Under complex scattering conditions, it is very difficult to capture clear object images hidden behind the media by modelling the inverse problem. With regard to dynamic scattering media, the challenge increases. For solving the inverse problem, we propose a new class-specific image reconstruction algorithm. The method based on deep learning classifies blurred scattering images according to scattering conditions and then recovers to clear images hidden behind the media. The deep learning network is used to learn the mapping relationship between the object and the scattering image rather than characterizing the scattering media explicitly or parametrically. 25000 scattering images are obtained under five sets of dynamic scattering condition to verify the feasibility of the proposed method. In addition, the generalizability of the method has been verified successfully. Compared with common CNN method, it’s confirmed that our algorithm has better performance in reconstructing higher-quality images. Furthermore, for a given scattering image with unknown scattering condition, the closest scattering condition information can be given by classification network, and then the corresponding clear image is restored by reconstruction network.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
As a long-standing problem in optics, there has been a soaring concern in scattering imaging through random media, such as atmosphere, underwater and biomedicine [1–7]. Imaging through scattering media is considered as a challenging topic in optics and has been attracted intensive attention for a long time . In traditional optical imaging systems, when propagating into the entrance pupil of the imaging system, it is required that the spherical wavelet emitted from the object must not be distorted seriously, otherwise the clear object image will not be captured by the lens imaging system.
However, when there are inhomogeneous media such as fogs, clouds and biological tissues in the imaging process, the traditional imaging methods will not work . The light propagating through scattering media suffers from many effects such as diffusion, multiple refraction and absorption . Due to the multiple scattering problem in the random media, the spherical wavelet emitted from the object is disturbed seriously and cannot be captured successfully through simple lens imaging system.
When acquired through scattering media, the captured scattering images are prone to suffer from degradation in visual effect, which leads to some serious errors in many applications . In order to obtain a clear object image as much as possible, many typical scattering imaging techniques have been proposed. In 1991, Wang et al. firstly proposed ballistic light measurement scheme to realize scattering imaging by measuring ballistic light . Yaqoob et al. proposed phase conjugation method that succeeded in imaging chicken breast tissue by generating phase conjugate light to track the beam trajectory . Subsequently, digital phase conjugation technology was applied to scattering imaging problem, and sub-wavelength imaging is achieved . Optical memory effect method was put forward, which is a non-contact and single-amplitude scattering imaging method. Bertolotti et al. combined angle scanning with optical memory effect to realize scattering imaging, appearing in the cover of Nature in the current period . Then, Vellekoop and Mosk took advantage of wave front shaping  to control the light beam to measure the transmission matrix [17, 18] of the scattering media, and then recovered the object hidden behind the scattering media by using the transmission matrix information. Nevertheless, it is difficult to measure the accurate transmission matrix.
Faced with the complex and strong scattering condition, it’s hard to capture clear object images hidden behind the scattering media using traditional imaging methods. For dynamic scattering media, it’s obvious that the challenge increases because of the time-varying scattering properties. In recent years, some attempts such as combining the above traditional techniques with faster SLM or all-optical feedback were made to explore the slowly evolving dynamic media [19, 20]. Moreover, these explorations have high demands with regard to the DMD technology and the processing speed. However, with current feedback control techniques, it’s hard to follow the changing characters of the dynamic scattering media when a décor related speckle field is generated by moving scatter centers on the millisecond timescale .
For better solving the dynamic scattering imaging problem, class-specific image reconstruction method based on deep learning through dynamic scattering media is proposed. The proposed method consists of classification network and GAN reconstruction network. First of all, five different concentrations of fat emulsion solution are used to simulate five dynamic scattering imaging conditions. 25000 degraded scattering images under five scattering conditions are collected as dataset to train the network. Secondly, the classification network is fed with scattering images with labels and then is trained to classify the input image according to the corresponding scattering condition. Thirdly, based on the classification output, the scattering images of the same category are put into the GAN reconstruction network. The GAN reconstruction network iteratively learns the mapping relationships between the input and the output signals of the scattering imaging systems. After being iteratively trained, the network automatically finds the optimal solution and realizes the image reconstruction.
