1. Introduction

Before reaching the photosensitive cells in the human eye, light travels through several retinal layers. Thanks to the high transparency of these layers, image formation is almost unperturbed. However, such a characteristic makes also observation of these structures extremely challenging. Indeed, usual retinal imaging techniques rely on backscattered light to provide contrast [1].

Optical Coherence Tomography (OCT) is the main technique used for the observation of the living retina. Despite its ability to distinguish between different retinal layers, standard OCT is not sufficient for the observation of many retinal microstructures en face [2, 3].

Monitoring retinal microstructures and their morphology is central in the analysis of eye pathologies. Indeed, an analysis of cells density could offer crucial information about the health of the retina. This information could then be used to diagnose diseases and dysfunctions at an early stage and for monitoring the effect of drugs during therapeutic treatment.

Because of that, research efforts are being dedicated to develop novel techniques for observing cellular structures with high contrast in the retina. One of these techniques is based on a confocal scanning system and a split detector. By subtracting the power measured on the two detectors, it is possible to observe microvasculature in the eye [4–7]. Another technique is based on the introduction of an offset in the aperture of a confocal scanning ophthalmoscope [8, 9]. Such a technique enhances the phase gradient of the image plane, allowing for the in-vivo observation of ganglion cells in monkeys [10]. The obtained images in humans show poor contrast mainly because of the limited optical power sent to the eye for safety reasons.

Another technique recently developed shows how OCT coupled with adaptive optics can be used to achieve high resolution images of the retina [11]. The organelles inside ganglion cells move within the cell’s soma and thus by averaging several images collected over a time span of about 10 minutes, it is possible to observe ganglion cell’s somas. However, the latter technique is not compatible with clinical use because the imaging session requires a short acquisition time, especially in the case of patients which are unable to fixate a target.

To overcome the high acquisition time, a recent technique, based on wide field oblique illumination of the retina, was introduced by the authors of this article [12]. The method used oblique illumination of a retinal sample (Fig. 1(a)) to provide phase contrast. Images of ganglions could be acquired in 25 seconds in contrast to 10 minutes for the work of Miller [11].

 

Fig. 1 (a) Method for obtaining oblique transmission illumination in an eye sample. Light passes through the retina and it is backscattered from the retina pigmented epithelium (RPE) and choroid. This backscattered light results in a new illumination source which acts effectively as a transmission-like imaging system as illustrated in (b) Transmission setup (from [15]) for phase contrast obtained through oblique illumination.

Download Full Size | PDF

In this article, we explain how phase contrast is obtained in the retina by using a reflection configuration (i.e. where both illumination and collection arm lays on the same side of the sample).Phase imaging of thick backscattering sample was originally demonstrated by Ford [13, 14]. This is not the case of the retina, which is a transparent layer located on top of a strongly backscattering layer. Because of this, we will consider the backscattered light as a new source for a transmission-like system. For the transmission configuration such as the one showed in Fig. 1(b), phase contrast has been obtained by oblique illumination [15–18]. Under the approximation of a weak object ($e^{i\varphi -\mu }\approx 1+i\varphi -\mu$) and by knowing the illumination function it is then possible to extract the quantitative phase image [15].

In section 2, we describe the model. We find that the gradient of the illumination function is a crucial parameter that affects the contrast. Hence, in section 3, we model a gradient illumination and simulate its impact on phase contrast. In section 4, we experimentally test the phase contrast on ex-vivo samples and compare with the model.

2. Theory

The phase contrast technique employed in our work is based on Fourier Filtering with incoherent illumination [15, 18], not on interferometry as it is most commonly used. Phase contrast is generated by the combined effect of a finite size aperture together with an oblique illumination. In the following we show, that it is reasonable to approximate the illumination function as being linear in the spatial frequency domain for retinal tissue. Thanks to this approximation, we furthermore demonstrate how phase contrast is proportional to the gradient of the illumination function.

