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

Problem statement Definitions Methods Experiments Conclusion Sufficient conditions for topological invariance of 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

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Rigid transformation Rigid transformation is a function T ab θ : R 2 → R 2 , such that � p ′ � p cos θ − q sin θ + a � � = q ′ p sin θ + q cos θ + b where a , b ∈ R , θ ∈ [0 , 2 π [ and ( p , q ) , ( p ′ , q ′ ) ∈ R 2 . DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 2/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Topology issues Problem statement Topology is not preserved due to digitization process from R 2 to Z 2 . DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 3/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Digital rigid transformation Digital rigid transformation is a function T ab θ : Z 2 → Z 2 such that � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ p sin θ + q cos θ + b ] where a , b ∈ R , θ ∈ [0 , 2 π [ and ( p , q ) , ( p ′ , q ′ ) ∈ Z 2 . DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 4/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Geometrical alteration: Distance and angle Distance Angle Before After Before After √ 1 2 90 ◦ 135 ◦ DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 5/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Geometrical alteration: Distance and angle Distance Angle Before After Before After √ 1 2 90 ◦ 135 ◦ √ 2 2 90 ◦ 90 ◦ DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 5/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Topological alteration The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a c a a a b a b b c e b d b c d b c d c d c d d DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Topological alteration The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a c a a a b a b b c e b d b c d b c d c d c d d c a d b a b c d DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Topological alteration The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a c a a a b a b b c e b d b c d b c d c d c d d c a a c e b d b d a b a b c d c d DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

Problem statement Definitions (Digital) Rigid Transformations Methods Geometrical and Topological Issues Experiments Conclusion Topological alteration The following configurations (up to symmetries) can generate a 2 × 2 sample under digital rigid transformations. a a c a a a b a b b c e b d b c d b c d c d c d d c a a a a c e b d c d b c d b b d a b a b a b a b c c c d d d c d DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 6/22

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

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

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

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

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

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

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Tools for topological invariance verification Methodological tools DRT graph, and DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 9/22

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Tools for topological invariance verification Methodological tools DRT graph, and simple points. DGCI’2013 – March 20th 2013 Sufficient conditions for topological invariance 9/22

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Critical transformation Digital rigid transformation T ab θ : Z 2 → Z 2 � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ 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

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Critical transformation Digital rigid transformation T ab θ : Z 2 → Z 2 � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ 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

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Critical transformation Digital rigid transformation T ab θ : Z 2 → Z 2 � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ 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

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Critical transformation Digital rigid transformation T ab θ : Z 2 → Z 2 � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ 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

Problem statement Definitions Topological Invariance Methods DRT Graph Experiments Simple Point Conclusion Critical transformation Digital rigid transformation T ab θ : Z 2 → Z 2 � [ p cos θ − q sin θ + a ] � p ′ � � = q ′ [ 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