Inverse Problems in Image Processing

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 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. 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 the Waveletbased 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. We routinely encounter digital color images that were previously JPEG-compressed. We aim to retrieve the various settings—termed JPEG 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.