Speckle spectrum autocorrelation imaging for complex strongly scattering scenarios

Si He; Xia Wang; Linhao Li

doi:10.3788/AI.2025.10006

1. Introduction

The acquisition of clear images in fields such as biomedical imaging, astronomical observation, remote sensing, and underwater detection is often hindered by the presence of scattering media, which introduces significant challenges due to the generation of scattered light. A key approach to overcoming these challenges involves utilizing unscattered ballistic photons. However, in the scenario where an object is hidden behind a strongly scattering medium, all photons emitted or reflected by the object undergo multiple random scattering events, resulting in the loss of their ballistic characteristics^[1–3]. Consequently, conventional optical imaging methods are limited to capturing speckle images and are unable to recover object features. In recent years, innovative imaging approaches such as speckle correlation imaging (SCI) have been proposed that capitalize on strongly scattered light rather than considering it as an obstacle^[1,2]. SCI exploits the optical memory effect (OME) of the scattering medium and considers the medium as a linear spatially invariant system within a specific incident angle range^[4–6]. To reconstruct the image of a hidden object, SCI initially reconstructs the Fourier amplitude spectrum of the object’s image (or object amplitude spectrum) from the speckle image autocorrelation and then recovers the Fourier phase spectrum of the object’s image (or object phase spectrum) using an iterative phase-retrieval algorithm (IPR)^[7]. SCI does not require the measurement of the media’s scattering effects, enabling noninvasive imaging and, thus, demonstrating significant application potentials. However, during its early development stages, SCI lacked the capability for stable reconstruction and was applicable only in scenarios involving no background light interference, with static objects smaller than the OME range, illuminated by narrowband light.

In recent years, numerous studies have focused on broadening the application of SCI to more complex scattering scenarios. To implement SCI under broadband illumination, spectral coding and compressed-sensing techniques have been employed to separate high-contrast speckle images within narrow spectral bands from low-contrast, multiplexed speckle signals^[8]. Alternatively, high-contrast object amplitude spectra can be extracted directly from broadband speckle images^[9]. To reconstruct multiple objects whose interobject distances exceed the OME range, the autocorrelation of each object can be extracted from three distinct sources: the cross-correlation of speckle images captured when objects are at different positions, speckle images captured at different imaging distances, and speckle images obtained under structured illumination^[10–12]. To enhance the reconstruction quality of SCI under background light interference, an optimization algorithm was reported that combined stripe background removal and low-rank and sparse decomposition to extract high-precision object autocorrelation^[13]. High-performance optimization algorithms were also developed to improve the object phase spectrum^[14]. In scenarios involving dynamic scattering media or moving objects, compressed sensing and IPRs have been employed on temporally coded speckle images for object reconstruction^[15]. The moving object can also be tracked using the optical transfer function (OTF) of the imaging system, which was estimated through blind deconvolution based on the reconstruction results of SCI^[16,17]. Moreover, researchers have achieved initial success in applying SCI to biomedical and super-resolution imaging^[18–22].

Despite the enhanced abilities of the aforementioned SCI-based techniques to address more complex scattering scenarios compared to the original SCI, they are still inherently limited by the deficiencies in phase spectrum reconstruction associated with the IPR: (i) the dependent reconstruction of phase spectrum relies on the object amplitude spectrum and is susceptible to scenario variations and noise interference; (ii) the iteration is prone to getting stuck in a locally optimal solution, necessitating numerous initial guesses and subsequently selecting an acceptable reconstruction result from numerous unstable solutions. Although bispectral analysis can replace the IPR for deterministic and higher-quality reconstructions, its flexible application in variable scattering scenarios has not yet been realized^[23,24]. Reconstruction models combining deep learning with physical priors of speckle autocorrelations have been reported to produce more precise object information than conventional SCI^[25–29]. However, the reconstruction quality of deep-learning-based methods degrades significantly as the scattering scenario varies. Furthermore, their application in noninvasive imaging is hindered by the need for invasive techniques in data acquisition, which is essential for network training. Consequently, it is crucial to devise a noninvasive imaging approach that incorporates a novel formalism for phase spectrum reconstruction to achieve high-stability and high-fidelity object reconstruction in variable and background light-interfered strongly scattering scenes.

In this study, we proposed a speckle spectrum autocorrelation imaging (SSAI) approach for reconstructing objects hidden behind strongly scattering media. SSAI features an independent and deterministic object phase spectrum reconstruction process that effectively eliminates interference from background light, dynamic media, and variable objects. Specifically, our study of the speckle spectrum autocorrelation demonstrated that it exhibits unique properties distinct from the speckle image autocorrelation used in SCI: it not only preserves information related to the object phase spectrum but also facilitates high-quality phase spectrum reconstruction. Initially, we demonstrated that the autocorrelation of the OTF of the speckle imaging system can be approximated as a real value. When the centroids of the speckle images coincide, the speckle spectrum autocorrelation and the object spectrum autocorrelation exhibit an approximately equivalent unit amplitude phasor. Subsequently, we investigated how perturbations originating from medium dynamics and object motion affect the established equivalence. To address these perturbations, we proposed a computational method designed to align the centroids and compute the spectrum autocorrelation of speckle images. Moreover, we discovered that the equivalence holds even when the size of the object’s ideal image is scaling. Building on these foundations, a recursive reconstruction method for the object phase spectrum and the object reconstruction strategy of SSAI were developed. The performance of SSAI was further validated with experiments including complex scattering scenarios encompassing dynamic media, moving objects, background light interference, and size-scaling objects.

