Neuronal structure detection using Persistent Homology
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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 Informática Mirian Andrés March 20, 2012 Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath, n J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 1/39
2 Outline 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 2/39
3 Outline Motivation 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 3/39
4 Motivation Motivation J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
5 Motivation Motivation J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
6 Motivation Motivation J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
7 Motivation Motivation J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
8 Motivation Motivation Peculiarity: the last three images are obtained from a stack J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 4/39
9 Outline Persistent Homology 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 5/39
10 Persistent Homology Intuitive Idea Intuitive idea Figure: La Seine a la Grande-Jatte. Seurat, Georges J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 6/39
11 Key ideas Persistent Homology Intuitive Idea Persistence key ideas: Provide an abstract framework to: Measure scales on topological features Order topological features in term of importance/noise J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 7/39
12 Key ideas Persistent Homology Intuitive Idea 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 Key ideas Persistent Homology Intuitive Idea 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 thresholds J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 7/39
14 Persistent Homology History History 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): 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): Doctoral thesis of Robins V. Robins. Toward computing homology from finite approximations. Topology Proceedings 24 (1999): J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 8/39
15 Simplicial Complexes Persistent Homology Notions 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 Simplicial Complexes Persistent Homology Notions 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 Simplicial Complexes Persistent Homology Notions 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 S n(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 An example Persistent Homology Notions 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 Chain Complexes Persistent Homology Notions Definition A chain complex C is a pair of sequences C = (C q, d q) q Z where: For every q Z, the component C q is an R-module, the chain group of degree q For every q Z, the component d q is a module morphism d q : C q C q 1, the differential map For every q Z, the composition d qd q+1 is null: d qd q+1 = 0 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 11/39
20 Chain Complexes Persistent Homology Notions Definition A chain complex C is a pair of sequences C = (C q, d q) q Z where: For every q Z, the component C q is an R-module, the chain group of degree q For every q Z, the component d q is a module morphism d q : C q C q 1, the differential map For every q Z, the composition d qd q+1 is null: d qd q+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 n i is the linear map n i : S n(k) S n 1 (K) defined by: n i ((v 0,..., v n)) = (v 0,..., v i 1, v i+1,..., v n). 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 C n(k) is the free Z module generated by the n-simplices of K. In addition, let (v 0,..., v n 1 ) be a n-simplex of K, the differential of this simplex is defined as: n d n := ( 1) i n i i=0 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 11/39
21 Persistent Homology Notions Homology Definition If C = (C q, d q) q Z is a chain complex: The image B q = im d q+1 C q is the (sub)module of q-boundaries The kernel Z q = ker d q C q is the (sub)module of q-cycles Given a chain complex C = (C q, d q) q Z : d q 1 d q = 0 B q Z q Every boundary is a cycle The converse is not generally true J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 12/39
22 Persistent Homology Notions Homology Definition If C = (C q, d q) q Z is a chain complex: The image B q = im d q+1 C q is the (sub)module of q-boundaries The kernel Z q = ker d q C q is the (sub)module of q-cycles Given a chain complex C = (C q, d q) q Z : d q 1 d q = 0 B q Z q Every boundary is a cycle The converse is not generally true Definition Let C = (C q, d q) q Z be a chain complex. For each degree n Z, the n-homology module of C is defined as the quotient module H n(c ) = Zn B n J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 12/39
23 Persistent Homology Notions Filtration 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 Persistent Homology Notions Filtration 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 K 0 K 1... K m = K J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 13/39
25 An example Persistent Homology Notions J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 14/39
26 Persistence Complexes Persistent Homology Notions Definition A persistence complex C is a family of chain complexes {C i } i 0 over a field together with chain maps f i : C i C i+1 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 15/39
27 Persistence Complexes Persistent Homology Notions Definition A persistence complex C is a family of chain complexes {C i } i 0 over a field together with chain maps f i : C i C i+1 A filtered complex K with inclusion maps for the simplices becomes a persistence complex... d 0 3 C 0 2 f 0 d 1 3 C 1 2 f 1 d 2 3 C 2 2 f 2... d 0 2 C 0 1 f 0 d 1 2 C 1 1 f 1 d 2 2 C 2 1 f 2... d 0 1 C 0 0 d 1 1 f 0 C 1 0 d 2 1 f 1 C 2 0 f 2... J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 15/39
