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\chapter{Image Restoration}
\section{Image Formation}
In image processing, an object is defined as a two-dimensional entity that emits or reflects radiant energy. Energy may be of a variety of forms, for example, optical energy, acoustic energy, or nuclear particles. An imaging system is a \textquotedblleft black box\textquotedblright ~that is capable of intercepting some or all of the radiant energy. It has the function of focusing the intercepted energy and making an image, that is, a representation of the original object which emits or reflects the energy. Schematically, the image formation process can be represented as in Figure \ref{fig:image}.
\begin{figure}[tb]
\centerbmp{4in}{3.5in}{image.bmp}
\caption{Image formation schema}
\label{fig:image}
\end{figure}
The description of the image formation process assumes spatial invariance and linearity in the process of image formation; or equivalently, changing the position of the object does not affect its image and the image formed from two objects simultaneously present in the object space is the sum of the images formed from individual objects. Under such assumptions, it is possible to describe image formation by the following equation,
\begin{equation}
\begin{array}{lllllll}
g(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x - \zeta,y - \eta)f(\zeta,\eta)~d\zeta d\eta , \\
\qquad f(\zeta, \eta) \geq 0 , \quad g(x, y) \geq 0
\end{array}
\label{eq:imageprocess}
\end{equation}
where $f(\zeta,\eta)$ represents the two-dimensional distribution of energy corresponding to the original object, $g(x,y)$ is the representation of $f(\zeta,\eta)$ obtained in the image space by the image-formation process, and $h(x,y,\zeta,\eta)$ is known as the {\it point spread function} (PSF). The operator $H = h(x,y,\zeta,\eta)$ maps an object $f(\zeta,\eta)$ into an image $g(x,y).$ According to the space-invariant character of image formation process, $H$ is defined as a convolution operator. Then the output is a function only on the difference of variables in the coordinate systems. Further, function $h$ is separable. Similarly, $h(x-\zeta, y-\eta) = h_1(x-\zeta)h_2(y-\eta),$ or $h(x, y,\zeta, \eta) = h_1(x,\zeta)h_2(y,\eta).$ If the point spread function $h$ is seperable, then the integrations can be sequentially performed and
\begin{align}
g(x,y) &=\int_{-\infty}^{\infty}h_1(x-\zeta)\left(\int_{-\infty}^\infty h_2(y-\eta)f(\zeta,\eta)d\eta \right)d\zeta \\\notag
&=\int_{-\infty}^{\infty}h_2(y-\eta)\left(\int_{-\infty}^\infty h_1(x-\zeta)f(\zeta,\eta)d\zeta \right)d\eta
\end{align}
or
\begin{align}
g(x,y) &=\int_{-\infty}^{\infty}h_1(x,\zeta)\left(\int_{-\infty}^\infty h_2(y,\eta)f(\zeta,\eta)d\eta \right) d\zeta \\\notag
&=\int_{-\infty}^{\infty}h_2(y,\eta)\left(\int_{-\infty}^\infty h_1(x,\zeta)f(\zeta,\eta)d\zeta \right) d\eta
\end{align}
Because the image is literally a flow of energy, it is necessary to use some recording mechanism. Thus, into (\ref{eq:imageprocess}) one must introduce a function $S$ that accounts for the sensing and recording of the energy flow. The mechanism of recording is always imperfect, that is, there are distortions in the recorded image. These distortions can be either predictable or random. The predictable or systematic distortions are encompassed in the function~$S$. The random distortion is accounted for a noise component $n.$ The result of adding these effects can be expressed by
\begin{equation}
g(x,y) = S \left(\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} h(x - \zeta,y - \eta)f(\zeta,\eta)d\zeta d\eta ~\right) + n(x,y).
\label{eq:imageprocess2}
\end{equation}
The statement of the image restoration problem is now a direct consequence of (\ref{eq:imageprocess2}): Given a recorded image $g(x,y),$ make an estimate of the original object $f(\zeta,\eta).$
\section{Mathematical Properties of Image Restoration}
A common method of analysis of Equation (\ref{eq:imageprocess2}) is to use operator theory. Given a function space $A$ on which $f$ is defined and a function space $B$ on which $g$ is defined, then the problem of the transformation or operator $T$ that maps $f$ to $g$ can be posed as $T: f \longrightarrow g.$
The problem of image restoration is then to find the inverse transformation $T^{-1}$ such that
$ T^{-1}: g \longrightarrow f.$
In a mathematical sense, a problem is well-posed if its solution exists uniquely and depends continuously on data, otherwise the problem is ill-posed. From this point of view, the problem of image restoration is ill-posed. Because a slight perturbation in $g$ can produce drastic change in $f.$ In other words, there exists $\epsilon,$ which can be made arbitrarily small in the norm of space $B$ such that $\delta = T^{-1} (g + \epsilon ) - f ,$ where the norm of $\delta$ in the space $A$ is not negligible.
The ill-conditioned aspect of image restoration can be demonstrated by means of the Riemann-Lebesgue Lemma \cite{Digital1,Digital2}.
\begin{lemma}[Riemann-Lebesgue Lemma]
Let f be a Riemann-integrable function defined on an interval $a\leq x\leq b$ of the real line. Then for any real $\beta$
\[
\lim_{|\alpha |\to \infty} \int^b_a f(x)\cos(\alpha x + \beta ) dx = 0.
