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SECTION 1

INTRODUCTION

Depth-resolved optical imaging is a key strategy for providing noninvasive information from within scattering medium. This task has spawned a broad spectrum of optical coherence imaging approaches, such as confocal laser microscopy [1] and optical coherence tomography (OCT) [2], [3], [4], [5]; the latter being an interferometric approach that utilizes low-coherent waves to discriminate light reflected from one given depth within the scattering medium against most of other scattered light.

Yet, no quantitative assessment of the limits to the capacity of optical coherence imaging to extract information through scattering media is available. The difficulty arises primarily from the complexity of light propagation in turbid media. The classical concepts of signal-to-noise ratio (SNR) and Shannon's information applied for evaluating the performance of optical imaging systems were mostly adequate in the specular reflection or no-scattering regime [2], [6]. However, assessment of such scattering limits is of fundamental interest, both due to the dramatic advances in optical coherence imaging through tissue [4], [5]—advances that rationalize the need to know where the limits are—and also because this evaluation will open up new avenues for the quantitative investigation of a wide variety of information processes being conducted through scattering environments, such as transdermal communication and atmospheric sensing.

In Section 2, we first combine the mutual information measure adopted from information theory and a statistical description for the quasi-singly and multiply scattered signals detected from a depth resolved coherent light imaging system to derive the scattering limits to the information capacity of such systems; that is, the quantitative behavior of the maximum amount of information as a function of the detected SNR. The term quasi-singly refers here to light that experiences single-backscattering events or few collisions with small scattering angles. Next, in Section 3, we use our approach to find the fundamental dependence of the information capacity on physical properties of the scattering medium and imaging depth, as well as on parameters of the imaging system. OCT is used here as a case study of our theoretical framework, but its application to a variety of coherent and incoherent imaging and communication strategies that conduct information through turbid media is straightforward and discussed in the end of Section 3.

SECTION 2

SCATTERING INFORMATION-THEORETIC LIMITS OF OPTICAL COHERENCE IMAGING

Mutual information is a measure of the amount of information that can be obtained about one event by observing another and is primarily used in communication theory [7]. The mutual information that establishes the information content of an image of an object embedded in a disordered medium is the mutual information between the detected backscattered signal and the signal originating quasi-directly from the object hidden at a particular depth within the sample as depicted in the inset of Fig. 1(a). To compute the mutual information for depth resolved optical coherence imaging through scattering media, we first establish a statistical description for the detected backscattered signal. Consider a light beam of wavelength Formula$\lambda$ illuminating a sample made up of packed scatterers and a large number of randomly distributed scatterers separated by distances Formula$d_{s}$. The incident field travels inside the sample in a random walk fashion and gives rise to a backscattered signal that in general comprises two classes of dominant waves: Waves that are quasi-singly backscattered from the desired sample depth and carry useful information, and multiply scattered waves emerging from a depth other than the desired one, thus distorting the useful information. In the context of low-coherence optical imaging, we note that recent experimental observations have suggested that the distortion of useful information arising from multiply scattered light is fairly severe as multiply scattered waves maintain a rather high degree of spatial coherence with the reference light [8].

Figure 1
Fig. 1. (a) Scattering limits to the information capacity of optical coherence imaging for fully formed speckle (lower curve) and speckle-free (middle curve) scenarios using a single detector. The Shannon limit is also shown (upper curve). The inset illustrates the detected backscattered signal comprising quasi-singly (black) and multiply (blue) scattered waves. (b) Scattering limits when diversity is employed to reduce speckle (two upper curves). Formula$N$ is the number of independent diversity branches.

