Okkeun Lee

Introduction

Contrast sensitivity function (CSF) is a subjective measurement of an ability of the visual system to detect a low contrast pattern stimuli. The stimuli considered usually are vertical sinusoidal Gabor patches of decreasing shades of black to grey. This use of sine wave gratings was first introduced in vision by Schade [1] and was subsequently used by early investigators to measure basic visual sensitivity [2]. The resulting measurement is used to validate the representation of the eyes' visual performance as it complements the visual acuity. CSF is synonyms with an audiogram where a person’s highest detectable pitch is measured and well as the ability to hear all lower pitches [3].

The CSF measurements are usually acquired with small patches of sinusoidal grating designed to fall within few central degrees of the visual field. It is well known that the CSF decreases as one measures contrast sensitivity at increasingly peripheral locations in the visual field. The reasons for such decreased CSF is attributed to a number of neural factors [4]. The human eye is structured such that the distribution of the cone mosaic falls off rapidly as a function of visual eccentricity, so that there are fewer sensors available to detect and encode the incoming stimuli. Towards the periphery the amount of retinal ganglion cells’ density falls as well. This structure of the cones is also a key factor in deciding the CSF. In particular, for this project, we make an effort to explore the role of inference engines in shaping the CSF.

Background

The human visual system’s CSF or the modulation transfer function (MTF) fundamentally characterises eye’s spatial frequency response and thus, one may think of it as a bandpass filter. This bandpass nature, however, is determined by a variety of factors such as the front-end optics, cone distribution geometry and the neural mechanism that is responsible for this interpretation which is typically inferred using the machine learning techniques such as the kNN and SVM [5].

The methods of CSF measurement usually employ sine-wave gratings of a fixed frequency. By varying the frequency, a set of stimuli are constructed. The response of the eye to these stimuli are then determined. This procedure is repeated for a large number of grating frequencies and the corresponding responses are recorded. A mechanism that emulates the brain inference is applied to these responses to find the resulting curve called the contrast sensitivity function and is illustrated in Fig. 1. [6].

Fig 1. The plot of CSF varies with retinal eccentricity.

From the measurements, contrast sensitivity score is determined such that it is equal to (1/threshold) for the given spatial form. The contrast sensitivity scores obtained for each of the sine-wave gratings examined are then plotted as a function of target spatial frequency yielding the contrast sensitivity function (CSF) [7]. Some typical CSF's are depicted in Figure 1 below shows the typical inverted-U shape of the CSF on logarithmic axes.

The Robson JG and Graham N [8] have demonstrated that the CSF attains its maximum while using foveal vision as indicated in Fig 2. They observed that the CSF reduces with increasing retinal eccentricity. Further, the decline of the CSF with eccentricity is greater for higher spatial frequencies but more gradual for lower spatial frequencies [9].

Fig 2. The plot of CSF as function of Log Spatial Frequency for various spatial locations in the retina.

Methods

In order to determine the CSF for various spatial frequency, we have used the readily available tool from the MATALB called The Image System Engineering Toolbox for Biology (ISETBio). The ISETBio is available as an open source project [10]. The toolbox is primarily used to design image sensors as well as for modeling image formation in biological systems and thus used widely for commercial and research purposes [11].

We have used ISETBio first to simulate the cone photoreceptor absorptions, which is the first stage processing in human vision and thus the CSF to some extent is determined by the photoreceptors. One of the factors that still remains unclear is the relation between size of the photoreceptors and the CSF as they are small in the fovea and much larger in the periphery . In order to include the cone mosaic distribution factors into account, we generated two portions of the retina- the center of the fovea, and slightly to the periphery using the “coneMosaic” method in ISETBio. We computed the CSF for six spatial frequencies from 1 to 1.5 in log space. We also varied the integration time in the retina as a factor in order to know how the integration factor affects the CSF. For each of the frequency, we generated a Gabor patch stimulus and obtained the visual response using the toolbox. The stimulus responses between the null and the Gabor patch stimulus was discriminated by using the inference engines from machine learning. In particular, we were interested to compare the performance of both the support vector machines (SVM) and k-nearest neighbours (kNN) algorithm in terms of their role in inferring the CSF.

SVM is a supervised learning method developed at AT&T Bell Laboratories by Vapnik with colleagues [12]. The way we used the SVM in our experiment is as follows: we marked the responses that belong to stimuli and null stimuli categories as one or zero, respectively. We then trained the SVM using the training samples and tested the model with the test set. We computed the percentage of correct detection of stimuli by the SVM and compare it against the given threshold to see if for the set of front-end optic parameters and the cone mosaics the contrast change is detected. We repeated this for various spatial frequencies and thus obtained the CSF curves.

