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Spatial wavelet analysis
Discrete
fMRITesting for active regions
Bootstrapping functional connectivity
Continuous
Lidar
Spatial wavelets
For now assume gridded data Zx,y, x=0,...,M-1; y=0,...,N-1 where N and M are dyadic integers. Recall from 1-d wavelets that we have a smoothing filter g and a differencing filter h. The two-dimensional wavelet convolves the image with four product filtershorizontal ghvertical hgdiagonal hhsmoother gg
Wx,y,1(h) =(gh∗Z2x,2y ) = gkhlZ2x+1−k mod M,2y+1−l mod N
l=0
L−1
∑k=0
L−1
∑
The filters
An image
Next step
Apply the same technique to the smoothed image from the previous step.
high freq= shortdistance
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fMRI
Functional Magnetic Resonance Imaging experiments aim to relate sensory stimuli to brain activity
Testing for active region
We are interested in testing correlation between brain activity and stimuli. Somewhat simplified, we consider pixels indexed by n, and for each pixel observe a time series y(n)=(y(n,t),t=1,...,N) of values. A simple model has and we estimate a contrast cT(n) by
The null hypothesis is that there is no activity, so cT(n) = 0.
y(n) =X(n) + ε(n)
cT (XTX)−1XTy(n)
Multiple testing
We want to test the null hypothesis for a large number V of pixels. A Bonferroni correction performs each test at level /V. Since the voxels may have substantial spatial dependence, this is likely extremely conservative (and has low power).
The wavelet advantage
Since the wavelet coefficients are (nearly) uncorrelated, we can test the corresponding contrasts using the wavelet coefficients, and set those coefficients that are not significant to zero, and then do the inverse transform to reconstruct the image
False detection rate
Instead of doing Bonferroni test, one can use the FDR = E(# false positives)/E(# positives)
FDR ≤ iff P(i) ≤ i /V where the P(i) are ordered P-values
Can be applied to either type of testing
Some results
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Bonferroni FDR
GLM
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Wavelets
Functional connectivity
Measure of spatio-temporal correlations between spatially distinct regions of cerebral cortex
Look at MRI data in regions of interest
Estimate correlation from (averaged) time series in regions
Regions of interest (r=.445)
Control region (r=.008)
Spatial wavestrap
Spatial bootstrap can be done using sufficiently separated blocks
Does not work if correlation range large
Alternatively:
•resample wavelet coefficients
•reconstruct image
•recalculate statistic of interest
Discrete wavelet packet transform
More general division of spatial frequencies
At each level, an image is divided into four subimages according to a quadtree (instead of horiz, vert, diag)
Instead choose where in the frequency spectrum to split
Wavelets and point process intensity estimation
A point process is a (finite) set of random locations.
Intensity:
(x)dx = Pr(point within dx of x)
Use wavelet reconstruction (deleting small components) of counts in smallest squares to estimate the intensity function
Data: emergency room visits of victims of urban violence
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circles: accidentssquares: assaults
Lidar
Light Detection and Ranging
40-150K pulses/sec
Mounted on airplane
Used to measure canopy heights
Multiple returns
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Lidar-derived canopy heights
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raw data topography
Continuous wavelet approach
Mexican hat wavelet
Dilated over scales
A sequence of Mexican hat wavelets are convolved with the lidar-derived crown height model. When the scale and location are “right” we get a good fit metric. Yields both crown diameter and height.
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ψ (x,y) = (1− x2 − y2 )exp(− 12 (x2 + y2 ))
ψ a,b (λ) =1
aψ
λ − b
a⎛⎝⎜
⎞⎠⎟
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Quality of lidar/wavelet estimates