In coherent light imaging systems such as confocal laser microscopy and OCT, the field amplitude of the backscattered signal detected from a sampling volume located at depth Formula$z$ within the sample can be expressed as Formula TeX Source $$\eqalignno{{\mmb y}(z) =&\, \left\vert{\mmb r}(z)e^{i\varphi(z)} + {\mmb n}(z) + {\mmb n}_{d} \right\vert\cr =&\, \left\vert \sum_{s = 1}^{S}{\mmb r}_{s}(z)e^{i\varphi_{s}(z)} + \sum_{m = 1}^{M}{\mmb a}_{m}(z)e^{i\theta_{m}(z)} + {\mmb n}_{d}\right\vert& \hbox{(1)}}$$ where the unknown parameters that establish depth information about the sample are the mean depth-dependent reflectance Formula$\langle r(z)\rangle= \rho(z)$ and phase Formula$\langle\varphi(z)\rangle = \gamma(z)$ which are treated here as random variables (RVs). The received signal is the sum of quasi-singly backscattered and multiply scattered complex-value optical fields, Formula$r(z)e^{i \varphi(z)}$ and Formula${\mmb n}(z)$, respectively, and the complex-value detection noise Formula$(n_{d})$. The scattered fields correspond to the superposition of multiple optical waves, Formula$\sum_{s}{\mmb r}_{s}e^{i\varphi_{s}}$ and Formula$\sum_{m}{\mmb a}_{m}e^{i\theta_{m}}$, traveling random paths in the scattering sample due to quasi-single and multiple collisions, respectively. We point out that the rather simple, 1-D (axial) stochastic model eq.(1) was shown to be adequate for the physical description of optical coherence imaging through turbid media [8], [9], [10]. As presented later in this work, the combination of model (1) with the mutual information measure is attractive computationally as it results in analytical conditional probability density functions (PDFs) that are necessary for calculating the information limits of coherent light imaging systems through scatterers.

We concentrate now on two limits to the capacity of optical coherence imaging to extract information through scattering media. The first limit, referred to as the speckle-free limit, deals with cases in which quasi-singly backscattered waves with identical phases are detected. For a constant Formula${\mmb r}(z)$ and a given mean depth reflectance Formula${\mmb \rho}(z) = \rho(z)$ of a dense scattering layer, it is readily verified that the received reflectance signal is Formula${\mmb y}(z) = \vert \rho(z) + {\mmb n}(z) + {\mmb n}_{d}\vert$ (Appendix A.1). The second limit, termed the fully formed speckle limit, considers circumstances where quasi-singly backscattered waves arrive at the detector with random, uniformly distributed phases on Formula$[0, 2\pi)$, thus giving rise to a fully developed speckle pattern [11]. For Formula${\mmb \rho}(z) = \rho(z)$ and packed scatterers separated by distances greater than Formula$\lambda$, we obtained the reflectance signal Formula${\mmb y}(z) = \vert\rho(z){\mmb x}(z) + {\mmb n}(z) + {\mmb n}_{d}\vert$ with Formula${\mmb x}(z)$ being a complex circular Gaussian RV with mean zero and variance Formula$(4/\pi)$ (Appendix A.2). Assuming that the number of multiple scattering events is large enough that the central limit theorem is applicable and for Formula$d_{s} \gg \lambda$, the multiply scattered field in the speckle-free and fully formed speckle scenarios can be approximated as a complex circular Gaussian RV with zero mean and variance Formula$2 \sigma_{N}^{2}$ [8], [11]. We note that the terms speckle-free and fully formed speckle are used here for speckle caused by quasi-single scattering and should not be confused with speckle of multiple scattering that exists in both scenarios. Lastly, Formula${\mmb n}_{d}$ is modeled as a complex circular Gaussian RV with mean zero and variance Formula$2 \sigma_{d}^{2}$, and Formula$\rho(z)$, Formula${\mmb x}(z)$, Formula${\mmb n}(z)$ and Formula${\mmb n}_{d}$ are assumed to be statistically independent of each other.

