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The Klein Bottle, a Continuous Dictionary for Distributions of High-Contrast Image Patches Jose Perea Mathematics Department, Stanford University June 25, 2010 Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June

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The Klein Bottle, a Continuous Dictionary for Distributions of High-Contrast Image Patches

Jose Perea

Mathematics Department, Stanford University

June 25, 2010

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 1 / 29

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The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk Geometric Inference Topological information = ⇒ Ingenuity = ⇒

  • Paramt. f ∈ Emb(T, Rn), f (T) ∼ X

Models: f ∈ Map(X, T)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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SLIDE 4

The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk Geometric Inference Topological information = ⇒ Ingenuity = ⇒

  • Paramt. f ∈ Emb(T, Rn), f (T) ∼ X

Models: f ∈ Map(X, T) Rewards

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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SLIDE 5

The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk Geometric Inference Topological information = ⇒ Ingenuity = ⇒

  • Paramt. f ∈ Emb(T, Rn), f (T) ∼ X

Models: f ∈ Map(X, T) Rewards

  • Better query strategies

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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SLIDE 6

The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk Geometric Inference Topological information = ⇒ Ingenuity = ⇒

  • Paramt. f ∈ Emb(T, Rn), f (T) ∼ X

Models: f ∈ Map(X, T) Rewards

  • Better query strategies
  • Solid theoretical

framework

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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SLIDE 7

The Data Analysis Pipeline

Topological Inference Point cloud X ⊆ Rn = ⇒ Persistent homology = ⇒ Betti numbers β1, . . . , βk Geometric Inference Topological information = ⇒ Ingenuity = ⇒

  • Paramt. f ∈ Emb(T, Rn), f (T) ∼ X

Models: f ∈ Map(X, T) Rewards

  • Better query strategies
  • Solid theoretical

framework

  • Your favorite

applications...

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29

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The Data Analysis Pipeline

A good examples... Natural Images On the local behavior of spaces of natural images, Carlsson et al. [08]

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29

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The Data Analysis Pipeline

A good examples... Natural Images On the local behavior of spaces of natural images, Carlsson et al. [08]

  • X ⊆ R9 : random sample of high-contrast 3x3 pixel patches from

natural scenes. |X| = 8 · 106.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29

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The Data Analysis Pipeline

A good examples... Natural Images On the local behavior of spaces of natural images, Carlsson et al. [08]

  • X ⊆ R9 : random sample of high-contrast 3x3 pixel patches from

natural scenes. |X| = 8 · 106.

  • After mean centering, contrast normalization and a linear change of

coordinates, can regard X ⊆ S7.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29

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The Data Analysis Pipeline

A good examples... Natural Images On the local behavior of spaces of natural images, Carlsson et al. [08]

  • X ⊆ R9 : random sample of high-contrast 3x3 pixel patches from

natural scenes. |X| = 8 · 106.

  • After mean centering, contrast normalization and a linear change of

coordinates, can regard X ⊆ S7.

  • 50% of the points in X have the topology of a Klein bottle, modeled

by the space K = {p(x, y) = c(ax + by) + d(ax + by)2, a2 + b2 = c2 + d2 = 1}.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29

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Rewards

As for rewards... Today:

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Rewards

As for rewards... Today: We’ll exploit the fact: K ∼ Natural space for distributions of high-contrast patches.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Rewards

As for rewards... Today: We’ll exploit the fact: K ∼ Natural space for distributions of high-contrast patches.

  • Let I be an image and Sk(I) a random sample of its high-contrast

k × k patches. Center them by subtracting their mean and normalize them by their D-norm.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Rewards

As for rewards... Today: We’ll exploit the fact: K ∼ Natural space for distributions of high-contrast patches.

  • Let I be an image and Sk(I) a random sample of its high-contrast

k × k patches. Center them by subtracting their mean and normalize them by their D-norm.

  • For small k, Sk(I) is contained within a small tubular neighborhood
  • f K. By means of projection, we get a random sample S′

k(I) ⊆ K.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Rewards

As for rewards... Today: We’ll exploit the fact: K ∼ Natural space for distributions of high-contrast patches.

  • Let I be an image and Sk(I) a random sample of its high-contrast

k × k patches. Center them by subtracting their mean and normalize them by their D-norm.

