Images play an important role in transferring information. In order to obtain more information from collected images, image enhancement techniques are commonly required to improve image quality. Traditional image enhancement methods can be roughly divided into two categories: 1) spatial domain methods, such as gray-level transformation, piecewise-linear transformation, and histogram equalization, etc. 2) frequency domain methods, which include high-pass filtering, high-frequency emphasizing filtering, and homomorphic filtering etc. However, these methods mentioned above are in general difficult to balance well among various requirements of image quality, such as contour enhancement, dynamic range, denoising and so on. In addition, the ability of traditional image enhancement methods is far behind the human visual system in almost all aspects.

Early physiological studies have revealed that the retinal ganglion cells have a receptive field (RF) consisting of concentric regions, i.e., an approximately circular center and an annular surround [1] [2] . DOG model was proposed by Rodieck to describe the classical receptive field (CRF) of the ganglion cells [1] . Li et al. found that the cells at some distance from the contrast borders were less affected, while the border enhancement might become quite stronger when the centre was close to the corner of a bright contour [2] . Ramachandran found that the luminance gradients of an area are essential for producing perception of three-dimensional visual scenes [3] . By analyzing the length-response functions of lateral geniculate neurons in the cat, Li et al. have demonstrated an extensive disinhibitory region (DIR, i.e., non-CRF) outside the classical inhibitory surround of the receptive field [4] . According to this finding, a three-Gaussian function model was proposed in [5] . By setting appreciate parameters of the three-Gaussian model, good fit could be obtained almost for all data that show disinhibitory phenomenon [5] . Functionally, the three-Gaussian model can not only enhance the edge information but also transmit brightness information with low spatial frequency [5] [6] .

However, the three-Gaussian model cannot dynamically adjust its parameters according to the local stimulus. In fact, the adaptation to the stimulus features (e.g., the luminance contrast) of the receptive field in the visual system has been deeply studied [7] -[12] . Some experiments [7] showed that the responses of retinal ganglion cells first increased abruptly, and then decayed exponentially to a lower value following the abrupt increase in stimulus contrast. Based on extracellular recordings from 69 LGN cells in the anesthetized cat, Nolt et al. found that the spatial summation within their receptive fields was dependent on the contrast of the stimuli presented. They reported that the contrast-dependency in the retinal ganglion cells directly resulted from a reduction in the size of the center mechanism due to an increase in contrast [8] . By characterizing the adaptation of neurons in the cat lateral geniculate nucleus (LGN) to changes in stimulus contrast and correlations, Lesica et al. found that the space constant of the excitatory center increased with a decreasing in stimulus contrast [9] . In addition, it has been shown that spatial summation in the primary visual cortex of the cat and monkey is strongly dependent on stimulus contrast; the area (length and width) over which responses summate generally increases as the stimulus contrast decreases; fitting summation curves with a DOG model shows that this contrast-dependent spatial summation seems to derive from a change in the actual size of the receptive field [10] -[12] .

To simulate the dynamic properties of the RF, in this work we present an adaptive three-Gaussian function model to automatically adjust its parameters according to the properties of local stimulus, i.e., local contrast and luminance, for image enhancement.

In this experiment, we compared our adaptive three-Gaussian method with the popular method of histogram equalization and the standard (non-adaptive) three-Gaussian model. Experimental results on several images are shown in Figure 4, Figure 5, Figure 6, and Figure 7. Note that the zoomed in view of each test image is also listed in Figures 4-7, respectively. From the figures, the results of the standard three-Gaussian model usually include more details than original images, but some regions are over-enhanced (especially in the high contrast place); in addition, the contrast of high-light and shading regions are not enhanced enough. Histogram equalization is efficient in adjusting global dynamic range of images, but it is difficult to obtain good local contrast. In addition, three-Gaussian model usually obtains better performance than histogram equalization.

Our adaptive three-Gaussian function model performs better in both enhancing the local contrast and adjusting global dynamic range. Meanwhile, the proposed method is capable of overcoming the phenomenon of over- enhancement. In addition, the performance of our new approach in edge enhancement is much better than the other two methods mentioned above, which can be clearly seen from Figures 4-7, especially from the zoomed in view of each test image.

For quantitative comparison, we employed EME (a measure of enhancement) [13] and SNR (Signal to Noise Ratio) [14] for performance evaluation of image enhancement. SNR is usually defined as the mean target signal to the standard deviation of the noise [13] . In this paper, we define E as the mean value of the all pixels in the

image and as the standard deviation of the all pixels in the image. Therefore, SNR is computed as

EME is computed as [14]

where V_min,k,l and V_max,k,l are respectively the minimum and maximum inside a certain block w(k,l) when the whole image is split into k₁k₂ blocks w(k,l) of equal sizes. c is a small constant that equals to 0.0001 to avoid dividing by zero. In general, a higher EME indicates a better enhancement in image details.

EME and SNR of four considered images shown in Figures 4-7 are listed in Table 1 and Table 2. Note that the EME and SNR were calculated from the whole images. From Table 1, the evaluation of EME shows that our adaptive three-Gaussian function model obtains the best performance on edge enhancement. From Table 2, we can see that our new approach achieves competitive performance compared with the standard three-Gaussian function model and histogram equalization in suppressing image noise. This indicates that our adaptive model can well balance the requirements of enhancing edges and inhibiting image noises.

It is generally accepted that the computational image processing methods are far behind the human visual system. They met difficulties to balance well among various requirements of image quality, e.g., contour enhancement and denoising which often cannot be well achieved at the same time. By seeking inspiration from the physiological findings, this paper proposes a physiologically based adaptive three-Gaussian model, which dynamically adjusts the parameters of the three-Gaussian model. The results on several real-world images show that the performance of our new model is better than the standard three-Gaussian function model, especially in overcoming over-enhancement and raising the contrast of highlight and shading regions. Our approach can keep the SNR of an image in an acceptable level; meanwhile, it can effectively enhance the edge profiles and local details of the image. Specifically, in the regions of low luminance, we increase the excitatory strength () in the regions with high local contrast, which helps enhance the edges with high contrast. Differently, we increase the inhibitory space constant () in the regions of low brightness, which helps improve the contrast of shading regions.

Our physiologically-based adaptive three-Gaussian function model only simulates the change of inhibitory space constant () and excitatory strength () based on the local contrast and local brightness, and don’t involve inhibitory strength () and excitatory space constant, () which should be improved in the future work. In addition, how to effectively suppress image noise is also an important future direction for us.

The authors would like to thank Professor Chaoyi Li for his valuable suggestions. The authors also thank the anonymous reviewers for their helpful comments. This work was supported by the 973 project (#2013CB329401) and the NSFC of China (#61375115 and #91120013).