The experiment results demonstrate that the class-specific method enhances the restoration performance compared with the reconstruction-only GAN network and the common CNN method. In the paper, the common CNN method is just the Generative part of the GAN network. Besides, the trained classification network provides the prior information of the scattering layer for next GAN reconstruction work. For a degraded scattering image with unknown scattering information, the trained classification network can even tell the closest scattering condition the image belongs to, and then the GAN network restores it to the clear one according to the scattering condition which is related to the output of the classification network. Furthermore, when with enough discrete fat emulsion solution concentrations, the network can be trained with scattering images under arbitrary scattering conditions and will solve the image reconstruction problem through arbitrary scattering media.
The structure of the paper is as follows: In Section 2, the architecture of the dynamic scattering imaging system and dataset acquisition process is introduced. Section 3 details the principle, constitution and concrete realization of the proposed method, including the experimental results. The further method evaluation and comparison are given in Section 4, and the conclusion is drawn in Section 5.
2. Imaging procedure
2.1. Imaging system
The experimental setup is shown in Fig. 1. A laser beam (He-Ne laser, wavelength: 633nm) expanded by beam expander illuminates the DMD (DLC9500P24, pixel count:1920×1080, pixel pitch:10.8μm). The DMD displays the object images on its surface. The reflected light passes through the scattering media and the lens, and then is ultimately captured by the CCD (2M360MCL, exposure time: 20ms).
DMD is a digital micromirror device. The object images are sequentially displayed onto the surface of the DMD. In the experiment, 5000 different digits from the MNIST are displayed as the object images. CCD is a charge coupled device which captures the light and converts optical images into digital signals. 128 × 128 valid pixels of CCD are used to collect the object images.
The scattering layer is made up of the fat emulsion solution. The fat emulsion solution is a highly scattering media with about 100um average particle diameter and is also widely used in optical experiments to simulate the scattering properties of biological tissues . Owing to the good scattering characters and fewer absorption to the laser with 632nm wavelength, fat emulsion solution is chosen to simulate the dynamic scattering media. In the experiment, 0ml/1.2ml/1.4ml/1.8ml/2.1ml fat emulsion solution are separately added to transparent glass container with 2500ml distilled water. The five different solution concentrations represent five kinds of scattering imaging condition with different optical thicknesses. Five scattering environments are named Case 1/Case 2/Case 3/Case 4/Case 5, respectively. For better simulating the dynamic scattering media, we keep the solution stirring during the experiment. The reflected light passing through the fat emulsion solution media will suffer from complex optical effects such as reflection, dispersion and absorption. The multiple scattering phenomenon randomly changes the transmission direction of the transmission light and will destroy the original spatial phase information, resulting in image degradation. Under each scattering condition, 5000 scattering images are captured through the corresponding scattering imaging system. Thus, the dataset with 25000 degraded scattering images under five scattering conditions is established.
2.2. Reconstruction-only GAN network based on deep learning
In the paper, the reconstruction-only GAN work begins with scattering images which are gray scale images. For scattering media with five different optical thicknesses, the same 5000 digits from MNIST are used to generate the five groups of scattering image dataset. 4000 degraded scattering images captured under each dynamic scattering condition are collected as the training set to train network, and 500 scattering images are used for validation. The rest 500 scattering images designed as the testing set to evaluate the performance of the network.
As a non-linear invertible problem, the deep learning method is adopted to solve the traditional scattering imaging problem so as to make up the disadvantages of traditional methods. This method makes full use of the powerful fitting ability of deep learning to fit the function mapping relationship of the imaging system with scattering media, and then derives the output from input.
The simple mathematical model of image system consisting of scattering media is as follows:equation 1 and 2, the network can learn the mapping relationship between scattering images and original object images to represent the forward function F inexplicitly, solving the inverse problem. The ultimate goal of this work is to make the look like I as same as possible.
In order to reconstruct the scattering images and improve the image quality, Generative Adversarial Network (GAN)  is chosen to restore the clear images with high image quality from the scattering images with low image quality. GAN used here is a supervised learning model with game training process, producing a fairly good output through the mutual game between two modules: the Generative Model and the Discriminative Model. The Generative model is intended to randomly generate observation data using given information. The Discriminative model requires input variables and is designed to predict the output by training model.
