## 3 Experiments Evaluation
### Video Compressive Sensing  
In video surveillance, background images are always modeled as a low-rank subspace and foreground moving objects are regarded as sparse signals. Then, the following \"Low-rank \(+\) Sparse\" model [17; 18] can well represent the video frame captured by stationary camera, which has been studied extensively in recent literature. \[I=Uw+s \tag{6}\]  
Here, we consider applying Algorithm 1 to video compressive sensing application [19], for example, the video frames are captured by single-pixel camera [20] in CS literature. We simulate video CS by performing noiselet operator \(\mathcal{A}\)[15] on each normal surveillance video frame. Then recovering the foreground moving objects is just the problem Equation (1) in this paper. We test our simulation on the two well-known datasets (Dataset can be downloaded from [[http://perception.i2r.a-star.edu.sg/bk_model/bk_index.html](http://perception.i2r.a-star.edu.sg/bk_model/bk_index.html)]([http://perception.i2r.a-star.edu.sg/bk_model/bk_index.html](http://perception.i2r.a-star.edu.sg/bk_model/bk_index.html))) \"airport\" and \"lobby\" in which the image dimension of the two datasets is _144 \(\times\) 176_. In order to perform CS measurement on each video frame, conforming to the noiselet convention, we first manually resize each frame as _128 \(\times\) 128_, which is the power of 2, thus, the ambient dimension of each video frame \(I_{{}_{t}}\) in our experiment is _128 \(\times\) 128_. Then, for each video frame \(I_{{}_{t}}\), we take the CS measurement \(v_{{}_{t}}=\mathcal{A}(I_{{}_{t}})\) with CS measurement ratio \(\rho_{{}_{cs}}\). As the two surveillance videos are static, our fixed low-rank subspace assumption holds. We obtain the fixed low-rank orthonormal matrix \(U\) as follows. For each dataset, we choose 200 video frames and perform the Robust PCA algorithm Inexact ALM (IALM) [21] on those frames to get the clean background images, which can be regarded as lying in a low-rank subspace. Then, the rank \(d\) orthonormal matrix \(U\) is obtained by performing SVD on those background images and keeping the columns corresponding to the largest \(d\) singular values. In our experiments we set \(d=5\). In order to show how CS measurement ratio affects the recovery results, for the moderate sparse dataset \"airport\" we set the sparsity \(\rho_{{}_{s}}=20\%\) and take CS measurement with ratios \(\rho_{{}_{cs}}=30\%\), \(\rho_{{}_{cs}}=20\%\), and \(\rho_{{}_{cs}}=15\%\), respectively; and for the truly sparse dataset \"lobby\" we set the sparsity \(\rho_{{}_{s}}=5\%\) and take CS measurement with ratios \(\rho_{{}_{cs}}=20\%\), \(\rho_{{}_{cs}}=5\%\), and \(\rho_{{}_{cs}}=1\%\), respectively. For \"airport\", Figure 6 shows that the moving foreground objects can be well estimated though there are some ghost-effects in the recovery images, for example the first column. This effect is because IALM can not totally remove foreground from background due to the objects moving slowly in the scene which does not strictly follow the PCP recovery theorem [17]. The second row of Figure 6 shows the background estimated by IALM in which there does exist some slight shadows. The FPS (frame per second) regarding the first 200 frames of \"airport\" is 2.16 fps at \(\rho_{{}_{cs}}=15\%\) in our Matlab implementation. For \"lobby\", from Figure 7, we can see that even if we just take \(\rho_{{}_{cs}}=1\%\) CS measurement (the bottom row), the recovered results are comparable to taking much more measurement \(\rho_{{}_{cs}}=20\%\) (the third row). More promisingly, while separating the total 1546 frames the proposed algorithm only takes 72.16 s, say the FPS is 21.42 fps for \"lobby\" at the extremely low CS measurement ratio \(\rho_{{}_{cs}}=1\%\). Note that though there is large illumination variation in this dataset the recovery performance is still stable. Figure 6: Foreground recovery from video compressive sensing on “airport” dataset.