28 Persistent Homology Notions Persistent Homology groups Definition If C = {C i } i 0 is a persistence complex: B i q = im di q+1 C i q Z i q = ker di q C i q J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
29 Persistent Homology Persistent Homology groups Notions Definition If C = {C } i i 0 is a persistence complex: Bq i = im di q+1 C q i Zq i = ker di q C q i Definition Let C = {C i } i 0 be a persistence complex associated with a filtration. The p-persistent kth homology group of C is: H i,p k = B i+p k Z i k Z i k J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
30 Persistent Homology Persistent Homology groups Notions Definition If C = {C } i i 0 is a persistence complex: Bq i = im di q+1 C q i Zq i = ker di q C q i Definition Let C = {C i } i 0 be a persistence complex associated with a filtration. The p-persistent kth homology group of C is: H i,p k = B i+p k Z i k Z i k Definition The p-persistent k-th Betti number β i,p k is the rank of H i,p k J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 16/39
31 Persistent Homology Persistent Homology groups Notions Persistence complex associated with a filtered complex K with inclusion maps... d 0 3 C 0 2 f 0 d 1 3 C 1 2 f 1 d 2 3 C 2 2 f 2... d 0 2 C 0 1 f 0 d 1 2 C 1 1 f 1 d 2 2 C 2 1 f 2... d 0 1 C 0 0 d 1 1 f 0 C 1 0 d 2 1 f 1 C 2 0 f 2... J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
32 Persistent Homology Persistent Homology groups Notions Persistence complex associated with a filtered complex K with inclusion maps... d 0 3 C 0 2 f 0 d 1 3 C 1 2 f 1 d 2 3 C 2 2 f 2... d 0 2 C 0 1 f 0 d 1 2 C 1 1 f 1 d 2 2 C 2 1 f 2... d 0 1 C 0 0 d 1 1 f 0 C 1 0 d 2 1 f 1 C 2 0 f 2... Z i k = ker di k C i k C i+p k B i+p k = im d i+p k C i+p k J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
33 Persistent Homology Persistent Homology groups Notions Persistence complex associated with a filtered complex K with inclusion maps... d 0 3 C 0 2 f 0 d 1 3 C 1 2 f 1 d 2 3 C 2 2 f 2... d 0 2 C 0 1 f 0 d 1 2 C 1 1 f 1 d 2 2 C 2 1 f 2... d 0 1 C 0 0 d 1 1 f 0 C 1 0 d 2 1 f 1 C 2 0 f 2... Z i k = ker di k C i k C i+p k B i+p k = im d i+p k C i+p k H i,p k = B i+p k Z i k Z i k J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 17/39
34 Persistent Homology Persistence: Interpretation Notions Captures the birth and death times of homology classes of the filtration as it grows J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
35 Persistent Homology Persistence: Interpretation Notions 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 Persistent Homology Persistence: Interpretation Notions 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 Persistent Homology Notions Persistence: 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. H 1 H 2 H 3 H 4 h H 1,0 f 2 1 H 1,1 f 3 2 H 1,2 f 4 3 h H 2 is born in H 2, and dies entering H 4 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 18/39
38 Barcodes Persistent Homology Notions 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 Barcodes Persistent Homology Notions 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 Barcodes Persistent Homology Notions 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: Definition 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) 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 Example Persistent Homology Notions J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
42 Example Persistent Homology Notions class 0 step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
43 Example Persistent Homology Notions class 1 0 step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
44 Example Persistent Homology Notions class 1 0 step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
45 Example Persistent Homology Notions class step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
46 Example Persistent Homology Notions class step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
47 Example Persistent Homology... Notions class step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
48 Example Persistent Homology Notions class step J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 20/39
49 Persistent Homology Notions Application: Point-Cloud datasets J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
50 Persistent Homology Notions Application: 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
51 Persistent Homology Notions Application: 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). J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 21/39
52 Outline The concrete problem 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 22/39
53 Dendritic spines The concrete problem 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
54 Dendritic spines The concrete problem 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
55 Dendritic spines The concrete problem 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
56 Dendritic spines The concrete problem 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 spines J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
57 Dendritic spines The concrete problem 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 spines J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 23/39
58 The problem The concrete problem Goal: Automatic measurement and classification of spines of a neuron J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 24/39
59 The concrete problem The 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 elements J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 24/39
60 Outline Using Persistent Homology in our problem 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 25/39
61 Using Persistent Homology in our problem Getting 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 26/39