For the imaging system, on the one hand, the goal of the Generative Model is to generate a new image as similar to the real object image as possible, which attempt to deceive the Discriminative Model during the training process. On the other hand, the goal of the Discriminative Model is to distinguish the generated image and the real image. Thus, the Generative Model and the Discriminative Model constitute a dynamic game training process.
Figure 2 manifests the flowchart of reconstruction processing. Generative Model learns from the 4000 input scattering images collected and produces the new output images. Both the restored images labeled as fake and the original images labeled as real are sent to Discriminative network to judge whether they are real or fake. The Generative Model further enhance its own learning skills by obtaining the feedback from the judgment results. The detailed information of network architecture is shown in the following Fig. 3.
The reconstruction work also includes two parts: training step and test step. In the training process, 4000 scattering images with 128 × 128 pixels are intended for training the network. In order to train the network, the scattering images and corresponding original object images are fed into the neural network, optimizing parameters that connect every two neurons in the two neighboring layers. For the details of the network structure, the input layer produces feature maps with from input data by using convolution operation. The convolution operation with 3 × 3 kernel size uses Relu activation function which means that input layer outputs 64 feature maps and size of each feature map is 128 × 128 pixels. For avoiding the gradient vanishing problem and the network degradation at the same time, we use the residual network with 16 layers to deepen the network depth and extract more abundant features. Through extracting the feature maps of input scattering images and optimizing the parameters ceaselessly, the output image can be obtained. The generated images with fake label are put into Discriminative network where the last layer is an activation layer. The activation layer using Sigmoid function outputs a probability value that not only identifies the real or fake of the input image but also solves the nonlinear problem. During the training processing, network parameters of one network update iteratively while the other network parameters remain unchanged until two parts reach a dynamic training balance. Thus, the dynamic game training process can make the Generative part reconstruct images as similar to the original images as possible and the Discriminative Network cannot tell real images and fake images apart.
The objective function that minimizes the loss function adopts adaptive moment estimation (Adam). The optimization target of Generator Network includes the similarity between restored images and original images and the ability to deceive Discriminative Network, while the optimization target Discriminative Network cannot tell the restored images and original images apart. The detail loss function expression of two sub-networks is:24]. The smaller the value of , the closer the distance between event q and event p. These cross entropies can express the deception ability of Generative part and accuracy of Discriminative Network by measuring the distance between the real output and the expected output in a way.
As the Discrimination part is a binary classification essentially, p represents one-dimensional expected classification output which is the image’s true label. And the q represents the Discrimination Network’s classification output, which contains the probability of judging the input as the original one and the restored one. In Generative part, the cross entropy is calculated to judge the restored image as real one. The smaller the value of , the better the deception ability and performance of Generative part. In Discrimination Network, the cross entropy shows the capacity to recognize the restored image as fake one and the original image as real one. The smaller the value of and , the higher the accuracy and performance of Discriminative Network. Both parties try their best to optimize their loss function so as to form a competitive confrontation.
2.3. Class-specific reconstruction network with classification
After building a reconstruction-only network for restoring scattering images, seeking for additional benefits for better performance becomes the top priority. What we propose here is to build class-specific reconstruction network. That means, limiting the training data to a specific imaging condition to improve the final performance. 25000 scattering images under five different scattering imaging conditions are collected. Five groups of scattering images with different scattering conditions are considered as five classes. Thus the class label information is provided manually by the scattering imaging condition.
In this work, we feed scattering images and corresponding labels related to the scattering conditions together into the classification network. The classification network includes convolutional layer, pooling layer and full-connected layer. In order to realize multi-class, softmax regression is designed for converting features learned by network into class probability, getting the final classification results. The classification network structure is shown in Fig. 4.
During the training process, the scattering images and labels are put into classification network to train network, which attempts to classify the images into five categories according to the scattering condition. For training GAN, the scattering images of the same category with the same scattering condition are put into the GAN to realize image reconstruction.
During the testing process, a scattering image with unknown information is first fed into trained classification network, and then the corresponding scattering condition is obtained according to the classification result. Thus, the scattering image is put into the GAN network trained by the images with same scattering condition to reconstruct.