  • For small k, Sk(I) is contained within a small tubular neighborhood
  • f K. By means of projection, we get a random sample S′

k(I) ⊆ K.

Idea: If hk(I) : K → R is the underlying PDF, Fourier Analysis on L2(K) yields a compact representation vk(I) of hk(I).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Rewards

As for rewards... Today: We’ll exploit the fact: K ∼ Natural space for distributions of high-contrast patches.

  • Let I be an image and Sk(I) a random sample of its high-contrast

k × k patches. Center them by subtracting their mean and normalize them by their D-norm.

  • For small k, Sk(I) is contained within a small tubular neighborhood
  • f K. By means of projection, we get a random sample S′

k(I) ⊆ K.

Idea: If hk(I) : K → R is the underlying PDF, Fourier Analysis on L2(K) yields a compact representation vk(I) of hk(I). Application: Texture discrimination and classification via {vk(I)}k.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29

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Outline

Outline

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K). 3 Estimating the coefficients vk(I) (w.r.t B) of hk(I).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K). 3 Estimating the coefficients vk(I) (w.r.t B) of hk(I). 4 The v-Invariant v(I) = (vk(I))k. Examples.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K). 3 Estimating the coefficients vk(I) (w.r.t B) of hk(I). 4 The v-Invariant v(I) = (vk(I))k. Examples. 5 Dissimilarity measures.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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SLIDE 24

Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K). 3 Estimating the coefficients vk(I) (w.r.t B) of hk(I). 4 The v-Invariant v(I) = (vk(I))k. Examples. 5 Dissimilarity measures. 6 Modeling image rotation.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Outline

Outline

1 Projection onto the Klein bottle K. 2 An orthonormal basis B for L2(K). 3 Estimating the coefficients vk(I) (w.r.t B) of hk(I). 4 The v-Invariant v(I) = (vk(I))k. Examples. 5 Dissimilarity measures. 6 Modeling image rotation.

All results were obtained using the MATLAB R2009b software.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29

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Projection onto K

Projection onto K

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 6 / 29

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Projection onto K

K = {p(x, y) = c(ax + by) + d(ax + by)2, a2 + b2 = c2 + d2 = 1}

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 7 / 29

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Projection onto K

K = {p(x, y) = c(ax + by) + d(ax + by)2, a2 + b2 = c2 + d2 = 1}

  • Horizontal coordinate:

predominant direction.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 7 / 29

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Projection onto K

K = {p(x, y) = c(ax + by) + d(ax + by)2, a2 + b2 = c2 + d2 = 1}

  • Horizontal coordinate:

predominant direction.

  • Vertical component:

linear versus quadratic contribution.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 7 / 29

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Projection onto K

K = {p(x, y) = c(ax + by) + d(ax + by)2, a2 + b2 = c2 + d2 = 1}

  • Horizontal coordinate:

predominant direction.

  • Vertical component:

linear versus quadratic contribution.

  • Given f : [−1, 1]2 → R,

its direction θ ∈ [0, π) should be so that f is, in average, as constant as possible along the (− sin θ, cos θ) direction.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 7 / 29

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Projection onto K

Theorem Let f : [−1, 1]2 − → R be differentiable and let Qf : R2 − → R be Qf (v) =

  • [−1,1]2

∇f , v2dxdy. Then:

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 8 / 29

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Projection onto K

Theorem Let f : [−1, 1]2 − → R be differentiable and let Qf : R2 − → R be Qf (v) =

  • [−1,1]2

∇f , v2dxdy. Then:

1 If f (x, y) = g(ax + by), a2 + b2 = 1, then Qf (a, b) ≥ Qf (v) for all

v = 1.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 8 / 29

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Projection onto K

Theorem Let f : [−1, 1]2 − → R be differentiable and let Qf : R2 − → R be Qf (v) =

  • [−1,1]2

∇f , v2dxdy. Then:

1 If f (x, y) = g(ax + by), a2 + b2 = 1, then Qf (a, b) ≥ Qf (v) for all

v = 1.

2 Let j : S1 → RP1 be the quotient map and assume the eigenvalues of

Af , the matrix representing Qf , are distinct. Then the direction map Dir(f ) = j

  • arg max

v=1 Qf (v)

  • is well defined and continuous in the C 1-topology.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 8 / 29

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Projection onto K

Theorem Let f : [−1, 1]2 − → R be differentiable and let Qf : R2 − → R be Qf (v) =

  • [−1,1]2

∇f , v2dxdy. Then:

1 If f (x, y) = g(ax + by), a2 + b2 = 1, then Qf (a, b) ≥ Qf (v) for all

v = 1.