2.1 Weak object transfer functions

The model we use here is based on the weak object approximation [15, 18]. Even if this approximation does not hold for thick samples, such as the retina, it has been showed that the resulting phase image obtained under this assumption is proportional to the real phase [19]. The phase error is smaller than 5% when a constant phase of 2.3 rad is added. Under the weak object approximation, and for spatially incoherent source we can write the Fourier transform (FT) of the intensity at the camera plane as:

2.2 Ramp approximation

As we will see in the following, it is convenient to analyze the case of a linear illumination function $S_l\left(u\right)$(linear in the spatial frequency domain):

This formulation shows that for an illumination of the form $S_l\left(u\prime \right)$, the two parameters q and m are not affecting the profile of the transfer functions, but are only rescaling it. Furthermore, we define r = m/q and introduce the normalized transfer functions $h\left(u\right)$and $g\left(u\right)$ as:

We further introduce$\widehat{J}\left(u,r\right)$, the image normalized by the background, as:

Such an expression shows that, for an illumination of the form of $S_l\left(u\prime \right)$, only the phase contrast $\widehat{\varphi }\left(u\right)$term is multiplied by a scaling factor r. The absorption contrast is unaffected by the linear illumination function. Therefore, by increasing this scaling factor r, we expect to obtain a better phase contrast image.

3. Simulations

3.1 Non-linear illumination

Equation (7) assumes a linear illumination function. In practice, the illumination caused by backscattering in a full retina sample (including choroid and RPE) will likely show some deviations from linearity. Thus, in this section we investigate the effect of a non-linear illumination on the transfer function $g$. To do this, we simulated the transfer functions in Matlab using Eqs. (2) without any approximation besides the weak object one. In this way, for any illumination function$S\left(u\prime \right)$ we obtained $G\left(u\right)$ and $B$. The definitions of $g\left(u\right)$ and $h\left(u\right)$in (6) requires the factor r, which was defined for a linear function. So, we introduce a more general definition of r that can be directly calculated from $S\left(u\prime \right)$ as:

In Fig. 2, we compare the transfer functions $g_i\left(u\right)$ obtained for different illumination functions$S_i\left(u\prime \right)$. The tested illumination functions are: linear functions with different slopes (Fig. 2(a)), linear function with added random noise (Fig. 2(b)), piecewise linear functions (Fig. 2(c)) and sinusoidal functions (Fig. 2(d)). It is worth noticing that in the calculation of the transfer function, $S\left(u\prime \right)$ is always multiplied by$P\left(u\prime \right)$. This means that the only values of $S\left(u\prime \right)$ affecting the final result are the ones for which the pupil function is non-zero. Furthermore, reflectance of human fundus (composed of retina and choroid) is estimated to be ~5% [20], while the difference of refractive index between retina and vitreous humor is estimated to be less than 0.023 [21], thus generating a much weaker reflection. Thus, reflection from this interface is estimated to be negligible. From the Figs. 2(e-h), we notice that all illumination functions$S\left(u\prime \right)$ resulted in similar $g_i\left(u\right)$ functions. The difference between the phase transfer functions$abs\left(g_i\left(u\right)-g_0\left(u\right)\right)$ is illustrated on the 3rd row of Fig. 2.$g_0$ is computed with the linear illumination function $S_0$. As expected, this difference is zero for all the tested linear illuminated functions however, it starts to increase when the illumination function deviates from the linear case. Figs. 2(o) and 2(p) show plots of the error between the transfer function with respect to $g_i\left(u\right)$ and $g_0\left(u\right)$. The illumination functions used for this analysis are:

 

Fig. 2 (a-d) illumination function in the case of linear, linear plus white noise, piecewise linear and sinusoidal profiles. b) and d) have been obtained for $\text{α}=1$. The dashed circle represents the size of the pupil function. (e-h) respective transfer functions. (i-n) subtractions of the obtained transfer function and the transfer function obtained for $S_0$ (perfectly linear case). (o-p) error between $S_i$ and $S_0$. and between $g_i$ and $g_0$ in case of sinusoidal and noisy illumination function. In o), the blue curve has been averaged and the dashed curves represent the mean value plus or minus the standard deviation.

Download Full Size | PDF

Even when the deviation from linearity in the illumination function is large (e.g. 35%), the error in the phase transfer function is still relatively small (6% in the same condition).

We conclude that the phase transfer function corresponding to a linear ramp illumination can be considered approximately valid also for illumination functions that differ from the ideal linear illumination function, which is what is expected in a real sample. We also observe that the parameter r is a multiplier of the phase contrast function. Hence maximizing this factor in the illumination is expected to bring better phase contrast images.

