Neuronal structure detection using Persistent                               Homology∗                                    J...
Outline     1    Motivation     2    Persistent Homology     3    The concrete problem     4    Using Persistent Homology ...
MotivationOutline     1    Motivation     2    Persistent Homology     3    The concrete problem     4    Using Persistent...
MotivationMotivationJ. Heras, G. Mata and J. Rubio    Neuronal structure detection using Persistent Homology   4/39
MotivationMotivationJ. Heras, G. Mata and J. Rubio    Neuronal structure detection using Persistent Homology   4/39
MotivationMotivationJ. Heras, G. Mata and J. Rubio    Neuronal structure detection using Persistent Homology   4/39
MotivationMotivationJ. Heras, G. Mata and J. Rubio    Neuronal structure detection using Persistent Homology   4/39
MotivationMotivation     Peculiarity: the last three images are obtained from a stackJ. Heras, G. Mata and J. Rubio    Neu...
Persistent HomologyOutline     1    Motivation     2    Persistent Homology     3    The concrete problem     4    Using P...
Persistent Homology   Intuitive IdeaIntuitive idea                     Figure: La Seine ` la Grande-Jatte. Seurat, Georges...
Persistent Homology   Intuitive IdeaKey ideas     Persistence key ideas:             Provide an abstract framework to:    ...
Persistent Homology   Intuitive IdeaKey ideas     Persistence key ideas:             Provide an abstract framework to:    ...
Persistent Homology   Intuitive IdeaKey ideas     Persistence key ideas:             Provide an abstract framework to:    ...
Persistent Homology   HistoryHistory             Biogeometry project of Edelsbrunner                    C. J. A. Delfinado ...
Persistent Homology   NotionsSimplicial Complexes     Definition     Let V be an ordered set, called the vertex set.     A ...
Persistent Homology   NotionsSimplicial Complexes     Definition     Let V be an ordered set, called the vertex set.     A ...
Persistent Homology   NotionsSimplicial Complexes     Definition     Let V be an ordered set, called the vertex set.     A ...
Persistent Homology       NotionsAn example                                            2                                  ...
Persistent Homology    NotionsChain Complexes     Definition     A chain complex C∗ is a pair of sequences C∗ = (Cq , dq )q...
Persistent Homology           NotionsChain Complexes     Definition     A chain complex C∗ is a pair of sequences C∗ = (Cq ...
Persistent Homology   NotionsHomology     Definition     If C∗ = (Cq , dq )q∈Z is a chain complex:             The image Bq...
Persistent Homology   NotionsHomology     Definition     If C∗ = (Cq , dq )q∈Z is a chain complex:             The image Bq...
Persistent Homology   NotionsFiltration of a Simplicial Complex     Definition     A subcomplex of K is a subset L ⊆ K that...
Persistent Homology   NotionsFiltration of a Simplicial Complex     Definition     A subcomplex of K is a subset L ⊆ K that...
Persistent Homology   NotionsAn exampleJ. Heras, G. Mata and J. Rubio             Neuronal structure detection using Persi...
Persistent Homology   NotionsPersistence Complexes     Definition                                                          ...
Persistent Homology                NotionsPersistence Complexes     Definition                                             ...
Persistent Homology   NotionsPersistent Homology groups     Definition              i     If C = {C∗ }i≥0 is a persistence ...
Persistent Homology     NotionsPersistent Homology groups     Definition              i     If C = {C∗ }i≥0 is a persistenc...
Persistent Homology     NotionsPersistent Homology groups     Definition              i     If C = {C∗ }i≥0 is a persistenc...
Persistent Homology                NotionsPersistent Homology groups     Persistence complex associated with a filtered com...
Persistent Homology                NotionsPersistent Homology groups     Persistence complex associated with a filtered com...
Persistent Homology                 NotionsPersistent Homology groups     Persistence complex associated with a filtered co...
Persistent Homology   NotionsPersistence: Interpretation     Captures the birth and death times of homology classes of the...
Persistent Homology   NotionsPersistence: Interpretation     Captures the birth and death times of homology classes of the...
Persistent Homology   NotionsPersistence: Interpretation     Captures the birth and death times of homology classes of the...
Persistent Homology        NotionsPersistence: Interpretation     Captures the birth and death times of homology classes o...
Persistent Homology   NotionsBarcodes     Definition     A P-interval is a half-open interval [i, j), which is also represe...
