Invariants concluded Lowe and Biederman

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Invariants (concluded) Lowe and Biederman
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Announcements

• No class Thursday. Attend Rao lecture.
• Double-check your paper assignments.

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Key Points

• Rigid rotation is 3x3 orthonormal matrix.
• 3-D Translation is 3x4 matrix.
• 3-D Translation Rotation is 3x4 matrix.
• Scaled Orthographic Projection Remove row three
  and allow scaling.
• Planar Object, remove column 3.
• Projective Transformations
• Rigid Rotation of Planar Object Represented by
  3x3 matrix.
• When we write in homogeneous coordinates,
  projection implicit.
• When we drop rigidity, 3x3 matrix is arbitrary.

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Projective
Rigid rotation and translation. Notation suggests
that first two columns are orthonormal, and
transformation has 6 degrees of freedom.
Projective Transformation Notation suggests that
transformation is unconstrained linear
transformation. Points in homogenous coordinates
are equivalent. Transformation has 8 degrees of
freedom, because its scale is arbitrary.
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Lines Parameterization

• Equation for line axbyc0.
• Parameterize line as l (a,b,c)T.
• p(x,y,1)T is on line if ltp,lgt0.

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Line Intersection

• The intersection of l and l is l x l (where x
  denotes the cross product).
• This follows from the fact that the cross product
  is orthogonal to both lines.

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Intersection of Parallel Lines

• Suppose l and l are parallel. We can write
  l(a,b,c), l (a,b,c). l x l
  (c-c)(b,-a,0). This equivalent to (b,-a,0).
• This point corresponds to a line through the
  focal point that doesnt intersect the image
  plane.
• We can think of the real plane as points (a,b,c)
  where c isnt equal to 0. When c 0, we say
  these points lie on the ideal line at infinity.
• Note that a projective transformation can map
  this to another line, the horizon, which we see.

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Invariants of Lines

• Notice that affine transformations are the
  subgroup of projective transformations in which
  the last row is (0, 0, 1).
• These map the line at infinity to itself.
• So parallel lines are affine invariants, since
  they continue to intersect at infinity.

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Invariance in 3D to 2D

• 3D to 2D Invariance isnt captured by
  mathematical definition of invariance because 3D
  to 2D transformations dont form a group.
• You cant compose or invert them.
• Definition Let f be a function on images. We
  say f is an invariant iff for every Object O, if
  I1 and I2 are images of O, f(I1)f(I2).
• This means we can define f(O) as f(I) for I any
  image of O. O and I match only if f(O)f(I).
• f is a non-trivial invariant if there exist two
  image I1 and I2 such that f(I1)f(I2).

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Non-Invariance in 3D to 2D

• Theorem Assume valid objects are any 3D point
  sets of size k, for some k. Then there are no
  non-trivial invariants of the images of these
  objects under perspective projection.

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Proof Strategy

• Let f be an invariant.
• Suppose two objects, A and B have a common image.
  Then f(I)f(J) if I and J are images of either A
  or B.
• Given any O0, Ok, we construct a series of
  objects, O1, , O(k-1), so that Oi and O(i1)
  have a common image for all i, and Ok and j have
  a common image.
• So for any pair of images, I, J, from any two
  objects, f(I) f(J).

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Constructing O1 Ok-1

• Oi has its first i points identical to the first
  i points of Ok, and the remaining points
  identical to the remaining points of O0.
• If two objects are identical except for one
  point, they produce the same image when viewed
  along a line joining those two points.
• Along that line, those two points look the same.
• The remaining points always look the same.

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Summary

• Planar objects give rise to rich set of
  invariants.
• 3-D objects have no invariants.
• We can deal with this by focusing on planar
  portions of objects.
• Or special restricted classes of objects.
• Or by relaxing notion of invariants.
• However, invariants have become less popular in
  computer vision due to these limitations.

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Lowe and Biederman

• Background
• Viewpoint Invariant Non-Accidental Properties.
• Lowe sees these as probabilistic.
• Biederman drops this.
• Primitive properties
• Composing them into units/geons.
• Use in Recognition.
• Speed search.
• Geons analogy to speech.
• Evidence for Value.
• Computational speed.
• Human psychology parts qualitative
  descriptions view invariance.

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Background

• Computational
• 2D approach to recognition.
• Lowe is reacting to Marr.
• Partly due to Lowe, recognition rarely involves
  reconstruction now. (But also 3D models more
  rare).
• State of the art
• Little recognition of 3D objects, grouping
  implicit.
• Speed, robustness a big concern.
• 2D recognition through search.
• Psychology
• Much more ambitious and specific than any prior
  theory of recognition (I believe).
• P.O. widely studied, rarely related to other
  tasks.
• Contrast.
• CS must account for low-level processing.
• Psych must account for categorization.

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Viewpoint Invariant NAPs

• Non-Accidental Property
• Happens rarely by chance
• More frequently by scene structure.
• p property, c chance, s structure.

Jepson and Richards consider this
Lowe focuses on this
This is high due to viewpoint invariance.

• Biederman downplays probabilistic inference.
• Not concerned with background, feature detection.

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Examples
(Copied from Lowe)
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Issues with Non-Accidental Properties

• Is it just Bayesian inference?
• Then why not model all information?
• This may fit Lowe
• Biederman relies more on certain inference.
• See also Feldman, Jepson, Richards.

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Viewpoint Invariance

• Match properties that are invariant to viewing
  conditions.
• Parallelism, symmetry, collinearity,
  cotermination, straightness.
• Lowe picks one side of property, Biederman
  stresses contrast. Why?
• How used?
• Lowe, correspondence of geometric features.
  Speed up search
• Description of parts for indexing.

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Geons

• Biederman, description of geons. Are they still
  view invariant when describing a geon?
• 3D shapes occluding contour depends on
  viewpoint. May be straight from one view, curved
  from another.
• Metric properties not truly invariant.
• Maybe more like quasi-invariants.

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Geons for Recognition

• Analogy to speech.
• 36 different geons.
• Different relations between them.
• Millions of ways of putting a few geons together.

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Empirical Support for Geons

• First, divide geons predictions
• Part structure is important in recognition.
• Perceptual grouping can be used for filling in.
• NAPs are used for indexing.
• View invariant descriptions.
• Qualitative descriptions.
• Second, what is alternative?
• View-based recognition with many examples.

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Empirical Support

• Recognition is fast. Fine metric judgments are
  slow.
• Does this disqualify other approaches?
• Recognition is view-invariant.
• Does this disqualify other approaches?
• Number of geon descriptions sufficient for number
  of categories we recognize.
• Argues plausibility, but no more.

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Empirical Support (2)

• 2-4 Geons needed for recognition. Complex
  objects no harder than simple ones.
• Line Drawings vs. Colored images. Color similar
  speed.

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Empirical Support (3) Degraded Objects

• Deleting contours that interfere with geon
  structure interferes more.
• Deleting Components worse than midsections.
• This argues for perceptual organization for
  interpolation/reconstruction. But for geons?
• Should we measure information deleted rather than
  contour length?

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Conclusions

• Maybe helpful to separate
• Perceptual organization/completion.
• View Invariance
• Part Structure.
• All three widely used in computer vision.
• Biedermans paper probably addresses
  view-invariance least.
• This became subject of much research.