Fundus photography with subpixel registration of correlated laser speckle images

The retina is located on the focal plane of the crystalline lens of the eyeball; thus, the retinal image is relayed to the conjugate plane (usually at infinity) of the crystalline lens. Typically, we place the lens L₁ in front of the pupil to generate a reverse image at the intermediate image plane, as shown in Fig. 1. This reverse image is further magnified by L₂ to produce an image on the sensor plane.

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To evaluate the imaging performance (marked by a solid line in Fig. 1), the imaging process is usually divided into two steps. The first step involves the optical path from the retina to the intermediate image plane, including the crystalline lens and a single lens L₁. The two lenses (crystalline lens and L₁) are respectively regarded as the objective and tube lens so that this optical train is equivalent to the mechanism of infinity-corrected microscopes.¹⁰⁾ Through this optical train, light rays emanating from either the central region or the periphery of the retina are gathered by the crystalline lens to produce parallel bundles of rays since the location of the retina (object plane) was at the front focal plane of the crystalline lens. The parallel bundle of light rays projected either along the optical axis or with oblique angles to the singlet L₁, is dependent on the retinal position. The lights gathered by L₁ are then focused at its focal plane (the intermediate image plane) to form a reverse image of the retina. Therefore, the magnification of this imaging step M₁ is calculated as the ratio of the focal length of L₁ (f₁), to the focal length of the crystalline lens (f_CL), as shown in

$$\begin{equation} |M_{1}| = \frac{f_{1}}{f_{\text{CL}}}. \end{equation} \tag{ 1 }$$

On the other hand, since the imaging process is equivalent to the telecentric system in the space of an object, the field of view (FOV) is constrained by a field diaphragm located at the intermediate image plane. Their relationship could be expressed as

$$\begin{equation} D_{\text{Obj}} = \frac{D}{M_{1}} = \frac{Df_{\text{CL}}}{f_{1}}, \end{equation} \tag{ 2 }$$

where D is the diameter of the field diaphragm and D_Obj is the corresponding diameter of the FOV at the object plane, as shown in Fig. 1. Equation (2) implies that the FOV has a reciprocal relationship with f₁. The maximum FOV at the object space is achieved when the diameter of the field diaphragm matches the diameter of L₁. Nevertheless, it is necessary to constrain the FOV via the field diaphragm to satisfy the paraxial approximation and alleviate the aberrations.

The second step involves the path from the intermediate image to the actual image plane (it can be an observer's eye or an imaging sensor). The magnification of this imaging step, M₂, can be simply determined by the magnification of L₂. As a result, the image size can be calculated as M₂D. The physical size of the image captured by the image sensor might sometimes be less than this result owing to the limitation of the sensor format. After considering both steps, we can attain the magnification M of the entire retinal imaging system as

$$\begin{equation} |M| = |M_{1}\times M_{2}| = \frac{Df_{\text{CL}}}{f_{1}}\times \frac{q}{p}, \end{equation} \tag{ 3 }$$

where the quantities p and q represent the object distance (from the intermediate image plane to the principal plane of L₂) and the image distance (from the principal plane of L₂ to the image plane).

To accomplish fundus photography, it is necessary to provide extra illumination with appropriate optics to light up the vitreous chamber. The most popular setup is based on the concept of Köhler illumination for eliminating the source image at the object plane.¹¹⁾ On the other hand, in order to eliminate glare reflection from the cornea, other specific requirements for the design of the light source (e.g., source shape and location) are considered.

Typically, the illumination optics for fundus photography requires at least three important elements, namely, a light source, a collector lens, and a condenser (the crystalline lens), as shown in Fig. 1. For simplicity, only the lights emanating from the periphery of the light source are illustrated (indicated by the green dash line in Fig. 1). The bundle of light rays transmitted from the collector lens produces a source image at the aperture diaphragm, which was then gathered by L₁ to illuminate onto the retina (object plane). This design prevents the production of a conjugate image of the light source at the object plane, which severely degrades the image quality. However, because the illumination optics share a common path with the imaging optics, the conjugate image of the aperture diaphragm at the pupil and the glare reflections from the cornea are likely gathered by the imaging optics. Compared with these signals, the back reflections from the retina are much weaker.^12,13) To minimize the glare reflections from the cornea, the common approach is to utilize an aperture diaphragm with an annular shape or a light source for off-axis illumination.^14–16) The rule of thumb is to maximize the use of the pupil with high efficiency, that is, the conjugate image of the annular light source is imaged at the outer portion of the pupil, as the illumination path. This implies that only the light ray bundle emanating from the central region of the pupil is captured by the imaging optics once the field diaphragm has been properly adjusted.