Figure 3. An example of a separable problem in a 2 dimensional space. The support vectors, marked with grey squares, define the margin of largest separation between the two classes.

Results

In our experiments, we vary the integration time of the cones and determined the CSF for the fixed location of the cone mosaic, that is at the fovea. The size of the cones mosaic is set at 149 X 149 with each pixel 2X2 um2 size. Figures 4 and 5 indicate the effect of integration time on the CSF. We observe from the Fig. 4 that we obtained the desired result for all the integration times except 60 msec for the kNN. That is, the CSF declines towards the higher spatial frequency as expected ( the overshoots are the spline interpolation artefacts and should be ignored). On the other hand, using the SVM inference engine, we were unable to detect such pattern except for the integration times 10 and 20 msec. We suspect that the potential problem could be some bugs in the use of SVM classifier or the contrast detection. We are in the process of debugging these results. Also, we could not see the exact role of integration times just from the Fig 5. That is, while in kNN it is clear that as the integration time increases the CSF increases and such pattern is not witness in the SVM graph, which we also would like to improve in the future.

Fig. 4. Plot of CSF vs. spatial frequency for various integration times using kNN classifier

Fig. 5. Plot of CSF vs. spatial frequency for various integration times using SVM binary classifier

Conclusions

We illustrated that the shape of the CSF is determined by the integration time of the photoreceptors in the retinal ganglion cells. However, we noted that integration time is one of the key factors that determine the CSF. While obtaining these curves, we assumed the default setting in the ISETBio, such as the fixed front end optics and cone mosaic distribution. With these fixed settings, we found that the CSF declines strongly as the frequency increases. Interestingly, as the theory suggested the value of the CSF for lower integration time is smaller than that of for the higher integration time. However, using the SVM, currently, we could not support this hypothesis as we have gotten the opposite results.

Limitations and Future Work

One of the major limitations of our project was that we generated the cone mosaic pattern randomly for each spatial frequency. First, this sets different cone mosaic pattern for each frequency. This is not ideal for a controlled experiment, as one person's eye would only have only one mosaic distribution. Therefore, the results in this study has to be inferred carefully as we did not account the fixed cone mosaic pattern. Second, we implicitly assume that the relative distribution of cone types matters more than the specific location of those cones in the mosaic.

Another limitation of the current appraoch is that we totally ignored the role of rods in shaping the CSF. While the cone distribution matters, we believe that the rod masaic may have their role as well to give a better picutre of the visual system.Thus, we may need to write scripts to include the rod distribution.

Finally, the script used to do the SVM and KNN inference engines for the CSF graphs indicated in this project are provided in the Appendix As a future work, we would like to add more functionality metioned in the previous paragraphs.

Reference

[1] Otto H. Schade, Optical and Photoelectric Analog of the Eye, J. Opt. Soc. Am. 46, 721-739 (1956)

[2] Westheimer, G. (1960). Modulation thresholds for sinusoidal light distributions on the retina. Journal of Physiology, 152, 67-74

[3] http://www.ssc.education.ed.ac.uk/resources/vi&multi/VIvideo/cntrstsn.html

[4] Jennings, J. A., & Charman, W. N. (1981). Off-axis image quality in the human eye. Vision Research, 21(4), 445–455.

[5] Nicolas P. Cottaris, Brian A. Wandell, Fred Rieke, David H. Brainard; A computational observer model of spatial contrast sensitivity: Effects of photocurrent encoding, fixational eye movements, and inference engine. Journal of Vision 2020;20(7):17

[6] https://foundationsofvision.stanford.edu/chapter-7-pattern-sensitivity/

[7] http://usd-apps.usd.edu/coglab/CSFIntro.html

[8] Robson JG, Graham N. Probability summation and regional variation in contrast sensitivity across the visual field. Vision Res. 1981;21:409–18.

[9] Skalicky S.E. (2016) Contrast Sensitivity. In: Ocular and Visual Physiology. Springer, Singapore. https://doi.org/10.1007/978-981-287-846-5_20.

[10] https://github.com/isetbio/isetbio

[11] http://isetbio.org/

[12] Cortes, Corinna; Vapnik, Vladimir N. (1995). "Support-vector networks". Machine Learning. 20 (3): 273–297.

Appendix

The codes for this project can be found on the following link Code_okkeun.zip