Persistent Homology   NotionsBarcodes     Definition     A P-interval is a half-open interval [i, j), which is also represe...
Persistent Homology   NotionsBarcodes     Definition     A P-interval is a half-open interval [i, j), which is also represe...
Persistent Homology   NotionsExampleJ. Heras, G. Mata and J. Rubio             Neuronal structure detection using Persiste...
Persistent Homology        NotionsExample                                                     2                           ...
Persistent Homology        NotionsExample                                                     2                           ...
Persistent Homology        NotionsExample                                                     2                           ...
Persistent Homology        NotionsExample                                                     2                           ...
Persistent Homology        NotionsExample                                                     2                           ...
Persistent Homology     NotionsExample                                                         ...                        ...
Persistent Homology            NotionsExample                                                     2                       ...
Persistent Homology   NotionsApplication: Point-Cloud datasetsJ. Heras, G. Mata and J. Rubio             Neuronal structur...
Persistent Homology   NotionsApplication: Point-Cloud datasets     Construct a filtration from the point-cloud dataset :   ...
Persistent Homology   NotionsApplication: Point-Cloud datasets     Construct a filtration from the point-cloud dataset :   ...
The concrete problemOutline     1    Motivation     2    Persistent Homology     3    The concrete problem     4    Using ...
The concrete problemDendritic spines     Dendritic spines (or spines):             A small membranous extrusion of the den...
The concrete problemDendritic spines     Dendritic spines (or spines):             A small membranous extrusion of the den...
The concrete problemDendritic spines     Dendritic spines (or spines):             A small membranous extrusion of the den...
The concrete problemDendritic spines     Dendritic spines (or spines):             A small membranous extrusion of the den...
The concrete problemDendritic spines     Dendritic spines (or spines):             A small membranous extrusion of the den...
The concrete problemThe problem     Goal:             Automatic measurement and classification of spines of a neuronJ. Hera...
The concrete problemThe problem     Goal:             Automatic measurement and classification of spines of a neuron     Fi...
Using Persistent Homology in our problemOutline     1    Motivation     2    Persistent Homology     3    The concrete pro...
Using Persistent Homology in our problemGetting the images     Optical sectioning:             Produces clear images of a ...
Using Persistent Homology in our problemGetting the images     Optical sectioning:             Produces clear images of a ...
Using Persistent Homology in our problemExample     Stack of images:     Important feature: the neuron persists in all the...
Using Persistent Homology in our problemExample     Z-projection:J. Heras, G. Mata and J. Rubio                   Neuronal...
Using Persistent Homology in our problem   Our methodOur method     Our method:         1   Reduce salt-and-pepper noise  ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsSimplicial Complexes from Digital Images...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsSimplicial Complexes from Digital Images...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Let ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection         ...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsA filtration of the Z-projection     Reme...
Using Persistent Homology in our problem      Our method: Dismiss irrelevant elementsA filtration of the Z-projection     R...
Using Persistent Homology in our problem      Our method: Dismiss irrelevant elementsA filtration of the Z-projection     R...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsFinal resultJ. Heras, G. Mata and J. Rub...
Using Persistent Homology in our problem   Our method: Dismiss irrelevant elementsFinal resultJ. Heras, G. Mata and J. Rub...
Conclusions and Further workOutline     1    Motivation     2    Persistent Homology     3    The concrete problem     4  ...
Conclusions and Further workConclusions and Further work     Conclusions:             Method to detect neuronal structure ...
Conclusions and Further workConclusions and Further work     Conclusions:             Method to detect neuronal structure ...
Conclusions and Further work               Neuronal structure detection using Persistent                               Hom...
