## 1 Introduction
### Wavelet-Based Matched-Filter for the Inverse Radon Transform  
The wavelet transform of a function, \(f\in\mathbb{R}^{2}\), is given by the set of scalar products of the function and the wavelets \(\Psi_{a,b}\) which are versions compressed or dilated by a factor \(a\) and translated by a vector \(\mathbf{b}\in\mathbb{R}^{2}\) of an analyzing wavelet \(\Psi\):  
\[T_{a,\mathbf{b}}(f)=\langle\Psi_{a,b}/f\rangle=\int_{\mathbb{R}^{2}}a^{-2}\Psi^{*} \left(\frac{\mathbf{a}-\mathbf{b}}{a}\right)f(\mathbf{\alpha})d^{2}\mathbf{\alpha} \tag{11}\]  
where \(\Psi^{*}\) is the transposed wavelet of \(\Psi\), and the braket \(\langle./.\rangle\) denotes the scalar product. The Radon transform of a perturbation \(f\) (Equations (2) and (10)) is the integral of \(f(\mathbf{x})\) over a set of all straight lines in a subspace of \(\mathbb{R}^{2}\), solution of \(\langle\mathbf{d}/\mathbf{\alpha}=\mathbf{\beta}\rangle\) with \(\mathbf{d}\in\mathbb{R}^{2}\) a vector of unit length (\(|\mathbf{d}|=1\)) identified with the angle of observation \(\theta\), with the incident \(\mathbf{n}_{0}\) and observation \(\mathbf{n}\) directions:  
\[g(\mathbf{\beta},\ \mathbf{n}_{0},\mathbf{n})=\int_{\mathbb{R}^{2}}f(\mathbf{\alpha})\delta( \langle\mathbf{n}.\mathbf{n}_{0}/\mathbf{\alpha}\rangle-\mathbf{\beta})d^{2}\mathbf{\alpha} \tag{12}\]  
Each projection in the form of straight lines (cf. Section 2.3) is obtained by rotation and dilatation/translation of the kernel \(K\). Equation (12) can thus be analyzed as a wavelet transform, with the delta distribution \(\delta\) as the analyzing wavelet such that:  
\[\Psi(\mathbf{\alpha})=\delta(\mathbf{n}_{0}.\mathbf{\alpha}) \tag{13}\]  
and:  
\[T_{a,\mathbf{b}}(f)=\langle\Psi_{a,\mathbf{b}}/f\rangle=\int_{\mathbb{R}^{2}}a^{-1} \delta(\mathbf{n}_{0}.\mathbf{\alpha}-\mathbf{n}.\mathbf{\alpha})f(\mathbf{\alpha})d^{2}\mathbf{ \alpha}=a^{-1}\,g(\mathbf{\beta},\ \mathbf{n}_{0},\mathbf{n})=a^{-1}\mathcal{R}f \tag{14}\]  
where \(\mathbf{b}=\mathbf{n}.\mathbf{\alpha}\)[34]. In the case of backscattering measurements (\(\mathbf{n}.\mathbf{n}_{0}<0\)), the associated function \(g(\mathbf{\beta},\ \mathbf{n}_{0},\mathbf{n})\) in the space domain (i.e., the impulse response of the object) is assimilated to a projection/signal in the time domain corresponding to the backpropagated waves, leading to a spatiotemporal sinogram as a function of the angle of incidence \(\phi\). In practice, the recorded signals are convolution filters between the impulse response of the object \(g(t,\theta,\phi)\) and that of the electro-acoustic device \(r(t,\theta,\phi)\). To take these filtering effects into account, and to \"achieve\" the impulse responses of the object, a wavelet-based coded-excitation method (WCE) has been used [26]. This method is decomposed into two steps. First, the transmitted signal is modeled as a wavelet adapted to the electro-acoustic configuration (center frequency \(f_{0}\), sampling frequency \(F_{e}\), number of samples \(N\)). This will be discussed in Section 3.6. Second, the spatiotemporal sinograms are filtered by wavelet functions, with \(a=2^{j}\) and \(b=\frac{q}{2^{j}}\), \((j,q)\in\mathbb{Z}^{2}\) :  
\[\psi_{j,q}(t)=2^{\frac{j}{2}}\psi\left[2^{j}(t-\frac{q}{2^{j}}.\frac{1}{F_{e} })\right] \tag{15}\]  
The function \(\psi_{j,q}(t)\) has orthonormal functions verifying the following properties:  
* \(\psi_{j,q}(t)\) is a function of the \(L^{2}\) space;
* The Fourier transform \(\hat{\psi}_{j,q}(v)\), has a compact support;\[- \psi_{j,d}(t)\] has a compact support: \[20.log\left|\frac{\psi_{j,d}(t)}{\max\bigl{(}\psi_{j,d}(t)\bigr{)}}\right|<-A\ dB \tag{16}\]  
where \(A\) is a coefficient left to the free choice of the user (\(A=10^{-3}\) in this work). The algorithm is detailed in [26]. The wavelet \(\psi(t)\) is the theoretical analyzing wavelet defined by [PERSON] [35; 36] and [PERSON] [37]. Its spectral modulus reaches a maximum at \(\bigl{(}\frac{2}{3}F_{e}\bigr{)}\), and has a bandwidth equal to two octaves \(\left[\frac{1}{3}F_{e};\frac{4}{3}F_{e}\right]\).