3.2 Monte Carlo simulations

In order to estimate the illumination function caused by backscattered light in the eye, we simulated light scattering process in the retina for a wavelength of633 nm. We used Monte Carlo simulations to trace photons in and out of the retina to obtain the emission direction of the backscattered photons. For each simulation all photons are delivered with the same angle on the same spot on the substrate, as if the light was composed of a single beam of zero diameter. Figure 3 shows the simulations for a 70° incoming beam on a substrate of Retina Pigmented Epithelium (RPE) laying on top of the choroidal layer. The distribution of the backscattered light is the spatial intensity measured in the far field of the scattering media. The key parameters for the simulation are listed in Table 1 ($g$, ${\mu }_s$ and ${\mu }_a$) and they have been obtained from literature in the case of 633 nm illumination of bovine RPE and choroid [22–24].

 

Fig. 3 Simulations for a 70° illumination of a RPE sample on choroid.(a) Profile of the backscattered light in the Fourier plane and (b) illustration of the illumination and backscattered light. We notice that the profile is approximately linear close to the center.

Download Full Size | PDF

Table 1. Scattering parameters for λ = 633 nm.

View Table

The scattered light intensity spectrum obtained from the simulations is shown in Fig. 3(a) and its central cross section is shown in Fig. 3(b) together with an illustration of the illumination and backscattering. We notice that the distribution is non-symmetrical, increasing the intensity in the direction of the incoming beam. The profile becomes approximately linear for small numerical aperture (reduced values of $k_x$ and $k_y$), which is close to the linear illumination approximation discussed in the previous section. Simulations for different incoming beams showed similar behavior for angles greater than 40°.In the next section we will show how this simulation relates to the experimental data.

4. Experimental validation

4.1 Experimental setup

In order to experimentally validate the proposed model, we built the optical setup shown in Fig. 4(a).It consists in two main imaging arms: one in which the camera plane is conjugated with the object plane (sample camera) and the second one in which the camera plane is conjugated with the Fourier plane (pupil camera), which gives directly the illumination profile in spatial frequency coordinates. The cameras used as sample camera and pupil camera are respectively DCC3240M from Thorlabs and camera C9100-23B from Hamamatsu.

 

Fig. 4 (a) Imaging system used for system validation. Illumination can be provided in reflection, using illumination S1, or in transmission, using illumination S2. The sensor of the Sample Camera is conjugated with the object plane, while the pupil camera is conjugated with its Fourier plane (FP). (b) pupil image in case of choroid sample illuminated in backreflection, (c) its cross section and (d) its odd part. The odd part appears to be more linear than its original profile.

Download Full Size | PDF

Illumination is provided with either the back-reflection illumination (S1) or with the transmission illumination (S2). Back-reflection illumination is obtained with a commercial light emitting diode (LED)with a dominant wavelength of 625 nm and an output power of 35 mW, as the light source. Transmission illumination is obtained by displaying a pattern on an active matrix organic light emitting diode (AMOLED) display, Samsung Galaxy J5.

The ex-vivo retinas used in this experiment are pigs’ eyes and a human eye, used for measuring the backscattering profile. The human eye sample has been obtained from the eye bank of Jules-Gonin eye Hospital, Lausanne, Switzerland, in conformity with the Swiss Federal law on transplantation. It has been collected 10 hours post mortem for research purposes, fixed in 4% paraformaldehyde solution (PFA). Samples have been fixed with 4% paraformaldehyde (PFA) for 24 hours and then stored in a 1% PFA solution at + 4°C. Thin samples coming from the pig’s eyes, such as retina on choroid, have been encapsulated between glass slides, while thicker samples have been immersed in phosphate buffered saline (PBS) to keep the sample stable during the measurement. A calibrated United State Air Force (USAF) phase target is used as a baseline for the quality of the obtained phase image. The USAF target is etched in glass with a grove depth of 200 nm.

Furthermore, all the images$I_s$coming from the sample camera have been normalized obtaining the normalized image $J_s$ calculated as follow:

4.2 Dependence of the phase contrast on the linear illumination slope

The main result from section 2 is that the fourier transform of the intensity in the camera is proportional to the phase transfer function (and so the phase contrast). The proportionality constant is the parameter $r=m/q$ (see Eq. (8)). The model has been developed for the transmission configuration, thus the model is first tested with source S2 (Fig. 4) to provide a trans-illumination of the sample. The illumination functions are programmed as linear (see Eq. (4) $S_l\left(u^{\prime }\right)$in which $q$ is kept constant and only the slope $m$is changed (and $r=m/q$).