2. Principles

The concept of SSAI is described in Fig. 1. Figure 1(a) shows the scenario of imaging with strongly scattered light. The light emitted by an object within the OME range enters the medium, undergoes strong scattering, travels through free space, reaches the surface of the detector, and is ultimately captured as a speckle image. The dynamic strongly scattering medium is considered a nonideal imaging element and, together with the detector, forms a speckle imaging system. Similar to the SCI approach, the detector frame integration time is restricted to ensure that the influence of scenario variations on the speckle image contrast can be disregarded. When the object light is spatially incoherent and in the narrowband, the relationship between the spectrum of the ideal image of the object (or object spectrum) and that of the speckle image (speckle spectrum) is expressed as $S (f) = H (f) G (f),$ (1)where $f$ is the two-dimensional spatial-frequency vector and $f = (u, v)$ ; $S$ is the speckle spectrum; $H$ is the OTF of the imaging system; and $G$ is the object spectrum. With $ϕ$ as the phase, $S$ and $G$ can be expressed as $S (f) = | S (f) | \exp [j ϕ_{S} (f)], G (f) = | G (f) | \exp [j ϕ_{G} (f)],$ (2)where $| S (f) |$ and $ϕ_{S} (f)$ are the amplitude and phase spectra of the speckle image, respectively, and $| G (f) |$ and $ϕ_{G} (f)$ are the amplitude and phase spectra of the object, respectively. The primary goal of SSAI is to reconstruct the object phase spectrum directly from the speckle spectrum without depending on the object amplitude spectrum.

Figure 1.Illustration of the proposed SSAI approach. (a) The scenario of imaging with strongly scattered light. The strongly scattering medium and the detector form a speckle imaging system. (b) Definition of the wavefront phase error in the speckle imaging system. (c) Implementation flowchart of the SSAI approach.

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Spectrum autocorrelation analysis, an approach based on random process theory, is employed in radio astronomy to reconstruct the Fourier phase of an object signal from the phase differences obtained through interferometric measurements^[30–33]. In this study, we exploited the statistical autocorrelation of the speckle image spectrum, or, simply, the speckle spectrum autocorrelation, to deterministically and independently reconstruct the object phase spectrum. If the centroids of the speckle images captured at different moments align, resulting in the ideal image of the object remaining position-invariant, then the speckle spectrum autocorrelation can be expressed as $E [S (f_{1}) S^{*} (f_{2})] = E [H (f_{1}) H^{*} (f_{2})] \cdot E [G (f_{1}) G^{*} (f_{2})] = E [H (f_{1}) H^{*} (f_{2})] G (f_{1}) G^{*} (f_{2}),$ (3)where $E (\dots)$ is the ensemble or statistical average of different speckle images and $E [H (f_{1}) H^{*} (f_{2})]$ is the statistical autocorrelation of the OTF, or, simply, OTF autocorrelation. The object spectrum autocorrelation $G (f_{1}) G^{*} (f_{2})$ is coupled with the OTF autocorrelation in the speckle spectrum autocorrelation $E [S (f_{1}) S^{*} (f_{2})]$ . Unlike the speckle image autocorrelation, which is a parameter in the spatial domain used in SCI, the speckle spectrum autocorrelation is a parameter in the Fourier frequency domain and retains information related to the object phase spectrum. An analysis of the form and values of the OTF autocorrelation is performed, and a comprehensive object reconstruction strategy within SSAI is presented in subsequent sections.

2.1. OTF autocorrelation and unit amplitude phasor equivalence

We considered that the light emitted by a point source on the object plane remains as an ideal spherical wave when it reaches the exit pupil of the speckle imaging system. Wavefront perturbation factors from the scattering medium and aberrations are regarded as wavefront deformation plates within the physically limiting aperture of the system. For the position vector $ξ$ of the exit pupil, assuming that the random wavefront phase error caused by the scattering medium is $η (ξ)$ , the wavefront phase error caused by other perturbing factors is $γ (ξ)$ , and the total wavefront amplitude error is $A (ξ)$ . If the pupil function of the system without wavefront errors is $p (ξ)$ , then the complex amplitude transmittance of the imaginary wavefront deformation plate is given by $P (ξ) = p (ξ) A (ξ) \exp {j [η (ξ) + γ (ξ)]},$ (4)where $P (ξ)$ is the generalized pupil function of the speckle imaging system. The definition of the wavefront phase error is illustrated in Fig. 1(b). To separate the wavefront phase error owing to the scattering medium from the other wavefront errors in Eq. (4), let $P (ξ) = p (ξ) A (ξ) \exp [j γ (ξ)] .$ (5)

The OTF of the system is the normalized autocorrelation of its scaled generalized pupil function^[34]. By omitting the normalization operation, which simplifies the expression without affecting the analysis of the value characteristics, the OTF can be expressed as $H (f) = \int_{- \infty}^{\infty} P (ξ) \exp [j η (ξ)] \cdot P^{*} (ξ - λ z f) \exp [- j η (ξ - λ z f)] d ξ,$ (6)where $λ$ is the light wavelength and $z$ is the distance from the exit pupil to the detection plane.

Assuming that the wavefront phase error owing to the scattering medium is uncorrelated with other wavefront errors, the OTF autocorrelation can be expressed as $E [H (f_{1}) H^{*} (f_{2})] = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E [P (ξ_{α}) P^{*} (ξ_{α} - λ z f_{1}) P^{*} (ξ_{β}) P (ξ_{β} - λ z f_{2})] \times E {\exp {j [η (ξ_{α}) - η (ξ_{α} - λ z f_{1}) - η (ξ_{β}) + η (ξ_{β} - λ z f_{2})]}} d ξ_{α} d ξ_{β},$ (7)where vectors $ξ_{α}$ and $ξ_{β}$ represent the arbitrary positions on the exit pupil. To facilitate the analysis of the value of OTF autocorrelation, we assumed $η (ξ)$ to be a stationary Gaussian random process and derived the following form of OTF autocorrelation^[35,36]: $E [H (f_{1}) H^{*} (f_{2})] = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E [P (ξ_{α}) P^{*} (ξ_{α} - λ z f_{1}) P^{*} (ξ_{β}) P (ξ_{β} - λ z f_{2})] \times \exp [- 2 σ^{2} + C (λ z f_{1}) + C (ξ_{α} - ξ_{β}) - C (ξ_{α} - ξ_{β} + λ z f_{2}) - C (ξ_{α} - ξ_{β} - λ z f_{1}) + C (ξ_{α} - ξ_{β} - λ z f_{1} + λ z f_{2}) + C (λ z f_{2})] d ξ_{α} d ξ_{β},$ (8)where $C$ is the covariance of $η (ξ_{i})$ and $η (ξ_{k})$ at any two positions $ξ_{i}$ and $ξ_{k}$ on the exit pupil and is expressed as $C (ξ_{i}, ξ_{k}) = E {[η (ξ_{i}) - \bar{η}] [η (ξ_{k}) - \bar{η}]} = C (ξ_{i} - ξ_{k}) = σ^{2} ρ (ξ_{i}, ξ_{k}) = σ^{2} ρ (ξ_{i} - ξ_{k}),$ (9)where $| ρ | \leq 1$ , $ρ$ represents the correlation coefficient of $η (ξ_{i})$ and $η (ξ_{k})$ , and $σ$ is the standard deviation of $η (ξ)$ . The derivation of Eq. (8) is presented in Appendix A.

