1 Introduction

Efficient technologies for underwater wireless transmission are desirable to facilitate ocean exploration and monitoring. The conversion of the technology from terrestrial to underwater links is difficult, since the channel parameters are radically different. The speed of the acoustic waves is much lower and the path loss is higher. Moreover, the bandwidth is limited, and thus limited data rates can be accommodated [1,2,3,4]. To avoid these limitations, two aspects should be taken into consideration: efficient compression technique and bandwidth efficient modulation. Compression enables transmission at a much lower bit rate. Orthogonal frequency division multiplexing efficiently uses the limited bandwidth and can combat the multipath propagation problem using less complex receivers [5].

The interest in exploring the underwater environment especially with image transmission has been increasing. In [6], error concealment and error correction algorithms are applied for quality-aware image transmission over underwater channel. In [7], OFDM with binary phase shift keying (BPSK) is used for underwater image transmission. Least square (LS) and minimum mean square error (MMSE) are used for equalization. Tomasi et al. [8] propose a system for transmitting encoded image and use forward error correction (FEC) for symbols protection. The multiple descriptions (MD) technique results in performance improvement compared to the basic allocation one. Ismaiel and Jiang [2] propose a system for SPIHT-based image transmission. Hierarchical quadrature amplitude modulation (16-HQAM) and Reed–Solomon (RS) are used to reduce the bit error rate using an unequal protection mechanism. Decision feedback equalizers (DFEs) are used to equalize the received data. In [4], they propose an OFDM system in which RS is used with HQAM to reduce bit error without using an equalizer. In [9], they propose a rate allocation scheme with SPIHT-based compression for performance improvement.

In this paper, an efficient source–channel coding is introduced to transmit images over underwater wireless channel. The target is reducing the bit error rate (BER) due to channel impairments, and reconstructing the images with high visual quality at the receiver. SPIHT algorithm is applied on WP coefficients. The spatial orientation trees (SOTs) are adapted to maintain the cross-subband similarities. For burst error correction, RS codes are used. OFDM is used in the proposed system to handle multipaths and to avoid complex channel equalization. DQPSK is used to avoid complex carrier tracking. To predict the quality of the received image, effective metrics are used. Multi-scale structural similarity index (MS-SSIM) is employed. It takes advantage of the human visual system characteristics, and can incorporate the details of the image at different levels. The peak signal-to-noise ratio (PSNR) and the weighted peak signal-to-noise ratio (WPSNR) are also used.

The rest of the paper is organized as follows: The underwater channel model is presented in Sect. 2. The proposed system is introduced in Sect. 3. Simulation setup and results are presented in Sect. 4. The paper is concluded in Sect. 5.

2 Underwater Acoustic Channel

The overall attenuation loss in underwater acoustic channel is given by [6, 10]

$$A\left( {d,f} \right) = d^{k} a\left( f \right)^{d}$$
(1)

where d is the distance between the transmitter and receiver, f is the signal frequency and k is the spreading coefficient. A value of k = 1.5 is usually used. The absorption loss a(f) results from the conversion of acoustic energy into heat. This absorption is proportional to the frequency squared and thus causes a frequency selective channel fading:

$$A\left( f \right) = \frac{{0.11f^{2} }}{{1 + f^{2} }} + \frac{{40f^{2} }}{{4100 + f^{2} }} + 2.75 \times 10^{ - 4} f^{2} + 0.003$$
(2)

where A(f) = 10 loga(f). The noise affecting the signal has a power spectral density given by:

$$N\left( f \right) = N_{t} \left( f \right) + N_{s} \left( f \right) + N_{w} \left( f \right) + N_{h} \left( f \right)$$
(3)

The right hand side of Eq. 3 is the superimposition of: turbulence (t), shipping (s), wind driven waves (w), and thermal noise (h). Thus the signal-to-noise ratio (SNR) of a signal transmitted from a distance d at a frequency f is given by:

$$SNR\left( {d,f} \right) = \frac{P}{{A\left( {d,f} \right)N\left( f \right)\Delta f}}$$
(4)

where P is the power of the transmitted signal and Δf is the receiver noise bandwidth. Doppler shifts occur due to the movement of underwater nodes. The Doppler phase is a function of the relative velocity υ and the ratio between the carrier frequency and symbol rate [11]:

$$\varphi_{d } = - 2\pi \frac{{f_{c} }}{R}\frac{\upsilon }{c - \upsilon }$$
(5)

where c is the speed of sound in water.

