# All You Need Is a Few Dots to Label CT Images for Organ Segmentation
## 3 Materials and Methods
### Proposed Method  
We propose a semi-automated tool that generates organ segmentation. Given a CT image, the radiologists annotate a few pixels--an anchor to specify the target organ. The pixels near the _anchor pixels_, which have similar characteristics, are candidates of the target organ to be segmented. To capture the segmentation region of the target organs, we utilize two types of information: _visual features_ and _locality_. In Figure 2, we provide a schematic representation of our network. Our proposed model consists of two sub-modules: Feature Encoder \(\mathcal{F}\) and Kernel Function \(\mathcal{K}\). The Feature Encoder \(\mathcal{F}\) extracts the feature vector for each pixel, which is used to calculate the visual similarity between pixels. We exploit a modified U-Net architecture to implement the Feature Encoder \(\mathcal{F}\). The Kernel Function \(\mathcal{K}\) is used to capture both _visual features_ and _locality_ information. In a CT image, let \(M_{c}\) be a set of anchor pixels that a user picks on the \(c\)-th organ. The Kernel Function \(\mathcal{K}\) computes the density for all pixels, which represents visual and positional similarity. As illustrated in Figure 2\(c\), the _kernel density_ is made of two types of densities:  
Figure 1: CT image examples that show different colors and textures for each organs. Figure 2: Model architecture that consists of 2 kernels. These kernels are used to compute the kernel density, which has a high value in the region of the target organ: (**a**) Feature similarity kernel density; (**b**) Gaussian kernel density; (**c**) Kernel density. feature similarity kernel density \(\mathcal{K}_{c}^{\text{Feature}}\) (shown in Figure 2a) and Gaussian kernel density \(\mathcal{K}_{c}^{\text{Gaussian}}\) (shown in Figure 2b). The _kernel density_\(\mathcal{K}_{c}\) of a pixel \(p\) is computed as follows:  
\[\mathcal{K}_{c}(p)=\sum_{m_{c}\in\mathbf{M}_{c}}\mathcal{K}_{c}^{\text{Feature }}(p,m_{c})\mathcal{K}_{c}^{\text{Gaussian}}(p,m_{c}) \tag{1}\]  
where \(p\) is an arbitrary pixel, and \(m_{c}\) is an anchor pixel of the \(c\)-th organ. \(\mathcal{K}_{c}^{\text{Feature}}\) means the feature similarity kernel density for the \(c\)-th organ. The feature similarity kernel density measures the feature similarity of two pixels. \(\mathcal{K}_{c}^{\text{Gaussian}}\) indicates the Gaussian kernel density for the \(c\)-th organ. The Gaussian kernel density calculates the positional similarity of two pixels. The kernel density has a maximum value at each anchor pixel \(m_{c}\). In the other case, the density value becomes high if a pixel \(p\) is closer to the anchor pixels \(\mathbf{M}_{c}\) and has a similar texture or color too. Therefore, we can figure out the region of the \(c\)-th organ, filtering pixels with a high density. If a pixel \(p\) has similar visual features to anchor pixels \(M_{c}\), the pixel \(p\) is more likely to belong to the \(c\)-th organ than other pixels. Using feature vectors \(\mathcal{F}(p)\) for the pixel \(p\) and \(\mathcal{F}(m_{c})\) for an anchor pixel \(m_{c}\), we can calculate the feature similarity kernel density by using the dot product between \(\mathcal{F}(p)\) and \(\mathcal{F}(m_{c})\) as follows:  
\[\mathcal{K}_{c}^{\text{Feature}}(p,m_{c})=\mathcal{F}(p)^{T}\cdot\mathcal{F}(m _{c}) \tag{2}\]  
where \(p\) is an arbitrary pixel, and \(m_{c}\) is an anchor pixel in the \(c\)-th organ. The operator \(\cdot\) means dot product. If two pixels \(p\) and \(m_{c}\) have similar features, the feature similarity kernel density has high value. Gaussian kernel quantified how close the pixel is to anchor points, that is, a pixel closer to anchor pixels has a higher density value. This density is required not to capture regions that have similar texture but are far from anchor points (in another organ). A pixel \(p\) should be closer to anchor pixels for being classified as a target label.