In Eq. (8), we observed that the terms after the time’s sign “ $\times$ ,” which are associated with the wavefront phase error $η$ owing to the medium, are already real-valued. Conversely, the ensemble average in front of the times sign, which is linked to other wavefront errors $P$ , remains complex-valued. Hence, if the phase error in $P$ , such as the error originating from the aberration of the imaging elements, can be neglected, then the OTF autocorrelation can be approximated as real.

When the phase error in $P$ cannot be neglected, we deduced from Laplace’s asymptotic method that the OTF autocorrelation can be approximated as $E [H (f_{1}) H^{*} (f_{2})] \approx \exp {2 σ^{2} [ρ (λ z \frac{f_{1} - f_{2}}{2}) - 1]} \cdot E [{| H (\frac{f_{1} + f_{2}}{2}) |}^{2}] .$ (10)

Please refer to Appendix B for the detailed derivation. Eq. (10) demonstrates that the OTF autocorrelation approximates to a real value; hence, its principal argument (or phase) can be approximated to zero. According to Eq. (3), the phase of the speckle spectrum autocorrelation is the sum of the phases of the OTF and the object spectrum autocorrelation. Therefore, the phase of the speckle spectrum autocorrelation is equal to that of the object spectrum autocorrelation, providing a theoretical foundation for reconstructing the object phase spectrum through speckle spectrum autocorrelation (as shown in Fig. 2). To facilitate subsequent reconstruction computations, we further found that the unit amplitude phasor of the speckle spectrum autocorrelation is equal to the unit amplitude phasor of the object spectrum autocorrelation, that is, $\frac{E [S (f_{1}) S^{*} (f_{2})]}{| E [S (f_{1}) S^{*} (f_{2})] |} = \frac{G (f_{1}) G^{*} (f_{2})}{| G (f_{1}) G^{*} (f_{2}) |} .$ (11)

Figure 2.Theoretical foundation for reconstructing the object phase spectrum in SSAI. The phase of the speckle spectrum autocorrelation is approximated to be equal to that of the object spectrum autocorrelation in the speckle imaging system.

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2.2. Spectrum autocorrelation computation of speckle images after centroid alignment

The previous analysis was based on the assumption that the centroids of the speckle images coincide. However, dynamic variations in the scattering medium or random movement of the object on the object plane cause random translation between the centroids of speckle images^[36]. For instance, in the experimental setup of this study, the scattering medium, represented by the ground glass, was not perfectly perpendicular to the optical axis because of mounting errors, resulting in noncoincident centroids in the speckle images captured during the rotation of the ground glass. Based on the analysis presented in Appendix C, if the centroids of the speckle images do not coincide, the autocorrelation of the speckle spectrum rapidly decreases to 1% of the autocorrelation when the centroids coincide. Thus, utilizing the unit amplitude phasor of the speckle spectrum autocorrelation as the unit amplitude phasor of the object spectrum autocorrelation introduces significant noise interference, which hinders object reconstruction. Calculating the speckle spectrum autocorrelation after centroid alignment is expected to significantly enhance reconstruction accuracy.

A formula for calculating the spectrum autocorrelation of speckle images with centroids uniformly translated to the coordinate origin was derived. The Fourier spectrum of the original speckle image $s (x)$ is expressed as $S (f) = F [s (x)] = \int s (x) \exp (- j 2 π x f) d x,$ (12)where $x$ represents a two-dimensional spatial position vector in the plane of the speckle image. The location of the centroid of $s (x)$ is expressed as $x_{c} = \frac{\int x s (x) d x}{\int s (x) d x} .$ (13)

The Fourier spectrum of the new speckle image obtained by translating $s (x)$ according to the centroid to the coordinate origin is given by $S_{c} (f) = F [s (x + x_{c})] = S (f) \exp (j 2 π x_{c} f) .$ (14)

Therefore, the spectrum autocorrelation of the speckle image after centroid translation to the coordinate origin is $E [S_{c} (f_{1}) S_{c}^{*} (f_{2})] = E {S (f_{1}) S^{*} (f_{2}) \exp [- j 2 π x_{c} (f_{2} - f_{1})]} .$ (15)

By representing $x_{c}$ with the parameters of the frequency domain, Eq. (15) can be further expressed as $E [S_{c} (f_{1}) S_{c}^{*} (f_{2})] = E {S (f_{1}) S^{*} (f_{2}) \exp [{j \frac{d ϕ_{S} (f)}{d f} |}_{f = 0} (f_{2} - f_{1})]} .$ (16)

Please refer to Appendix D for a detailed derivation of Eq. (16). Equation (16) can be approximated as $E [S_{c} (f_{1}) S_{c}^{*} (f_{2})] \approx E {S (f_{1}) S^{*} (f_{2}) \exp [j ϕ_{S} (f_{2} - f_{1})]} .$ (17)

Eq. (17) provides a simplified formulation for computing the speckle spectrum autocorrelation, which inherently achieves centroid alignment of the speckle images.