2 Let j : S1 → RP1 be the quotient map and assume the eigenvalues of

Af , the matrix representing Qf , are distinct. Then the direction map Dir(f ) = j

  • arg max

v=1 Qf (v)

  • is well defined and continuous in the C 1-topology. Moreover, the

maximum is attained at the eigenvectors of Af with largest eigenvalue.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 8 / 29

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Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

1 Let QP : R2 −

→ R be as before, and let λ1 ≤ λ2 be its eigenvalues.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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SLIDE 37

Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

1 Let QP : R2 −

→ R be as before, and let λ1 ≤ λ2 be its eigenvalues.

2 If λ2 − λ1 ≥ tk (for an experimentally determined threshold tk) let

(a, b) be a unitary vector in the eigenspace Eλ2. Direction.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

1 Let QP : R2 −

→ R be as before, and let λ1 ≤ λ2 be its eigenvalues.

2 If λ2 − λ1 ≥ tk (for an experimentally determined threshold tk) let

(a, b) be a unitary vector in the eigenspace Eλ2. Direction.

3 (ax + by) and (ax + by)2 are perpendicular with respect to

f , g =

  • [−1,1]2

f (x, y)g(x, y)dxdy

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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SLIDE 39

Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

1 Let QP : R2 −

→ R be as before, and let λ1 ≤ λ2 be its eigenvalues.

2 If λ2 − λ1 ≥ tk (for an experimentally determined threshold tk) let

(a, b) be a unitary vector in the eigenspace Eλ2. Direction.

3 (ax + by) and (ax + by)2 are perpendicular with respect to

f , g =

  • [−1,1]2

f (x, y)g(x, y)dxdy so one can write the degree 2 polynomial approximation P ∼ r1(ax + by) + r2(ax + by)2 where r1 = P,ax+by

ax+by2 and r2 = P,(ax+by)2 (ax+by)22 . Let r = (r1, r2).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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SLIDE 40

Projection onto K

Projection P → c(ax + by) + d(ax + by)2, discrete case... Given a k × k patch P : [−1, 1]2 − → R, approximate ∇P and

  • · dxdy.

1 Let QP : R2 −

→ R be as before, and let λ1 ≤ λ2 be its eigenvalues.

2 If λ2 − λ1 ≥ tk (for an experimentally determined threshold tk) let

(a, b) be a unitary vector in the eigenspace Eλ2. Direction.

3 (ax + by) and (ax + by)2 are perpendicular with respect to

f , g =

  • [−1,1]2

f (x, y)g(x, y)dxdy so one can write the degree 2 polynomial approximation P ∼ r1(ax + by) + r2(ax + by)2 where r1 = P,ax+by

ax+by2 and r2 = P,(ax+by)2 (ax+by)22 . Let r = (r1, r2). 4 If r ≥ sk (threshold) let (c, d) = r/r.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 9 / 29

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An orthonormal basis B for L2(K)

An orthonormal basis for L2(K)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 10 / 29

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An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Then the orbit space is the usual model K = [0, π] × [0, 2π]/ ∼ for the Klein bottle.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Then the orbit space is the usual model K = [0, π] × [0, 2π]/ ∼ for the Klein bottle.

  • The map

ϕ : S1 × S1 − → K (α, β) → pα,β pα,β(x, y) = c(ax + by) + d(ax + by)2, (here a + bi = eiα) is µ-invariant (ϕ ◦ µ(g, z, w) = ϕ(z, w)) and induces a homeomorphism ϕ∗ : K − → K from the orbit space.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Then the orbit space is the usual model K = [0, π] × [0, 2π]/ ∼ for the Klein bottle.

  • The map

ϕ : S1 × S1 − → K (α, β) → pα,β pα,β(x, y) = c(ax + by) + d(ax + by)2, (here a + bi = eiα) is µ-invariant (ϕ ◦ µ(g, z, w) = ϕ(z, w)) and induces a homeomorphism ϕ∗ : K − → K from the orbit space. Samples on K ⇒ samples on K = [0, π] × [0, 2π]/ ∼.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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SLIDE 46

An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Then the orbit space is the usual model K = [0, π] × [0, 2π]/ ∼ for the Klein bottle.

