1. Watershed Based Visualization of High-Density EEG Coherence
Michael ten Caat1,2, Natasha M. Maurits2,3, Jos B.T.M. Roerdink1,2
1. Institute for Mathematics and Computing Science, University of Groningen
2. BCN Neuroimaging Center, University of Groningen
3. Department of Neurology, University Medical Center Groningen
ISMM, Rio de Janeiro, October 2007

2. Outline Introduction electroencephalography (EEG) coherence
Data-driven visualization:
Existing: maximal clique based (MCB)
Novel: watershed based (WB)
Comparison MCB and WB
Conclusion, Discussion, Questions

3. EEG coherence
EEG measures electrical brain activity using up to 512 electrodes
Synchronous electrical activity between brain regions is assumed to imply functional relationships between these regions
Measure for this synchrony:
EEG coherence
Coherence: similarity of a pair of signals, as a
function of frequency (values between 0 and 1)
Generally computed per frequency band

4. Data Participant listened to high (15%) and low (85%) tones
High tones were counted
20 brain responses to high tones were recorded in 20 segments, using 128 EEG electrodes (high-density EEG)
Significance threshold for coherence: θ = 1 - p1/(L-1)
(p = probability, L = no. segments)
Here, for L = 20 and p = 0.05 : θ = 0.15
High-density EEG refers to EEG recorded with large number of electrodes.

5. Conventional data-driven visualization
1-3 Hz
significance threshold
coh > 0.15
top 10%
coh > 0.37
top 1%
coh > 0.91
All electrodes and all significant coherences
Result: visually cluttered edges

6. Data-driven ROI: Functional Unit (FU)
1-3 Hz
Functional Unit (FU): spatially connected clique
(clique = maximally connected subgraph)
ROI
Volume is the motivation to define a functional unit as a spatially connected clique.
Volume conduction: activity from one source recorded by multiple electrodes [Holsheimer & Feenstra, 1977]

7. Functional Unit Map

8. Functional Unit Map

9. 1. Define neighborship electrodes
Voronoi
diagram
of (planar
projection of)
electrode
positions
For example, the blue vertex surrounded by fourVoronoi-edges (red) has four spatial Voronoi neighbors (red)
For each point in a cell, the closest electrode is the electrode in that cell

10. 2. Find spatially connected cliques
1-3 Hz
Larger ROIs are assumed to correspond to stronger source signals and are therefore considered to be more interesting.
=> Detect spatially connected maximal cliques.
Functional unit: spatially connected maximal clique

11. 3. Determine functional units
FU visualized as connected set of cells with identical gray value
1-3Hz

12. 3. Determine functional units
FU visualized as connected set of cells with identical gray value
1-3Hz
The sum of all coherences in a Voronoi-connected maximal clique is referred to as total strength.
a - The Voronoi-connected maximal cliques are sorted by their total strength, from high to low.
b - The vertex set corresponding to the one with the highest total strength is labeled W1.
c - The vertices in W1 are assigned the label 1 and are removed from the remaining (maximal) cliques.
d - If the remaining cliques are not Voronoi-connected any
more, they are split into Voronoi-connected components.
e - For every changed clique the total strength is recomputed,
and the clique is inserted in an appropriate location in the
sorted list of cliques.
This procedure (a - e) is repeated, until all vertices received a label, or until all maximal cliques have been considered.
This results in vertex sets labeled W_1, ...,W_M. Every vertex set W_i (i 2 {1, 2, ...,M}) is considered to be a functional unit, being a clique consisting of Voronoi-connected vertices.