2.3. Unit amplitude phasor equivalence in a scenario with an object of scaling ideal images

In addition to the dynamic scattering media and moving object, we further investigated whether the equivalence relationship of unit amplitude phasors in Eq. (11) holds in another variable scattering scenario. We considered the case in which different speckle images corresponded to differently scaled ideal images of the hidden object, which can simulate scattering scenarios wherein the object is moving along the optical axis of the system or scaling in the object plane. For convenience, the following is analyzed in one-dimensional space: Assuming that the ideal image of the object before scaling is $g (x)$ and its domain of definition is $x \in [0, T)$ , as well as the ideal image of the object after scaling is $g (a x)$ and its domain of definition is $x \in [0, T / a) (a > 0)$ , according to the definition of the continuous Fourier transform of a finite duration signal^[37], the Fourier transform of $g (x)$ is expressed as $G (f) = \int_{0}^{T} g (x) \exp (- j 2 π f x) d x, f \in {k / T, k \in Z},$ (18)where $Z = {\dots, - 2, - 1, 0, 1, 2, \dots}$ . According to the stretch theorem for the continuous Fourier transform of a finite duration signal^[37], if the Fourier transform of $g = {g (x); x \in [0, T)}$ is ${G (k / T); k \in Z}$ , then the Fourier transform of the scaled signal $g_{a} = {g (a x); x \in [0, T / a)}$ is $G_{a} (\frac{k}{T / a}) = {\frac{1}{a} G (\frac{1}{a} \frac{k}{T / a}); k \in Z} = {\frac{1}{a} G (\frac{k}{T}); k \in Z} .$ (19)

We observed that, if the spectrum value of the ideal image of the object before scaling at frequency $k / T$ is $G (k / T)$ , the spectrum value of the ideal image of the object after scaling at frequency $\frac{k}{T / a}$ is $\frac{1}{a} G (\frac{k}{T})$ . In other words, the spectral values of the ideal image of the object after scaling are proportional to the spectral values before scaling, with a scaling factor of $a$ . Therefore, before and after scaling, the range of the object phase spectrum remains unchanged, and only the domain of the object spectrum undergoes modification. However, when object reconstruction is performed on a computer platform, computations in the frequency domain utilize a fast Fourier transform with a unified domain. Therefore, even if speckle images correspond to differently scaled ideal images of the object, the reconstruction process scales the domain of the object spectrum to maintain consistency. Furthermore, the domain of the object phase spectrum remains consistent. In summary, in the computational reconstruction process, the scaling of the ideal image of the object does not alter the object phase spectrum. This implies that, even if speckle images correspond to differently scaled ideal images of the object, the unit amplitude phasor of the speckle spectrum autocorrelation can still be regarded as an approximation of the unit amplitude phasor of the object spectrum autocorrelation.

2.4. Recursive reconstruction of the object phase spectrum and the object reconstruction strategy

Based on the previous analysis and demonstrations, we observed that, when handling a variable scattering scenario with a dynamic scattering medium, an object moving on the object plane, or an object with a scaling ideal image, the object spectrum autocorrelation shares the equivalent unit amplitude phasor as the spectrum autocorrelation of speckle images after centroid alignment, that is, $\frac{G (f_{1}) G^{*} (f_{2})}{| G (f_{1}) G^{*} (f_{2}) |} = \frac{E [S_{c} (f_{1}) S_{c}^{*} (f_{2})]}{| E [S_{c} (f_{1}) S_{c}^{*} (f_{2})] |} .$ (20)

We further deduced that $\frac{G (f_{1})}{| G (f_{1}) |} = \frac{G (f_{2})}{| G (f_{2}) |} \frac{E [S_{c} (f_{1}) S_{c}^{*} (f_{2})]}{| E [S_{c} (f_{1}) S_{c}^{*} (f_{2})] |} .$ (21)

This equation indicates that the unit amplitude phasor of the unknown object spectrum at the frequency vector $f_{1}$ , denoted as $\frac{G (f_{1})}{| G (f_{1}) |}$ , can be reconstructed by utilizing the known unit amplitude phasor of the object spectrum at the frequency vector $f_{2}$ , denoted as $\frac{G (f_{2})}{| G (f_{2}) |}$ , and the known unit amplitude phasor of the spectrum autocorrelation of the speckle images after centroid alignment at frequency vectors $f_{1}$ and $f_{2}$ , denoted as $\frac{E [S_{c} (f_{1}) S_{c}^{*} (f_{2})]}{| E [S_{c} (f_{1}) S_{c}^{*} (f_{2})]}$ . Therefore, the unit amplitude phasor of the object spectrum at any frequency vector can be reconstructed recursively according to Eq. (21) with an initial condition that $\frac{G (f_{2} = 0)}{| G (f_{2} = 0) |} = 1$ .

Figure 3 illustrates the recursive reconstruction path. In the discrete frequency domain, starting with $f_{1} = (1, 0)$ , the unit amplitude phasor of the object spectrum at each $f_{1}$ was reconstructed according to $| f_{1} |$ , from small to large in a counterclockwise direction. The choice of reconstruction path is flexible and not limited to the one adopted here. For example, one could follow a clockwise path or begin reconstruction from any arbitrary spatial frequency. However, we recommend that, at each step of the recursive reconstruction, the unit amplitude phasor at the frequency vector be estimated using all previously reconstructed frequency vectors, and the final value be taken as the average of all such estimates. This approach can significantly reduce the impact of noise. During the reconstruction, every frequency vector for which the unit amplitude phasor of the object spectrum had been reconstructed is denoted as $f_{2}$ . The final unit amplitude phasor of the object spectrum at $f_{1}$ was then determined as the average of the unit amplitude phasors reconstructed at $f_{1}$ based on $f_{2}$ . After reconstructing the unit amplitude phasor of the object spectrum $\frac{G (f)}{| G (f) |}$ , the object phase spectrum can be obtained as $ϕ_{G} (f) = Arg (\frac{G (f)}{| G (f) |}),$ (22)where $Arg (\dots)$ represents the operator that takes the principal argument.

Figure 3.Recursive reconstruction path of the unit amplitude phasor of the object spectrum in SSAI. Each grid point represents a frequency point within the discrete frequency domain, denoted by the frequency vector $f = (u, v)$ . The arcs depict the recursive reconstruction path. The arrows on the arcs indicate the order of reconstruction. The reconstruction begins at $(u, v) = (1, 0)$ , and the unit amplitude phasor at each $f$ is reconstructed according to $| f |$ from small to large in the counterclockwise direction. Here, only the early stages of the recursive path are displayed.