3 The Proposed System

To develop a system for wireless image transmission through a challenging channel, appropriate compression and transmission techniques should be employed. The wavelet packet decomposition is a generalization of the dyadic decomposition, where detail subbands can be further decomposed. Thus, a multitude of choices are available for image representation, and the best one with respect to the target compression efficiency can be used. SPIHT algorithm [12], provides energy compaction property, cross-subband similarity and decaying of coefficients across subbands. These properties motivated us to use wavelet packet decomposition with SPIHT for image compression (SPIHT on WP). The targets are higher energy compaction, higher coding efficiency, meanwhile preserving the image textures, and reconstructing high quality images at the receiver under the challenging channel conditions. The system block diagram is shown in Fig. 1.

Fig. 1
figure 1

System block diagram: a transmitter, b receiver

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For finding the best tree of WP decomposition, the full growth approach is employed. A transform basis is adaptively selected. To compare WP basis, Shannon entropy of wavelet coefficients is used as cost function. For best basis selection, an adaptive search using single spatial tree algorithm [13, 14] is applied. The WP-based SPIHT is compared with the base-line SPIHT. Using wavelet packet decomposition, the parent–child assignments are adapted in a way that preserves the cross-level similarities. The approaches in [14,15,16] are used to resolve the different parental conflicts. The spatial relationship among wavelet coefficients in different subbands is defined using the spatial orientation tree structure. The significance of coefficients is predicted based on the significance of the parent coefficients.

For error detection and correction Reed–Solomon codes are used, due to their effectiveness in burst-error correction. The incorrect byte is replaced by the correct one, whether the error is caused by one or all bits. Correcting t symbols errors requires no more than 2t parity symbols. Improvement in error performance can be achieved by increasing the redundancy, at the expense of increased bandwidth and implementation complexity. OFDM is used to handle multipaths and to avoid complex channel equalization. By having a low bit error rate per subcarrier, the intersymbol interference (ISI) is reduced considerably. Also, by using guard intervals the ISI is reduced even more.

Errors at the receiver after the fast Fourier transform (FFT) operation may occur, thus synchronization is required. The start of the FFT window is found by correlating a reference symbol known at the receiver with the received symbols. Phase errors result from the movement of the transmitter or receiver together with carrier offset. To avoid complex carrier tracking at the receiver, and to reduce the system complexity, differential quadrature phase shift keying is used. Differential coding is done on the parallel data, since it works when there is a phase difference from two succeeding symbols from the same subcarrier.

4 Simulation Setup and Results

The proposed OFDM system for underwater wireless image transmission is simulated and tested under different channel conditions. The effects of noise, multipaths and phase errors are well investigated. For using SPIHT on WP, biorthogonal 9/7 transform is used for wavelet packets decomposition. The number of decomposition levels is 5. Fully grown tree is employed, and all possible bases are searched for best basis selection. For using SPIHT, 9/7-tap filters are used to construct five-level pyramids. Results are obtained without arithmetic coding. The performance is evaluated by analyzing the quality of the reconstructed image at the receiver versus BER. Both 0.5 and 1 bpp are used as decoding bit rates (bits per pixel).

For a given message of length k < n, the (n, k) Reed–Solomon codes add parity bits at the end of the message and use them to correct the errors. In this work, n = 200 and k = 128. The size of the original image is 256 × 256 = 512 × 128 = 65,536 bytes. After adding parity bytes, the size will be 512 × 200 = 102,400 bytes. For the compressed image, the size after applying RS is 64 × 200 = 12,800 bytes with 1 bpp bit rate and 32 × 200 = 6400 bytes with 0.5 bpp.

The OFDM parameters used in this work are given in Table 1, and are calculated according to [17]. A base system is first designed with 8 kHz bandwidth and 1600 subcarriers. The bandwidth is then scaled by using the same value of symbol duration and guard interval, and changing the number of subcarriers. The transmission data rate scales with the bandwidth since the utilization factor is unchanged. The total symbol time including the guard interval is 240 ms. Without compression, the total time required for transmission, including the reference symbol is 61.68 s. The bit rate is approximately 13 kbps. The total transmission time is reduced to 24.24 s with compression. This gives a bit rate of approximately 4 and 2 kbps for 1 and 0.5 bpp compression, respectively.

