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\begin{document}
\title{Multiscale Edge Detection of Images from Its Radon Transform
\footnote{The work is supported by The Hong Kong Polytechnic University
Research Grant No. 351/051 and NNSF of China}
}
\author{T.M.~Shih  \\
Department of Applied Mathematics, \\
The Hong Kong Polytechnic University \\
Kowloon, Hong Kong 
\and
Z.X.~Su \\
Institute of Mathematical Sciences, \\Dalian University of Technology \\
Dalian 116024, P.R. China
\and
F.T.~Chau\\
Department of Applied Biology \& Chemical Technology, \\The Hong Kong 
Polytechnic University,\\ Kowloon, Hong Kong
}

\date{}
\maketitle
Tomographic image reconstruction aims to recovering the image 
$f(x,y) \in L^2(R^2) \cap L^1(R^2)$ from its Radon tranform
\begin{equation}
Rf(s,\theta) = \int^{+\infty}_{-\infty} f(s \cos \theta - t \sin \theta,
s \sin \theta + t \cos \theta) dt, s \in R, 0 \le \theta \le \pi.
\end{equation}

By the Wvalet-Vaguelette Decomposition (W.V.D) method introduced by Donoho
in 1992, $W f(x,y), the wavelet transform of the image $f(x,y), can be 
computed directly from $Rf$ as follows,
\begin{equation}
\langle f, \psi_\lambda \rangle = [Rf, \gamma_\lambda], 
\end{equation}

where $\{\gamma_\lambda\}$ forms a near-orhtogonal basis and is called
Vaguelettes which are dilations and translations of a single ``mother''
Vaguelette.  $Wf(x,y)$ can also be obtained by other methods, such as the
Convolution Back Projection reconstruction method, and now the 
corresponding formulae are as follows:
\begin{eqnarray}
\langle f,\psi_\lambda\rangle &=& F^{-1}_2 \{ | \omega | [\frac{1}{2^j}
(F_2 \psi_{(0,0,\epsilon)})(-\frac{\omega}{2^j})]F_2B(Rf)\}
(\frac{k_x}{2^j},\frac{k_y}{2^j})\nonumber \\
 &=& BF_1^{-1}\{F_2\bar{\psi}_{\lambda}(r \cos \theta, r \sin \theta)
F_1(Rf)\}(\frac{k_x}{2^j},\frac{k_y}{2^j}) \label{e3}
\end{eqnarray}

where $F_1$ and $F_2$ denote the one dimensional and two dimensional 
Fourier transform respectively, $\psi_\lambda$ is the two dimensional
wavelet, $\lambda = (j, (k_x, k_y), \epsilon)$, and 
\[
B(g) = \int_0^\pi g(\theta) d \theta
\]
is the back projection operator.  With the aid of (\ref{e3}), algorithms for
denoising and edge detection of the image can be employed at the same
time of the reconstruction process.  In this paper, multiscale edge 
detection algorithms and further compression image reconstructions 
where there is noise in the input data.  The complexity of the algorithm
is of the same order as that of the Convolution Back Projection 
algorithm, which is the typical fast reconstruction method.  Different
algorithms are tested on the Shepp-Logan Phantom, which is a typical
example in this area, and numerical results are compared. \\

{\bf Keyword:} Wavelet Transform, Tomographic image reconstruction,  \\
Radon transform, Wavelet-Vaguelette Decomposition, Edge detection, 
Compression. 
\end{document}