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The object reconstruction strategy of SSAI is presented in the flowchart shown in Fig. 1(c). Before object reconstruction, the original speckle images were preprocessed to avoid introducing errors from the image capture process into the reconstruction and to enhance the utilization of the speckle information. In the experiments comparing the object reconstruction performance with the SCI method, the same preprocessing steps were uniformly applied in this work to ensure consistency. The preprocessing steps are as follows:

(a)The original speckle images were divided by their low-pass-filtered versions to obtain spatial normalization, thus attenuating the effects of the halo-like envelope on the reconstruction^[2].
(b)Each normalized speckle image was segmented into overlapped subimages to fully utilize the speckle information.
(c)A Hanning window was applied to every subimage to attenuate the spectrum leakage effect caused by the image edge.

After preprocessing the speckle images, the spectrum autocorrelation of the speckle images after centroid alignment was calculated using Eq. (17), where the statistical average was approximated by averaging over several speckle images. The object phase spectrum was then retrieved based on recursive reconstruction. Subsequently, the object amplitude spectrum was reconstructed based on the autocorrelation of the preprocessed speckle images in accordance with SCI^[1,2]. Finally, the object spectrum was recovered by combining the object phase spectrum with the object amplitude spectrum, followed by obtaining the reconstructed object image using an inverse Fourier transform.

Furthermore, to quantitatively evaluate the reconstruction quality of the SSAI approach, we utilized a detector equipped with an imaging lens to capture the original object image in the absence of the scattering medium. We then computed the structural similarity (SSIM) between the reconstructed and original object images. As the original and reconstructed images have different magnifications and object positions, the conventional SSIM calculation method is not directly applicable and must be modified. The procedure for calculating the modified SSIM (mSSIM) in this study is as follows:

(a)Extract the smallest rectangular region from the original object image that fully contains the object.
(b)Scale this rectangular region proportionally multiple times to generate masks of different sizes, ensuring that the mask sizes gradually transition from smaller to larger than the object region in the reconstructed image.
(c)Slide each mask, pixel by pixel, over the reconstructed object image, which has been zero-padded around the edges, and compute the conventional SSIM for the overlapping regions.
(d)Take the maximum value among these conventional SSIM values as the mSSIM between the reconstructed image and the original object image, accurately representing the similarity between them.

3. Experiments and Results

To demonstrate the validity of the proposed SSAI approach, we captured speckle images using the experimental setup shown in Fig. 4. The beam from the He–Ne laser (Edmund 1125P; 5 mW and 632.8 nm) was expanded and incident on a high-speed rotating diffuser to generate pseudo-thermal light. After beam diameter adjustment by passing through an iris and a convex lens, the pseudo-thermal light was used to illuminate the object pattern loaded onto a digital micromirror device (DMD; Texas Instruments, DLP5500). The resulting object light was then incident on a slowly rotating ground glass (Bobang Quartz, 220 grit), which was chosen as the dynamic strongly scattering medium, with a decorrelation time of 6 s. Finally, after passing through an iris (diameter of approximately 2 mm), the diffused object light was captured as speckle images using a lensless detector (Andor, iXon Ultra 897; 512 pixel × 512 pixel). The exposure time for each speckle image was 0.5 s. The distance between the ground glass and the object was approximately 29 cm, that between the ground glass and the iris was approximately 2.5 cm, and that between the ground glass and the detector surface was approximately 14 cm. Next, we sequentially present the reconstruction results of stationary, moving, background light-interfered stationary, background light-interfered moving, and size-scaling objects using SSAI. All reconstructions were performed on a computer equipped with an i7 8700K CPU and 64 GB of DDR4 RAM.

Figure 4.Experimental setup of SSAI. BE, beam expander; RD, rotating diffuser; L, convex lens; DMD, digital micromirror device; DSM, dynamic scattering medium.

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3.1. Reconstruct a stationary object hidden behind the dynamic scattering medium

First, we reconstructed a stationary object hidden behind the dynamic scattering medium. During the experiment, the DMD displayed six different object patterns in the sequence, including the number “6,” the English letter “G,” and the Chinese characters “友,” “智,” “礼,” and “和.” Original images of the objects captured by the detector with an imaging lens in the absence of the scattering medium are illustrated in Fig. 5(a). When the imaging lens was removed and the scattering medium was introduced, the captured speckle images are shown in Fig. 5(b). A total of 100 speckle images of each object were captured in succession. The original speckle images were preprocessed prior to reconstruction, as described in Sec. 2.4. In all the experiments performed in this study, each normalized speckle image was segmented into nine 256 pixel × 256 pixel subimages with a 50% overlap between adjacent subimages.

Figure 5.Reconstruction results of stationary objects hidden behind a dynamic scattering medium via the SSAI approach proposed in this paper. (a) The original images of six different objects: “6,” “G,” “友,” “智,” “礼,” and “和.” (b) An unprocessed 512 pixel × 512 pixel speckle image of each object. (c) Reconstruction results via the popular SCI approach. (d) Reconstruction results via SSAI. mSSIM: modified structural similarity between the original object image and the reconstruction result.

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In the following section, we compare the reconstruction results of SSAI with those of SCI using 100 original 512 pixel × 512 pixel speckle images. Owing to the indeterminacy of phase spectrum reconstruction, several reconstruction attempts using SCI failed. Figure 5(c) shows the results of each object with the highest quality and correct orientation among the 200 reconstruction attempts with SCI. In subjective visual perception, only the structural features of three objects, namely, “6,” “G,” and “礼,” were reconstructed correctly by SCI. The object images reconstructed using the proposed SSAI approach are shown in Fig. 5(d). The maximum values of $| f_{1} |$ and $| f_{2} |$ were set to 80 and 10, respectively, during the reconstruction process to balance efficiency and quality. The structural features of all six objects were reconstructed with high quality in visual perception using SSAI. The mSSIM values of the SCI and SSAI approaches are shown in Fig. 5. The mSSIM score in the SSAI group was significantly higher than that in the SCI group, with an improvement of up to 41%. The evaluation results of the mSSIM were consistent with those of subjective visual perception. In addition, experimental observations indicate that the SSAI method is capable of accurately reconstructing the directional information of the object, which the SCI method fails to achieve. This can be attributed to the fact that the speckle spectrum autocorrelation retains the directional features of the object, whereas the speckle image autocorrelation does not. As a result, SSAI exhibited a significantly higher confidence than SCI in reconstructing the stationary object hidden behind the dynamic scattering medium.

