## 3 Advantage  
The IDWA interpolation assumes uniform correlation in the neighboring data. In many situations, this may not be true due to the fast and anisotropy changes in the neighborhood. A sparsity-based interpolation method does not require high correlation of the neighboring data. As long as the data are sparse in a chosen dictionary, it will work. We created a toy example image like this: the left half of the image is a smooth image, while the right half is also smooth, but there is a sharp boundary. As shown in Figure 3, we can choose an artificial image for the left so that it is very sparse under DCT or wavelet domain. We could even simply linearly combine a few bases to form the left image. We construct a right half image similarly. Now suppose we only sample some pixels on the left half, and right half, we should be able to reconstruct the entire image nicely (e.g., the sparsity constraints select the basis functions that we used to generate the left image), but IDWA will blur the boundary. The sharp edge information is very hard for IDWA to capture because the weights of neighbor values depend on the distances between an interpolated node and its neighbors. In the following, the K-nearest neighbor algorithm (KNN) [20,21] is chosen as an IDWA method for the 2-D case. The weight for each neighbor is computed by the inverse distance from the neighbor to the target missing samples. We use normalized root mean squared error (RMSE) to assess the accuracy of estimation which is defined as:  
\[\text{RMSE}(\mathbf{f},\mathbf{\hat{f}})=\frac{\sqrt{\sum_{i=1}^{N}\left(f_{i }-\hat{f}_{i}\right)^{2}}}{N} \tag{12}\]  
Figure 2: Basis functions of 2-D DCT and Wavelet, with size 64 \(\times\)64. **(a)**, **(b)** and **(c)** are DCT basis waveform according to its low, middle and high frequency component, respectively; **(d)**, **(e)** and **(f)** are Wavelet basis waveform according to its low, middle and high frequency component, respectively. where \(f_{i}\) and \(\hat{f}_{i}\) stand for \(i\)th (\(i=1,2,\cdots,N\)) entry of the original data vector \(\mathbf{f}\) and the recovered data vector \(\hat{\mathbf{f}}\), respectively. This normalization in RMSE allows for the comparison of estimation accuracy between different data sets. **Figure 3.** A toy example on boundary recovery. **(a)** Complete data, **(b)** Available data, **(c)** KNN interpolation, RMSE_KNN = 8.71 \(\times\) 10\({}^{-2}\), **(d)** Sparsity-based interpolation, RMSE_DCT = 4.31 \(\times\)10\({}^{-5}\). (a)(b)(c)(d)(d)(b)(c)(d)  
**4. A Fast Iterative Thresholding Algorithm to Solve the Sensor Data Recovery**  
Consider an optimization task that mixes \(\ell_{2}\) and \(\ell_{1}\) expressions in the form:  
\[F\left(\mathbf{x}\right)=\frac{1}{2}\left|\mathbf{f}_{a}-\mathbf{A}\mathbf{x} \right|_{2}^{2}+\lambda\left|\mathbf{x}\right|_{1} \tag{13}\]  
where \(F:\mathbb{R}^{N}\mapsto\mathbb{R}\) is a function of the vector \(\mathbf{x}\). This is a relaxed variant of the problem posed in Equation (6), and the parameter \(\lambda\) governs the tradeoff between the data consistency and the sparsity of \(\mathbf{x}\). In recent years, a family of iterative thresholding algorithms has gradually been built to address the above optimization task in a computationally effective way [22, 23, 24]. [PERSON] and [PERSON] [25] proves the convergence of iterative thresholding and they guarantees that the solution is the global minimizer for convex \(F\left(\mathbf{x}\right)\). The core idea is to minimize the function \(F\left(\mathbf{x}\right)\) iteratively [19], and Equation (7) can be simply solved by iterative thresholding:  
\[\mathbf{x}_{i+1}=S_{\lambda/c}\left(\frac{1}{c}\mathbf{A}^{T}\left(\mathbf{x} -\mathbf{A}\mathbf{x}_{i}\right)+\mathbf{x}_{i}\right) \tag{14}\]  
where the parameter c will be chosen such that \(c\mathbf{I}-\mathbf{A}^{T}\mathbf{A}>0\) and \(S_{\lambda/c}\left(\tau\right)\) is a soft thresholding operator to shrinkage each entry \(\tau_{j}\) of vector \(\tau\) according to:  
\[S_{\lambda/c}\left(\tau_{j}\right)=\left[