Figure 5 shows the results for a calibrated USAF phase target illuminated by source S2. This source produces a non-uniform intensity pattern in the Fourier domain (a linear ramp), which then produces the phase contrast (similarly to what shown in Fig. 1). The power density at the sample is estimated to be ~12 μw/cm². Here, the parameter r has been normalized in such a way that the maximum value is equal to 1.

 

Fig. 5 Experimental validation of the model. (a) Phase image of the USAF target and the region of interest highlighted with a rectangle. (b-d) pupil function for 3 different illumination cases obtained with a 0.25 NA objective. (e-g) bars of the region of interest in the USAF target for the same illumination condition. We observe that, as the illumination function becomes steeper (and so r larger), the contrast increases. (h) normalized contrast versus normalized r value. Each data point is obtained with a different value of r. Contrast values are obtained from an average of 100 images. The colored triangles represent the values from the corresponding images. We observe that the experimental data follow the theoretical curve except for small values of r. (i) cross section of the target images.

Download Full Size | PDF

4.3 Phase imaging of retina sample

In section 4.1 we have shown that oblique illumination of RPE and choroid provides linear backscattered light, and in 4.2 we have shown that under this illumination, the intensity image has a contrast proportional to r. From these observations, we would expect to observe the same results in a thick eye sample (i.e. retina on a choroidal layer). However, it is important to notice that, by using an illumination configuration as shown in Fig. 1(b), light passes twice through the retina. We modeled the retina as perfectly transparent, however, we can expect that a certain amount of light will be first scattered by the retina features. Such a signal is expected to introduce some disturbance in the linearity of the backscattered illumination spectrum.

We performed the experiment by shining light on the eye sample and measuring both r(in the pupil camera image) and the contrast in the intensity image (in the sample camera image). Contrast has been estimated for the same cross section as (max-min)/(max + min), where max and min are respectively the maximum and minimum values of the analyzed cross section. The slope of the illumination function is changed by modifying the incoming illumination angle.

The measurement has been repeated for 10 sample areas for different illumination angles ranging between 45 and 80° (i.e. different parameter r).For representation purposes, data has been normalized with respect to the maximum value of contrast and of r. Results are shown in Fig. 6. Overall, the trend is that a higher image contrast is obtained when the parameter r is increased. However, the variability in contrast in different areas (error bars) is probably due to the contribution of direct scattered light from the retinal structures. Increasing the parameter r, corresponding to a steeper illumination angle, maximizes image contrast.

 

Fig. 6 (a) phase image of the retinal sample and the section (green bar) used to calculate the contrast. (b) Pupil image. (c) Normalized contrast versus normalized r. The magnitude error in contrast comes from different analyzed areas of the same sample. This is expected to come from the contribution of back diffracted light which varies across the sample. The curve also shows that maximizing r brings, in general, a better contrast.

Download Full Size | PDF

6. Conclusion

In this article, we studied how to improve phase contrast in imaging of thick retina samples. By modeling the illumination of a retina sample in reflection and verifying with experiments, we found that the light backscattered from the RPE has approximately a linear profile. By using a linear illumination function in the weak object approximation model, we found that the contrast is proportional to the ratio between the slope and the mean value of the illumination function, here called r. Simulations showed that the phase transfer function is almost insensitive to deviation from a linear illumination function, thus providing a robust phase image against variation in the illumination function.

Finally, we tested experimentally ex-vivo retina samples (RPE + choroid) in a transmission and in reflection configuration. The transmission case shows results in excellent agreement with the proposed model. In reflection, light passes twice through the same sample, which adds an amount of scattered light form the retinal structures. We showed that this affects the linearity of the phase contrast enhancement. However, without a priori information, the trend is conserved: moving towards higher values of r tends to increase the contrast of the phase objects.

The main challenges of applying such a technique in real eyes are connected to the eye movement and optical aberrations. Indeed, human eyes move even when fixating a target. Eye aberrations distort the image, thereby decreasing the resolution. In an optimized system, the optical aberration is corrected with an eye tracker together with an adaptive optic system. The motion challenge can be overcome with a short (<10ms) camera acquisition. This is described in more details in [12].