Image Restoration Based on I-Divergence Minimization
Ge Wang, Ph.D.
ge-wang@uiowa.edu
February 26, 1998
ABSTRACT
Image restoration is a major branch of image processing, which refers to computerized recovery of an ideal image from its blurred version. Recent theoretical progress suggests that the expectation maximization formula for emission computed tomography can be adapted to invert a linear system monotonically, nonnegatively and optimally in the sense of minimizing I-divergence. In this lecture, we will explain the rationale for use of I-divergence, outline the mathematical foundation of I-divergence minimization, formulate a spiral CT image restoration algorithm, report our experiments with cochlear images, and discuss further research directions.
OUTLINE
  1. INTRODUCTION - Image restoration, linear system
  2. I-DIVERGENCE CHARACTERIZATION - Discrepancy established using an axiomatic approach
  3. ITERATIVE DEBLURRING - EM-like formula; monotonic, nonnegative, optimal convergence
  4. SPIRAL CT IMAGE RESTORATION - Spatially invariant linear system with a Gaussian kernel; super-resolution for cochlear implantation
  5. DISCUSSIONS - Regularization, acceleration, blind deblurring
1. INTRODUCTION
Image blurring model
Linear system
Blurred image = Blurring operator (Ideal image)
Deblurred image = Deblurring operator (Blurred image)
Linear equations
or
Well-known techniques
Rank, determinant, Gaussian elimination
Least square approximation: Minimize
2. I-DIVERGENCE CHARACTERIZATION
Deblurring problem
,
Measurement; Blurring kernel
Function to be recovered
Deblurring approach - Fitting P
P Prior estimate of c
R
D(Q,P) Discrepancy measure
Q*
Euler-Lagrange equation for Q*
generally asymmetric
Q* must satisfy the Euler-Lagrange equation
, where are constants
If we regard R as input and Q* as output, we have
, or
Axiomatic system
  1. (identical)
  2. (nonnegative)
  3. (strictly convex)
  4. (directed orthogonality)
  5. For and subject to the same data, (scale invariance)
Lemma:
Theorem 1: Axioms 1-4 lead to , where , , and is continuous and positive
Examples
Mean square distance: ,
I-divergence: ,
Theorem 2: Axioms 1-5 characterize I-divergence
3. ITERATIVE DEBLURRING
Recall the problem
Deblurring approach - Fitting
Minimizing , where are predicted data
Deblurring formula
It has been recently recognized by Snyder et al. that the expectation maximization (EM) formula for emission computed tomography (CT) is applicable to deterministic linear nonnegative inverse problems
, where , and are current and updated guesses of, , k=0, 1, ...
Nonnegative, monotonic, and optimal convergence
Monotonic convergence
Nonnegative solution
Minimum I-divergence between measurement and prediction
Assumptions by Snyder et al.
  1. for all
  2. is not always equal to zero
  3. is summable
  4. for all
  5. is summable with respect to for all
Limitation: too strong
Our assumptions
  1. for all
  2. is summable
  5. for all
  6. is summable with respect to and
Remarks on our assumptions
4. SPIRAL CT IMAGE RESTORATION
Spiral CT model
Spiral CT image
Gaussian PSF with a standard deviation
Actual image
Spiral CT image restoration algorithm: Unconstrained
, Current and updated guesses
Experiments
Cochlear cross-section phantom: synthesized based on histologic and CT images consists of three types of structures: fluid, tissue and bone
Fitting error
Deblurred PSF:
5. DISCUSSIONS
Regularization
Deblurring kernel; Sieve kernel; Resolution kernel
Acceleration
Ordered subsets, parallel-processing
Blind deblurring
Recall the spiral CT model
PSF can also be unknown; double iteration loops
References
  1. C. L. Byrne and L. K. Jones: An axiomatic approach to certain inverse problems. Proc. SPIE 1351:50-55, 1990
  2. D. L. Snyder, T. J. Schulz, J. A. O'Sullivan: Deblurring subject to nonnegativity constraints. IEEE Trans. on Signal Processing 40:1143-1150, 1992
  3. D. L. Snyder and M. I. Miller and L. J. Thomas and D. G. Politte: Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography. IEEE Trans. on Medical Imaging 6:228-238, 1987
  4. G. Wang, M. W. Vannier, M. W. Skinner, M. G. P. Cavalcanti, G. Harding: Spiral CT image deblurring for cochlear implantation. To appear in IEEE Trans. on Medical Imaging
ACKNOWLEDGMENTS
Collaborators:
MW Vannier, MW Skinner, DL Snyder
B Brunsden, GH Esselman
GW Harding, BA Bohne
MGP Cavalcanti
Grants:
NIH/NIDDK (R29 DK50184)
NIH/NIDCD (R03 DC02798)
Whitaker Foundation (Biomedical Engineering)