# Spatially Adaptive Regularization in Image Segmentation
## 4 Results and Comparisons
For the remaining images, the values of \(\lambda\) and \(\mu\) are set by trial and error, following the empirical rule reported in [PERSON]'s code. The values of \(\lambda_{min}\) and \(\lambda_{max}\) used in the spatially adaptive approaches are chosen, so that the corresponding \(\lambda\) in the original CEN model is in \([\lambda_{min},\lambda_{max}]\), with few exceptions. The associated values of \(\mu\) are set as in the non-adaptive case. The values of \(\lambda\), \(\lambda_{min}\), \(\lambda_{max}\) and \(\mu\) used for each image are specified in Tables 2-4. The values of \(\lambda_{ij}\) in the adaptive models are initialized, as specified in (9), (11), and (12). As described in Section 2.3, in the strategy based on thresholding those values change at each iteration \(k\) of Algorithm 1. It is worth \begin{table}
\begin{tabular}{l c} \hline \hline
**Image** & **Size (pixels)** \\ \hline bacteria & \(233\times 256\) \\ bacteria2 & \(380\times 391\) \\ brain & \(210\times 210\) \\ cameraman & \(204\times 204\) \\ flowerbed & \(321\times 481\) \\ ninetyeight & \(300\times 225\) \\ squirrel & \(167\times 230\) \\ tiger & \(321\times 481\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test images and their sizes. noting that our adaptive approaches also simplify the choice of the regularization parameter, which may be a time-consuming task. The initial approximation \(u^{0}\) is set equal to the given image \(\bar{u}_{i,j}\), which takes its values in \([0,1]\), as specified in Section 1. This is used to compute the starting values of \(c_{1}\) and \(c_{2}\), and \(s_{i,j}=\left(c_{1}-\bar{u}_{i,j}\right)^{2}-\left(c_{2}-\bar{u}_{i,j}\right)^{2}\) for all \((i,j)\). In the original non-adaptive code, the value of \(\lambda\) is scaled using the difference between the maximum and the minimum value of \(s_{i,j}\); the same scaling is applied to \(\lambda_{min}\) and \(\lambda_{max}\) in the implementations of the adaptive approaches. We run the tests on an Intel Core i7 processor with clock frequency of 2.6 GHz, 8 GB of RAM, and a 64-bit Linux system. For each of the six images bacteria, bacteria2, brain, flowerbed, ninetyeight, and squirrel, we show the results that were obtained with the spatially adaptive strategy yielding the best segmentation for that image. The corresponding (unscaled) values of \(\lambda\), \(\lambda_{min}\), \(\lambda_{max}\), the value of \(\mu\), as well as the number of outer iterations and the mean number of GS iterations per outer iteration are reported in Table 2. Note that, for squirrel, we use two values of \(\lambda_{min}\), one equal to the value of \(\lambda\) in the CEN model and the other greater than that, obtained by trial and error. Both values of \(\lambda_{min}\) produce the same segmentation, but the larger value of \(\lambda_{max}\) reduces the number of outer and GS iterations. The segmentations corresponding to the data in Table 2 are shown in Figures 1-3. For squirrel, we display the CTD segmentation computed by using the larger value of \(\lambda_{min}\). We see that, on selected problems, CTD reduces the number of outer and GS iterations with respect to the CEN model; on the other hand, the setup of the regularization parameters is computationally more expensive. In terms of iterations, there is no clear winner between CEN and MM and between CEN and THR. The models based on the spatially adaptive techniques are slightly more expensive than the CEN model in this case too. A significant result is that the segmentations obtained with the adaptive techniques appear better than those obtained with the non-adaptive model, i.e., the spatially adaptive models can better outline boundaries between objects and foreground. This is clearly visible by looking at the segmentations of brain, bacteria2 (see the upper right corner), ninetyeight, bacteria (see the shape of the bacteria). It is also worth noting that the adaptive model based on THR removes textural details that do not belong to the flowerbed in the homonymous image.