Prabahan Basu

A new technique for the restoration of low resolution text images

Many problems in security and surveillance require the resolution expansion of a captured image. This paper introduces a new system for expanding the resolution of text images. The system inputs are single instances of low resolution text images which must be restored and transformed into data readable text. To achieve acceptable results, such systems require a robust restoration subsystem. The sub-system proposed in this paper creates a bimodal image which favors polygonal representations, consistent with the characteristics of scripts of most non-cursive languages. The resolution expanded image is generated by iteratively solving a non linear optimization equation in the Radon domain with the imposition of image domain constraints. To the best of our knowledge, the idea of Radon domain restoration is new.

On estimation error using maximum entropy density estimates

Purpose – In many problems involving decision‐making under uncertainty, the underlying probability model is unknown but partial information is available. In some approaches to this problem, the available prior information is used to define an appropriate probability model for the system uncertainty through a probability density function. When the prior information is available as a finite sequence of moments of the unknown probability density function (PDF) defining the appropriate probability model for the uncertain system, the maximum entropy (ME) method derives a PDF from an exponential family to define an approximate model. This paper, aims to investigate some optimality properties of the ME estimates.Design/methodology/approach – For n>m, when the exact model can be best approximated by one of an infinite number of unknown PDFs from an n parameter exponential family. The upper bound of the divergence distance between any PDF from this family and the m parameter exponential family PDF defined by the M...

Error Bounds for Maximum-Entropy Estimates

5.1 Introduction In a parametric, or model-based, estimation approach, deriving error bounds on density estimates would be straightforward; the uncertainty of the parameter estimates would fully determine the uncertainty in the overall density estimate. However, in the absence of a parametric model, there is no well-defined approach for finding the goodness of a density estimator. Consequently, we now detail the construction of error bounds and confidence intervals for ME density estimates. Key to our approach is the representation of densities by finite-order exponential families . Using exponential families as probability models of uncertain systems offers many advantages. Often, if two independent random variables have densities belonging to the same exponential family, their joint density will also be a member of this family. As such, exponential family models are easily updated as new information becomes available. Moreover, if f ( x ) is infinitely differentiable, then, modulo certain pathological cases, it may be approximated with arbitrary precision by some density from an exponential family. In this light, the ME method is highly desirable; its estimates are always members of exponential families. The linearly independent statistics sequence used to define the exponential family may include the sequence of monomials { x k } and the sequences of orthogonal polynomials. We assume the unknown density is from a canonical exponential family of an unknown but finite orderm, and only n moments on n linearly independent statistics { T k ( x )} are known. We consider two cases: m ≤ n and m > n . In the first case, we have an overdetermined model. In the second underdetermined case, even when m is known, there may still be an infinite number of densities from an m -parameter exponential family having the same set of prescribed n moments.

The Inverse Problem

2.1 Introduction Given a list of effects, the problem of determining cause has intrigued philosophers, mathematicians and engineers throughout recorded history. Problems of this type are formally referred to as inverse problems. Inverse problems pose a particularly difficult challenge: no solution is guaranteed to be unique or stable. The solution is unique only if for some reason known to the observer the given list of effects can be due to one and only one cause. We are concerned here with the inverse problem as it relates to signal and image restoration. In this context of linear time-invariant (LTI) systems, it is common to use the terms inverse problem and deconvolution interchangeably. The problem here may be stated as that of estimating the true signal given a distorted and noisy version of the true signal. 2.2 Signal Restoration In general, the goal of signal recovery is to find the best estimate of a signal that has been distorted. Although the mathematics is the same, we would like to distinguish between signal restoration and signal reconstruction. In the first problem, the research is concerned with obtaining a signal that has been distorted by a measuring device whose transfer function is available. Such a problem arises in image processing, wherein the distorting apparatus could be a lens or an image grabber. In the second problem, the scientist is faced with the challenge of reconstructing a signal from a set of its projections, generally corrupted by noise. This problem arises in spectral estimation, tomography, and image compression. In the image-compression problem, a finite subset of projections of the original signal are given, perhaps on the orthonormal cosine basis, and the original signal is desired. Generally, to go about the problem of signal recovery, a mathematical model of the signal-formation system is needed. Different models are available; simple linear models are easy to work with but do not reflect the real world. More realistic models are complex and may be used at some additional computational cost.

Information Theory Preliminaries

1.1 Scope The quality of any decision depends both upon the amount of relevant information available as well as the quality of the decision-making process itself. In this book, we investigate decision making in the context of detection and estimation problems that are statistically modeled according to some assumed probability distribution. Assumptions become necessary to help find true solutions when facing problems where available information is insufficient. Appropriate assumptions provide guidance when selecting the best solution from a given set of candidates. However, new assumptions impart new information. Consequently, making the minimum number of assumptions consistent with available evidence is vital; making these assumptions using principled methods is also key. Given the important role of information in these problems, we appeal to information theory as a principled approach to making appropriate assumptions and assigning meaningful models. Much of this book deals with the application of information-theoretic notions to problems of estimation. Following a brief review of information theory basics, we introduce the inverse problem and its role in signal restoration. Next, we review both classical and recent methods for solving the inverse problem. Part I of this monograph explores density-estimation techniques. We first review one such technique-maximum-entropy (ME) estimation. Then, we present a novel result which establishes confidence bounds on these estimates. In Part II, we introduce a framework for solving inverse problems to find unique solutions that maintain fidelity with observed data. We conclude the book with some examples and applications of the proposed algorithms to common problems encountered in signal processing.