Restoration of longitudinal laser tomography target image from inhomogeneous medium degradation under common conditions

WenJun Yi; Ping Wang; MeiCheng Fu; JiChun Tan; Jubo Zhu; XiuJian Li

doi:10.1364/OE.25.015687

1. Introduction

Range-gated laser imaging [1] is a prominent technique for underwater detection [2–4], target recognition [5,6], and three-dimensional imaging [7–9], which can effectively improve the signal-to-noise ratio (SNR) of target images by removing backscattering noises. Unfortunately, the layer-cut target images still suffer from inhomogeneous degradations when the targets are obscured by inhomogeneous medium layers, and the inhomogeneous influence is difficult to be perfectly removed through blind image restoration algorithms such as the filtering algorithms [10–14] and the total variation methods [15–19].

We developed a longitudinal laser tomography (LLT) system [20,21] to capture both the target images and the backscattering images of turbid media by longitudinally scanning along the laser transmission path to form image sequences. As the photons backscattered from turbid medium clusters usually carry plentiful physical information of the turbid media, the degraded target images can be restored by using the backscattering image through an image restoration method [21]. Our previous work extracts prior information with help of the target image, and consequently it has the limitation that it only works well under the conditions of localized turbid distributions and fairly uniform targets.

Herein, we develop a new image restoration method for the LLT, which can estimate the degradation caused by inhomogeneous turbid media by using prior parameters and backscattering medium images. Furthermore, as the new method extracts prior information from pre-calibration rather than from a target image, it can realize an unsupervised restoration process, and meanwhile overcome the limitation of the previous method. Even if the turbid medium is spread out over the field of view and the target is highly variable spatially, the image restoration method can work properly.

2. Principles of image restoration

The schematic diagram of a LLT system is shown in Fig. 1. The transceiver system consists of a nanosecond pulsed laser and an intensified charge-coupled device (ICCD) camera. By finely tuning gate delays of the ICCD camera, echo signals from various ranges will be captured by the ICCD camera to generate image sequences, including both the target image and the backscattering image of the turbid medium. Based on the backscattering images together with some prior parameters, the degradation of the target image will be estimated for further image restoration.

Fig. 1 Schematic diagram of the LLT system.

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The degradation model of the degraded target image can be simplified as

U_{d} = V \cdot U + N .

Here,

U_{d}

represents the degraded target image,

U

represents the ideal image without degeneration,

V

is the degradation matrix determined by turbid media along the laser transmission path, and

N

is the additive noise of the system respectively [20]. The key point of the target image restoration is the estimation of the degradation matrix

V

.

According to Ref [21], when the gate range for the turbid medium is set to cover the entire layer of the turbid medium as shown in Fig. 1, a certain element $(i, j)$ of the degradation matrix $V$ corresponding to the target image pixel $I_{t a r} (i, j)$ can be simplified as:

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

Here,

ρ_{S}

is the reflectivity of the entire turbid medium layer;

S

is the Light Detection and Ranging (LIDAR) ratio [22] of the turbid medium, i.e., the extinction-to-backscattering ratio

S = K_{e} / β

, where

K_{e}

and

β

are respectively the extinction coefficient and the backscattering coefficient of the turbid medium. The LIDAR ratio

S

depends on the complex refractive index and the size distribution of the medium particles rather than the total number density of the particles. In other words, for a given medium, its LIDAR ratio will remain constant even when the medium particles get spread or gathered.

Herein, the reflectivity $ρ_{S}$ of the turbid medium can be calibrated by a reference target with uniform reflectivity $\bar{ρ_{0}}$ that can be pre-measured. Basically, the gray value of the medium image is proportional to the reflectivity of the turbid medium, therefore the medium reflectivity $ρ_{S}$ are supposed to be obtained by the pre-calibration of the transceiver system as following: a LLT experiment in absence of turbid media can be performed by using a reference target plate with uniform reflectivity $\bar{ρ_{0}}$ as the imaging target, and the perpendicular-incidence reflectivity $\bar{ρ_{0}}$ of this reference target plate needs to be measured in advance. The average gray value of the reference target image $I_{0}$ can be represented as:

\bar{I_{0}} = C E_{0} \bar{ρ_{0}} / Z_{0}^{2},

where

E_{0}

is the single pulse energy for the pre-calibration,

C

is a constant determined by the system parameters, and

Z_{0}

is the distance between the reference target and the camera respectively. In the LLT system, the target distance is provided by

Z_{0} = c τ_{0} / 2

, in which

c

denotes the speed of light and

τ_{0}

denotes the gate delay of the target signal.

