PLATELETS FOR MULTISCALE ANALYSIS IN PHOTON-LIMITED IMAGING
Rebecca Willett and Robert Nowak, ECE Department, Rice University
Approximation Rates
Wedgelet Partitions
Photon-Limited Imaging
Photon-limited imaging
encompasses applications
such as PET, SPECT,
confocal microscopy,
astronomical imaging, and IR imaging.
New alternatives to traditional wavelet
analysis are required to tackle
C
Haar Wavelet Partition
Wedgelet Partition
Poissonian Observations
Image Curves
Inverse Problems
Beyond Wavelets
2
C
Original Image
MPLE Algorithm
2
Error decay rates:
-1/2)
 Fourier: O(m
-1
 Wavelets: O(m )
-1
 Wedgelets: O(m )
-2
 Platelets: O(m )
Data

Const Estimate

Wedge Estimate

Platelet Estimate


Maximum Penalized
Likelihood Estimation
Platelet Approximations
Goal: Maximize the penalized likelihood
Wedged Platelet
Estimate
Inherit from
finer scale
Nuclear Medicine
5
4.5
4
3
2.5
2
1.5
1
Platelets
Wedgelets
0.5
50
5
4.5
4.5
4
4
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
100
150
Terms = 1164
200
150
200
250
Terms = 2068
5
50
100
250
is the likelihood
is a smoothing parameter
is the number of parameters in
So the MPLE is
Confocal Microscopy
2.6
MLE (min error level shown in dots)
Piecewise Constant
Platelet
2.4
2.2
50
100
150
200
250
2
Terms = 510
Error
Platelets are localized atoms at various
scales, locations, and orientations that
can produce highly accurate, piecewise
linear approximations to images
consisting of smooth regions separated
by smooth boundaries. Platelets
generalize Donoho’s wedgelets and are
capable of concisely representing both
smooth surfaces and edges.
Haar
3.5
1.8
1.6
Supported by NSF, ARO, and ONR
http://www.ece.rice.edu/~willett/
Estimate depends on partition and
parameters.
1.4
1.2
2
4
6
8
Iteration
10
12
14