invariants (concluded); lowe and biederman. announcements no class thursday. attend rao lecture....
TRANSCRIPT
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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
11
02,31,3
2,21,2
2,11,1
3,32,31,3
3,22,21,2
3,12,11,1
y
x
trr
trr
trry
x
trrr
trrr
trrr
z
y
x
z
y
x
Rigid rotation and translation.
Notation suggests that first two columns are orthonormal, and
transformation has 6 degrees of freedom.
1111
1
wvwu
w
v
u
y
x
hg
fed
cba
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: ax+by+c=0.• Parameterize line as l = (a,b,c)T.
• p=(x,y,1)T is on line if <p,l>=0.
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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 doesn’t intersect the image plane.
• We can think of the real plane as points (a,b,c) where c isn’t 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” isn’t captured by mathematical definition of invariance because 3D to 2D transformations don’t form a group.– You can’t 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(i+1) 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.
)|()|(
)()|(
)(
)()|()|(
cpPspP
sPspP
pP
sPspPpsP
Lowe focuses on this
Jepson and Richards consider this
• Biederman downplays probabilistic inference.
•Not concerned with background, feature detection.
This is high due to viewpoint invariance.
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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 shape’s 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.• Biederman’s paper probably addresses
view-invariance least.– This became subject of much research.