In the LLT measurements, as the gate range for the backscattering image of the turbid medium covers the entire medium layer as shown in Fig. 1, the intensity of a certain pixel $(i, j)$ in the backscattering image can be represented as:

I_{S} (i, j) = C E_{S} ρ_{S} (x, y) / Z_{S}^{2},

Here,

E_{S}

is the single pulse energy, and

Z_{S}

is the distance of the medium cluster. As shown in Fig. 1, the medium point positions

(x, y, Z_{S})

and the pixel coordinates

(i, j)

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

{\begin{matrix} - i d_{p i x} / f = x / Z_{S} \\ - j d_{p i x} / f = y / Z_{S} \end{matrix}

, where

d_{p i x}

is the pixel size and

f

is the focal length of the optical receiving system. The medium reflectivity

ρ_{S}

can be obtained by:

ρ_{S} (x, y) = \frac{E_{0} Z_{S}^{2} \bar{ρ_{0}}}{E_{S} Z_{0}^{2} \bar{I_{0}}} I_{S} (i, j) .

Moreover, in Eq. (2), there is still an unknown value, i.e., the LIDAR ratio $S$ , which is supposed to be obtained by pre-measurement, or by pre-estimation through prior information of the turbid medium. By substituting $S$ and the calibrated $ρ_{S}$ into Eq. (2), the estimated degradation matrix can be achieved as:

\tilde{V} (i, j) = \exp [- 2 S \frac{E_{0} Z_{S}^{2} \bar{ρ_{0}}}{E_{S} Z_{0}^{2} \bar{I_{0}}} I_{S} (i, j)] .

The entire restoration process is shown in Fig. 2.

Fig. 2 Flowchart of the proposed restoration method.

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3. Simulations

A three-dimensional distribution of the turbid medium for the simulation is designed as Fig. 3, in which a turbid medium layer is located between the target and the transceiver system; this medium layer causes the inhomogeneous degradation as well as provides the backscattering images. The gated medium layer and the target layer are denoted with dotted bordered rectangles in Fig. 3.

Fig. 3 Turbid medium distribution in the simulation.

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The particle size distribution of the medium particles obeys the following Diermendjian distribution [23],

n (r) d r = a r^{μ} \exp (- b r^{ν}) d r .

Here,

r

is the particle radius,

a

,

b

,

μ

and

ν

are the four tunable Diermendjian parameters [24]; in the simulation, the four parameters are set to be

a = 1 .0 85 \times 10^{- 2}

,

b = 1 / 24

,

μ = 8

and

ν = 3

, and the total particle number density can be calculated as

N = \int_{0}^{\infty} n (r) d r = 100 c m^{- 3}

. The complex refractive index of the turbid medium is set to be

m = 1 .334-1 .32 \times 10^{- 9} i

(in reference of the value of the water droplet at the wavelength of 532nm) [25]. According to the atmospheric parameters above, the extinction coefficient

K_{e}

and the backscattering coefficient

β

can be calculated as

1 .128 \times 10^{- 2} m^{- 1}

and

5 .896 \times 10^{- 4} m^{- 1} \cdot s r^{- 1}

respectively.

3.1 Pre-estimation of LIDAR ratio and pre-calibration of system parameters

For water clouds, the LIDAR ratio has a near constant value close to 19sr at 532 nm [26–29]. More specifically, five groups of Diermendjian parameters for water clouds, including those for the LLT simulation experiment (D2), are listed in the first four columns of Table 1, and their corresponding functions are plotted in Fig. 4; $r_{m}$ represents mode radius, i.e., the maximum point of the function. Since the medium particle is water droplet, i.e., the complex refractive index is $m = 1 .334-1 .32 \times 10^{- 9} i$ , the LIDAR ratios $S$ for the five groups of Diermendjian parameters are calculated as listed in the last column of Table 1. Obviously, in spite of different mode radius $r_{m}$ , the LIDAR ratios remain close to 19sr. Due to the good robustness of the exponential factors in the degradation matrix equations [21], $\tilde{S} = 19 s r$ is chosen as the estimated LIDAR ratio for the degradation matrix establishment. Generally, for common turbid media, the LIDAR ratios can be estimated according to the medium types and the approximate ranges of the median particle size.

Table 1. Five groups of Diermendjian parameters and the corresponding LIDAR ratio $S$

View Table

Fig. 4 Diermendjian functions of the five groups of parameters in Table 1.

