[PPT] - Sufficient conditions for topological invariance of 2D images under PowerPoint Presentation

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Problem statement Definitions Methods Experiments Conclusion

Sufficient conditions for topological invariance

f 2D images under rigid transformations

Phuc NGO Yukiko KENMOCHI Nicolas PASSAT Hugues TALBOT

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 1/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Rigid transformation

Rigid transformation is a function Tabθ : R2 → R2, such that p′ q′

=

p cos θ − q sin θ + a p sin θ + q cos θ + b

where a, b ∈ R, θ ∈ [0, 2π[ and (p, q), (p′, q′) ∈ R2.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 2/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Topology issues

Problem statement Topology is not preserved due to digitization process from R2 to Z2.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 3/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Digital rigid transformation

Digital rigid transformation is a function Tabθ : Z2 → Z2 such that p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

where a, b ∈ R, θ ∈ [0, 2π[ and (p, q), (p′, q′) ∈ Z2.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 4/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Geometrical alteration: Distance and angle

Distance Before After 1 √ 2 Angle Before After 90◦ 135◦

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 5/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Geometrical alteration: Distance and angle

Distance Before After 1 √ 2 √ 2 2 Angle Before After 90◦ 135◦ 90◦ 90◦

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 5/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Topological alteration

The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a a b b b c c c d d d a c a c b d d b e a b c d

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Topological alteration

The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a a b b b c c c d d d a c a c b d d b e a b c d

a b c d a b c d

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Topological alteration

The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a a b b b c c c d d d a c a c b d d b e a b c d

a b c d a b c d

a c b d e a b c d

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Topological alteration

The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a a b b b c c c d d d a c a c b d d b e a b c d

a b c d a b c d a b c d a b c d a b c d a b c d

a c b d e a b c d

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Motivations

Our questions Do binary images exist that preserve their topology under any rigid transformations?

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 7/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Motivations

Our questions Do binary images exist that preserve their topology under any rigid transformations? Can we verify topological preservation for a given image?

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 7/22

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Problem statement Definitions Methods Experiments Conclusion (Digital) Rigid Transformations Geometrical and Topological Issues

Motivations

Our questions Do binary images exist that preserve their topology under any rigid transformations? Can we verify topological preservation for a given image? What the conditions for images to preserve their topology?

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 7/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Topological invariance

Definition An image I is topologically invariant if its transformed images under any rigid transformations have the same homotopy-type as I.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 8/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Topological invariance

Definition An image I is topologically invariant if its transformed images under any rigid transformations have the same homotopy-type as I.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 8/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Topological invariance

Definition An image I is topologically invariant if its transformed images under any rigid transformations have the same homotopy-type as I.

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Tools for topological invariance verification

Methodological tools DRT graph, and

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Tools for topological invariance verification

Methodological tools DRT graph, and simple points.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 9/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

A critical rigid transformation moves at least one integer point into either a vertical or a horizontal half-grid point.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Critical transformation

Digital rigid transformation Tabθ : Z2 → Z2 p′ q′

=

[p cos θ − q sin θ + a] [p sin θ + q cos θ + b]

Definition

Tipping surfaces are the surfaces associated to the critical transformations in the parameter space (a, b, θ).

Φpqk :

R2 − → R (b, θ) − → a = k + 1

2 + q sin θ − p cos θ

(vertical)

Ψpql :

R2 − → R (a, θ) − → b = l + 1

2 − p sin θ − q cos θ

(horizontal) for p, q, k, l ∈ Z.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 10/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Example of tipping surfaces

(a) Tipping surfaces (b) Orthogonal sections All tipping surfaces Φpqk and Ψpql with p, q ∈ [0, 4] and k, l ∈ [0, 5].

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 11/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

DRT graph

Definition A discrete rigid transformation graph (DRT graph) is a graph G = (V , E) such that each vertex in V corresponds to a DRT, each edge in E connects two vertices sharing a tipping surface.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 12/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion Topological Invariance DRT Graph Simple Point

Simple point

Definition A simple point is a pixel whose value can be modified without changing the digital topology of the associated image.

x y t z The simplicity can be tested locally in constant time using 3× 3 neighbourhood.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 13/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Principles of the naive method

We use DRT graph to generate incrementally all transformed images,

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Principles of the naive method

We use DRT graph to generate incrementally all transformed images, simplicity test to verify the topological invariance.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 14/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Principles of the naive method

We use DRT graph to generate incrementally all transformed images, simplicity test to verify the topological invariance.

The DRT graph has a high complexity of O(N9), for N × N is the image size.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 14/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Local property

Property Let p ∈ Z2 be a pixel and q ∈ N8(p), we have T −1(q) ∈ N20(T −1(p)), N8(p) N20(T −1(p))

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 15/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Method using local configurations

Algorithm generate all 20-neighborhood samples;

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Method using local configurations

Algorithm generate all 20-neighborhood samples; generate the set P of the samples whose center point is topologically invariant;

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Method using local configurations

Algorithm generate all 20-neighborhood samples; generate the set P of the samples whose center point is topologically invariant; verify the topological invariance for each pixel of the given image using P.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 16/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Method using local configurations

Algorithm generate all 20-neighborhood samples; generate the set P of the samples whose center point is topologically invariant; verify the topological invariance for each pixel of the given image using P.

The verification is in linear time with respect to the image size.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 16/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Samples in the Look-Up-Tables

Among 124 260 binary configurations up to symmetries, there are 10 644 samples in P4 for (4, 8)-adjacency.

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Samples in the Look-Up-Tables

Among 124 260 binary configurations up to symmetries, there are 10 644 samples in P4 for (4, 8)-adjacency. 10 752 samples in P8 for (8, 4)-adjacency.

DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 17/22

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Problem statement Definitions Methods Experiments Conclusion DRT Graph Approach LUT Approach

Samples in the Look-Up-Tables

Among 124 260 binary configurations up to symmetries, there are 10 644 samples in P4 for (4, 8)-adjacency. 10 752 samples in P8 for (8, 4)-adjacency.

Sufficient conditions of topologically invariant images.

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for half-planes

All centered half-planes generated in images of size 21 × 21.

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

Digitized disks of radius 11 whose center is anywhere inside the center pixel.

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

Digitized disks of radius 11 whose center is anywhere inside the center pixel.

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for disks

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological invariance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological invariance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological invariance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological invariance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion Half-planes Disks Letters

Experiment for letters: Topological variance

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Problem statement Definitions Methods Experiments Conclusion

Conclusion

Results: study on the relationships between geometry and topology in the framework of computer imagery, proposition of sufficient conditions and a linear algorithm to evaluate the topological invariance of digital images, classification of points in binary images.

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Problem statement Definitions Methods Experiments Conclusion

Conclusion

Perspectives: repairing images to be topologically invariant, proving the necessary conditions, finding topologically invariant conditions, similarly to regularity for digitization, extending to well-composed sets, labelled images, etc.

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Problem statement Definitions Methods Experiments Conclusion

Thank you for your attention

We welcome your questions, suggestions and comments! I’m looking for a post-doc position

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Problem statement Definitions Methods Experiments Conclusion

Annexes

5
4
3
2
1

1 2 3 4 5

5
4
3
2
1

1 2 3 4 5

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Problem statement Definitions Methods Experiments Conclusion

Annexes

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