13. 3. Determine functional units
FU visualized as connected set of cells with identical gray value
Adjacent FUs different gray values
Geographic center of an FU indicated by cross inside circle
1-3Hz

14. 4. Show only FUs larger than 5 cells
1-3Hz

15. 5. Inter-FU coherence Calculate average coherence S between FUs
Lines connect FUs if S > significance threshold
Color mapped to value S
1-3Hz

16. Time complexity
Because maximal clique detection has time complexity O(3^{n/3})

17. Goal
Because maximal clique detection has time complexity O(3^{n/3})

18. Watershed Based FU Maps
Initialization:
Define coherence of vertex v as average coherence between v and its Voronoi neighbors
Select all local maxima as markers
Label markers and associate markers with basins
Initialize edge queues with edges between markers and their spatial neighbors (i.e. Voronoi neighbors)
Sort edges in decreasing order of value
Extend basins as follows, see:
Illustration
Pseudocode

19. Watershed Based FU maps
basin B and
Voronoi-neighbors
queue: edges
between B and
Voronoi neighbors
candidates
connected
to all w in B
Electrode
(Imaginary example)
Left: blue rectangles are basin vertices, green crosses are Voronoi neighbors of this basin (4-connected)
Middle: goldish lines are edges between basin vertices and Voronoi neighbors (put in a sorted edge queue from high to low values)
Right: red rectangles are candidates that may extend the basin; imagine that every red rectangle is connected each of the blue rectangles in the underlying graph.
B = basin (clique)
V = {v | v is Voronoi neighbor of a basin vertex}
C = candidates = { v | v connected to all w in B}
queue = {ei(vi,wi) = | vi = , wi = : ek > em iff k < m}

20. Watershed Based FU maps
dequeue first edge
from ‘queue’ ;
w not in C
dequeue first edge;
w in C
add w to B w w w
Electrode
Left: return and remove the first edge from the queue (i.e. the edge with the highest value) between a blue rectangle and a green cross.
The vertex corresponding to the green cross is not in the candidates set. Nothing happens.
Middle: the next edge is between a basin vertex (blue rectangle) and a candidate vertex (red rectangle).
The candidate vertex (w) is
B = basin (clique)
V = {v | v is Voronoi neighbor of a basin vertex}
C = candidates = { v | v connected to all w in B}
queue = {ei(vi,wi) = | vi = , wi = : ek > em iff k < m}

21. Watershed Based FU maps
add Voronoi neighbors
v of w to V ;
add e(w,v) to queue
remove vertices
not connected to w
from C w w w
Electrode
Left is same as last ‘graph’ from previous slide.
Middle: edges between w and spatial neighbors of w (green crosses) are added to queue (goldish lines)
Right: imagine that some red rectangles are not connected to w in the underlying graph, and remove these.
B = basin (clique)
V = {v | v is Voronoi neighbor of a basin vertex}
C = candidates = { v | v connected to all w in B}
queue = {ei(vi,wi) = | vi = , wi = : ek > em iff k < m}

22. Watershed Based FU Maps
1 while queue not empty do
2 e(v,w) ← dequeue(queue) (* v in basinL(v) *)
3 if L(w) = 0 (* w not in any basin *)
4 if w in candidatesL(v) then
5 candidatesL(v) ← intersect(candidatesL(v),adjcohw)
6 L(w) ← L(v)
7 basinL(v) ← insertVertexSort(w,basinL(v))
8 for all v* in adjvorw do
if L(v*)=0 then insertEdgeSort(e(w,v*),queue) end if
10 end for
11 end if
12 end if
13 end while
basin_i vertices in basin (sorted by vertex nr)
e(v,w) edge between vertex v and w
L(v) basin label of vertex v
adjcohw vertices adjacent to w in the coherence graph (sorted by nr)
adjvorw Voronoi neighbors of w (sorted by nr)
queue list of edges sorted in decreasing order; e(v,w) is added v is put into a basin for every unlabeled Voronoi neighbor w of v
As long as the edge queue is not empty
Return and remove the first edge e(v,w) from the queue
Vertex v was labeled when this edge was inserted, w was still unlabeled. If w still unlabeled (i.e. w still not in any basin) then
If w is connected to each basin vertex then
Intersect the candidates with the neighbors of w (in the coherence graph)
Assign to w the same basin label as v
Add w to the basin which contains v
For each Voronoi neighbor v* of w
If v* still unlabeled (i.e. not in any basin yet) then add e(w,v*) to the queue.

