Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Avoiding artifacts in spectral white matter fiber clustering and embedding Demian Wassermann & Rachid Deriche Odyss´ ee project - INRIA DMRI ARC kick off meeting 2007 Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress “Tractography applied to the tensor field in diffusion tensor imaging (DTI) results in sets of streamlines which can be associated with major fiber tracts. If fibers are reconstructed and visualized individually through the complete white matter, the display gets easily cluttered making it difficult to get insight in the data.” [ ? ] Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Spectral embedding algorithms 1 Spectral embedding and clustering pipeline 2 overview Step 1: From data to affinity Step 2: From affinity to embedding Step 3: From embedding to clustering Conclusions 3 Takehome There are other techniques Work in progress 4 Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Why spectral embedding Why spectral embedding Spectral embedding algorithms provide spectral representation of the data perform non-linear embedding into an euclidean representation are simple to implement can be solved efficiently by standard linear algebra software very often they outperform traditional embedding algorithms allow to perform statistics and clustering in an embedding space which is simpler Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Why spectral embedding Why spectral embedding Spectral embedding algorithms provide spectral representation of the data perform non-linear embedding into an euclidean representation are simple to implement can be solved efficiently by standard linear algebra software very often they outperform traditional embedding algorithms allow to perform statistics and clustering in an embedding space which is simpler Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Why spectral embedding Why spectral embedding Spectral embedding algorithms provide spectral representation of the data perform non-linear embedding into an euclidean representation are simple to implement can be solved efficiently by standard linear algebra software very often they outperform traditional embedding algorithms allow to perform statistics and clustering in an embedding space which is simpler Previous works applying spectral embedding and clustering to fiber tracts: [ ? ],[ ? ],[ ? ]. Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Spectral embedding Hypothesis Spectral embedding Hypothesis Isometry of the embedding: after a distance is defined 1 between tracts, the learned manifold should preserve the distance relation. Uniform sampling of the elements: the density of the 2 extracted tracts changes if and only if these tracts belong to anatomically different bundles. Convexity of the original space: if two elements are in the 3 data set, almost all of the intermediate tracts obtained by interpolation are in the data set. Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress overview Spectral embedding and clustering pipeline data affinity matrix embedding clustering Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress overview data affinity matrix embedding clustering Given a set of elements as data, Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress overview 0 1 ’affinity.txt’ matrix 0.9 data 50 0.8 100 0.7 150 0.6 affinity matrix 200 0.5 250 0.4 0.3 300 embedding 0.2 350 0.1 400 0 0 50 100 150 200 250 300 350 400 clustering and a symmetrical similarity measure, an affinity matrix A is obtained, A ij is the similarity between element i and j , Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress overview -0.015 data -0.01 -0.005 second eigenvector affinity matrix 0 0.005 embedding 0.01 0.015 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 first eigenvector clustering then, and euclidean representation (embedding) of the data is obtained. Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress overview data affinity matrix embedding clustering Finally, a clustering algorithm is ap- plied in the euclidean representation and then the clusters are used to group the data Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 1: From data to affinity Given a set of N fibers, f 1 , · · · , f N , and the data representation of two fibers f i := f i 1 , f i 2 . . . , f i | f i | ∈ R 3 affinity matrix f j := f j 1 , f j 2 . . . , f j | f j | ∈ R 3 Several distance metrics d ( f i , f j ) have embedding been proposed between them, for in- stance, Brun : ff 1 − f j | fi | − f j | fi | − f j 1 − f j clustering � f i 1 � + � f i | fj | � , � f i 1 � + � f i min | fj | � 2 O’Donnell : k − fj l min k ( � fj k min l ( � fi l − fi P l � ) P k � ) + | fi | | fj | 2 Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 1: From data to affinity Finally, the affinity matrix is usually de- data fined as: � − d 2 ( f i , f j ) /σ 2 � A ij := exp affinity matrix where σ is a scale space parameter embedding clustering Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 2: From affinity to embedding The most used method in order to per- data form the clustering is N-cuts which is equivalent to the Laplacian Eigenmaps embedding. affinity matrix D := rowSum ( A ) D − 1 2 AD − 1 W := 2 embedding L := rowSum ( W ) − W Calculate the eigenvectors v 1 , · · · , v N of clustering L , where the corresponding eigenvalues are sorted 0 = λ 0 ≤ · · · ≤ λ N . Take the embedding function e ( · ) for the fiber f i : e ( f i ) = ( v 2 i , v 3 i , · · · , v d i ) , d << N Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 2: From affinity to embedding Synthetical results data affinity matrix embedding -0.015 -0.01 clustering -0.005 second eigenvector 0 0.005 0.01 0.015 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 first eigenvector Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 2: From affinity to embedding But what if the hypothesis are not met? data Uniform sampling affinity matrix embedding clustering Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 2: From affinity to embedding But what if the hypothesis are not met? data Uniform sampling affinity matrix embedding clustering third eigenvector third eigenvector second eigenvector first eigenvector Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 2: From affinity to embedding Does real data meet the hypothesis? data affinity matrix embedding clustering Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...
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