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In the pre-calibration in absence of turbid media, a reference target plate with perpendicular-incidence reflectivity of $\bar{ρ_{0}} = 0.5 s r^{- 1}$ was considered, the single pulse energy was $E_{0} = 1 m j$ and the distance between the reference target and the transceiver system was $Z_{0} = 100 m$ . The reference target image $I_{0}$ is shown in Fig. 5, and its average gray value is denoted by $\bar{I_{0}}$ . The pre-calibration parameters such as $E_{0}$ , $Z_{0}$ , $\bar{ρ_{0}}$ , $\bar{I_{0}}$ together with the LIDAR ratio $\tilde{S} = 19 s r$ all served as the prior parameters for the following image restoration.

Fig. 5 Recovery process of the simulation images. $\tilde{V}$ represents the estimated degradation matrix; $E_{0}$ , $Z_{0}$ and $\bar{ρ_{0}}$ represent the single pulse energy, the target distance and the reference target reflectivity in the pre-calibration, respectively; $\bar{I_{0}}$ denotes the average gray value of the reference target image.

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3.2 Simulation Verification

The simulated images and the recovery process of the target image are depicted in Fig. 5. The degradation matrix $\tilde{V}$ can be estimated by the backscattering image $I_{S}$ together with the prior parameters such as $E_{0}$ , $Z_{0}$ , $\bar{ρ_{0}}$ , $\bar{I_{0}}$ and $\tilde{S}$ , and then the recovery target image $U$ in Eq. (1) can be solved through a proper variational model [20].

The captured turbid-medium-free target image in Fig. 5 served as the approximate ideal target image $I_{i d e a l}$ , and the structural similarity (SSIM) indexes with a real number between 0 and 1 is used for the image quality assessment [30]. Obviously, the proposed method can improve the SSIM indexes remarkably, which indicates that the proposed method is effective in eliminating the influence of the turbid medium.

3.3 Homomorphic filtering recovery for comparison

As we know, the homomorphic filtering [12] is a common technique to extract high-frequency information by logarithmic transformation and frequency domain filtering. Here, we compare the proposed method with the blind digital image restoration algorithms based on the homomorphic filtering. The Butterworth filter functions [31] for the homomorphic filtering algorithm and their corresponding results for the single degraded target image (Fig. 7(b)) are shown in Figs. 6 and 7 respectively. The results indicate that, the homomorphic filtering can visually weaken the influence of the turbid medium to some extent. However, the homomorphic filtering will change the relative gray value, i.e., the relative reflectivity of the target image at the same time, which leads to lower SSIM indexes. Taking Fig. 7 as an example, the homomorphic filtering eliminates the shadows (yellow circle in Fig. 7(d)) caused by the turbid medium; however, it changes the gray values of the sub-windows (yellow, green and red circles) to almost the same level, which is far from the true reflectivity of the targets and consequently results in lower SSIM indexes. According to Fig. 7(c), obviously the proposed method can reveal the degradation area of the target image, avoiding changing the relative reflectivity of the other parts. The simulation results indicate that the proposed method is much better than the homomorphic filtering for inhomogeneous degradation restoration of target images.

Fig. 6 Butterworth filter functions of homomorphic filtering.

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Fig. 7 Recovery results of homomorphic filtering. (a) Ideal target image (b) Degraded target image (c) Recovery image by the proposed method; (d-f) show homomorphic filtering results of various Butterworth filter functions as shown in Fig. 6.

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4. Image restoration experiments and analysis

4.1 Pre-calibration measurements

The LLT experimental setup and the signal timing were similar with those in Ref [21]. A reference target plate with uniform reflectivity $\bar{ρ_{0}}$ was captured by the LLT system in absence of turbid media for pre-calibration. The perpendicular-incidence reflectivity $\bar{ρ_{0}}$ of the reference target plate was pre-measured by an optical power meter at various ranges, according to the following Eq. (8):

P_{r} = \bar{ρ_{0}} \frac{A_{r}}{Z_{r}^{2}} P_{i}

Here,

P_{i}

is the incident power,

P_{r}

is the received power, and the incident direction and the receiving direction are both normal to the surface of the reference target;

A_{r}

is the receiving area of the optical power meter, and

Z_{r}

is the distance between the reference target and the optical power meter. The average reflectivity was obtained as

\bar{ρ_{0}} = 0 {.4643sr}^{- 1}

by linear fitting of

P_{r} / P_{i}

versus

A_{r} / Z_{r}^{2}

, as shown in Fig. 8.

Fig. 8 Linear fitting of $P_{r} / P_{i}$ versus $A_{r} / Z_{r}^{2}$

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The reference target image $I_{0}$ captured by the turbid-medium-free LLT experiment is shown in Fig. 9, and its average gray value is denoted by $\bar{I_{0}}$ . In the turbid-medium-free LLT experiment, the single pulse energy $E_{0}$ and the target distance $Z_{0}$ , along with $\bar{ρ_{0}}$ and $\bar{I_{0}}$ , served as the prior parameters for the following image restoration process.

