Ph.D. Thesis: Inverse Problems In Image Processing
Thesis in PDF,
Defence talk (2.00 pm CST, June 3rd, 2003) in PDF (400 KB), in Compressed Postscript (600 KB).
Inverse problems involve estimating parameters or data from
inadequate observations; the observations are often noisy and contain
incomplete information about the target parameter or data due to
physical limitations of the measurement devices. Consequently, solutions
to inverse problems are non-unique. To pin down a solution,
we must exploit the underlying structure of the desired solution
set. In this thesis, we formulate novel solutions to three image
processing inverse problems: deconvolution, inverse halftoning, and
JPEG compression history estimation for color images.
- Deconvolution: Deconvolution aims to extract crisp images from blurry
observations. We propose an efficient, hybrid Fourier-Wavelet
Regularized Deconvolution (ForWaRD) algorithm that comprises
blurring operator inversion followed by noise attenuation via scalar
shrinkage in both the Fourier and wavelet domains. The Fourier
shrinkage exploits the structure of the colored noise inherent in
deconvolution, while the wavelet shrinkage exploits the piecewise
smooth structure of real-world signals and images. ForWaRD yields
state-of-the-art mean-squared-error (MSE) performance in practice.
Further, for certain problems, ForWaRD guarantees an optimal rate of
MSE decay with increasing resolution.
- Inverse Halftoning:
Halftoning is a technique used to render gray-scale images using only
black or white dots. Inverse halftoning aims to recover the shades of
gray from the binary image and is vital to process scanned
images. Using a linear approximation model for halftoning, we propose
Inverse Halftoning via Deconvolution (WInHD)
algorithm. WInHD exploits the piece-wise smooth structure of
real-world images via wavelets to achieve good inverse halftoning
performance. Further, WInHD also guarantees a fast rate of MSE decay
with increasing resolution.
- JPEG Compression History Estimation: We routinely
encounter digital color images that were previously
JPEG-compressed. We aim to retrieve the various settings---termed
compression history---employed during previous JPEG operations. This
information is often discarded en-route to the image's current
representation. We discover that the previous JPEG compression's
quantization step introduces lattice structures into the image. Our
study leads to a fundamentally new result in lattice theory---nearly
orthogonal sets of lattice basis vectors contain the lattice's
shortest non-zero vector. We exploit this insight along with other
known, novel lattice-based algorithms to effectively uncover the
image's compression history. The estimated compression history
significantly improves JPEG recompression.