We can now calculate the maximum mutual information Formula$I_{\rm max}({\mmb y}(z); \rho(z))$ to assess the maximum amount of information that is provided by optical coherence imaging through turbid media. Formally, Formula$I_{\rm max}({\mmb y}(z); \rho(z))$ is given by [7] Formula TeX Source $$I_{max}(y; \rho) = \max_{f(\rho) : \langle \rho^{2}\rangle = R}\int \limits_{0}^{\infty}\int\limits_{0}^{\infty}f(\rho)f(y\vert \rho) \times log_{2}\left\{{f(y\vert \rho) \over \int_{0}^{\infty}f(\rho^{\prime})f(y\vert \rho^{\prime})d\rho^{\prime}}\right\} dy\;d\rho\eqno{\hbox{(2)}}$$ where for brevity of notation the depth dependence Formula$z$ is suppressed. In Eq. (2), Formula$f(\rho)$ is the PDF of Formula${\mmb \rho}$ and Formula$f(y\vert\rho)$ is the conditional PDF of Formula${\mmb y}$ given ρFormula$\,= \rho$. For the speckle-free limit, Formula$f(y\vert\rho)$ is the Rice distribution with parameters Formula$v=\rho$ and Formula$\sigma^{2} = \sigma_{N}^{2} + \sigma_{d}^{2}$, whereas for the fully formed speckle limit Formula$f(y\vert\rho)$ is the Rayleigh distribution with parameter Formula$\sigma^{2} = (2/\pi)\rho^{2} + \sigma_{N}^{2} + \sigma_{d}^{2}$. The maximization in Eq. (2) is performed over all possible PDFs of Formula$f(\rho)$ obeying the mean depth reflectivity constraint Formula$\langle\rho^{2}\rangle = R$. The nature of the PDFs that achieve the information limits has been intensively investigated in the context of communications systems and thorough analyses can be found in [12] and [13].

The behavior of Eq. (2) for the two limiting scenarios described above may be computed numerically as a function of the SNR (Appendix B). A graphical illustration of these two limit curves is shown in Fig. 1(a). Note that Formula${\rm SNR} = R/(2\sigma_{N}^{2} + 2\sigma_{d}^{2})$ which accounts for both detection noise and multiple scattering noise. The speckle-free curve sets the upper bound to the maximum amount of extractable information and is asymptotically linear in high SNR (measured in decibels [14]). The fully formed speckle curve sets the lower bound due to the harmful fades that speckle introduces into the information carrying signal, and is asymptotically logarithmic in high SNR (measured in decibels [15]). This notable effect of speckle on the information capacity is clearly observed in Fig. 1(a): Fully developed speckle significantly reduces the information capacity of optical coherence imaging systems; up to a threefold reduction is observed for Formula${\rm SNR} = 15{-}30\ {\rm dB}$. To combat speckle without scarifying spatial or temporal resolution, it is imperative to use diversity imaging; for instance, by combining independent speckled signals that image the same sampling volume from different angles [9]. It can be shown that in the fully formed speckle case, Formula$f(y\vert\rho)$ for diversity imaging using Formula$N$ independent branches is the Nakagami- Formula$N$ distribution (Appendix A.3). Fig. 1(b) presents the scattering limits for diversity imaging in the scenario of fully developed speckle. The figure shows a twofold to threefold increase in the information capacity for 30 independent imaging branches Formula$(N)$ at Formula${\rm SNR} = 5{-}30\ {\rm dB}$. Interestingly, at a given SNR the enhancement of information with Formula$N$ is nonlinear and saturates as Formula$N$ increases. This nonlinear behavior can be understood by noting that Formula$dI_{\rm max}/dN$ at a specific SNR is not a constant; for example: Formula$[I_{\rm max}(N = 30 @ {\rm SNR} = 30\ {\rm dB}) - I_{\rm max}(N = 10 @ {\rm SNR} = 30\ {\rm dB})]/[30{-}10]$ is much smaller than Formula$[I_{\rm max}(N = 10 @ {\rm SNR} = 30\ {\rm dB}) - I_{\rm max}(N = 1 @ {\rm SNR} = 30\ {\rm dB}))/(10{-}1)$. In addition, it was shown that Formula$I_{\rm max}$ is logarithmic in Formula$N$ for Formula$N \gg 1$ [16]. We point out that in cases where phase information can be retrieved in addition to amplitude information, the maximum amount of extractable information considerably increases and follows the Shannon limit Formula${\rm log}_{2}(1 + {\rm SNR})$ shown in Fig. 1(a) [7].