It is worth mentioning that the classification network is not only capable of classifying scattering images under five known concentration solutions, but also can give the distribution interval values of scattering solution that an unknown scattering image belongs to. Generally speaking, when two scattering solutions’ concentrations are not far from each other, the scattering characteristic is accordingly close to the other. Thus, the classification network can classify the unknown image into a close class, which can tell the distribution interval values of scattering solution that the unknown image belongs to. With more concentration classes, the distribution interval of scattering solution the unknown image belongs to can be more precise.
3. Experimental results
Due to the agitation and the irregular motion of inner particles, the scattering media is time-varying and the speckles are randomly distributed, which results in more complex scattering effects compared with static scattering media. For describing the dynamic property of the scattering media, six continuous shots of the same object under the same dynamic scattering condition are captured and analyzed. PCC (Pearson correlation coefficient) and speckle intensity correlation are used to evaluate the correlation between different speckle shots.
Figure 5 shows the six continuous shots of the two objects (the digit 4 and the pure white image) under the two different dynamic scattering conditions (Case 2 and Case 5). The speckle pattern of pure white object is polygon due to the diaphragm. After setting the first shot as reference, the cross-correlation and PCC between the first shot and the next five shots are calculated. From the Fig. 5, it’s obvious that the cross-correlation and PCC value changes due to the dynamic scattering media. Besides, the higher the fat emulsion solution’s concentration, the smaller the PCC value.
Multiple shots of the same object under dynamic scattering condition, Case 5, are put into class-specific reconstruction network. The reconstruction results are shown in Fig. 6. Seen from the Fig. 6, the network can predict the same object and scattering condition.
For demonstrating the feasibility of proposed method, 25000 scattering images are collected under five different scattering conditions as a scattering dataset to train neural network. The different scattering conditions are modeled by the fat emulsion solution with different concentrations. The transmitted light passing through the fat emulsion solution media will suffer complex optical effects such as reflection, dispersion and absorption and will degrade the quality of images detected by CCD. The five scattering conditions are named as Case 1, Case 2, Case 3, Case 4 and Case 5 respectively. The solution concentration in the Case 5 is the highest and the solution concentration in Case 1 is the lowest. Under different scattering conditions, the scattering images suffer from different image quality degradation in different degrees. In other words, the images’ quality degrades the most in Case 5. From the scattering images in Fig. 7, the phenomenon such as noise, brightness down and fuzzy edge can be sensed intuitively.
At first, the GAN is chosen to restore the images from the scattering images. 25000 scattering images are put into GAN to train the network all together. The final reconstruction results after training processing are shown in Fig. 7.
In the Fig. 7, the column (a)–(e) show a portion of scattering images collected under five scattering conditions. It is found out that with higher fat emulsion solution concentration, the scattering image will present higher degree of image degradation and lower image quality. Column (f)–(j) represent the corresponding restored images. The restored results demonstrate that under better scattering conditions, GAN can perform well directly. While with worse scattering conditions just like Case 5, the restored images cannot provide any valid information to help us recognize the object. By and large, the performance of using GAN merely is not satisfactory enough.
For better restored performance, classification network is introduced to realize the class-specific reconstruction. The classification network classifies the 25000 scattering images into five classes according to the scattering conditions. After obtaining the corresponding category, the scattering images with the same scattering condition are put into the GAN to train the network and then to reconstruct.
Seen from Fig. 8, there is no doubt that the class-specific reconstruction method we proposed can reconstruct the scattering images to clear object images with high quality even when the scattering images are too blurred to recognize the object under terrible scattering condition. Compared with using GAN network merely, the GAN combing classification network enhances the performance apparently. It is well-known that restoring a clear image from scattering image is an inverse problem. In general, solving the inverse problem needs to build a complex model. The proposed method can automatically handle the scattering effects. So, there is no need to do any additional special processing for that and the class-specific reconstruction method can directly learn from the detected scattering images.
4. Evaluation and discussion
The classification network can point out the scattering condition that the scattering images belong to. The scattering condition can be used as the prior information to get reconstructed image with higher image quality. If there is no classification network before reconstructing, the reconstructed images cannot perform well.
