thing.

What is a good model?
An approximation to the real thing that captures
the most important qualities of that
thing. (most important varies by application)
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I. Functions with bounded derivative

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I. Functions with bounded derivative

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I. Functions with bounded derivative

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I. Functions with bounded derivative

To get a good model for these functions, can just
take first few frequencies!
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II. Functions with point discontinuities

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III. Functions with curve discontinuities

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III. Functions with curve discontinuities

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III. Functions with curve discontinuities

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Thank you!
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?-entropy infinite dimensions

• L8 (a, b)
• compact set Lipschitz functions
• Theorem (Kolmogorov)

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Idea of proof

• Subdivide domain into equal intervals of width no
  more than .
• Construct balls with boundaries consisting of
  piecewise linear functions with slopes of K.

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?-entropy infinite dimensions

• 2. Spaces of plane curves
• compact set curves with Lipschitz tangent
  angle, arclength parameterized

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Metric orientation

• Given two curves, ?1, ?2 , the distance between
  them is

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The importance of orientation

• On left, random orientation, correct magnitude.
• Images taken from Eero Simoncelli created using
  steerable pyramids.

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The importance of orientation

• On left, random orientation, correct magnitude.
• On right, correct orientation, random magnitude
• Images taken from Eero Simoncelli created using
  steerable pyramids.

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?-entropy for curves

• Theorems (KL)
• General curves
• Closed curves where
• Conclusion The space of Lipschitz functions and
  the space of curves with Lipschitz tangent angle
  are the same size.

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Translation into compression

• Enumerate the balls in any cover to obtain its
  compression rate.
• B1 ? 0000 B2 ? 0001
• B3 ? 0010 B4 ? 0011
• B15? 1110 B16? 1111
• For a minimal covering, this number of bits is
  log2 N?, i.e., the ?-entropy of the space.
• Note
• finite number of balls
• same number of bits for each element in the space.

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Relaxing compactness adaptive encoding

• Back to the question at hand
• How can we get a handle on whole space?
• Answer by adaptively encoding each element.

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?-entropy -vs adaptive encoding integers

• ? 1/2
• ?-entropy X -1/2, n1/2
• bit length log2n for each
• integer 0 m n.
• adaptive encoding X -1/2, 8)
• bit length log2m for any
• positive m.

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Adaptive encoding function spaces

• Theorem (KL) One can encode Lipschitz functions
  over a bounded interval so that, for a fixed ?
  and any ? gt 0, a function f requires no more
  than
• bits to be represented to within ? in L8, where
• C(f, ?) and ? are independent of ?.

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Adaptive encoding spaces of curves

• Theorem (KL) There is an adaptive encoding
  scheme for boundary curves in the metric ? with
  asymptotic bit-rate

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Application Shape model selection

• Boundary curve
• Area
• Landmark points
• Feature vectors
• Structural descriptions
• Which shape model is best for a given shape?
• The one requiring the fewest bits to describe.

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Competing shape models

• Boundary curve of shape (edge-based)
• Blums medial axis pair (m,r) (region-based)

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Inspiration from childhood, I
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Inspiration from childhood, II
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Encoding for curves

• Boundary curve bit-rate
• Compare to medial axis bit-rate

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Characterization of curves

• Theorem (KL) The medial axis is more efficient
  than the boundary curve on a domain I whenever I
  ? Ik so that for each k
• or

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Empirical Results 2322 shapes
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Conclusion

• The medial axis wins for naturally occurring
  shapes!