  • The map

ϕ : S1 × S1 − → K (α, β) → pα,β pα,β(x, y) = c(ax + by) + d(ax + by)2, (here a + bi = eiα) is µ-invariant (ϕ ◦ µ(g, z, w) = ϕ(z, w)) and induces a homeomorphism ϕ∗ : K − → K from the orbit space. Samples on K ⇒ samples on K = [0, π] × [0, 2π]/ ∼.

  • Fourier Analysis on L2(S1 × S1) ⇒ orthonormal basis for L2(K).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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SLIDE 47

An orthonormal basis B for L2(K)

  • Let µ be the Z/2Z-action on S1 × S1 given by 1 · (z, w) = (−z, w).

Then the orbit space is the usual model K = [0, π] × [0, 2π]/ ∼ for the Klein bottle.

  • The map

ϕ : S1 × S1 − → K (α, β) → pα,β pα,β(x, y) = c(ax + by) + d(ax + by)2, (here a + bi = eiα) is µ-invariant (ϕ ◦ µ(g, z, w) = ϕ(z, w)) and induces a homeomorphism ϕ∗ : K − → K from the orbit space. Samples on K ⇒ samples on K = [0, π] × [0, 2π]/ ∼.

  • Fourier Analysis on L2(S1 × S1) ⇒ orthonormal basis for L2(K).
  • Let L2(K, C) be the set of µ-invariant functions f : S1 × S1 −

→ C so that f , f K = 1 (2π)2 2π π f (x, y) · f (x, y)dxdy < ∞.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 11 / 29

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SLIDE 48

An orthonormal basis B for L2(K)

Theorem Let f ∈ L2(K, R) be a PDF. Then

N

  • m=0

N

  • n=−N
  • f (n, m)φnm

= 1 2π2 +

N

  • m=1

am(2 cos my) +

N0

  • n=1

bn(2 cos 2nx) + cn(2 sin 2nx) +

N

  • n,m=1

dnm

  • 2

√ 2 cos(nx) · sin

  • my + π

4 (1 + (−1)n)

  • +

N

  • n,m=1

enm

  • 2

√ 2 sin(nx) · sin

  • my + π

4 (1 + (−1)n)

  • converges to f in the L2-norm as N → ∞.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 12 / 29

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SLIDE 49

An orthonormal basis B for L2(K)

Theorem Let f ∈ L2(K, R) be a PDF. Then

N

  • m=0

N

  • n=−N
  • f (n, m)φnm

= 1 2π2 +

N

  • m=1

am(2 cos my) +

N0

  • n=1

bn(2 cos 2nx) + cn(2 sin 2nx) +

N

  • n,m=1

dnm

  • 2

√ 2 cos(nx) · sin

  • my + π

4 (1 + (−1)n)

  • +

N

  • n,m=1

enm

  • 2

√ 2 sin(nx) · sin

  • my + π

4 (1 + (−1)n)

  • converges to f in the L2-norm as N → ∞. Moreover, the functions on the

right hand side make up an orthonormal basis B for L2(K, R).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 12 / 29

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SLIDE 50

Estimating the Coefficients vk (I) of hk (I) w.r.t. B

Estimating the Coefficients vk(I) of hk(I)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 13 / 29

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SLIDE 51

Estimating the Coefficients vk (I) of hk (I) w.r.t. B

Let {P1, . . . , PN} ⊆ K, with Pr = (Xr, Yr), be the projection of a random sample of high-contrast k × k patches from an image I.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 14 / 29

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SLIDE 52

Estimating the Coefficients vk (I) of hk (I) w.r.t. B

Let {P1, . . . , PN} ⊆ K, with Pr = (Xr, Yr), be the projection of a random sample of high-contrast k × k patches from an image I. For each n, m ∈ N we have unbiased estimators

  • am

= 1 2Nπ2

N

  • r=1

cos(mYr)

  • bn

= 1 2Nπ2

N

  • r=1

cos(2nXr) , cn = 1 2Nπ2

N

  • r=1

sin(2nXr)

  • dnm

= 1 √ 2Nπ2

N

  • r=1

cos(nXr) · sin

  • mYr + π

4 (1 + (−1)n)