According to a previous work, the imaging system could achieve the largest illumination efficiency for acquiring maximum reflected power when the area of the illuminating annulus was equal to the area of the imaging pupil.¹⁵⁾ Nevertheless, this inference was based on an aberration-free system and efficiency might be deteriorated by any aberration in optical elements or the oculus. For example, the geometric shape and location of the illuminating annulus on the pupil may be significantly distorted by the aberrations of optical elements, resulting in undesired back reflections from the cornea. Such aberrations could be eliminated effectively by increasing the inner radius of the illuminating annulus but sacrificing illumination efficiency. On the other hand, ocular aberrations also affect the image quality of fundus photography because the point spread function of the crystalline lens is degraded.^17–23) Therefore, the optimal pupil diameter of the oculus was suggested to be 2 mm in most practical devices in order to minimize the ocular aberrations.^15,16,23) Although a smaller imaging pupil may alleviate the image degradation to maintain the image quality, the resolving power is simultaneously reduced owing to the increase in f-number. To increase the resolving power of the optical system, several approaches had been proposed, such as wavefront correction with the adaptive optics system,^24,25) iterative shift-and-add technique,²⁶⁾ image deconvolution via estimation of the point spread function,^27,28) and impedance matching.²⁹⁾ Recently, a binocular fundus camera with a larger pupil diameter has also been developed.³⁰⁾ These methods aimed to improve the image quality via post-processing or reducing the f-number, and have successfully achieved a cellular-level resolution. Nevertheless, additional perturbations induced by the movement of the testing sample or the device itself might increase computational complexity. Rather than portable usage,³¹⁾ this kind of device is more suitable for stationary usage, providing great potential for further improvement.

When a rough surface is illuminated by a coherent light source, a large number of interference dots called speckles are generated owing to the severe scattering phenomenon. Although these interference dots seemed to be annoying in most holographic purposes, many studies have revealed that high-order information can be extracted from an underlying randomly distributed pattern.^3–5) By analyzing the statistical properties of such an interference phenomenon, the speckle contrast (K) of a speckle pattern could be defined as

$$\begin{equation} K(x,y)\equiv \frac{\sigma_{I}}{\langle I\rangle}, \end{equation} \tag{ 4 }$$

where the angle brackets 〈·〉 denote the ensemble average so that 〈I〉 is the first moment (mean) and σ_I is its second moment (standard deviation) of a selected window.³⁾ The speckle contrast is widely used as a trait to characterize fluid dynamics. From the definition of K, local regions without vessels are expected to have a lower speckle contrast K than those with vessels since any moving particle might lead to a blurred speckle pattern during one exposure of an imaging sensor. This characteristic enabled us to reconstruct a contrast image to monitor the blood flow in tissues.^4,5,9,32)

Typically, there are two approaches to calculating the speckle contrast K, spatial and temporal approaches, using different methods of statistical analysis,⁴⁾ as shown in Fig. 2. The first approach requires the pixel values of N frames to calculate the speckle contrast, $K_{\text{t}}(x,y)$, of every specific point $(x,y)$; this approach is known as laser speckle temporal contrast analysis (LSTCA). This implies that it is necessary to take N exposures into consideration to obtain one contrast image, which is a challenge in real-time monitoring, especially in cases of low sample irradiance such as in fundus photography. An alternative way is to calculate the speckle contrast using a kernel, h, which is obtained with a snapshot. This is called laser speckle spatial contrast analysis (LSSCA). LSSCA has a superior real-time response to LSTCA with the sacrifice of spatial resolution. Either spatial or temporal resolution would be considered as a trade-off. For LSSCA, the kernel size predominately degrades the spatial resolution of images. The statistical characteristics of local speckle images sampled on the basis of kernel h are also affected by kernel size owing to insufficient sampling.³³⁾ The kernel size in practical LSCI-related devices is commonly 5 × 5 or 7 × 7 in order to maintain sufficient sampling, which severely compromises the spatial resolution of contrast images.³⁴⁾ Moreover, previous research carried out by Cheng et al. also indicated that the performance of LSSCA is susceptible to stationary speckles induced by static tissues or other scattered media in the optical path.³⁵⁾ On the other hand, for LSTCA, the first critical issue is to guarantee that the speckle patterns acquired by the same pixel are statistically independent.³⁾ The condition usually holds in most coherent sources since the coherence time is usually much less than the exposure time Δt. The second issue is the dependence of the exposure time on the flow velocity of an object. For example, a short exposure time enables us to acquire clear speckle patterns in both regions with and without blood vessels. In contrast, a long exposure time might result in blurry speckle patterns in the region with blood vessels since the lights are scattered by moving particles, allowing us to reconstruct a contrast image by LSCI. For the flow velocity on the order of 7–10 cm/s, the suggested exposure time is from several milliseconds to dozens of milliseconds.^4,5,35)

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Since there exists a fundamental tradeoff between the spatial and temporal resolutions, in this work, we chose LSTCA to preserve the highest possible spatial resolution. The image quality was improved using correlated laser speckles.