1. Spatial wavelet analysis Discrete fMRI Testing for active regions Bootstrapping functional connectivity Continuous Lidar

2. Spatial wavelets For now assume gridded data Z x,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 filters horizontalgh verticalhg diagonalhh smoothergg

3. The filters

4. An image

5. Next step Apply the same technique to the smoothed image from the previous step. high freq = short distance

7. fMRI Functional Magnetic Resonance Imaging experiments aim to relate sensory stimuli to brain activity

8. 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 c T  (n) by The null hypothesis is that there is no activity, so c T  (n) = 0.

9. 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).

10. 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

11. 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

12. Some results Bonferroni FDR GLM Wavelets

13. 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)

14. 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

15. 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

18. 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

19. circles: accidents squares: assaults

20. Lidar Light Detection and Ranging 40-150K pulses/sec Mounted on airplane Used to measure canopy heights

21. Multiple returns

22. Lidar-derived canopy heights raw data topography

23. 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.

25. Quality of lidar/wavelet estimates