Avoiding artifacts in spectral white matter fiber clustering and - - PowerPoint PPT Presentation

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

1

Spectral embedding algorithms

2

Spectral embedding and clustering pipeline

• verview

Step 1: From data to affinity Step 2: From affinity to embedding Step 3: From embedding to clustering

3

Conclusions Takehome There are other techniques

4

Work in progress

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

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Isometry of the embedding: after a distance is defined between tracts, the learned manifold should preserve the distance relation.

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Uniform sampling of the elements: the density of the extracted tracts changes if and only if these tracts belong to anatomically different bundles.

3

Convexity of the original space: if two elements are in the 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

• verview

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

• verview

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

• verview

data affinity matrix embedding clustering

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 ’affinity.txt’ matrix

and a symmetrical similarity measure, an affinity matrix A is obtained, Aij 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

• verview

data affinity matrix embedding clustering

• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

• 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 second eigenvector first eigenvector

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

• verview

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

data affinity matrix embedding clustering Given a set of N fibers, f 1, · · · , f N, and the representation of two fibers f i := f i

1, f i 2 . . . , f i |f i| ∈ R3

f j := f j

1, f j 2 . . . , f j |f j| ∈ R3

Several distance metrics d(f i, f j) have been proposed between them, for in- stance, Brun :

min  f i

1−f j 1+f i |fi |−f j |fj |,f i |fi |−f j 1+f i 1−f j |fj |

ff 2

O’Donnell :

P k minl (fi k −fj l ) |fi |

+

P l mink (fj l −fi k ) |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

data affinity matrix embedding clustering Finally, the affinity matrix is usually de- fined as: Aij := exp

• −d2(f i, f j)/σ2

where σ is a scale space parameter

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

data affinity matrix embedding clustering The most used method in order to per- form the clustering is N-cuts which is equivalent to the Laplacian Eigenmaps embedding. D := rowSum(A) W := D− 1

2 AD− 1 2

L := rowSum(W) − W Calculate the eigenvectors v1, · · · , vN of L, where the corresponding eigenvalues are sorted 0 = λ0 ≤ · · · ≤ λN. Take the embedding function e(·) for the fiber f i: e(f i) = (v2

i , v3 i , · · · , vd 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

data affinity matrix embedding clustering Synthetical results

• 0.015
• 0.01
• 0.005

0.005 0.01 0.015

• 0.005

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 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

data affinity matrix embedding clustering But what if the hypothesis are not met? Uniform sampling

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

data affinity matrix embedding clustering But what if the hypothesis are not met? Uniform sampling

third eigenvector first eigenvector second eigenvector third 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

data affinity matrix embedding clustering Does real data meet the hypothesis?

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 3: From embedding to clustering

data affinity matrix embedding clustering “Finally a clustering algorithm is applied” Which clustering algorithm? Usually k-means. How many clusters are we expecting to find? In order to answer that we use the eigenvalue gap heuristic.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 3: From embedding to clustering

data affinity matrix embedding clustering “Finally a clustering algorithm is applied” Which clustering algorithm? Usually k-means. How many clusters are we expecting to find? In order to answer that we use the eigenvalue gap heuristic.

0.00355 0.0036 0.00365 0.0037 0.00375 0.0038 0.00385 0.0039 0.00395 10 5 1 eigenvalue

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 3: From embedding to clustering

data affinity matrix embedding clustering When the hypothesis are met

0.00355 0.0036 0.00365 0.0037 0.00375 0.0038 0.00385 0.0039 0.00395 10 5 1 eigenvalue

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 3: From embedding to clustering

data affinity matrix embedding clustering When the hypothesis are not met

0.004 0.0041 0.0042 0.0043 0.0044 0.0045 0.0046 0.0047 0.0048 0.0049 0.005 10 5 1 eigenvalue

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Step 3: From embedding to clustering

data affinity matrix embedding clustering Does real data meet the hypothesis?

0.01 0.011 0.012 0.013 0.014 0.015 0.016 10 5 1 eignevalue

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Takehome

Takehome #1 The spectral embedding and clustering algorithms rely on three hypothesis Takehome #2 Open question: Do white matter fiber tracts meet the hypothesis?

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Takehome

Takehome #1 The spectral embedding and clustering algorithms rely on three hypothesis Takehome #2 Open question: Do white matter fiber tracts meet the hypothesis? Conjecture White matter fiber tracts do not meet the hypothesis.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Takehome

Conjecture White matter fiber tracts do not meet the hypothesis.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Takehome

Conjecture White matter fiber tracts do not meet the hypothesis. Solution Use or develop techniques with a different hypothesis set

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress Takehome

Conjecture White matter fiber tracts do not meet the hypothesis. Solution Use or develop techniques with a different hypothesis set Example The diffusion maps embedding technique does not require uniform sampling.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress There are other techniques

“The diffusion maps embedding technique does not require uniform sampling.”

1e-08 1.1e-08 1.2e-08 1.3e-08 1.4e-08 1.5e-08 1.6e-08 1.7e-08 1.8e-08 1.9e-08 2e-08 10 5 1 eigenvalue

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Work in progress

Use or develop techniques with a different hypothesis set. Perform a whole white matter fiber tract clustering.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Work in progress

Use or develop techniques with a different hypothesis set. Perform a whole white matter fiber tract clustering.

• btain real data

embedding and clustering validate

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Work in progress

Use or develop techniques with a different hypothesis set. Perform a whole white matter fiber tract clustering.

• btain real data

embedding and clustering validate we need real data with a provided ground truth

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Questions?

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Questions? Thank you

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

• A. Brun, H.J. Park, H. Knutsson, M. E. Shenton, and C.F

. Westin. Clustering fiber traces using normalized cuts. In Medical Image Computing and Computer-Assisted Intervention MICCAI 2004, volume 3216 of Lecture Notes in Computer Science, pages 518–529. Springer/Verlag, 2004.

• M. Maddah, A. U. J. Mewes, S. Haker, W. E. L. Grimson,

and S. K. Warfield. Automated atlas-based clustering of white matter fiber tracts from dtmri. In Medical Image Computing and Computer-Assisted Intervention MICCAI 2005, volume 3749 of Lecture Notes in Computer Science, pages 188–195. Springer Berlin / Heidelberg, 2005.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...

Spectral embedding algorithms Spectral embedding and clustering pipeline Conclusions Work in progress

Lauren O’Donnell and Carl-Fredrik Westin. High-dimensional white matter atlas generation and group analysis. In Rasmus Larsen, Mads Nielsen, and Jon Sporring, editors, MICCAI, volume 4191 of Lecture Notes in Computer Science, pages 243–251. Springer, 2006.

Wassermann & Deriche - Odys´ ee - INRIA Spectral embedding & clustering ...