Figure 9 shows the comparison of results between reconstruction-only GAN network without classification and class-specific reconstruction method. From the Fig. 9, it is obviously found in Case 1,4 and 5 that reconstructed images using classification network have higher quality than reconstruction-only without classification network. Especially in Case 5, the results without classification network lose too much valid information and cannot recognize the original object.
Besides, for a scattering image with unknown information (the scattering condition is not included in the five known experimental scattering conditions), classification can give the closest category similar to the original condition. That means, when we get the large amount scattering images under enough kinds of different scattering condition, we can reconstruct any scattering images to clear object images.
Given the scattering images shown in the first row of the Fig. 10, the classification network classifies them into Case 4. According to the output of the classification network, we fed the scattering images into the GAN network trained by the images with the same category, Case 4. The ultimate restoration results are given in the fifth row of the Fig. 10. As a consequence, we can draw a conclusion that the concentration of these scattering images’ scattering condition is closest to Case 4 and is between Case 3(1.4ml fat emulsion into 2500ml water) and Case 5(2.1ml fat emulsion into 2500ml water). In fact, the true concentration is adding 1.7ml fat emulsion into 2500ml water, which demonstrates the right output of our classification network.
In addition, for verifying the performance of our method, the scattering images shown in the first row of the Fig. 10 are fed into other four GAN networks trained with scattering images under other four scattering conditions. It’s no doubt that the restoration quality performs the best in Case 4, which indicates the feasibility and good performance of the proposed method.
In order to illustrate the advantage of proposed class-specific reconstruction method, the results compared with traditional CNN restored method are given in Fig. 11. In the paper, the common CNN method just contains the Generative part of the GAN network. It is seen from the results that CNN performs as well as our method under low concentration scattering solution. As for the higher concentration scattering solution and poorer scattering condition such as Case 4 and Case 5, CNN’s performance is barely satisfactory. The edge of image restored by CNN is blurrier than that restored by the proposed method. Over all, our method can reconstruct the clear object images even when CNN network doesn’t work under the same scattering images and the same training environment.
Seen from the Fig. 12, it’s obvious that the images which are not digits can be restored to clearer images with higher quality. The result demonstrates the reason why these non-digit images can be restored by the trained network is due to the generalizability of the proposed method rather than the memory effect.
For evaluating the restored results quantitatively and describing the reconstruction performance explicitly, the average Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index (SSIM)  of test images sets are calculated and shown in Table 1 and Table 2. The PSNR and SSIM are widely used to measure the performance of image reconstruction method. The definition of PSNR is
Structural similarity method evaluates the image quality from the following aspects: brightness, contrast and structure. The concrete expression of SSIM between X and Y is
The Table 1 and Table 2 show the average PSNR and SSIM values of different methods under five scattering imaging cases. With the help of Fig. 13, it’s clearly illustrated that the PSNR and SSIM of class-specific reconstruction method is the highest and that without classification is lowest. In Case 4 and Case 5, the PSNR and SSIM of proposed method is higher than the CNN method. For SSIM, the proposed method performs stably compared with CNN method and no classification method. Especially in Case 5, the SSIM of proposed method can still achieve 0.79 while that of other methods reduce to beyond 0.6. Based on the above evaluations and results, there is no doubt that the class-specific reconstruction method works better than two methods especially under the poor scattering imaging condition.
For indicating the generalizability of the proposed method, some speckles which original images are not the digits from MNIST are captured. The corresponding reconstruction results are given in Fig. 12.
In this work, the class-specific image reconstruction method based on deep learning for dynamic scattering media is proposed. The method exploits classification network and GAN network to realize the image reconstruction. Restoring the clear object image through scattering media is regarded as an inverse problem. Due to the complicated random properties of dynamic scattering media, deep learning network is adopted and designed to automatically learn the mapping relationship between the object and the scattering image. It avoids modeling the complex inverse problem to characterize the scattering system explicitly or parametrically. The classification section can provide prior information to enhance the performance of the subsequent GAN reconstruction section. Compared with the common CNN method, the proposed method boosts in the reconstruction performance and improve the image quality. Furthermore, the closest scattering condition the unknown scattering image belongs to can be pointed out by classification network and restored by GAN reconstruction network. With more classes of scattering solution concentration, the distribution interval of condition of unknown image can be given more precisely. Besides, the generalizability has been verified successfully. In the future, different scattering media and more classes with same scattering media can be considered, which can be applied to atmospheric imaging and biomedical imaging.