SECTION 3

INFORMATION LIMITS OF OCT THROUGH SCATTERING MEDIA

Central to our approach is the ability to relate the scattering limits to physical characteristics of the scattering medium and imaging depth, as well as to parameters of the imaging system. The key to this ability is Monte Carlo modeling of light propagation in turbid media, which is a powerful tool for solving complex boundary-value scattering problems for which analytic solutions are not available [17], [18], [19]. In this modeling process, photons entering the medium, propagating inside it and collected by the diffraction-limited optics are classified either as quasi-singly backscattered photons or multiply scattered photons. The photons path in the medium is tracked such that the depth within the sample which the photons appear to emerge from can be recorded. This allows the SNR to be estimated as a function of depth and then to be mapped into quanta of information using the curves shown in Fig. 1. The important end result is therefore a depth-resolved profile of the maximum amount of information provided by a particular design of the imaging system through a given scattering medium.

As an illustrative case, we employ the methodology outlined above to find the following new results for OCT. As mentioned in Section 2, eq. (1) accounts only for the extraction of axial information from within the sample—traditionally, a major task of OCT. More sophisticated OCT models that take into account also extraction of transversal information [20] may be useful for evaluating the enhanced information content of OCT. Our Monte Carlo modeling of the OCT signal in scattering media follows that reported, e.g., in [18] and [19] (Appendix C). First, we address a fundamental question regarding the ability of light of shorter coherence length to improve the amount of information imaged by OCT from within a homogeneous scattering medium in the presence of fully developed speckle. Our approach reveals that this ability is limited as shown in Fig. 2. This finding follows by noting that although a shorter coherence length increases the number of sampling volumes along the sample depth, the SNR and consequently the information at each of the reduced sampling volumes decreases. We found that for depths greater than Formula$\sim$0.4 mm (in this example) the information per sampling volume decreases approximately inversely with the increased number of volumes, therefore impeding a significant increase of information even when light of shorter coherence length is introduced. This result highlights the significance of the mutual information measure in assessing the performance of imaging through scattering media as it accounts for the tradeoff between SNR and spatial resolution, and of using diversity in OCT to alleviate speckle.

Figure 2
Fig. 2. Dependence of the amount of maximum extractable information from nondiversity OCT on depth and coherence length in the presence of fully formed speckle. In the Monte Carlo simulations, the scattering and anisotropy coefficients of the medium were Formula$\mu_{s} = 5\ {\rm mm}^{-1}$ and Formula$g = 0.9$, respectively. Absorption was neglected. Formula$n_{\rm medium} = 1.4$ and Formula$n_{\rm ambient} = 1$. The collection angle and detection radius of the OCT system were 7 ° and 15 Formula$\mu{\rm m}$, respectively. The observation length was 40 Formula$\mu{\rm m}$.

To demonstrate the applicability of our method also to heterogeneous scattering media, we evaluated the maximum amount of information conveyed by an OCT signal about the epidermal-dermal interface of skin. Fig. 3(a) shows that for the speckle-free case, more than one bit of information can be gained for epidermis thicknesses Formula$(t_{\rm epidermis})$ lower than 0.2 mm compared to the maximum information achieved at Formula$t_{\rm epidermis} = 0.5\ {\rm mm}$. In contrast, when speckle is fully formed, the maximum information content roughly plateaus over Formula$0.1 \leq t_{\rm epidermis} \leq 0.5\ {\rm mm}$ even though there is a sixfold difference in SNR across this thickness range. As illustrated in Fig. 3(b) diversity imaging effectively combats speckle in this scenario; an increase of nearly twofold to threefold in the extractable amount of information is obtained by employing 10 and 30 independent diversity branches. Diversity imaging therefore offers the potential for the superior identification of the epidermis layer, which is an important task in monitoring skin health, aging, and photodamage [21].

