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

QuickTime™ and a decompressor

are needed to see this picture.

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

QuickTime™ and a decompressor

are needed to see this picture.

Bonferroni FDR

GLM

QuickTime™ and a decompressor

are needed to see this picture.

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

QuickTime™ and a decompressor

are needed to see this picture. QuickTime™ and a decompressor

are needed to see this picture.

circles: accidentssquares: assaults

Lidar

Light Detection and Ranging

40-150K pulses/sec

Mounted on airplane

Used to measure canopy heights

Multiple returns

QuickTime™ and a decompressor

are needed to see this picture.

Lidar-derived canopy heights

QuickTime™ and a decompressor

are needed to see this picture.

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.

QuickTime™ and a decompressor

are needed to see this picture.

ψ (x,y) = (1− x2 − y2 )exp(− 12 (x2 + y2 ))

ψ a,b (λ) =1

aψ

λ − b

a⎛⎝⎜

⎞⎠⎟

QuickTime™ and a decompressor

are needed to see this picture.

Quality of lidar/wavelet estimates