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Neuronal Detection using Persistent Homology

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Neuronal Detection using Persistent Homology

  1. 1. Neuronal structure detection using Persistent Homology∗ J. Heras, G. Mata and J. Rubio Department of Mathematics and Computer Science, University of La Rioja Seminario de Inform´tica Mirian Andr´s a e March 20, 2012 ∗ Partially supported by Ministerio de Educaci´n y Ciencia, project MTM2009-13842-C02-01, and by European o Commission FP7, STREP project ForMath, n. 243847J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 1/39
  2. 2. Outline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 2/39
  3. 3. MotivationOutline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 3/39
  4. 4. MotivationMotivationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
  5. 5. MotivationMotivationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
  6. 6. MotivationMotivationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
  7. 7. MotivationMotivationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
  8. 8. MotivationMotivation Peculiarity: the last three images are obtained from a stackJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
  9. 9. Persistent HomologyOutline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 5/39
  10. 10. Persistent Homology Intuitive IdeaIntuitive idea Figure: La Seine ` la Grande-Jatte. Seurat, Georges aJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 6/39
  11. 11. Persistent Homology Intuitive IdeaKey ideas Persistence key ideas: Provide an abstract framework to: Measure scales on topological features Order topological features in term of importance/noiseJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 7/39
  12. 12. Persistent Homology Intuitive IdeaKey ideas Persistence key ideas: Provide an abstract framework to: Measure scales on topological features Order topological features in term of importance/noise How long is a topological feature persistent? As long as it refuses to die . . .J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 7/39
  13. 13. Persistent Homology Intuitive IdeaKey ideas Persistence key ideas: Provide an abstract framework to: Measure scales on topological features Order topological features in term of importance/noise How long is a topological feature persistent? As long as it refuses to die . . . Roughly speaking: Homology detects topological features (connected components, holes, and so on) Persistent homology describes the evolution of topological features looking at consecutive thresholdsJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 7/39
  14. 14. Persistent Homology HistoryHistory Biogeometry project of Edelsbrunner C. J. A. Delfinado and H. Edelsbrunner. An incremental algorithm for Bettin numbers of simplicial complexes on the 3-sphere. Computer Aided Geometry Design 12 (1995):771–784. Work of Frosini, Ferri and collaborators P. Frosini and C. Landi. Size theory as a topological tool for computer vision. Pattern Recognition and Image Analysis 9 (1999):596–603. Doctoral thesis of Robins V. Robins. Toward computing homology from finite approximations. Topology Proceedings 24 (1999):503–532.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 8/39
  15. 15. Persistent Homology NotionsSimplicial Complexes Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V .J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 9/39
  16. 16. Persistent Homology NotionsSimplicial Complexes Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . Definition Let α and β be simplices over V , we say α is a face of β if α is a subset of β.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 9/39
  17. 17. Persistent Homology NotionsSimplicial Complexes Definition Let V be an ordered set, called the vertex set. A simplex over V is any finite subset of V . Definition Let α and β be simplices over V , we say α is a face of β if α is a subset of β. Definition An ordered (abstract) simplicial complex over V is a set of simplices K over V satisfying the property: ∀α ∈ K, if β ⊆ α ⇒ β ∈ K Let K be a simplicial complex. Then the set Sn (K) of n-simplices of K is the set made of the simplices of cardinality n + 1.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 9/39
  18. 18. Persistent Homology NotionsAn example 2 5 3 4 0 6 1 V = (0, 1, 2, 3, 4, 5, 6) K = {∅, (0), (1), (2), (3), (4), (5), (6), (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3), (3, 4), (4, 5), (4, 6), (5, 6), (0, 1, 2), (4, 5, 6)}J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 10/39
  19. 19. Persistent Homology NotionsChain Complexes Definition A chain complex C∗ is a pair of sequences C∗ = (Cq , dq )q∈Z where: For every q ∈ Z, the component Cq is an R-module, the chain group of degree q For every q ∈ Z, the component dq is a module morphism dq : Cq → Cq−1 , the differential map For every q ∈ Z, the composition dq dq+1 is null: dq dq+1 = 0J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 11/39