Figure 3
Fig. 3. (a) Maximum amount of information content conveyed about the epidermal-dermal reflectance as a function of the epidermis thickness using nondiversity OCT and (b) diversity OCT. For diversity OCT only the fully formed speckle is considered. In the Monte Carlo simulations, skin was modeled as a dual-layered medium with the following properties: variable finite epidermal layer Formula$(\mu_{s} = 2\ {\rm mm}^{-1}, g = 0.9, n = 1.36)$ and semi-infinite dermal layer Formula$(\mu_{s} = 8\ {\rm mm}^{-1}, g = 0.9, n = 1.43)$. Coherence length, collection angle, and detection radius of the OCT system were 7°, 15 Formula$\mu{\rm m}$, and 25 Formula$\mu{\rm m}$, respectively.

In closing, we point out that the extension of our information-theoretic framework to additional coherent and incoherent imaging approaches simply requires the use of an appropriate Monte Carlo modeling of the imaging signal in disordered media. For example, the Monte Carlo model described by Schmitt et al. [22] is suitable for investigating the information capacity of confocal laser microscopy through scattering media, whereas the Monte Carlo model developed by Blanca and Saloma [23] is appropriate for two photon fluorescence imaging. We note that in the latter incoherent imaging scenario, the classical Shannon limit for additive white Gaussian real-value channels should be used [6]. We anticipate that the methods developed in this work will spawn practical realizations of a superior, information-theoretic-based design of optical information systems to approach the limits predicted here. In particular, spatial coding of the field illuminating the scattering sample may be one such strategy.

APPENDIX A

STATISTICAL MODELS FOR THE BACKSCATTERED REFLECTANCE SIGNAL

Speckle-Free Model

Presume that quasi-singly scattered waves emerge from the sampling volume of a scattering layer of mean reflectance Formula$\rho$ with identical phases Formula${\mmb \varphi}_{s} = \gamma$ such that speckle is completely suppressed. The detected quasi-singly scattered electric field Formula${\mmb r}e^{i \varphi} = \sum_{s = 1}^{S}{\mmb r}_{s}e^{i\varphi_{s}}$ equals then to Formula$\rho e^{i\gamma}$ where Formula${\mmb r} = \sum_{s = 1}^{S}{\mmb r}_{s}$ and Formula$\varphi$ are degenerate RVs which take the values Formula$\rho$ and Formula$\gamma$, respectively. Here, Formula$\rho$ and Formula$\gamma$, which constitute useful depth information about the sample, are unknown parameters and are modeled as the RVs Formula$\rho$ and Formula$\gamma$.

Under the assumptions that the loss of spatial coherence and depolarization effects due to scattering of light in turbid media are negligible and that scatterers are randomly distributed in the medium and are separated by distances much larger than a wavelength, the multiply scattered field detected from the sampling volume is modeled as Formula${\mmb n} = \sum_{m = 1}^{M}{\mmb a}_{m}e^{i\theta_{m}}$ where Formula$M$ is the number of multiply scattered waves contributing to the signal, Formula${\mmb a}_{m}$ are independent identically distributed (iid) RVs with finite mean and variance and Formula$\theta_{m}$ are iid RVs uniformly distributed on Formula$[0, 2\pi)$ [8]. Formula${\mmb a}_{m}$ and Formula$\theta_{m}$ are considered statistically independent of each other. Although Formula$M$ is finite, we assumed it is sufficiently large such that the central limit theorem is applicable and the multiply scattered field is approximately a complex circular Gaussian RV with mean zero and variance Formula$2\sigma_{N}^{2}$ [11]. Using the derived statistical properties of the quasi-singly backscattered wave and the multiply scattered field, we obtained the backscattered reflectance signal for a given mean reflectance Formula${\mmb \rho} = \rho$ in the speckle-free scenario; explicitly, Formula${\mmb y} = \vert\rho + {\mmb n} + {\mmb n}_{d}\vert$ with Formula${\mmb n}_{d}$ being the detection noise which is modeled as a complex circular Gaussian RV with mean zero and variance Formula$2\sigma_{d}^{2}$. Also, Formula$\rho$, Formula${\mmb n}$ and Formula${\mmb n}_{d}$ are assumed to be statistically independent of each other. It is readily verified now that in this case Formula$f(y\vert\rho)$ is the Rice distribution with parameters Formula$v = \rho$ and Formula$\sigma^{2} = \sigma_{N}^{2} + \sigma_{d}^{2}$ [24].