Furthermore, when reconstructing an object using 100 original speckle images, the SSAI method takes an average of 176 s per reconstruction. In contrast, to achieve the highest reconstruction quality attainable by the SCI method, it typically requires 100 reconstruction attempts, with a total computation time of 160 s. Under these conditions, the computation times of the two methods are comparable.

We further investigate the performance of SSAI in reconstructing stationary objects using different numbers of speckle images. As shown in Figs. 6(a)–6(e), the reconstructed object images correspond to varying numbers ( $F$ ) of input speckle images, while the original object images are displayed in Fig. 6(f). It can be observed that, as the number of speckle images increases, the structural features of the reconstructed object become progressively clearer, and the background noise is significantly reduced. Notably, the objects “6” and “友” can be accurately reconstructed using only 6 and 10 original 512 pixel × 512 pixel speckle images, respectively. This observation suggests that, although, by definition, speckle spectrum autocorrelation typically requires statistical averaging over a large number of speckle images, the sample-averaged estimation from a limited number of images can still yield high-quality reconstructions. Considering the ergodic nature of speckle patterns, if the experimental system can be optimized to capture a complete and accurate speckle intensity distribution within a single frame, the SSAI method holds promise for object reconstruction from a single speckle image. It is worth noting that reconstructing an object using 10 original speckle images with the SSAI method takes only 18 s. This suggests that, when aiming for high-quality reconstruction, SSAI has the potential to achieve significantly lower computation time than SCI.

Figure 6.Reconstruction results of stationary objects via SSAI when using different numbers of speckle images. (a)-(e) Reconstruction results. The letter $F$ below each reconstructed image indicates the number of original 512 pixel × 512 pixel speckle images used for the reconstruction. (f) The original images of two different objects.

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3.2. Reconstruct a moving object hidden behind the dynamic scattering medium

Second, we reconstruct moving objects hidden behind the dynamic scattering medium. As shown in Fig. 7(a), the object was moved from the coordinate origin to six other positions 0.5 or 1 mm away from the origin in the horizontal and vertical directions of the object plane during the experiment. A total of 15 speckle images were captured at each position, and 105 speckle images were captured for each object. The preprocessing of speckle images and the reconstruction parameter settings were the same as those used in the previous reconstruction of stationary objects. The reconstruction results of six different objects via SSAI are illustrated in Figs. 7(b)–7(g) and are of high quality in subjective visual perception. In addition, the mSSIM between the reconstruction result via SSAI and the original object image still demonstrated high values, decreasing by up to only 0.08, compared with that when the stationary object was reconstructed. The results showed that SSAI can reconstruct a moving object accurately while ignoring the varying position of the object.

Figure 7.Reconstruction results of moving objects hidden behind a dynamic scattering medium via SSAI. (a) The object is moved to seven different positions on the object plane when speckle images are captured. (b)–(g) The reconstruction results of six different objects via SSAI.

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Furthermore, we investigated the performance of the SSAI method under various object displacement conditions by reconstructing the object using speckle images acquired before and after the displacement. In the experiment, the object was horizontally moved from the origin to positions at $- 1$ , $- 0.5$ , 0.5, and 1 mm, and speckle images were recorded at each position. For each displacement, 15 speckle images acquired at the displaced position were combined with 15 images acquired at the original position, and the SSAI method was applied to reconstruct the object. The reconstruction results are shown in Figs. 8(a), 8(b), 8(d), and 8(e). For example, Fig. 8(a) presents the reconstructed image using 15 speckle images captured when the object was at $- 1 mm$ and 15 images from the original position. As a reference, Fig. 8(c) shows the reconstruction result obtained using 30 speckle images acquired at the original position with the object fixed. Figure 8(f) displays the original images of two different objects.

Figure 8.Reconstruction results of the object via SSAI under different horizontal object displacements, based on speckle images captured before and after the displacement. (a)–(e) Reconstruction results. The object displacements from left to right are $- 1$ , $- 0.5$ , 0, 0.5, and 1 mm, respectively. (f) The original images of two different objects.

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The reconstructed results demonstrate that SSAI can accurately recover the structural features of the object under different displacement conditions. In addition, for the same object, slight variations in intensity distribution can be observed across different reconstructions. For instance, for the object “G,” the intensity appears relatively uniform in Fig. 8(b), whereas in Fig. 8(e), the left half of the character exhibits slightly higher intensity than the right half. This variation is likely caused by changes in the influence of the aperture within the speckle imaging system after the object is displaced, which alters the spatial distribution of the scattered light received by the detector. However, this phenomenon does not affect object recognition. Overall, SSAI exhibits robust and stable reconstruction performance under varying object displacement conditions.

3.3. Reconstruct a stationary object hidden behind the dynamic scattering medium under background light interference

Third, we reconstructed stationary objects hidden behind the dynamic media under different intensities of background light. Unlike the previous reconstructions without background light interference, the patterns loaded onto the DMD had gray values at pixels outside the object region. Thus, the illuminated light reflected from the nonobject region accompanied the object light in the scattering medium and then became the background light during detection. The intensity of the background light was adjusted by changing the gray values of the pixels in the nonobject area of the DMD. The signal intensity $I_{sig}$ was determined by calculating the average difference between the response value of each pixel of the detector and the average dark current noise when only the object light was present without any background light. The background intensity $I_{back}$ is the average response value of each pixel of the detector when only the background light is present without any object light. Subsequently, for a speckle image captured in the presence of both object light and background light, the signal-to-background ratio (SBR) is $10 \lg (I_{sig} / I_{back})$ . Fifty speckle images were captured in the experiment for three different objects under four SBR conditions. The preprocessing of speckle images and the reconstruction parameter settings were consistent with those of the previous reconstructions without background light.

