## 2 Problem Formulation
Let **h** = **x** - \(\tilde{\textbf{x}}\), then **Ah** = 0 meaning that **h** is a nonzero vector in the nullspace of **A**, and **h** has at most \(k+\tilde{k}\) nonzero entries. Because the sparsity-based interpolation method seeks the \(\ell_{{}_{1}}\) minimization solution under the constraint of available data consistency **A\(\textbf{x}\)** = **A\(\tilde{\textbf{x}}\)**, the error of interpolated signal solution \(\tilde{\textbf{f}}=\boldsymbol{\Phi}\tilde{\textbf{x}}\)  
and **h** contains \(k+\tilde{k}\leq d\) nonzero entries, and \(a\) has many nonzeros. When \(\left\|\textbf{{u}}\right\|_{2}\) is smaller than expected error level, then we can say that we get a reasonable interpolation result. [PERSON] _et al_[19] gave deterministic conditions that guarantee a successful exact recovery. It states the condition as strict \(k\) -balancedness of null space of **A**, where \(\textbf{{x}}^{*}\) is the sparsest solution to \(\textbf{{f}}=\textbf{{\Phi}}\textbf{{x}}\) with \(k\) nonzeros. Thus the following factors play important roles in recovery performance:  
(1) The sparser a vector **x** is, the more likely a null space \(\left\{\textbf{{v}}\in\mathbb{R}^{n}:\textbf{{A}}\textbf{{v}}=0\right\}\) will be strict k-balanced. (2) Let \(\textbf{{f}}_{a}\in\mathbb{R}^{m}\), then the smaller \(m\) is, the less likely a null space will be strict \(k\) -balanced since the null space becomes larger. (3) The available data correspond to **A**, and the missing data correspond to **B. ** In other words, the missing node locations decide the rows of **B**. (4) If \(\textbf{{\Phi}}\) is a basis, the row vectors of **B** spans the null space. However, the conditions for exact recovery are not verifiable in polynomial time. In this paper, we aim to reasonably interpolate the missing data, not necessarily to achieve exact recovery, so an important question is how to choose a good basis for data of sensor networks to get a more reasonable interpolation result? From an application point of view, a WSN consists of spatially distributed autonomous sensors to cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, or pressure. Generally speaking, these physical phenomena are more often fields [13], so the network data are usually smooth. Due to the limited number of sensors, the data often have low resolution. Since the discrete cosine transform (DCT) can expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies, the DCT is an appropriate basis to sparsify WSN data which are smooth and in low-resolution. On contrary, natural images usually contain crisp boundaries or strong edges at localized regions. These image features can be sparsely represented by the localized basis such as wavelets. According to Equation (8), let \(\textbf{{A}}_{ij}\) denote the \(i\)th entry of \(\textbf{{A}}_{j}\) (\(j\in S_{\textbf{{x}}}\)). Then, \(\textbf{{A}}_{ij}\) is the weight of \(x_{j}\) for the linear combination \(\sum\limits_{j=1}^{d}\textbf{{A}}_{ij}x_{j}\), where \(i\) corresponds to the \(i\)th available sample in \(\textbf{{f}}_{a}\). \(x_{j}\) has no way to be estimated if all its weights are zero. Let \(\textbf{{\Phi}}_{j}\) be the \(j\)th column of dictionary \(\textbf{{\Phi}}\). If \(\textbf{{\Phi}}_{j}\) has compact support and the missing block overlap with the compact support, then most of the entries in the \(\textbf{{A}}_{j}\) will be zeros. In this case, \(\textbf{{A}}_{j}\) cannot provide enough constraints that \(x_{i}\) must satisfy. In another word, more nonzeros in \(\textbf{{A}}_{i}\) can provide more information for \(x_{i}\) because of more constraint equations. So, a dictionary with non compact support is preferred. If both DCT and wavelet transform can sparsify the data, DCT is a better choice since the wavelet basis functions are localized as Figure 2 shows, while DCT basis functions always have non-zero in a large range.