# Metric Operations

## Clustering

Metric values of nodes that do not meet the cluster parameters are set to zero.

• Surface - The surface in the Main Window is used for determining the neighbors of nodes.
• Metric Column - Metric column on which clustering is performed.
• Minimum Surface Area - Minimum surface area of clusters that are retained.
• Positive Maximum - The most positive value for clusters of positive regions.
• Positive Minimum - The least positive value for clusters of positive regions.
• Negative Minimum - The least negative value for clusters of negative regions.
• Negative Maximum - The most negative value for clusters of negative regions.

## Smoothing

Metric smoothing replaces a node's metric (or surface shape) value using a weighted, or unweighted, average of the node's metric value with the metric values of the node's neighbors. Smoothing essentially removes the noise (reduces any local minima or maxima). Smoothing can also be thought of as a low-pass filter.

• Surface - The surface in the Main Window is used for determining the neighbors of nodes.
• Metric Column - Metric column on which clustering is performed.
• Algorithm
• Average Neighbors - A node's new value is some average of its current metric value and the metric values from the node's immediate neighbors.
• Full Width Half Maximum - Smoothing is performed until a Full Width Half-Maximum (FWHM) is reached.
• Weighted Average Neighbors - Similar to Average Neighbors except that nearby nodes receive a higher weighting than node's further away.
• Strength - Determines influence of neighbors when smoothing. A value of one is 100% neighbors and a value of zero is 0% neighbors (no smoothing).
• Iterations - Iterations of smoothing that are performed.
• FWH - The desired Full Width Half-Maximum value used by the FWHM algorithm.

Several algorithms use inter-neighbor distances. It is important that the Fiducial (Anatomical) Surface is used for smoothing.

### Average Neighbors

Average neighbors smooths a node by averaging its metric value with the metric values of the node's immediate neighbors.

For node Mx with N neighbors Mi, one iteration of smoothing is:

• $M_x = (strength * \frac{\sum_{i=1}^N M_i}{N}) + ( (1.0 - strength) * M_x)$

### Dilation

Dilation spreads the metric values without altering any non-zero metric values.

For node Mx with a value of zero and N NON-ZERO neighbors Mi, one iteration of smoothing is:

• $M_x = \frac{\sum_{i=1}^N M_i}{N} \forall M_x$ = 0

### Full Width Half Maximum (FWHM)

The FWHM smoothing algorithm is based upon Hagler DJ Jr, Saygin AP, Sereno MI. Smoothing and cluster thresholding for cortical surface-based group analysis of fMRI data. Neuroimage. (2006) 33:1093–1103. PubMed.

For node Mx with N neighbors Mi, one iteration of smoothing is average of the node and its neighbors:

• $M_x = \frac{((\sum_{i=1}^N M_i) + Mx)}{N + 1}$

The number of iterations provided by the user is the maximum number of iterations of smoothing that are performed using the Full Width Half Maximum Algorithm. Prior to each iteration of smoothing, the Full Width Half Maximum is estimated using equations below. If the calculated FWHM exceeds the FWHM provided by user, smoothing ceases.

• dv = Average Euclidean (straight-line) Inter-Neighbor Distance for all nodes.
• var(ds) = Variance of node and neighbor metric differences.
• var(s) = Variance of node metric values
• FWHMsurf = $dv \sqrt{\frac{-2ln(2)}{ln(1 - \frac{var(ds)}{2 var(s)})}}$

### Weighted Average Neighbors

Weighted Average Neighbors is similar to Average Neighbors except that neighbors closer to the node being smoothed receive a higher weighting than neighbors further away.

Let:

• Di = distance(Mx,Mi)
• $D = \sum_{i=1}^N D_i$
• $W_i = 1.0 - (\frac{D_i}{D})$
• $W = \sum_{i=1}^N W_i$

For node Mx with N neighbors Mi, one iteration of smoothing is :

• $M_x = (strength * \sum_{i=1}^N (M_i * (W_i / W))) + ( (1.0 - strength) * M_x)$

### A Smoothing Analysis

This example uses data from Jason Hill's neonate study. The average neighbors algorithm is used with 10 iterations and strength of 1.0.

 Before Smoothing Iterations=4, Strength=0.5 Iterations=10, Stength=1.0

To further quantify the amount of smoothing this represents, we created a metric map with zero values across the entire inter24 fiducial surface, right hemisphere. Next, we set the values of a handful of isolated nodes in known regions of interest to 10.0. Then, we smoothed the map 10 iterations at 1.0 strength. The resulting map and histogram are shown in the table below:

 Isolated 10.0 valued nodes after 10 iterations smoothing at strength 1.0 Histogram after 10 iterations smoothing at strength 1.0

Note the scale (0-0.01). The peak of the smoothed map was 0.26, so we thresholded it at half that value, 0.13, and found the area of the resulting clusters:

Surface: FIDUCIAL inter24_PALS-inter24_Rhem_Fiducial_AVERAGE_AFFTRANS_Pass2.R.Fiducial_MidVoxel.coord
Topology: OPEN Human.OPEN_sphere_6.73730.topo
69740 of 73730 nodes in region of interest
Region of Interest Surface Area: 29032.1
Region Mean Distance Between Nodes: 0.69286

Threshold Column Num-Nodes Area Area Corrected COG-X COG-Y COG-Z
0.13000 1 31 15.64660 21.67346 40.05918 4.35479 -21.19949
0.13000 1 31 17.97602 25.84802 46.11361 -31.20967 -8.25840
0.13000 1 31 15.67979 17.26593 30.08122 -13.26249 11.01580
0.13000 1 31 11.51662 17.91896 40.75745 -2.32964 35.74964
0.13000 1 31 19.16867 19.47389 35.19091 -9.88959 27.80424


The corrected area of the largest cluster was 25.8. Dividing by pi and taking the square root gives:

Area=πRadius2=3.14Radius2=25.8