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Sensitive shapes
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To learn more

• ?-entropy
• www.acm.caltech.edu/kathryn
• www-stat.stanford.edu/donoho
• Adaptive encoding
• www. mdl-research.org/jorma.rissanen
• Boundary curves
• www.acm.caltech.edu/kathryn
• www.dam.brown.edu/people/eitans or mumford
• Medial axis
• www.lems.brown.edu (Ben Kimia Peter Giblin)
• www.stat.ucla.edu/sczhu
• www.acm.caltech.edu/kathryn

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Goals Recognize, Classify, Approximate 2D shape
?
?
?
Recognize
Approximate
?
?
or
Classify
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Perceptual similarity I (Goldmeier)
All perceptual figures from Goldmeier.
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Perceptual similarity I
Proposed rule 1 Greatest number of same parts
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Perceptual similarity I
Proposed rule 1 Number of same parts.
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Perceptual similarity II
Proposed rule 2 Relations between parts
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Perceptual similarity II
Proposed rule 1 Number of same parts Proposed
rule 2 Relations between parts
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Perceptual similarity III
Proposed rule 1 Number of same parts Proposed
rule 2 Relations between parts Proposed
rule 3 Relations between parts form and
so on
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Goals Recognize, Classify, Approximate 2D shape
?
?
?
Recognize
Approximate
?
?
or
Classify
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Some shape neighborhoods
Image courtesy of David Mumford.
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Translation to mathematics

• Recognition occurs in neighborhoods of varying
  size
• ?
• Metric spaces of shapes
• Note d(x,y) is a metric when it is non-negative
  and
• d(x,y) 0 iff xy.
• d(x,y) is less than or equal to d(x,z) d(z,y).

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Goldmeier caveat

• Human perception is slippery
• -- not actually a metric.
• Not symmetric.
• Doesnt satisfy triangle inequality.

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Measuring shape space

• Idea
• Count the number of ?-balls required to cover the
  space.
• Problem
• Space is really large.

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Current/Future Work

• Repeat for surfaces--best modeled by multiple 2D
  views or with 3D model?
• Explore texture--best modeled by randomness or a
  deterministic pattern?
• Apply to image segmentation--which regions are
  shape and which are texture?
• Other 2D/3D/texture representations

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Mathematical setting

• Shape region in R2 bounded by some
  curve (or collection of curves) with
  some degree of smoothness.
• Define a shape representation based on perceptual
  knowledge.
• Based on representation, define suitable shape
  metric(s).
• What nice properties does a particular
  representation give?

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Shapes as Closed Curves

• Simplifying assumption
• Shapes are bounded by a single simple, closed
  curve of some degree of smoothness.
• Properties of shape space
• Non-linear.
• Locally linear--an infinite-dimensional manifold.
• Contractible.

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Local linearity

• Given the boundary curve ?(s) to a shape, with
  normal vector n(s), nearby shapes can be
  described as

where a(s) lives in some linear function space
and is sufficiently small in the associated
metric (norm). Local coordinates around ? take
.
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Contractibility
Images taken from Kimia, Tannenbaum, Zucker, and
from Mumford.
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?-entropy for curves

• Idea
• Find a good linear function space associated to
  space of curves--
• the space of tangent angle functions.
• If the tangent angle functions are Lipschitz,
  then curves will have bounded curvature.
• If tangent angles functions are Lipschitz, we can
  use Kolmogorovs result.
• Compact classes
• Spaces of curves of length bounded by L and
  curvature bounded by K.

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?-entropy for curves

• Step 1 Apply Kolmogorovs theorem to
• X Lipschitz tangent angle functions on

This gives an entropy result for the tangent
angle functions of .
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?-entropy for curves

• Step 1 Apply Kolmogorovs theorem to
• X Lipschitz tangent angle functions on

• Step 2 Lift to a covering on curve primitives,
  and refine to give an ?-covering there in the
  metric ?.

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Measuring shape space

• Solution
• Kolmogorovs ?-entropy for compact classes.

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Repeat for medial axis

• Note
• The asymptotic bit rate for encoding a boundary
  curve ? depends only on the bit rate for ??.
• If we can recover an ?-approximation for ?? from
  medial quantities, we have the desired bit rate
  for the medial axis.

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Results

• All but three shapes are better modeled by the
  medial axis.

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Wonderful properties of the medial axis

• tangent angle to axis branch m.
• angle between tangent to m and outward
  normal to ?.
• Explicit formulas relating nth order quantities
  of one to the other.