Table 1 OFDM parameters
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To simulate noise in the channel, additive white Gaussian noise (AWGN) with different signal-to-noise (S/N) ratios is added to the transmitted signal. Phase errors are simulated by multiplying the transmitted signal with exp(iθ), where θ is the phase error, which increases from 0 to 2π during the total time of transmission. To simulate multipaths, five multipath arrivals are considered; the direct path and four delayed paths with different arrival strengths, as in Table 2. The metrics used for objective evaluation are: PSNR, WPSNR [18] and MS-SSIM [19]. With WPSNR, the contrast sensitivity function (CSF) is used to weight the spatial frequencies of a given image. MS-SSIM is used to incorporate image details at different resolutions.

Table 2 Amplitude and time delay of different paths
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Table 3 shows the effect of bit error when transmitting the original uncompressed Lena image. Table 4 shows the compressed image obtained using SPIHT and SPIHT on WP with their quality metrics. Table 7 gives the quality metrics of the decompressed image at the receiver at different values of BER. Tables 5, 6 and 8 show the quality metrics of the Dolphin image. Although the difference in quality metrics obtained using both compression techniques is slight, results show that SPIHT reconstructed images at the receiver have degraded image quality. Bits corruption may introduce significant image distortion. Errors on critical bits may lead to failure in decoding and image reconstruction. The wavelet packet transform has a no dyadic nature. This transform allows adapting the basis to the image content and to the purpose of the transformation. Applying SPIHT on WP and adapting the parent–child assignments lead to preserving the textures of the reconstructed images at the receiver and thus images with high visual quality are obtained. The sensitivity to bit errors decreases noticeably compared to SPIHT. The proposed system has an error resilience property that curtails the severity of the damage. The execution time of applying SPIHT on WP is at least two times longer than SPIHT.

Table 3 Quality metrics of uncompressed Lena image at different BER values
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Table 4 Quality metrics of compressed Lena image using both SPIHT and SPIHT on WP
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Table 5 Quality metrics of uncompressed dolphin image at different BER values
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Table 6 Quality metrics of compressed dolphin image using both SPIHT and SPIHT on WP
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Table 7 Quality metrics of decompressed Lena image at the receiver at different BER values
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Table 8 Quality metrics of decompressed dolphin image at the receiver at different BER values
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The final stages in the proposed system are smoothing and interpolation. The algorithm in [20] is applied on the received image. Smoothing is used to reduce noise while keeping the most important imprints. Interpolation is used to assign values to missing data. The algorithm uses non parametric regression. It allows robust and fast smoothing using a discrete cosine transform. The amount of smoothing is chosen by the minimization of the generalized cross-validation score. The algorithm is first applied on the received image, in case of transmitting original uncompressed image. The results are given in Tables 9 and 10. Comparing with the results before smoothing (Tables 3, 5), noticeable improvement in the image quality metrics can be obtained using the smoothing technique. The effect of smoothing is also studied when transmitting WP-based SPIHT compressed images. The algorithm is applied on the decompressed images at the receiver, and the results are given in Table 11. In this case, the image quality may be slightly improved.

Table 9 Quality metrics of uncompressed Lena image after smoothing
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Table 10 Quality metrics of uncompressed dolphin image after smoothing
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Table 11 Quality metrics of WP-based SPIHT decompressed images after smoothing
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5 Conclusion

An efficient underwater system for wireless image transmission has been proposed. Compression is essential, and the images need to be coded within the limited bandwidth. The best tree of wavelet packet decomposition is found using the full growth approach, and SPIHT is applied on WP coefficients. The cross subband similarities have been preserved by adapting the parent–child relationships and solving the parental conflicts. For error detection and correction, Reed–Solomon has been applied. OFDM has been used with DQPSK for bit stream transmission. The SPIHT on WP and SPIHT algorithms have been compared using effective quality metrics. SPIHT results show high sensitivity to bit errors, and an uncontrolled degradation of the reconstructed quality may occur. The proposed system, on the other hand, is sufficiently error-resilient, and images can be reconstructed at the receiver with high visual quality.