The reconstruction results are shown in Fig. 9. Figures 9(a) and 9(b) show the original images of the objects “友,” respectively. Figures 9(a-1) and 9(b-1) show the 512 pixel × 512 pixel speckle images of the two objects captured under four SBR conditions, respectively. The reconstruction results of these objects using SCI and SSAI are shown in Figs. 9(a-2), 9(b-2) and 9(a-3), 9(b-3), respectively. First, the reconstruction results of the object “友” were observed, with the SBR ranging from $- 0.42$ to $- 9.60 dB$ . Consistent with the absence of background light interference, the object “友” could not be reconstructed via SCI in any of the four SBR conditions. However, both the subjective visual perception and mSSIM values indicated that SSAI achieved high-quality reconstruction when the SBR was above $- 5.52 dB$ . When the SBR was decreased to $- 9.60 dB$ , SSAI still reconstructed the object structure completely. Next, the reconstruction results of object “2” were observed, with the SBR ranging from $- 0.8 0$ to $- 1 1.2 1 dB$ . The object structure could only be completely reconstructed by SCI at an SBR higher than $- 5.3 8 dB$ . However, SSAI was still able to completely reconstruct the object structure when the SBR was decreased to $- 11.21 dB$ , significantly surpassing the operational SBR limit of SCI. In other words, SSAI not only functions under the SBR conditions where SCI fails but also exhibits superior performance in both quantitative evaluations and perceptive quality when SCI is effective. Benefiting from its recursive phase spectrum reconstruction framework that reduces noise interference, the SSAI method demonstrates strong robustness against background light interference.

Figure 9.Reconstruction results of stationary objects hidden behind a dynamic scattering medium under background light interference via SSAI. (a) and (b) show the original images of the objects “友” and “2,” respectively. (a-1) and (b-1) show the 512 pixel × 512 pixel speckle images of the two objects captured under four SBR conditions, respectively. Reconstruction results of these objects via SCI and SSAI are shown in (a-2), (b-2) and (a-3), (b-3) respectively.

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3.4. Reconstruct a moving object hidden behind the dynamic scattering medium under background light interference

Fourth, we carried out an experiment to reconstruct moving objects hidden behind the dynamic scattering medium under background light interference. In this experiment, the approach of introducing and controlling background light was the same as the previous experiment, and the movement of the object was identical to the experiment reconstructing moving objects without background light interference. It should be noted that the detector remained fixed during the data collection process and can only capture a portion of the scattered light formed by the object light. When the object moved to different positions, the total energy and distribution of the scattered light from the object received by the detector changed. Consequently, the SBR of the speckle image also varied. The object to be reconstructed in this experiment was the Chinese character “田.” We set the pixels in the nonobject area on the DMD to two different grayscale values in the experiment, thus generating two different intensity levels of background light. Under each background light intensity setting, we moved the object to seven different positions following Fig. 7(a). Thirty frames of speckle images were captured at each position. Speckle images obtained at the seven different positions were used for reconstruction simultaneously. The preprocessing of speckle images and the reconstruction parameter settings in this experiment were consistent with those of the previous experiments.

Figure 10 illustrates the speckle images captured under the interference of two different intensities of background light and the object images reconstructed using the SCI approach and the SSAI approach. The object to be reconstructed was the Chinese character “田.” Figure 10(a-1) shows a speckle image captured at each position when the object moved to seven different positions under the weaker intensity of the background light. The numbers on the horizontal axis represent the object position IDs. The average SBR of the speckle images at each object position is annotated in Fig. 10(a-1). The average SBR of all speckle images collected at the object positions was $- 6.60 dB$ . Figure 10(a-2) shows the original object image. Figure 10(a-3) shows the object image reconstructed through the SCI approach, with an mSSIM of 0.41. Figure 10(a-4) presents the object image reconstructed using the SSAI approach, with an mSSIM of 0.86. Under the stronger intensity of the background light, Figs. 10(b-1), 10(b-2), 10(b-3), and 10(b-4), respectively, show the speckle images captured at the seven object positions, the original object image, the reconstruction result of the SCI approach, and the reconstruction result of the SSAI approach. The average SBR of all speckle images was $- 8.61 dB$ . The mSSIM of the object image reconstructed by the SCI approach is only 0.34, while the mSSIM of the object image reconstructed by the SSAI approach remains high at 0.79. A comprehensive analysis of the experimental results under the interference of two different intensities of background light shows that the SCI approach fails to reconstruct the object structure, while the SSAI approach can achieve high-fidelity object reconstruction. Therefore, when reconstructing a moving object hidden behind the dynamic scattering media under background light interference, the SSAI approach still demonstrates significantly better performance than the SCI approach.

Figure 10.Reconstruction results of moving objects hidden behind a dynamic scattering medium under background light interference via SSAI. (a-1) and (b-1) show speckle images captured at seven different object positions under two different intensities of background light interference, respectively. The average SBR of the speckle images captured at each object position is annotated under each speckle image. The value of mean SBR indicates the average SBR of all speckle images. (a-2) and (b-2) show the original object image. The reconstructed results via SCI under the two different intensities of background light are shown in (a-3) and (b-3). The reconstructed results via SSAI under the two different intensities of background light are shown in (a-4) and (b-4).

