Reconstruction of target image from inhomogeneous degradations through backscattering medium images using self-calibration

WenJun Yi; Haibo Liu; Ping Wang; MeiCheng Fu; JiChun Tan; XiuJian Li

doi:10.1364/OE.25.007392

1. Introduction

Imaging through optically inhomogeneous turbid mediums is indispensable in many fields such as astronomical observation, underwater observation, biological imaging and security surveillance [1–5]. Range-gated laser imaging [6] is a prominent approach for suppressing and eliminating the backscattered photons from turbid mediums with the view to improve the signal-to-noise ratio (SNR) of the target images. The applications of this technique are plenty, but not limited to underwater detection [7–9], target recognition [10,11], and three-dimensional imaging [12–14]. However, the photons that are backscattered from the turbid medium clusters and carry plenty of physical information about the medium and the light beams are suppressed. As a result, the inhomogeneous degradations of the range-gated target images are not entirely eliminable by the blind digital image restoration algorithms such as filtering algorithms [15–19] and total variation methods [20–24]. Imaging methods such as ghost imaging [25,26], wavefront shaping [27–29], and speckle correlations [30,31] are also incapable of sufficiently dealing with the issue of obtaining high quality target images through rapidly changing inhomogeneous turbid mediums such as moving cloud/mist in air and flowing muddy masses under water.

In a previous report, we proposed a longitudinal laser tomography (LLT) system [32] to capture the target images and the images of the nonuniform attenuator (an opaque solid layer) that can successfully eliminate light beam degradation and improve the quality of the target image. In this method, the degradation was mainly caused by the nonuniform attenuator, and the backscattering images were produced by photons backscattered from the turbid medium lying close to the target. As a result, this method would not work for the common turbid medium conditions in which the distribution of the turbid medium is not in the vicinity of the target. Here, by broadening the working conditions of the LLT system, we propose an approach to estimate the degradation caused by the turbid medium clusters and retrieve the true target images under common turbid medium conditions.

2. Principles of the method

As shown in Fig. 1, using the LLT system comprising of a nanosecond pulsed laser and an intensified charge-coupled device (ICCD) camera, the raw images—including the backscattering images of the turbid medium clusters and the target images—are captured. The signal timings for the LLT system are shown in Fig. 1. By suitably tuning the gating ranges of the ICCD camera, various groups of photons from different spatial layers on the optical path will be captured to form the raw images. The thickness of each layer is determined by the corresponding duration of laser pulse and camera exposure. It should be noted that the gated backscattering images contain physical information about the turbid medium. As far as possible, these images should be collected simultaneously along with the target images for rapidly changing turbid mediums, which means that all the raw images should be captured within a short time.

Fig. 1 Schematic diagram of the longitudinal laser tomography experimental set up. The gate delay satisfies $τ_{n} = n τ_{1}$ , where $τ_{1}$ is the gate step and $n = 1, 2, 3 \cdot \cdot \cdot$ . For clarity, yellow and blue are used to indicate the backscattering pulse and the target reflecting pulse, respectively, although their wavelengths are the same as the incident pulse. See Visualization 1 for details.

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2.1 Modeling of target image degradation

While propagating through a turbid medium, some of the photons interact with the medium, and are eventually scattered and absorbed. This inevitably degrades the quality of the light beam along with the influences of ordinary diffraction and interference. We introduce a degradation matrix $V$ , which can be used to describe the target image degradation model as shown in Eq. (1) [32].

u_{0} = V \cdot u + n .

Here,

u_{0}

denotes the captured raw target image,

u

denotes the ideal target image (free from degeneration), and

n

denotes the additive noise imposed on the image. According to the basic physical process of image formation, the degradation matrix

V

, which is multiplied with the ideal target image

u

, contains information of the light beam evolution through the turbid medium cluster and is in the form of a matrix. Since the additive noise

n

is usually removed by digital image processing methods, the perfect target image can be restored if a suitable degradation matrix

V

is applied.

2.2 Establishment of the degradation matrix

According to the Light Detection and Ranging (LIDAR) principles [33–36], for the middle-range and long-range LLT usually with a range of hundreds or thousands of meters, some of the photons reflected from the target with reflectivity of $ρ_{t}$ will be captured by a pixel $(i, j)$ of the ICCD camera [32],

I_{t a r} (i, j) = \frac{E_{t} D^{2} d_{p i x}^{2} ρ_{t} (X_{t}, Y_{t}, Z_{t})}{θ^{2} f^{2} Z_{t}^{2}} \exp [- 2 \int_{0}^{Z_{t}} K_{e} (x, y, z) d z] .