  20. 20. Persistent Homology NotionsChain Complexes Definition A chain complex C∗ is a pair of sequences C∗ = (Cq , dq )q∈Z where: For every q ∈ Z, the component Cq is an R-module, the chain group of degree q For every q ∈ Z, the component dq is a module morphism dq : Cq → Cq−1 , the differential map For every q ∈ Z, the composition dq dq+1 is null: dq dq+1 = 0 Definition Let K be an (ordered abstract) simplicial complex. Let n ≥ 1 and 0 ≤ i ≤ n be two integers n and i. Then the face operator ∂in is the linear map ∂in : Sn (K) → Sn−1 (K) defined by: n ∂i ((v0 , . . . , vn )) = (v0 , . . . , vi−1 , vi+1 , . . . , vn ). The i-th vertex of the simplex is removed, so that an (n − 1)-simplex is obtained. Definition Let K be a simplicial complex. Then the chain complex C∗ (K) canonically associated with K is defined as follows. The chain group Cn (K) is the free Z module generated by the n-simplices of K. In addition, let (v0 , . . . , vn−1 ) be a n-simplex of K, the differential of this simplex is defined as: n i n dn := (−1) ∂i i=0J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 11/39
  21. 21. Persistent Homology NotionsHomology Definition If C∗ = (Cq , dq )q∈Z is a chain complex: The image Bq = im dq+1 ⊆ Cq is the (sub)module of q-boundaries The kernel Zq = ker dq ⊆ Cq is the (sub)module of q-cycles Given a chain complex C∗ = (Cq , dq )q∈Z : dq−1 ◦ dq = 0 ⇒ Bq ⊆ Zq Every boundary is a cycle The converse is not generally trueJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 12/39
  22. 22. Persistent Homology NotionsHomology Definition If C∗ = (Cq , dq )q∈Z is a chain complex: The image Bq = im dq+1 ⊆ Cq is the (sub)module of q-boundaries The kernel Zq = ker dq ⊆ Cq is the (sub)module of q-cycles Given a chain complex C∗ = (Cq , dq )q∈Z : dq−1 ◦ dq = 0 ⇒ Bq ⊆ Zq Every boundary is a cycle The converse is not generally true Definition Let C∗ = (Cq , dq )q∈Z be a chain complex. For each degree n ∈ Z, the n-homology module of C∗ is defined as the quotient module Zn Hn (C∗ ) = BnJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 12/39
  23. 23. Persistent Homology NotionsFiltration of a Simplicial Complex Definition A subcomplex of K is a subset L ⊆ K that is also a simplicial complex.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 13/39
  24. 24. Persistent Homology NotionsFiltration of a Simplicial Complex Definition A subcomplex of K is a subset L ⊆ K that is also a simplicial complex. Definition A filtration of a simplicial complex K is a nested subsequence of complexes K0 ⊆ K1 ⊆ . . . ⊆ Km = KJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 13/39
  25. 25. Persistent Homology NotionsAn exampleJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 14/39
  26. 26. Persistent Homology NotionsPersistence Complexes Definition i A persistence complex C is a family of chain complexes {C∗ }i≥0 over a field together i+1 with chain maps f i : C∗ → C∗ iJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 15/39
  27. 27. Persistent Homology NotionsPersistence Complexes Definition i A persistence complex C is a family of chain complexes {C∗ }i≥0 over a field together i+1 with chain maps f i : C∗ → C∗ i A filtered complex K with inclusion maps for the simplices becomes a persistence complex . . . 0 1 2 d3 d3 d3 / C2 / C2 / ... 0 1 0 f 1 f 2 f2 C2 0 1 2 d2 d2 d2 0 C1 f0 / C1 1 f1 / C1 2 f2 / ... 0 1 2 d1 d1 d1 / C0 / C0 / ... 0 1 0 f 1 f 2 f2 C0J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 15/39
  28. 28. Persistent Homology NotionsPersistent Homology groups Definition i If C = {C∗ }i≥0 is a persistence complex: i i i Bq = im dq+1 ⊆ Cq i i i Zq = ker dq ⊆ CqJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
  29. 29. Persistent Homology NotionsPersistent Homology groups Definition i If C = {C∗ }i≥0 is a persistence complex: i i i Bq = im dq+1 ⊆ Cq i i i Zq = ker dq ⊆ Cq Definition i Let C = {C∗ }i≥0 be a persistence complex associated with a filtration. The p-persistent kth homology group of C is: i Zk i,p Hk = i+p Bk i ∩ ZkJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
  30. 30. Persistent Homology NotionsPersistent Homology groups Definition i If C = {C∗ }i≥0 is a persistence complex: i i i Bq = im dq+1 ⊆ Cq i i i Zq = ker dq ⊆ Cq Definition i Let C = {C∗ }i≥0 be a persistence complex associated with a filtration. The p-persistent kth homology group of C is: i Zk i,p Hk = i+p Bk i ∩ Zk Definition i,p i,p The p-persistent k-th Betti number βk is the rank of HkJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
  31. 31. Persistent Homology NotionsPersistent Homology groups Persistence complex associated with a filtered complex K with inclusion maps . . . 0 1 2 d3 d3 d3 / C2 / C2 / ... 0 1 0 f 1 f 2 f2 C2 0 1 2 d2 d2 d2 0 C1 f0 / C1 1 f1 / C1 2 f2 / ... 0 1 2 d1 d1 d1 0 C0 f0 / C0 1 f1 / C0 2 f2 / ...J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
  32. 32. Persistent Homology NotionsPersistent Homology groups Persistence complex associated with a filtered complex K with inclusion maps . . . 0 1 2 d3 d3 d3 / C2 / C2 / ... 0 1 0 f 1 f 2 f2 C2 0 1 2 d2 d2 d2 0 C1 f0 / C1 1 f1 / C1 2 f2 / ... 0 1 2 d1 d1 d1 0 C0 f0 / C0 1 f1 / C0 2 f2 / ... i i i i+p Zk = ker dk ⊆ Ck ⊆ Ck i+p i+p i+p Bk = im dk ⊆ CkJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
  33. 33. Persistent Homology NotionsPersistent Homology groups Persistence complex associated with a filtered complex K with inclusion maps . . . 0 1 2 d3 d3 d3 / C2 / C2 / ... 0 1 0 f 1 f 2 f2 C2 0 1 2 d2 d2 d2 0 C1 f0 / C1 1 f1 / C1 2 f2 / ... 0 1 2 d1 d1 d1 0 C0 f0 / C0 1 f1 / C0 2 f2 / ... i i i i+p Zk = ker dk ⊆ Ck ⊆ Ck i+p i+p i+p Bk = im dk ⊆ Ck i Zk i,p Hk = i+p Bk i ∩ ZkJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
