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
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Geometric Inference Paramt. f ∈ Emb ( T , R n ), f ( T ) ∼ X Topological information = ⇒ Ingenuity = ⇒ Models: f ∈ Map ( X , T ) Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Geometric Inference Paramt. f ∈ Emb ( T , R n ), f ( T ) ∼ X Topological information = ⇒ Ingenuity = ⇒ Models: f ∈ Map ( X , T ) Rewards Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Geometric Inference Paramt. f ∈ Emb ( T , R n ), f ( T ) ∼ X Topological information = ⇒ Ingenuity = ⇒ Models: f ∈ Map ( X , T ) Rewards • Better query strategies Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 2 / 29
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Geometric Inference Paramt. f ∈ Emb ( T , R n ), f ( T ) ∼ X Topological information = ⇒ Ingenuity = ⇒ 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
The Data Analysis Pipeline Topological Inference Point cloud Betti numbers X ⊆ R n = ⇒ Persistent homology = ⇒ β 1 , . . . , β k Geometric Inference Paramt. f ∈ Emb ( T , R n ), f ( T ) ∼ X Topological information = ⇒ Ingenuity = ⇒ 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
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
The Data Analysis Pipeline A good examples... Natural Images On the local behavior of spaces of natural images , Carlsson et al. [08] • X ⊆ R 9 : random sample of high-contrast 3x3 pixel patches from natural scenes. | X | = 8 · 10 6 . Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29
The Data Analysis Pipeline A good examples... Natural Images On the local behavior of spaces of natural images , Carlsson et al. [08] • X ⊆ R 9 : random sample of high-contrast 3x3 pixel patches from natural scenes. | X | = 8 · 10 6 . • After mean centering, contrast normalization and a linear change of coordinates, can regard X ⊆ S 7 . Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29
The Data Analysis Pipeline A good examples... Natural Images On the local behavior of spaces of natural images , Carlsson et al. [08] • X ⊆ R 9 : random sample of high-contrast 3x3 pixel patches from natural scenes. | X | = 8 · 10 6 . • After mean centering, contrast normalization and a linear change of coordinates, can regard X ⊆ S 7 . • 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 , a 2 + b 2 = c 2 + d 2 = 1 } . Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 3 / 29
Rewards As for rewards... Today: Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29
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
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 S k ( 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
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 S k ( 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 , S k ( I ) is contained within a small tubular neighborhood of 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
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 S k ( 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 , S k ( I ) is contained within a small tubular neighborhood of K . By means of projection, we get a random sample S ′ k ( I ) ⊆ K . Idea: If h k ( I ) : K → R is the underlying PDF, Fourier Analysis on L 2 ( K ) yields a compact representation v k ( I ) of h k ( I ). Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29
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 S k ( 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 , S k ( I ) is contained within a small tubular neighborhood of K . By means of projection, we get a random sample S ′ k ( I ) ⊆ K . Idea: If h k ( I ) : K → R is the underlying PDF, Fourier Analysis on L 2 ( K ) yields a compact representation v k ( I ) of h k ( I ). Application: Texture discrimination and classification via { v k ( I ) } k . Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 4 / 29
Outline Outline Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). 3 Estimating the coefficients v k ( I ) (w.r.t B ) of h k ( I ). Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). 3 Estimating the coefficients v k ( I ) (w.r.t B ) of h k ( I ). 4 The v-Invariant v ( I ) = ( v k ( I )) k . Examples. Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). 3 Estimating the coefficients v k ( I ) (w.r.t B ) of h k ( I ). 4 The v-Invariant v ( I ) = ( v k ( I )) k . Examples. 5 Dissimilarity measures. Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 5 / 29
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). 3 Estimating the coefficients v k ( I ) (w.r.t B ) of h k ( I ). 4 The v-Invariant v ( I ) = ( v k ( 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
Outline Outline 1 Projection onto the Klein bottle K . 2 An orthonormal basis B for L 2 ( K ). 3 Estimating the coefficients v k ( I ) (w.r.t B ) of h k ( I ). 4 The v-Invariant v ( I ) = ( v k ( 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
Projection onto K Projection onto K Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 6 / 29
Projection onto K K = { p ( x , y ) = c ( ax + by ) + d ( ax + by ) 2 , a 2 + b 2 = c 2 + d 2 = 1 } Jose Perea (Stanford University) The Klein Bottle, a Continuous Dictionary June 25, 2010 7 / 29
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