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3.5. Reconstruct a size-scaling object hidden behind the dynamic scattering medium

Finally, we reconstructed the size-scaling objects hidden behind the dynamic scattering medium using SSAI. We chose three objects—“友,” “2,” and “S”—for experimental validation. As shown in Fig. 11(a), the object was made to maintain its center position constant to undergo constant aspect ratio scaling on the object plane during the acquisition of speckle images. The object was scaled to three sizes with heights of 50, 60, and 70 DMD pixels in the vertical direction. The medium remained dynamic but did not interfere with the background light. For reconstruction, 50 speckle images were captured of each object in three different sizes. The process and parameter settings of speckle image preprocessing and phase spectrum reconstruction were consistent with those of previous experiments. However, only one speckle image was arbitrarily selected to reconstruct the amplitude spectrum of the object. The reconstruction results are shown in Fig. 11(b), where the structural features of all three objects are clearly reconstructed. Therefore, SSAI has the potential to be directly applied to scattering scenarios when the object moves along the optical axis of the imaging system or when the object scales on the object plane, thereby showing good scalability.

Figure 11.Reconstruction results of size-scaling objects hidden behind a dynamic scattering medium via SSAI. (a) The object is scaled to three different sizes at different moments during the capture of speckle images. (b) Reconstructed images of objects “友,” “2,” and “S.”

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4. Discussion and Conclusion

To date, the major challenge in utilizing strongly scattering light for noninvasive imaging has been related to stable and precise reconstruction of the object phase spectrum. Although SCI has established a high-quality formalism for reconstructing the object amplitude spectrum, the deficiencies of its phase spectrum reconstruction formalism, IPR, have significantly restricted the capabilities of object reconstruction. SCI-based techniques face significant challenges in adapting to complex scenarios with strong background light interference and variable elements. In this study, we proposed a novel SSAI approach for imaging with strongly scattered light. SSAI can achieve independent and deterministic reconstruction of the object phase spectrum based on the spectrum autocorrelation of centroid-aligned speckle images. This approach effectively mitigates interference arising from background light, dynamics of scattering media, object movement, and object scaling in complex scattering scenarios.

In the validation experiments, we compared the performances of SSAI and SCI in reconstructing stationary objects hidden behind a dynamic medium, both with and without background light interference. SSAI demonstrated superior performance over SCI in terms of stability, fidelity, and resistance to interference, as evidenced by visual perception and the quantitative metric mSSIM. SSAI can also function in scenarios where SCI fails. Remarkably, we found that SSAI can completely reconstruct the object with an SBR as low as $- 11.21 dB$ , significantly surpassing the operational SBR limit of SCI. Furthermore, SSAI successfully addresses the challenge of unstable reconstruction of the direction and position of an object, a problem commonly encountered with SCI. Furthermore, we showcased the high-fidelity reconstruction of moving, background light-interfered moving, and size-scaling objects hidden behind a dynamic scattering medium using SSAI. These results underline the high scalability of SSAI, highlighting its adaptability to complex scattering scenarios.

Although SSAI has shown encouraging results in its implementation, it is still constrained by some performance limitations. In the recursive phase spectrum reconstruction of SSAI, all reconstruction errors at low frequencies tended to accumulate and propagate into the reconstruction at higher frequencies. To achieve high-quality object reconstruction in scattering scenarios with more severe disturbances, the recursive reconstruction problem should be reformulated as an optimization problem that minimizes the mismatch between the known phase of the speckle spectrum autocorrelation and the Fourier phase of the unknown object.

Empirically, we observed that SSAI requires a minimum of 10 speckle images for high-fidelity object reconstruction. Because strongly scattered light can result in speckles with ergodicity^[2,36], it is theoretically feasible to calculate the speckle spectrum autocorrelation from a single speckle image that accurately records the speckle intensity distribution. Future work may explore the advancement of SSAI in practical applications using a limited number of speckle images or even a single frame. Key elements include enlarging the speckle sampling area, improving the resolution of speckle detail, and mitigating noise effects. Equipping the detector with imaging lenses can facilitate the capture of a wider range of speckle data, but may compromise the resolution of fine speckle details, necessitating a trade-off. Another feasible method might involve employing a detector with higher resolution and sensitivity.

Several interesting aspects of the SSAI method remain to be explored. For example, investigating its applicability in non-uniform scattering media is of considerable significance. In such media, the statistical properties of light scattering can vary significantly across different regions, making it difficult to represent the overall scattering effect with a single OTF. We propose that it may be feasible to selectively detect the scattered light from different regions of the medium using appropriately designed apertures. If the scattering effect within each region can be characterized by an individual OTF, then the SSAI method may be capable of reconstructing the corresponding object areas from the acquired speckle images. The reconstruction results from different regions could then be combined to form a complete image of the object.

In addition, as the average number of scattering events within the medium increases, the field of view over which the SSAI method based on the OME can reconstruct the object becomes narrower. Then investigating the applicability of SSAI with respect to the average number of scattering events is also of significant importance. It is also worth noting that, since the Fourier spectrum contains information about the object’s grayscale variation, SSAI is theoretically capable of reconstructing grayscale objects with continuous intensity variations. Determining the range of grayscale variation to which the SSAI method is applicable remains an important topic for further investigation.

Looking ahead, the furtherance of SSAI is not limited to its currently demonstrated capabilities. Considering that SSAI and SCI can utilize the same imaging system and speckle data and that both require reconstruction of the object Fourier spectrum, SSAI has the potential to substitute the object reconstruction formalism in SCI-based techniques, thereby enhancing the performance of these techniques. Furthermore, as mentioned in the introduction section, methods developed initially to extend the applications of SCI, such as reconstruction under broadband illumination, reconstruction of multiple objects separated beyond the OME range, fluorescence microscopy, and super-resolution imaging, can be transferred to SSAI. Evidently, SSAI is not only applicable to the complex scattering scenarios demonstrated in this study but also scalable to a significantly broader range of scenarios. Notably, no imaging lenses were employed in the implementation of SSAI. As a lensless imaging approach, SSAI can potentially be applied in electromagnetic wavebands or wave domains where lens fabrication remains a challenge. SSAI bridges the gap between strongly scattered light and imaging hidden objects and is anticipated to find wide applicability in fields such as biomedical imaging, astronomical observations, remote sensing, and underwater detection.

Category: Research Article

Received: Mar. 19, 2025

Accepted: May. 28, 2025

Published Online: Jun. 26, 2025

The Author Email: Xia Wang (angelniuniu@bit.edu.cn)

DOI:10.3788/AI.2025.10006