## 2 Proposed Superpixel-Based Model
### Image Compensation  
The operations described above can be regarded to be a superpixel version of bilateral filtering. It is clear that the image details with a scale smaller than the superpixel size will be over-smoothed incorrectly. To precisely maintain useful image details, we introduce a compensation operation. It should be noted that compensation is a commonly used strategy for fine-tuning ultrasound image details [14]. However, differing from others, our compensation strategy is HVS-inspired, which is described below. As the reconstructed image can be regarded as the low frequency component of the input ultrasound image, we can subtract the low frequency component from the input image to get the high frequency component of the input image:  
Figure 2: Illustration of the image reconstruction. A represents the current pixel to be reconstructed and C1, C2, C3, and C4 represent the four superpixel centers in a \(3\times 3\) squared grid around A.  
\[I_{high}(x,y)=I(x,y)-I_{low}(x,y). \tag{3}\]  
This high frequency component, \(I_{high}(x,y)\), contains both the useful details and the unwanted high-frequency noise, which are difficult to separate from each other. In this work, we propose a HVS-inspired solution to separate and remove the unwanted high-frequency noise from the useful details of \(I_{high}(x,y)\). Our solution is based on the following evidence from the field of visual neuroscience: (1) As the first stage of the HVS, the retina processes the visual information, from the bipolar cells to the ganglion cells, along the ON and OFF pathways. (2) The ON and OFF pathways process the bright and dark components, respectively. (3) The responses of the bipolar and ganglion cells in the retina show center-surround opponent patterns within their receptive fields (RFs). Inspired by these biological findings, we first separate the high-frequency component into bright and dark parts, which are written as  
\[I_{b}(x,y)=I_{high}(x,y),\ \ \text{where}\ I_{high}(x,y)>=0 \tag{4}\]  
\[I_{d}(x,y)=-I_{high}(x,y),\ \ \text{where}\ I_{high}(x,y)<0. \tag{5}\]  
Then, we use the non-classical receptive field (nCRF) model to separately process these two parts of the high frequency component. Suggested by the physiological finding that some retinal ganglion cells (RGCs) in cats respond quite weakly to the stimulus of dispersedly distributed dots, in comparison to compactly distributed dots (e.g., where the dispersedly distributed dots are compacted into a line) [23], we have built a nCRF model based on RGCs, in our previous works, for the tasks of image de-hazing and color constancy [24; 25]. As for the RF's spatial structure, a nCRF model of ON-type contains an excitatory center and an inhibitory surround with a disinhibition property (see Figure 3)--that is, the RF surround is composed of many sub-units, and these sub-units first inhibit each other; then, the inhibited sub-units in the RF surround inhibit the neuronal response elicited by the stimulus in the RF center. The neuronal processing can be formulated as follows. We, first, define a two-dimensional (2D) Gaussian function with a scale of \(\sigma\) as  
\[g(x,y;\sigma)=\frac{1}{2\pi\sigma^{2}}\exp(-(x^{2}+y^{2})/(2\sigma^{2})). \tag{6}\]  
For the bright part of the high-frequency component (i.e., \(I_{b}(x,y)\)), the ON-type nCRF model is employed to compute the neuronal response as  
\[S_{on}(x,y)=\mathbb{H}\{I_{b}(x,y)\otimes(g(x,y;\sigma_{c})-g(x,y;\sigma_{s}))\}, \tag{7}\]  
Figure 3: Illustration of the spatial structure of the non-classical receptive field (nCRF) model of ON-type, containing an excitatory center and an inhibitory surround with multiple sub-units.