  34. 34. Persistent Homology NotionsPersistence: Interpretation Captures the birth and death times of homology classes of the filtration as it growsJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
  35. 35. Persistent Homology NotionsPersistence: Interpretation Captures the birth and death times of homology classes of the filtration as it grows Definition Let K 0 ⊆ K 1 ⊆ . . . ⊆ K m = K be a filtration. A homology class α is born at K i if it is not in the image of the map induced by the inclusion K i−1 ⊆ K i .J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
  36. 36. Persistent Homology NotionsPersistence: Interpretation Captures the birth and death times of homology classes of the filtration as it grows Definition Let K 0 ⊆ K 1 ⊆ . . . ⊆ K m = K be a filtration. A homology class α is born at K i if it is not in the image of the map induced by the inclusion K i−1 ⊆ K i . If α is born at K i it dies entering K j if the image of the map induced by K i−1 ⊆ K j−1 does not contain the image of α but the image of the map induced by K i−1 ⊆ K j does.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
  37. 37. Persistent Homology NotionsPersistence: Interpretation Captures the birth and death times of homology classes of the filtration as it grows Definition Let K 0 ⊆ K 1 ⊆ . . . ⊆ K m = K be a filtration. A homology class α is born at K i if it is not in the image of the map induced by the inclusion K i−1 ⊆ K i . If α is born at K i it dies entering K j if the image of the map induced by K i−1 ⊆ K j−1 does not contain the image of α but the image of the map induced by K i−1 ⊆ K j does. H1 H2 H3 H4 h 2 4 f1 f3 H 1,0 3 H 1,1 f2 H 1,2 h ∈ H 2 is born in H 2 , and dies entering H 4J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
  38. 38. Persistent Homology NotionsBarcodes Definition A P-interval is a half-open interval [i, j), which is also represented by its endpoints (i, j) ∈ Z × (Z ∪ ∞)J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 19/39
  39. 39. Persistent Homology NotionsBarcodes Definition A P-interval is a half-open interval [i, j), which is also represented by its endpoints (i, j) ∈ Z × (Z ∪ ∞) Relation between P-intervals and persistence: A class that is born in H i and never dies is represented as (i, ∞) A class that is born in H i and dies entering in H j is represented as (i, j)J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 19/39
  40. 40. Persistent Homology NotionsBarcodes Definition A P-interval is a half-open interval [i, j), which is also represented by its endpoints (i, j) ∈ Z × (Z ∪ ∞) Relation between P-intervals and persistence: A class that is born in H i and never dies is represented as (i, ∞) A class that is born in H i and dies entering in H j is represented as (i, j) Definition Finite multisets of P-intervals are plotted as disjoint unions of intervals, called barcodes.J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 19/39
  41. 41. Persistent Homology NotionsExampleJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  42. 42. Persistent Homology NotionsExample 2 5 0 3 4 1 6 class 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  43. 43. Persistent Homology NotionsExample 2 5 0 3 4 1 6 class 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  44. 44. Persistent Homology NotionsExample 2 5 0 3 4 1 6 class 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  45. 45. Persistent Homology NotionsExample 2 5 0 3 4 1 6 class 3 2 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  46. 46. Persistent Homology NotionsExample 2 5 0 3 4 1 6 class 3 2 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  47. 47. Persistent Homology NotionsExample ... class 3 2 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11 12J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  48. 48. Persistent Homology NotionsExample 2 5 4 0 1 3 6 class 6 5 4 3 2 1 0 step 0 1 2 3 4 5 6 7 8 9 10 11J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
  49. 49. Persistent Homology NotionsApplication: Point-Cloud datasetsJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
  50. 50. Persistent Homology NotionsApplication: Point-Cloud datasets Construct a filtration from the point-cloud dataset : Cech Complexes Vietoris-Rips Complexes Voronoi Diagram and the Delaunay Complex Alpha Complexes ... Idea of proximityJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
  51. 51. Persistent Homology NotionsApplication: Point-Cloud datasets Construct a filtration from the point-cloud dataset : Cech Complexes Vietoris-Rips Complexes Voronoi Diagram and the Delaunay Complex Alpha Complexes ... Idea of proximity G. Singh et. al. Topological analysis of population activity in visual cortex. Journal of Vision 8:8(2008). http://www.journalofvision.org/content/8/8/11.fullJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
  52. 52. The concrete problemOutline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 22/39
  53. 53. The concrete problemDendritic spines Dendritic spines (or spines): A small membranous extrusion of the dendrites that contains the molecules and organelles involved in the postsynaptic processing of the synaptic informationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