We lastly note that superior information about the tissue is gained when the mean phase Formula$\gamma$ is retrieved from the backscattered field signal Formula${\mmb y} = \rho e^{i\gamma} + {\mmb n} + {\mmb n}_{d}$ in addition to the mean reflectance Formula$\rho$. It is straightforward to show that Formula$f(y\vert\rho,\gamma)$ follows then the complex Gaussian distribution with mean Formula$\rho e^{i\gamma}$ and variance Formula$2\sigma_{N}^{2}$.

Fully Formed Speckle Model

Assume that quasi-singly scattered waves emerge from the sampling volume of a scattering layer of mean reflectance Formula$\rho$ with independent uniformly distributed phases Formula$\varphi_{s}$ on Formula$[0, 2\pi)$ such that speckle is fully formed. The detected quasi-singly scattered field Formula${\mmb r}e^{i\varphi}$ is modeled then as Formula$\sum_{s = 1}^{S}{\mmb r}_{s}e^{i\varphi_{s}}$ where Formula$S$ is the number of quasi-singly scattered waves contributing to the signal and Formula${\mmb r}_{s}$ are iid RVs, statistically independent of Formula$\varphi_{s}$, with finite mean and variance. Presuming that Formula$S$ is sufficiently large, we obtained through the application of the central limit theorem that the quasi-singly scattered field Formula${\mmb r}e^{i\varphi}$ has a complex Gaussian distribution with mean zero and variance Formula$(4/\pi)\rho^{2}$. We also note that Formula${\rm E}({\mmb r}) = \rho$.

Employing similar arguments to those yielding the model of multiply scattered light in the speckle-free case, we determined that the multiply scattered field in the fully formed speckle case Formula$({\mmb n})$ is also a complex circular Gaussian RV with mean zero and variance Formula$2\sigma_{N}^{2}$.

Recalling now that the unknown mean reflectance Formula$\rho$ is treated as the RV Formula${\mmb \rho}$, the backscattered reflectance signal for a given mean reflectance Formula${\mmb \rho} = \rho$ in the fully formed speckle case follows the model Formula${\mmb y} = \vert\rho {\mmb x} + {\mmb n} + {\mmb n}_{d}\vert$ where Formula${\mmb x}$ is a complex circular Gaussian RV with mean zero and variance Formula$4/\pi$ and Formula${\mmb n}_{d}$ is the detection noise modeled as a complex circular Gaussian RV with mean zero and variance Formula$2\sigma_{d}^{2}$. Formula$\rho$, Formula${\mmb x}$, Formula${\mmb n}$ and Formula${\mmb n}_{d}$ are assumed to be statistically independent of each other. Now, it is readily verified that Formula$f(y\vert\rho)$ is the Rayleigh distribution with parameter Formula$\sigma^{2} = (2/\pi)\rho^{2} + \sigma_{N}^{2} +\sigma_{d}^{2}$ [24].