Here,

I_{t a r} (i, j)

is the target-reflected energy received by the pixel

(i, j)

,

Z_{t}

is the distance between the ICCD camera and the target,

E_{t}

is the energy of a single pulse used in the measurement,

θ

is the divergence angle of the laser beam,

d_{p i x}

is the pixel size,

D

is the aperture width of the receiver,

f

is the focal length of the optical receiving system, and

K_{e} (x, y, z)

is the attenuation coefficient of the turbid mediums at position

(x, y, z)

, which takes into account the energy loss by scattering and absorption in the turbid medium. Moreover, the reflection point positions

(X_{t}, Y_{t}, Z_{t})

and the pixel coordinates

(i, j)

satisfy the conditions of projective transformation, i.e.,

{\begin{matrix} i d_{p i x} = - f X_{t} / Z_{t} \\ j d_{p i x} = - f Y_{t} / Z_{t} \end{matrix}

, as shown in Fig. 2. The target image without the influences of the turbid medium is defined as the ideal target image:

Fig. 2 Coordinate system of longitudinal laser tomography set up.

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I_{i d e a l} (i, j) = \frac{E_{t} D^{2} d_{p i x}^{2} ρ_{t} (X_{t}, Y_{t}, Z_{t})}{θ^{2} f^{2} Z_{t}^{2}} .

Obviously, the quality of the ideal target image depends on the target’s reflectivity distribution and the parameters of the transceiver system rather than the atmospheric conditions. For the pixel $(i, j)$ of the raw target image, the degradation of the corresponding light beam intensity produced by the turbid medium can be represented as $V (i, j)$ , which is an element of the degradation matrix V:

V (i, j) = \frac{I_{t a r} (i, j)}{I_{i d e a l} (i, j)} = \exp [- 2 \int_{0}^{Z_{t}} K_{e} (x, y, z) d z] .

The key point in constructing the degradation matrix is to obtain the values of $\int_{0}^{Z_{t}} K_{e} (x, y, z) d z$ (or the attenuation coefficient $K_{e} (x, y, z)$ and the corresponding distribution parameters of the turbid medium).

2.3 Obtaining key information about the degradation matrix by applying sub-window segmentation

First, in order to obtain each element of the degradation matrix, the LIDAR ratio [36], i.e., the extinction-to-backscattering ratio $S = K_{e} / β$ , is extracted from the LLT images, where $β$ is the backscattering coefficient of the turbid medium. The LIDAR ratio $S$ depends on the complex refractive index and the particle size distribution of the turbid medium rather than the number density of the particles [36]. As shown in Fig. 1, when the gate range and the camera exposure time are tuned to cover the entire turbid medium, the relationship $\int_{0}^{Z_{t}} β (x, y, z) d z \approx ρ_{S} (x, y)$ exists, where $ρ_{S} (x, y)$ can be approximated as the total equivalent reflectivity of the turbid medium cluster. Therefore, each element of the degradation matrix can be simplified as:

V (i, j) = \exp [- 2 S ρ_{S} (x, y)] .

Herein, to ensure perfect timeliness for the extraction of $ρ_{S}$ and $S$ , we put forward an estimation approach based on calibration using an observed target image and a backscattering image of the turbid medium. Basically, the gray value of the backscattering image is proportional to the total equivalent reflectivity of the turbid medium clusters, i.e., $I_{S} (i, j) = C E_{S} ρ_{S} (x, y) / Z_{s}^{2}$ , where $E_{S}$ is the single pulse energy in the measurement, $Z_{S}$ is the distance of the medium cluster from the camera, and $C$ is a constant determined by the transceiver system parameters. Therefore, Eq. (5) can be rewritten as:

V (i, j) = \exp [- 2 S \frac{Z_{S}^{2}}{C E_{S}} I_{S} (i, j)] \equiv \exp [- 2 K_{S} I_{S} (i, j)] .

Here,

K_{S} = S Z_{S}^{2} / (C E_{S})

contains both the LIDAR ratio

S

and the transceiver system parameters.

Assuming that in a measurement, the ideal target image $I_{i d e a l}$ and the degraded target image $I_{t a r}$ are both captured by the transceiver system, the exponential factor $K_{S}$ can be calculated as:

K_{S} = - \frac{\ln (I_{t a r} / I_{i d e a l})}{2 I_{S}} .

