Electrical Impedance Tomography (EIT) is a radiation-free imaging method [1]. A commonly used imaging strategy is called time-difference reconstruction. It attempts to reveal the conductivity distribution changes inside the human body between two time points through the electrical data obtained via electrodes attached on skin. In time-difference lung EIT typically 16 electrodes are placed equidistantly on a horizontal chest plane. For each EIT frame, currents are successively injected into the human body through adjacent electrodes. A conductivity image is then reconstructed by collecting the voltage measurements recorded from the remaining electrodes.

We denote the conductivity changes of the domain between two time steps by Δ*s* and the measured voltage changes on the electrodes by a vector Δ*V*. Employing the finite element model (FEM) with *M* elements, the conductivity change Δ*s* is represented by a *M* × 1 vector. For time-difference imaging, the reconstruction problem can be linearized [2]. Approximately, there exists the following relation:

$${\mathit{\text{J}}}^{\mathit{\text{E}}}\cdot \mathrm{\Delta}\mathit{\text{s}}\approx \mathrm{\Delta}\mathit{\text{V}}$$(1)

where *J*^{E} denotes the Jacobian matrix of elements calculated at the constant conductivity 1:

$${\mathit{\text{J}}}_{\mathit{\text{ij}}}^{\mathit{\text{E}}}={\frac{\mathrm{\Delta}{\mathit{\text{V}}}_{\mathit{\text{i}}}}{\mathrm{\Delta}{\mathit{\text{s}}}_{\mathit{\text{j}}}}|}_{1}$$(2)

This Jacobian matrix of elements could be calculated by EIDORS toolbox [3]. The Jacobian matrix *J*^{E} is ill-conditioned because the degree of freedom of this inverse problem is too large. Solving the conductivity changes from the above relation leads to an ill-posed inverse problem. Especially, the solution Δ*s* from equation (eq 1) is unstable. To circumvent this difficulty, additional prior information is introduced to restrict the flexibility of the solution [2]. In this study, we employ a prior illustrating the sparsity hypothesis. Explicitly, we solve Δ*s* from the following regularized optimization problem instead of (eq 1):

$$\widehat{\mathrm{\Delta}\mathit{\text{s}}}={\text{argmin}}_{\mathrm{\Delta}s}\frac{1}{2}{\parallel \mathrm{\Delta}\mathit{\text{V}}-{\mathit{\text{J}}}^{\mathit{\text{E}}}\cdot \mathrm{\Delta}\mathit{\text{s}}\parallel}_{2}^{2}+\alpha {\parallel \mathit{\text{R}}\cdot \mathrm{\Delta}\mathit{\text{s}}\parallel}_{1}$$(3)

where *α* is a regularization parameter that controls the trade-off between the regularization term and the fidelity term. According to sparse regularization theory, the regularization term ${\parallel \mathit{\text{R}}\cdot \mathrm{\Delta}\mathit{\text{s}}\parallel}_{1}$ promotes a solution $\widehat{\mathrm{\Delta}\mathit{\text{s}}}$ with $\mathit{\text{R}}\cdot \widehat{\mathrm{\Delta}\mathit{\text{s}}}$ being sparse.

Triangular finite element meshes are commonly used in EIT reconstructions. Triangular meshes can simulate irregular domains as well as the electrical properties around electrodes without approximation. Wavelet transforms have been widely used in medical imaging such as Computed Tomography or Magnetic Resonance Imaging and shown a great success in these imaging techniques. However, canonical wavelet transforms are not appropriate to apply on general finite element meshes. Instead, in this article we used spectral graph wavelet transforms and view the triangular meshes as graphs.

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