\]
\label{lem:Rie}
\end{lemma}
Because $\cos (\alpha x + \beta) = \cos \beta \cos (\alpha x ) - \sin \beta \sin (\alpha x),$ the statement of the previous lemma is in fact equivalent to the following two statements
$$
\underset{|\alpha |\to \infty}{\lim} \int^b_a f(x)\cos(\alpha x ) dx = 0, \quad \mbox{and} \quad \underset{|\alpha |\to \infty}{\lim} \int^b_a f(x)\sin(\alpha x ) dx = 0.
$$
Therefore if $h(\zeta, \eta)$ is an integrable function on $[a,b] \times [a,b]$ then
\begin{equation}
\lim_{ \beta \to \infty} \lim_{ \alpha \to \infty } \int_a^b \int_a^b h(\zeta, \eta) \sin(\alpha \zeta) \sin(\beta \eta) d\zeta d\eta = 0.
\label{eq:transform4}
\end{equation}
Now, consider function $h(\zeta, \eta, x, y).$ Then from (\ref{eq:transform4}) it follows that
\begin{equation}
\lim_{ \beta \to \infty} \lim_{ \alpha \to \infty } \int_a^b \int_a^b h(x,y, \zeta, \eta) \sin(\alpha \zeta) \sin(\beta \eta) d\zeta d\eta = 0.
\label{eq:transform5}
\end{equation}
Equation (\ref{eq:transform5}) can be used in image restoration problem as follows:
\begin{equation}
\begin{split}
\underset{ \alpha, \beta \to \infty}{\lim} &\int_a^b \int_a^b h(x, y, \zeta, \eta) \left(f(\zeta, \eta) + \sin(\alpha \zeta) \sin(\beta \eta)\right) d\zeta d\eta \\
&= \int_a^b \int_a^b h(x, y, \zeta, \eta) f(\zeta, \eta) d\zeta d\eta+\\
&\phantom{=====} \underset{ \alpha, \beta \to \infty}{\lim} \int_a^b \int_a^b h(x, y, \zeta, \eta) \sin(\alpha \zeta) \sin(\beta \eta)] d\zeta d\eta \\
&= g(x,y).
\end{split}
\label{eq:transform6}
\end{equation}
In other words, a sinusoid of infinite frequency can be added to the object distribution $f$ and the resulting sum is identical to the image distribution $g.$
The demonstration of (\ref{eq:transform6}) is very general since it requires only integrability of the function $h.$ Given a small value $\delta >0,$ there exists a value $A,$ and $\epsilon $ such that
\[
\int_a^b \int_a^b h(x,y, \zeta, \eta) \sin(\alpha \zeta) \sin(\beta \eta) ~d\zeta d\eta = \epsilon,
\]
where $|\epsilon| < \delta,$ and $\alpha, ~\beta \geqslant A.$ Thus,
\begin{multline}
\int_a^b \int_a^b h(x,y, \zeta, \eta) \left(f(\zeta, \eta) + \sin(\alpha \zeta) \sin(\beta \eta)\right) ~d\zeta d\eta =\\
\int_a^b \int_a^b h(x,y, \zeta, \eta) f(\zeta, \eta) d\zeta d\eta + \epsilon = g(x,y) + \epsilon,
\label{eq:transform8}
\end{multline}
for all values of $\alpha, ~\beta \geqslant A.$ Therefore, if an infinitely small value $\epsilon$ is chosen and added to the image distribution $g,$ it cannot be separated, in the sense of image restoration, from an original object distribution that has an additional component of a sinusoid with frequency $\alpha \geq A.$ Since $\epsilon$ can be made infinitely small, then a trivial perturbation in the image cannot be distinguished from a finite, nontrivial perturbation in the original object distribution. Thus, the statement that image restoration is an {\it ill-conditioned} problem is justified~\cite{Digital1,Digital2}.
\section{Popularly Used Methods}
There are numerous approaches to image restoration: Fourier and wavelet-transform methods which involve computation in the frequency domain \cite{Gonzalez}, statistical methods (Wiener filtering) \cite{Digital1}, regularized least squares \cite{Hunt}, etc. However, these methods are not well suited for capturing sharp edges, because linear filters are not able to remove an impulsive noise imposed on an image without blurring its edges and small details \cite{Tony1}.
These shortcomings led to the development of nonlinear methods in image processing \cite{nonlinearbook}. Recently, there has been a new movement towards a more PDE-based approach; see for example the article by Alvarez and Morel \cite{Alvarez} and the books by Morel and Solimini \cite{Morel} and Ter Haar Romeny \cite{Romeny}. The new models are motivated by a more systematic approach to restoring images with sharp edges. The image is denoised according to a nonlinear anisotropic diffusion PDE, designed to diffuse less near edges. The PDEs are often designed to possess certain desirable geometrical properties such as affine invariance and causality. Total Variation based image restoration methods belong to this new class of models \cite{Rudin}. The restoration is obtained by minimization of the Total Variation functional, subject to constraints which relate the solution to the measured image, and the noise level \cite{Tony1}.
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