Nevertheless, in a practical imaging system, it is impossible to capture the target images with and without the influences of the turbid medium at the same time. By segmenting two sub-windows in the degraded target image, i.e., one for the severely degraded part and another for the slightly degraded or even unaffected part, which can be inferred to represent the original target approximately, $K_{S}$ can be calculated as:

K_{S} = - \frac{\ln ({\bar{W}}_{1} / \bar{W_{0}})}{2 \bar{W_{S}}} .

Here,

W_{0}

is the sub-window of the slightly degraded or even unaffected part of the observed target image,

W_{1}

is the sub-window of the severely degraded part, and

W_{S}

is the sub-window in the backscattering image corresponding to

W_{1}

;

\bar{W_{0}}

,

\bar{W_{1}}

, and

\bar{W_{S}}

are the average gray values of the corresponding sub-windows.

Moreover, for performing a new measurement with this technique, if the medium components remain the same, i.e., the LIDAR ratio $S$ remains unchanged, another round of calibration of $K_{S}$ will be unnecessary, and the exponential factors ${K^{'}}_{S}$ can be derived from the previously obtained $K_{S}$ . Consequently, the new degradation matrix can be represented as:

V^{'} (i, j) = \exp [- 2 \frac{{Z^{'}}^{2}_{S}}{Z_{S}^{2}} \frac{E_{S}^{2}}{{E^{'}}^{2}_{S}} K_{S} {I^{'}}_{S} (i, j)],

where

{E^{'}}_{S}

is the pulse energy,

{Z^{'}}_{S}

is the distance between the turbid medium and the camera, and

{I^{'}}_{S}

is the backscattering image of the turbid medium. In the LLT system, the distance of the turbid medium from the camera will be

Z_{S} = c τ_{S} / 2

, in which

c

denotes the speed of light and

τ_{S}

denotes the delay in signal propagation.

2.4 Algorithm for image recovery

Based on the TV-L¹ variational model [22,23], the following image recovery algorithm is established [32].

\min_{u} \int_{Σ} [| \nabla u | + γ | u_{0} - \tilde{v} \cdot u |],

where

γ

represents the regularization parameter,

u_{0}

represents the captured raw target image, u represents the restored target image, and

\tilde{v}

represents the estimated degradation matrix . The flowchart of the recovery process is presented in Fig. 3.

Fig. 3 Flowchart of the restoration method.

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3. Experiments and analysis

The LLT experimental setup is shown in Fig. 1, in which the turbid medium was created by a fog/smoke generator, and the synchronizing signals for the ICCD camera were provided by a Si photodetector. The laser pulses were generated by a pulsed Nd:YAG laser having a wavelength of 532 nm and pulse width of 20 ns. The output images of the ICCD camera (an AndoriStar ICCD camera) were composed of 340 × 340 pixels. The laser repetition frequency and the frame frequency of the ICCD camera were both 10 Hz, which allowed a 100 ms time interval between the capture of the backscattering image and the target image. The signal timings for the longitudinal tomography imaging system are shown in Fig. 1. In the experiments, the gate step $τ_{1}$ and the exposure time were both 20 ns. The gate delays corresponding to the backscattering image and the target image are $τ_{i} = 2 Z_{S} / c$ and $τ_{j} = 2 Z_{t} / c$ , respectively, where $Z_{S}$ ( $20 m \leq Z_{S} \leq 27 m$ ) is the distance between the turbid mediums and the transceiver system, $Z_{t} = 30 m$ is the target distance, and $c$ denotes the speed of light.

3.1 Experimental results

The target image and the backscattering image of the turbid medium clusters captured by the LLT system are shown in Figs. 4 and 5. The establishment of the degradation matrix and the recovery of the target are both depicted in Fig. 4. The sub-windows $W_{0}$ , $W_{1}$ , and $W_{S}$ are segmented from the observed target image and the backscattering image. According to Eq. (8), the calibration parameter $K_{S}$ can be calculated as 0.0043 for the first group of experimental images. The degradation matrix $\tilde{V}$ can be estimated with the calibration parameter $K_{S}$ and the denoised backscattering image $I_{S}$ ; then, through the proposed variational model in Eq. (10), the retrieved target image $U$ can be solved using the captured target image $U_{0}$ and the estimated degradation matrix $\tilde{V}$ . For a comparison between the observed and retrieved target images, the turbid-medium-free target image $I_{r e f}$ serves as the reference image.