Fig. 9 Recovery process of the first experimental image group.

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4.2 Restoration results

The LLT experiments with turbid media were performed, and the captured images along with the establishment of the degradation matrix are depicted in Fig. 9.

In the LLT experiments, the turbid medium particles were water droplets sprayed by an ultrasonic nebulizer that has a vibration frequency of 1.7MHz and can convert averagely 7ml of water per minute into water droplets via ultrasonic atomization [32]. According to Lang’s formula [32], the median particle diameter of the water droplets can be calculated as 2.92 $μ m$ ; combining with the analyses in Sec. 3.1, $\tilde{S} = 19 s r$ can be chosen as the estimated LIDAR ratio. As shown in Fig. 9, the degradation matrix $\tilde{V}$ can be estimated by the backscattering image $I_{S}$ together with the prior parameters such as $E_{0}$ , $Z_{0}$ , $\bar{ρ_{0}}$ , $\bar{I_{0}}$ and $\tilde{S}$ ; through the proposed variational model, the retrieved target images $U$ can be solved by inputting the estimated degradation matrix $\tilde{V}$ and the degraded target image $U_{0}$ .

By considering the captured turbid-medium-free target images in Fig. 9 as the approximate ideal target image $I_{i d e a l}$ , the SSIM indexes are listed in the corresponding images. Obviously, the proposed method can improve the SSIM indexes remarkably, which indicates that the proposed method is effective in eliminating the inhomogeneous degradations caused by the turbid medium. The same procedures can be performed for the restorations of another two groups of experimental images, as shown in Figs. 10 and 11, which come to the same conclusions.

Fig. 10 Recovery results of the second experimental image group. (a) Turbid-medium-free target image (b) Degraded target image (c) Backscattering image (d) Denoised backscattering image (e) Degradation matrix visualization (f) Retrieved target image.

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Fig. 11 Recovery results of the third experimental image group.(a) Turbid-medium-free target image (b) Degraded target image (c) Backscattering image (d) Denoised backscattering image (e) Degradation matrix visualization (f) Retrieved target image.

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5. Conclusion

Based on LLT, assisted by the backscattering images and some prior parameters from pre-calibration, a novel and feasible image restoration method was developed to estimate the degradation caused by inhomogeneous turbid media, and further to eliminate the inhomogeneous degradation through a proper total variation model. The proposed method can work under common conditions and overcome the limitation of the previous method that only works well under the conditions of localized turbid attenuations and fairly uniform targets. The recovery process is unsupervised, which can improve the real-time property.

The proposed restoration method is based on the physical signal relevance between the target layer and the turbid medium layer, instead of a blind image restoration algorithm. Simulations and experiments are both conducted to verify the validity and feasibility of the proposed method. The results demonstrated that, the proposed method can effectively eliminate the interferences of inhomogeneous turbid media to achieve the real target images.

As the proposed method can reveal the true target behind inhomogeneous turbid media, it can reduce the false recognition rate for target recognition and identification. The proposed method potentially will be applied in target acquisition for observations in air and under water, military reconnaissance, and fire rescue.

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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Group	a	$μ$	$ν$	b	$r_{m}$	S
D1	5.5556	8	3	0.3333	$2 μ m$	19.92sr
D2	$1 .0 85 \times 10^{- 2}$	8	3	$4 .167 \times 10^{- 2}$	$4 μ m$	19.13sr
D3	$2 .823 \times 10^{- 4}$	8	3	$1 .23 5 \times 10^{- 2}$	$6 μ m$	19.21sr
D4	$2.119 \times 10^{- 5}$	8	3	$5 .2 08 \times 10^{- 3}$	$8 μ m$	19.18sr
D5	$2.844 \times 10^{- 6}$	8	3	$2 .667 \times 10^{- 3}$	$10 μ m$	18.89sr

Restoration of longitudinal laser tomography target image from inhomogeneous medium degradation under common conditions

Abstract

1. Introduction

2. Principles of image restoration

3. Simulations

3.1 Pre-estimation of LIDAR ratio and pre-calibration of system parameters

3.2 Simulation Verification

3.3 Homomorphic filtering recovery for comparison

4. Image restoration experiments and analysis

4.1 Pre-calibration measurements

4.2 Restoration results

5. Conclusion

Funding

Acknowledgment

References and links

Cited By

Figures (11)

Tables (1)

Equations (8)

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