We finally note that in the fully formed speckle model, phase information Formula$\gamma$ cannot be retrieved from the backscattered field signal Formula${\mmb y} = \rho \cdot e^{i\gamma}{\mmb x} + {\mmb n} + {\mmb n}_{d}$ due to the multiplicative nature of speckle which randomly maps Formula$\gamma$ with uniform probability onto Formula$[0, 2\pi)$.

Diversity Imaging Model

In diversity imaging schemes, the random nature of light propagation in turbid media is exploited to reduce the severity of speckle by proper combining multiple backscattered reflectance signals emerging from the same sampling volume. Under the fully formed speckle assumption, we expressed the resultant diversity signal for a given mean reflectance Formula${\mmb \rho} = \rho$ as Formula${\mmb y} = C\sqrt{\sum_{l = 1}^{N}{\mmb y}_{l}^{2}} = C\sqrt{\sum_{l = 1}^{N} \vert \rho {\mmb x}_{l} + {\mmb n}_{l} + {\mmb n}_{d_{l}}\vert^{2}}$ where Formula$N$ is the number of diversity branches, Formula${\mmb y}_{l}$ is the backscattered reflectance signal on the Formula$l$th diversity branch and Formula$C$ is a constant which guarantees that Formula${\rm E}({\mmb y}) = \rho$ for Formula${\mmb n}_{l} = 0$ and Formula${\mmb n}_{d} = 0$. Assuming now that the backscattered reflectance signals on the different diversity branches are statistically independent of each other, we obtained that Formula$f(y\vert\rho)$ is the Nakagami- Formula$N$ distribution with parameter Formula$\sigma^{2} = C^{2}(2/\pi)\rho^{2} + C^{2}\sigma_{N}^{2} + C^{2}\sigma_{d}^{2}$ [24].

APPENDIX B

NUMERICAL EVALUATION OF Formula$I_{\rm max}({\mmb y};\rho)$

We computed the maximum mutual information given in Eq. (2) by carrying out a numerical constrained optimization with respect to the probabilities and mass points' locations of Formula$f(\rho)$. Specifically, we wrote Eq. (2) in discrete form by substituting the discrete probability function Formula$f(\rho) = \sum_{k = 1}^{K}p_{k}\delta(\rho - \rho_{k})$, replacing the analytic Formula$f(y\vert\rho)$ with its sampled version Formula$f(y_{j}\vert\rho_{k})$ and approximating the integrals using either trapezoid or Simpson quadrature sums [25]. Then, for each SNR value, we maximized the mutual information between Formula${\mmb y}$ and Formula$\rho$ over the probabilities Formula$\{p_{k}\}$ and their locations Formula$\{\rho_{k}\}$ subject to the probability constraints Formula$p_{k} \ >\ 0$ (for all Formula$k$) and Formula$\sum_{k = 1}^{K}p_{k} = 1$, and the mean depth reflectivity constraint Formula$\sum_{k = 1}^{K}p_{k}\rho_{k}^{2} = R$. The maximization procedure was initialized to different distributions (positive normal, Rayleigh and Nakagami- Formula$N$ distributions) and was started with previous estimations of Formula$f(\rho)$ as well. We finally point out that for each SNR value the number of mass points Formula$K$ was iteratively increased until the variation of the mutual information became insignificant (< 4%).

APPENDIX C

MONTE CARLO MODELING OF LIGHT PROPAGATION IN TURBID MEDIA AND ESTIMATION OF THE MAXIMUM INFORMATION CONTENT PROFILES IN OCT