Fig. 4 Recovery process of the first experimental image group. $K_{S}$ is the calibration parameter extracted from the target image and the medium image, while $\tilde{V}$ represents the estimated degradation matrix.

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Fig. 5 Recovery process of the second experimental image group.

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By considering the turbid-medium-free target images in Fig. 4 as the approximate ideal target image, the structural similarity (SSIM) indexes [37] for the degraded area (as marked in Fig. 4 by dotted bordered rectangles) of the observed and retrieved target images are listed along with the corresponding images. The SSIM index is a real number lying between 0 and 1, and value 1 can only be reached in the case of two identical images. A larger SSIM index implies higher structural similarity. The proposed method can improve the values of SSIM indexes remarkably. The experimental results indicate that the proposed method can effectively eliminate the influences of the inhomogeneous turbid medium to achieve the true target image.

The experimental raw images shown in Fig. 5 are captured with the same fog/smoke generator parameters as those shown in Fig. 4. Therefore, the calibration parameter $K_{S 1}$ can be derived from the previous parameter $K_{S}$ as obtained from Fig. 4. For a better illustration of the improvement, the 3D surfaces of the grid targets are presented in Fig. 5, where the height specifies the gray scale instead of the target depth. The experimental results illustrate that, the proposed method can effectively reveal the true target image and can be used to represent the reflectivity distribution through inhomogeneous turbid mediums.

3.2 Robustness check

Since the degradation matrix is obtained from the backscattering image of the turbid medium along with crucial calibration parameters such as the exponential factor $K_{S}$ , an error in the calibration will affect the robustness of the image recovery method.

The exponential factor corresponding to the observed target image along with the turbid-medium-free target image is calculated to be $K_{S 0} = 0.0046$ , in accordance with Eq. (7). This value is considered more accurate than the value obtained from a single observed target image and can be approximated as the standard value. The recovery results for various values of $K_{S}$ are listed in Figs. 6(a)–6(i), in which Fig. 6(e) displays the recovery result with the standard value $K_{S 0}$ and the relative errors of $K_{S}$ for the rest of the recovery images vary between −40% and + 40%. It is obvious that if the relative errors increase, despite the elimination of the inhomogeneous influence from the target images, the recovery images will be of poor quality, leading to a reduction in the corresponding SSIM indexes. An excessively large $K_{S}$ causes the degraded part of the target image to become too bright, while an extremely small $K_{S}$ fails to eliminate the degradation effects significantly. Moreover, when the relative errors of the exponential factor $K_{S}$ are less than 20% (Figs. 6(c)–6(g)), the corresponding SSIM indexes are greater than 0.8 and the recovery effects are acceptable. This indicates that the proposed estimation approach for the degradation matrix shows good robustness for the calibration parameters.

Fig. 6 Recovery results for the exponential factor $K_{S}$ with various relative errors.

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4. Conclusion

A novel and feasible method was put forward to reconstruct a true target image free from degradations caused by inhomogeneous turbid medium. This technique utilized raw images obtained from LLT, particularly the ones generated by photons backscattered from the turbid medium clusters. The results demonstrated that, based on the backscattering medium images that represent light beam degradation induced by the turbid medium, the influence of the inhomogeneous turbid medium can be approximately estimated and subsequently eliminated.

This method can remove the interferences of the inhomogeneous turbid medium from the target images and reveal the true reflectivity distribution of the targets behind the turbid medium, which makes it helpful in reduction of false recognition rate in target recognition and identification. The proposed method will be beneficial in target acquisition for observations in air and under water, military reconnaissance, and fire rescue. It should be pointed out that, the sub-window extraction of the proposed method may have to be assisted with human operation and certain prior information when the target is highly variable spatially.

Funding

National Natural Science Foundation of China (NSFC) NO.613192, 61070040, 61108089, 61205087 and 61107005.

Acknowledgment

The authors acknowledge Yanyun Ma for lending the measurement equipment for the experiments.

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Reconstruction of target image from inhomogeneous degradations through backscattering medium images using self-calibration

Abstract

1. Introduction

2. Principles of the method

2.1 Modeling of target image degradation

2.2 Establishment of the degradation matrix

2.3 Obtaining key information about the degradation matrix by applying sub-window segmentation

2.4 Algorithm for image recovery

3. Experiments and analysis

3.1 Experimental results

3.2 Robustness check

4. Conclusion

Funding

Acknowledgment

References and links

Supplementary Material (1)

Cited By

Figures (6)

Equations (10)

Optics Express