  • enm

= 1 √ 2Nπ2

N

  • r=1

sin(nXr) · sin

  • mYr + π

4 (1 + (−1)n)

  • converging a.s. to the coefficients w.r.t B, of the underlying PDF

hk(I) : K − → R.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 14 / 29

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SLIDE 53

The v-Invariant v(I) = (vk (I))k

The v-Invariant

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 15 / 29

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SLIDE 54

The v-Invariant v(I) = (vk (I))k

Let I be a digital image in gray scale, and for each k = 3, 9, 15, 21, 27:

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 16 / 29

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SLIDE 55

The v-Invariant v(I) = (vk (I))k

Let I be a digital image in gray scale, and for each k = 3, 9, 15, 21, 27:

1 Let S′ k(I) be a random sample of size 60,000 of its k × k patches.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 16 / 29

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SLIDE 56

The v-Invariant v(I) = (vk (I))k

Let I be a digital image in gray scale, and for each k = 3, 9, 15, 21, 27:

1 Let S′ k(I) be a random sample of size 60,000 of its k × k patches. 2 Keep the top 30% highest-contrast patches from S′ k(I) and let

Sk(I) ⊆ K be the resulting projection.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 16 / 29

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SLIDE 57

The v-Invariant v(I) = (vk (I))k

Let I be a digital image in gray scale, and for each k = 3, 9, 15, 21, 27:

1 Let S′ k(I) be a random sample of size 60,000 of its k × k patches. 2 Keep the top 30% highest-contrast patches from S′ k(I) and let

Sk(I) ⊆ K be the resulting projection.

3 From the random sample Sk(I), let

vk(I) = ( a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) be the estimated coefficients of hk(I) w.r.t B. vk(I) ∈ R40 (degree ≤ 6) has shown good results.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 16 / 29

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SLIDE 58

The v-Invariant v(I) = (vk (I))k

Let I be a digital image in gray scale, and for each k = 3, 9, 15, 21, 27:

1 Let S′ k(I) be a random sample of size 60,000 of its k × k patches. 2 Keep the top 30% highest-contrast patches from S′ k(I) and let

Sk(I) ⊆ K be the resulting projection.

3 From the random sample Sk(I), let

vk(I) = ( a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) be the estimated coefficients of hk(I) w.r.t B. vk(I) ∈ R40 (degree ≤ 6) has shown good results. Definition The v-Invariant for I at scale k is the vector vk(I).

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 16 / 29

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SLIDE 59

A First Example, Straw

Examples

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 17 / 29

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SLIDE 60

A First Example, Straw

v3(Straw) = (v1, . . . , v40)

= ⇒

Projected sample

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 18 / 29

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SLIDE 61

A First Example, Straw

v3(Straw) = (v1, . . . , v40)

= ⇒

Projected sample

⇐ ⇒

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 18 / 29

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SLIDE 62

A First Example, Straw

A better visualization: Let vk(I) = ( a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) and let hk(I) : K − → R be the function whose coefficients w.r.t. B are:

  • a1, . . . ,

am, bn, cn, dnm, enm, . . . , eRS, 0 . . .

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 19 / 29

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SLIDE 63

A First Example, Straw

A better visualization: Let vk(I) = ( a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) and let hk(I) : K − → R be the function whose coefficients w.r.t. B are:

  • a1, . . . ,

am, bn, cn, dnm, enm, . . . , eRS, 0 . . . The Heat-map for hk(I) ⇐ ⇒

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 19 / 29

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SLIDE 64

Examples from Real World Textures Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 20 / 29

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SLIDE 65

Dissimilarity Measures

Dissimilarity Measures

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 21 / 29

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SLIDE 66

Dissimilarity Measures

L2-like distance Let d2(I, J) =

  • 5
  • i=1
  • hki(I) −

hki(J)

  • 2

L2

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 22 / 29

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SLIDE 67

Dissimilarity Measures

L2-like distance Let d2(I, J) =

  • 5
  • i=1
  • hki(I) −

hki(J)

  • 2

L2 =

  • 5
  • i=1
  • vki(I) − vki(J)
  • 2

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 22 / 29

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SLIDE 68

Dissimilarity Measures

L2-like distance Let d2(I, J) =

  • 5
  • i=1
  • hki(I) −

hki(J)