The propagation of light in turbid media was modeled as follows: photon packets were injected orthogonally onto the tissue at the origin. Once launched, the photons were propagated inside the sample where they potentially experienced scattering, absorption and inner reflections according to the physical laws of light propagation in turbid media [17]. In brief, these laws provide a statistical description for scattering, absorption and reflection of light in the medium. Scattering is characterized by the mean free path and the scattering angles. The former is the average distance photons travel in the tissue between scattering events and equals to the inverse of the tissue scattering coefficient Formula$\mu_{s}$, whereas the later are defined by the uniform Formula$[0, 2 \pi)$ phase function for the azimuthal angle and the Henyey–Greenstein phase function, characterized by the anisotropy factor Formula$g$, for the polar angle. Absorption was modeled by the absorption coefficient Formula$\mu_{a}$, and inner reflections are characterized by the refractive index mismatches between the different layers inside the medium. For simplicity, the specular reflection from the ambient-tissue interface was neglected. The Monte Carlo model was validated in our previous work [26] where we compared the radial diffuse reflectance PDF of a semi-infinite random medium illuminated by a pencil beam entering the sample at the origin to the theoretical PDF.

Once photons exited the top surface of the sample and resided within the detection radius Formula$r_{d}$ and collection angle Formula$\alpha$, they were classified as quasi-singly backscattered photons and multiply scattered photons using the requirement for low-coherence imaging. Explicitly, in homogeneous scattering media photons that fulfilled the condition Formula$\vert L_{\rm photon} - 2\cdot n_{\rm medium}\cdot z_{\rm max}\vert \leq l_{c}$ were classified as quasi-singly backscattered photons emerging from depth Formula$z_{\rm max}$ within the sample; otherwise the photons were classified as multiply scattered photons that appear to emerge from depth Formula$L_{\rm photon}/2n_{\rm medium}$. Here, Formula$L_{\rm photon}$ is the optical path length traveled by the photons inside the sample, Formula$n_{\rm medium}$ is the refractive index of the scattering medium, Formula$z_{\rm max}$ denotes the maximum depth reached by the photons over the course of their propagation and Formula$l_{c}$ represents the coherence length of the optical source employed in the imaging system. We point out that the condition for low-coherence imaging in inhomogeneous tissue is slightly modified with respect to that derived in the homogeneous scattering medium case due to the different refractive indices of the medium layers.

Having estimated the depth profiles of the quasi-singly backscattered photons and those of the multiply scattered photons, the depth-dependent SNR was evaluated and then the corresponding value of the maximum amount of depth reflectance information was computed using the speckle-free and fully formed speckle curves shown in Fig. 1(a). We note that typically Formula$\sigma_{N}^{2} \gg \sigma_{d}^{2}$ due to the high detection sensitivity of practical imaging systems. Therefore, the SNR is effectively the ratio of quasi-singly to multiply scattered light.

Fig. 2 was calculated by partitioning the scattering medium into five regions of SNR values (in dB units), namely Formula${\rm SNR}\ <\ 0$, Formula$0\ <\ {\rm SNR}\ <\ 10$, Formula$10\ <\ {\rm SNR}\ <\ 20$, Formula$20\ <\ {\rm SNR}\ <\ 30$ and Formula${\rm SNR}\ >\ 30$, which were then mapped into areas of maximum amount of reflectance information using the fully formed speckle curves displayed in Fig. 1(a). Fig. 3(a) and (b) were computed by first simulating the detected SNR at the epidermal-dermal junction of a skin model for numerous epidermis thicknesses that were subsequently mapped into the corresponding maximum information values using the maximum information curves displayed in Fig. 1(a) and (b), respectively.

All simulations were executed on Intel Pentium CPUs (1.66–3 GHz, 1–2 GB of RAM). The simulation of each depth profile used Formula$20^{7}{-}20^{80}$ photon packets and lasted between 6–12 hours depending on the characteristics of the turbid medium, the number of incident photons and the specifications of the computer on which the specific simulation was running.

ACKNOWLEDGMENT

Thanks to M. Colice, A. Gordon, A. Desjardins, B. Karamata, R. Motaghianand, and B. Park for helpful discussions.

Footnotes

First published Online. The work of A. Bilenca was supported by the European Commission under the Marie Curie Fellowship. Corresponding author: A. Bilenca (e-mail: bilenca@lehigh.edu).

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A. Bilenca

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T. Lasser

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B. E. Bouma

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R. A. Leitgeb

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G. J. Tearney

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