National Natural Science Foundation of China (NSFC) (Grants No: 61471239, 61631014); Hi-Tech Research and Development Program of China (2013AA122901).
The authors declare that there are no conflicts of interest related to this article.
1. A. Ishimaru, “Wave propagation and scattering in random media and rough surfaces,” Proc. IEEE 79, 1359–1366 (1991). [CrossRef]
2. A. Ishimaru, Wave propagation and scattering in random media(Academy Press, 1978).
3. J. Goodman, W. Huntley Jr, D. Jackson, and M. Lehmann, “Wavefront-reconstruction imaging through random media,” Appl. Phys. Lett. 8, 311–313 (1966). [CrossRef]
5. T. Ando, R. Horisaki, and J. Tanida, “Speckle-learning-based object recognition through scattering media,” Opt. Express 23, 33902–33910 (2015). [CrossRef]
6. G. Satat, M. Tancik, and R. Raskar, “Towards photography through realistic fog,” in Computational Photography (ICCP), 2018 IEEE International Conference on, (IEEE, 2018), pp. 1–10.
7. Y. Li, H. Lu, K.-C. Li, H. Kim, and S. Serikawa, “Non-uniform de-scattering and de-blurring of underwater images,” Mob. Networks Appl. 23, 352–362 (2018). [CrossRef]
8. M. Lyu, H. Wang, G. Li, and G. Situ, “Exploit imaging through opaque wall via deep learning,” arXiv preprint arXiv:1708.07881 (2017).
9. J. Tian, Z. Murez, T. Cui, Z. Zhang, D. Kriegman, and R. Ramamoorthi, “Depth and image restoration from light field in a scattering medium,” in 2017 IEEE International Conference on Computer Vision (ICCV), (IEEE, 2017), pp. 2420–2429.
10. A. K. Pediredla, S. Zhang, B. Avants, Y. Fan, S. Nagayama, Z. Chen, C. Kemere, J. Robinson, and A. Veeraraghavan, “Deep imaging in scattering media with single photon selective plane illumination microscopy (spim),” J. Biomed. Opt. 21, 126009 (2016). [CrossRef]
11. R. Lan, H. Wang, S. Zhong, Z. Liu, and X. Luo, “An integrated scattering feature with application to medical image retrieval,” Comput. & Electr. Eng. 69, 669–675 (2018). [CrossRef]
14. T. R. Hillman, T. Yamauchi, W. Choi, R. R. Dasari, M. S. Feld, Y. Park, and Z. Yaqoob, “Digital optical phase conjugation for delivering two-dimensional images through turbid media,” Sci. Reports 3, 1909 (2013). [CrossRef]
16. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283 (2012). [CrossRef]
17. S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104,100601 (2010). [CrossRef] [PubMed]
18. H. B. de Aguiar, S. Gigan, and S. Brasselet, “Enhanced nonlinear imaging through scattering media using transmission-matrix-based wave-front shaping,” Phys. Rev. A 94, 043830 (2016). [CrossRef]
19. M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919 (2013). [CrossRef]
20. D. B. Conkey, A. M. Caravaca-Aguirre, and R. Piestun, “High-speed scattering medium characterization with application to focusing light through turbid media,” Opt. Express 20, 1733–1740 (2012). [CrossRef] [PubMed]
21. E. Tajahuerce, V. Durán, P. Clemente, E. Irles, F. Soldevila, P. Andrés, and J. Lancis, “Image transmission through dynamic scattering media by single-pixel photodetection,” Opt. Express 22, 16945–16955 (2014). [CrossRef] [PubMed]
23. I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative adversarial nets,” in Advances in Neural Information Processing Systems, (Neural Information Processing Systems Foundation, 2014), pp. 2672–2680.
24. J. Shore and R. Johnson, “Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,” IEEE Transactions on Inf. Theory 26, 26–37 (1980). [CrossRef]
25. A. Hore and D. Ziou, “Image quality metrics: Psnr vs. ssim,” in 2010 20th International Conference on Pattern Recognition (IEEE, 2010), pp. 2366–2369.