  54. 54. The concrete problemDendritic spines Dendritic spines (or spines): A small membranous extrusion of the dendrites that contains the molecules and organelles involved in the postsynaptic processing of the synaptic information Related to learning and memoryJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
  55. 55. The concrete problemDendritic spines Dendritic spines (or spines): A small membranous extrusion of the dendrites that contains the molecules and organelles involved in the postsynaptic processing of the synaptic information Related to learning and memory Associated with Alzheimer’s and Parkinson’s diseases, autism, mental retardation and fragile X SyndromeJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
  56. 56. The concrete problemDendritic spines Dendritic spines (or spines): A small membranous extrusion of the dendrites that contains the molecules and organelles involved in the postsynaptic processing of the synaptic information Related to learning and memory Associated with Alzheimer’s and Parkinson’s diseases, autism, mental retardation and fragile X Syndrome The dendrites of a single neuron can contain hundreds to thousands of spinesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
  57. 57. The concrete problemDendritic spines Dendritic spines (or spines): A small membranous extrusion of the dendrites that contains the molecules and organelles involved in the postsynaptic processing of the synaptic information Related to learning and memory Associated with Alzheimer’s and Parkinson’s diseases, autism, mental retardation and fragile X Syndrome The dendrites of a single neuron can contain hundreds to thousands of spinesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
  58. 58. The concrete problemThe problem Goal: Automatic measurement and classification of spines of a neuronJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 24/39
  59. 59. The concrete problemThe problem Goal: Automatic measurement and classification of spines of a neuron First step: Detect the region of interest (the dendrites) Main problem: noise salt-and-pepper non-relevant biological elementsJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 24/39
  60. 60. Using Persistent Homology in our problemOutline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 25/39
  61. 61. Using Persistent Homology in our problemGetting the images Optical sectioning: Produces clear images of a focal planes deep within a thick sample Reduces the need for thin sectioning Allows the three dimensional reconstruction of a sample from images captured at different focal planes http://loci.wisc.edu/files/loci/8OpticalSectionB.swfJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 26/39
  62. 62. Using Persistent Homology in our problemGetting the images Optical sectioning: Produces clear images of a focal planes deep within a thick sample Reduces the need for thin sectioning Allows the three dimensional reconstruction of a sample from images captured at different focal planes http://loci.wisc.edu/files/loci/8OpticalSectionB.swf Procedure: 1 Get a stack of images using optical sectioning 2 Obtain its maximum intensity projectionJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 26/39
  63. 63. Using Persistent Homology in our problemExample Stack of images: Important feature: the neuron persists in all the slicesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 27/39
  64. 64. Using Persistent Homology in our problemExample Z-projection:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 27/39
  65. 65. Using Persistent Homology in our problem Our methodOur method Our method: 1 Reduce salt-and-pepper noise 2 Dismiss irrelevant elements as astrocytes, dendrites of other neurons, and so onJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 28/39
  66. 66. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Salt-and-pepper noise: Produced when captured the image from the microscopeJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
  67. 67. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Salt-and-pepper noise: Produced when captured the image from the microscope Solution: 1 Low-pass filter: Decrease the disparity between pixel values by averaging nearby pixels uniform, Gaussian, median, maximum, minimum, mean, and so on In our case (after experimentation): median filter with a radius of 10 pixelsJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
  68. 68. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Salt-and-pepper noise: Produced when captured the image from the microscope Solution: 1 Low-pass filter: Decrease the disparity between pixel values by averaging nearby pixels uniform, Gaussian, median, maximum, minimum, mean, and so on In our case (after experimentation): median filter with a radius of 10 pixels 2 Threshold: Discrimination of pixels depending on their intensity A binary image is obtained In our case (after experimentation): Huang’s methodJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
  69. 69. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Original image:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
  70. 70. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise After low-pass filter:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