62 Using Persistent Homology in our problem Getting 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 Procedure: 1 Get a stack of images using optical sectioning 2 Obtain its maximum intensity projection J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 26/39
63 Using Persistent Homology in our problem Example Stack of images: Important feature: the neuron persists in all the slices J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 27/39
64 Example Using Persistent Homology in our problem Z-projection: J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 27/39
65 Using Persistent Homology in our problem Our method Our method Our method: 1 Reduce salt-and-pepper noise 2 Dismiss irrelevant elements as astrocytes, dendrites of other neurons, and so on J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 28/39
66 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our method: Reducing salt-and-pepper noise Salt-and-pepper noise: Produced when captured the image from the microscope J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
67 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
68 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our method: Reducing salt-and-pepper noise Salt-and-pepper noise: Produced when captured the image from the microscope Solution: 1 Low-pass filter: 2 Threshold: 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 Discrimination of pixels depending on their intensity A binary image is obtained In our case (after experimentation): Huang s method J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 29/39
69 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our method: Reducing salt-and-pepper noise Original image: J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
70 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our 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 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our method: Reducing salt-and-pepper noise After threshold: J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
72 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our method: Reducing salt-and-pepper noise Undesirable elements: J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 30/39
73 Using Persistent Homology in our problem Our method: Reducing salt-and-pepper noise Our 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 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements Simplicial Complexes from Digital Images J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 32/39
75 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements Simplicial 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 D 0 D 1... D m = D a filtration of an image induces a filtration of simplicial complexes J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 32/39
76 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements 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 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements Let us construct a filtration from the processed Z-projection: 1 D = D m is the processed Z-projection J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
78 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements A 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
79 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements A 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
80 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements A 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
81 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements A 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 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 33/39
82 Using Persistent Homology in our problem Our method: Dismiss irrelevant elements A 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 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements D 0 D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 34/39
84 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements Remember: the neuron persists in all the slices J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
85 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements Remember: the neuron persists in all the slices x 0 x 1 x 2 x 3. D 0 D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
86 Using Persistent Homology in our problem A filtration of the Z-projection Our method: Dismiss irrelevant elements Remember: the neuron persists in all the slices x 0 x 1 x 2 x 3. D 0 D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 The components of the neuron persist all the life of the filtration J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 35/39
87 Final result Using Persistent Homology in our problem Our method: Dismiss irrelevant elements J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 36/39
88 Final result Using Persistent Homology in our problem Our method: Dismiss irrelevant elements J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 36/39
89 Outline Conclusions and Further work 1 Motivation 2 Persistent Homology 3 The concrete problem 4 Using Persistent Homology in our problem 5 Conclusions and Further work J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 37/39
90 Conclusions and Further work Conclusions and Further work Conclusions: Method to detect neuronal structure based on persistent homology J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 38/39
91 Conclusions and Further work Conclusions 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 Theory J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 38/39
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 Informática Mirian Andrés March 20, 2012 Partially supported by Ministerio de Educación y Ciencia, project MTM C02-01, and by European Commission FP7, STREP project ForMath, n J. Heras, G. Mata and J. Rubio Neuronal structure detection using Persistent Homology 39/39
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