1. Multiscale Likelihood Analysis and Inverse Problems in Imaging
Rob Nowak
Rice University
Electrical and Computer Engineering
Supported by DARPA, INRIA, NSF, ARO, and ONR

2. Image Analysis Applications
remote sensing
brain mapping
time
image restoration
network tomography

3. Maximum Likelihood Image Analysis
unknown
object
statistical
model
measurements
maximize
likelihood
physics
data
prior knowledge
Maximum
likelihood
estimate

4. Analysis of Gamma-Ray Bursts
unknown
object
statistical
model
measurements
Maximum
likelihood
estimate
maximize
physics
data
prior knowledge
photon
counting
piecewise
smoothness
estimate of
underlying intensity

5. Image Deconvolution unknown object statistical model measurements
Maximum
likelihood
estimate
maximize
physics
data
prior knowledge
noise &
blurring
image model

6. Brain Tomography unknown object statistical model measurements Maximum
likelihood
estimate
maximize
physics
data
prior knowledge
counting &
projection
anatomy &
physiology

7. Wavelet-Based Multiresolution Analysis
high resolution
image
mid resolution +
prediction errors
low resolution +
prediction errors
prediction errors  wavelet coefficients
most wavelet coefficients are zero  sparse representation

8. Wavelets and Denoising
Noisy image
Wavelet transform
Denoised
reconstruction
thresholding estimator
‘keep’ large coefficents, ‘kill’ small coefficients
near minimax optimality over range of function spaces
extensions to non-Gaussian noise models are non-trivial

9. Wavelets and Inverse Problems
linear
noise
operator
Two common approaches :
Linear reconstruction, followed by nonlinear denoise
image is easy to model in reconstruction domain,
but noise is not
Denoise raw data, followed by linear reconstruction
noise is easy to model in observation domain,
but image is not
Exception: If linear operator K is approximately diagonalized
by a wavelet transform, then both image and noise are easy
to model in both domains

10. Example: Image Restoration
convolution
noise
operator
Deblurring in FFT Domain O(N log N) :
Denoising in DWT Domain O(N) :

11. Wavelets and Inverse Problems
Fast exact methods for special operators
Donoho (1995) wavelet-vaguelette decomposition for
scale-homogeneous operators (e.g., Radon transform)
Rougé (1995), Kalifa & Mallat (1999) wavelet-packet
decompositions for certain scale-inhomogeneous convolutions
General Methods: Heavy computation or approximate solution
Liu & Moulin (1998), Yi & Nowak (1999), Neelamani et al (1999),
Belge et al (2000), Jalobeanu et al (2001), Rivaz et al (2001)
General Methods: Fast exact solutions via EM
Nowak & Kolaczyk (1999) Poisson inverse problems
Figueiredo & Nowak (2001) Gaussian inverse problems

12. Maximum Penalized Likelihood Estimation (MPLE)
= log likelihood function
Gaussian Model:
Poisson Model:

13. Image Complexity Images have structure Not a totally unorganized city
Not a completely
random brain
Images have structure

14. Location Uncertainty in Images
Location information is central to image analysis :
When are there
jumps / bursts ?
Where are the
edges ?
Where are the
hotspots ?
From limited data we can detect & locate key “structure”
“images” lie on a low-dimensional manifold
… but we don’t know which manifold

15. Complexity Penalized Likelihood Analysis
Let  = wavelet/multiscale coefficients
find  with
high likelihood
but keep model
as simple as possible
Goal: Balance trade-off between model complexity
and fitting to the data
Examples:
gamma-ray bursts: penalty  # jumps / bumps
satellite images
& brains: penalty  # boundaries / edges

16. function and maximizing this function
Basic Approach
Multiscale maximum penalized likelihood estimator (MPLE) is easy to compute if K = I
If we could directly observe the image then we would have a simple denoising problem
MPLE is very difficult to compute in indirect case – no analytic solution exists in general case (K  I)
Expectation Maximization (EM) Idea:
introduce (unobserved) direct image data
iterate between estimating direct data likelihood
function and maximizing this function
Expectation step involves simple linear filtering !
Maximization step involves simple denoising !

17. Key Ingredients
Fast multiscale analysis and thresholding-type estimators for directly observed Gaussian or Poisson data
Selection of the “right” unobserved data space
Multiscale likelihood factorizations : generalization of conventional “energy-based” MRA to “information-based” MRA
Re-interpretation of the inverse problem as combination of two imaging experiments – a fictitious direct (K=I) observation and real observed (K  I) observation

18. Multiscale Likelihood Factorizations
Assume direct observation model:
noise (possibly nonGaussian)
y =   
Orthonormal Wavelet Decomposition:
multiscale decomposition into
orthogonal energy components I I I
Multiscale Likelihood Factorization:
g(y | x) =  f (y | y ,  )
c(I) I I I
Haar analysis
factorization into independent
“information” components