  71. 71. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise After threshold:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
  72. 72. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Undesirable elements:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
  73. 73. Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noiseOur method: Reducing salt-and-pepper noise Preprocessing is applied to all the slices of the stack:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 31/39
  74. 74. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsSimplicial Complexes from Digital ImagesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 32/39
  75. 75. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsSimplicial Complexes from Digital Images A monochromatic image D: set of black pixels a subimage of D is a subset L ⊆ D a filtration of an image D is a nested subsequence of images D0 ⊆ D1 ⊆ . . . ⊆ Dm = D a filtration of an image induces a filtration of simplicial complexesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 32/39
  76. 76. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection:J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  77. 77. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projectionJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  78. 78. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection 2 D m−1 consists of the connected components of D m such that its intersection with the first slide is not emptyJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  79. 79. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection 2 D m−1 consists of the connected components of D m such that its intersection with the first slide is not emptyJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  80. 80. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection 2 D m−1 consists of the connected components of D m such that its intersection with the first slide is not empty 3 D m−2 consists of the connected components of D m−1 such that its intersection with the second slide is not emptyJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  81. 81. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection 2 D m−1 consists of the connected components of D m such that its intersection with the first slide is not empty 3 D m−2 consists of the connected components of D m−1 such that its intersection with the second slide is not emptyJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  82. 82. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection 2 D m−1 consists of the connected components of D m such that its intersection with the first slide is not empty 3 D m−2 consists of the connected components of D m−1 such that its intersection with the second slide is not empty 4 ...J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
  83. 83. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection ⊆ ⊆ ⊆ ⊆ D0 D1 D2 D3 ⊆ ⊆ ⊆ ⊆ D4 D5 D6 D7 D8J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 34/39
  84. 84. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Remember: the neuron persists in all the slicesJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
  85. 85. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Remember: the neuron persists in all the slices x0 x1 x2 x3 . . . D0 D1 D2 D3 D4 D5 D6 D7 D8J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
  86. 86. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsA filtration of the Z-projection Remember: the neuron persists in all the slices x0 x1 x2 x3 . . . D0 D1 D2 D3 D4 D5 D6 D7 D8 The components of the neuron persist all the life of the filtrationJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
  87. 87. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsFinal resultJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 36/39
  88. 88. Using Persistent Homology in our problem Our method: Dismiss irrelevant elementsFinal resultJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 36/39
  89. 89. Conclusions and Further workOutline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further workJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 37/39
  90. 90. Conclusions and Further workConclusions and Further work Conclusions: Method to detect neuronal structure based on persistent homologyJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 38/39
  91. 91. Conclusions and Further workConclusions and Further work Conclusions: Method to detect neuronal structure based on persistent homology Further work: Implementation of an ImageJ plug-in Intensive testing Software verification? Measurement and classification of spines Persistent Homology ↔ Discrete Morse TheoryJ. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 38/39
  92. 92. Conclusions and Further work Neuronal structure detection using Persistent Homology∗ J. Heras, G. Mata and J. Rubio Department of Mathematics and Computer Science, University of La Rioja Seminario de Inform´tica Mirian Andr´s a e March 20, 2012 ∗ Partially supported by Ministerio de Educaci´n y Ciencia, project MTM2009-13842-C02-01, and by European o Commission FP7, STREP project ForMath, n. 243847J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 39/39

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