  • 2

L2 =

  • 5
  • i=1
  • vki(I) − vki(J)
  • 2

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 22 / 29

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SLIDE 69

Dissimilarity Measures

The Evolution function: ε(I) =

4

  • i=1

vki(I) − vki+1(I)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 23 / 29

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SLIDE 70

Dissimilarity Measures

The Evolution function: ε(I) =

4

  • i=1

vki(I) − vki+1(I)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 23 / 29

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SLIDE 71

Image Rotation

Image Rotation

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 24 / 29

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SLIDE 72

Image Rotation

Let I τ be the image obtained from I by a rotation of τ degrees. I = ⇒ I

π 2 Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 25 / 29

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SLIDE 73

Image Rotation

Let I τ be the image obtained from I by a rotation of τ degrees. I = ⇒ I

π 2

Proposition Let T ∈ SL2(R) be rotation by τ ∈ [−π, π] and let vk(I) = (a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) vk(I τ) = (aτ

1, . . . , aτ m, bτ n, cτ n , dτ nm, eτ nm, . . . , eτ RS)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 25 / 29

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SLIDE 74

Image Rotation

Let I τ be the image obtained from I by a rotation of τ degrees. I = ⇒ I

π 2

Proposition Let T ∈ SL2(R) be rotation by τ ∈ [−π, π] and let vk(I) = (a1, . . . , am, bn, cn, dnm, enm, . . . , eRS) vk(I τ) = (aτ

1, . . . , aτ m, bτ n, cτ n , dτ nm, eτ nm, . . . , eτ RS)

Then hk(I τ)(x, y) = hk(I)(x − τ, y) for all (x, y) ∈ K and aτ

m = am

, cτ

n

dτ

n

  • = T 2n

cn dn , dτ

nm

eτ

nm

  • = T n

dnm enm .

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 25 / 29

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SLIDE 75

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 76

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 77

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric. Minimizers τ ∗ for (1) come from roots of a complex polynomial, depending only on {vk(I)}k and {vk(J)}k.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 78

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric. Minimizers τ ∗ for (1) come from roots of a complex polynomial, depending only on {vk(I)}k and {vk(J)}k. Applications...

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 79

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric. Minimizers τ ∗ for (1) come from roots of a complex polynomial, depending only on {vk(I)}k and {vk(J)}k. Applications...

  • Rotation invariance of vk(·)

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 80

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric. Minimizers τ ∗ for (1) come from roots of a complex polynomial, depending only on {vk(I)}k and {vk(J)}k. Applications...

  • Rotation invariance of vk(·)
  • Compute relative rotation angle

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 81

Image Rotation

Yields a way of computing d2(I τ, J) =

  • 5
  • i=1

vki(I τ) − vki(J)2 Theorem The R-distance dR(I, J) = min

τ

d2(I τ, J) (1) defines a (pseudo-)metric. Minimizers τ ∗ for (1) come from roots of a complex polynomial, depending only on {vk(I)}k and {vk(J)}k. Applications...

  • Rotation invariance of vk(·)
  • Compute relative rotation angle
  • Classification of images with unknown viewpoint

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 26 / 29

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SLIDE 82

Image Rotation

Unknown viewpoint... Several texture classes

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 27 / 29

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SLIDE 83

Image Rotation

Unknown viewpoint... Several texture classes For each texture class: apply surface rotation, unknown light source a total of 56 (or 28) poses per texture class.

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 27 / 29

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SLIDE 84

Image Rotation

Computing the pairwise distance dR(I, J) between the 252 images:

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 28 / 29

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SLIDE 85

Conclusion

Two Ideas

1 Distributions on the Klein bottle have been around for quite a long

time in disguise (responses to filter banks, texton dictionaries), with no mathematical framework, but good results. The v-Invariant gives both the mathematical framework and a novel, compact representation.

2 The topological data analysis story is not complete without geometric

inference: It’d be great to have a framework for going from topological inference to geometric models!

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 29 / 29

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SLIDE 86

Conclusion

Two Ideas

1 Distributions on the Klein bottle have been around for quite a long

time in disguise (responses to filter banks, texton dictionaries), with no mathematical framework, but good results. The v-Invariant gives both the mathematical framework and a novel, compact representation.

2 The topological data analysis story is not complete without geometric

inference: It’d be great to have a framework for going from topological inference to geometric models!

Thank you!

Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 29 / 29