23. Watershed Based FU Maps
initialization queue: O(n2 log n)
1 while queue not empty do
2 e(v,w) ← dequeue(queue) (* v in basinL(v) *)
3 if L(w) = 0 (* w not in any basin *)
4 if w in candidatesL(v) then
5 candidatesL(v) ← intersect(candidatesL(v),adjcohw)
6 L(w) ← L(v)
7 basinL(v) ← insertVertexSort(w,basinL(v))
8 for all v* in adjvorw do
if L(v*)=0 then insertEdgeSort(e(w,v*),queue) end if
10 end for
11 end if
12 end if
13 end while
O(m) = O(n2) : m = # edges; n = # vertices = # electrodes
O(log n) : binary search in sorted set candidates
O(n) : intersection two vertex sets of size at most n
O(n log m) = O(n log n) : insert at most n edges in sorted edge list
at most
n times }
Initialization of queue: occurs before the pseudocode shown here. Its time complexity is O(n^2\log{n}).
Main loop (is this pseudo code): Total time complexity at first sight O(n^3\log{n}).
Last three steps executed at most n times.
Total time complexity actually O(n^2\log{n}).
Total time-complexity: O(n2 log n)

24. FU Map: maximal clique based, watershed based method
1-3 Hz
Dataset
max.
clique
based
water-
shed
based
Comparison. First 1-3Hz frequency band.
Similarity between FU map for MCB and WB method (per dataset)
Similar FU locations
Similar inter-FU connections
Largest difference for dataset 2: larger number of inter-FU connections for WB method – come back to that in next slide.

25. FU Map: maximal clique based, watershed based method
13-20 Hz
FU Map: maximal clique based, watershed based method
1-3 Hz, dataset 2
max. clique based,
p=.05
Comparing the middle with the left FU map, the middle one shows over-segmentation (accepting the MCB as the gold standard).
Six inter-FU connections for MCB method for p=.05 (left) similar to inter-FU connections with higher value for WB method (see right).
Even though adapting a threshold is not the fanciest you can do, this indicates that the inter-FU connections with the highest value (i.e. the most important ones) correspond between the MCB and the WB method.
(The discussion will mention a different solution.)
watershed based,
p=.05
watershed based,
p=.001
‘oversegmentation’

26. FU Map: maximal clique based, watershed based method
13-20 Hz
Dataset
max.
clique
based
water-
shed
based
For higher frequency band (13-20Hz) even more similarity between MCB and WB FU maps.

27. Interpretation Results
Simultaneous activity
- at a more global scale for a lower EEG frequency
- at a more local scale for a higher EEG frequency
[Nunez et al, Electroenceph Clin Neurophysiol 103, 5 (1997)]
In accordance, we observe less inter-FU connections for higher frequencies
Connections between anterior and posterior FUs are probably associated with the two most important sources of activity for this type of data [Comerchero & Polich, Clin Neurophysiol 110, 1 (1999)]
Correspondingly, we observe connections between anterior and posterior FUs
This slide is not required for the ISMM presentation.
However, it motivates the use of the FU map in general.
Given enough time, I would present this slide.

28. Performance maximal clique based and watershed based method
Time complexity
Avg. FU detection time
Maximal clique
based
O(3n/3)
24 s
Watershed
O(n2 log n)
0.06 s
For the datasets used, there was hardly any difference in FU detection between datasets.
(In our paper submitted to IEEE TVCG, the difference is less consistent but the WB method is always (up to 100,000 times) faster.)
n = no. electrodes (up to 512)
Average speedup: 400 x

29. Conclusion & Discussion
Functional unit (FU) map
FU = data-driven ROI = spatially connected clique
data-driven EEG coherence visualization
FU detection
- Existing method
Maximal clique based (MCB)
Time complexity O(3n/3)
- Novel method
Watershed based (WB)
Time complexity O(n2 log n)
Result:
- WB method on average 400x faster than MCB method
- FU maps: MCB and WB similar, but
Oversegmentation:
adapting significance threshold increases similarity
future: merge spatially neighboring FUs if together one FU

30. Future Work
Case studies: aging, mental fatigue
Group analysis