19. Summary of Likelihood MRA Results
Multiresolution Analysis (MRA): Set of sufficient conditions for multiscale factorization of likelihood
Efficient Algorithms for MPLE: Linear-complexity algorithms for analogues of “denoising” methods
Risk Bounds: Risk rates like
for BV and Besov objects estimated from Gaussian or Poisson data

20. Wavelet MRA vs. Likelihood MRA
Function Space MRA
Hierarchy of nested subspaces
Orthonormal Basis in
Scaling between subspaces
Translations within subspace
Likelihood MRA
Hierarchy of data partitions
Statistical independence in
data space
Likelihood function reproduces
under summation
Conditional likelihood of
child given parent is a
single parameter density

21. Multiscale Data Aggregation
basic Haar multiscale analysis

22. Multiscale Likelihood Factorizations
original
likelihood
multiscale
factorization

23. Examples  Haar wavelet coefficient function of Haar wavelet & scaling
coefficients

24. Denoising and Information Extraction
Wavelet Denoising:
“keep”
“kill”
Multiscale Information Extraction:
“keep”
“kill”

25. Penalized Likelihood Estimation
Algorithm:
Set penalty to count the number of non-trivial multiscale coefficient
Then optimization reduces to a set of N separate Generalized Likelihood Ratio tests which can be computed in O(N) operations
analog of hard thresholding

26. Risk Analysis Hellinger loss: Bound on Hellinger risk:
(follows from Li & Barron ’99)
KL distance
Choice of Penalty:

27. Risk Analysis – Upper Bounds
Theorem (Nowak & Kolaczyk ’01) : If underlying function / intensity x
belongs to BV (=1) or Besov space , then
Gaussian model:
Poisson model:
Multinomial model (density estimation):
n = number of samples

28. Risk Analysis – Lower Bounds
Theorem (Nowak & Kolaczyk ’01)
Proof technique similar in spirit to “method
of hyperrectangles” (Donoho et al ’90)
Multiscale likelihood factorization plays key role
in deriving minimax bounds in Poisson and
multinomial cases
Key Point:
Multiscale MPLE

29. Gamma-Ray Burst Analysis
photon
counts
burst
time
Compton Gamma-Ray Observatory
Burst and Transient Source Experiment (BATSE)
One burst (10’s of seconds) emits
as much energy as our entire Milky
Way does in one hundred years !
x-ray “after glow”

30. Gamma-Ray Burst Analysis (Poisson data)
If we know where jumps are located, then
N = number of measurements
If we don’t know where jumps are located, then
we can still achieve
Simultaneously detect
jump locations and
estimate intensity
log N
log N / N
1 / N
Optimal estimate computed via multiresolution
sequence of GLRTs in O(N) operations

31. BATSE Trigger 845 piecewise linear multiscale PMLE
piecewise polynomial
multiscale PMLE

32. Inverse Problems Observation model:
noise (possibly nonGaussian)
Goal: Apply the same multiscale PMLE methodology
Difficulty: Likelihood function does not admit multiscale
factorization due to presence of K
Example:

33. EM Algorithm: Basic Idea

34. EM Algorithm We do not actually measure z, therefore must be estimated
Given observed data y and an estimate
of define
The EM algorithm alternates between
E Step: Computing
M Step:
Monotonicity:

35. Gaussian Model Observation model: Wavelet parameterization:
Reformulation with unobserved “direct” image:
“inner noise”:
“outer noise”:

36. EM Algorithm – Gaussian Case
Intialize:
E Step (linear filtering) O(N log N) :
M Step (basic hard-thresholding) O(N) :
“complete data” likelihood
log  linear function of z

37. Example – Image Deblurring
BSNR = 40dB
original
blurred+noise
unrestored SNR = 15dB
restored SNR = 21dB

38. Example – Image Deblurring
original
initialization of EM with
“aggressive” linear
Wiener restoration
Wiener restored SNR = 18dB
EM restored SNR = 22dB

39. Example – Satellite Image Restoration
original
© CNES
simulated observed
unrestored SNR = 18dB
EM restored SNR = 26dB

40. Poisson Model Observation model:
Multiscale parameterization of unobserved image:
Complete data likelihood function:
“inner noise”:
“outer noise”:

41. EM Algorithm – Poisson Case
Intialize:
E Step (linear filtering) O(N log N) :
M Step (N independent GLRTs) O(N) :
“complete data” likelihood
log  linear function of z

42. Example – Tomographic Reconstruction
Shepp-Logan
phantom
sinogram
projection process
photon-counting

43. Example – Tomographic Reconstruction

44. Open Problems What can we say about optimality ? Key Challenges
Minimax optimal
brain reconstruction
Information-theoretic
restoration / segmentation
in remote sensing
Key Challenges
indirect measurements
adaptation of risk bounds, uniqueness
complex image manifolds
images edges are curves

45. Future Work determination of convergence rates and
log N
log N / N
1 / N
determination of convergence rates and
performance bounds in inverse problems
development of practical algorithms that
achieve near-optimal rates
multiscale image segmentation schemes
based on recursive partitioning
natural image modeling and alternative
representations (eg edgelets, curvelets)